Murray Wiki - User contributions [en]

CDS 212 Fall 2010

2011-07-08T06:44:37Z

Sojoudi:

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructor'''
* [http://cds-web1.cds.caltech.edu/~doyle2/wiki/index.php?title=Main_Page John Doyle], doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
*[http://www.cds.caltech.edu/~sojoudi Somayeh Sojoudi], sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 16 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/LinearNotes.pdf Slides] for lecture 15 are now posted.
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" |
* Stability of nonlinear systems
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" | {{FBS}} Ch 4 [http://www.cds.caltech.edu/~utopcu/VerInCtrl/lecture-4.pdf SOS]
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 9, Fall 2010|HW 9]]
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212, Homework 8, Fall 2010

2010-11-25T23:52:41Z

Sojoudi:

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #8
| issued = 12 Nov 2010
| due = 30 Nov 2010
}}

=== Reading ===
*[http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Notes 1]
*[http://www.cds.caltech.edu/~sojoudi/RobustnessDetCond.pdf Notes 2]

=== Problems ===
<ol>
<li>
Suppose <amsmath>M</amsmath> is a real matrix. Consider 3 cases where there is just one
type of uncertainty, either real repeated scalar, complex repeated
scalar, or complex full block. The exact answer for the minimum norm
<amsmath>\Delta</amsmath> that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using
standard linear algebra. Compare this with the LMI upper bound and show
that they are equal.
</li>

<li>
Suppose <amsmath>M</amsmath> is a complex matrix that is rank one, so that <amsmath>M=xy^T</amsmath> where
<amsmath>x</amsmath> and <amsmath>y</amsmath> are vectors. Assume there is one block of each type of
uncertainty. Again compute the analytic answer and compare with the LMI
solution.
</li>

<li>
Suppose <amsmath>M</amsmath> is a full complex matrix. Use the robust control toolbox
to write a short program to set up and compute <amsmath>\mu(M)</amsmath> for the block
uncertainty in this handout. Compute upper and lower bounds for some
random matrices of moderate size.
</li>

<li>
Suppose <amsmath>M</amsmath> is a real matrix and there is no real repeated scalar, just
the complex repeated scalar and full block. Suppose the complex repeated
scalar is treated as if it were a z transform variable for a discrete
time system. Compare the LMI conditions for <amsmath>det(I-M\Delta)=0</amsmath> with LMIs
that would arise in computing whether the discrete time <amsmath>H_\infty</amsmath> norm is
less than 1 (discrete version of KYP).
</li>

</li>

CDS 212, Homework 9, Fall 2010

2010-11-25T23:51:23Z

Sojoudi:

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use issos.)</li>
<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath> 
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath> (wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salle's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:38:56Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use issos.)</li>
<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath> 
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath> (wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salle's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:31:07Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use issos.)</li>
<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath> 
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:28:49Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use issos.)</li>
<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:05:13Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
<li> How can we use the results of the first two parts of this question to conclude that <amsmath>p</amsmath> is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:04:32Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
<li> How can we use the results of the first two parts of this question to conclude that $p$ is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that <amsmath>V</amsmath> satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:03:00Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
<li> How can we use the results of the first two parts of this question to conclude that $p$ is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T09:01:47Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
<li> How can we use the results of the first two parts of this question to conclude that $p$ is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~{}~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)</li>
<li> If you cannot find a quadratic Lyapunov function, try a 4th degree one. If you cannot find a 4th degree Lyapunov function, then increase the degree of the candidate Lyapunov functions until you find one. (Hint: 4th degree should work.)
</li>
</ol>
</li>

<li> Use the data in http://www.cds.caltech.edu/\~{}utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables <amsmath>V</amsmath> and <amsmath>f.</amsmath> If you care, <amsmath>V</amsmath> is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by <amsmath>\dot{x} = f(x).</amsmath> Compute a lower bound on the optimal value of the following optimization problem.
<center><amsmath>
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} ~~~~\mu\\
\text{subject to}~~~~~\{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
</amsmath></center> (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)
</li>
<li> Consider the system
<center><amsmath>
\begin{aligned}
&\dot{x}_1 = -x_2\\
&\dot{x}_2 = -f(x_2) - g(x_1),
\end{aligned}
</amsmath></center>where the functions <amsmath>f</amsmath> and <amsmath>g</amsmath> satisfy the following conditions:
<ol>
<li> <amsmath>f</amsmath> and <amsmath>g</amsmath> are continuous.</li>
<li> <amsmath>f(0)=g(0)=0.</amsmath> <amsmath>\sigma f(\sigma) >0</amsmath> and <amsmath>\sigma g(\sigma)>0</amsmath> whenever <amsmath>\sigma\neq 0.</amsmath>
<li> <amsmath>\int_0^\sigma g(\xi) d\xi \rightarrow \infty</amsmath> as <amsmath>|\sigma| \rightarrow \infty.</amsmath>
</ol>
Using
<center><amsmath>
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
</amsmath></center>
as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)
</li>

CDS 212, Homework 9, Fall 2010

2010-11-23T08:45:17Z

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #9 | issued = 23 Nov 2010 | due = 2 Dec 2010 }} …'

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #9
| issued = 23 Nov 2010
| due = 2 Dec 2010
}}

=== Problems ===
<ol>
<li>
Prove that a quadratic polynomial is positive semidefinite if and only of it is sum-of-squares.
</li>

<li> Let <amsmath>p(x_1,x_2) = x_1^2x_2^4 + x_1^4x_2^2 + 1 - 3x_1^2x_2^2.</amsmath>
<ol type="a">
<li> Is <amsmath>p</amsmath> sum-of-squares? </li>
<li> Is <amsmath>p(x_1,x_2)\cdot(x_1^2+x_2^2)</amsmath> sum-of-squares?</li>
<li >Can you intuitively interpret the difference between the results in the first two parts? (Hint: Remember to use {\tt issos}.)</li>
<li> How can we use the results of the first two parts of this question to conclude that $p$ is positive semidefinite (even though it is not sum-of-squares)? </li>
</ol>
</li>

<li> Generalized S-procedure: Given polynomials <amsmath>f</amsmath> and <amsmath>g,</amsmath> if there exists a positive semidefinite polynomial <amsmath>s</amsmath> such that <amsmath>f-gs</amsmath> is positive semidefinite, then <amsmath>\{x \in\mathbb{R}^n~:~g(x) \geq 0\} \subseteq \{x\in \mathbb{R}^n~:~f(x) \geq 0\}.</amsmath>
Let <amsmath>f_1(x) = x_1+x_2+1,</amsmath> <amsmath>f_2(x) = 19-14x_1+3x_1^2-14x_2+6x_1x_2+3x_2^2,</amsmath> <amsmath>f_3(x) = 2x_1-3x_2,</amsmath> <amsmath>f_4(x) = 18-32x_1+12x_1^2+48x_2-36x_1x_2+27x_2^2,</amsmath> and <amsmath>f = (1+f_1^2f_2)(30+f_3^2f_4).</amsmath>
<ol type="a">
<li> Compute a lower bound on the global minimal value of <amsmath>f.</amsmath></li>
<li> Compute a lower bound on the minimal value of <amsmath>f</amsmath> over the set <amsmath>\{x \in \mathbb{R}^2 ~:~1-(1-x_1)^2 - (1-x_2)^2 \geq 0\}.</amsmath> (Hint: Use the generalized S-procedure and sosopt to set up a SOS program to solve this problem. Try polynomial multipliers <amsmath>s</amsmath>(wherever you need them) of different degrees.)
</ol>
</li>

<li> If there exists a polynomial that satisfies
<center><amsmath>
\begin{array}{c}
V(x) - \epsilon x^Tx \in \Sigma[x],~~~~V(0) = 0,\\
-\frac{\partial V}{\partial x} f(x) - \epsilon x^Tx \in \Sigma[x],
\end{array}
</amsmath></center>
then the system <amsmath>\dot{x} = f(x),</amsmath> with <amsmath>f(0)=0,</amsmath> is globally asymptotically stable around the origin. Let's take <amsmath>\epsilon = 10^{-6}</amsmath> and
<center><amsmath>
f(x) =\left[ \begin{array}{c} -x_2-1.5x_1^2-0.5x_1^3\\
3x_1-x_2\end{array}\right].
</amsmath></center>
<ol type="a">
<li> Can you construct a quadratic Lyapunov function that satisfies the above conditions? (Hint: You can try to modify the last piece of the demo file at http://www.cds.caltech.edu/\~{}~utopcu/VerInCtrl/lecture4Demo.m which is on global stability analysis.)
\item If you cannot find a quadratic Lyapunov function, try a 4th
degree one. If you cannot find a 4th degree Lyapunov function, then
increase the degree of the candidate Lyapunov functions until you
find one. (Hint: 4th degree should work.)
\end{itemize}

\item Use the data in http://www.cds.caltech.edu/\~{}utopcu/VerInCtrl/assignment4Data.mat for this exercise. This file contains variables $V$ and $f.$ If you care, $V$ is a Lyapunov function (obtained through some analysis that we will cover later in this course) computed for a system governed by $\dot{x} = f(x).$ Compute a lower bound on the optimal value of the following optimization problem.

\begin{equation}
\begin{array}{c}
\displaystyle{\max_{\mu > 0}} [[User:Sojoudi|Sojoudi]] 08:45, 23 November 2010 (UTC)\mu \\
\text{subject to}08:45, 23 November 2010 (UTC) \{ x~:~ V(x) \leq 0.01\} \subseteq \{ x~:~ \frac{\partial V}{\partial x} \cdot f(x) \leq -\mu V\}.
\end{array}
\end{equation} (Hint: Generalized S-procedure and SOS relaxations for polynomial nonnegativity.)

\item Consider the system \[
\begin{array}{l}
\dot{x}_1 = -x_2\\
\dot{x}_2 = -f(x_2) - g(x_1),
\end{array}
\] where the functions $f$ and $g$ satisfy the following conditions:
\begin{itemize}
\item $f$ and $g$ are continuous.
\item $f(0)=g(0)=0.$ $\sigma f(\sigma) >0$ and $\sigma g(\sigma)>0$ whenever $\sigma\neq 0.$
\item $\int_0^\sigma g(\xi) d\xi \rightarrow \infty$ as $|\sigma| \rightarrow \infty.$
\end{itemize} Using
\[
V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1} g(\xi) d\xi
\] as a Lyapunov function candidate, show that the origin is a globally asymptotically stable equilibrium point for this system. (Hint: Using the above conditions and La Salla's invariance principle, show that $V$ satisfies the conditions for certifying global asymptotic stability.)

CDS 212 Fall 2010

2010-11-23T08:20:52Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructor'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 16 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/LinearNotes.pdf Slides] for lecture 15 are now posted.
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" |
* Stability of nonlinear systems
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" | {{FBS}} Ch 4 [http://www.cds.caltech.edu/~utopcu/VerInCtrl/lecture-4.pdf SOS]
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 9, Fall 2010|HW 9]]
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-19T08:04:36Z

Sojoudi:

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructor'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 16 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/LinearNotes.pdf Slides] for lecture 15 are now posted.
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" |
* Stability of nonlinear systems
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" | {{FBS}} Ch 4 [http://www.cds.caltech.edu/~utopcu/VerInCtrl/lecture-4.pdf SOS]
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-19T07:59:21Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 16 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/LinearNotes.pdf Slides] for lecture 15 are now posted.
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" |
* Stability of nonlinear systems
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" | {{FBS}} Ch 4 [http://www.cds.caltech.edu/~utopcu/VerInCtrl/lecture-4.pdf SOS]
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-19T07:56:22Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 16 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/LinearNotes.pdf Slides] for lecture 15 are now posted.
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" |
* Stability of nonlinear systems
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" | [http://www.cds.caltech.edu/~utopcu/VerInCtrl/lecture-4.pdf SOS]
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212, Homework 8, Fall 2010

2010-11-17T14:44:00Z

Sojoudi:

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #8
| issued = 12 Nov 2010
| due = 25 Nov 2010
}}

=== Reading ===
*[http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Notes 1]
*[http://www.cds.caltech.edu/~sojoudi/RobustnessDetCond.pdf Notes 2]

=== Problems ===
<ol>
<li>
Suppose <amsmath>M</amsmath> is a real matrix. Consider 3 cases where there is just one
type of uncertainty, either real repeated scalar, complex repeated
scalar, or complex full block. The exact answer for the minimum norm
<amsmath>\Delta</amsmath> that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using
standard linear algebra. Compare this with the LMI upper bound and show
that they are equal.
</li>

<li>
Suppose <amsmath>M</amsmath> is a complex matrix that is rank one, so that <amsmath>M=xy^T</amsmath> where
<amsmath>x</amsmath> and <amsmath>y</amsmath> are vectors. Assume there is one block of each type of
uncertainty. Again compute the analytic answer and compare with the LMI
solution.
</li>

<li>
Suppose <amsmath>M</amsmath> is a full complex matrix. Use the robust control toolbox
to write a short program to set up and compute <amsmath>\mu(M)</amsmath> for the block
uncertainty in this handout. Compute upper and lower bounds for some
random matrices of moderate size.
</li>

<li>
Suppose <amsmath>M</amsmath> is a real matrix and there is no real repeated scalar, just
the complex repeated scalar and full block. Suppose the complex repeated
scalar is treated as if it were a z transform variable for a discrete
time system. Compare the LMI conditions for <amsmath>det(I-M\Delta)=0</amsmath> with LMIs
that would arise in computing whether the discrete time <amsmath>H_\infty</amsmath> norm is
less than 1 (discrete version of KYP).
</li>

</li>

CDS 212 Fall 2010

2010-11-17T08:02:39Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 16 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/LinearNotes.pdf Slides] for lecture 15 are now posted.
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T08:11:33Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf MIMO] [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T08:09:09Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* MIMO robust control, Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:45:14Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
|
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:40:27Z

Sojoudi: /* Textbook */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [DP] 
| G. Dullerud and F. Paganini, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:37:03Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8 [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf CvxOpt1] [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf CvxOpt2]
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:28:51Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP]
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:26:54Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs] Ch 2

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:22:40Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4 [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf LMIs]

| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T07:20:09Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| style="border-bottom:3px solid gray;" | DP Ch 4
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | DP Ch 4,7
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| DP Ch 8
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-15T06:54:28Z

Sojoudi: /* Lecture Schedule */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" |
*Fundamental limits
*Realization theory, controllability, observability
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4), DP Ch 2, 4
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Lyapunov equation and stability conditions
* LMIs
| rowspan=2 style="border-bottom:3px solid gray;" | DP Ch 4
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |
* KYP lemma
* Model reduction
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
|
* Uncertain systems
* Convex optimization
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212, Homework 8, Fall 2010

2010-11-12T22:50:25Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #8
| issued = 12 Nov 2010
| due = 25 Nov 2010
}}

=== Reading ===
*[http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Notes 1]
*[http://www.cds.caltech.edu/~sojoudi/RobustnessDetCond.pdf Notes 2]

=== Problems ===
<ol>
<li>
Suppose <amsmath>M</amsmath> is a real matrix. Consider 3 cases where there is just one
type of uncertainty, either real repeated scalar, complex repeated
scalar, or complex full block. The exact answer for the minimum norm
<amsmath>\Delta</amsmath> that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using
standard linear algebra. Compare this with the LMI upper bound and show
that they are equal.
</li>

<li>
Suppose <amsmath>M</amsmath> is a complex matrix that is rank one, so that <amsmath>M=xy^T</amsmath> where
<amsmath>x</amsmath> and <amsmath>y</amsmath> are vectors. Assume there is one block of each type of
uncertainty. Again compute the analytic answer and compare with the LMI
solution.
</li>

<li>
Suppose <amsmath>M</amsmath> is a full complex matrix. Use the robust control toolbox
to write a short program to set up and compute <amsmath>\mu(M)</amsmath> for the block
uncertainty in this handout. Compute upper and lower bounds for some
random matrices of moderate size.
</li>

<li>
Suppose <amsmath>M</amsmath> is a real matrix and there is no real repeated scalar, just
the complex repeated scalar and full block. Suppose the complex repeated
scalar is treated as if it were a z transform variable for a discrete
time system. Compare the LMI conditions for <amsmath>det(I-M\Delta)=0</amsmath> with LMIs
that would arise in computing whether the discrete time <amsmath>H_\infty</amsmath> norm is
less than 1 (discrete version of KYP).
</li>

</li>

CDS 212, Homework 8, Fall 2010

2010-11-12T22:48:50Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #8
| issued = 12 Nov 2010
| due = 25 Nov 2010
}}

=== Reading ===
*[http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Notes 1]
*[http://www.cds.caltech.edu/~sojoudi/RobustnessDetCond.pdf Notes 2]

=== Problems ===
<ol>
<li>
Suppose <amsmath>M</amsmath> is a real matrix. Consider 3 cases where there is just one
type of uncertainty, either real repeated scalar, complex repeated
scalar, or complex full block. The exact answer for the minimum norm
delta that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using
standard linear algebra. Compare this with the LMI upper bound and show
that they are equal.
</li>

<li>
Suppose <amsmath>M</amsmath> is a complex matrix that is rank one, so that <amsmath>M=xy^T</amsmath> where
<amsmath>x</amsmath> and <amsmath>y</amsmath> are vectors. Assume there is one block of each type of
uncertainty. Again compute the analytic answer and compare with the LMI
solution.
</li>

<li>
Suppose <amsmath>M</amsmath> is a full complex matrix. Use the robust control toolbox
to write a short program to set up and compute <amsmath>\mu(M)</amsmath> for the block
uncertainty in this handout. Compute upper and lower bounds for some
random matrices of moderate size.
</li>

<li>
Suppose <amsmath>M</amsmath> is a real matrix and there is no real repeated scalar, just
the complex repeated scalar and full block. Suppose the complex repeated
scalar is treated as if it were a z transform variable for a discrete
time system. Compare the LMI conditions for <amsmath>det(I-M\Delta)=0</amsmath> with LMIs
that would arise in computing whether the discrete time <amsmath>H_\infty</amsmath> norm is
less than 1 (discrete version of KYP).
</li>

</li>

CDS 212, Homework 8, Fall 2010

2010-11-12T22:45:33Z

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #8 | issued = 12 Nov 2010 | due = 25 Nov 2010 }}…'

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #8
| issued = 12 Nov 2010
| due = 25 Nov 2010
}}

=== Problems ===
<ol>
<li>
Suppose <amsmath>M</amsmath> is a real matrix. Consider 3 cases where there is just one
type of uncertainty, either real repeated scalar, complex repeated
scalar, or complex full block. The exact answer for the minimum norm
delta that makes <amsmath>det(I-M\Delta)=0</amsmath> can easily be computed exactly using
standard linear algebra. Compare this with the LMI upper bound and show
that they are equal.
</li>

<li>
Suppose <amsmath>M</amsmath> is a complex matrix that is rank one, so that <amsmath>M=xy^T</amsmath> where
<amsmath>x</amsmath> and <amsmath>y</amsmath> are vectors. Assume there is one block of each type of
uncertainty. Again compute the analytic answer and compare with the LMI
solution.
</li>

<li>
Suppose <amsmath>M</amsmath> is a full complex matrix. Use the robust control toolbox
to write a short program to set up and compute <amsmath>\mu(M)</amsmath> for the block
uncertainty in this handout. Compute upper and lower bounds for some
random matrices of moderate size.
</li>

<li>
Suppose <amsmath>M</amsmath> is a real matrix and there is no real repeated scalar, just
the complex repeated scalar and full block. Suppose the complex repeated
scalar is treated as if it were a z transform variable for a discrete
time system. Compare the LMI conditions for <amsmath>det(I-M\Delta)=0</amsmath> with LMIs
that would arise in computing whether the discrete time <amsmath>H_\infty</amsmath> norm is
less than 1 (discrete version of KYP).
</li>

</li>

CDS 212 Fall 2010

2010-11-12T22:34:55Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 4], [http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-12T22:33:21Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro.pdf 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3],[http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 4], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-12T22:32:32Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 11 Nov 2010: Notes from lecture 14: [http://www.cds.caltech.edu/~sojoudi/MITBoydCvxOpt.pdf 1], [http://www.cds.caltech.edu/~sojoudi/MITBoydintro 2], [http://www.cds.caltech.edu/~sojoudi/MuNotes.pdf 3],[http://www.cds.caltech.edu/~sojoudi/Nov11Overview.pdf 4], [http://www.cds.caltech.edu/~sojoudi/SpinSatellite.pdf 5].
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-10T18:22:00Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 9 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/DetCondUpperBound.pdf Slides] for lecture 13 (PD-Ch 8) are now posted.
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted.
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted.

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212, Homework 7, Fall 2010

2010-11-10T01:24:53Z

Sojoudi:

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #7
| issued = 9 Nov 2010
| due = 18 Nov 2010
}}

=== Problems ===
<ol>
<li>
Show that <amsmath>E(s) = D+C(sI-A)^{-1}B</amsmath> has <amsmath>H_\infty</amsmath> norm <amsmath>< \gamma</amsmath> if the
following LMI is satisfied:
<center><amsmath>
\left[\begin{array}{ccccccc} A^TP+PA& PB& C^T\\ B^TP& -\gamma^2 I& D^T\\ C& D& -I\end{array}\right]\leq 0,
</amsmath></center>
for some <amsmath>P>0.</amsmath>
</li>

<li>
Formulate the model fitting problem <amsmath> min ||(G-\hat{G})||_{H_\infty}</amsmath> where <amsmath>\hat{G}=\hat{D}
+ \hat{C} (sI - \hat{A})^{-1}\hat{B}</amsmath> with <amsmath>\hat{A}</amsmath> and <amsmath>\hat{B}</amsmath> given and <amsmath>\hat{C}</amsmath> and <amsmath>\hat{D}</amsmath> to
be optimized as an LMI. Write a MATLAB/cvx code for this problem.
</li>

<li>
Consider the system <amsmath>G(s) =\frac{P(s)}{(s+0.1)}</amsmath> where <amsmath>P(s)</amsmath> is a 10th order Pade approximation to a 1 second delay. Calculate the Hankel singular values for this system (using balancmr). Output the truncated balanced truncations of orders 1:10. (note that balancmr can produce a set of output ss systems). and compare the norm of the error with the upper and lower bounds.
</li>

<li>
For <amsmath>G(s)</amsmath> as above calculate the optimal Hankel norm approximations. Note the Hankel singular values of the error system and comment. Note that the better error bound on the <amsmath>H_\infty</amsmath> norm requires a non-zero <amsmath>D</amsmath>-term but the hankelmr function does not output this. By examining the Nyquist plot of the error in an example demonstrate that there exists such a <amsmath>D</amsmath>-term. Note how the poles positions vary with the order of the approximation.
</li>

<li>
Use cvx to examine improvements to the above <amsmath>H_\infty</amsmath> norm errors that can be achieved by optimizing the <amsmath>C</amsmath> and <amsmath>D</amsmath> terms with the <amsmath>A</amsmath> and <amsmath>B</amsmath> terms from the balanced and Hankel-norm approximants.
</li>
</li>

CDS 212, Homework 7, Fall 2010

2010-11-09T06:32:10Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #7
| issued = 9 Nov 2010
| due = 18 Nov 2010
}}

=== Problems ===
<ol>
<li>
Show that <amsmath>E(s) = D+C(sI-A)^{-1}B</amsmath> has <amsmath>H_\infty</amsmath> norm <amsmath>< \gamma</amsmath> if the
following LMI is satisfied:
<center><amsmath>
\left[\begin{array}{ccccccc} A^TP+PA& PB& C^T\\ B^TP& -\gamma^2 I& D^T\\ C& D& -I\end{array}\right]\leq 0,
</amsmath></center>
for some <amsmath>P>0.</amsmath>
</li>

<li>
Formulate the model fitting problem <amsmath> min ||(G-\hat{G})||_{H_\infty}</amsmath> where <amsmath>\hat{G}=\hat{D}
+ \hat{C} (sI - \hat{A})^{-1}\hat{B}</amsmath> with <amsmath>\hat{A}</amsmath> and <amsmath>\hat{B}</amsmath> given and <amsmath>\hat{C}</amsmath> and <amsmath>\hat{D}</amsmath> to
be optimized as an LMI. Write a MATLAB/cvx code for this problem.
</li>

<li>
Consider the system <amsmath>G(s) =\frac{P(s)}{(s+0.1)}</amsmath> where <amsmath>P(s)</amsmath> is a 10th order Pade approximation to a 1 second delay. Calculate the Hankel singular values for this system (using balancmr). Output the truncated balanced truncations of orders 1:10. (note that balancmr can produce a set of output ss systems). and compare the norm of the error with the upper and lower bounds.
</li>

<li>
For <amsmath>G(s)</amsmath> as above calculate the optimal Hankel norm approximations. Note the Hankel singular values of the error system and comment. Note that the better error bound on the <amsmath>H_\infty</amsmath> norm requires a non-zero <amsmath>D</amsmath>-term but the hankelmr function does not output this. By examining the Nyquist plot of the error in an example demonstrate that there exists such a <amsmath>D</amsmath>-term. Note how the poles positions vary with the order of the approximation.
</li>

<li>
Use cvx to examine improvements to the above <amsmath>H_\infty</amsmath> norm errors that can be achieved by optimizing the <amsmath>C</amsmath> and <amsmath>D</amsmath> terms with the <amsmath>A</amsmath> and <amsmath>B</amsmath> terms from the balanced and Hankel-norm approximants.
</li>
</li>

CDS 212, Homework 7, Fall 2010

2010-11-09T06:27:39Z

Sojoudi: Created page with '{{CDS 212 draft HW}} {{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #7 | issued = 9 Nov 2010 | due = 18 Nov 2010 }} …'

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #7
| issued = 9 Nov 2010
| due = 18 Nov 2010
}}

=== Problems ===
<ol>
<li>
Show that <amsmath>E(s) = D+C(sI-A)^{-1}B</amsmath> has <amsmath>H_\infty</amsmath> norm <amsmath>< \gamma</amsmath> if the
following LMI is satisfied:
<center><amsmath>
\left[\begin{array}{ccccccc} A^TP+PA& PB& C^T\\ B^TP& -\gamma^2 I& D^T\\ C& D& -I\end{array}\right]\leq 0,
</amsmath></center>
for some <amsmath>P>0.</amsmath>
</li>

<li>
Formulate the model fitting problem <amsmath> min ||(G-\hat{G})||_{H_\infty}</amsmath> where <amsmath>\hat{G}=\hat{D}
+ \hat{C} (sI - \hat{A})^{-1}\hat{B}</amsmath> with <amsmath>\hat{A}</amsmath> and <amsmath>\hat{B}</amsmath> given and <amsmath>\hat{C}</amsmath> and <amsmath>\hat{D}</amsmath> to
be optimized as an LMI. Write MATLAB/cvx code for this problem.
</li>

<li>
Consider the system <amsmath>G(s) =\frac{P(s)}{(s+0.1)}</amsmath> where <amsmath>P(s)</amsmath> is a 10th order Pade approximation to a 1 second delay. Calculate the Hankel singular values for this system (using balancmr). Output the truncated balanced truncations of orders 1:10. (note that balancmr can produce a set of output ss systems). and compare the norm of the error with the upper and lower bounds.
</li>

<li>
For <amsmath>G(s)</amsmath> as above calculate the optimal Hankel norm approximations. Note the Hankel singular values of the error system and comment. Note that the better error bound on the <amsmath>H_\infty</amsmath> norm requires a non-zero <amsmath>D</amsmath>-term but the hankelmr function does not output this. By examining the Nyquist plot of the error in an example demonstrate that there exists such a <amsmath>D</amsmath>-term. Note how the poles positions vary with the order of the approximation.
</li>

<li>
Use cvx to examine improvements to the above <amsmath>H_\infty</amsmath> norm errors that can be achieved by optimizing the <amsmath>C</amsmath> and <amsmath>D</amsmath> terms with the <amsmath>A</amsmath> and <amsmath>B</amsmath> terms from the balanced and Hankel-norm approximants.
</li>
</li>

CDS 212, Homework 6, Fall 2010

2010-11-07T18:47:46Z

Sojoudi: /* Reading */

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 2 Nov 2010
| due = 11 Nov 2010
}}

=== Reading ===
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7cadd4d3ff3995a6e887e15c1a57b578&searchtype=a KYP paper:] (On the Kalman—Yakubovich—Popov lemma, Anders Rantzer, 1996)

=== Problems ===
<ol>
<li>
Proof of Lemma 5 in the KYP paper:
<ol type="a">
<li> Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).</li>
<li> Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.</li>
</ol>
(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)
</li>

<li>
Consider the following state space equation:
<center><amsmath>
\dot{x}=ax+bu, \quad y=cx
</amsmath></center>
where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:
<ol type="a">
<li>Frequency analysis.</li>
<li>KYP Lemma.</li>
</ol>
</li>
<li> Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students); each team needs to hand in only a single write-up.
</li>
</li>

CDS 212, Homework 6, Fall 2010

2010-11-05T20:21:01Z

Sojoudi: /* Reading */

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 2 Nov 2010
| due = 11 Nov 2010
}}

=== Reading ===
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1528893102&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=77673afc1097c2809dd28b860844f55e&searchtype=a KYP paper:] (On the Kalman—Yakubovich—Popov lemma, Anders Rantzer)

=== Problems ===
<ol>
<li>
Proof of Lemma 5 in the KYP paper:
<ol type="a">
<li> Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).</li>
<li> Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.</li>
</ol>
(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)
</li>

<li>
Consider the following state space equation:
<center><amsmath>
\dot{x}=ax+bu, \quad y=cx
</amsmath></center>
where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:
<ol type="a">
<li>Frequency analysis.</li>
<li>KYP Lemma.</li>
</ol>
</li>
<li> Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students); each team needs to hand in only a single write-up.
</li>
</li>

CDS 212, Homework 6, Fall 2010

2010-11-05T20:20:40Z

Sojoudi: /* Reading */

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 2 Nov 2010
| due = 11 Nov 2010
}}

=== Reading ===
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1528893102&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=77673afc1097c2809dd28b860844f55e&searchtype=a KYP paper] (On the Kalman—Yakubovich—Popov lemma, Anders Rantzer)

=== Problems ===
<ol>
<li>
Proof of Lemma 5 in the KYP paper:
<ol type="a">
<li> Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).</li>
<li> Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.</li>
</ol>
(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)
</li>

<li>
Consider the following state space equation:
<center><amsmath>
\dot{x}=ax+bu, \quad y=cx
</amsmath></center>
where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:
<ol type="a">
<li>Frequency analysis.</li>
<li>KYP Lemma.</li>
</ol>
</li>
<li> Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students); each team needs to hand in only a single write-up.
</li>
</li>

CDS 212, Homework 6, Fall 2010

2010-11-05T17:39:53Z

Sojoudi: /* Reading */

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 2 Nov 2010
| due = 11 Nov 2010
}}

=== Reading ===
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1528893102&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=77673afc1097c2809dd28b860844f55e&searchtype=a KYP paper]

=== Problems ===
<ol>
<li>
Proof of Lemma 5 in the KYP paper:
<ol type="a">
<li> Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).</li>
<li> Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.</li>
</ol>
(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)
</li>

<li>
Consider the following state space equation:
<center><amsmath>
\dot{x}=ax+bu, \quad y=cx
</amsmath></center>
where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:
<ol type="a">
<li>Frequency analysis.</li>
<li>KYP Lemma.</li>
</ol>
</li>
<li> Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students); each team needs to hand in only a single write-up.
</li>
</li>

CDS 212 Fall 2010

2010-11-05T06:48:07Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 4 Nov 2010: [http://www.cds.caltech.edu/~sojoudi/ModelRed.pdf Slides] form Prof. Keith Glover's Lecture and the [http://www.cds.caltech.edu/~sojoudi/Tutorial-Modelred.pdf tutorial paper] on Hankel norm approximations are now posted.
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212 Fall 2010

2010-11-03T08:45:36Z

Sojoudi: /* Announcements */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 2 Nov 2010: A reference for lectures 9 and 10: [http://www.stanford.edu/~boyd/lmibook/lmibook.pdf Linear Matrix Inequalities in System and Control Theory, Stephen Boyd.]
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212, Homework 6, Fall 2010

2010-11-03T07:52:40Z

Sojoudi: /* Problems */

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 2 Nov 2010
| due = 11 Nov 2010
}}

=== Reading ===
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1524979904&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8c10ce1dc59d5d8995645582d27731ca&searchtype=a KYP paper]

=== Problems ===
<ol>
<li>
Proof of Lemma 5 in the KYP paper:
<ol type="a">
<li> Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).</li>
<li> Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.</li>
</ol>
(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)
</li>

<li>
Consider the following state space equation:
<center><amsmath>
\dot{x}=ax+bu, \quad y=cx
</amsmath></center>
where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:
<ol type="a">
<li>Frequency analysis.</li>
<li>KYP Lemma.</li>
</ol>
</li>
<li> Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students); each team needs to hand in only a single write-up.
</li>
</li>

CDS 212, Homework 6, Fall 2010

2010-11-03T07:49:41Z

Sojoudi: Created page with '{{CDS homework | instructor = J. Doyle | course = CDS 212 | semester = Fall 2010 | title = Problem Set #5 | issued = 2 Nov 2010 | due = 11 Nov 2010 }} === Reading === [htt…'

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 2 Nov 2010
| due = 11 Nov 2010
}}

=== Reading ===
[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4X-3VTSW0Y-2&_user=10&_coverDate=06%2F03%2F1996&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1524979904&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8c10ce1dc59d5d8995645582d27731ca&searchtype=a KYP paper]

=== Problems ===
<ol>
<li>
Proof of Lemma 5 in the KYP paper:
<ol type="a">
<li> Use part (i) of Lemma 3 to prove part (iii) of this lemma in the KYP paper. Note that you do not need to prove part (i).</li>
<li> Use part (iii) of Lemma 3 in the KYP paper to prove Lemma 5.</li>
</ol>
(Note: a short proof is given in the paper for both parts (a) and (b); but you must provide a detailed proof for full credit.)
</li>

<li>
Consider the following state space equation:
<center><amsmath>
\dot{x}=ax+bu, \quad y=cx
</amsmath></center>
where a,b,c are some scalers and <amsmath>a<0</amsmath>. Find a necessary and sufficient condition in terms of a, b and c such that <amsmath>||H||_\infty<1</amsmath>, using two different methods:
<ol type="a">
<li>Frequency analysis.</li>
<li>KYP Lemma.</li>
</ol>
</li>
<li> Write up a complete proof for the KYP Lemma (you need to reproduce the proof given in the class with all details to make sure you have got the main ideas). For this problem, students can work in small groups (up to 3 students), and that each team needs to hand in only a single write-up.
</li>
</li>

CDS 212 Fall 2010

2010-10-28T20:38:15Z

Sojoudi: /* Grading */

{| width=100%
|-
| colspan=2 align=center |
Feedback Control Theory__NOTOC__
|- valign=top
| width=50% |
'''Instructors'''
* John Doyle, doyle@cds.caltech.edu
* Lectures: Tu/Th, 2:30-4 pm, 314 Annenberg
| width=50% |
'''Teaching Assistants'''
* Somayeh Sojoudi, sojoudi@cds.caltech.edu
* Richard Murray, murray@cds.caltech.edu
|}

=== Course Description ===
Introduction to modern feedback control theory with emphasis on the role of feedback in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. State-space methods, time and frequency domain, stability and stabilization, realization theory. Time-varying and nonlinear models. Uncertainty and robustness.

===Announcements ===
* 7 Oct 2010: [http://www.cds.caltech.edu/~sojoudi/Chap4.pdf Slides] for lecture 4 (DFT-Ch 4) are now posted
* 4 Oct 2010: Office Hours: Wed 4-5pm, 314 Annenberg
* 29 Sep 2010: [http://www.cds.caltech.edu/~sojoudi/2.1_BioBodeDetails.pdf Slides] for lecture 1 are now posted

=== Textbook ===

The two primary texts for the course (available via the online bookstore) are
{|
|- valign=top
| align=right |  [DFT] 
| J. Doyle, B. Francis and A. Tannenbaum, ''Feedback Control Theory'', Dover, 2009 (originally published by Macmillan, 1992). Available online at http://www.control.utoronto.ca/people/profs/francis/dft.html.
|- valign=top
| align=right |  [PD] 
| F. Paganini and G. Dullerud, ''A Course in Robust Control Theory'', Springer, 2000.
|}

The following additional texts may be useful for some students:
{|
|- valign=top
| align=right |  [FBS] 
| K. J. Astrom and R. M. Murray, ''Feedback Systems: An Introduction for Scientists and Engineers'', Princeton University Press, 2008. Available online at http://www.cds.caltech.edu/~murray/amwiki.
|}

=== Lecture Schedule ===

{| width=100% border=1 cellspacing=0 cellpadding=5
|-
| '''Week'''
| '''Date'''
| '''Trunk'''
| '''Reading'''
| '''Homework'''
| '''Branch'''
|- valign=top
| 1
| 28 Sep  30 Sep
| Norms for signals and systems
| {{DFT}} Ch 1, 2  DP Ch 3
| [[CDS 212, Homework 1, Fall 2010|HW 1]]
|
|- valign=top
| 2
| 5 Oct+ 7 Oct
| Feedback, stability and performance
| {{DFT}} Ch 3 ({{FBS}} 9.1-9.3) ({{FBS}} 11.1-11.2)
| [[CDS 212, Homework 2, Fall 2010|HW 2]]
|
* 6 Oct: Mung Chiang (Princeton), An Axiomatic Theory of Fairness
* 6 Oct: Mung Chiang (Princeton), Can Random Access Be Optimal?
|- valign=top
| 3
| 12 Oct+ 14 Oct+
| Uncertainty and robustness
| {{DFT}} Ch 4 ({{FBS}} 12.1‑12.3)
| [[CDS 212, Homework 3, Fall 2010|HW 3]]
|
* 12 Oct: Raff D'Andrea (ETHZ), Some applications of distributed estimation and control
|- valign=top
| style="border-bottom:3px solid gray;" | 4
| style="border-bottom:3px solid gray;" | 19 Oct 21 Oct+
| style="border-bottom:3px solid gray;" | Fundamental limits
| style="border-bottom:3px solid gray;" | {{DFT}} Ch 6 ({{FBS}} 11.4, 12.4)
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 4, Fall 2010|HW 4]]
| style="border-bottom:3px solid gray;" |
|- valign=top
| 5
| 26 Oct+ 28 Oct*
| Stability in state space
* Realization theory, controllability, observability
* Lyapunov equation and stability conditions
| rowspan=2 style="border-bottom:3px solid gray;" | PD, Ch 2, 4, 5
| [[CDS 212, Homework 5, Fall 2010|HW 5]]
|
|- valign=top
| style="border-bottom:3px solid gray;" | 6
| style="border-bottom:3px solid gray;" | 2 Nov* 4 Nov*
| style="border-bottom:3px solid gray;" |Model reduction, LMIs
* Balanced realizations
* KYP lemma
| style="border-bottom:3px solid gray;" | [[CDS 212, Homework 6, Fall 2010|HW 6]]
| rowspan=2 style="border-bottom:3px solid gray;" |
* Keith Glover
** Model reduction
** Loop shaping
|- valign=top
| 7
| 9 Nov 11 Nov
| Nonlinear systems
* Stability of nonlinear systems
* L2-gain for nonlinear systems
| {{FBS}}, Ch 4
| [[CDS 212, Homework 7, Fall 2010|HW 7]]
|- valign=top
| 8
| 16 Nov+ 18 Nov
| rowspan=2 style="border-bottom:3px solid gray;" | Sum-of-squares
* Semi-algebraic sets
* Semidefinite programming
* Sum-of-squares
| rowspan=2 style="border-bottom:3px solid gray;" |
| rowspan=2 style="border-bottom:3px solid gray;" | [[CDS 212, Homework 8, Fall 2010|HW 8]]
|
* IPAM: robust optimization
|- valign=top
| style="border-bottom:3px solid gray;" | 9
| style="border-bottom:3px solid gray;" | 23 Nov+
| style="border-bottom:3px solid gray;" |
* Pablo Parrilo?
|- valign=top
| 10
| 30 Nov 2 Dec
| Links with nformation theory and statistical mechanics
|
|
|
* IPAM: applications of optimization
|}

=== Grading ===
The ﬁnal grade will be based on homework and a ﬁnal exam:
* Homework (75%) - There will be 9 one-week problem sets, due each Thursday by 5pm in the TA's mailbox on the third floor of Annenberg. Each student may hand in at most one homework late (no more than 5 days).
* Final exam (25%) - The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.

=== Collaboration Policy ===
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.

No collaboration is allowed on the ﬁnal exam.

=== Additional References (Optional)===

{| width=100% border=1 cellspacing=0 cellpadding=2
|-
| '''Date'''
| '''Reading'''
|- valign=top
| 28 Sep 
|[http://www.cds.caltech.edu/~sojoudi/AldersonDoyle-tsmca-July2010.pdf AldersonDoyle-tsmca (Paper)],[http://www.cds.caltech.edu/~sojoudi/Glycolysis_Main.pdf Glycolysis (Paper)], [http://www.cds.caltech.edu/~sojoudi/SuppInfo.pdf SuppInfo], [http://www.cds.caltech.edu/~sojoudi/1NetCmplxIntro.pdf 1NetCmplxIntro (Slides)]
|- valign=top
| 5 Oct 
|[http://www.cds.caltech.edu/~sojoudi/layering.pdf layering (Slides)]
|- valign=top
| 19 Oct 
|[http://www.cds.caltech.edu/~sojoudi/BioMetabModeling.pdf BioMetabModeling (Slides)],[http://www.cds.caltech.edu/~sojoudi/FinalBodyCaptions.pdf Glycolysis (Paper)],[http://www.cds.caltech.edu/~sojoudi/Figures.pdf Figures],[http://www.cds.caltech.edu/~sojoudi/Chap6.pdf Chap6 (Slides)]
|}

== Old Announcements ==

[[Category:Courses]]

CDS 212, Homework 5, Fall 2010

2010-10-26T22:07:13Z

Sojoudi:

{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 26 Oct 2010
| due = 4 Nov 2010
}}

=== Reading ===
* [PD], Chapter 4

=== Problems ===
<ol>
<li>[PD 4.1] 
Suppose <amsmath>A, X</amsmath> and <amsmath>C</amsmath> satisfy <amsmath>A^*X+XA+C^*C=0.</amsmath> Show that any two of the following implies the third:
<ol type="i">
<li> <amsmath>A</amsmath> Hurwitz.</li>
<li><amsmath>(C,A)</amsmath> observable.</li>
<li><amsmath>X>0</amsmath></li>
</ol>
</li>
<li>
Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with
<center><amsmath>
F=\left[\begin{array}{ccc} A&0\\C&0 \end{array}\right], G=\left[\begin{array}{ccc} B\\0 \end{array}\right],
</amsmath></center>
is controllable if and only if
<center><amsmath>
\left[\begin{array}{ccc} A&B\\C&0 \end{array}\right]
</amsmath></center>
is a full row rank matrix.
</li>
<li>[PD 4.4] 
Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix
</li>
<center><amsmath>
M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]
</amsmath></center>
cannot be used for the same purpose, since its singular values are unrelated to those of <amsmath>X_c</amsmath>. In particular, construct examples (<amsmath>A\in\mathcal{C}^{2\times 2}, B\in\mathcal{C}^{2\times 1}</amsmath> suffices) such that
<ol type="a">
<li><amsmath>X_c=I</amsmath>, but <amsmath>\underline{\sigma}(M_c)</amsmath> is arbitrarily small.</li>
<li><amsmath>M_c=I</amsmath>, but <amsmath>\underline{\sigma}(X_c)</amsmath> is arbitrarily small.</li>
</ol>
</li>
<li> Prove the Schur complement inequality
<center><amsmath>
\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]> 0 \Longleftrightarrow A-BC^{-1}B^T>0,\quad C>0
</amsmath></center>
</li>
<li>We know that the discrete-time system <amsmath>x[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
</li>
<li>Consider the following optimization problem:
<center><amsmath>
\max_{Q,\alpha}\; \alpha
</amsmath></center>
subject to
<center><amsmath>
AQ+QA^T+\alpha Q< 0, \quad Q>0
</amsmath></center>
Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>

CDS 212, Homework 5, Fall 2010

2010-10-26T19:44:55Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 26 Oct 2010
| due = 4 Nov 2010
}}

=== Reading ===
* [PD], Chapter 4

=== Problems ===
<ol>
<li>[PD 4.1] 
Suppose <amsmath>A, X</amsmath> and <amsmath>C</amsmath> satisfy <amsmath>A^*X+XA+C^*C=0.</amsmath> Show that any two of the following implies the third:
<ol type="i">
<li> <amsmath>A</amsmath> Hurwitz.</li>
<li><amsmath>(C,A)</amsmath> observable.</li>
<li><amsmath>X>0</amsmath></li>
</ol>
</li>
<li>
Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with
<center><amsmath>
F=\left[\begin{array}{ccc} A&0\\C&0 \end{array}\right], G=\left[\begin{array}{ccc} B\\0 \end{array}\right],
</amsmath></center>
is controllable if and only if
<center><amsmath>
\left[\begin{array}{ccc} A&B\\C&0 \end{array}\right]
</amsmath></center>
is a full row rank matrix.
</li>
<li>[PD 4.4] 
Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix
</li>
<center><amsmath>
M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]
</amsmath></center>
cannot be used for the same purpose, since its singular values are unrelated to those of <amsmath>X_c</amsmath>. In particular, construct examples (<amsmath>A\in\mathcal{C}^{2\times 2}, B\in\mathcal{C}^{2\times 1}</amsmath> suffices) such that
<ol type="a">
<li><amsmath>X_c=I</amsmath>, but <amsmath>\underline{\sigma}(M_c)</amsmath> is arbitrarily small.</li>
<li><amsmath>M_c=I</amsmath>, but <amsmath>\underline{\sigma}(X_c)</amsmath> is arbitrarily small.</li>
</ol>
</li>
<li> Prove the Schur complement inequality
<center><amsmath>
\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]> 0 \Longleftrightarrow A-BC^{-1}B^T>0,\quad C>0
</amsmath></center>
</li>
<li>We know that the discrete-time system <amsmath>x[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
</li>
<li>Consider the following optimization problem:
<center><amsmath>
\max_{Q,\alpha}\; \alpha
</amsmath></center>
subject to
<center><amsmath>
AQ+QA^T+\alpha Q< 0, \quad Q>0
</amsmath></center>
Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>

CDS 212, Homework 5, Fall 2010

2010-10-26T08:41:38Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 26 Oct 2010
| due = 4 Nov 2010
}}

=== Reading ===
* [PD], Chapter 4

=== Problems ===
<ol>
<li>[PD 4.1] 
Suppose <amsmath>A, X</amsmath> and <amsmath>C</amsmath> satisfy <amsmath>A^*X+XA+C^*C=0.</amsmath> Show that any two of the following implies the third:
<ol type="i">
<li> <amsmath>A</amsmath> Hurwitz.</li>
<li><amsmath>(C,A)</amsmath> observable.</li>
<li><amsmath>X>0</amsmath></li>
</ol>
</li>
<li>
Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with
<center><amsmath>
F=\left[\begin{array}{ccc} A&0\\C&0 \end{array}\right], G=\left[\begin{array}{ccc} B\\0 \end{array}\right],
</amsmath></center>
is controllable if and only if
<center><amsmath>
\left[\begin{array}{ccc} A&B\\C&0 \end{array}\right]
</amsmath></center>
is a full row rank matrix.
</li>
<li>[PD 4.4] 
Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix
</li>
<center><amsmath>
M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]
</amsmath></center>
cannot be used for the same purpose, since its singular values are unrelated to those of <amsmath>X_c</amsmath>. In particular, construct examples (<amsmath>A\in\mathcal{C}^{2\times 2}, B\in\mathcal{C}^{2\times 1}</amsmath> suffices) such that
<ol type="a">
<li><amsmath>X_c=I</amsmath>, but <amsmath>\underline{\sigma}(M_c)</amsmath> is arbitrarily small.</li>
<li><amsmath>M_c=I</amsmath>, but <amsmath>\underline{\sigma}(X_c)</amsmath> is arbitrarily small.</li>
</ol>
</li>
<li> Prove the Schur complement inequality
<center><amsmath>
\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]> 0 \Longleftrightarrow A-BC^{-1}B^T>0,\quad C>0
</amsmath></center>
</li>
<li>We know that the discrete-time system <amsmath>X[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
</li>
<li>Consider the following optimization problem:
<center><amsmath>
\max_{Q,\alpha}\; \alpha
</amsmath></center>
subject to
<center><amsmath>
AQ+QA^T+\alpha Q< 0, \quad Q>0
</amsmath></center>
Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>

CDS 212, Homework 5, Fall 2010

2010-10-26T08:40:16Z

Sojoudi: /* Problems */

{{CDS 212 draft HW}}
{{CDS homework
| instructor = J. Doyle
| course = CDS 212
| semester = Fall 2010
| title = Problem Set #5
| issued = 26 Oct 2010
| due = 4 Nov 2010
}}

=== Reading ===
* [PD], Chapter 4

=== Problems ===
<ol>
<li>[PD 4.1] 
Suppose <amsmath>A, X</amsmath> and <amsmath>C</amsmath> satisfy <amsmath>A^*X+XA+C^*C=0.</amsmath> Show that any two of the following implies the third:
<ol type="i">
<li> <amsmath>A</amsmath> Hurwitz.</li>
<li><amsmath>(C,A)</amsmath> observable.</li>
<li><amsmath>X>0</amsmath></li>
</ol>
</li>
<li>
Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with
<center><amsmath>
F=\left[\begin{array}{ccc} A&0\\C&0 \end{array}\right], G=\left[\begin{array}{ccc} B\\0 \end{array}\right],
</amsmath></center>
is controllable if and only if
<center><amsmath>
\left[\begin{array}{ccc} A&B\\C&0 \end{array}\right]
</amsmath></center>
is a full row rank matrix.
</li>
<li>[PD 4.4] 
Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix
</li>
<center><amsmath>
M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]
</amsmath></center>
cannot be used for the same purpose, since its singular values are unrelated to those of <amsmath>X_c</amsmath>. In particular, construct examples (<amsmath>A\in\mathcal{C}^{2\times 2}, B\in\mathcal{C}^{2\times 1}</amsmath> suffices) such that
<ol type="a">
<li><amsmath>X_c=I</amsmath>, but <amsmath>\underline{\sigma}(M_c)</amsmath> is arbitrarily small.</li>
<li><amsmath>M_c=I</amsmath>, but <amsmath>\underline{\sigma}(X_c)</amsmath> is arbitrarily small.</li>
</ol>
</li>
<li> Prove the Schur complement inequality
<center><amsmath>
\left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]> 0 \Longleftrightarrow A-BC^{-1}B^T>0,\quad C>0
</amsmath></center>
</li>
<li>We know that the discrete-time system <amsmath>X[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
</li>
<li>Consider the following optimization problem:
<center><amsmath>
\max_{Q,\alpha}\; \alpha
</amsmath></center>
such that
<center><amsmath>
AQ+QA^T+\alpha Q< 0, \quad Q>0
</amsmath></center>
Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.</li>