Murray Wiki - User contributions [en]

Problem 2 (CDS101) Problem 1 (CDS110)

2008-12-03T07:03:53Z

Soto:

The plant in Problem 2b (CDS101) Problem 1b (CDS110) actually meets most (if not all) the specifications. This was unintentional. You need to reduce the steady state error even more (while making sure all other specifications are also met), and include all relevant plots and code to get full credit.
--[[User:Soto|Luis Soto]] 23:03, 2 December 2008 (PST)
[[Category: CDS 101/110 FAQ - Homework 8]]
[[Category: CDS 101/110 FAQ - Homework 8, Fall 2008]]

Problem 2 (CDS101) Problem 1 (CDS110)

2008-12-03T07:03:17Z

Soto:

The plant in Problem 2b (CDS101) Problem 1b (CDS110) actually meets most (if not all) the specifications. This was unintentional. You need to reduce the steady state error even more (while making sure all other specifications are also met), and include all relevant plots and code to get full credit.
--[[User:Soto|Soto]] 23:03, 2 December 2008 (PST)
[[Category: CDS 101/110 FAQ - Homework 8]]
[[Category: CDS 101/110 FAQ - Homework 8, Fall 2008]]

In Problem 2b what does Mp mean?

2008-11-14T05:16:51Z

Soto:

Mp represents the percent overshoot (if applicable) of the step response. Refer to Figure 5.9 on pg. 151 for details.

--[[User:Soto|Luis Soto]] 21:16, 13 November 2008 (PST)
[[Category: CDS 101/110 FAQ - Homework 6]]
[[Category: CDS 101/110 FAQ - Homework 6, Fall 2008]]

In Problem 2b what does Mp mean?

2008-11-14T05:16:15Z

Soto:

Mp represents the percent overshoot (if applicable) of the step response. Refer to Figure 5.9 on pg. 151 for details.

--[[User:Soto|Soto]] 21:16, 13 November 2008 (PST)
[[Category: CDS 101/110 FAQ - Homework 6]]
[[Category: CDS 101/110 FAQ - Homework 6, Fall 2008]]

CDS 101/110a, Fall 2008

2008-11-13T05:20:19Z

Soto: /* Announcements */

{{cds101-fa08}}
This is the homepage for CDS 101 (Analysis and Design of Feedback Systems) and CDS 110 (Introduction to Control Theory) for Fall 2008. __NOTOC__

<table width=100%>
<tr valign=top>
<td>
'''Instructors'''
* [[Main Page|Richard Murray]], murray@cds.caltech.edu
* Doug MacMynowski, macmardg@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 74 JRG
* Office hours (RMM): Fridays, 3-4 pm (by appt)
* Prior years: [[CDS 101/110a, Fall 2006|FA06]], [[CDS 101/110a, Fall 2007|FA07]]
</td><td>
'''Teaching Assistants''' (cds110-tas@cds)
* Julia Braman, head TA
* Gentian Buzi, Shuo Han, Max Merfeld, Luis Soto
* Office hours: Fri 4-5, Sun 4-5 in 114 STL
'''Course Ombuds'''
* Clara O'Farrell and Albert Wu
</td></tr>
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 10 Nov 08: [[CDS 101/110 - Loop Analysis|Week 7 - Loop Analysis]]
** {{cds101 handouts|hw6-fa08.pdf|HW #6}} is posted; due 17 Nov
* 10 Nov 08: Midterm is graded and {{cds101 local|solnMT-fa08.pdf|solutions}} are posted. CDS 101 average = 36/40, CDS 110 average = 41/50.
* 3 Nov 08: [[CDS 101/110 - Transfer Functions|Week 6 - Transfer Functions]]
** {{cds101 handouts|hw5-fa08.pdf|HW #5}} is posted; due 10 Nov
* 3 Nov 08: HW 4 has been graded; averages are for 101: 15/20, 6 hrs; for 110: 35.5/40, 7 hrs.

= Course Syllabus =
<table align=right border=1 width=20% cellpadding=6>
<tr><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] </li>
<li> [[#Lectures and Recitations|Lectures/Recitations]] </li>
<li> [[#Collaboration Policy|Collaboration Policy]] </li>
<li> [[#Software|Software]] </li>
<li> [[#Course Text and References|Course Text]] </li>
<li> [[#Course_Schedule|Course Schedule]]</li>
</ul>
</table>
CDS 101/110 provides an introduction to feedback and control in physical,
biological, engineering, and information sciences. Basic principles of
feedback and its use as a tool for altering the dynamics of systems and
managing uncertainty. Key themes throughout the course will include
input/output response, modeling and model reduction, linear versus nonlinear
models, and local versus global behavior. The course has several variants:

* CDS 101 is a 6 unit (2-0-4) class intended for advanced students in science and engineering who are interested in the principles and tools of feedback control, but not the analytical techniques for design and synthesis of control systems.

* CDS 110 is a 12 unit class (3-0-9) that provides a traditional first course in control for engineers and applied scientists. It assumes a stronger mathematical background, including working knowledge of linear algebra and ODEs. Familiarity with complex variables (Laplace transforms, residue theory) is helpful but not required.

* CDS 210 is a special section of CDS 110, that will be an advanced version of the course for CDS graduate students and others interested in a more theoretical approach to the material. CDS 210 will have an additional Friday lecture and a separate set of homework sets.

=== Lectures and Recitations ===
The main course lectures are on MW from 2--3 pm in 74 Jorgansen. CDS 101 students are not required to attend the Wednesday lectures, although they are welcome to do so. In addition to the main lectures, a series of problem solving (recitation) sessions are run by the course teaching assistants and given on Fridays from 2--3 p m. The recitation session locations will be determined in the first week of classes and will be posted on the course web page.

The TAs will hold office hours on Fridays from 4-5 pm and Sundays from 4-6 pm in 114 Steele
(CDS library).


=== Grading ===
The final grade will be based on homework sets, a midterm exam, and a final exam:

*''Homework (50%):'' Homework sets will be handed out weekly and due on Mondays by 5 pm to the box outside of 109 Steele. Students are allowed three grace periods of two days each that can be used at any time (but no more than 1 grace period per homework set). Late homework beyond the grace period will not be accepted without a note from the health center or the Dean. MATLAB code and SIMULINK diagrams are considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).

* ''Midterm exam (20%):'' A midterm exam will be handed out at the beginning of midterms period (29 Oct) and due at the end of the midterm examination period (4 Nov). The midterm exam will be open book and computers will be allowed (though not required).

* ''Final exam (30%):'' The final exam will be handed out on the last day of class (5 Dec) and due at the end of finals week. It will be an open book exam and computers will be allowed (though not required).

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult
outside reference materials, other students, the TA, or the
instructor, but you cannot consult homework solutions from
prior years and you must cite any use of material from outside
references. All solutions that are handed in should be written up
individually and should reflect your own understanding of the subject
matter at the time of writing. MATLAB scripts and plots are
considered part of your writeup and should be done individually (you
can share ideas, but not code).

No collaboration is allowed on the midterm or final exams.

=== Software ===
Computer exercises will be assigned as part of the regular homeworks. The
exercises are designed to be done in MATLAB, using the Control Toolbox and
SIMULINK. Caltech has a site license for this software and it may be obtained
from [http://software.caltech.edu IMSS] (Caltech students only). An online tutorial is available at
<center>
http://www.engin.umich.edu/group/ctm/basic/basic.html
</center>

=== Course Text and References ===

The primary course text is [[AM:Main Page|''Feedback Systems: An Introduction for Scientists and Engineers'']] by {{Astrom}} and Murray (2008). This book is available in the Caltech bookstore and via download from the [[AM:Main Page|companion web site]]. The following additional references may also be useful:

* A. D. Lewis, ''A Mathematical Approach to Classical Control'', 2003. [http://penelope.mast.queensu.ca/math332/notes.shtml Online access].

In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.

* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', McGraw-Hill, 1986.
* G. F. Franklin, J. D. Powell, and A. Emami-Naeni, ''Feedback Control of Dynamic Systems'', Addison-Wesley, 2002.

=== Course Schedule ===
A detailed course schedule is available on the [[CDS 101/110a, Fall 2008 - Course Schedule|course schedule]] page (also shown on the "menu bar" at the top of each course page).

== Old Announcements ==
* 21 Aug 08: created course homepage
* 28 Sep 08: [[CDS 101/110 - Introduction and Review|Week 1 - Introduction and Review]]
** Please take the {{cds101 handouts|bgsurvey.pdf|background survey}} and turn in at class on Wed
** Wed lecture is for ''all'' students (including CDS 101 - this week only)
** Fri: MATLAB/SIMULINK sessions from 2-4p and 4-6 pm in 328 SFL; bring a laptop with MATLAB installed if you have one
** {{cds101 handouts|hw1-fa08.pdf|HW #1}} is posted; due 6 Oct
* 6 Oct 08: [[CDS 101/110 - Dynamic Behavior|Week 2 - Dynamic Behavior]]
** [[CDS 101/110a, Fall 2008 - Recitation Schedule|Recitations]] begin this week: Fridays, 2-3 pm (see schedule for locations)
** {{cds101 handouts|hw2-fa08.pdf|HW #2}} is posted (updated 8 Oct); due 13 Oct
* 10 Oct 08: HW 1 has been graded; averages are for 101: 26.5/30, 3.2 hrs; for 110: 37/40, 7.6 hrs; for 210: 44/50, 7.3 hrs. {{cds101 local|soln1-fa08.pdf|Solutions for HW #1}} have been posted (only available from Caltech network).
* 13 Oct 08: [[CDS 101/110 - Linear Systems|Week 3 - Linear Systems]]
** {{cds101 handouts|hw3-fa08.pdf|HW #3}} is posted; due 20 Oct
* 20 Oct 08: HW 2 has been graded; averages are for 101: 19/20, 5.5 hrs; for 110: 36.5/40, 10 hrs; for 210: 34/40.
** This week only, the CDS 210 section will be on Thursday (10/23) from 2-3pm
* 20 Oct 08: [[CDS 101/110 - State Feedback|Week 4 - State Feedback]]
** {{cds101 handouts|hw4-fa08.pdf|HW #4}} is posted; due 27 Oct
* 27 Oct 08: HW 3 has been graded; averages are for 101: 17.6/20, 6 hrs; for 110: 36/40, 9 hrs; for 210: 37/40, 9 hrs.
* 27 Oct 08: [[CDS 101/110 - Output Feedback|Week 5 - Output Feedback]]
** CDS 101/110: the [[CDS 101/110 Midterm, Fall 2008|midterm]] will be handed out after class on 29 Oct (Wed); after that, it will be available outside 102 STL.
** CDS 210: {{cds101 handouts|hwM-fa08.pdf|HW #M}} is posted; due 3 Nov
** CDS 210 students- note the additional recitation section scheduled on Mondays from 1-2. More info on the recitation page.

[[Category: Courses]] [[Category: 2008-09 Courses]]

How do you actually find the Jordan canonical form of a matrix?

2008-10-16T19:43:48Z

Soto:

Computing the matrix exponential exp^(At) is easy if A is already in diagonal form (i.e., all elements not on the diagonal are zero). The nxn matrix A can be transformed to a similar diagonal nxn matrix D if and only if A has n linearly independent eigenvectors. This is of great consequence because it simplifies the work required to solve a system of linear ODEs or when using the convolution integral.

If the nxn matrix A does not have n linearly independent eigenvectors, it cannot be represented by a similar diagonal matrix. A similar matrix has the same eigenvalues and eigenvectors as A. We then try to find what are called "generalized eigenvectors" until we have a total of n eigenvectors. Then we can transform matrix A to its Jordan form J = inv(T)*A*T, where T is an invertible nxn matrix with the n eigenvectors (including the generalized eigenvectors) as columns.

If A does not have n linearly independent eigenvectors, then J will not be diagonal, but will be in a block-diagonal form that separates the different "normal modes" of the system into Jordan blocks. To find the matrix exponential of J, one finds the matrix exponential of each Jordan block separately.

For definitions and simple examples you can go to
http://www.maths.surrey.ac.uk/explore/emmaspages/option3.html

or consult the book by Perko titled "Differential Equations and Dynamical Systems".
--[[User:Soto|Luis Soto]] 18:51, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-2]]
[[Category: CDS 101/110 FAQ - Lecture 3-2, Fall 2008]]

How do you actually find the Jordan canonical form of a matrix?

2008-10-16T01:56:32Z

Soto:

Computing the matrix exponential exp^(At) is easy if A is already in diagonal form (i.e., all elements not on the diagonal are zero). The nxn matrix A can be transformed to a similar diagonal nxn matrix D if and only if A has n linearly independent eigenvectors. This is of great consequence because it simplifies the work required to solve a system of linear ODEs or when using the convolution integral.

If the nxn matrix A does not have n linearly independent eigenvectors, it cannot be represented by a similar diagonal matrix. A similar matrix has the same eigenvalues and eigenvectors as A. We then try to find what are called "generalized eigenvectors" until we have a total of n eigenvectors. Then we can transform matrix A to its Jordan form J = inv(P)*A*P, where P is an invertible nxn matrix with the n eigenvectors (including the generalized eigenvectors) as columns.

If A does not have n linearly independent eigenvectors, then J will not be diagonal, but will be in a block-diagonal form that separates the different "normal modes" of the system into Jordan blocks. To find the matrix exponential of J, one finds the matrix exponential of each Jordan block separately.

For definitions and simple examples you can go to
http://www.maths.surrey.ac.uk/explore/emmaspages/option3.html

or consult the book by Perko titled "Differential Equations and Dynamical Systems".
--[[User:Soto|Luis Soto]] 18:51, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

How do you actually find the Jordan canonical form of a matrix?

2008-10-16T01:52:35Z

Soto:

Computing the matrix exponential $\exp^{At}$ is easy if A is already in diagonal form (i.e., all elements not on the diagonal are zero). The nxn matrix A can be transformed to a similar diagonal nxn matrix D if and only if A has n linearly independent eigenvectors. This is of great consequence because it simplifies the work required to solve a system of linear ODEs or when using the convolution integral.

If the nxn matrix A does not have n linearly independent eigenvectors, it cannot be represented by a similar diagonal matrix. A similar matrix has the same eigenvalues and eigenvectors as A. We then try to find what are called "generalized eigenvectors" until we have a total of n eigenvectors. Then we can transform matrix A to its Jordan form J = \inv(P)*A*P, where P is an invertible nxn matrix with the n eigenvectors (including the generalized eigenvectors) as columns.

If A does not have n linearly independent eigenvectors, then J will not be diagonal, but will be in a block-diagonal form that separates the different "normal modes" of the system into Jordan blocks. To find the matrix exponential of J, one finds the matrix exponential of each Jordan block separately.

For definitions and simple examples you can go to
http://www.maths.surrey.ac.uk/explore/emmaspages/option3.html

or consult the book by Perko titled "Differential Equations and Dynamical Systems" available in the CDS library.
--[[User:Soto|Luis Soto]] 18:51, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

How do you actually find the Jordan canonical form of a matrix?

2008-10-16T01:52:01Z

Soto:

Computing the matrix exponential $\exp^{At}$ is easy if A is already in diagonal form (i.e., all elements not on the diagonal are zero). The nxn matrix A can be transformed to a similar diagonal nxn matrix D if and only if A has n linearly independent eigenvectors. This is of great consequence because it simplifies the work required to solve a system of linear ODEs or when using the convolution integral.

If the nxn matrix A does not have n linearly independent eigenvectors, it cannot be represented by a similar diagonal matrix. A similar matrix has the same eigenvalues and eigenvectors as A. We then try to find what are called "generalized eigenvectors" until we have a total of n eigenvectors. Then we can transform matrix A to its Jordan form J = \inv(P)*A*P, where P is an invertible nxn matrix with the n eigenvectors (including the generalized eigenvectors) as columns.

If A does not have n linearly independent eigenvectors, then J will not be diagonal, but will be in a block-diagonal form that separates the different "normal modes" of the system into Jordan blocks. To find the matrix exponential of J, one finds the matrix exponential of each Jordan block separately.

For definitions and simple examples you can go to

http://www.maths.surrey.ac.uk/explore/emmaspages/option3.html

or consult the book by Perko titled "Differential Equations and Dynamical Systems" available in the CDS library.
--[[User:Soto|Luis Soto]] 18:51, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

How do you actually find the Jordan canonical form of a matrix?

2008-10-16T01:51:36Z

Soto:

Computing the matrix exponential \exp^{At} is easy if A is already in diagonal form (i.e., all elements not on the diagonal are zero). The nxn matrix A can be transformed to a similar diagonal nxn matrix D if and only if A has n linearly independent eigenvectors. This is of great consequence because it simplifies the work required to solve a system of linear ODEs or when using the convolution integral.

If the nxn matrix A does not have n linearly independent eigenvectors, it cannot be represented by a similar diagonal matrix. A similar matrix has the same eigenvalues and eigenvectors as A. We then try to find what are called "generalized eigenvectors" until we have a total of n eigenvectors. Then we can transform matrix A to its Jordan form J = \inv(P)*A*P, where P is an invertible nxn matrix with the n eigenvectors (including the generalized eigenvectors) as columns.

If A does not have n linearly independent eigenvectors, then J will not be diagonal, but will be in a block-diagonal form that separates the different "normal modes" of the system into Jordan blocks. To find the matrix exponential of J, one finds the matrix exponential of each Jordan block separately.

For definitions and simple examples you can go to

http://www.maths.surrey.ac.uk/explore/emmaspages/option3.html

or consult the book by Perko titled "Differential Equations and Dynamical Systems" available in the CDS library.
--[[User:Soto|Luis Soto]] 18:51, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

How do you actually find the Jordan canonical form of a matrix?

2008-10-16T01:51:18Z

Soto:

Computing the matrix exponential \exp^{At} is easy if A is already in diagonal form (i.e., all elements not on the diagonal are zero). The nxn matrix A can be transformed to a similar diagonal nxn matrix D if and only if A has n linearly independent eigenvectors. This is of great consequence because it simplifies the work required to solve a system of linear ODEs or when using the convolution integral.

If the nxn matrix A does not have n linearly independent eigenvectors, it cannot be represented by a similar diagonal matrix. A similar matrix has the same eigenvalues and eigenvectors as A. We then try to find what are called "generalized eigenvectors" until we have a total of n eigenvectors. Then we can transform matrix A to its Jordan form J = \inv(P)*A*P, where P is an invertible nxn matrix with the n eigenvectors (including the generalized eigenvectors) as columns.

If A does not have n linearly independent eigenvectors, then J will not be diagonal, but will be in a block-diagonal form that separates the different "normal modes" of the system into Jordan blocks. To find the matrix exponential of J, one finds the matrix exponential of each Jordan block separately.

For definitions and simple examples you can go to

http://www.maths.surrey.ac.uk/explore/emmaspages/option3.html

or consult the book by Perko titled "Differential Equations and Dynamical Systems" available in the CDS library.
--[[User:Soto|Soto]] 18:51, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

Do complex matrices also have a Jordan canonical form?

2008-10-16T01:02:25Z

Soto:

Yes. However, in this course we will only be dealing with matrices having real elements.
--[[User:Soto|Luis Soto]] 18:02, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

Do complex matrices also have a Jordan canonical form?

2008-10-16T01:02:02Z

Soto:

Yes. However, in this course we will only be dealing with matrices having real elements.
--[[User:Soto|Soto]] 18:02, 15 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

What is a plant in the context of this class?

2008-10-14T05:37:11Z

Soto:

A plant for this course will refer to the system or process we are interested in controlling or studying its dynamics. For example, if we want to drive at a reference speed using the cruise control, the motion of the car is the process (plant) we would like to control. The cruise control system implements a control law to modify the dynamics of the car and cruise at the reference speed.
--[[User:Soto|Luis Soto]] 22:36, 13 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

What is a plant in the context of this class?

2008-10-14T05:36:17Z

Soto:

A plant for this course will refer to the system or process we are interested in controlling or studying its dynamics. For example, if we want to drive at a reference speed using the cruise control, the motion of the car is the process (plant) we would like to control. The cruise control system implements a control law to modify the dynamics of the car and cruise at the reference speed.
--[[User:Soto|Soto]] 22:36, 13 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 3-1]]
[[Category: CDS 101/110 FAQ - Lecture 3-1, Fall 2008]]

Do we have to consider complex states when trying to find equilibrium points analytically?

2008-10-13T19:33:14Z

Soto:

If you want to find the equilibrium points analytically for problem 1b of HW2, you can write a 5th order polynomial in terms of z1 or in terms of z2. You will get 2 complex roots and 3 real. Since you are solving for equilibrium points, which are states where dz/dt = 0 and all states have to be real to make physical sense (could represent position, velocity, population density, etc.), you only use the real roots of the 5th order polynomial. In general, we always ignore the complex roots.
Phaseplot, pplane7 and other numerical packages will only plot real states.

Of course, if eigenvalues is what you are trying to find, then you have to consider complex eigenvalues as well. Remember that wherever the eigenvalues of the system dynamics matrix A are located in the complex plane will determine stability and affect system performance.
--[[User:Soto|Luis Soto]] 12:32, 13 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Homework 2]]
[[Category: CDS 101/110 FAQ - Homework 2, Fall 2008]]

Do we have to consider complex states when trying to find equilibrium points analytically?

2008-10-13T19:32:11Z

Soto:

If you want to find the equilibrium points analytically for problem 1b of HW2, you can write a 5th order polynomial in terms of z1 or in terms of z2. You will get 2 complex roots and 3 real. Since you are solving for equilibrium points, which are states where dz/dt = 0 and all states have to be real to make physical sense (could represent position, velocity, population density, etc.), you only use the real roots of the 5th order polynomial. In general, we always ignore the complex roots.
Phaseplot, pplane7 and other numerical packages will only plot real states.

Of course, if eigenvalues is what you are trying to find, then you have to consider complex eigenvalues as well. Remember that wherever the eigenvalues of the system dynamics matrix A are located in the complex plane will determine stability and affect system performance.
--[[User:Soto|Soto]] 12:32, 13 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Homework 2]]
[[Category: CDS 101/110 FAQ - Homework 2, Fall 2008]]

In lecture 1.2, y(x) was used as a function of the state variables. Is y a generic function of vector x?

2008-10-02T01:58:43Z

Soto:

In control theory, we usually use y to represent the output of a system. This output can be a subset of the state variables x1, x2, x3, etc. depending on the # of states and the states that are accessible for measurement by the sensors. For example, if we have sensor matrix C = [1 0], this could mean that only the first state is accessible by the sensors (or that we don't care about the dynamics of the second state in a 2-D system) and the output will be y = Cx = [1 0]*[x1 x2]' = x1. Thus, in this case the output y is a function of only state x1. You will learn later in the course that an "observer" can be used to estimate states that are not accessible for measurement by the sensors.
--[[User:Soto|Luis Soto]] 18:56, 1 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 1-2]]
[[Category: CDS 101/110 FAQ - Lecture 1-2, Fall 2008]]

In lecture 1.2, y(x) was used as a function of the state variables. Is y a generic function of vector x?

2008-10-02T01:58:02Z

Soto:

In control theory, we usually use y to represent the output of a system. This output can be a subset of the state variables x1, x2, x3, etc. depending on the # of states and the states that are accessible for measurement by the sensors. For example, if we have sensor matrix C = [1 0], this could mean that only the first state is accessible by the sensors (or that we don't care about the dynamics of the second state in a 2-D system) and the output will be y = Cx = [1 0]*[x1 x2]' = x1. Thus, in this case the output y is a function of only state x1. You will learn later in the course that an "observer" can be used to estimate states that are not accessible for measurement by the sensors.
--[[User:Soto|Soto]] 18:56, 1 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 1-2]]
[[Category: CDS 101/110 FAQ - Lecture 1-2, Fall 2008]]

In lecture 1.2, y(x) was used as a function of the state variables. Is y a generic function of vector x?

2008-10-02T01:56:08Z

Soto:

In control theory, we usually use y to represent the output of a system. This output can be a subset of the state variables x1, x2, x3, etc. depending on the # of states and the states that are accessible for measurement by the sensors. For example, if we have sensor matrix C = [1 0], this could mean that only the first state is accessible by the sensors (or that we don't care about the dynamics of the second state in a 2-D system) and the output will be y = Cx = [1 0]*[x1 x2]' = x1. Thus, in this example the output y is a function of only state x1. You will learn later in the course that an "observer" can be used to estimate states that are not accessible for measurement by the sensors.
--[[User:Soto|Soto]] 18:56, 1 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 1-2]]
[[Category: CDS 101/110 FAQ - Lecture 1-2, Fall 2008]]

Can I turn my homework by email (pdf file)?

2008-10-02T01:42:18Z

Soto:

No. A printed hardcopy should be turned in.
--[[User:Soto|Luis Soto]] 18:39, 1 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 1-2]]
[[Category: CDS 101/110 FAQ - Lecture 1-2, Fall 2008]]

Can I turn my homework by email (pdf file)?

2008-10-02T01:42:00Z

Soto:

No. A printed hardcopy should be turned in.
--[[User: Luis Soto|Soto]] 18:39, 1 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 1-2]]
[[Category: CDS 101/110 FAQ - Lecture 1-2, Fall 2008]]

Can I turn my homework by email (pdf file)?

2008-10-02T01:39:21Z

Soto:

No. A printed hardcopy should be turned in.
--[[User:Soto|Soto]] 18:39, 1 October 2008 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 1-2]]
[[Category: CDS 101/110 FAQ - Lecture 1-2, Fall 2008]]

CDS 110b, Winter 2008

2008-03-18T05:23:45Z

Soto: /* Announcements */

{{cds110b-wi08}}
<table align=right border=1 width=20% cellpadding=6>
<tr ><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br></li>
<li> [[#Collaboration Policy|Collaboration Policy]] <br></li>
<li> [[#Course Text and References|Course Text and References]] <br></li>

</ul>
</td></tr></table>
This is the homepage for CDS 110b, Introduction to Control Theory for Winter 2008. __NOTOC__ [[Category:Courses]]

<table width=80%>
<tr valign=top>
<td>
'''Instructor'''
* [[User:Murray|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 206 Thomas
* Office hours: Fridays, 3-4 pm, 109 Steele
<td>
'''Teaching Assistants''' ([mailto:cds110-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Luis Soto
* Office hours: Sun 3-4 pm (110 Steele), Tue 3-4 pm (110 Steele)
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 17 Mar 08: HW #8 is graded
** Non-project track: avg = 36, hours = 12
** Project track: avg = 18.9, hours = 7
* 13 Mar 08: {{cds110b-wi08 pdfs|soln7.pdf|HW #7 solutions}} are now posted
* 6 Mar 08: {{cds110b-wi08 pdfs|hw8.pdf|HW #8}} is posted (due 14 Mar 08)
* 5 Mar 08: {{cds110b-wi08 pdfs|soln6.pdf|HW #6 solutions}} are now posted
* 3 Mar 08: [[CDS 110b: Robust Performance|Week 9 - Robust Performance]]
* 3 Mar 08: HW #6 is graded
** Non-project track: avg = 23, hours = 15.5
** Project track: avg = 8, hours: 4.5
* 27 Feb 08: {{cds110b-wi08 pdfs|hw7.pdf|HW #7}} is now posted (due 5 Mar 08)
* 25 Feb 08: [[CDS 110b: Sensor Fusion|Week 8 - Sensor Fusion]]
* 25 Feb 08: HW #5 is graded ({{cds110b-wi08 pdfs|soln5.pdf|solutions}})
** Non-project track: avg = 36.3, hours = 7
** Project track: avg = 17.5, hours: 4
* 20 Feb 08: [[CDS 110b: Kalman Filters|Week 7 - Kalman Filters]]
** {{cds110b-wi08 pdfs|hw6.pdf|HW #6}} is now posted (due 27 Feb 08)

== Course Syllabus ==

'''Course Desciption and Goals:''' CDS 110b focuses on intermediate topics in control theory, including state estimation using Kalman filters, optimal control methods and modern control design techniques. Upon completion of the course, students will be able to design and analyze control systems of moderate complexity. Students may optionally participate in a course project in lieu of taking the midterm and final. Students participating in the course project will learn how to implement and test control systems on a modern experimental system.

* [http://listserv.cds.caltech.edu/mailman/listinfo/cds110-students cds110-students mailing list] - all students in the class should be signed up on this list ([http://listserv.cds.caltech.edu/pipermail/cds110-students/ archive])

=== Grading ===
The final grade will be based on homework sets, a midterm exam and a final exam:
* '''Homework: 50%''' <br> Homework sets will be handed out weekly and will generally be due one week later at 5 pm to the box outside of 109 Steele. <font color=blue>Students are allowed three grace periods of two days each which can be used at any time (but no more than 1 grace period per homework set).</font> Additional extensions on homework will only be allowed under exceptional circumstances and require prior permission for the instructor.<br>
* '''Midterm: 20%''' <br> A midterm exam will be handed out at the beginning of midterms week and due at the end of the midterm examination period. The midterm exam will be open book.<br>
* '''Final: 30%''' <br>The final exam will be handed out on the last day of class due at the end of finals week. It will be an open book exam.<br>

Note: students working on the [[#Course Project|course project]] will not be required to take the midterm or final. Instead, two project reports will be due documenting the experimental work performed as part of the class. In addition, students working on the course project are only required to complete the first 2 problems on each homework set.

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. All solutions that are handed in should reflect your understanding of the subject matter at the time of writing. MATLAB scripts and plots are considered part of your writeup and should be done individually. Use of written solutions from prior years or other sources is not allowed.

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The main course text is
* R. M. Murray, ''[[AM:Supplement: Optimization-Based Control|Optimization-Based Control]]'', Preprint, 2008.

You may find the following texts useful as well:
* K. J. {{Astrom}} and R. M. Murray, ''[[AM:Main Page|Feedback Systems]]'', Princeton University Press, 2008.
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', Dover, 2004.
* F. L. Lewis and V. L. Syrmos, ''Optimal Control'', Second Edition, Wiley-IEEE, 1995. ([http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Google Books])
* A. D. Lewis, ''[http://penelope.mast.queensu.ca/math332/notes.shtml A Mathematical Approach to Classical Control]'', 2003.
* J. Doyle, B. Francis, A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.html ''Feedback Control Theory''], Macmillan, 1992.

<hr>
<span id=Old_Announcements />'''Old Announcements'''
* 16 Feb 08: {{cds110b-wi08 pdfs|mtsolns.pdf|Midterm solutions}} are posted. Average = 37/50, std = 6
* 13 Feb 08: {{cds110b-wi08 pdfs|hw5.pdf|HW #5}} is posted (due 20 Feb 08)
* 11 Feb 08: [[CDS 110b: Stochastic Systems|Week 6 - Stochastic Systems]]
* 10 Feb 08: HW #4 is graded;
** Non-project track: avg = 30, hours = 10
** Project track: avg = 19, hours: 5
* 8 Feb 08: [[CDS 110b: State Estimation|Week 5 - State Estimation]]
* 8 Feb 08: {{cds110b-wi08 pdfs|soln4.pdf|HW #4 solutions}} are now available.
* 6 Feb 08: {{cds110b-wi08 pdfs|soln3.pdf|HW #3 solutions}} are now available.
** Non-project track: avg = 32 +/- 8, hours = 11.5
** Project track: avg = 17 +/-2, hours: 5
* 3 Feb 08: {{cds110b-wi08 pdfs|soln2.pdf|HW #2 solutions}} are now available.
* 30 Jan 08: HW #2 is graded; Avg score = 28/30 +/- 2, average time = 8 hours.
* 30 Jan 08: [[CDS 110b: Receding Horizon Control|Week 4 - Receding Horizon Control]]
** Homework 4 (due 6 Feb 08): {{obc08|problems 3.1, 3.3, 3.2}} (students working on course project do first two problems only)
* 29 Jan 08: Office hours today at 3-4pm will be held in 114 STL.
* 26 Jan 08: an updated version of {{cds110b-wi08 pdfs|optimal-26Jan08.pdf|Chapter 2}} of the course text has been posted (small fixes)
* 24 Jan 08: {{cds110b-wi08 pdfs|soln1.pdf|HW #1 solutions}} are available; Avg score = 36 +/- 2, average time = 16.5 hours (!).
* 23 Jan 08: [[CDS 110b: Linear Quadratic Regulators|Week 3 - LQR]]
** {{cds110b-wi08 pdfs|hw3.pdf|Homework 3}} (due 30 Jan 08)
* 14 Jan 08: [[CDS 110b: Optimal Control|Week 2 - Optimal Control]]
** Homework 2 (due 22 Jan 08): {{obc08|problems 2.3, 2.4a-d, 2.6}}
* 7 Jan 08: [[CDS 110b: Two Degree of Freedom Control Design|Week 1 - Two Degree of Freedom Control Design]]
** Homework 1 (due 14 Jan 08): {{obc08|problems 1.2, 1.3, 1.4 and 1.5}}
* 13 Dec 07: initial web page created; this is still in DRAFT form

[[Category:Courses]] [[Category:2007-08 Courses]]

CDS 110b, Winter 2008

2008-03-18T05:22:22Z

Soto: /* Announcements */

{{cds110b-wi08}}
<table align=right border=1 width=20% cellpadding=6>
<tr ><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br></li>
<li> [[#Collaboration Policy|Collaboration Policy]] <br></li>
<li> [[#Course Text and References|Course Text and References]] <br></li>

</ul>
</td></tr></table>
This is the homepage for CDS 110b, Introduction to Control Theory for Winter 2008. __NOTOC__ [[Category:Courses]]

<table width=80%>
<tr valign=top>
<td>
'''Instructor'''
* [[User:Murray|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 206 Thomas
* Office hours: Fridays, 3-4 pm, 109 Steele
<td>
'''Teaching Assistants''' ([mailto:cds110-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Luis Soto
* Office hours: Sun 3-4 pm (110 Steele), Tue 3-4 pm (110 Steele)
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 17 Mar 08: HW #8 is graded
** Non-project track: avg = 36, hours = 12
** Project track: avg = 18.9, hours: 7
* 13 Mar 08: {{cds110b-wi08 pdfs|soln7.pdf|HW #7 solutions}} are now posted
* 6 Mar 08: {{cds110b-wi08 pdfs|hw8.pdf|HW #8}} is posted (due 14 Mar 08)
* 5 Mar 08: {{cds110b-wi08 pdfs|soln6.pdf|HW #6 solutions}} are now posted
* 3 Mar 08: [[CDS 110b: Robust Performance|Week 9 - Robust Performance]]
* 3 Mar 08: HW #6 is graded
** Non-project track: avg = 23, hours = 15.5
** Project track: avg = 8, hours: 4.5
* 27 Feb 08: {{cds110b-wi08 pdfs|hw7.pdf|HW #7}} is now posted (due 5 Mar 08)
* 25 Feb 08: [[CDS 110b: Sensor Fusion|Week 8 - Sensor Fusion]]
* 25 Feb 08: HW #5 is graded ({{cds110b-wi08 pdfs|soln5.pdf|solutions}})
** Non-project track: avg = 36.3, hours = 7
** Project track: avg = 17.5, hours: 4
* 20 Feb 08: [[CDS 110b: Kalman Filters|Week 7 - Kalman Filters]]
** {{cds110b-wi08 pdfs|hw6.pdf|HW #6}} is now posted (due 27 Feb 08)

== Course Syllabus ==

'''Course Desciption and Goals:''' CDS 110b focuses on intermediate topics in control theory, including state estimation using Kalman filters, optimal control methods and modern control design techniques. Upon completion of the course, students will be able to design and analyze control systems of moderate complexity. Students may optionally participate in a course project in lieu of taking the midterm and final. Students participating in the course project will learn how to implement and test control systems on a modern experimental system.

* [http://listserv.cds.caltech.edu/mailman/listinfo/cds110-students cds110-students mailing list] - all students in the class should be signed up on this list ([http://listserv.cds.caltech.edu/pipermail/cds110-students/ archive])

=== Grading ===
The final grade will be based on homework sets, a midterm exam and a final exam:
* '''Homework: 50%''' <br> Homework sets will be handed out weekly and will generally be due one week later at 5 pm to the box outside of 109 Steele. <font color=blue>Students are allowed three grace periods of two days each which can be used at any time (but no more than 1 grace period per homework set).</font> Additional extensions on homework will only be allowed under exceptional circumstances and require prior permission for the instructor.<br>
* '''Midterm: 20%''' <br> A midterm exam will be handed out at the beginning of midterms week and due at the end of the midterm examination period. The midterm exam will be open book.<br>
* '''Final: 30%''' <br>The final exam will be handed out on the last day of class due at the end of finals week. It will be an open book exam.<br>

Note: students working on the [[#Course Project|course project]] will not be required to take the midterm or final. Instead, two project reports will be due documenting the experimental work performed as part of the class. In addition, students working on the course project are only required to complete the first 2 problems on each homework set.

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. All solutions that are handed in should reflect your understanding of the subject matter at the time of writing. MATLAB scripts and plots are considered part of your writeup and should be done individually. Use of written solutions from prior years or other sources is not allowed.

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The main course text is
* R. M. Murray, ''[[AM:Supplement: Optimization-Based Control|Optimization-Based Control]]'', Preprint, 2008.

You may find the following texts useful as well:
* K. J. {{Astrom}} and R. M. Murray, ''[[AM:Main Page|Feedback Systems]]'', Princeton University Press, 2008.
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', Dover, 2004.
* F. L. Lewis and V. L. Syrmos, ''Optimal Control'', Second Edition, Wiley-IEEE, 1995. ([http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Google Books])
* A. D. Lewis, ''[http://penelope.mast.queensu.ca/math332/notes.shtml A Mathematical Approach to Classical Control]'', 2003.
* J. Doyle, B. Francis, A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.html ''Feedback Control Theory''], Macmillan, 1992.

<hr>
<span id=Old_Announcements />'''Old Announcements'''
* 16 Feb 08: {{cds110b-wi08 pdfs|mtsolns.pdf|Midterm solutions}} are posted. Average = 37/50, std = 6
* 13 Feb 08: {{cds110b-wi08 pdfs|hw5.pdf|HW #5}} is posted (due 20 Feb 08)
* 11 Feb 08: [[CDS 110b: Stochastic Systems|Week 6 - Stochastic Systems]]
* 10 Feb 08: HW #4 is graded;
** Non-project track: avg = 30, hours = 10
** Project track: avg = 19, hours: 5
* 8 Feb 08: [[CDS 110b: State Estimation|Week 5 - State Estimation]]
* 8 Feb 08: {{cds110b-wi08 pdfs|soln4.pdf|HW #4 solutions}} are now available.
* 6 Feb 08: {{cds110b-wi08 pdfs|soln3.pdf|HW #3 solutions}} are now available.
** Non-project track: avg = 32 +/- 8, hours = 11.5
** Project track: avg = 17 +/-2, hours: 5
* 3 Feb 08: {{cds110b-wi08 pdfs|soln2.pdf|HW #2 solutions}} are now available.
* 30 Jan 08: HW #2 is graded; Avg score = 28/30 +/- 2, average time = 8 hours.
* 30 Jan 08: [[CDS 110b: Receding Horizon Control|Week 4 - Receding Horizon Control]]
** Homework 4 (due 6 Feb 08): {{obc08|problems 3.1, 3.3, 3.2}} (students working on course project do first two problems only)
* 29 Jan 08: Office hours today at 3-4pm will be held in 114 STL.
* 26 Jan 08: an updated version of {{cds110b-wi08 pdfs|optimal-26Jan08.pdf|Chapter 2}} of the course text has been posted (small fixes)
* 24 Jan 08: {{cds110b-wi08 pdfs|soln1.pdf|HW #1 solutions}} are available; Avg score = 36 +/- 2, average time = 16.5 hours (!).
* 23 Jan 08: [[CDS 110b: Linear Quadratic Regulators|Week 3 - LQR]]
** {{cds110b-wi08 pdfs|hw3.pdf|Homework 3}} (due 30 Jan 08)
* 14 Jan 08: [[CDS 110b: Optimal Control|Week 2 - Optimal Control]]
** Homework 2 (due 22 Jan 08): {{obc08|problems 2.3, 2.4a-d, 2.6}}
* 7 Jan 08: [[CDS 110b: Two Degree of Freedom Control Design|Week 1 - Two Degree of Freedom Control Design]]
** Homework 1 (due 14 Jan 08): {{obc08|problems 1.2, 1.3, 1.4 and 1.5}}
* 13 Dec 07: initial web page created; this is still in DRAFT form

[[Category:Courses]] [[Category:2007-08 Courses]]

CDS 110b, Winter 2008

2008-02-25T21:56:08Z

Soto: /* Announcements */

{{cds110b-wi08}}
<table align=right border=1 width=20% cellpadding=6>
<tr ><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br></li>
<li> [[#Collaboration Policy|Collaboration Policy]] <br></li>
<li> [[#Course Text and References|Course Text and References]] <br></li>

</ul>
</td></tr></table>
This is the homepage for CDS 110b, Introduction to Control Theory for Winter 2008. __NOTOC__ [[Category:Courses]]

<table width=80%>
<tr valign=top>
<td>
'''Instructor'''
* [[User:Murray|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 206 Thomas
* Office hours: Fridays, 3-4 pm, 109 Steele
<td>
'''Teaching Assistants''' ([mailto:cds110-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Luis Soto
* Office hours: Fri 3-4 pm (214B Thomas), Sun 3-4 pm (110 Steele)
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 25 Feb 08: HW #5 is graded;
** Non-project track: avg = 36.3, hours = 7
** Project track: avg = 17.5, hours: 4
* 20 Feb 08: [[CDS 110b: Kalman Filters|Week 7 - Kalman Filters]]
** {{cds110b-wi08 pdfs|hw6.pdf|HW #6}} is now posted (due 27 Feb 08)
* 16 Feb 08: {{cds110b-wi08 pdfs|mtsolns.pdf|Midterm solutions}} are posted. Average = 37/50, std = 6
* 13 Feb 08: {{cds110b-wi08 pdfs|hw5.pdf|HW #5}} is posted (due 20 Feb 08)
* 11 Feb 08: [[CDS 110b: Stochastic Systems|Week 6 - Stochastic Systems]]
* 10 Feb 08: HW #4 is graded;
** Non-project track: avg = 30, hours = 10
** Project track: avg = 19, hours: 5
* 8 Feb 08: [[CDS 110b: State Estimation|Week 5 - State Estimation]]

== Course Syllabus ==

'''Course Desciption and Goals:''' CDS 110b focuses on intermediate topics in control theory, including state estimation using Kalman filters, optimal control methods and modern control design techniques. Upon completion of the course, students will be able to design and analyze control systems of moderate complexity. Students may optionally participate in a course project in lieu of taking the midterm and final. Students participating in the course project will learn how to implement and test control systems on a modern experimental system.

* [http://listserv.cds.caltech.edu/mailman/listinfo/cds110-students cds110-students mailing list] - all students in the class should be signed up on this list ([http://listserv.cds.caltech.edu/pipermail/cds110-students/ archive])

=== Grading ===
The final grade will be based on homework sets, a midterm exam and a final exam:
* '''Homework: 50%''' <br> Homework sets will be handed out weekly and will generally be due one week later at 5 pm to the box outside of 109 Steele. <font color=blue>Students are allowed three grace periods of two days each which can be used at any time (but no more than 1 grace period per homework set).</font> Additional extensions on homework will only be allowed under exceptional circumstances and require prior permission for the instructor.<br>
* '''Midterm: 20%''' <br> A midterm exam will be handed out at the beginning of midterms week and due at the end of the midterm examination period. The midterm exam will be open book.<br>
* '''Final: 30%''' <br>The final exam will be handed out on the last day of class due at the end of finals week. It will be an open book exam.<br>

Note: students working on the [[#Course Project|course project]] will not be required to take the midterm or final. Instead, two project reports will be due documenting the experimental work performed as part of the class. In addition, students working on the course project are only required to complete the first 2 problems on each homework set.

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. All solutions that are handed in should reflect your understanding of the subject matter at the time of writing. MATLAB scripts and plots are considered part of your writeup and should be done individually. Use of written solutions from prior years or other sources is not allowed.

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The main course text is
* R. M. Murray, ''[[AM:Supplement: Optimization-Based Control|Optimization-Based Control]]'', Preprint, 2008.

You may find the following texts useful as well:
* K. J. {{Astrom}} and R. M. Murray, ''[[AM:Main Page|Feedback Systems]]'', Princeton University Press, 2008.
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', Dover, 2004.
* F. L. Lewis and V. L. Syrmos, ''Optimal Control'', Second Edition, Wiley-IEEE, 1995. ([http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Google Books])
* A. D. Lewis, ''[http://penelope.mast.queensu.ca/math332/notes.shtml A Mathematical Approach to Classical Control]'', 2003.
* J. Doyle, B. Francis, A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.html ''Feedback Control Theory''], Macmillan, 1992.

<hr>
<span id=Old_Announcements />'''Old Announcements'''
* 8 Feb 08: {{cds110b-wi08 pdfs|soln4.pdf|HW #4 solutions}} are now available.
* 6 Feb 08: {{cds110b-wi08 pdfs|soln3.pdf|HW #3 solutions}} are now available.
** Non-project track: avg = 32 +/- 8, hours = 11.5
** Project track: avg = 17 +/-2, hours: 5
* 3 Feb 08: {{cds110b-wi08 pdfs|soln2.pdf|HW #2 solutions}} are now available.
* 30 Jan 08: HW #2 is graded; Avg score = 28/30 +/- 2, average time = 8 hours.
* 30 Jan 08: [[CDS 110b: Receding Horizon Control|Week 4 - Receding Horizon Control]]
** Homework 4 (due 6 Feb 08): {{obc08|problems 3.1, 3.3, 3.2}} (students working on course project do first two problems only)
* 29 Jan 08: Office hours today at 3-4pm will be held in 114 STL.
* 26 Jan 08: an updated version of {{cds110b-wi08 pdfs|optimal-26Jan08.pdf|Chapter 2}} of the course text has been posted (small fixes)
* 24 Jan 08: {{cds110b-wi08 pdfs|soln1.pdf|HW #1 solutions}} are available; Avg score = 36 +/- 2, average time = 16.5 hours (!).
* 23 Jan 08: [[CDS 110b: Linear Quadratic Regulators|Week 3 - LQR]]
** {{cds110b-wi08 pdfs|hw3.pdf|Homework 3}} (due 30 Jan 08)
* 14 Jan 08: [[CDS 110b: Optimal Control|Week 2 - Optimal Control]]
** Homework 2 (due 22 Jan 08): {{obc08|problems 2.3, 2.4a-d, 2.6}}
* 7 Jan 08: [[CDS 110b: Two Degree of Freedom Control Design|Week 1 - Two Degree of Freedom Control Design]]
** Homework 1 (due 14 Jan 08): {{obc08|problems 1.2, 1.3, 1.4 and 1.5}}
* 13 Dec 07: initial web page created; this is still in DRAFT form

[[Category:Courses]] [[Category:2007-08 Courses]]

CDS 110b, Winter 2008

2008-02-11T00:24:29Z

Soto: /* Announcements */

{{cds110b-wi08}}
<table align=right border=1 width=20% cellpadding=6>
<tr ><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br></li>
<li> [[#Collaboration Policy|Collaboration Policy]] <br></li>
<li> [[#Course Text and References|Course Text and References]] <br></li>

</ul>
</td></tr></table>
This is the homepage for CDS 110b, Introduction to Control Theory for Winter 2008. __NOTOC__ [[Category:Courses]]

<table width=80%>
<tr valign=top>
<td>
'''Instructor'''
* [[User:Murray|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 206 Thomas
* Office hours: Fridays, 3-4 pm, 109 Steele
<td>
'''Teaching Assistants''' ([mailto:cds110-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Luis Soto
* Office hours: Fri 3-4 pm (214B Thomas), Sun 3-4 pm (110 Steele)
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 10 Feb 08: HW #4 is graded;
** Non-project track: avg = 30, hours = 10
** Project track: avg = 19, hours: 5
* 8 Feb 08: {{cds110b-wi08 pdfs|soln4.pdf|HW #4 solutions}} are now available.
* 6 Feb 08: {{cds110b-wi08 pdfs|soln3.pdf|HW #3 solutions}} are now available.
** Non-project track: avg = 32 +/- 8, hours = 11.5
** Project track: avg = 17 +/-2, hours: 5
* 3 Feb 08: {{cds110b-wi08 pdfs|soln2.pdf|HW #2 solutions}} are now available.
* 30 Jan 08: HW #2 is graded; Avg score = 28/30 +/- 2, average time = 8 hours.
* 30 Jan 08: [[CDS 110b: Receding Horizon Control|Week 4 - Receding Horizon Control]]
** Homework 4 (due 6 Feb 08): {{obc08|problems 3.1, 3.3, 3.2}} (students working on course project do first two problems only)

== Course Syllabus ==

'''Course Desciption and Goals:''' CDS 110b focuses on intermediate topics in control theory, including state estimation using Kalman filters, optimal control methods and modern control design techniques. Upon completion of the course, students will be able to design and analyze control systems of moderate complexity. Students may optionally participate in a course project in lieu of taking the midterm and final. Students participating in the course project will learn how to implement and test control systems on a modern experimental system.

* [http://listserv.cds.caltech.edu/mailman/listinfo/cds110-students cds110-students mailing list] - all students in the class should be signed up on this list ([http://listserv.cds.caltech.edu/pipermail/cds110-students/ archive])

=== Grading ===
The final grade will be based on homework sets, a midterm exam and a final exam:
* '''Homework: 50%''' <br> Homework sets will be handed out weekly and will generally be due one week later at 5 pm to the box outside of 109 Steele. <font color=blue>Students are allowed three grace periods of two days each which can be used at any time (but no more than 1 grace period per homework set).</font> Additional extensions on homework will only be allowed under exceptional circumstances and require prior permission for the instructor.<br>
* '''Midterm: 20%''' <br> A midterm exam will be handed out at the beginning of midterms week and due at the end of the midterm examination period. The midterm exam will be open book.<br>
* '''Final: 30%''' <br>The final exam will be handed out on the last day of class due at the end of finals week. It will be an open book exam.<br>

Note: students working on the [[#Course Project|course project]] will not be required to take the midterm or final. Instead, two project reports will be due documenting the experimental work performed as part of the class. In addition, students working on the course project are only required to complete the first 2 problems on each homework set.

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. All solutions that are handed in should reflect your understanding of the subject matter at the time of writing. MATLAB scripts and plots are considered part of your writeup and should be done individually. Use of written solutions from prior years or other sources is not allowed.

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The main course text is
* R. M. Murray, ''[[AM:Supplement: Optimization-Based Control|Optimization-Based Control]]'', Preprint, 2008.

You may find the following texts useful as well:
* K. J. {{Astrom}} and R. M. Murray, ''[[AM:Main Page|Feedback Systems]]'', Princeton University Press, 2008.
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', Dover, 2004.
* F. L. Lewis and V. L. Syrmos, ''Optimal Control'', Second Edition, Wiley-IEEE, 1995. ([http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Google Books])
* A. D. Lewis, ''[http://penelope.mast.queensu.ca/math332/notes.shtml A Mathematical Approach to Classical Control]'', 2003.
* J. Doyle, B. Francis, A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.html ''Feedback Control Theory''], Macmillan, 1992.

<hr>
<span id=Old_Announcements />'''Old Announcements'''
* 29 Jan 08: Office hours today at 3-4pm will be held in 114 STL.
* 26 Jan 08: an updated version of {{cds110b-wi08 pdfs|optimal-26Jan08.pdf|Chapter 2}} of the course text has been posted (small fixes)
* 24 Jan 08: {{cds110b-wi08 pdfs|soln1.pdf|HW #1 solutions}} are available; Avg score = 36 +/- 2, average time = 16.5 hours (!).
* 23 Jan 08: [[CDS 110b: Linear Quadratic Regulators|Week 3 - LQR]]
** {{cds110b-wi08 pdfs|hw3.pdf|Homework 3}} (due 30 Jan 08)
* 14 Jan 08: [[CDS 110b: Optimal Control|Week 2 - Optimal Control]]
** Homework 2 (due 22 Jan 08): {{obc08|problems 2.3, 2.4a-d, 2.6}}
* 7 Jan 08: [[CDS 110b: Two Degree of Freedom Control Design|Week 1 - Two Degree of Freedom Control Design]]
** Homework 1 (due 14 Jan 08): {{obc08|problems 1.2, 1.3, 1.4 and 1.5}}
* 13 Dec 07: initial web page created; this is still in DRAFT form

[[Category:Courses]] [[Category:2007-08 Courses]]

CDS 110b, Winter 2008

2008-01-31T05:40:34Z

Soto: /* Announcements */

{{cds110b-wi08}}
<table align=right border=1 width=20% cellpadding=6>
<tr ><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br></li>
<li> [[#Collaboration Policy|Collaboration Policy]] <br></li>
<li> [[#Course Text and References|Course Text and References]] <br></li>

</ul>
</td></tr></table>
This is the homepage for CDS 110b, Introduction to Control Theory for Winter 2008. __NOTOC__ [[Category:Courses]]

<table width=80%>
<tr valign=top>
<td>
'''Instructor'''
* [[User:Murray|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 206 Thomas
* Office hours: Fridays, 3-4 pm, 109 Steele
<td>
'''Teaching Assistants''' ([mailto:cds110-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Luis Soto
* Office hours: Fri 3-4 pm (214B Thomas), Sun 3-4 pm (110 Steele)
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 30 Jan 08: HW #2 is graded; Avg score = 28/30 +/- 2, average time = 8 hours.
* 30 Jan 08: [[CDS 110b: Receding Horizon Control|Week 4 - Receding Horizon Control]]
** Homework 4 (due 6 Feb 08): {{obc08|problems 3.1, 3.3, 3.2}} (students working on course project do first two problems only)
* 29 Jan 08: Office hours today at 3-4pm will be held in 114 STL.
* 26 Jan 08: an updated version of {{cds110b-wi08 pdfs|optimal-26Jan08.pdf|Chapter 2}} of the course text has been posted (small fixes)
* 24 Jan 08: {{cds110b-wi08 pdfs|soln1.pdf|HW #1 solutions}} are available; Avg score = 36 +/- 2, average time = 16.5 hours (!).
* 23 Jan 08: [[CDS 110b: Linear Quadratic Regulators|Week 3 - LQR]]
** {{cds110b-wi08 pdfs|hw3.pdf|Homework 3}} (due 30 Jan 08)
* 14 Jan 08: [[CDS 110b: Optimal Control|Week 2 - Optimal Control]]
** Homework 2 (due 22 Jan 08): {{obc08|problems 2.3, 2.4a-d, 2.6}}
* 7 Jan 08: [[CDS 110b: Two Degree of Freedom Control Design|Week 1 - Two Degree of Freedom Control Design]]
** Homework 1 (due 14 Jan 08): {{obc08|problems 1.2, 1.3, 1.4 and 1.5}}
* 13 Dec 07: initial web page created; this is still in DRAFT form

== Course Syllabus ==

'''Course Desciption and Goals:''' CDS 110b focuses on intermediate topics in control theory, including state estimation using Kalman filters, optimal control methods and modern control design techniques. Upon completion of the course, students will be able to design and analyze control systems of moderate complexity. Students may optionally participate in a course project in lieu of taking the midterm and final. Students participating in the course project will learn how to implement and test control systems on a modern experimental system.

* [http://listserv.cds.caltech.edu/mailman/listinfo/cds110-students cds110-students mailing list] - all students in the class should be signed up on this list ([http://listserv.cds.caltech.edu/pipermail/cds110-students/ archive])

=== Grading ===
The final grade will be based on homework sets, a midterm exam and a final exam:
* '''Homework: 50%''' <br> Homework sets will be handed out weekly and will generally be due one week later at 5 pm to the box outside of 109 Steele. <font color=blue>Students are allowed three grace periods of two days each which can be used at any time (but no more than 1 grace period per homework set).</font> Additional extensions on homework will only be allowed under exceptional circumstances and require prior permission for the instructor.<br>
* '''Midterm: 20%''' <br> A midterm exam will be handed out at the beginning of midterms week and due at the end of the midterm examination period. The midterm exam will be open book.<br>
* '''Final: 30%''' <br>The final exam will be handed out on the last day of class due at the end of finals week. It will be an open book exam.<br>

Note: students working on the [[#Course Project|course project]] will not be required to take the midterm or final. Instead, two project reports will be due documenting the experimental work performed as part of the class. In addition, students working on the course project are only required to complete the first 2 problems on each homework set.

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. All solutions that are handed in should reflect your understanding of the subject matter at the time of writing. MATLAB scripts and plots are considered part of your writeup and should be done individually. Use of written solutions from prior years or other sources is not allowed.

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The main course text is
* R. M. Murray, ''[[AM:Supplement: Optimization-Based Control|Optimization-Based Control]]'', Preprint, 2008.

You may find the following texts useful as well:
* K. J. {{Astrom}} and R. M. Murray, ''[[AM:Main Page|Feedback Systems]]'', Princeton University Press, 2008.
* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', Dover, 2004.
* F. L. Lewis and V. L. Syrmos, ''Optimal Control'', Second Edition, Wiley-IEEE, 1995. ([http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Google Books])
* A. D. Lewis, ''[http://penelope.mast.queensu.ca/math332/notes.shtml A Mathematical Approach to Classical Control]'', 2003.
* J. Doyle, B. Francis, A. Tannenbaum, [http://www.control.utoronto.ca/people/profs/francis/dft.html ''Feedback Control Theory''], Macmillan, 1992.

<hr>
<span id=Old_Announcements />'''Old Announcements'''

[[Category:Courses]] [[Category:2007-08 Courses]]

CDS 110b: Optimal Control

2008-01-23T04:07:45Z

Soto: /* Frequently Asked Questions */

{{cds110b-wi08}}
This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle. __NOTOC__

* {{cds110b-wi08 pdfs|L2-1_optimal.pdf|Lecture notes: optimal control}}
* Homework 2 (due 22 Jan @ 5 pm): {{obc08|problems 2.3, 2.4a-d, 2.6}}

== References and Further Reading ==
* R. M. Murray, ''Optimization-Based Control''. Preprint, 2008: {{cds110b-wi08 pdfs|optimal_14Jan08.pdf|Chapter 2 - Optimal Control}}
* {{cds110b-pdfs|LS95-optimal.pdf|Excerpt from LS95 on optimal control}} - This excerpt is from [http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Lewis and Syrmos, 1995] and gives a derivation of the necessary conditions for optimaliity. A few pages have been left out from the middle that contained some additional examples (which you can find in similar books in the library, if you are interested). Other parts of the book can be searched via [http://books.google.com Google Books] and purchased online.
* [http://www.statslab.cam.ac.uk/~rrw1/oc/L13.pdf Notes on Pontryagin's Maximum Principle] - these come from a set of [http://www.statslab.cam.ac.uk/~rrw1/oc/index.html lecture notes on optimization and control] by [http://www.statslab.cam.ac.uk/~rrw1/ Richard Weber] at Cambridge University. The notes are based on dynamic programming (DP) and uses a slightly different notation than we used in class.

== Frequently Asked Questions ==
'''Q: Could you please explain what the psi function is or what it means if psi(x(T))=0 versus what it means if psi(x(T))=x(T)?'''

<blockquote>
<p>The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0. If we take x_f=0, then psi(x(T))=x(T).</p>

<p>Luis Soto, 22 Jan 08</p>
</blockquote>
'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''

<blockquote>
<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(t_f) of the optimal solution won't be exactly 0,
but will be close enough for the intent of this problem.</p>

<p>Luis Soto, 21 Jan 08</p>
</blockquote>

'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''

<blockquote>
<p>Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts. </p>

<p>Julia Braman, 18 Jan 08</p>
</blockquote>

'''Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for <math>u</math> obtained?'''

<blockquote>
<p>Pontryagin's Maximum Principle says that <math>u</math> has to be chosen to minimise the Hamiltonian <math>H(x,u,\lambda)</math> for given values of <math>x</math> and <math>\lambda</math>. In the example, <math>H = 1 + ({\lambda}^TA)x + ({\lambda}^TB)u</math>. At first glance, it seems that the more negative <math>u</math> is the more <math>H</math> will be minimised. And since the most negative value of <math>u</math> allowed is <math>-1</math>, <math>u=-1</math>. However, the co-efficient of <math>u</math> may be of either sign. Therefore, the sign of <math>u</math> has to be chosen such that the sign of the term <math>({\lambda}^TB)u</math> is negative. That's how we come up with <math>u = -sign({\lambda}^TB)</math>.</p>

<p>Shaunak Sen, 12 Jan 06</p>
</blockquote>

'''Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that <math>T</math> is the final time and <sup><math>T</math></sup> (superscript T) is a transpose operation. Am I correct in my assumption?'''

<blockquote>
<p>Yes, you are correct.</p>

<p>Jeremy Gillula, 07 Jan 05</p>
</blockquote>

CDS 110b: Optimal Control

2008-01-23T04:02:57Z

Soto: /* Frequently Asked Questions */

{{cds110b-wi08}}
This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle. __NOTOC__

* {{cds110b-wi08 pdfs|L2-1_optimal.pdf|Lecture notes: optimal control}}
* Homework 2 (due 22 Jan @ 5 pm): {{obc08|problems 2.3, 2.4a-d, 2.6}}

== References and Further Reading ==
* R. M. Murray, ''Optimization-Based Control''. Preprint, 2008: {{cds110b-wi08 pdfs|optimal_14Jan08.pdf|Chapter 2 - Optimal Control}}
* {{cds110b-pdfs|LS95-optimal.pdf|Excerpt from LS95 on optimal control}} - This excerpt is from [http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Lewis and Syrmos, 1995] and gives a derivation of the necessary conditions for optimaliity. A few pages have been left out from the middle that contained some additional examples (which you can find in similar books in the library, if you are interested). Other parts of the book can be searched via [http://books.google.com Google Books] and purchased online.
* [http://www.statslab.cam.ac.uk/~rrw1/oc/L13.pdf Notes on Pontryagin's Maximum Principle] - these come from a set of [http://www.statslab.cam.ac.uk/~rrw1/oc/index.html lecture notes on optimization and control] by [http://www.statslab.cam.ac.uk/~rrw1/ Richard Weber] at Cambridge University. The notes are based on dynamic programming (DP) and uses a slightly different notation than we used in class.

== Frequently Asked Questions ==
'''Q: Could you please explain what the psi function is or what it means if psi(x(T))=0 versus what it means if psi(x(T))=x(T)?'''

<blockquote>
<p>The psi function represents a general form of terminal constraint for the state variables. It gives a way of indicating which states have a terminal cost attached to them. For example, by defining psi_i(x(T))=x_i(T)-x_i,f for i=1,2,...n, we can impose terminal costs on all states (a fully constrained case) by letting p=n (n being the # of states). When we optimize over time and want x(T)=x_f, then x(T)-x_f=0, and so psi(x(T))=0.</p>

<p>Luis Soto, 22 Jan 08</p>
</blockquote>
'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''

<blockquote>
<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(t_f) of the optimal solution won't be exactly 0,
but will be close enough for the intent of this problem.</p>

<p>Luis Soto, 21 Jan 08</p>
</blockquote>

'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''

<blockquote>
<p>Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts. </p>

<p>Julia Braman, 18 Jan 08</p>
</blockquote>

'''Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for <math>u</math> obtained?'''

<blockquote>
<p>Pontryagin's Maximum Principle says that <math>u</math> has to be chosen to minimise the Hamiltonian <math>H(x,u,\lambda)</math> for given values of <math>x</math> and <math>\lambda</math>. In the example, <math>H = 1 + ({\lambda}^TA)x + ({\lambda}^TB)u</math>. At first glance, it seems that the more negative <math>u</math> is the more <math>H</math> will be minimised. And since the most negative value of <math>u</math> allowed is <math>-1</math>, <math>u=-1</math>. However, the co-efficient of <math>u</math> may be of either sign. Therefore, the sign of <math>u</math> has to be chosen such that the sign of the term <math>({\lambda}^TB)u</math> is negative. That's how we come up with <math>u = -sign({\lambda}^TB)</math>.</p>

<p>Shaunak Sen, 12 Jan 06</p>
</blockquote>

'''Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that <math>T</math> is the final time and <sup><math>T</math></sup> (superscript T) is a transpose operation. Am I correct in my assumption?'''

<blockquote>
<p>Yes, you are correct.</p>

<p>Jeremy Gillula, 07 Jan 05</p>
</blockquote>

CDS 110b: Optimal Control

2008-01-21T23:23:47Z

Soto: /* Frequently Asked Questions */

{{cds110b-wi08}}
This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle. __NOTOC__

* {{cds110b-wi08 pdfs|L2-1_optimal.pdf|Lecture notes: optimal control}}
* Homework 2 (due 22 Jan @ 5 pm): {{obc08|problems 2.3, 2.4a-d, 2.6}}

== References and Further Reading ==
* R. M. Murray, ''Optimization-Based Control''. Preprint, 2008: {{cds110b-wi08 pdfs|optimal_14Jan08.pdf|Chapter 2 - Optimal Control}}
* {{cds110b-pdfs|LS95-optimal.pdf|Excerpt from LS95 on optimal control}} - This excerpt is from [http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Lewis and Syrmos, 1995] and gives a derivation of the necessary conditions for optimaliity. A few pages have been left out from the middle that contained some additional examples (which you can find in similar books in the library, if you are interested). Other parts of the book can be searched via [http://books.google.com Google Books] and purchased online.
* [http://www.statslab.cam.ac.uk/~rrw1/oc/L13.pdf Notes on Pontryagin's Maximum Principle] - these come from a set of [http://www.statslab.cam.ac.uk/~rrw1/oc/index.html lecture notes on optimization and control] by [http://www.statslab.cam.ac.uk/~rrw1/ Richard Weber] at Cambridge University. The notes are based on dynamic programming (DP) and uses a slightly different notation than we used in class.

== Frequently Asked Questions ==
'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''

<blockquote>
<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(t_f) of the optimal solution won't be exactly 0,
but will be close enough for the intent of this problem.</p>

<p>Luis Soto, 21 Jan 08</p>
</blockquote>

'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''

<blockquote>
<p>Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts. </p>

<p>Julia Braman, 18 Jan 08</p>
</blockquote>

'''Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for <math>u</math> obtained?'''

<blockquote>
<p>Pontryagin's Maximum Principle says that <math>u</math> has to be chosen to minimise the Hamiltonian <math>H(x,u,\lambda)</math> for given values of <math>x</math> and <math>\lambda</math>. In the example, <math>H = 1 + ({\lambda}^TA)x + ({\lambda}^TB)u</math>. At first glance, it seems that the more negative <math>u</math> is the more <math>H</math> will be minimised. And since the most negative value of <math>u</math> allowed is <math>-1</math>, <math>u=-1</math>. However, the co-efficient of <math>u</math> may be of either sign. Therefore, the sign of <math>u</math> has to be chosen such that the sign of the term <math>({\lambda}^TB)u</math> is negative. That's how we come up with <math>u = -sign({\lambda}^TB)</math>.</p>

<p>Shaunak Sen, 12 Jan 06</p>
</blockquote>

'''Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that <math>T</math> is the final time and <sup><math>T</math></sup> (superscript T) is a transpose operation. Am I correct in my assumption?'''

<blockquote>
<p>Yes, you are correct.</p>

<p>Jeremy Gillula, 07 Jan 05</p>
</blockquote>

CDS 110b: Optimal Control

2008-01-21T23:22:51Z

Soto: /* Frequently Asked Questions */

{{cds110b-wi08}}
This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle. __NOTOC__

* {{cds110b-wi08 pdfs|L2-1_optimal.pdf|Lecture notes: optimal control}}
* Homework 2 (due 22 Jan @ 5 pm): {{obc08|problems 2.3, 2.4a-d, 2.6}}

== References and Further Reading ==
* R. M. Murray, ''Optimization-Based Control''. Preprint, 2008: {{cds110b-wi08 pdfs|optimal_14Jan08.pdf|Chapter 2 - Optimal Control}}
* {{cds110b-pdfs|LS95-optimal.pdf|Excerpt from LS95 on optimal control}} - This excerpt is from [http://books.google.com/books?ie=UTF-8&hl=en&vid=ISBN0471033782&id=jkD37elP6NIC Lewis and Syrmos, 1995] and gives a derivation of the necessary conditions for optimaliity. A few pages have been left out from the middle that contained some additional examples (which you can find in similar books in the library, if you are interested). Other parts of the book can be searched via [http://books.google.com Google Books] and purchased online.
* [http://www.statslab.cam.ac.uk/~rrw1/oc/L13.pdf Notes on Pontryagin's Maximum Principle] - these come from a set of [http://www.statslab.cam.ac.uk/~rrw1/oc/index.html lecture notes on optimization and control] by [http://www.statslab.cam.ac.uk/~rrw1/ Richard Weber] at Cambridge University. The notes are based on dynamic programming (DP) and uses a slightly different notation than we used in class.

== Frequently Asked Questions ==
'''Q: In Problem 2.4d, are the boundary conditions for the differentially-flat trajectory correct?'''

<blockquote>
<p>Please ignore the boundary conditions given in part 2.4d for the differentially-flat trajectory and instead use x(0)=1 for the initial condition and x(1)=0 for the condition at final time t=1. Moreover, use c=100 instead of c=1. Note: the x(tf) of the optimal solution won't be exactly 0,
but will be close enough for the intent of this problem.</p>

<p>Luis Soto, 21 Jan 08</p>
</blockquote>

'''Q: In problem 2.4(d) of the homework, to what positive value should the parameter b be set?'''

<blockquote>
<p>Use b = 1 for part d when solving for and comparing the two trajectories found symbolically in previous parts. </p>

<p>Julia Braman, 18 Jan 08</p>
</blockquote>

'''Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for <math>u</math> obtained?'''

<blockquote>
<p>Pontryagin's Maximum Principle says that <math>u</math> has to be chosen to minimise the Hamiltonian <math>H(x,u,\lambda)</math> for given values of <math>x</math> and <math>\lambda</math>. In the example, <math>H = 1 + ({\lambda}^TA)x + ({\lambda}^TB)u</math>. At first glance, it seems that the more negative <math>u</math> is the more <math>H</math> will be minimised. And since the most negative value of <math>u</math> allowed is <math>-1</math>, <math>u=-1</math>. However, the co-efficient of <math>u</math> may be of either sign. Therefore, the sign of <math>u</math> has to be chosen such that the sign of the term <math>({\lambda}^TB)u</math> is negative. That's how we come up with <math>u = -sign({\lambda}^TB)</math>.</p>

<p>Shaunak Sen, 12 Jan 06</p>
</blockquote>

'''Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that <math>T</math> is the final time and <sup><math>T</math></sup> (superscript T) is a transpose operation. Am I correct in my assumption?'''

<blockquote>
<p>Yes, you are correct.</p>

<p>Jeremy Gillula, 07 Jan 05</p>
</blockquote>

HW8 Problem 1 Hints

2007-12-05T05:36:45Z

Soto:

Problem 1a) The "Gang of Four" refers to the transfer functions at the bottom of page 315. Note that some of these rational expressions may not have zeros after substituting the plant and controller, and consequently, there may not be an analytical formula for the zeros. If this is the case, say so in your answer. Don't forget to also give analytical formulas for the poles for each member of the "Gang of Four."

Problem 1b) The relationship between gain crossover frequency, phase margin, resonance peak, and percentage overshoot vs. k can be determined by analytical methods or by using the suggested commands. Note that the command 'stepinfo' will be useful to plot percentage overshoot vs k. You will need to extract the value from the corresponding structure by using the appropriate syntax, i.e. S(i,j).Overshoot. Note that the resonance peak you are asked to plot is from the closed-loop bode plot and not from the step response. Finally, you can also calculate explicit formulas such as for the gain crossover frequency as a function of k, if you prefer. Don't forget to use log scale for the values of k.

Problem 1c) should be straight forward once you do part b)

Problem 1d) Use the relationship between phase margin and k from part 1b) to determine what k values correspond to 30deg, 45deg, and 60deg phase margin and plot the step response for those particular k values.
--[[User:Soto|Soto]] 21:36, 4 December 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 8]]
[[Category: CDS 101/110 FAQ - Homework 8, Fall 2007]]

Why would it matter if the loop transfer function had high amplification of radio waves if nothing else could respond to such high frequency inputs?

2007-11-27T06:14:37Z

Soto:

An important topic of robust control is to design controllers that make the closed-loop system more robust to disturbances and to sensor noise. Thus, we seek to reduce the maximum value of the load sensitivity transfer function PS = P/(1+PC) and the noise sensitivity transfer function CS = C/(1+PC). Of course there may be instances when we want to amplify certain inputs at a particular range of frequencies. In this case we want the open loop transfer function to have high amplification in that frequency range. The point is to attenuate undesirable effects cause by exogenous inputs and for L to have high gain at certain frequencies if this gives a desirable outcome (i.e., we want L to have high gain up to a certain frequency for good reference signal tracking).
--[[User:Soto|Soto]] 22:14, 26 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Lecture 9-1]]
[[Category: CDS 101/110 FAQ - Lecture 9-1, Fall 2007]]

HW 7 Prob 1 Comments

2007-11-26T17:51:47Z

Soto:

For Problem 1c you need to design only ONE controller (ignore the plural 'controllers' in the HW wording for 1c). The controller should be designed using the Ziegler-Nichols rules for the step response method.
You need to plot the step response and the frequency response (bode plot) of your final design (i.e., the closed-loop system) You also need to compute the gain & phase margins of the loop transfer function L.

The Ziegler-Nichols step response method rules are found in Table 10.1a on pg. 301. You must draw the steepest tangent line to the step response of the plant until it crosses the two axes. You can then approximate the values for "a" and "tao" by visual or numerical approximation. You might need to change the y-axis range of the step response plot to determine the y-intercept.

As usual for all HW questions, give a title to your plots, label axes, and turn in code along with your plots.
--[[User:Soto|Soto]] 14:20, 25 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2007]]

HW 7 Prob 1 Comments

2007-11-26T17:49:45Z

Soto:

For Problem 1c you need to design only ONE controller (ignore the plural 'controllers' in the HW wording for 1c). The controller should be designed using the Ziegler-Nichols rules for the step response method.
You need to plot the step response and the frequency response (bode plot) of your final design (i.e., the closed-loop system) You also need to compute the gain & phase margins of the loop transfer function L.

The Ziegler-Nichols step response method rules are found in Table 10.1a on pg. 301. Remember that you need to plot the unit step response for the plant. You must draw the steepest tangent line to the step response until it crosses the two axes. You can then approximate the values for "a" and "tao" by visual or numerical approximation. You might need to change the y-axis range of the plot to determine the y-intercept.

As usual for all HW questions, give a title to your plots, label axes, and turn in code along with your plots.
--[[User:Soto|Soto]] 14:20, 25 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2007]]

HW 7 Prob 1 Comments

2007-11-26T07:34:54Z

Soto:

For Problem 1c you need to design two controllers. The first one is
designed using the Ziegler-Nichols rules for the step response method.
You need to plot the step response of the closed-loop and the frequency
response of the loop transfer function L. Then you are asked to design a
controller using the Ziegler-Nichols rules for the frequency response
method. You also need to plot the step response of the closed-loop and
the frequency response of the loop transfer function L (this is what is
in the solutions). Both the step response method and frequency response
method rules are found in table 10.1 on pg. 301.

Remember that when you use the step-response Ziegler-Nichols method, you need to plot the unit step response for the plant. You must draw the steepest tangent line to the step response until it crosses the two axes. You can then approximate the values for "a" and "tao" by visual or numerical approximation.
As usual for all HW questions, give a title to your plots, label axes, and turn in code along with your plots.
--[[User:Soto|Soto]] 14:20, 25 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2007]]

HW 7 Prob 1 Comments

2007-11-25T22:22:20Z

Soto:

Remember that when you use the Ziegler-Nichols method, you need to plot the unit step response for the plant. You must draw the steepest tangent line to the step response until it crosses the two axes. You can then approximate the values for "a" and "tao" by visual or numerical approximation. Knowing these values will then allow you to determine the values of othe parameters found on Table 10.1 on pg. 301 of AM07. You should also plot the step response of the closed-loop system and determine the phase and gain margins.
As usual for all HW questions, give a title to your plots, label axes, and turn in code along with your plots.
--[[User:Soto|Soto]] 14:20, 25 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2007]]

HW 7 Prob 1 Comments

2007-11-25T22:20:49Z

Soto:

Remember that when you use the Ziegler-Nichols method, you need to plot the unit step response for the plant. You must draw the steepest tangent line to the step response until it crosses the two axes. You can then approximate the values for "a" and "tao" by visual or numerical approximation. Knowing these values will then allow you to determine the values of othe parameters found on Table 10.1 on pg. 301 of AM08. You should also plot the step response of the closed-loop system and determine the phase and gain margins. As usual for all HW questions, give a title to your plots, label axes, and turn in code along with your plots.
--[[User:Soto|Soto]] 14:20, 25 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2007]]

HW 7 Prob 1 Comments

2007-11-25T22:17:43Z

Soto:

Remember that when you use the Ziegler-Nichols method, you need to plot the unit step response for the plant. You must draw the steepest tangent line to the step response until it crosses the two axes. You can then approximate the values for "a" and "tao" by visual or numerical approximation. Knowing these values will then allow you to determine the values of othe parameters found on Table 10.1 on pg. 301 of AM08. You should also plot the step response of the closed-loop system and determine the phase and gain margins. As usual for all HW questions, give a title to your plots, label axes, and turn in code along with your plots.
----~~

<ncl>CDS 101/110 FAQ - Homework 7, Fall 2007</ncl>

CDS 101/110a, Fall 2007

2007-11-19T23:34:41Z

Soto: /* Announcements */

{{cds101-fa07}}
<table align=right border=1 width=20% cellpadding=6>
<tr><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br>
<li> [[#Collaboration Policy|Collaboration Policy]] <br>
<li> [[#Course Text and References|Course Texts]] <br>
<li> [[#Course_Schedule|Course Schedule]]<br>
<li> [[#Course Project|Course Project]]
</ul>
</table>
This is the homepage for CDS 101 (Analysis and Design of Feedback Systems) and CDS 110 (Introduction to Control Theory) for Fall 2007. __NOTOC__

<table width=80%>
<tr valign=top>
<td width=50%>
'''Instructor'''
* [[Main Page|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 74 JRG
* Office hours: Fridays, 3-4 pm (by appt)
* Prior years: [http://www.cds.caltech.edu/~murray/courses/cds101/fa03 FA03], [http://www.cds.caltech.edu/~murray/courses/cds101/fa04 FA04], [[CDS 101/110a, Fall 2006|FA06]]
<td>
'''Teaching Assistants''' ([mailto:cds101-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Elisa Franco, Sawyer Fuller, George Hines, Luis Soto
* Office hours: Sundays, 4-5; Tuesdays, 4-5 in 114 STL
'''Course Ombuds'''
* Vanessa Carson and Matthew Feldman
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 19 Nov 07: Homework #5 is graded
** CDS 110 average: 36.3/40, <math>\sigma</math> = 3.2
** CDS 101 average: 17.9/20, <math>\sigma</math> = 1.1
* 19 Nov 07: [[CDS 101/110, Week 8 - PID Control]]
* 12 Nov 07: [[CDS 101/110, Week 7 - Loop Analysis]]
** {{cds101 handouts|hw6.pdf|HW #6}} is now posted; due 19 Nov @ 5 pm
* 12 Nov 07: Midterms are graded
** CDS 110 average: 44/55, <math>\sigma</math> = 7.6
** CDS 101 average: 29/35, <math>\sigma</math> = 4.0
* 9 Nov 07: [[Media:EF11092007Recitation.pdf | Notes ]] for Section 4 (theory) recitation only - Laplace transforms.
* 5 Nov 07: [[CDS 101/110, Week 6 - Transfer Functions]]
** {{cds101 handouts|hw5.pdf|HW #5}} is now posted; due 12 Nov @ 5 pm

== Course Syllabus ==

CDS 101/110 provides an introduction to feedback and control in physical,
biological, engineering, and information sciences. Basic principles of
feedback and its use as a tool for altering the dynamics of systems and
managing uncertainty. Key themes throughout the course will include
input/output response, modeling and model reduction, linear versus nonlinear
models, and local versus global behavior.

CDS 101 is a 6 unit (2-0-4) class intended for advanced students in science
and engineering who are interested in the principles and tools of feedback
control, but not the analytical techniques for design and synthesis of control
systems. CDS 110 is a 9 unit class (3-0-6) that provides a traditional first
course in control for engineers and applied scientists. It assumes a stronger
mathematical background, including working knowledge of linear algebra and
ODEs. Familiarity with complex variables (Laplace transforms, residue theory)
is helpful but not required.

=== Grading ===
The final grade will be based on homework sets, a midterm exam, and a final exam:

*''Homework (50%):'' Homework sets will be handed out weekly and due on Mondays by 5 pm to the box outside of 109 Steele. A two day grace period is allowed to turn in your homework. Late homework beyond the grace period will not be accepted without a note from the health center or the Dean. MATLAB code and SIMULINK diagrams are considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).

* ''Midterm exam (20%):'' A midterm exam will be handed out at the beginning of midterms period (31 Oct) and due at the end of the midterm examination period (6 Nov). The midterm exam will be open book and computers will be allowed (though not required).

* ''Final exam (30%):'' The final exam will be handed out on the last day of class (7 Dec) and due at the end of finals week. It will be an open book exam and computers will be allowed (though not required).

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult
outside reference materials, other students, the TA, or the
instructor, but you cannot consult homework solutions from
prior years and you must cite any use of material from outside
references. All solutions that are handed in should be written up
individually and should reflect your own understanding of the subject
matter at the time of writing. MATLAB scripts and plots are
considered part of your writeup and should be done individually (you
can share ideas, but not code).

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The primary course text is [[AM:Main Page|''Feedback Systems: An Introduction for Scientists and Engineers'']] by {{Astrom}} and Murray (2008). This book is available in the Caltech bookstore and via download from the [[AM:Main Page|companion web site]]. The following additional references may also be useful:

* A. D. Lewis, ''A Mathematical Approach to Classical Control'', 2003. [http://penelope.mast.queensu.ca/math332/notes.shtml Online access].

In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.

* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', McGraw-Hill, 1986.
* G. F. Franklin, J. D. Powell, and A. Emami-Naeni, ''Feedback Control of Dynamic Systems'', Addison-Wesley, 2002.

=== Course Schedule ===
The course is scheduled for MWF 2-3 pm in 74 Jorgenson. CDS 101 meets on Monday and Friday only. A detailed course schedule is available on the [[CDS 101/110a, Fall 2007 - Course Schedule|course schedule]] page.

== Old Announcements ==
* 20 Aug 07: created wiki page for CDS 101/110a, Fall 2007
* 1 Oct 07: [[CDS 101/110, Week 1 - Introduction to Feedback and Control]]
* 8 Oct 07: [[CDS 101/110, Week 2 - System Modeling]]
* 15 Oct 07: {{cds101 handouts|soln1.pdf|Solutions to homework #1}} are now available
** CDS 110: Average score = 35.7/40 (<math>\sigma</math> = 3.4); average time = 6.2 hours
** CDS 101: Average score = 18.7/20 (<math>\sigma</math> = 1.6); average time = 3.4 hours
* 15 Oct 07: [[CDS 101/110, Week 3 - Dynamic Behavior]]
* 22 Oct 07: {{cds101 handouts|soln2.pdf|Solutions to homework #2}} are now available
** CDS 110: Average score = 22.4/30 (<math>\sigma</math> = 4.3); average time = 9.7 hours
** CDS 101: Average score = 14.8/20 (<math>\sigma</math> = 3.6); average time = 8.1 hours
* 22 Oct 07: [[CDS 101/110, Week 4 - Linear Systems]]
* 29 Oct 07: HW # 3 is graded and the {{cds101 handouts|soln3.pdf|solutions}} are now posted
** CDS 110: Average score = 30.5/40 (<math>\sigma</math> = 6.49); average time = 11.8 hours
** CDS 101: Average score = 19.3/20 (<math>\sigma</math> = 0.75).
* 1 Nov 07: [[Media:Sawyer_reviewnotes.pdf|Midterm review]] notes are up
* 1 Nov 07: Midterms are outside 102 Steele. Due back by Tuesday 5pm
* 1 Nov 07: No recitations Friday; Midterm review is Friday 2 Nov at the normal recitation hour, 2-3p, in 74 Jorgenson
* 2 Nov 07: HW #4 is graded and the {{cds101 handouts|soln4.pdf|solutions}} are now posted
** CDS 110: Average score = 34.5/40 (<math>\sigma</math> = 4.6); average time = 9.9 hours
** CDS 101: Average score = 17.0/20 (<math>\sigma</math> = 3.4)
* 3 Nov 07: [[CDS 101/110, Week 5 - State Feedback]] web page is now updated

[[Category: Courses]] [[Category: 2007-08 Courses]]

CDS 101/110a, Fall 2007

2007-11-19T23:29:51Z

Soto: /* Announcements */

{{cds101-fa07}}
<table align=right border=1 width=20% cellpadding=6>
<tr><td>
<center>'''Contents'''</center>
<ul>
<li> [[#Grading|Grading]] <br>
<li> [[#Collaboration Policy|Collaboration Policy]] <br>
<li> [[#Course Text and References|Course Texts]] <br>
<li> [[#Course_Schedule|Course Schedule]]<br>
<li> [[#Course Project|Course Project]]
</ul>
</table>
This is the homepage for CDS 101 (Analysis and Design of Feedback Systems) and CDS 110 (Introduction to Control Theory) for Fall 2007. __NOTOC__

<table width=80%>
<tr valign=top>
<td width=50%>
'''Instructor'''
* [[Main Page|Richard Murray]], murray@cds.caltech.edu
* Lectures: MWF, 2-3 pm, 74 JRG
* Office hours: Fridays, 3-4 pm (by appt)
* Prior years: [http://www.cds.caltech.edu/~murray/courses/cds101/fa03 FA03], [http://www.cds.caltech.edu/~murray/courses/cds101/fa04 FA04], [[CDS 101/110a, Fall 2006|FA06]]
<td>
'''Teaching Assistants''' ([mailto:cds101-tas@cds.caltech.edu cds110-tas@cds])
* Julia Braman, Elisa Franco, Sawyer Fuller, George Hines, Luis Soto
* Office hours: Sundays, 4-5; Tuesdays, 4-5 in 114 STL
'''Course Ombuds'''
* Vanessa Carson and Matthew Feldman
</table>

== Announcements ==
<table align=right border=0><tr><td>[[#Old Announcements|Archive]]</table>
* 19 Nov 07: Homework #5 is graded
** CDS 110 average: 36.3, <math>\sigma</math> = 3.2
** CDS 101 average: 17.9, <math>\sigma</math> = 1.1
* 19 Nov 07: [[CDS 101/110, Week 8 - PID Control]]
* 12 Nov 07: [[CDS 101/110, Week 7 - Loop Analysis]]
** {{cds101 handouts|hw6.pdf|HW #6}} is now posted; due 19 Nov @ 5 pm
* 12 Nov 07: Midterms are graded
** CDS 110 average: 44/55, <math>\sigma</math> = 7.6
** CDS 101 average: 29/35, <math>\sigma</math> = 4.0
* 9 Nov 07: [[Media:EF11092007Recitation.pdf | Notes ]] for Section 4 (theory) recitation only - Laplace transforms.
* 5 Nov 07: [[CDS 101/110, Week 6 - Transfer Functions]]
** {{cds101 handouts|hw5.pdf|HW #5}} is now posted; due 12 Nov @ 5 pm

== Course Syllabus ==

CDS 101/110 provides an introduction to feedback and control in physical,
biological, engineering, and information sciences. Basic principles of
feedback and its use as a tool for altering the dynamics of systems and
managing uncertainty. Key themes throughout the course will include
input/output response, modeling and model reduction, linear versus nonlinear
models, and local versus global behavior.

CDS 101 is a 6 unit (2-0-4) class intended for advanced students in science
and engineering who are interested in the principles and tools of feedback
control, but not the analytical techniques for design and synthesis of control
systems. CDS 110 is a 9 unit class (3-0-6) that provides a traditional first
course in control for engineers and applied scientists. It assumes a stronger
mathematical background, including working knowledge of linear algebra and
ODEs. Familiarity with complex variables (Laplace transforms, residue theory)
is helpful but not required.

=== Grading ===
The final grade will be based on homework sets, a midterm exam, and a final exam:

*''Homework (50%):'' Homework sets will be handed out weekly and due on Mondays by 5 pm to the box outside of 109 Steele. A two day grace period is allowed to turn in your homework. Late homework beyond the grace period will not be accepted without a note from the health center or the Dean. MATLAB code and SIMULINK diagrams are considered part of your solution and should be printed and turned in with the problem set (whether the problem asks for it or not).

* ''Midterm exam (20%):'' A midterm exam will be handed out at the beginning of midterms period (31 Oct) and due at the end of the midterm examination period (6 Nov). The midterm exam will be open book and computers will be allowed (though not required).

* ''Final exam (30%):'' The final exam will be handed out on the last day of class (7 Dec) and due at the end of finals week. It will be an open book exam and computers will be allowed (though not required).

=== Collaboration Policy ===

Collaboration on homework assignments is encouraged. You may consult
outside reference materials, other students, the TA, or the
instructor, but you cannot consult homework solutions from
prior years and you must cite any use of material from outside
references. All solutions that are handed in should be written up
individually and should reflect your own understanding of the subject
matter at the time of writing. MATLAB scripts and plots are
considered part of your writeup and should be done individually (you
can share ideas, but not code).

No collaboration is allowed on the midterm or final exams.

=== Course Text and References ===

The primary course text is [[AM:Main Page|''Feedback Systems: An Introduction for Scientists and Engineers'']] by {{Astrom}} and Murray (2008). This book is available in the Caltech bookstore and via download from the [[AM:Main Page|companion web site]]. The following additional references may also be useful:

* A. D. Lewis, ''A Mathematical Approach to Classical Control'', 2003. [http://penelope.mast.queensu.ca/math332/notes.shtml Online access].

In addition to the books above, the textbooks below may also be useful. They are available in the library (non-reserve), from other students, or you can order them online.

* B. Friedland, ''Control System Design: An Introduction to State-Space Methods'', McGraw-Hill, 1986.
* G. F. Franklin, J. D. Powell, and A. Emami-Naeni, ''Feedback Control of Dynamic Systems'', Addison-Wesley, 2002.

=== Course Schedule ===
The course is scheduled for MWF 2-3 pm in 74 Jorgenson. CDS 101 meets on Monday and Friday only. A detailed course schedule is available on the [[CDS 101/110a, Fall 2007 - Course Schedule|course schedule]] page.

== Old Announcements ==
* 20 Aug 07: created wiki page for CDS 101/110a, Fall 2007
* 1 Oct 07: [[CDS 101/110, Week 1 - Introduction to Feedback and Control]]
* 8 Oct 07: [[CDS 101/110, Week 2 - System Modeling]]
* 15 Oct 07: {{cds101 handouts|soln1.pdf|Solutions to homework #1}} are now available
** CDS 110: Average score = 35.7/40 (<math>\sigma</math> = 3.4); average time = 6.2 hours
** CDS 101: Average score = 18.7/20 (<math>\sigma</math> = 1.6); average time = 3.4 hours
* 15 Oct 07: [[CDS 101/110, Week 3 - Dynamic Behavior]]
* 22 Oct 07: {{cds101 handouts|soln2.pdf|Solutions to homework #2}} are now available
** CDS 110: Average score = 22.4/30 (<math>\sigma</math> = 4.3); average time = 9.7 hours
** CDS 101: Average score = 14.8/20 (<math>\sigma</math> = 3.6); average time = 8.1 hours
* 22 Oct 07: [[CDS 101/110, Week 4 - Linear Systems]]
* 29 Oct 07: HW # 3 is graded and the {{cds101 handouts|soln3.pdf|solutions}} are now posted
** CDS 110: Average score = 30.5/40 (<math>\sigma</math> = 6.49); average time = 11.8 hours
** CDS 101: Average score = 19.3/20 (<math>\sigma</math> = 0.75).
* 1 Nov 07: [[Media:Sawyer_reviewnotes.pdf|Midterm review]] notes are up
* 1 Nov 07: Midterms are outside 102 Steele. Due back by Tuesday 5pm
* 1 Nov 07: No recitations Friday; Midterm review is Friday 2 Nov at the normal recitation hour, 2-3p, in 74 Jorgenson
* 2 Nov 07: HW #4 is graded and the {{cds101 handouts|soln4.pdf|solutions}} are now posted
** CDS 110: Average score = 34.5/40 (<math>\sigma</math> = 4.6); average time = 9.9 hours
** CDS 101: Average score = 17.0/20 (<math>\sigma</math> = 3.4)
* 3 Nov 07: [[CDS 101/110, Week 5 - State Feedback]] web page is now updated

[[Category: Courses]] [[Category: 2007-08 Courses]]

Question 1

2007-11-08T20:47:56Z

Soto:

Question 1a) To find Hyr which is the transfer function between the input r and the output y, you need to write down equations that involve r and y by using the block diagram. You need to determine what the ratio y/r equals to. The block diagram does not label all signals, as this is not necessary. However, feel free to label signals that are not labeled if this helps you to write down the equations. Be careful of the direction of the arrows and the signs in the summation junctions. You could start by writing y = ...

Question 1b) To find the transfer function given a state-space realization, you can use the formula for conversion from ss to tf: H(s) = C[(sI - A)^-1]*B + D.

Question 1c) To find Hzr, we need to determine what the ratio z/r equals to. You could start by writing z = ...

--[[User:Soto|Luis Soto]] 21:10, 7 November 2007 (PST)
[Category: CDS 101/110 FAQ - Homework 5]
[[Category: CDS 101/110 FAQ - Homework 5, Fall 2007]]

Question 1

2007-11-08T05:15:07Z

Soto:

Question 1a) To find Hyr which is the transfer function between the input r and the output y, you need to write down equations that involve r and y by using the block diagram. You need to determine what the ratio y/r equals to. The block diagram does not label all signals, as this is not necessary. However, feel free to label signals that are not labeled if this helps you to write down the equations. Be careful of the direction of the arrows and the signs in the summation junctions. You could start by writing y = ...

Question 1b) To find the transfer function given a state-space realization, you can use the formula for conversion from ss to tf: H(s) = C[(sI - A)^-1]*B + D.

Question 1c) To find Hzr, we need to determine what the ratio z/r equals to. You could start by writing z = ...

--[[User:Soto|Luis Soto]] 21:10, 7 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 5]]
[[Category: CDS 101/110 FAQ - Homework 5, Fall 2007]]

HW

2007-11-08T05:13:47Z

Soto:

HW

2007-11-08T05:10:49Z

Soto:

Question 1a) To find Hyr which is the transfer function between the input r and the output y, you need to write down equations that involve r and y by using the block diagram. You need to determine what the ratio y/r equals to. The block diagram does not label all signals, as this is not necessary. However, feel free to label signals that are not labeled if this helps you to write down the equations. Be careful of the direction of the arrows and the signs in the summation junctions. You could start by writing y = ...

Question 1b) To find the transfer function given a state-space realization, you can use the formula for conversion from ss to tf: H(s) = C[(sI - A)^-1]*B + D.

Question 1c) To find Hzr, we need to determine what the ratio z/r equals to. You could start by writing z = ...

--[[User:Soto|Soto]] 21:10, 7 November 2007 (PST)
[[Category: CDS 101/110 FAQ - Homework 5]]
[[Category: CDS 101/110 FAQ - Homework 5, Fall 2007]]

HW

2007-11-08T05:10:31Z

Soto:

Question 1a) To find Hyr which is the transfer function between the input r and the output y, you need to write down equations that involve r and y by using the block diagram. You need to determine what the ratio y/r equals to. The block diagram does not label all signals, as this is not necessary. However, feel free to label signals that are not labeled if this helps you to write down the equations. Be careful of the direction of the arrows and the signs in the summation junctions. You could start by writing y = ...

Question 1b) To find the transfer function given a state-space realization, you can use the formula for conversion from ss to tf: H(s) = C[(sI - A)^-1]*B + D.

Question 1c) To find Hzr, we need to determine what the ratio z/r equals to. You could start by writing z = ...

'--[[User:Soto|Soto]] 21:10, 7 November 2007 (PST)'
[[Category: CDS 101/110 FAQ - Homework 5]]
[[Category: CDS 101/110 FAQ - Homework 5, Fall 2007]]

HW

2007-11-08T05:05:52Z

Soto:

Is it standard practice to use degrees to represent the phase in frequency response analysis?

2007-10-23T01:36:16Z

Soto:

It usually is standard practice to use degrees to represent the phase or phase margin, although you can certainly use radians (use the formula Phase = 2*pi*(deltaT/T) or Phase = 360*(deltaT/T), as long as you are consistent. For example, in electronic amplifiers, phase margin is the difference, measured in degrees, between the phase angle of the amplifier's output signal and -360° (wikipedia).
--[[User:Soto|Soto]] 18:36, 22 October 2007 (PDT)
[[Category: CDS 101/110 FAQ - Lecture 4-1]]
[[Category: CDS 101/110 FAQ - Lecture 4-1, Fall 2007]]

CDS 101/110a, Fall 2007 - Recitation Schedule

2007-10-17T23:48:26Z

Soto: /* Section 1: Feedback and Control in Nature */

{{cds101-fa07}}{{righttoc}}
The purpose of the recitation sections is to provide additional insight into the material for the week, including answer questions on specific topics of interests to the students in that section. The TAs leading the recitation will generally work through one problem from the homework set (same problem in each section) so that students can see what is expected on the homeworks and how the tools from the course can be applied. (Note: students must still work through and turn in the problem that the TAs work through and what you turn in must reflect your understanding of the problem.)

Recitations for CDS 101/110a will be on Fridays from 2-3 pm unless otherwise noted. Each recitation session is tuned for a slightly different audience and we have made initial assignments based on the course you are taking, the option you are in, and your class standing (So, Jr, Sr, G1, G2, etc).

=== Section 1: Feedback and Control in Nature ===

This section is designed for students interested in the application of ideas from feedback and control to systems in nature. It is also suitable for students who do not have lots of prior coursework in linear algebra, ordinary differential equations or complex variables. All students in CDS 101 are iniitially assigned to this section.

*TA: Luis Soto

*Students:
Arkya Dhar,
Stephan Duewel,
Alberto Izarraraz,
Lauren LeBon,
Ophelia Venturelli

*Location: 110 STL

=== Section 2: Ae/ME/BE practical ===

This section is intended for students who are interested in the application of feedback and control to mechanical and electro-mechanical systems such as airplanes, cars, robots, etc. TA: Sawyer Fuller

=== Section 3: EE/CS ===

This section is intended for students who are interested in the application of feedback and control to electrical and information systems, such as queuing systems, economics and computers. TA: Julia Braman

=== Section 4: Theory ===

This section is intended for more advanced students who would like a more theoretical description of some of the tools of the class. This section will not go through a problem from the homework in much detail, but will instead discuss more advanced approaches to the topics being considered for that week. TA: Elisa Franco

=== Section 5: Off Schedule ===

This section will be held at on Fridays at 8PM in 214 Steele. It will focus on engineering applications of feedback and control and provide brief introductions to some application topics that may not be covered in class. TA: George Hines