Murray Wiki - User contributions [en]

Winter 2009 Meeting Schedule

2008-12-22T07:34:46Z

Han: /* Fri */

__NOTOC__
Please sign up for a time to meet. Note that some meetings are for different lengths of time and different frequencies.
{| width=100% border=1
|- valign=top
| width=20% |
==== Mon ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|2:00p|Dionysios}}
{{agenda item||}}
{{agenda item||}}
{{agenda item|5:00p|Open}}
{{agenda item|6:00p|Open}}
{{agenda end}}
|

==== Tue ====
{{agenda begin}}
{{agenda item|1:30p|Open (short)}}
{{agenda item|2:00p|Open}}
{{agenda item||}}
{{agenda item||}}
{{agenda item|5:00p|Open}}
{{agenda item|6:00p|Open}}
{{agenda end}}
|

==== Wed ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|2:00p|Julia}}
{{agenda item||}}
{{agenda item||}}
{{agenda item|5:00p|Elisa}}
{{agenda item|6:00p|Ufuk}}
{{agenda end}}
|

==== Thu ====
{{agenda begin}}
{{agenda item|1:30p|Open (short)}}
{{agenda item|2:00p|Yizhar}}
{{agenda item||}}
{{agenda item|4:30p|Odd: Francisco}}
{{agenda item||Even: Open}}
{{agenda item|6:00p|Open}}
{{agenda end}}
|

==== Fri ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|2:00p|Shuo}}
{{agenda item|3:00p|Andrea}}
{{agenda item|4:00p|Open}}
{{agenda item|5:00p|Open}}
{{agenda item||}}
{{agenda end}}
|}

David Hill, Dec 08

2008-12-02T07:17:51Z

Han:

David Hill from Australian National University will be visiting Caltech on 4-5 Dec (Thu-Fri).

=== Schedule ===

4 Dec (Thu)
* 7:30 am - breakfast with Richard
* 8:45 am - Francisco/Sawyer
* 9:30 am - Ufuk
* 10:15 am - open
* 10:45 am - seminar prep
* 11:00 am - Seminar, 114 Steele
* 12:00 pm - Lunch with CDS faculty
* 1:30 pm - open
* 2:15 pm - Steven Low
* 3:00 pm - Shuo (and probably Andrea?)
* 3:45 pm - open

5 Dec (Fri)
* Appointments on request (send e-mail to Richard)
* 1:30 pm - Mani Chandy

=== Seminar Abstract ===
<center>
'''Feedback Networks'''

David J Hill<br>
Federation Fellow<br>
Research School of Information Sciences and Engineering<br>
The Australian National University (ANU)
</center>
The field of systems and control has gone through stages based around focus on
preferred models for systems and appropriate structures for controllers, including linear systems and state variable feedback, time-varying parametric systems and adaptive control, nonlinear systems and optimal control and so on. However, important control tasks in living systems and modern infrastructure technology actually take the form of systems with network structure controlled by distributed control, with switching, time-delays and other complexities, i.e. control of
networks by networks or feedback networks. Examples in engineering include power grids, road traffic control and Internet congestion control. This model opens up a plethora of new systems and control science questions, which are being studied by separate communities in science and engineering.

This seminar will describe recent work in this area mainly focussing on work in the Lab for Networks and Control at ANU. The emphasis is on the role of structure, i.e. the various graphs in the networks (system, sensing and controller), stability-related questions and self-organising mechanisms for control.

=== Biography ===
David J Hill received the BE and BSc degrees from the University of Queensland, Australia, in 1972 and 1974, respectively. He received the PhD degree in Electrical Engineering from the University of Newcastle, Australia, in 1976. He is currently a Professor and Australian Research Council Federation Fellow in the Research School of Information Sciences and Engineering at The Australian National University. He has held academic and substantial visiting positions at the universities of Melbourne, California (Berkeley), Newcastle (Australia), Lund (Sweden), Sydney and Hong Kong (City University). He holds honorary professorships at the University of Sydney, University of Queensland (Australia), South China University of Technology, City University of Hong Kong, Wuhan University and Northeastern University (China). His research interests are in network systems science, stability analysis, nonlinear control and applications. He is a Fellow of the Institution of Engineers, Australia, the Institute of Electrical and Electronics Engineers, USA and the Australian Academy of Science; he is also a Foreign Member of the Royal Swedish Academy of Engineering Sciences.

If there is a common pole and zero (say at a) in the closed loop transfer function, should I say there's one pole at a and also one zero at a?

2008-12-01T01:48:41Z

Han:

No. They cancel each other.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 6]]
[[Category: CDS 101/110 FAQ - Homework 6, Fall 2008]]

How do I evaluate a certain transfer function at desired frequencies numerically?

2008-12-01T01:45:49Z

Han:

There are several ways to do it other than writing out the transfer function explicitly:

1. If you'd like to evaluate G(s) at a single frequency f (f is real, in Hz), you can use: evalfr(G,i*f)

2. For a range of angular frequencies w (w is real, in rad/s), either:
* Use H = freqresp(G,w) and use squeeze(H) to get an array; H is complex
* Use [mag phase] = bode(G,w) to get the magnitude and phase separately; use squeeze() accordingly.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

How do I evaluate a certain transfer function at desired frequencies numerically?

2008-12-01T01:45:07Z

Han:

There are several ways to do it other than writing out the transfer function explicitly:

1. If you'd like to evaluate G(s) at a single frequency f (f is real, in Hz), you can use: evalfr(G,i*f)

2. For a range of angular frequencies w (w is real, in rad/s):
* Use H = freqresp(G,w) and use squeeze(H) to get an array; H is complex
* Use [mag phase] = bode(G,w) to get the magnitude and phase separately; Use squeeze() accordingly.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

When evaluating the Bode integral, I am not getting even close to the ideal result (something negative)?

2008-12-01T01:33:11Z

Han:

This can be possible if you happen to have a poor design. The sensitivity function will not die down at very high frequencies. In this case, you can either try to extend the upper limit of your integral or use a different design with larger phase margin.

Also remember that the logarithm in the integrand is to the base <math>e\,</math>, not 10!

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

When evaluating the Bode integral, I am not getting even close to the ideal result (something negative)?

2008-12-01T01:33:02Z

Han:

In MATLAB, use feedback() command to obtain the closed loop transfer function. Do not use L/(1 L).

2008-12-01T01:24:57Z

Han:

Always use feedback() command to compute the closed loop transfer functions. L/(1+L) sometimes doesn't do the cancellation properly.

The Gang of Four can be obtained by:

S: feedback(1,L)

T: feedback(L,1)

PS: feedback(P,C)

CS: feedback(C,P)

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

A typo in equation (6.24).

2008-11-23T00:36:20Z

Han:

For <math>\zeta<1\,</math>, the corresponding equation should be:

<math>y(t)=k\Big(1-e^{-\zeta\omega_0t}\cos{\omega_dt}-\frac{\zeta}{\sqrt{1-\zeta^2}}e^{-\zeta\omega_0t}\sin{\omega_dt}\Big)</math>.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

Typo in CDS110 problem 1(a).

2008-11-23T00:31:04Z

Han: Typo in CDS110 problem 1(a). moved to A typo in equation (6.24).

#redirect [[A typo in equation (6.24).]]

A typo in equation (6.24).

2008-11-23T00:31:04Z

Han: Typo in CDS110 problem 1(a). moved to A typo in equation (6.24).

The closed loop transfer function <math>G(s)\,</math> (which means <math>H_{yr}(s)\,</math>) should be <math>\frac{\omega_0^2}{s^2+2\zeta\omega_0s+\omega_0^2}\,</math>. The last term of the denominator is <math>\omega_0^2\,</math> rather than <math>\omega_0\,</math>, and the second term is missing an additional <math>s\,</math>.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

CDS 101/110 - Loop Shaping

2008-11-21T00:26:47Z

Han:

{{cds101-fa08}}

{{righttoc}}
== Overview ==
The learning objectives for this week are:
* Students should be to design a simple compensator with given performance and robustness (phase margin) specifications
* Students should be analyze and understand the overall performance of the system using the Gang of Four
* Students should compute the limits on the performance that arise from right half plane poles and zeros

'''Monday:''' Loop Shaping ({{cds101 handouts|L8-1_loopsyn.pdf|Slides}}, {{cds101 mp3|cds101-2008-11-17.mp3|MP3}})

This lecture describes how to design a control system by converting the performance specifications to constraints on the loop transfer function, and then shaping the loop transfer function to satisfy the constraints. Sensitivity functinos are defined and tradeoffs between different input/output transfer functions are discussed.

* {{cds101 handouts|L8-1_loopsyn_h.pdf|Lecture handout}}
* MATLAB handouts: {{cds101 matlab|L8_1_dfan.m}}

'''Wednesday:''' Performance Limits ({{cds101 handouts|L8-2_limits.pdf|Slides}}, {{cds101 mp3|cds101-2008-11-19.mp3|MP3}})

This lecture investigates some of the limits of performance for feedback systems, including the effects of right half plane poles and zeros on the closed loop system performance. A magnetic levitation system and lateral control of the Caltech ducted fan are used to illustrate the basic concepts.

* {{cds101 handouts|L8-2_limits_h.pdf|Lecture handout}}
* MATLAB handouts: {{cds101 matlab|L8_2_maglev.m}}

'''Friday:''' Recitation sections

== Reading ==

* {{AM08|Chapter 11 - Loop Shaping}}
** CDS 101: Read sections 11.1, 11.3-11.4 [45 min]
** CDS 110: Read sections 11.1, 11.2-11.5 [60 min]
** CDS 210: Skim AM08 Ch 11.1-11.4, read AM08 11.5, DFT Ch 4 and 6 [90 min]

== Homework ==

* {{cds101 handouts|hw7-fa08.pdf|Homework #7}} (due 24 Nov 08): {{cds101 handouts|hw7-101-fa08.pdf|CDS 101}}, {{cds101 handouts|hw7-110-fa08.pdf|CDS 110}}, {{cds101 handouts|hw7-210-fa08.pdf|CDS 210}}

== FAQ ==
'''Monday'''
<ncl>CDS 101/110 FAQ - Lecture 8-1, Fall 2008</ncl>
'''Wednesday'''
<ncl>CDS 101/110 FAQ - Lecture 8-2, Fall 2008</ncl>
'''Homework'''
<ncl>CDS 101/110 FAQ - Homework 7, Fall 2008</ncl>

A typo in equation (6.24).

2008-11-21T00:23:29Z

Han:

CDS 101/110a, Fall 2008

2008-11-20T21:24:42Z

Han:

{{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>
* 17 Nov 08: [[CDS 101/110 - Loop Shaping|Week 7 - Loop Shaping]]
** {{cds101 handouts|hw7-fa08.pdf|HW #7}} is posted; due 24 Nov
* 15 Nov 08: HW 5 has been graded and {{cds101 local|soln5-fa08.pdf|solutions}} are posted; averages are for 101: 16/20, 4 hours; for 110: 34/40, 11 hours.
* 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.

= 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://www.mast.queensu.ca/~andrew/teaching/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.
* 3 Nov 08: HW 4 has been graded; averages are for 101: 15/20, 6 hrs; for 110: 35.5/40, 7 hrs.
* 3 Nov 08: [[CDS 101/110 - Transfer Functions|Week 6 - Transfer Functions]]
** {{cds101 handouts|hw5-fa08.pdf|HW #5}} is posted; due 10 Nov

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

A typo in equation (6.24).

2008-11-20T03:43:10Z

Han:

The closed loop transfer function <math>G(s)\,</math> (which means <math>H_{yr}(s)\,</math>) should be <math>\frac{\omega_0^2}{s^2+2\zeta\omega_0+\omega_0^2}\,</math>. The last term of the denominator is <math>\omega_0^2\,</math> rather than <math>\omega_0\,</math>.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 7]]
[[Category: CDS 101/110 FAQ - Homework 7, Fall 2008]]

What is G and what is H? What is the difference in the way they are defined?

2008-11-04T00:09:34Z

Han:

Both <math>G(s)\,</math> and <math>H(s)\,</math> are notations for transfer functions. Sometimes we use <math>G(i\omega)\,</math> (or <math>H(i\omega)\,</math>) to refer to the "frequency response" if the system is asymptotically stable; it can be obtained by evaluating <math>G(s)\,</math> at <math>s=i\omega\,</math>.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 6-1]]
[[Category: CDS 101/110 FAQ - Lecture 6-1, Fall 2008]]

Do we always need to know everything about the states? The Segway example sort of suggests not.

2008-10-28T02:40:29Z

Han:

If you choose to design a controller using state feedback, you need to know/estimate the full states of the system. The Segway/inverted pendulum example shows that you can estimate the full states from the output in certain cases, even if you do not have direct access to the states.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 5-1]]
[[Category: CDS 101/110 FAQ - Lecture 5-1, Fall 2008]]

Does lim (t to infty) E(x-x hat) = 0 imply that there will be less disturbance over time?

2008-10-28T02:39:11Z

Han:

<math>E(x)\,</math> here denotes the expected value of <math>x\,</math>, where <math>x\,</math> is a random variable. <math>\lim_{t\to\infty} E(x-\hat x)=0\,</math> only implies that the mean estimation error will converge to zero over time. Disturbance, however, will affect the variance of the error, which is given by <math> E(x-\hat x)^2\,</math>. In CDS 110b, we will learn how to design observers that minimize this estimation variance if the disturbance can be modeled as Gaussian noise.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 5-1]]
[[Category: CDS 101/110 FAQ - Lecture 5-1, Fall 2008]]

What do kr and r represent?

2008-10-24T05:12:14Z

Han:

In state feedback controller design (<math>u=-Kx+k_rr\,</math>), properly choosing the feedback gain <math>K\,</math> can only guarantee the stability of the closed loop system. However, to ensure that the output of the system (for example, the vehicle's speed in the cruise control example) tracks the reference input <math>r\,</math>, we need to introduce an addition term <math>k_rr\,</math> to offset the steady state output. For linear systems, the control input is proportional to <math>r\,</math> so that we can write <math>u = k_rr\,</math>, where <math>k_r\,</math> is the associated gain.

Because our textbook mainly deals with single output systems, <math>k_r\,</math> and <math>r\,</math> are both scalars in the book. However, they can become vectors in general if the system has more than one output.

--Shuo

[[Category: CDS 101/110 FAQ - Homework 4]]
[[Category: CDS 101/110 FAQ - Homework 4, Fall 2008]]

Do complex matrices also have a Jordan canonical form?

2008-10-16T03:19:01Z

Han:

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 3-2]]
[[Category: CDS 101/110 FAQ - Lecture 3-2, Fall 2008]]

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

2008-10-16T03:18:45Z

Han:

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 3-2]]
[[Category: CDS 101/110 FAQ - Lecture 3-2, Fall 2008]]

I keep getting mixed up on whether the diagonalized form of A is T^(-1)AT or TAT^(-1). Is there an easy way to remember the correct form?

2008-10-16T03:17:20Z

Han:

This is the way I use to remember which form to use. Note the <math>T\,</math> matrix consists of all the eigenvectors, <math>v_i\,</math>, of <math>A\,</math>: <math>T=(v_1|v_2|\cdots|v_n)\,</math>. Using the definition of (right) eigenvectors: <math>Av_i=\lambda_i v_i\,</math>, we have:

<math>
AT = A(v_1|v_2|\cdots|v_n) = (Av_1|Av_2|\cdots|Av_n) = (\lambda_1v_1|\lambda_2v_2|\cdots|\lambda_nv_n)
\,</math>
<math>= (v_1|v_2|\cdots|v_n)\begin{pmatrix}
\lambda_1 \\
& \lambda_2 \\
& & \ddots \\
& & & \lambda_n
\end{pmatrix}=T\Lambda
\,</math>

Right multiply each side by <math>T^{-1}\,</math>: <math>A=T\Lambda T^{-1}\,</math>, which is the correct form.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 3-2]]
[[Category: CDS 101/110 FAQ - Lecture 3-2, Fall 2008]]

Could you please make it clear what is an 'A' and what is a 'Lambda' (matrix)?

2008-10-16T02:53:06Z

Han:

The 'A' matrix is the same 'A' matrix in the state-space model (<math>\dot x=Ax\,</math>). The <math>\Lambda\,</math> matrix is a diagonal matrix whose diagonal consists of all the eigenvalues of A.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 3-2]]
[[Category: CDS 101/110 FAQ - Lecture 3-2, Fall 2008]]

What does "up to permutations" mean?

2008-10-16T02:49:50Z

Han:

In the context of Jordan forms mentioned in today's lecture, it means we don't consider that we will have a new Jordan form by simply exchanging the Jordan blocks.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 3-2]]
[[Category: CDS 101/110 FAQ - Lecture 3-2, Fall 2008]]

In problem 3 (CDS 110), please use a different V2(x).

2008-10-08T06:46:23Z

Han:

Use <math>V_2(x) = \frac{1}{2}x_1^2 + \frac{1}{2}(x_2+\frac{b}{c-a}x_1)^2\,</math> (change the "<math>-\,</math>" to "<math>+\,</math>").

--Shuo

[[Category: CDS 101/110 FAQ - Homework 2]]
[[Category: CDS 101/110 FAQ - Homework 2, Fall 2008]]

You mentioned "aggressive" dynamics a couple of times. Please define. How/when is it useful?

2008-10-06T23:48:35Z

Han:

In the context of dynamical systems, "aggressive" can refer to fast changing, but usually unstable dynamics. The X-29 aircraft mentioned in week 1 is a good example of this. Having intrinsic aggressive dynamics is desired because we can exploit its fast varying nature to generate more agile maneuvers.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 2-1]]
[[Category: CDS 101/110 FAQ - Lecture 2-1, Fall 2008]]

What is the significance of having eigenvalues that are 0? I think I heard you say "in that case you don't know anything". Does that mean you cannot determine if the system is stable or asymptotically stable?

2008-10-06T23:37:37Z

Han:

In today's lecture we were trying to analyze (rather than design) the stability of a given system, so the eigenvalues are already fixed. In the case that one or more eigenvalues are zero, you can say the following things depending on the type of system:

* Linear system, can be transformed into diagonal form (4.8):
System is stable (in the sense of Lyapunov) if all other eigenvalues are strictly negative.

* Linear system, can be transformed into the block diagonal form on page 106:
System is stable (in the sense of Lyapunov) if the real parts all other eigenvalues are strictly negative.

* Linear system, cannot be transformed into the above two forms:
In general cannot determine the stability.

* Linearized version of a nonlinear system:
In general cannot determine the stability.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 2-1]]
[[Category: CDS 101/110 FAQ - Lecture 2-1, Fall 2008]]

I'm a 101/110 student. Can I sit in on the 210 lectures on Fridays without being enrolled in 210?

2008-10-06T23:11:38Z

Han:

You are welcome to any recitation session you like. The reason that we used a sign-up sheet is to estimate the number of students in each session to make sure that the room sizes are OK etc.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 2-1]]
[[Category: CDS 101/110 FAQ - Lecture 2-1, Fall 2008]]

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:35:40Z

Han: In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m moved to In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

#redirect [[In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m]]

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:35:40Z

Han: In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m moved to In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

The "MATLAB code" mentioned in the problem should be the SIMULINK model "cruise_ctrl.mdl".

--Shuo

[[Category: CDS 101/110 FAQ - Homework 1]]
[[Category: CDS 101/110 FAQ - Homework 1, Fall 2008]]

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:35:31Z

Han:

The "MATLAB code" mentioned in the problem should be the SIMULINK model "cruise_ctrl.mdl".

--Shuo

[[Category: CDS 101/110 FAQ - Homework 1]]
[[Category: CDS 101/110 FAQ - Homework 1, Fall 2008]]

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:33:58Z

Han:

The "MATLAB code" mentioned in the problem should be the SIMULINK model "cruise_ctrl.mdl".

--Shuo

{{pound|title=In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise_ctrl.mdl, NOT cruisedyn.m}}

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise\ ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:31:49Z

Han: In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise\ ctrl.mdl, NOT cruisedyn.m moved to In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m

#redirect [[In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m]]

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:31:49Z

Han: In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise\ ctrl.mdl, NOT cruisedyn.m moved to In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m

The "MATLAB code" mentioned in the problem should be the SIMULINK model "cruise_ctrl.mdl".

--Shuo

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:31:29Z

Han: In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise ctrl.mdl, NOT cruisedyn.m moved to In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise\ ctrl.mdl, NOT cruisedyn.m

The "MATLAB code" mentioned in the problem should be the SIMULINK model "cruise_ctrl.mdl".

--Shuo

In problem 1 (CDS 110)/problem 2 (CDS 101), download cruise-ctrl.mdl, NOT cruisedyn.m

2008-10-01T01:30:49Z

Han:

The "MATLAB code" mentioned in the problem should be the SIMULINK model "cruise_ctrl.mdl".

--Shuo

What is an example of an open loop system

2008-09-30T09:38:56Z

Han:

There are many of them. One example is to drive a car with your eyes closed.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 1-1]]
[[Category: CDS 101/110 FAQ - Lecture 1-1, Fall 2008]]

Please turn the AC higher.

2008-09-30T00:35:24Z

Han:

We will talk to the building administrator and find out what can be done.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 1-1]]
[[Category: CDS 101/110 FAQ - Lecture 1-1, Fall 2008]]

What is an example of an open loop system

2008-09-30T00:06:05Z

Han:

There are many of them. One example is to drive a car with your eye closed.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 1-1]]
[[Category: CDS 101/110 FAQ - Lecture 1-1, Fall 2008]]

Is it true that feedback systems are in closed loop by definition?

2008-09-30T00:00:49Z

Han:

Correct. The controller can be considered as "System 1" whereas the plant (e.g. vehicle) as "System 2" on page 5 of the slides.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 1-1]]
[[Category: CDS 101/110 FAQ - Lecture 1-1, Fall 2008]]

Is a high gain good in the speed control example?

2008-09-29T23:53:57Z

Han:

First, one needs to be cautious about defining what "good" means when designing control laws. There are trade-offs between various performance measures, including steady-state error, disturbance rejection, robustness, and other things we did not mention in today's lecture like response time. Therefore, it is impossible to optimize all of them.

In this example, a high gain is good at reducing the steady-state error (<math>v_\mathrm{des}-v</math>) and rejecting external disturbances coming from <math>F_\mathrm{hill}</math>. However, it will not be desirable if one does not want the magnitude of <math>F_\mathrm{eng}</math> to be large.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture 1-1]]
[[Category: CDS 101/110 FAQ - Lecture 1-1, Fall 2008]]

Is a high gain good in the speed control example?

2008-09-29T23:53:33Z

Han:

First, one needs to be cautious about defining what "good" means when designing control laws. There are trade-offs between various performance measures, including steady-state error, disturbance rejection, robustness, and other things we did not mention in today's lecture like response time. Therefore, it is impossible to optimize all of them.

In this example, a high gain is good at reducing the steady-state error (<math>v_\mathrm{des}-v</math>) and rejecting external disturbances coming from <math>F_\mathrm{hill}</math>. However, it will not be desirable if one does not want the magnitude of <math>F_\mathrm{eng}</math> to be large.

--Shuo

[[Category: CDS 101/110 FAQ - Lecture w-m]]
[[Category: CDS 101/110 FAQ - Lecture w-m, Fall 2008]]

What do you mean by "gain" in the speed control example? What relation to sensing does it have?

2008-09-29T23:36:49Z

Han:

In this example, we let the engine force, <math>F_\mathrm{eng}</math>, be proportional to <math>(v_\mathrm{des}-v)</math>. The "gain" is the proportionality constant <math>k_p</math>. The use probably follows from that in [http://en.wikipedia.org/wiki/Gain electronic amplifiers].

We do not consider sensing problems in this example. Instead we assume perfect sensing/measurement of the output, which is the vehicle's velocity, <math>v</math>. The gain is only related to the control law.

--Shuo
[[Category: CDS 101/110 FAQ - Lecture 1-1]]
[[Category: CDS 101/110 FAQ - Lecture 1-1, Fall 2008]]

Group Schedule, Fall 2008

2008-09-24T13:16:23Z

Han:

This page contains information about various upcoming events that are of interest to the group. __NOTOC__
{| width=100%
|- valign=top
| width=50% |
* [[Fall 2008 Meeting Schedule]]
* [http://www.cds.caltech.edu/~murray/calendar.html Richard's calendar]
| width=50% |
* [[Group Schedule, Summer 2008]]
|}

== Group Meetings ==
Group meetings are at noon in 114 Steele. Visitors are welcome (but be prepared to get signed up to give a talk!).
{| width=100% border=1
|- valign=top
| width=30% |
{{agenda begin}}
{{agenda item|Date|Speaker}}
{{agenda item|1 Oct (Wed)|Nok}}
{{agenda item|6-10 Oct|No meeting}}
{{agenda item|14 Oct (Tue)|Shuo}}
{{agenda item|20 Oct (Mon)|Open}}
{{agenda end}}
| width=30% |
{{agenda begin}}
{{agenda item|Date|Speaker}}
{{agenda item|28 Oct (Tue)|Open}}
{{agenda item|4 Nov (Tue)|Open}}
{{agenda item|11 Nov (Tue)|Open}}
{{agenda item|17 Nov (Mon)|Open}}
{{agenda end}}
| width=30% |
{{agenda begin}}
{{agenda item|Date|Speaker}}
{{agenda item|25 Nov (Tue)|Open}}
{{agenda item|3 Dec (Wed)|Hold: visiting students}}
{{agenda item|9 Dec (Tue)|Hold for schedule changes}}
{{agenda end}}
|}

Fall 2008 Meeting Schedule

2008-09-24T13:10:36Z

Han: /* Wed */

__NOTOC__
Sign up for a time to meet. Note that different slots are for different amounts of time. Here are some guides:
* 30 minute slot - weekly meeting; good if you are busy with mainly classes
* 60 minute slot - standard weekly meeting
* 90 minute slot - ''biweekly'' meeting. Pick even or odd weeks (week of 29 Sep = odd)
{| width=100% border=1
|- valign=top
| width=20% |
==== Mon ====
{{agenda begin}}
{{agenda item||}}
{{agenda item||}}
{{agenda item||}}
{{agenda item|4:30p|Open}}
{{agenda item|5:30p|Open}}
{{agenda item|6:30p|Andrea}}
{{agenda end}}
| width=20% |

==== Tue ====
{{agenda begin}}
{{agenda item|1:30p|Open}}
{{agenda item|2:30p|Open (short)}}
{{agenda item|3:00p|Odd: Open}}
{{agenda item||Even: Open}}
{{agenda item|6:00p|Odd: Open}}
{{agenda item||Even: Open}}
{{agenda end}}
| width=20% |

==== Wed ====
{{agenda begin}}
{{agenda item||}}
{{agenda item||}}
{{agenda item||}}
{{agenda item|4:30p|Shuo}}
{{agenda item|5:30p|Open}}
{{agenda item|6:30p|Open}}
{{agenda end}}
| width=20% |

==== Thu ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|3:30p|Open (short)}}
{{agenda item|4:00p|Odd: Open}}
{{agenda item||Even: Open}}
{{agenda item|5:30p|Open}}
{{agenda item|6:30p|Open}}
{{agenda end}}
| width=20% |

==== Fri ====
{{agenda begin}}
{{agenda item||}}
{{agenda item||}}
{{agenda item|3:00p|Open}}
{{agenda item|4:00p|Open}}
{{agenda item|5:00p|Open}}
{{agenda item||}}
{{agenda end}}
|}

Summer 2008 Meeting Schedule

2008-06-08T00:53:25Z

Han: /* Mon */

__NOTOC__
Pick a time that works..
{| width=100% border=1
|- valign=top
| width=20% |
==== Mon ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|2:00p|Julia}}
{{agenda item|3:00p|Shuo}}
{{agenda item|4:00p|Open}}
{{agenda item|5:00p|Open}}
{{agenda end}}
| width=20% |

==== Tue ====
{{agenda begin}}
{{agenda item|12:00p|Group meeting}}
{{agenda item|2:00p|Open}}
{{agenda item|3:00p|Open}}
{{agenda item|4:00p|Synbio SURF}}
{{agenda item|5:00p|Open}}
{{agenda end}}
| width=20% |

==== Wed ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|2:00p|Open}}
{{agenda item|3:30p|Alice SURF}}
{{agenda item|4:30p|Open}}
{{agenda item|5:30p|Open}}
{{agenda end}}
| width=20% |

==== Thu ====
{{agenda begin}}
{{agenda item||}}
{{agenda item|2:00p|Open}}
{{agenda item|3:00p|Open}}
{{agenda item|4:00p|Open}}
{{agenda item|5:00p|Nok}}
{{agenda item|6:00p|iGEM SURF?}}
{{agenda end}}
|}

Vinutha Kallem, April 2008

2008-04-28T17:39:57Z

Han: /* Tuesday */

Vinutha Kallem is a PhD student at Johns Hopkins who is visiting on 28-29 April 2008. __NOTOC__

=== Schedule ===
{| width=100%
|- valign=top
| width=50% |
==== Monday ====
{{agenda begin}}
{{agenda item|9:30a|Richard}}
{{agenda item|10:00a|Julia}}
{{agenda item|10:45a|Seminar prep}}
{{agenda item|11:00a|Seminar}}
{{agenda item|12:15p|Lunch: Nok et al}}
{{agenda item|1:30p|Mani Chandy}}
{{agenda item|2:00p|Nok}}
{{agenda item|3:15p|Ling}}
{{agenda item|4:00p|Elisa}}
{{agenda item|4:45p|Sawyer}}
{{agenda end}}
| width=50% |

==== Tuesday ====
{{agenda begin}}
{{agenda item|9:30a|Dionysios}}
{{agenda item|10:15a|Shuo}}
{{agenda item|11:00a|Erik Winfree}}
{{agenda item|11:30p|Richard}}
{{agenda item|12:00p|Lunch: Mary et al}}
{{agenda item|1:30p|Depart for airport}}
{{agenda end}}
|}

=== Abstract ===

TASK-INDUCED REDUCTION WITH APPLICATIONS TO NEEDLE STEERING

Vinutha Kallem<br>
Mechanical Engineering Department<br>
Johns Hopkins University<br>

Monday, April 28, 2008<br>
11:00 AM to 12:00 PM<br>
Steele Bldg. Room 114 (CDS Library)<br>

What if sensitive organs prevents a physician from accessing a percutaneous target using a straight, rigid needle? One promising solution involves steering flexible bevel-tip needles. These needles introduce exciting robotics and control systems challenges because the needle tip evolves on a Lie group, and the system exhibits a high degree of nonholonomy.

In this work, we present image-guided controllers for steerable needles to improve the accuracy of needle insertions. We build upon a previously proposed needle steering model to develop nonlinear observer-based controllers to drive the needle tip to a desired subspace. These controllers are designed to work in conjunction with subspace planners for the needle tip to reach a desired location in human tissue. We show that the tasks of these controllers induces symmetry, thus resulting in a reduced system which greatly simplifies controller and observer design. We propose a method to perform such reductions for generic nonholonomic kinematic systems on Lie groups with left-invariant vector fields. This technique is used to develop controllers for curve-following of a unicycle and subspace-following in needle steering. We show that this "task-induced" reduction lifts to mechanical systems.

Wei Kang, 26 Feb 08

2008-02-20T08:27:35Z

Han:

This page is the agenda for the visit by Wei Kang from NPS.

== Tuesday ==

* 10:00: Richard, 109 Steele
* 10:30: Open
* 11:15: Open
* 12:00: Lunch time seminar
* 1:30: Open
* 2:15: Shuo
* 3:00: Open
* 3:30: Open
* 4:00: Richard

Jan/Feb 2008 Meetings

2008-01-29T11:02:03Z

Han: /* Tue, 5 Feb */

The list below has times that I am available to meet between 28 Jan and 8 Feb. Please pick a time that works and fill in your name. If none of the times work, send me e-mail (or find someone else who has a slot that does work and see if you can switch). __NOTOC__

<br>
<table width=100% border=1>
<tr valign=top><td width=20%>
==== Mon, 28 Jan ====
{{agenda begin}}
{{agenda item|9:15a|Julia}}
{{agenda item|6:00p|Mary}}
{{agenda end}}

</td><td width=20%>

==== Tue, 29 Jan ====
{{agenda begin}}
{{agenda item|2:00p|Francisco}}
{{agenda item|3:00p|Sayan}}
{{agenda end}}
</td><td width=20%>

==== Wed, 30 Jan ====
{{agenda begin}}
{{agenda item|9:15a|John}}
{{agenda item|10:00a|Dionysios}}
{{agenda end}}
</td><td width=20%>

==== Tue, 5 Feb ====
{{agenda begin}}
{{agenda item|11:00a|Shuo}}
{{agenda item|3:00p|Johan}}
{{agenda item|4:00p|Elisa}}
{{agenda item|5:00p|Ling}}
{{agenda end}}
</td><td width=20%>

==== Wed, 6 Feb ====
{{agenda begin}}
{{agenda item|10:00a|Yizhar}}
{{agenda item|4:00p|Pete}}
{{agenda item|5:00p|Nok}}
{{agenda item|6:00p|Open}}
{{agenda end}}

</td></tr>
</table>

Jan/Feb 2008 Meetings

2008-01-29T11:01:52Z

Han: /* Wed, 6 Feb */

The list below has times that I am available to meet between 28 Jan and 8 Feb. Please pick a time that works and fill in your name. If none of the times work, send me e-mail (or find someone else who has a slot that does work and see if you can switch). __NOTOC__

<br>
<table width=100% border=1>
<tr valign=top><td width=20%>
==== Mon, 28 Jan ====
{{agenda begin}}
{{agenda item|9:15a|Julia}}
{{agenda item|6:00p|Mary}}
{{agenda end}}

</td><td width=20%>

==== Tue, 29 Jan ====
{{agenda begin}}
{{agenda item|2:00p|Francisco}}
{{agenda item|3:00p|Sayan}}
{{agenda end}}
</td><td width=20%>

==== Wed, 30 Jan ====
{{agenda begin}}
{{agenda item|9:15a|John}}
{{agenda item|10:00a|Dionysios}}
{{agenda end}}
</td><td width=20%>

==== Tue, 5 Feb ====
{{agenda begin}}
{{agenda item|11:00a|Open}}
{{agenda item|3:00p|Johan}}
{{agenda item|4:00p|Elisa}}
{{agenda item|5:00p|Ling}}
{{agenda end}}
</td><td width=20%>

==== Wed, 6 Feb ====
{{agenda begin}}
{{agenda item|10:00a|Yizhar}}
{{agenda item|4:00p|Pete}}
{{agenda item|5:00p|Nok}}
{{agenda item|6:00p|Open}}
{{agenda end}}

</td></tr>
</table>