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Lecture 3.1: Stability and
Performance
14 October 2002
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This
lecture provides and introduction to stability and performance of (nonlinear)
control systems. A formal definition of stable systems is given and phase portraits
are introduced to help visualize the concepts. Local and global behavior of
nonlinear systems is discussed, using a damped pendulum and the preditor-prey
problem as examples. Performance of control systems is presented for both transient
(step respones) and steady state (frequency domain) specifications.
Mud card responses [advanced
search]:
- Are limit cycles always necessarily found around equilibrium points?
- What is ε (Lecture 3.1 slide 7)?
- Was (77,92) the limit cycle on slide 10 of lecture 3.1? If so, why was it unstable?
- Can you explain the phase \phi more (i.e., what's its physical meaning)?
- Today's lecture had lots of definitions and theorems; can you use more examples in the lectures?
- Why don't you update (i.e., FAQs, solutions etc) on the CDS110 page,
while you are updating the 101 page?
- What was backwards in slide 6?
- In slide 5, the length of the vectors is determined by ___? The direction of the vectors is determined by ____?
- Is there a MATLAB command that produces the streamline and vector plots like those shown in the beginning of lecture?
- Why is it called a "phase" portraits?
- The solution of the odes in the 5th problem was kind of ugly, so I used Maple to do it. It was just copying, so what was the point?
- How do we create a function like "dosc"?
- How computationally difficult will the tests be?
- There's a limit to how many people can use MATLAB in the ITS lab - can this be increased?
- Some of the questions on homework #2 required lecture #3.1 to solve. Why did you do this?
- Is there a more convenient method for finding settling time in MATLAB other than
the find() function?
- What is the difference between disturbance rejection and using a simple filter?
- What happens when your disturbances have similar frequencies to the reference you would like to track? (Is there still a way to reject disturbances?)
- For asymptotic stability, does the point get reached or just orbited forever
and get closer?
Handouts from lecture
The following materials were handed out in lecture. These have been updated to
include any corrections.
Required reading
Supplemental reading
This homework set covers stability and performance through a series of application
examples. The first problem provides a set of three real-world models in which
the student must identify the equilibrium points and determine stability of
the equilibrium points (through simulation). The second problem explores performance
specification in the conext of the cruise control example, including step response
and frequency response.
Modifications to the homework (link above is always the latest version):
- 19 Oct 02: minor changes to fix typos and clarify some statements
- Step response in 2(a) should be from 55 to 65 mph
- Frequency in 2(b) should be 1 Hz (about 6 rad/sec)
- Problem 3(a) had some information which was not correct. It has been
modified and the initial condition has been changed from y(0) = -1 to
y(0) = 1.
- 16 Oct 02: handed out remaining problems (CDS 110)
- 14 Oct 02: handed out first two problems
Frequently asked questions on homework and TA hints: