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Lecture 4.1: Linear Systems
21 October 2002
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This
lecture gives an introduction to linear control systems. The main properties
of linear systems are given and the matrix exponential is used to provide a
formula for the output response given an initial condition and input signal.
Linearization of nonlinear systems as an approximation of the dynamics is also
covered.
Mud card responses [advanced
search]:
- On slide 10 of lecture 4.1, where did the vo equation come from?
- If we need to know Kirchhoff's Laws, can you summarize them?
- Mistakes on slide 7 of handouts of lecture 4.1
- For the Wednesday lecturees, could the board examples be made available has handouts or put on the web?
- Does "exp" stable mean asymptotically stable?
- I'm understand the concepts from class, but have trouble solving the problems on the homework. Can you spend more time on tools to solve the problems; or schedule a recitation section?
- How do you make phase plots in MATLAB for nonlinear systems? Can we see any MATLAB code used to make graphs in the lecture notes?
- (correction)
According to your definition of lpdf, you need a function \alpha(||x||) such
that V(x)>=\alpha(||x||), \forall x \in B_\epsilon. Do I need to find an
\alpha(.)? If not, what's the alternate definition of pdf?
- Will Laplace transform tables be allowed on the midterm exam?
- Could you make sure to provide ample documentation for MATLAB examples in
the future?
- Can you provide more worked examples using/finding Lyapunov functions?
- Can you draw an I/O block diagram for x''-2zwx'-w2=u(t)?
- Can you try harder to catch errors and make corrections on the HW earlier?
- How will the frequency response of x''-2zwx'-w2=u(t) when u(t)=0
relate to that when u(t)=A sinWt?
- When would I ever use Lyapunav funs when eig() is much faster?
- How would you make the BODE plot (using MATLAB) of the system (inverted
pendulum) on slide 15?
- When can you say H(s)|s=jw is the frequency response of the system?
- On slide 12 of today's lecture notes, where did you get the matrices A,B,C,D from? Are they the same as the A,B,C,D in the integral?
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 linear control systems. The first problem asks for
stability, step and frequency response for some common examples of linear systems.
The second problem considers stabilization of an inverted pendulum on a cart,
using the local linaerization. The remaining problems (for CDS 110 students)
include derivation of discrete time linear systems response functions and derivation
of the linearized model for the Caltech ducted fan.
Modifications to the homework (link above is always the latest version):
- 26 Oct 02: version 1.1 release
- Fixed small typo in problem 1a (see FAQ below)
- 21 Oct 02: problem set release
Frequently asked questions on homework and TA hints: