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Lecture 5.1: Controllability
and State Space Feedback
28 October 2002
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This
lecture introduces the concept of controllability and explores the use of state
space feedback for control of linear systems. Controllability is defined as
the ability to move the system from one condition to another over finite time.
The controllability matrix test is given to check if a linear system is controllable,
and the test is applied to several examples. The concept of (linear) state space
feedback is introduced and the ability to place eigenvalues of the closed loop
system arbitrarily is related to controllability. A cart and pendulum system
and the preditor prey problem are used as examples.
Mud card responses [advanced
search]:
- Why do they call it "step response"?
- Is full rank equivalent to nonsingular?
- On slide 6/7 and in homework 4, problem 2, is q labeled correctly to be zero in the "up" position?
(I thought one of the TAs said it should be zero in the down position).
- For a matrix P, how exactly is the expression P>0 defined?
- What is the physical interpretation of arbitrarily changing the eigenvalues? Is it possible to do in a real system?
- Does controllability generalize to discrete systems? Do you have to be able to reach the desired state in 1 time step?
- The lecture seemed very fast.
- Why is the following theorem true: "Linear system is controllable iff the controllability matrix is full rank" ?
- On slide 8, is the curve x1(t) unique?
- How do we set initial conditions in Simulink?
- Do people use simulink in the "Real World"?
- I can't make it to the review session this Friday. Could you tape that please?
- If a linearization is controllable, is the full nonlinear case always controllable also? Does this apply for any initial conditions or only a limited region?
- What topics are covered by the midterm? How much linear algebra, theory, or applications are on the midterm?
- Are non-course linear algebra books or differential equation books OK for the midterm?
- When / where do we pick up the midterm if we are taking CDS 101 and we don't come to the Wednesday lecture? Also where do we buy blue books?
- On HW3, I couldn't find the eqm. points besides (0,0) and (77,95) in the
predator prey problem - what are they and how do we find them?
- What does "full rank" mean when applied to the integral in the convolution integral?
- Please explain full rank, definition. What it entails.
- Are degenerate states in the controllability matrix always physically realizable?
- If a system is not controllable, how does one find what "isn't controllable", i.e. what states can't be achieved. Is it the null space of the controllablility matrix?
- What do Q and R represent(in slide 15)?
Handouts from lecture
The following materials were handed out in lecture. These have been updated to
include any corrections.
Required reading
Supplemental reading
- B. Friedland, Sections 5.1-5.4 (controllability only)
- A. D. Lewis,
A Mathematical Introduction to Feedback Control, Chapter 2. [pdf]
No Homework
There is no homework set this week, due to the
midterm exam.