Description

Ding Zhao
24-677 Special Topics: Linear Control Systems
• At the beginning of each question you will find the key words for the knowledge that the exercise will help you to practice.
• Right after each week’s recitation, we have a half-hour homework Q&A session. I request you to work on the assignment early and bring your questions to this session to take advantage of this support.
• We designed an Autograder to provide you instantaneous feedback for most of the questions. Submit hw6 theory.py under ”Programming Assignment 6” and a your derivations in .pdf format to ”Homework 6”. We will manually check all of the answers marked as wrong by Autograder to make sure you get the points you deserve.
• Note the we do not have programming questions in HW6.
Exercise 1. Controllability and Observability
Is the state equation

Controllable? Observable?
Exercise 2. Controllability and Observability
Is the state equation

Controllable? Observable?
Exercise 3. Jordan form test
Is the Jordan-form state equation controllable and observable?
2
0
0

x˙ = 0
 0

0
0 1
2
0
0
0
0
0 0
0
2
0
0
0
0 0
0
0
2
0
0
0 0
0
0
0
1
0
0 0
0
0
0
1
1
0 0  2
0  2
0  1
0x +  3
0 −1
0  1
1 1 1
1
1
2
0
0
0 0
1
1 1u
1
1
0
2
1
1 1
1
1 3
2
1 −1
0
1 1
0
1

Exercise 4. Jordan form test
Is it possible to find a set of bij and a set of cij such that the state equation
1 1 0 0 0 b11 b12
0 1 0 0 0 b21 b22
x˙ = 0 0 1 1 0x + b31 b32u
   
0 0 0 1 0 b41 b42 0 0 0 0 1 b51 b52
y x
is controllable? Observable?
Exercise 5. PBH test
Consider the LTI system ˙x = Ax + Bu with:

Using PBH test, find the eigenvalue(s) corresponding to the uncontrollable mode(s) of the system.
Exercise 6. Controllable decomposition
Reduce the state equation

to a controllable form. Is the reduced state equation observable? (Autograder will not be used for this question, it shall be graded manually.)
Exercise 7. kalman decomposition
Decompose the state equation

to a form that is both controllable and observable.
(Autograder will not be used for this question, it shall be graded manually)