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 hw7 theory.py under ”Programming Assignment 7” and a your derivations in .pdf format to ”Homework7”. We will manually check all of the answers marked as wrong by Autograder to make sure you get the points you deserve.
• Note: we do not have programming questions in HW7.
• Remember to submit derivations of all questions in your pdf submission Exercise 1. Asymptotic stability and Lyapunov stability.
For each of the systems given below, determine whether it is Lyapunov stable, whether it is asymptotic stable.
(a)

(b)

Exercise 2. Operator Norms Find the Norms of the matrices:
1. L2 norm of :
2. For A = calculate the Nuclear norm, ||A||N and Frobenius norm, ||A||F.
3. L1, L2 and L∞ norm of
Exercise 3. Operator Norms
The LASSO cost is given as:
βˆlasso = argminβ∈Rv {K}
where,

Given,
= 20 and N = 2.
Calculate the cost value K. Write calculation steps.



 Exercise 4. Given A = 



2 0 0 0 0 0
0 1 1 0 0 0
0 0 1 1 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 −1 


, find e
 At for t = 2




Exercise 5. Stability of a CT LTV system. Is the homogeneous equation
x˙ x
for t0 ≥ 0, stable i.s.L? Asymptotically stable?
(Autograder will not be used for this question, it shall be graded manually)
Exercise 6.
Prove that,
.
where σ1 > σ1 > … > σn are the singular values of A.
(Autograder will not be used for this question, it shall be graded manually.)
Exercise 7. Stabilizability
Decompose the state equation

to a controllable form. Is the reduced state equation observable, stabilizable, detectable?
(Autograder will not be used for this question, it shall be graded manually.)
Additional Exercise (Optional)
”William L. Brogan. Modern Control Theory”, Example 10.5 (Page 359) and Example 10.6 (page 360).