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 hw5 theory.py, hw5 script.py, hw5 output.npy under ”Programming Assignment 5” and a your derivations in .pdf format to ”Homework 5”. We will manually check all of the answers marked as wrong by Autograder to make sure you get the points you deserve.
Exercise 1. DT Dynamics
Consider the following system, Given and the input u(k) = 1, k =
0,1,···∞. Find y(5).

Exercise 2. Discretizing, DT Dynamics
Discretize the system below using a sample time of T = 1s.

Exercise 3. CT Dynamics
Given the following system, find y(5).

Assume x(0) = 0 and u is a unit step input.
Exercise 4. Linear Discrete-Time State Equations Solve for x(3) where x = [x1 x2 x3]T.

When put in matrix form x(k + 1) = Ax(k), the system matrix is
Exercise 5. State transition matrix
Find the state transition matrix of

(Autograder will not be used for this question. It shall be graded manually.)
Exercise 6. CT Dynamics
An oscillation can be generated by

Show that the solution is

(Autograder will not be used for this question. It shall be graded manually.)