Problem 2Consider the system described by the following difffference equation.y(k) + y(k-1) + 2y(k - 2) = 3u(k) + u(k-1) + u(k-2)1. Derive the transfer function for the system2. Write the system in the controllable state space form3. Is the system stable for zero input. Find the eigenvalues of matrix A to answer thisquestion.4. Is the system controllable? Show your work.5. Is the system observable? Show your work.6. Design a feedback controller using pole placement to place both eigenvalues of theclosed loop system at -0.5.7. Use function sys = ss(A,B,C,D,Ts) to create the state space model in Matlab, where A, B,C, and Dare appropriate matrices and Ts is the sampling rate (use 0.1), and then use step toprint out the step response for 5 time units.8. Use the function eig() to find the eigenvalues of A in Matlab, and also for the closedloop system after pole placement.9. Use the function [K,S,e] = dlqr(A,B,Q,R,N) to design an LQR optimal control law. Here,useQ = [10 0;0 10];R = 100;N = [0;0];Ts = 0.01;Find the gain K.10. Obtain the eigenvalues of the closed loop lqr system.

Answered: Problem 2 Consider the system described…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

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Problem 2
Consider the system described by the following difffference equation.
y(k) + y(k-1) + 2y(k - 2) = 3u(k) + u(k-1) + u(k-2)
1. Derive the transfer function for the system
2. Write the system in the controllable state space form
3. Is the system stable for zero input. Find the eigenvalues of matrix A to answer this
question.
4. Is the system controllable? Show your work.
5. Is the system observable? Show your work.
6. Design a feedback controller using pole placement to place both eigenvalues of the
closed loop system at -0.5.
7. Use function sys = ss(A,B,C,D,Ts) to create the state space model in Matlab, where A, B,
C, and D
are appropriate matrices and Ts is the sampling rate (use 0.1), and then use step to
print out the step response for 5 time units.
8. Use the function eig() to find the eigenvalues of A in Matlab, and also for the closed
loop system after pole placement.
9. Use the function [K,S,e] = dlqr(A,B,Q,R,N) to design an LQR optimal control law. Here,
use
Q = [10 0;0 10];
R = 100;
N = [0;0];
Ts = 0.01;
Find the gain K.
10. Obtain the eigenvalues of the closed loop lqr system.

Expert Solution

Step 1: State the given data.

Since you have posted a question with multiple sub parts, we will provide the solution only to the first three sub parts as per Q&A guidelines. Please repost the remaining subparts separately.

Consider the system described by the following difference equation.

The difference equation of the system is given by

$y (k) + y (k - 1) + 2 y (k - 2) = 3 u (k) + u (k - 1) + u (k - 2);$

We need to find
1. The transfer function for the system.
2. The system in the controllable state space form.
3. We need to check whether the system stable for zero input or not by finding the eigenvalues of matrix A.

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