1 8 -5 6 -21 eigenvalues of the controller:-5,-5+8j,-5-8j) eigenvalues of the ob

Question

1 8 -5 6 -21 eigenvalues of the controller:-5,-5+8j,-5-8j) eigenvalues of the observer: (-10,-105j,-10-5j. a. Identify the transfer function of the system. b. Compute the controllability matrix of the system. Is the system controllable? c. Compute the observability matrix of the system. Is the system observable? d. Suppose state feedback is possible. Design a state feedback controller K such that the eigenval- e. Suppose the state feedback controller is used to track a step reference. Find the pre-compensator f. Suppose state feedback is not possible and we wish to use an output feedback controller. Design g. Write the state space equations of the closed loop system with output feedback. What are the h. We wish to implement the output feedback controller as follows: ues of the closed loop systems are ±8j. N such that the steady state error is zero. a Luenberger controller such that the eigenvalues of the observer are-10 eigenvalues of the closed loop system. 5j. Y(s)

Explanation / Answer

I have soved in MATLAB. MATLAB code is give below followd by the results.

clc;
close all;
clear all;

A = [1 8 -5;3 -4 9;6 -2 1];
B = [4 ;2 ;0];
C = [1 -1 0];
D = [0];

s=tf('s');

[num,den] = ss2tf(A,B,C,D);

G = tf(num,den);
Control_Matrix = [B A*B A^2*B ]
det(Control_Matrix)

Obsvr_Matrix = [C ; C*A ;C*A^2 ]
det(Obsvr_Matrix)
P = [-5 -5+8j -5-8j];
K = PLACE(A,B,P)

Results:

Control_Matrix =

4 20 -48
2 4 224
0 20 132

ans =

-23008 (system is controllable)

Obsvr_Matrix =

1 -1 0
-2 12 -14
-50 -36 104

ans =

-164 (system is observable)

P =

-5.0000 -5.0000 + 8.0000i -5.0000 - 8.0000i

K =

2.7017 1.0967 1.8790 (state feedback gains for required placement of poles)

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