Question

5 For a system: Y() 10.4s? +47s +160 U(s) 5+148° +568 +160 use Matlab to do: (a) obtain the state-space representation of the system. (b) transfer the state-space representation into Modal canonical form. (c) find the eigenvalues of the system matrix A, determine the system stability (d) find the controllability and observability matrixes. Determine the controllability and observability.

Answer 1

MATLAB CODE:

clc
close all
num = [10.4 47 160];
den = [1 14 56 160];
G = tf(num,den);
% a)
[A,B,C,D] = tf2ss(num,den)

% b)
csys = canon(G,'modal')

% c)
Roots = eig(A)
if real(Roots)>0
disp('Given system is unstable')
else
disp('Given system is stable')
end

% d)

Contr = ctrb(A,B)
if det(Contr)==0
disp('Given system is controllable')
else
disp('Given system is Not controllable')
end

obser = obsv(A,C)
if det(obser)==0
disp('Given system is observable')
else
disp('Given system is Not observable')
end

OUTPUT

For a ;

Command Window -14 -56 -160 10.4000 47.0000 160.0000

For b :

Command Window csys = x1 x2 x3 x1 -10 x2 -2 3.464 x3 -3.464 -2 ul x1 -6.947 x2 2.036 x3 -4.437 x1 x2 x3 -1.383 0.1194 -0.1243 y1 ul y1 Continuous-time state-space model.

For c :

Command Window Roots = -10.0000 + 0.0000i -2.0000 + 3.4641i -2.0000 - 3.4641i Given system is stable Contr = -14 140 -14 !!

For d :
Command Window Contr 140 -14 -14 Given system is Not controllable obser = 1.0e+04 * 0.0047 0.0010 0.0160 -0.0099 -0.0422 -0.1664 0.3858 0.0958 1.5776 Given system is Not observable >> %!

THANK YOU.

Answer 2

clc
close all
num = [1 2 15 10];
den = [1 4 8 10];
G = tf(num,den);
% a)
[A,B,C,D] = tf2ss(num,den)

% b)
csys = canon(G,'modal')

% c)
Roots = eig(A)
if real(Roots)>0
disp('Given system is unstable')
else
disp('Given system is stable')
end

% d)

Contr = ctrb(A,B)
if det(Contr)==0
disp('Given system is controllable')
else
disp('Given system is Not controllable')
end

obser = obsv(A,C)
if det(obser)==0
disp('Given system is observable')
else
disp('Given system is Not observable')
end