Consider a system with the state and output equations: [ q\'1 ] [ 2 0 0 ] [ q1]
ID: 2988576 • Letter: C
Question
Consider a system with the state and output equations:
[ q'1 ] [ 2 0 0 ] [ q1] [ 0 1 ] [ q1 ]
[ q'2 ] = [ 1 -2 0 ] [ q2 ] + [ 1 0 ] [ x1(t) ] and [ y1 ] = [ 1 2 0 ] [ q2 ] + [ 0 -1] [ x1(t) ]
[ q'3 ] [ 0 0.5 -1 ] [ q3 ] [1 -1 ] [ x2(t) ] [ y2 ] [ 0 1 -2 ] [ q3 ] [ 1 0] [ x2(t) ]
a. Solve for the vector q(t) and the transfer function matrix H(s), and display the results.
b. Determine the output y2(t) in response to the input x1(t).
Explanation / Answer
MATLAB code ->
a = [2,0,0;1,-2,0;0,0.5,-1];
b = [0,1;1,0;1,-1];
c = [1,2,0;0,1,-2];
d = [0,-1;1,0];
sys_ss = ss(a,b,c,d);
disp('a:')
H = tf(sys_ss)
H21 = [cell2mat(H(2,1).num);cell2mat(H(2,1).den)];
syms s t
F = (H21(1,1)*s^2 + H21(1,2)*s + H21(1,3)) / (H21(2,1)*s^2 + H21(2,2)*s + H21(2,3));
disp('b:')
y2_x1 = ilaplace(F, s, t)
clear a b c d sys_ss H H21 s t F y2_x1
Output ->
a:
H =
From input 1 to output...
2
1: -----
s + 2
s^2 + 2 s - 2
2: -------------
s^2 + 3 s + 2
From input 2 to output...
-s^2 + s + 8
1: ------------
s^2 - 4
2 s^2 + s - 8
2: -------------------
s^3 + s^2 - 4 s - 4
Continuous-time transfer function.
b:
y2_x1 =
2*exp(-2*t) - 3*exp(-t) + dirac(t)