Question

Use matlab to obtain a state-space representation of C(s) G(s)= E(s) 25° +4.88² +9.65 +16 S+5 Using the state-space model and matlab command ‘Isim()”.obtain and plot the response c(t) for a step input e(t) = 2 sin(0.01t).

Answer 1

Solution:

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MATLAB Script

clc;
clear;
close all;
%% To obtain the state space representation
num = [1, 5];
den = [2, 4.8, 9.6, 16];

[A, B, C, D] = tf2ss(num, den);

disp('The state space representation of G(s) is:');

G = ss(A, B, C, D)

%% To obtain and plot the response c(t) for a step input e(t) = 2

t = 0:0.0001:40;
c = 2*step(G, t);

plot(t, c, 'k', 'linewidth',2);
grid;
xlabel('Time, t (sec)');
ylabel('c(t)');
title('Response of the system to an input e(t) = 2');

%% To obtain and plot the response c(t) for e(t) = 2.sin(0.01t)
T = 2*pi/0.01;
t = 0:T/1000:5*T;
e = 2*sin(0.01*t);

c = lsim(G, e, t);

figure
subplot(2,1,1)
plot(t, e, 'k', 'linewidth',2);
grid;
xlabel('Time, t (sec)');
ylabel('e(t)');
title('Input e(t) = 2.sin(0.01t)');

subplot(2,1,2)
plot(t, c, 'k', 'linewidth',2);
grid;
xlabel('Time, t (sec)');
ylabel('c(t)');
title('Response of the system to an input e(t) = 2.sin(0.01t)');

After executing we get

In the command window

The state space representation of G(s) is:

G =

A =
x1 x2 x3
x1 -2.4 -4.8 -8
x2 1 0 0
x3 0 1 0

B =
u1
x1 1
x2 0
x3 0

C =
x1 x2 x3
y1 0 0.5 2.5

D =
u1
y1 0

Continuous-time state-space model.

>>

Response of the system to an input e(t) = 2 0.9 0.8 0.7 Mm 0.6 량 0.5 04 0.3 02. 0.1 0 0 5 10 15 25 30 35 40 20 Time, t (sec)

Input e(t) = 2.sin(0.01t) 1 eo M -1 -2 0 500 1000 1500 2000 2500 3000 3500 Time, t (sec) Response of the system to an input e(t) = 2.sin(0.01) 0.6 0.4 0.2 -0.2 -0.4 -0.6 500 1000 1500 2000 2500 3000 3500 Time, t (sec)