Modern Control Engineering

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202 Chapter 5 / Transient and Steady-State Response Analyses

EXAMPLE 5–7 Consider the system

Using MATLAB, obtain the response curves y(t)when the input uis given by

1. u=unit-step input


2. u=e–t

Assume that the initial state is x(0)= 0.

A possible MATLAB program to produce the responses of this system to the unit-step input

Cu=1(t)Dand the exponential input Cu=e–tDis shown in MATLAB Program 5–13. The result-

ing response curves are shown in Figures 5–29(a) and (b), respectively.

y =[1 0]B

x 1

x 2

R

B

x# 1

x# 2

R= B

- 1

- 1

0.5

0

RB

x 1

x 2

R + B

0

1

Ru

MATLAB Program 5–13

t = 0:0.1:12;

A = [-1 0.5;-1 0];

B = [0;1];

C = [1 0];

D = [0];

% For the unit-step input u = 1(t), use the command "y = step(A,B,C,D,1,t)".

y = step(A,B,C,D,1,t);

plot(t,y)

grid

title('Unit-Step Response')

xlabel('t Sec')

ylabel('Output')

% For the response to exponential input u = exp(-t), use the command

% "z = lsim(A,B,C,D,u,t)".

u = exp(-t);

z = lsim(A,B,C,D,u,t);

plot(t,u,'-',t,z,'o')

grid

title('Response to Exponential Input u = exp(-t)')

xlabel('t Sec')

ylabel('Exponential Input and System Output')

text(2.3,0.49,'Exponential input')

text(6.4,0.28,'Output')

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