Modern Control Engineering

Section 8–3 / Design of PID Controllers with Frequency-Response Approach 581

Therefore, the designed system satisfies all the requirements. Thus, the designed system

is acceptable. (Note that there exist infinitely many systems that satisfy all the require-

ments. The present system is just one of them.)

Next, we shall obtain the unit-step response and the unit-ramp response of the de-

signed system. The closed-loop transfer function is

Note that the closed-loop zeros are located at

The closed-loop poles are located at

Notice that the complex-conjugate closed-loop poles have the damping ratio of 0.5237.

MATLAB Program 8–5 produces the unit-step response and the unit-ramp response.

s=-0.1897

s=-2.4052-j3.9119

s=-2.4052+j3.9119

s=-4, s=-0.2

C(s)

R(s)

=

5 s^2 + 21 s+ 4

s^3 + 5 s^2 + 22 s+ 4

MATLAB Program 8–5

% Unit-step response

num = [5 21 4];

den = [1 5 22 4];

t = 0:0.01:14;

c = step(num,den,t);

plot(t,c)

grid

title('Unit-Step Response of Compensated System')

xlabel('t (sec)')

ylabel('Output c(t)')

% Unit-ramp response

num1 = [5 21 4];

den1 = [1 5 22 4 0];

t = 0:0.02:20;

c = step(num1,den1,t);

plot(t,c,'-',t,t,'--')

title('Unit-Ramp Response of Compensated System')

xlabel('t (sec)')

ylabel('Unit-Ramp Input and Output c(t)')

text(10.8,8,'Compensated System')