Modern Control Engineering

614 Chapter 8 / PID Controllers and Modified PID Controllers

Figure 8–43

(continued) (c)

yr

(t)

t (sec)

Response to Unit-Acceleration Reference Input

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Input

Output

The response exhibits the maximum overshoot of 21%and the settling time is approximately

1.6 sec. Figures 8–43(b) and (c) show the ramp response and acceleration response. The steady-

state errors in both responses are zero. The response to the step disturbance was satisfactory. Thus,

the designed controllers and given by Equations (8–12) and (8–13), respectively, are

satisfactory.

If the response characteristics to the unit-step reference input are not satisfactory, we need to

change the location of the dominant closed-loop poles and repeat the design process. The domi-

nant closed-loop poles should lie in a certain region in the left-halfsplane (such as 2 a6,

2 b6, 6c 12 ). If the computational search is desired, write a computer program (sim-

ilar to MATLAB Program 8–8) and execute the search process. Then a desired set or sets of val-

ues ofa, b,andcmay be found such that the system response to the unit-step reference input

satisfies all requirements on maximum overshoot and settling time.

Example Problems and Solutions

A–8–1. Describe briefly the dynamic characteristics of the PI controller, PD controller, and PID

controller.

Solution.The PI controller is characterized by the transfer function

The PI controller is a lag compensator. It possesses a zero at and a pole at s=0.Thus,

the characteristic of the PI controller is infinite gain at zero frequency. This improves the

steady-state characteristics. However, inclusion of the PI control action in the system increases the

s=- 1 Ti

Gc(s)=Kpa 1 +

1

Ti s

b

Gc1(s) Gc2(s)

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