1
Sheet 8 Solution: Lead,Lag,Lead-Lag Compensators and PID Controllers:
Problem 1: Consider the system shown in figure 5.8 (a) the feedforward transfer function is:
G(s) = 𝟒
𝒔(𝒔+𝟐)
Figure 5.8 (a): Control System
2
Figure 5.8 (b): The root locus and closed location of system
5.8 (c)
3
Figure 5.8 (c): The lead compensator design
4
Figure 5.8 (d): The Control system with lead compensator
5
Figure 5.8 (e): The root locus of compensated and uncompensated systems
Problem 2: Consider the system shown in figure 5.8 (a) the feedforward transfer function is:
G(s) = 𝟏.𝟎𝟔
𝒔(𝒔+𝟏)(𝒔+𝟐)
Figure 5.8 (a): Control System
6
Figure 5.8 (b): Root Locus of the original system
7
Figure 5.8 (c): The block diagram of the compensated system
8
Figure 5.8 (d): The root locus of compensated and uncompensated systems
Figure 5.8 (e): The step response of compensated and uncompensated systems
9
Figure 5.8 (f): The ramp response of compensated and uncompensated systems
Problem 3: Consider the system shown in figure E.1 the feedforward transfer function is:
G(s) = 𝟒
𝒔(𝒔+𝟎.𝟓)
Figure E.1: Control system
10
11
12
13
Figure E.2: (a) root locus plot of compensated system (b) root locus plot near the origin
14
Figure E.3: Transient response curves of the compensated and uncompensated systems
(a) Unit-step response curves; (b) Unit ramp response curves
15
Given:
Problem 4: Consider the system shown in figure E.1
Figure E.1: Control system with PID controller
16
Given:
Problem 5: The open loop step response for transfer function Gp(s) is given in figure E.1
Kp
17
Figure E.1: Open loop step response
based on figure E.1
