Modern Control Engineering

Section 6–8 / Lag-Lead Compensation 337

The compensated system will have the open-loop transfer function

Because of the cancellation of the (s+0.5)terms, the compensated system is a third-order system.

(Mathematically, this cancellation is exact, but practically such cancellation will not be exact be-

cause some approximations are usually involved in deriving the mathematical model of the sys-

tem and, as a result, the time constants are not precise.) The root-locus plot of the compensated

system is shown in Figure 6–56(a). An enlarged view of the root-locus plot near the origin is shown

in Figure 6–56(b). Because the angle contribution of the phase lag portion of the lag–lead

compensator is quite small, there is only a small change in the location of the dominant closed-

loop poles from the desired location, The characteristic equation for the com-

pensated system is

or

Hence the new closed-loop poles are located at

The new damping ratio is z=0.491. Therefore the compensated system meets all the required per-

formance specifications. The third closed-loop pole of the compensated system is located at

Since this closed-loop pole is very close to the zero at the effect of this pole

on the response is small. (Note that, in general, if a pole and a zero lie close to each other on the

negative real axis near the origin, then such a pole-zero combination will yield a long tail of small

amplitude in the transient response.)

s=-0.2078. s=-0.2,

s=-2.4123;j4.2756

=(s+2.4123+j4.2756)(s+2.4123-j4.2756)(s+0.2078)= 0

s^3 +5.0325s^2 +25.1026s+5.008

s(s+5.02)(s+0.01247)+25.04(s+0.2)= 0

s=-2.5;j4.33.

Gc(s)G(s)=

25.04(s+0.2)

s(s+5.02)(s+0.01247)

Root-Locus Plot of Compensated System

Imag Axis

• 10 – 5 0 5 10
  Real Axis
  (a)


• 2

2

8

6

0

4

10

• 10
  
  
  • 4
  
  
  • 6
  
  
  • 8
  
  
  
  

Root-Locus Plot of Compensated System near the Origin

RealAxis

Imag Axis

• 0.5 –0.4 –0.3 –0.2 –0.1 0


• 0.05

0.05

0.2

• 0.15


• 0.25

0.15

0

0.1

• 0.1

0.25

• 0.2

(b)

Figure 6–56
(a) Root-locus plot of the compensated system; (b) root-locus plot near the origin.