Modern Control Engineering

340 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method

0

jv

s

B A

P

55 °

j 5

j 4

j 3

j 2

j 1

• j 4


• j 3


• j 2


• j 1


• 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 112

Figure 6–58

Determination of the

desired pole-zero

location.

By simple calculations, we find that if we choose , then

and if we choose , then

Since is one of the time constants of the lag–lead compensator, it should not be too large. If

can be acceptable from practical viewpoint, then we may choose. Then

Thus, the lag–lead compensator becomes

The compensated system will have the open-loop transfer function

No cancellation occurs in this case, and the compensated system is of fourth order. Because the

angle contribution of the phase lag portion of the lag–lead network is quite small, the dominant

closed-loop poles are located very near the desired location. In fact, the location of the dominant

closed-loop poles can be found from the characteristic equation as follows: The characteristic

equation of the compensated system is

which can be simplified to

=(s+2.4539+j4.3099)(s+2.4539-j4.3099)(s+0.1003)(s+3.8604)= 0

s^4 +8.8685s^3 +44.4219s^2 +99.3188s+9.52

(s+8.34)(s+0.0285)s(s+0.5)+ 40 (s+2.38)(s+0.1)= 0

Gc(s)G(s)=

40(s+2.38)(s+0.1)

(s+8.34)(s+0.0285)s(s+0.5)

Gc(s)=(10)a

s+2.38

s+8.34

ba

s+0.1

s+0.0285

b

1

bT 2

=

1

3.503* 10

=0.0285

T 2 = 10 T 2 = 10

T 2

17 magnitude 7 0.99, - 1 ° 6 angle 60 °

T 2 = 10

17 magnitude 7 0.98, -1.5° 6 angle 60 °

T 2 = 5

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