Modern Control Engineering

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

or

Hence

The open-loop transfer function of the designed system then becomes

The closed-loop transfer function of the compensated system becomes

Note that in the present case the zero of the lead compensator will cancel a pole of the plant, re-

sulting in the second-order system, rather than the third-order system as we designed using Method 1.

The static velocity error constant for the present case is obtained as follows:

Notice that the system designed by Method 1 gives a larger value of the static velocity error con-

stant. This means that the system designed by Method 1 will give smaller steady-state errors in fol-

lowing ramp inputs than the system designed by Method 2.

For different combinations of a zero and pole of the compensator that contributes 40.894°, the

value of Kvwill be different. Although a certain change in the value of Kvcan be made by alter-

ing the pole-zero location of the lead compensator, if a large increase in the value of Kvis desired,

then we must alter the lead compensator to a lag–lead compensator.

Comparison of step and ramp responses of the compensated and uncompensated systems.

In what follows we shall compare the unit-step and unit-ramp responses of the three systems: the

original uncompensated system, the system designed by Method 1, and the system designed by

Method 2. The MATLAB program used to obtain unit-step response curves is given in

=limsS 0 sc

9

s(s+ 3 )

d= 3

Kv=limsS 0 sGc(s)G(s)

C(s)

R(s)

=

9

s^2 + 3 s+ 9

Gc(s)G(s)=0.9

s+ 1

s+ 3

10

s(s+ 1 )

=

9

s(s+ 3 )

Gc(s)=0.9

s+ 1

s+ 3

Kc=^2

s(s+ 3 )

10

2

s=-1.5+j2.5981

=0.9

jv

–4 –3 –2 –1 1

j 2

j 1

j 3

• j 2


• j 1

s

Desired

closed-loop pole

Compensator

pole

Compensator

zero

60° 120°

Figure 6–44

Compensator pole

and zero.

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