Modern Control Engineering

854 Chapter 10 / Control Systems Design in State Space

Assume that is stable and its upper bound is known. Assume also that and Gare related by

=G+ a

Obtain the condition that the controller Kmust satisfy for robust stability. Also, obtain a gener-

alized plant diagram for this system.

Solution.Let us obtain the transfer function between point Aand point Bin Figure 10–57(a).

Redrawing Figure 10–57(a), we obtain Figure 10–57(b). Then the transfer function between points

AandBcan be obtained as

Define

Then Figure 10–57(b) can be redrawn as Figure 10–57(c). By using the small-gain theorem, the con-

dition for the robust stability of the closed-loop system can be obtained as

(10–180)

Since it is impossible to model precisely, we need to find a scalar transfer function

such that

for all v

and use this instead of a. Then, the condition for the robust stability of the closed-loop

system can be given by

(10–181)

If Inequality (10–181) holds true, then it is evident that Inequality (10–180) also holds true. So this

is the condition to guarantee the robust stability of the designed system. In Figure 10–57(e), a

in Figure 10–57(d) was replaced by.

To summarize, if we make the norm of the transfer function from wtozto be less than

1, the controller Kthat satisfies Inequality (10–181) can be determined.

Figure 10–57(e) can be redrawn as that shown in Figure 10–57(f), which is the generalized

plant diagram for the system considered.

Note that for this problem the matrix that relates the controlled variable zand the exoge-

nous disturbance wis given by

Noting that u(s)=K(s)y(s)and referring to Equation (10–128), is given by the elements

of the Pmatrix as follows:

To make this equal to we may choose P 11 =0, P 12 =Wa, P 21 =I,and

P 22 =G. Then, the Pmatrix for this problem can be obtained as

P= B

0

I

Wa

• G

R

£(s) WaK(I+GK)-^1 ,

£(s)=P 11 +P 12 K(I-P 22 K)-^1 P 21

£(s)

z=£(s)w=(WaTa)w=[WaK(I+GK)-^1 ]w

£

Hq

WaI

¢

7 WaTa (^7) q 61
Wa(jv) ¢
s{¢a(jv)} 6 Wa(jv)
¢a Wa(jv)
7 ¢aTa (^7) q 61
K(1+GK)-^1 =Ta

K

1 +GK

=K(1+GK)-^1

G ¢

¢a G

Openmirrors.com