Modern Control Engineering

Consider the system shown in Figure 10–43(a). Let us examine the transfer function between

pointAand point B. Notice that Figure 10–43(a) can be redrawn as shown in Figure 10-43(b). The

transfer function between point Aand point Bcan be given by

Define

(10–121)

Using Equation (10–121) we can redraw Figure 10–43(b) as Figure 10–43(c). Applying the small-

gain theorem to the system consisting of and Tas shown in Figure 10–43(c), we obtain the

condition for stability to be

(10–122)

In general, it is impossible to precisely model Therefore, let us use a scalar transfer function

such that

where is the largest singular value of.

Consider, instead of Inequality (10–122), the following inequality:

(10–123)

If Inequality (10–123) holds true, Inequality (10–122) will always be satisfied. By making

the norm of to be less than 1, we obtain the controller Kthat will make the system

stable.

Suppose that we cut the line at point Ain Figure 10–43(a). Then we obtain Figure 10–43(d).

Replacing by , we obtain Figure 10–43(e). Redrawing Figure 10–43(e), we obtain Figure

10–43(f). Figure 10–43(f) is called a generalized plant diagram.

Referring to Equation (10–121),Tis given by

(10–124)

Then Inequality (10–123) can be rewritten as

(10–125)

Clearly, for a stable plant model G(s), K(s)=0 will satisfy Inequality (10–125). However,

K(s)=0 is not the desirable transfer function for the controller. To find an acceptable trans-

fer function for K(s), we may add another condition—for example, that the resulting system will

have robust performance such that the system output follows the input with minimum error, or

another reasonable condition. In what follows we shall obtain the condition for robust

performance.

ß

WmK(s)G(s)

1 +K(s)G(s)

ß

q

61

T=

KG

1 +KG

¢m WmI

Hq WmT

7 WmT (^7) q 61
s{¢m(jv)} ¢m(jv)
s{¢m(jv)} 6 Wm(jv)
Wm(jv)
¢m.
7 ¢mT (^7) q 61
¢m

(1+KG)-^1 KG=T

KG

1 +KG

=( 1 +KG)-^1 KG

810 Chapter 10 / Control Systems Design in State Space

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