Modern Control Engineering

Robust Performance. Consider the system shown in Figure 10–44. Suppose that we want

the output y(t)to follow the input r(t)as closely as possible, or we wish to have

Since the transfer function Y(s)/R(s)is

we have

Define

whereSis commonly called the sensitivity function and Tdefined by Equation (10–124) is called

the complementary sensitivity function. In this robust performance problem we want to make

the norm of Ssmaller than the desired transfer function or which can be

written as

(10–126)

Combining Inequalities (10–123) and (10–126), we get

whereT+S=1,or

(10–127)

Our problem then becomes to find K(s)that will satisfy Inequality (10–127). Note that depend-

ing on the chosen Wm(s)andWs(s) there may be many K(s) that satisfy Inequality (10–127), or

may be no K(s)that satisfies Inequality (10–127). Such a robust control problem using Inequality

(10–127) is called a mixed-sensitivity problem.

Figure 10–45(a) is a generalized plant diagram, where two conditions (robust stability and ro-

bust performance) are specified. A simplified version of this diagram is shown in Figure 10–45(b).

∑

Wm(s)

K(s)G(s)

1 +K(s)G(s)

Ws(s)

1

1 +K(s)G(s)

∑

q

61

g

WmT

WsS

g

q

61

7 Ws S (^7) q 61
Hq Ws -^17 S (^7) q 6 Ws-^1

1

1 +KG

=S

E(s)

R(s)

=

R(s)-Y(s)

R(s)

= 1 -

Y(s)

R(s)

=

1

1 +KG

Y(s)

R(s)

=

KG

1 + KG

lim

tSq

[r(t)-y(t)] = lim

tSq

e(t) S 0

812 Chapter 10 / Control Systems Design in State Space

re y

+– K(s) G(s)

Figure 10–44

Closed-loop system.

Openmirrors.com