Modern Control Engineering

If both the robust stability and robust performance conditions are required, the control sys-

tem must satisfy the condition given by Inequality (10–127), rewritten as

(10–134)

For the Pmatrix, we combine Equations (10–133) and (10–131) and get

(10–135)

If we construct P(s)as given by Equation (10–135), then the problem of designing a control

system to satisfy both robust stability and robust performance conditions can be formulated by

using the generalized plant represented by Equation (10–135). As mentioned earlier, such a

problem is called a mixed-sensitivity problem. By using the generalized plant given by Equation

(10–135) we are able to determine the controller K(s)that satisfies Inequality (10–134). The

generalized plant diagram for the system considered in Example 10–14 becomes as shown in

Figure 10–47.

HInfinity Control Problem. To design a controller Kof a control system to sat-

isfy various stability and performance specifications, we utilize the concept of the gen-

eralized plant.

As mentioned earlier a generalized plant is a linear model consisting of a model of

the plant and weighting functions corresponding to the specifications for the required

performance. Referring to the generalized plant shown in Figure 10–48, the Hinfinity

control problem is a problem to design a controller Kthat will make the norm of

the transfer function from the exogenous disturbance wto the controlled variable zless

than a specified value.

Hq

P= C

Ws

0

I

• WsG
  WmG
  
  
  • G
  
  
  
  

S

∑

Wm

KG

1 +KG

Ws

1

1 +KG

∑ 61

816 Chapter 10 / Control Systems Design in State Space

K

w

u

z 2

y

O

I

WmG

−G

z 1 Ws −WsG

Figure 10–47

Generalized plant of

the system discussed

in Example 10–15.

Openmirrors.com