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5.2 Feedback-Linearization Controllers 289

the closed-loop system. This design was briefly discussed in Chapter 2,
Example 2.11.4. It is in a different spirit from the other design of this chapter,
because it relies on classical frequency-domain SISO concepts. The general
structure is shown in Figure 5.2.8. A two-DOF robust controller was designed
and simulated in [Sugie et al. 1988] and will be presented next. Let the plant
be given by (5.2.5) and consider the following factorization:

G(s)=N(s)D (s),
-1

where

D(s)=s , N(s)=I. (5.2.37)
2

The following result presents a two-DOF compensator which will robustly
stabilize (in the ∞ sense) the error system.

THEOREM 5.2–5: Consider the two-DOF structure of Figure 5.2.8. Let
K 1 (S) be a stable system and K 2 (s) be a compensator to stabilize G(s). Then
the controller

2
u=s K 1v+K 2(K 1v-q) (1)
will lead to the closed-loop system

q=K1v (2)
and the closed-loop error system (5.2.13) will be ∞ stable.

Proof:
With simple block-diagram manipulations, it may be shown that the closed-
loop system is

q=K 1v.
The actual robustness analysis is involved and will be omitted, but a particular
design and its robustness are discussed in the next example.
Note from (2) that K 1 (s) is used to obtain the desired closed-loop transfer
function. It should then be stable, and to guarantee a zero steady-state error,
we choose v=qd and make sure that the dc gain K 1 (0)=1. Finally, we would
like K 1(S) to be exactly proper (i.e., zero relative degree). K 2(s), on the other
hand, will assure the robustness of the closed-loop system. Therefore, K 2(s)
should stabilize G(s) and provide suitable stability margins. It should contain

Copyright © 2004 by Marcel Dekker, Inc.