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9.11. NONLINEAR L2-GAIN CONTROL 251
L, signals:
Similarly, in this case the problem is to find a stabilizing controller that minimizes the
L, gain of the (closed-loop) system mapping d -> z. For linear time-invariant systems,
we saw that the L. gain is the Ll norm of the transfer function mapping d -- z, and
the theory behind the synthesis of these controllers is known as Ll optimization. The
L optimization theory was proposed in 1986 by M. Vidyasagar [89]. The first solution
of the Li-synthesis problem was obtained in a series of papers by M. A. Dahleh and
B. Pearson [16]-[19]. See also the survey [20] for a comprehensive treatment of the Ll
theory for linear time-invariant systems.
All of these references deal exclusively with linear time-invariant systems. In the
remainder of this chapter we present an introductory treatment of the .C2 control problem
for nonlinear systems.
We will consider the standard configuration of Figure 6.4 and consider a nonlinear
system of the form
i = f (x, u, d)
V) y = 9(x, u, d) (9.45)
z = h(x,u,d)
where u and d represent the control and exogenous input, respectively, and d and y represent
the measured and regulated output, respectively. We look for a controller C that stabilizes
the closed-loop system and minimizes the ,C2 gain of the mapping from d to z. This problem
can be seen as an extension of the H. optimization theory to the case of nonlinear plants,
and therefore it is often referred to as the nonlinear Ham-optimization problem.
9.11 Nonlinear G2-Gain Control
Solving the nonlinear G2-gain design problem as described above is very difficult. Instead,
we shall content ourselves with solving the following suboptimal control problem. Given a
"desirable" exogenous signal attenuation level, denoted by ryl, find a control Cl such that
the mapping from d -* w has an .C2 gain less than or equal to ryl. If such a controller Cl
exists, then we can choose rye < yl and find a new controller C2 such that the mapping
from d -> w has an G2 gain less than or equal to 'y2. Iterating this procedure can lead to a
controller C that approaches the "optimal" (i.e., minimum) -y.
We will consider the state feedback suboptimal L2-gain optimization problem. This
problem is sometimes referred to as the full information case, because the full state is
assumed to be available for feedback. We will consider a state space realization of the form