NonlinearBook10pt November 20, 2007
604 CHAPTER 10
bounded energy disturbance mod els. As in the pure H
∞
case, the mixed-
norm H
2
/H
∞
problem can be compared to a game-theoretic framework
involving a Nash differential game problem [279].
Using a n onlinear game-theoretic framework the authors in [22, 31]
replace the algebraic Riccati equation arising in linear H
∞
theory with
a particular Hamilton-Jacobi-Bellman equation (the Isaacs equation) to
obtain a nonlinear equivalent to the H
∞
analysis and synthesis control
problem. Sufficient conditions for the existence of stabilizing solutions of the
Isaacs equation are given in [438–440] in terms of the existence of a linear
(sub)optimal H
∞
controller for the linearized (about a given equilibriu m
point) nonlinear controlled system. In parallel research, the authors in
[213–216] use nonlinear dissipativity theory [77,188,189,191,320,456,457] for
nonlinear affine systems with appropriate storage functions and quadratic
supply rates to obtain nonexpansive (gain bounded) closed-loop s ystems.
Although a nonlinear equivalent to H
∞
analysis and synthesis has been
developed it is important to note that the methods and results discussed
in [22,31,213–216,438–440] are independent of optimality considerations. In
this chapter, we develop an optimality-based theory for disturbance rejection
for nonlinear systems with bounded exogenous d isturbances. T he key
motivation for developing an optimal and inverse optimal nonlinear control
theory that additionally gu arantees disturbance rejection is th at it provides
a class of candidate disturbance rejection controllers parameterized by the
cost functional that is minimized. In the case of linear systems, optimality-
based theories have proven extremely successful in numerous applications.
Specifically, to fully address the trade-offs between H
2
and H
∞
performance,
the optimality-based linear-quadratic control problem was merged with H
∞
methods to address the m ixed-norm H
2
/H
∞
control problem [49, 238].
In order to address the op timality-based disturbance rejection non-
linear control problem we extend the nonlinear-nonquadratic, continuous-
time controller analysis and synthesis framework presented in Chapter 8.
Specifically, using nonlinear dissipativity th eory with appropriate storage
functions and supply rates we transform the nonlinear disturbance rejection
problem into an optimal control pr ob lem. This is accomplished by pr operly
modifying the cost functional to account for exogenous disturbances so that
the solution of the modified optimal nonlinear control problem serves as th e
solution to the distu rbance rejection problem.
The framework guarantees that the closed-loop nonlinear input-output
map is dissipative with respect to general supply rates. Specializing to
quadratic sup ply rates involving net system energy flow and weighted input
and output energy, the results guarantee passive and nonexpansive (gain