NonlinearBook10pt November 20, 2007
ROBUST NONLINEAR CONTROL 669
Proposition 11.14. Let
G(s) ∼
A − I
n
I
n
0
and assume that there exists P = P
T
> 0 such that
0 > A
T
P + P A + Ω(P ). (11.92)
Then the linear dynamical system given by the transfer fun ction G(s) with
input u and output y is exponentially dissipative with respect to the supply
rate r(u, y) = −y
T
Ω(P )y − 2y
T
P u.
Proof. It follows from (11.92) that there exists a scalar ε > 0 such
that 0 ≥ A
T
P + P A + εP + Ω(P ). The result now follows immediately f rom
Theorem 6.2 with B = −I
n
, C = I
n
, D = 0, Q = −Ω(P ), R = 0, S = − P ,
W = 0, and L = (−A
T
P − P A − εP − Ω(P ))
1/2
.
In light of Propositions 11.13 an d 11.14, it follows fr om Theorem 6.2,
with G = G(s), G
c
= ∆A(·), Q = −R
c
= −Ω(P ), R = Q
c
= 0, S =
S
c
= −P , and σ = 1, that if (11.91) and (11.92) hold, then A + ∆A is
asymptotically stable for all ∆A ∈ ∆
A
. This of course establishes that Ω-
bound theory for robust stability analysis is a special case of dissipativity
theory. This exposition thus demonstrates that all (parameter-independent)
guaranteed cost bounds developed in the literature including the boun ded
real bound [11, 147, 337, 356], the positive r eal bound [8, 147], the shifted
bounded real bound [435], the shifted positive real bound [435], the implicit
small gain bound [161], the absolute value bound [83], the linear bound
[41,50,221,243], the inverse bound [50], the d ouble commutator bound [436],
the shifted linear bound [53], and the shifted inverse bound [53] are a special
case of dissipativity theory.
11.4 Robust Optimal Control for Nonlinear Uncertain Systems
In this section, we consider a control problem for nonlinear uncertain
dynamical s ystems involving a notion of optimality with respect to an
auxiliary cost which guarantees a bound on the worst-case value of a
nonlinear-nonquadratic cost criterion over a prescribed uncertainty set. The
optimal robust feedback controllers are derived as a direct consequence of
Theorem 11.1 and provide a generalization of the Hamilton-Jacobi-Bellman
conditions for time-invariant, infinite-horizon problems for addressing robust
feedback controllers of nonlinear uncertain systems. To address the robust
optimal control problem let D ⊂ R
n
be an open set and let U ⊂ R
m
, where
0 ∈ D and 0 ∈ U. Furthermore, let F ⊂ {F : D × U → R
n
: F (0, 0) = 0}.