Nonlinear Control Sy.. - Free

252 CHAPTER 9. DISSIPATIVITY
x = a(x) + b(x)u + g(x)d, u E II8"`, d E R''
z = [h(x)]
where a, b, g and h are assumed to be Ck, with k > 2.
(9.46)
Theorem 9.6 The closed-loop system of equation (9.46) has a finite C2 gain < -y if and
only if the Hamilton-Jacobi inequality 9{ given by
H
x_ i
8xa(x) [.g(x)gT(x)
+ 2 8 - b(x)bT (x)
J
has a solution 0 > 0. The control law is given by
ll
(Ocb)T
+ hT (x)h(x) < 0 (9.47)
T
u= -bT (x) (a,) (x). (9.48)
Proof: We only prove sufficiency. See Reference [85] for the necessity part of the proof.
Assume that > 0 is a solution of L. Substituting u into the system equations (9.46),
we obtain
()T
a(x) - b(x)bT(x) +g(x)d (9.49)
z = L -bT(x)
Substituting (9.49) and (9.50) into the Hamilton-Jacobi inequality (9.36) with
results in
f (x) = a(x) - b(x)bT (x) ()T (x)
h(x)
[_bT(X)
( )T ]
H = ax (a(x)_b(x)bT(x)_) ) + 2ry2
+2 [h(X)T
T T
hx
8xb(x), [bT(X) (g)T ]
axg(x)gT (x) (8
(9.50)
which implies (9.47), and so the closed-loop system has a finite G2 gain ry. 0