Nonlinear Control Sy.. - Free

234 CHAPTER 9. DISSIPATIVITY
for all x, where
and
q(x) > 0 vx $ 0, 0(0) = 0 (9.14)
a-f (x) = hT (x)Qh(x) - LT (x)L(x) (9.15)
1
19T (ax)T
STh(x) - WTL(x) (9.16)
R = WTW (9.17)
R = R + jT (x)S + STj (x) +7T(x)Q7 (x)
(9.18)
S(x) = Qj(x) + S. (9.19)
Proof: To simplify our proof we assume that j (x) = 0 in the state space realization (9.13)
(see exercise 9.3). In the case of linear systems this assumption is equivalent to assuming
that D = 0 in the state space realization, something that is true in most practical cases.
With this assumption we have that
k = R, and S = S,
We prove sufficiency. The necessity part of the proof can be found in the Appendix. Assuming
that S, L and W satisfy the assumptions of the theorem, we have that
w(u,y) = yTQY+UTRu+2yTSu
= hT Qh + uT Ru + 2hTSu substituting (9.13)
f (x) + LTL] + UTRu + 2hTSu substituting (9.15)
aof (x) +LTL+uTRu+2uTSTh
a46f(x)+LTL+uTWTWu+2uT[2gT(O )T +WTL]
substituting (9.17) and (9.16)
a[f(x)+gu]+LTL+uTWTWu+2uTWTL
ox + (L + Wu)T (L + Wu)
_ 0+ (L + Wu)T (L + Wu) (9.20)