Nonlinear Control Sy.. - Free

A.6. CHAPTER 9 329
that is
ff ffo
T w(t) dt > -
o Jt,
w(t) dt
J
The right-hand side of the last inequality depends only on xo, whereas u can be chosen
arbitrarily on [0, T]. Hence there exists a bounded function C : R' --4 IR such that
JO
which implies that 0a is bounded. By Theorem 9.1 the available storage is itself a storage
function, i.e.,
it
w(s) ds > 4a(x(t)) - 0a(xo) Vt > 0
which, since 0a is differentiable by the assumptions of the theorem, implies that
Substituting (9.13), we have that
d(x,u)
d(x,u) +w(u,y)
a0axx)
fw - d dtx) > 0.
[f(x) +g(x)u] +yTQy+2yTSu+uTRu
00"2x) f(x) - a a(x)g(x)u+hTQh+2hTSu+UTRu
we notice that d(x, u), so defined, has the following properties: (i) d(x, u) > 0 for all x, u;
and (ii) it is quadratic in u. It then follows that d(x, u) can be factored as
which implies that
This completes the proof.
d(x,u) = [L(x) + Wu]T [L(x) + Wu]
= LTL+2LTWu+uTWTWu
aSa
ax
1 T aSa
2 ax
T
w(t) dt > C(xo) > -oc
R = WTW
f = hT (x)Qh(x) - LT (x)L(x)
-g - = S
Th(x) - WTL(x).