Nonlinear Control Sy.. - Free

322 APPENDIX A. PROOFS
To this end we consider a new ISS Lyapunov function candidate W, defined as follows
W = p o V = p(V(x)), where (A.32)
p(s) = f q(t) dt (A.33)
0
where pt+ -+ R is a positive definite, smooth, nondecreasing function.
We now look for an inequality of the form (A.30) in the new ISS Lyapunov function
W. Using (A.32) we have that
Now define
that is,
W = aW f(x,u) = P (V(x))V(x,n)
= q[V(x)) [a(IIuII)-a(IIxII)l.
9=aoa-1o(2o)
0(s) = d(a-1(2a)).
We now show that the right hand-side of (A.34) is bounded by
To see this we consider two separate cases:
q[9(IIuIUa(IIuII) - 1q[V(x)1a(IIxII)-
(i) a(IIuII) < za(IIxII): In this case, the right hand-side of (A.34) is bounded by
-2q[V(x)]a(IIxII)
(ii) 2a(IIxII) < a(IIuII): In this case, we have that V(x) 5 a(IIxII) < 9(IIuII).
From (i) and (ii), the the right-hand side of (A.34) is bounded by (A.36).
(A.34)
(A.35)
(A.36)
The rest of the proof can be easily completed if we can show that there exist & E 1C,,.
such that
q[9(r)]o(r) - 2q[a(s))a(s) < a (r) - a(s) Vr,s > 0. (A.37)
To this end, notice that for any /3, /3 E 1C,,. the following properties hold:
Property 1: if /3 = O(13(r)) as r -+ oo, then there exist a positive definite, smooth, and
nondecreasing function q:
q(r)/3(r) < 4(r) Vr E [0, oo).