Nonlinear Control Sy.. - Free

186 CHAPTER 7. INPUT-TO-STATE STABILITY
Alternative Definition: A variation of Definition 7.1 is to replace equation (7.3) with
the following equation:
Iix(t)II 0, 0 < T < t. (7.4)
The equivalence between (7.4) and (7.3) follows from the fact that given 0 > 0 and
y > 0, max{Q, y} < Q + -y < {2$, 2y}. On occasions, (7.4) might be preferable to
(7.3), especially in the proof of some results.
It seems clear that the concept of input-to-state stability is quite different from that
of stability in the sense of Lyapunov. Nevertheless, we will show in the next section that
ISS can be investigated using Lyapunov-like methods. To this end, we now introduce the
concept of input-to-state Lyapunov function (ISS Lyapunov function).
Definition 7.2 A continuously differentiable function V : D -> Ift is said to be an ISS
Lyapunov function on D for the system (7.1) if there exist class AC functions al, a2, a3,
and X such that the following two conditions are satisfied:
al(IIXII) 0 (7.5)
aa(x) f (x, u) < -a3(IIxII) Vx E D, u E Du : IIxII >_ X(IIuII). (7.6)
V is said to be an ISS Lyapunov function if D = R", Du = R, and al, a21 a3 E 1Coo .
Remarks: According to Definition 7.2, V is an ISS Lyapunov function for the system (7.1)
if it has the following properties:
(a) It is positive definite in D. Notice that according to the property of Lemma 3.1,
given a positive definite function V, there exist class IC functions al and a2 satisfying
equation (7.5).
(b) It is negative definite in along the trajectories of (7.1) whenever the trajectories are
outside of the ball defined by IIx*II = X(IIuPI).
7.3 Input-to-State Stability (ISS) Theorems
Theorem 7.1 (Local ISS Theorem) Consider the system (7.1) and let V : D -a JR be an
ISS Lyapunov function for this system. Then (7.1) is input-to-state-stable according to