Nonlinear Control Sy.. - Free

7.2. DEFINITIONS 185
Setting u = 0, we obtain the autonomous LTI system ± = -x, which clearly has an asymptotically
stable equilibrium point. However, when the bounded input u(t) = 1 is applied,
the forced system becomes ± = x3, which results in an unbounded trajectory for any initial
condition, however small, as can be easily verified using the graphic technique introduced in
Chapter 1.
7.2 Definitions
In an attempt to rescue the notion of "bounded input-bounded state", we now introduce
the concept of input-to-state stability (ISS).
Definition 7.1 The system (7.1) is said to be locally input-to-state-stable (ISS) if there
exist a ICE function 0, a class K function -y and constants k1, k2 E ][i;+ such that
IIx(t)II 0, 0 o IUT(t)II = IluTllc_ < k2i 0 <
T < t. It is said to be input-to-state stable, or globally ISS if D = R", Du = Rm and (7.3)
is satisfied for any initial state and any bounded input u.
Definition 7.1 has several implications, which we now discuss.
Unforced systems: Assume that i = f (x, u) is ISS and consider the unforced system
x = f (x, 0). Given that y(0) = 0 (by virtue of the assumption that -y is a class IC
function), we see that the response of (7.1) with initial state xo satisfies.
Ilx(t)II < Q(Ilxoll, t) Vt > 0, Ilxoll < ki,
which implies that the origin is uniformly asymptotically stable.
Interpretation: For bounded inputs u(t) satisfying Ilullk < b, trajectories remain
bounded by the ball of radius 3(Ilxoll,t) +y(b), i.e.,
lx(t) I1 oo, and the trajectories approach the
ball of radius y(5), i.e.,
lim Ilx(t)II