Nonlinear Control Sy.. - Free

194 CHAPTER 7. INPUT-TO-STATE STABILITY
Once inside this region, the trajectory is trapped inside Std, because of the condition on V.
According to the discussion above, the region Std seems to depend on the composition
of and It would then appear that it is the composition of these two functions
that determines the correspondence between a bound on the input function u and a bound
on the state x. The functions a(.) and a(.) constitute a pair [a(.), referred to as an
ISS pair for the system (7.1). The fact that Std depends on the composition of
a given system, the pair [a(.), o(.)] is perhaps nonunique. Our next
theorem shows that this is in fact the case.
In the following theorem we consider the system (7.1) and we assume that it is globally
input-to-state-stable with an ISS pair [a, o].
a(IIxII) < V(x) < a(IIxII) Vx E 1R", (7.16)
VV(x) f(x,u) < -a(IIx1I) +o(IIu1I) Vx E 1R",u E 1W7 (7.17)
for some a, a, d E and or E 1C.
if
Similarly
if
We will need the following notation. Given functions x(.), R -> IR, we say that
x(s) = O(y(s)) as s --+ oo+
lim 1x(s)1 < oo.
8-100 ly(s)I
x(s) = O(y(s)) as s -+ 0+
lim Ix(s)I < 00.
s-+0 I y(s) I
Theorem 7.7 Let (a, a) be a supply pair for the system (7.1). Assume that & is a 1C,,.
function satisfying o(r) = O(&(r)) as r -a oo+. Then, there exist & E 1C,,. such that (a,&)
is a supply pair.
Theorem 7.8 Let (a, a) be a supply pair for the system (7.1). Assume that & is a 1C,,.
function satisfying a(r) = O(a(r)) as r -a 0+. Then, there exist & E 1C. such that (&, ii)
is a supply pair.
Proof of theorems 7.7 and 7.8: See the Appendix.
Remarks: Theorems 7.7 and 7.8 have theoretical importance and will be used in the next
section is connection with the stability of cascade connections if ISS systems. The proof of
theorems 7.7 and 7.8 shows how to construct the new ISS pairs using only the bounds a
and a, but not V itself.