Nonlinear Control Sy.. - Free

CHAPTER 7. INPUT-TO-STATE STABILITY
al(s) = 2a2
there exist al : [al, all = [al, 1&2] is an ISS pair for the system El.
Proof: A direct application of Theorems 7.7 and 7.8.
We now state and prove the main result of this section.
Theorem 7.9 Consider the cascade interconnection of the systems El and E2. If both
systems are input-to-state-stable, then the composite system E
is input-to-state-stable.
Proof: By (7.20)-(7.21) and Lemma 7.1, the functions V1 and V2 satisfy
Define the ISS Lyapunov function candidate
for the composite system. We have
VV1 f(x,z) < -al(IIxII)+2612(IIZII)
VV2 9(z,u) < -a2(IIzII) +&2(IIUII)
V = V1 + V2
V ((x, z), u) = VV1 f (X, z) + VV2 f (Z, u)
1
-a1(IIxII) - 2a2(IIZII)+a2(IIuII)
It the follows that V is an ISS Lyapunov function for the composite system, and the theorem
is proved.
This theorem is somewhat obvious and can be proved in several ways (see exercise
7.3). As the reader might have guessed, a local version of this result can also be proved.
For completeness, we now state this result without proof.
Theorem 7.10 Consider the cascade interconnection of the systems El and E2. If both
systems are locally input-to-state-stable, then the composite system E
is locally input-to-state-stable.
x
E:u -+ z