ABSTRACT
One of the main features of the ISS property is that it behaves well under composition: a cascade (Figure 45.7) of ISS systems is again ISS, see [21]. This section sketches how the cascade result can also be seen as a consequence of the dissipation characterization of ISS, and how this suggests a more general feedback result. For more details regarding the rich theory of ISS small-gain theorems, and their use in nonlinear feedback design, the references should be consulted. Consider a cascade as follows:
z˙ = f (z, x), x˙ = g(x, u),
where each of the two subsystems is assumed to be ISS. Each system admits an ISS-Lyapunov function Vi . But, moreover, it is always possible (see [27]) to redefine the Vi ’s so that the comparison functions for both are matched in the following way:
V˙1(z, x) ≤ θ(|x|)− α(|z|), V˙2(x, u) ≤ θ˜(|u|)− 2θ(|x|).
