Nonlinear Control Sy.. - Free

7.5. CASCADE-CONNECTED SYSTEMS
U
E2
z
Figure 7.1: Cascade connection of ISS systems.
7.5 Cascade-Connected Systems
Throughout this section we consider the composite system shown in Figure 7.1, where El
and E2 are given by
E1
x
195
E1 : ± = f (x, z) (7.18)
E2 : z = 9(z, u) (7.19)
where E2 is the system with input u and state z. The state of E2 serves as input to the
system E1.
In the following lemma we assume that both systems E1 and E2 are input-to-statestable
with ISS pairs [al, 01] and [a2, o2], respectively. This means that there exist positive
definite functions V1 and V2 such that
VV1 f(x,z) < -al(1Ix1I)+a1(I1z11) (7.20)
VV2 9(Z, U) G -a2(IIZIj) +J2(jIUID) (7.21)
The lemma follows our discussion at the end of the previous section and guarantees the
existence of alternative ISS pairs [&1, Si] and [&2i Q2] for the two systems. As it turns out,
the new ISS-pairs will be useful in the proof of further results.
Lemma 7.1 Given the systems E1 and E2, we have that
(i) Defining
a2
{
a2(s) fors "small"
a2(S) for s "large"
then there exist &2 such that (&2, a2) is an ISS pair for the system E2.