Nonlinear Control Sy.. - Free

170 CHAPTER 6. INPUT-OUTPUT STABILITY
Figure 6.7: The Feedback System S.
In the following definition, we introduce vectors u, e, and y, defined as follows:
u(t) = [
uul (t)
2 (t) I I e(t) - [
eel (t)
2 (t) I , y(t)
yi (t)
Y 2
]
(6.25)
Definition 6.12 Consider the feedback interconnection of subsystems H1 and H2. For this
system we introduce the following input-output relations. Given Ui, U2 E Xe , i, j = 1, 2,
and with u, e, and y given by (6.25) we define
E = {(u,e) E Xe x Xe and e satisfies (6.23) and (6.24)} (6.26)
F = {(u, y) E Xe x Xe and y satisfies (6.23) and (6.24)}. (6.27)
In words, E and F are the relations that relate the inputs u2 with ei and y2, respectively.
Notice that questions related to the existence and uniqueness of the solution of equations
(6.23) and (6.24) are taken for granted. That is the main reason why we have chosen to
work with relations rather than functions.
Definition 6.13 A relation P on Xe is said to be bounded if the image under P of every
bounded subset of XX E dom(P) is a bounded subset of Xe .
In other words, P is bounded if Pu E X for every u c X , u E dom(P).
Definition 6.14 The feedback system of equations (6.23) and (6.24) is said to be bounded
or input-output-stable if the closed-loop relations E and F are bounded for all possible u1, u2
in the domain of E and F.