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160 CHAPTER 6. INPUT-OUTPUT STABILITY
u(t)
u(t) =UT(t)
H
H
y(t) = Hu(t)
y(t) = HUT(t)
Figure 6.2: Experiment 1: input u(t) applied to system H. Experiment 2: input fl(t) = UT(t)
applied to system H.
We end this discussion by pointing out that the causality condition (6.8) is frequently
expressed using the projection operator PT as follows:
PTH = PTHPT (i.e., PT(Hx) = PT[H(PTx)] dx E Xe and VT E W1 .
It is important to notice that the notion of input-output stability, and in fact, the
notion of input-output system itself, does not depend in any way on the notion of state.
In fact, the internal description given by the state is unnecessary in this framework. The
essence of the input-output theory is that only the relationship between inputs and outputs
is relevant. Strictly speaking, there is no room in the input-output theory for the existence
of nonzero (variable) initial conditions. In this sense, the input-output theory of systems
in general, and the notion of input-output stability in particular, are complementary to
the Lyapunov theory. Notice that the Lyapunov theory deal with equilibrium points of
systems with zero inputs and nonzero initial conditions, while the input-output theory
considers relaxed systems with non-zero inputs. In later chapters, we review these concepts
and consider input-output systems with an internal description given by a state space
realization.
We may now state the definition of input-output stability.
Definition 6.6 A system H : Xe -> Xe is said to be input-output X-stable if whenever the
input belongs to the parent space X, the output is once again in X. In other words, H is
X-stable if Hx is in X whenever u E X.