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6.2. INPUT-OUTPUT STABILITY 159
Thus XT E Xe VT E R+. However x ¢ X since IXT I = oo.
Remarks: In our study of feedback systems we will encounter unstable systems, i.e., systems
whose output grows without bound as time increases. Those systems cannot be described
with any of the LP spaces introduced before, or even in any other space of functions
used in mathematics. Thus, the extended spaces are the right setting for our problem. As
mentioned earlier, our primary interest is in the spaces £2 and G,. The extension of the
space LP , 1 < p < oo, will be denoted Lpe. It consists of all the functions u(t) whose
truncation belongs to LP .
6.2 Input-Output Stability
We start with a precise definition of the notion of system.
Definition 6.5 A system, or more precisely, the mathematical representation of a physical
system, is defined to be a mapping H : Xe -> Xe that satisfies the so-called causality
condition:
[Hu(')]T = Vu E Xe and VT E R. (6.8)
Condition (6.8) is important in that it formalizes the notion, satisfied by all physical systems,
that the past and present outputs do not depend on future inputs. To see this, imagine
that we perform the following experiments (Figures 6.2 and 6.3):
(1) First we apply an arbitrary input u(t), we find the output y(t) = Hu(t), and from
here the truncated output yT(t) = [Hu(t)]T. Clearly yT = [Hu(t)]T(t) represents the
left-hand side of equation (6.8). See Figure 6.3(a)-(c).
(2) In the second experiment we start by computing the truncation u = uT(t) of the input
u(t) used above, and repeat the procedure used in the first experiment. Namely, we
compute the output y(t) = Hu(t) = HuT(t) to the input u(t) = uT(t), and finally we
take the truncation yT = [HuT(t)]T of the function y. Notice that this corresponds
to the right-hand side of equation (6.8). See Figure 6.3(d)-(f).
The difference in these two experiments is the truncated input used in part (2). Thus,
if the outputs [Hu(t)]T and [HUT(t)]T are identical, the system output in the interval
0 < t < T does not depend on values of the input outside this interval (i.e., u(t) for t > T).
All physical systems share this property, but care must be exercised with mathematical
models since not all functions behave like this.