2.3 LTI Continuous-Time Systems 153
2.3.9 Bounded-Input Bounded-Output Stability
Stability characterizes useful systems. A stable system is such that well-behaved outputs are obtained
for well-behaved inputs. Of the many possible definitions of stability, we consider here bounded-
input bounded-output (BIBO) stability.
Bounded-input bounded-output (BIBO) stabilityestablishes that for a bounded (i.e., well-behaved) inputx(t)
the output of a BIBO stable systemy(t)is also bounded.This means that if there is a finite boundM<∞such
that|x(t)|<M(you can think of it as an envelope[−M,M]inside which the input is in) the output is also
bounded.
An LTI system with an absolutely integrable impulse response—that is,
∫∞
−∞
|h(t)|dt<∞ (2.23)
is BIBO stable. A simpler way, using the Laplace transform, to test the BIBO stability of a system is given later.
For a bounded input, the outputy(t)of an LTI system is represented by a convolution integral that is
bounded as follows:
|y(t)|=
∣ ∣ ∣ ∣ ∣ ∣
∫∞
−∞
x(t−τ)h(τ)dτ
∣ ∣ ∣ ∣ ∣ ∣ ≤
∫∞
−∞
|x(t−τ)||h(τ)|dτ
≤M
∫∞
−∞
|h(τ)|dτ
≤ML<∞
whereLis the bound for
∫∞
−∞|h(τ)|dτ, or equivalently the impulse response is absolutely integrable.
nExample 2.16
Consider the BIBO stability and causality of RLC circuits. Consider, for instance, a series RL circuit
whereR= 1 andL=1 H, and a voltage sourcevs(t), which is bounded. Discuss why such a
system would be causal and stable.
Solution
RLC circuits are naturally stable. As you know, inductors and capacitors simply store energy and
so LC circuits simply exchange energy between these elements. Resistors consume energy, which
