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CLOSED LOOP CONTROL 77
Input - Bounded Output - Bounded

Dyn. sys.

(a) BIBO stability
x
o
?
Dyn. sys. x(t) x e

(b) Stability about an equilibrium point
FIGURE 2.28: Definition of two different stability notions.

2.8.1 Bounded Input–Bounded Output Stability
Definition: a dynamic system (linear or nonlinear) is said to be bounded input–bounded
output (BIBO) stable if for every bounded input, the output is bounded.
For a linear time invariant (LTI) system, the output to any input can be calculated as,
t
y(t) = h( )u(t − )d
∫
−∞
where h(t) is the impulse response of the LTI system. If the input is bounded, there must
exist a constant M such that
|u(t)| ≤ M < ∞
Hence
| t |
| |
|∫ |
|y(t)| = | h( )u(t − )d |
| −∞ |
t
≤ |h( )||u(t − )|d
∫
−∞
t
≤ M |h( )|d
∫
−∞
For the LTI system to be BIBO stable, y(t) must be bounded for all t as t → ∞. Therefore,
for y(t)
∞
lim |y(t)| ≤ M |h(z)|dz
t→∞ ∫
−∞
to be bounded, the following expression must be bounded,
∞
|h(z)|dz
∫ −∞
This requires that
h(t) → 0 as t → ∞

In conclusion, if a LTI system is BIBO stable, this means that its impulse response goes
to zero as time goes to infinity. The opposite is also true. If the impulse response of a LTI
system goes to zero as time goes to infinity, the LTI system is BIBO stable (Figure 2.29).