When a system's outputs are bounded for every bounded input
In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded.
A signal is bounded if there is a finite value formula_1 such that the signal magnitude never exceeds formula_2, that is
For discrete-time signals: formula_3
For continuous-time signals: formula_4
Time-domain condition for linear time-invariant systems.
Continuous-time necessary and sufficient condition.
For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse response, formula_5 , be absolutely integrable, i.e., its L1 norm exists.
 formula_6
Discrete-time sufficient condition.
For a discrete time LTI system, the condition for BIBO stability is that the impulse response be absolutely summable, i.e., its formula_7 norm exists.
formula_8
Proof of sufficiency.
Given a discrete time LTI system with impulse response formula_9 the relationship between the input formula_10 and the output formula_11 is
formula_12
where formula_13 denotes convolution. Then it follows by the definition of convolution
formula_14
Let formula_15 be the maximum value of formula_16, i.e., the formula_17-norm.
formula_18
formula_19 (by the triangle inequality)
 formula_20
If formula_21 is absolutely summable, then formula_22 and
formula_23
So if formula_21 is absolutely summable and formula_25 is bounded, then formula_26 is bounded as well because formula_27.
The proof for continuous-time follows the same arguments.
Frequency-domain condition for linear time-invariant systems.
Continuous-time signals.
For a rational and continuous-time system, the condition for stability is that the region of convergence (ROC) of the Laplace transform includes the imaginary axis. When the system is causal, the ROC is the open region to the right of a vertical line whose abscissa is the real part of the "largest pole", or the pole that has the greatest real part of any pole in the system. The real part of the largest pole defining the ROC is called the abscissa of convergence. Therefore, all poles of the system must be in the strict left half of the s-plane for BIBO stability.
This stability condition can be derived from the above time-domain condition as follows:
formula_28
where formula_29 and formula_30
The region of convergence must therefore include the imaginary axis.
Discrete-time signals.
For a rational and discrete time system, the condition for stability is that the region of convergence (ROC) of the z-transform includes the unit circle. When the system is causal, the ROC is the open region outside a circle whose radius is the magnitude of the pole with largest magnitude. Therefore, all poles of the system must be inside the unit circle in the z-plane for BIBO stability.
This stability condition can be derived in a similar fashion to the continuous-time derivation:
formula_31
where formula_32 and formula_33.
The region of convergence must therefore include the unit circle.
Further reading.
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References.
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