Stability is an crucial property of systems, as unstable systems are often unusable or even dangerous in the real-world.
There are two primary kinds of system stability: external (BIBO) stability and internal (asymptotic) stability.
External (BIBO) Stability
If every bounded input results in a bounded output, a system is said to be externally stable. This type of stability is also called BIBO (bounded-input/bounded output) stability.
BIBO stability examines how the system responds to external stimuli when starting from a resting state.
For an LTIC system, we can see whether a system is BIBO stable by checking whether its impulse response
Proof
The output
of an LTI system can be expressed as the convolution of the input with the system’s impulse response : Taking the absolute value of both sides and applying the triangle inequality:
Moreover, if
is bounded, then , and For the system to be BIBO stable,
must be bounded for all bounded . This is guaranteed if:
Though an easier way is to check the asymptotic stability of the system (which implies BIBO stability for an LTIC system).
Internal (Asymptotic) Stability
If in the absence of an external input, a system remains in a particular state indefinitely, then we call that state an equilibrium state. Now if we apply a small perturbation to the system,
- If the system is asymptotically stable, the response will decay back to the equilibrium state as time passes.
- If the system is asymptotically unstable, the response will grow over time, moving further away from the equilibrium state.
- Otherwise the system is marginally stable
For an LTIC system, it is asymptotically stable if and only if all characteristic roots (also called poles) are in the left-hand plane (
Conversely, an LTIC system is asymptotically unstable if and only if at least one of the following conditions exist:
- at least one root is in the right-hand plane (RHP) of the complex plane
- there are repeated roots on the imaginary axis
If neither the conditions for asymptotic stability nor asymptotic instability are satisfied, the system is classified as marginally stable.
Intuitively, a root at the right-hand side of the complex plane or a repeated root means that the zero-input response of the system will grow exponentially (and is thus unstable). And if all roots are on the left-hand side, the response will decay exponentially (and is therefore stable).
What's the stability of the system
? We have the characteristic polynomial
with characteristic roots
and . Since all roots are on the LHP, the system is asymptotically stable, and it is also BIBO stable.
Relationship between BIBO and Asymptotic Stability
see also: zero-input response
External stability is determined by applying an input with zero initial conditions (zero-state response), while internal stability is determined by applying the nonzero initial conditions and no external input (zero-input response). As a result, these two kinds of stability are also called the zero-state stability and zero-input stability, respectively.
For LTIC systems, internal stability implies external stability, though the converse is not always true. For nonlinear system, the relation can be more complicated and require separate analysis.
Implications of Stability
Practical signal processing systems must be asymptotically stable!
Marginally stable systems, though BIBO unstable, do have one important application as the oscillator.