11.3: Stability

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Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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Stability in Linear Systems

In linear systems, exponential stability occurs when all the real exponents of \(e\) are strictly negative. The signals decay within an exponential envelope. If one exponent is \(0\), the response never decays or grows in amplitude; this is called marginal stability. If at least one real exponent is positive, then one element of the response grows without bound, and the system is unstable.

Stability \(\Leftrightarrow\) Poles in LHP

In the context of partial fraction expansions, the relationship between stability and pole locations is especially clear. The unit step function \(1(t)\) has a pole at zero; the exponential \(e^{-at}\) has a pole at \(-a\), and so on. All of the other pairs exhibit the same property: A system is stable if and only if all of the poles occur in the left half of the complex plane. Marginally stable parts correlate with a zero real part, and unstable parts to a positive real part.

General Stability

There are two definitions of general stability, which apply to systems with input \(u(t)\) and output \(y(t)\).

Exponential. If \(u(t) = 0\) and \(y(0) = y_o\), then \(|y(t)| < \alpha e^{-\gamma t},\) for some finite \(\alpha\) and \(\gamma > 0\). The output asymptotically approaches zero, within a decaying exponential envelope.
Bounded-Input-Bounded-Output (BIBO). If \(y(0) = 0,\) and \(|u(t) < \gamma, \, \gamma > 0\) and finite, then \(|y(t)| < \alpha, \, \alpha > 0\) and finite.

In linear time-invariant systems, the two definitions are identical. Exponential stability is easy to check for linear systems, but for nonlinear systems, BIBO stability is usually easier to achieve.