Marginal stability

In the theory of dynamical systems, and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Marginal stability is sometimes referred to as neutral stability.

A homogeneous continuous linear time-invariant system is marginally stable if and only if the real part of every pole in the system's transfer-function is non-positive, one or more poles have zero real part, and all poles with zero real part are simple roots (i.e. the poles on the imaginary axis are all distinct from one another). In contrast, all the poles have strictly negative real parts, the system is instead asymptotically stable. If one or more poles have positive real parts, the system is unstable.

If the system is in state space representation, marginal stability can be analyzed by deriving the Jordan normal form: if and only if the Jordan blocks corresponding to poles with zero real part are scalar is the system marginally stable.

A homogeneous discrete time linear time-invariant system is marginally stable if and only if the greatest magnitude of any of the poles of the transfer function is 1, and the poles with magnitude equal to one are all distinct. That is, the transfer function's spectral radius is 1. If the spectral radius is less than 1, the system is instead asymptotically stable.

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