Marginal stability

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

A discrete linear time-invariant system is marginally stable if and only if the transfer function's spectral radius is 1. That is, the greatest magnitude of any of the eigenvalues (or poles) of the transfer function is 1. The values of the poles must also be distinct. If the spectral radius is less than 1, the system is instead asymptotically stable.

Practical consequences

A marginally stable system is one that, if given an impulse of finite magnitude as input, will not "blow up" and give an unbounded output. However, oscillations in the output will persist indefinitely, and so there will, in general, be no final steady-state output. If the system is given a step as an input, the system's output will increase indefinitely, with the system effectively acting as an integrator on the input, and so a marginally stable system is not a Bounded Input/Bounded Output system (the information in this para must be verified from other sources).

A system having imaginary poles, i.e having zero real part in the pole(s), will produce sustained oscillations in the output. For example a undamped second order system such as suspension system of your car (mass-spring-damper), from where damper has been removed and spring is ideal i.e. no friction is there, then in theory your car will oscillate forever when you will hit a bump. A system with pole at the origin is also marginally stable but in this case there will be no oscillation in the response as the imaginary part is also zero (jw = 0 means w = 0 rad/sec).

See also

• Lyapunov stability