System Stability

This web page provides links to discussions and illustrations of some different types of stability for linear continuous-time systems.

Now consider the relationship between Lyapunov and BIBO stability. We know that a system is Lyapunov stable if its eigenvalues are the left-half plane, and we know that a system is BIBO stable if its poles are in the left-half plane. We also know that a system's poles are a subset of its eigenvalues. Therefore it is possible for a system's poles to all be in the left-half plane while some of its eigenvalues are in the right-half plane. This means that it is possible for a system to be both BIBO stable and Lyapunov unstable. As an example, consider the following system:

The eigenvalues of the A matrix are at 1 and -2. Therefore, the system is Lyapunov unstable (one of its eigenvalues is in the right-half plane. However, a calculation of the transfer function shows that the transfer function G(s) = 1 / (s + 2). All the poles are in the left-half plane (at s = - 2) so the system is BIBO stable.

For this system, the output will always remain bounded (BIBO stable) but the states may increase to infinity (Lyapunov unstable). Notice that the state equation is 2nd order while the transfer function is 1st order. The system has an unobservable mode that is unstable.

Below is a simulation of this system. Try different initial conditions and inputs. You will find that the output remains bounded, but for certain initial conditions and inputs the states increase to infinity. Some initial conditions and inputs result in bounded states, but this does not prove anything about Lyapunov stability.