Periodic Lyapunov Research Articles

Inertia theorems for the periodic Lyapunov difference equation and periodic Riccati difference equation

The periodic Lyapunov difference equation (PLDE) and periodic Riccati difference equation (PRDE) are dealt with. The inertia (i.e., the number of positive, null, and negative eigenvalues) of any symmetric periodic solution of such equations is linked with the pattern of eigenvalues of the monodromy matrix associated with the open-loop (for PLDE) or closed-loop (for PRDE) underlying systems. Different results are obtained by imposing requirements with decreasing strength to the original system. Precisely, assumptions of observability, reconstructibility, and detectability are successively introduced. Some results are also given for the particular case of positive semidefinite periodic solutions.

Analysis of the Periodic Lyapunov and Riccati Equations via Canonical Decomposition

A new proof is given of the following result: a periodic Riccati equation admits a unique positive semidefinite periodic solution and the solution is stabilizing if and only if the underlying system is stabilizable and detectable. The proof hinges on the decomposition of the Riccati equation induced by the system canonical decomposition. The solution structure is therefore naturally pointed out.

Stabilizability and detectability of linear periodic systems

In this paper, four characterizations of stabilizability and detectability of linear periodic systems are considered. Two of them look as natural extensions of the classical definitions given for time-invariant systems. The remaining two are modal characterizations which turn out to be useful in the analysis of the periodic Lyapunov and Riccati equations. It is shown that all these notions of stabilizability (and detectability) are in fact equivalent to each other. One of the various definitions calls for the existence of the Kalman canonical decomposition of periodic systems. This issue is addressed in the Appendix.

On the improved Lyapunov functions for periodic systems

A new method to generate a periodic Lyapunov function that gives improved stability limits for a given periodic system is given here. This is an extension of the method given by the authors [1] for autonomous systems. Example illustrates the application of the method.

An improved stability criterion for the damped Mathieu equation

An improved stability criterion is derived for the damped Mathieu equation using periodic Lyapunov functions.