Theoretical and numerical comparisons of looped functionals and clock-dependent Lyapunov functionsThe case of periodic and pseudo-periodic systems with impulses

摘要

The stability of uncertain periodic and pseudo-periodic systems with impulses is analyzed in the looped-functional and clock-dependent Lyapunov function frameworks. These alternative and equivalent ways for characterizing discrete-time stability have the benefit of leading to stability conditions that are convex in the system matrices, hence suitable for robust stability analysis. These approaches, therefore, circumvent the problem of computing the monodromy matrix associated with the system, which is known to be a major difficulty when the system is uncertain. Convex stabilization conditions using a non-restrictive class of state-feedback controllers are also provided. The obtained results readily extend to uncertain impulsive periodic and pseudo-periodic systems, a generalization of periodic systems that admit changes in the period' from one pseudo-period to another. The obtained conditions are expressed as infinite-dimensional semidefinite programs, which can be solved using recent polynomial programming techniques. Several examples illustrate the approach, and comparative discussions between the different approaches are provided. A major result obtained in the paper is that despite being equivalent, the approach based on looped functional reduces to the one based on clock-dependent Lyapunov functions when a particular structure for the looped functional is considered. The conclusion is that the approach based on clock-dependent Lyapunov functions is preferable because of its lower computational complexity and its convenient structure enabling control design. Copyright (c) 2015 John Wiley & Sons, Ltd.