Stability of Switched Linear Discrete-Time Descriptor Systems with Explicit Calculation of a Common Quadratic Lyapunov Sequence

Abstract-In this paper, the stability of a switched linear regular descriptor system is considered. It will be shown that if a certain simultaneous triangularization condition on the subsystems is fulfilled and all the subsystems are stable then the switched system is stable under arbitrary switching. The result involves different descriptor matrices and extends to the singular case well-known results from the standard one. Furthermore, an explicit construction of a common Lyapunov sequence for a set of discrete-time regular linear descriptor subsystems is performed. The main novelty of the proposed approach is that the common Lyapunov sequence can be easily computed in comparison with previous works which either presented computationally-demanding methods or did not construct the common Lyapunov sequence explicitly

Lie Algebraic Stability Analysis for Switched Systems With Continuous-Time and Discrete-Time Subsystems

Abstract—We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all continuous-time subsystems are Hurwitz stable, all discrete-time subsystems are Schur stable, and furthermore the obtained Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. A numerical example is provided to demonstrate the result. Index Terms—Arbitrary switching, common quadratic Lyapunov functions, continuous-time, discrete-time, exponential stability, Lie algebra, switched systems. I

Extended Lie algebraic stability analysis for switched systems with continuous-time and discrete–time subsystems

We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area