Towards robust Lie-algebraic stability conditions for switched linear systems (original) (raw)

Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time SubsystemsWe 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.

This paper investigates some conditions that can provide stabilizability for linear switched systems with polytopic uncertainties via their closed loop linear quadratic state feedback regulator. The closed loop switched systems can stabilize unstable open loop systems or stable open loop systems but in which there is no solution for a common Lyapunov matrix. For continuous time switched linear systems, we show that if there exists solution in an associated Riccati equation for the closed loop systems sharing one common Lyapunov matrix, the switched linear systems are stable. For the discrete time switched systems, we derive a Linear Matrix Inequality (LMI) to calculate a common Lyapunov matrix and solution for the stable closed loop feedback systems. These closed loop linear quadratic state feedback regulators guarantee the global asymptotical stability for any switched linear systems with any switching signal sequence.

– In this paper, sufficient conditions are proposed to investigate the robust stability of arbitrary switched linear systems with uncertain parameters belongs to the known intervals. In addition, a method is then established to determine the maximum intervals of parameters' variations which guarantee robust exponential stability of uncertain switched linear systems under arbitrary switching. In the proposed method, the known information about the parametric structure of uncertainties is considered; therefore it will result in less conservative stability margins. A generalization of the method is also provided to determine stability bounds on perturbations of entries in subsystem matrices, when subsystems are subjected to independent perturbations. Numerical examples are included to illustrate the effectiveness of the results, and compare them with the previous results. It is shown that the proposed methods provide stability intervals on the uncertain parameter for all switched linear systems which admit a common quadratic Lyapunov function for the nominal system.

SUMMARYWe report conditions on a switching signal that guarantee that solutions of a switched linear system converge asymptotically to zero. These conditions apply to continuous, discrete‐time and hybrid switched linear systems, those having both stable subsystems and mixtures of stable and unstable subsystems. Copyright © 2012 John Wiley & Sons, Ltd.