Robust exponential stability and stabilisation of parametric uncertain switched linear systems under arbitrary switching

Stability of switched linear systems under arbitrary switching signals is an important requirement, when the switching mechanism is unknown, or too complicated to be useful in the stability analysis. In this study, robust exponential stability and exponential stabilisation of parametric uncertain switched linear systems are investigated under arbitrary switching. First, sufficient conditions are proposed to ensure the existence of a common quadratic Lyapunov function (CQLF) for arbitrary switched linear systems with uncertain parameters belong to known intervals. Then, an estimation of stability intervals for uncertain parameters is provided via a theorem. To enlarge the estimated stability intervals, an offline optimisation algorithm is also proposed. Finally, the derived results for robust exponential stability are used to stabilise the uncertain switched linear systems which are not stable under arbitrary switching signals. For this purpose, a method is proposed to design a single state feedback gain in the way that the closed-loop switched linear system is robustly exponentially stable under arbitrarily fast switching signals. Numerical examples are included to demonstrate the effectiveness of the results.