Feedback stabilization of switching discrete-time systems via Lie-algebraic techniques

HERNAN HAIMOVICH; JULIO H. BRASLAVSKY; FLAVIA FELICIONI

IEEE TRANSACTIONS ON AUTOMATIC CONTROL

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC

Año: 2011 vol. 56 p. 1129 - 1135

0018-9286

This technical note addresses the stabilization of switching discrete- time linear systems with control inputs under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems, which is ensured when the closed-loop component subsystem matrices are stable and generate a solvable Lie algebra. The present work develops an iterative algorithm that seeks the feedback maps required for stabilization based on the previous Lie-algebraic condition. The main theoretical contribution of the technical note is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution. The core of the proposed algorithm is a common eigenvector assignment procedure, which is executed at every iteration. We also show how the latter procedure can be numerically implemented and provide a key structural condition which, if satisfied, greatly simplifies the required computations.