On Lie-algebraic-solvability-based feedback stabilization of systems with controller-driven sampling

HERNAN HAIMOVICH; ESTEBAN OSELLA; JULIO BRASLAVSKY

Milan

Congreso; 18th IFAC World Congress; 2011

International Federation of Automatic Control

We address state-feedback stabilization of discrete-time switched systems (DTSSs) that describe a continuous-time linear time-invariant (LTI) system sampled at varying rates. We consider a setting in which the controller, in addition to applying feedback, selects and varies the sampling rate. We refer to this situation as controller-driven sampling. Our feedback control design approach relies on Lie-algebraic stability results that involve the solvability of the Lie algebra generated by the DTSS´s subsystem closed-loop evolution matrices. In matrix terms, Lie-algebraic solvability is equivalent to the simultaneous triangularization of the generating matrices by means of a single similarity transformation. The present paper continues previous work based on the fact that even if the corresponding Lie algebra may be not solvable, closed-loop stability can still be achieved by approximate" triangularization. A first contribution of the present paper is to show that the Lie-algebraic stabilization problem arising for the class of DTSSs considered can be much less restrictive than for DTSSs of arbitrary form. A second contribution is the identification of a potential pitfall in the application of the approximate triangularization strategy to the class of DTSSs considered, and a suggestion to avoid it.