A switched state feedback law for the stabilization of LTI systems

Abstract

Inspired by prior work in the design of switched feedback controllers for second order systems, we develop a switched state feedback control law for the stabilization of LTI systems of arbitrary dimension. The control law operates by switching between two static gain vectors in such a way that the state trajectory is driven onto a stable n − 1 dimensional hyperplane (where n represents the system dimension). We begin by brief y examining relevant geometric properties of the phase portraits in the case of two-dimensional systems to develop intuition, and we then show how these geometric properties can be expressed as algebraic constraints on the switched vector f elds that are applicable to LTI systems of arbitrary dimension. We then derive necessary and suff cient conditions to ensure stabilizability of the resulting switched system (characterized primarily by simple conditions on eigenvalues), and describe an explicit procedure for designing stabilizing controllers. We then show how the newly developed control law can be applied to the problem of minimizing the maximal Lyapunov exponent of the corresponding closed-loop state trajectories, and we illustrate the closed-loop transient performance of these switched state feedback controllers via multiple examples.3 4

Global Uniform Asymptotic Stability of a Class of Switched Linear Systems with an Infinite Number of Subsystems

“…Santarelli studied switched state feedback control law for the stabilization of LTI systems of arbitrary dimension [13]. A unified approach to stability analysis for switched linear descriptor systems under arbitrary switching in both continuous-time and discrete-time domains was studied in [14].…”

Analysis and synthesis of randomly switched systems with known sojourn probabilities

“…Secondly, in contrast to arbitrarily switching, there is a class of switched systems where an event to trigger a switching to another subsystem can be completely defined by the trajectory of the system state variables. Interested readers may refer to a brief overview presented in the introduction of a recent paper [11] and the references therein. Thirdly, the stability analysis can be based on the knowledge of the minimum time or average time of the system dynamics staying in subsystems.…”