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 linear time-invariant (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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> represents the system dimension). We begin by briefly 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 fields that are applicable to LTI systems of arbitrary dimension. We then derive necessary and sufficient conditions to ensure stabilizability of the resulting switched system using the proposed approach (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.