Nonsmooth control design for stabilizing switched linear systems by left eigenstructure assignment

Novel design algorithms for exponential stability of switched linear systems using the left eigenstructure assignment approach by state feedback control are proposed in this article. For given switching constraints in the state space R, a state feedback controller for single-input switched systems is designed, based on the fact that closed loop system solutions are enforced to converge towards the invariant hyperplane attractor in R, as defined by the imposed common left eigenvector. The latter is appropriately constructed to guarantee the simultaneous stabilization of all constituent linear sub-systems, and to avoid the intersection of the switching manifolds with its invariant attractor set. For arbitrary switching, a set of n linearly independent common left eigenvectors in R is selected, introducing n hyperplane attractors, and n(n− 1) switching control hyperplane manifolds. The attrators are sequentially “turned on”, in accordance with the switching event generated by the Filippov solutions upon hitting the underlying control manifolds. Thereby, an additional state feedback control action for a second input is designed, using the nonsmooth Lyapunov stability criteria to avoid sliding modes and guarantee exponential stability for the Filippov solutions.