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

Abstract

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 n , 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 n , 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 n 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.

Robust control of switched linear systems

“…To this end, we show that condition (5) and condition (6) {I, ... , n-1}, and PV n = �£2V n , implying that P has n-1 eigenvalues equal to � corresponding to each left eigenvector Wi, and one eigenvalue equal to �£2 corresponding to the right eigenvector V n . As a consequence, condition (5) follows immediately if £2 < 1, a l = �£2 and a 2 = � in (14).…”

“…This specific class of switched systems has been motivated by the simple state-feedback control algorithms for single-input and multi-input systems based on the com mon left eigenstructure assignment approach, leading to constructive stability analysis methods, see [4], [5], [8]. In a further step we adopt results (see [6], [7]) on converse Lyapunov theorems about robust asymptotic stability of the solutions of differential inclusions involving a non-empty upper-semicontinuous compact and convex set-valued map in IRn.…”

Stability and control of the nonlinear system for tractor with N trailer in the presence of slip

“…The most common and useful method for studying the stability of the theory of nonlinear control systems, which is described by the title of the Lyapunov stability function in various forms, is known in terms of the kinematics of the problem and the system. 1,2 This study included two methods, called “linearization method” 3 and “direct method.” In linearization method, using the linearization of the near-system nonlinear system of equilibrium points, and using the methods of checking the stability of linear systems, it investigates the stability of the point of equilibrium. In the direct method, the definition of a quasi-energy function for the general stability system examines the nonlinear system.…”

Stabilization of a Tractor with n Trailers in the Presence of Wheel Slip Effects

“…In the direct method, the definition of a quasi-energy function is used for the system to examine the nonlinear system stability. 13,14 The Lyapunov theory is one of the most important techniques for stability analysis of the nonlinear systems in modern and classical control. An important point in Lyapunov stability is that all of the Lyapunov theorems determine sufficient conditions for analyzing stability.…”