Stability analysis and state feedback control design of discrete-time systems with a backlash

This paper considers the class of discrete-time nonlinear systems resulting from the connection of a linear system with a backlash operator. By conveniently exploiting the properties of the backlash, a class of candidate Lyapunov functions with quadratic terms and Lur´e type terms, derived from generalized sector conditions, is introduced. Using this class of Lyapunov functions, the stability of the time-shifted system is investigated. Additionally, the set of equilibrium points, which can be estimated, may be not reduced to the origin, since the backlash operator contains a dead-zone. Sufficient convex conditions, formulated in terms of semi-definite programming, are provided for the stability analysis and for the design of a linear stabilizing state-feedback controller. Numerical simulations illustrate the results and some computational issues.