Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters

Abstract

New results for an established for the global asymptotic stability of the equilibrium x=0 of nth order discrete-time systems with state saturations, x(k+1)=sat(Ax(k)), utilizing a class of positive definite and radially unbounded Lyapunov functions, v. When v is a quadratic form, necessary and sufficient conditions are obtained under which positive definite matrices H can be used to generate a Lyapunov function v(w)=w/sup T/Hw with the properties that v(Aw(k)) is negative semidefinite, and that v(sat(w))<v(w(k)) is negative semidefinite, and that v(sat(w))<v(w) under appropriate restrictions on w. This Lyapunov function is then used in the stability analysis of systems described by x(k+1)=sat(Ax(k)). For nth-order fixedpoint digital filters, previous results are reviewed, and the above results are used to establish conditions for the nonexistence of limit cycles in such filters that are easier to apply and less conservative than previous results.<<ETX>>