Exponential Stabilization of Discrete Nonlinear Time-Varying Systems

Abstract We consider a discrete nonlinear control time-varying system x(k + 1) = f(k, x(k), u(k)), k ∈ ℕ, x ∈ ℝn, u ∈ ℝr. A control process of this system is a pair (x(k), u(k))k ∈ N consisting of a control (u(k))k∈N and some solution (x(k))k∈N of the system with this control. We assume that the control process is defined for all k ∈ N. We have obtained sufficient conditions for uniform and non-uniform (with respect to the initial moment) exponential stabilization of the control process with any pregiven decay of rate. Exponential convergence to zero of the deviation of both the state vector and the control vector is guaranteed. The result is based on the property of uniform complete controllability (in the sense of Kalman) for a system of linear approximation.