Abstract
Semilinear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic semilinear discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Moreover, the converse Lyapunov theorem on Lyapunov (nonasymptotic) stability is proved for complex and real linear periodic discrete-time systems. Finally, examples of using the obtained results are presented.
Highlights
Consider a time-varying nonlinear control discrete-time system: x(t + 1) = f (t, x(t), u(t)). (1)Here x ∈ Rn is the state, u ∈ Rr is the control input, t ∈ Z, and f (t, 0, 0) ≡ 0
We investigate the problem of uniform global asymptotic stabilization of system (1): one needs to construct a feedback control u(t) = u(t, x(t)) with u(t, 0) ≡ 0 in system (1) such that the equilibrium x = 0 of the closedloop system x(t + 1) = f (t, x(t), u(t, x(t)))
Where F (t, 0) ≡ 0 and F (t, ·) is continuous, is called: (a) stable, if for any ε > 0 and for any t0 ∈ Z there exists a δ = δ(t0, ε) > 0 such that x0 ≤ δ implies that x(t, t0, x0) ≤ ε for all t ≥ t0; here x(t) = x(t, t0, x0) is the solution of (2) with an initial condition x(t0) = x0 (b) uniformly stable, if for any ε > 0 there exists a δ = δ(ε) > 0 such that for any t0 ∈ Z the inequality x0 ≤ δ implies that x(t, t0, x0) ≤ ε for all t ≥ t0
Summary
In [4], sufficient conditions for global asymptotic stabilization of bilinear discrete time-invariant systems were obtained. In [5], sufficient conditions were obtained for global asymptotic stabilization of affine discrete time-invariant systems (1) (f (t, x, u) = f (x) + g(x)u), see [6]. In the paper [7], sufficient conditions for global asymptotic stabilization were obtained for general discrete time-invariant nonlinear systems (1) (f (t, x, u) ≡ f (x, u)). The aim of this paper is to extend the results of work [6] on global asymptotic stabilization of affine time-invariant systems to time-varying semi-linear periodic discrete-time systems and to obtain sufficient conditions for uniform global asymptotic stabilization. We will use the Krasovsky–La Salle invariance principle for periodic discrete-time systems
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