Abstract
Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.
Highlights
Remarkable progress has been made in recent years on the design of predictor feedback laws for nonlinear delay systems (Bekiaris-Liberis & Krstic, 2012; Bekiaris-Liberis & Krstic, 2013a, 2013b; Karafyllis, 2011; Karafyllis & Jiang, 2011; Karafyllis & Krstic, 2012a, 2013a, XXXXb; Krstic, 2004, 2008, 2009, 2010)
We focus on a class of nonlinear systems that is different from the class of globally Lipschitz systems: the systems with a compact absorbing set
The contribution of our paper is twofold: (1) Predictor feedback is designed and stability is proved for the class of nonlinear delay systems with a compact absorbing set under appropriate assumptions (Theorem 2.2)
Summary
Remarkable progress has been made in recent years on the design of predictor feedback laws for nonlinear delay systems (Bekiaris-Liberis & Krstic, 2012; Bekiaris-Liberis & Krstic, 2013a, 2013b; Karafyllis, 2011; Karafyllis & Jiang, 2011; Karafyllis & Krstic, 2012a, 2013a, XXXXb; Krstic, 2004, 2008, 2009, 2010). (1) Predictor feedback is designed and stability is proved for the class of nonlinear delay systems with a compact absorbing set under appropriate assumptions (Theorem 2.2). The disadvantages of the dynamic predictor are the difficulty of implementation (one has to approximate numerically the solution of the IDEs or the equivalent distributed delay differential equations) and that it works only for certain classes of nonlinear systems (globally Lipschitz systems and systems with a compact absorbing set). Assumption H1 guarantees that for every initial condition x(0) ∈ Rn and for every measurable and essentially bounded input u : R+ → U the solution x(t) of (1.1) enters the compact set S = {x ∈ Rn : V (x) ≤ R} after a finite transient period, i.e., there exists T ∈ C0(Rn; R+) such that x(t) ∈ S, for all t ≥ T (x(0)). The existence of functions a1 : Rn → Rl, a2 : Rl → Rm satisfying the assumptions of Theorem 2.4 is a restrictive assumption, which can be verified in certain cases (see Example 4.2)
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