A separation principle for the control of a class of nonlinear systems

Atassi, A.N.; Khalil, Hassan K.

doi:10.1109/9.920793

Cited by 86 publications

(34 citation statements)

References 13 publications

(33 reference statements)

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“…They showed, under the additional assumption that either A is compact or all solutions exist for all backward time, that the estimate (5) for the differential inclusion (10) implies the existence of a smooth converse Lyapunov function. In [1], the authors combined the ideas of [18] with the idea of Kurzweil [15] establishing the existence of a smooth converse Lyapunov function for the differential inclusion (10) in the case of the existence of a compact set A, a neighborhood G of A and function ω : G → R ≥0 that is locally Lipschitz, positive definite with respect to A and proper with respect to G and a function β ∈ KL such that, for all x • ∈ G, the solutions of (10) satisfy…”

Section: X(t)| ≤ K|x(0)| Exp(−λt)mentioning

confidence: 99%

“…Also without loss of generality, we can assume that ω 1 is locally Lipschitz on G. Indeed, if ω 1 is only continuous on G then ω 1 can be smoothed on G using Lemma 15 and Lemma 17. In particular, we first get a functionω 1 , continuous on G and smooth on G \ {x : ω 1 (x) = 0} and satisfying |ω 1 …”

Section: Robust Kl-stability =⇒ Forward Completeness Smooth Conversementioning

confidence: 99%

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A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions

2000

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Abstract. We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-KL estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-KL estimate, exists if and only if the class-KL estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class-KL estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature. AMS Subject Classification. 34A60, 34D20, 34B25.Received November 3, 1999. Revised May 24, 2000. Basic definitions• Given a set A, A stands for the closure of A,Å stands for the interior set of A, coA stands for the closed convex hull of A and ∂A stands for the boundary of A. • The notation x → ∂A ∞ indicates a sequence of points x belonging to A converging to a point on the boundary of A or, if A is unbounded, having the property |x| → ∞.• Given a closed set A ⊂ R n and a point x ∈ R n , |x| A denotes the distance from x to A. • A function α : R ≥0 → R ≥0 is said to belong to class-K (α ∈ K) if it is continuous, zero at zero, and strictly increasing. It is said to belong to class-K ∞ if, in addition, it is unbounded. • A function β : R ≥0 × R ≥0 → R ≥0 is said to belong to class-KL if, for each t ≥ 0, β(·, t) is nondecreasing and lim s→0 + β(s, t) = 0, and, for each s ≥ 0, β(s, ·) is nonincreasing and lim t→∞ β(s, t) = 0.The requirements imposed for a function to be of class-KL are slightly weaker than usual. In particular, β(·, t) is not required to be continuous or strictly increasing. See, also, Remark 3.

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Section: X(t)| ≤ K|x(0)| Exp(−λt)mentioning

confidence: 99%

Section: Robust Kl-stability =⇒ Forward Completeness Smooth Conversementioning

confidence: 99%

A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions

2000

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“…Over the past twenty years, high-gain observers have been studied in a wide range of nonlinear control problems, including stabilization, regulation, tracking, and adaptive control, in, e.g., [42][43][44]. More extensive references can be found in the survey paper [46].…”

Section: Standard Output Regulationmentioning

confidence: 99%

Output Regulation and Active Disturbance Rejection Control: Unified Formulation and Comparison

Chen¹,

Xu²

2015

Asian Journal of Control

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Output regulation theory aims to design a controller for achieving reference tracking and disturbance rejection while maintaining system stability. Different from the stabilization problem about an equilibrium point, the output regulation problem is capable of characterizing more complicated steady-state trajectories induced by reference and/or disturbance. Many efforts have been made to reveal how a steady-state trajectory can be characterized, estimated, and hence compensated by a controller such that output regulation can be asymptotically achieved. When the steady-state trajectory is approximately treated as a constant "quantity", the standard output regulation implies an approximate version within which output regulation is practically achieved. It is revealed in this paper that such an approximate version of output regulation includes the later developed active disturbance rejection control (ADRC) method as a special case.

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“…Fruitful results have been reported for a variety of research issues on Lipscthiz nonlinear systems such as stabilization and control [1][2][3], estimation and filtering [4,5], and fault diagnosis [6,7]. Meanwhile, time delays always exist in practical processes due to the distributed nature of the system, material transport, and communication lag, which may cause the system performance degradation, and even instability [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Estimation and Compensation for Lipschitz Nonlinear Discrete-Time Systems Subjected to Unknown Measurement Delays

Gao

2015

IEEE Trans. Ind. Electron.

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Abstract-Unknown measurement delays usually degrade system performance, and even damage the system under output feedback control, which motivates us to develop an effective method to attenuate or offset the adverse effect from the measurement delays. In this paper, an augmented observer is proposed for discrete-time Lipschitz nonlinear systems subjected to unknown measurement delays, enabling a simultaneous estimation for system states and perturbed terms caused by the output delays. On the basis of the estimates, a sensor compensation technique is addressed to remove the influence from the measurement delays to the system performance. Furthermore, an integrated robust estimation and compensation technique is proposed to decouple constant piece-wise disturbances, attenuate other disturbances/noises, and offset the adverse effect caused by the measurement delays. The proposed methods are applied to a two-stage chemical reactor with delayed recycle, and an electro-mechanical servo system, which demonstrates the effectiveness of the present techniques.Index Terms-Discrete-time systems, measurement delays, nonlinear systems, observers, robustness, signal compensation. I. INTRODUCTIONIPSCHITZ nonlinear systems have attracted continuous interest during the last decades, since many nonlinearities in engineering systems can be characterized, at least locally, as Lipschitz form. Fruitful results have been reported for a variety of research issues on Lipscthiz nonlinear systems such as stabilization and control [1][2][3], estimation and filtering [4,5], and fault diagnosis [6,7]. Meanwhile, time delays always exist in practical processes due to the distributed nature of the system, material transport, and communication lag, which may cause the system performance degradation, and even instability [8][9][10][11]. Therefore, research on nonlinear systems Manuscript received July 26, 2014; revised March 7, 2015; accepted March 27, 2015. Copyright © 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permission@ieee,org.The work is partially supported by the Alexander von Humboldt Foundation (GRO/1117303 STP) and the sabbatical leave scheme at the E&E faculty in the University of Northumbria.Z. Gao is with the Faculty of Engineering and Environment (E&E), University of Northumbria at Newcastle, Newcastle upon Tyne, NE1 8ST, UK (+441912437832; e-mail: zhiwei.gao@northumbria.ac.uk).subjected to delays has received much more attention. In [12], state and output feedback stabilization was discussed for Lipschitz nonlinear systems with delays. In [13], a distributed ∞ filter was proposed for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks. By using linear matrix inequality technique, a robust observer was presented in [14] for nonlinear discrete-time systems with time-delays. In [15], an adaptive observer was addressed for Lipschitz nonlinear systems with time-varying de...

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A separation principle for the control of a class of nonlinear systems

Cited by 86 publications

References 13 publications

A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions

A smooth Lyapunov function from a class-${\mathcal{KL}}$ estimate involving two positive semidefinite functions

Output Regulation and Active Disturbance Rejection Control: Unified Formulation and Comparison

Estimation and Compensation for Lipschitz Nonlinear Discrete-Time Systems Subjected to Unknown Measurement Delays

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