Technical communiqueObserver analysis and synthesis for perturbed Lipschitz systems under noisy time-varying measurements☆
Introduction
Nowadays, in a large variety of control systems sensing is done discretely in time and information is sent through communication networks to a computer for state estimation. Aperiodic communications are an intrinsic characteristic of a networked environment (Hespanha et al., 2007, Hetel et al., 2017, Zhang et al., 2001). Therefore, there is an increasing interest in the development of estimation methods which take into account aperiodic sampling. Here, we will deal with the estimation problem for a perturbed Lipschitz nonlinear system. In the unperturbed case, the problem has been addressed using various approaches. In Deza, Busvelle, Gauthier, and Rakotopara (1992) and Nadri, Hammouri, and Grajales (2013) a high-gain approach is considered relying on the computation of normal forms. The works (Dinh et al., 2015, Mazenc et al., 2015) use the approximation of reachable sets based on convex embeddings. Impulsive models and time-varying Lyapunov functions have been used in Chen et al., 2013, Etienne et al., 2017, Raff and Allgower, 2007. A classical issue when considering observer design is the problem of “robustness”, i.e. how to characterize the degradation of the estimation quality when noise and perturbation affect, respectively, the output and the plant. Despite the high relevance of this problem, there is a lack of robust analysis methods for aperiodically sampled observation schemes.
In this article, for the case of Lipschitz nonlinear systems, we will study the Input-to-State Stability (ISS) properties of a classical Luenberger observer subject to an aperiodically sampled implementation. Unlike the existing approaches, relaying on the computation of normal forms, convex embeddings or time-varying Lyapunov functions for impulsive systems, here we propose a radically different method, based on an input/output interconnection approach. The effect of sampling in the estimation loop is captured by a reset integrator operator. The observer design problem is studied by exploiting conditions inspired by the Dissipativity Theory (Brogliato, Lozano, Maschke, & Egeland, 2007). The approach allows to analyse the effect of perturbation (in terms of ISS) on the observation error dynamics.
The remainder of this paper is organized as follows. Section 2 is dedicated to the problem statement. In Section 3, preliminaries on the reset integrator are provided. In Section 4, generic conditions for robust observer analysis are given. In Section 5, numerically tractable conditions for observer design and robustness analysis are provided in terms of LMI’s. At last, in Section 6, we illustrate our approach by numerical simulations.
Notation: For a square symmetric matrix , () denotes the largest (smallest) eigenvalue; , () means that is positive definite (negative definite). In a symmetric matrix, the symbol denotes the elements that can be induced by symmetry. For a matrix , . corresponds to the set of non-negative real numbers. Given , the set denotes the unit simplex,
The space of functions which are quadratically integrable over the interval is denoted as while denotes the space of functions which are quadratically integrable over every bounded interval of . For a bounded function , we write .
Section snippets
Problem statement
We consider a nonlinear Lipschitz system of the form where is the state with initial condition and is the input applied to the system. Here , , , , and . The nonlinear term satisfies the Lipschitz condition: , for all , for some . It is assumed that is piecewise continuous and bounded. denotes the sequence of sampling times, We assume
Preliminaries
Consider the observation error . Since the observer (2) uses sampled measurements and , the error dynamics also depends on the perturbation induced by the sampling . Using (1), (2), the error dynamics in the sampled case is given by where . The idea of the input/output interconnection approach is to characterize the sampling induced perturbation by an operator which has as
Generic conditions for convergence
Consider a more general class of nonlinear systems where is Lipschitz continuous in all arguments, is the sampling induced error, is a bounded perturbation with and is a continuous function defined over a closed set that may account for uncertainties or exogenous inputs. System (3) can be rewritten in the form of (8) with , where . Before presenting a generic result
Tractable conditions
In order to transform Problem 1 into a numerically tractable one, the structure of the nonlinear function can be used. Since is Lipschitz, it is possible to find a finite set of matrices , such that for all —see for instance (Zemouche, Boutayeb, & Bara, 2008) where a constructive procedure is given.
Problem 2 Find and a continuously differentiable function , with , such that the
Example: DC drive
Consider the example of a single-link direct-drive manipulator actuated by a permanent magnet DC brush motor (Mazenc et al., 2015) . For , , , considering the specific from (Omran et al., 2016) and using Proposition 1 with and as additional LMI variables, one finds the observer gain . Applying Proposition 2, one finds that the system is ISS with parameters , . The results
Conclusion
In this work the problems of observer synthesis and robustness analysis of an observer for a nonlinear Lipschitz system with sampled measurements have been investigated. The case where the state is perturbed and where noisy measurements are sampled aperiodically has been studied. Our analysis is based on the use of a reset integrator operator, which captures the effect of the error induced by sampling. Both theoretical and numerically tractable conditions were provided. An LMI criterion for the
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Cited by (9)
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2022, Journal of the Franklin InstituteCitation Excerpt :Because the measurement noises can deteriorate the observation performance, it is of importance to analyze how the measurement noises influence the estimation accuracy. Recently, the researchers of [32] studied the observer synthesis problem of Lipschitz nonlinear systems with discrete noisy measurements. By applying the dissipativity theory [33], the robust performance of the observation error dynamics with respect to measurement noises and continuous disturbance is analyzed in the framework of input-to-state stability.
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2023, 2023 European Control Conference, ECC 2023Local robust stability on compact set for nonlinear systems with continuous time controller against to aperiodic sampling and disturbance
2023, IET Control Theory and ApplicationsEllipsoid-Based Interval Estimation for Lipschitz Nonlinear Systems
2022, IEEE Transactions on Automatic ControlActive Disturbance Rejection Control for Uncertain Nonlinear Systems With Sporadic Measurements
2022, IEEE/CAA Journal of Automatica Sinica
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This work has been partially supported by the ANR, France project DIGITSLID. The material in this paper was partially presented at the 20th World Congress of the International Federation of Automatic Control, July 9–14, 2017, Toulouse, France. This paper was recommended for publication in revised form by Associate Editor Juan I. Yuz under the direction of Editor André L. Tits.