Elsevier

Automatica

Volume 106, August 2019, Pages 406-410
Automatica

Technical communique
Observer analysis and synthesis for perturbed Lipschitz systems under noisy time-varying measurements

https://doi.org/10.1016/j.automatica.2019.04.003Get rights and content

Abstract

In this paper the observer synthesis problem is studied for nonlinear Lipschitz systems with noisy time-varying sampling and bounded state perturbations. To establish criteria for robust convergence of the observer, we model the impact of sampling by a reset integrator operator. First, generic conditions for the input-to-state stability of a sampled-data system are presented. Second, it is shown how to derive a tractable numerical criterion for the synthesis of a sampled-data Luenberger observer. Then, new conditions for robustness analysis of a known observation gain are given.

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 P, λmax(P) (λmin(P)) denotes the largest (smallest) eigenvalue; P>0, (P<0) means that P is positive definite (negative definite). In a symmetric matrix, the symbol denotes the elements that can be induced by symmetry. For a matrix A, He(A)A+A. R0 corresponds to the set of non-negative real numbers. Given rN, the set Δr denotes the unit simplex, ΔrλR0r:i=1rλi=1.

The space of functions f:[t0,t1)Rm which are quadratically integrable over the interval [t0,t1) is denoted as L2[t0,t1) while L2e denotes the space of functions which are quadratically integrable over every bounded interval of R0. For a bounded function f:R0Rm, we write |f|=suptR0|f(t)|.

Section snippets

Problem statement

We consider a nonlinear Lipschitz system of the form ẋ(t)=Ax(t)+Bu(t)+Gϕ(Hx(t))+δ(t),t0,νk=Cx(tk)+σ(tk),kN, where xRn is the state with initial condition x(0)=x0 and uRl is the input applied to the system. Here ARn×n, BRn×l, CRq×n, GRn×m, HRm×n and ϕ:RmRm. The nonlinear term ϕ satisfies the Lipschitz condition: |ϕ(a)ϕ(b)|Kϕ|ab|, for all (a,b)Rm×Rm, for some Kϕ>0. It is assumed that u(.) is piecewise continuous and bounded. (tk)kN denotes the sequence of sampling times, We assume

Preliminaries

Consider the observation error z=xxˆ. Since the observer (2) uses sampled measurements x(tk) and xˆ(tk), the error dynamics also depends on the perturbation induced by the sampling w(t)=z(t)z(tk). Using (1), (2), the error dynamics in the sampled case is given by ż(t)=(A+LC)z(t)+Ψ(x(t),z(t))+LCw(t)+Lσ(tk)+δ(t),t[tk,tk+1), where Ψ(x,z)=G(ϕ(Hx)ϕ(HxHz)). The idea of the input/output interconnection approach is to characterize the sampling induced perturbation w by an operator which has as

Generic conditions for convergence

Consider a more general class of nonlinear systems ż=f(z,w,Γ,v),z(0)Rn,w(t)=z(tk)z(t),t[tk,tk+1),kN,where f:Rn×Rn×Rn×VRn is Lipschitz continuous in all arguments, w is the sampling induced error, Γ is a bounded perturbation with |Γ|=Γ̄ and v:R0V is a continuous function defined over a closed set V that may account for uncertainties or exogenous inputs. System (3) can be rewritten in the form of (8) with f(z,w,Γ,v)=(A+LC)z+LCw+Ψ(v,z)+Γ, where Γ=Lσ+δ. Before presenting a generic result

Tractable conditions

In order to transform Problem 1 into a numerically tractable one, the structure of the nonlinear function Ψ(x,z)=G(ϕ(Hx)ϕ(HxHz)) can be used. Since ϕ is Lipschitz, it is possible to find a finite set of p matrices ΨiRn×n,iR={1,2,,r}, such that Ψ(x,z)Conv{Ψiz}iR for all x,zRn—see for instance (Zemouche, Boutayeb, & Bara, 2008) where a constructive procedure is given.

Problem 2

Find α,c1,c2>0,p1,γ0 and a continuously differentiable function V:RnR0, with c1|z|pV(z)c2|z|p,zRn, 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) ẋ=x22sin(x1)3x2+x3ux2x3, νk=x1(tk)+0.2sin(1000tk),0,0. For α=1, δ(t)=0, h̄=0.4, considering the specific Π=XY(2h̄π)2Xfrom (Omran et al., 2016) and using Proposition 1 with X>0 and Y>0 as additional LMI variables, one finds the observer gain L=[2.2,0.16,0.45]. Applying Proposition 2, one finds that the system is ISS with parameters γ=2.7, c1=1. 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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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.

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