Elsevier

Automatica

Volume 103, May 2019, Pages 294-301
Automatica

Brief paper
An unknown input interval observer for LPV systems under L2-gain and L-gain criteria

https://doi.org/10.1016/j.automatica.2019.02.012Get rights and content

Abstract

This paper investigates the design of an unknown input interval observer, for linear parameter varying systems. The proposed method provides a solution to the decoupling conditions under L2-gain and L-gain performance level. It is shown that the design of the unknown input interval observer parameters can be formulated as a SDP problem. The major advantages with regard to other solutions is that first, it does not require the explicit knowledge of a (time-varying) state transformation, for the estimation error dynamics to satisfy the Meztler property. Rather, it is shown how such a property can be formulated as a part of the SDP optimization process; Second, decoupling and L2/L constraints are formulated on the thickness of the interval length in spite of half the interval length. The proposed methodology is illustrated on a numerical academic example.

Introduction

Several set membership/interval observer-based solutions have been proposed in the literature to solve the so-called “guaranteed” state estimation problem for Linear Parameter Varying (LPV)/quasi-LPV systems (Chebotarev et al., 2015, Efimov et al., 2012, Wang et al., 2015). The design of an interval observer is generally based on the monotone system theory (Farina & Rinaldi, 2000) to get the cooperativity property of the observation error dynamics. The difficulty to obtain this property was relaxed for a large class of systems using a time invariant state transformation (Raissi, Efimov, & Zolghadri, 2012) and a time varying state transformation (Mazenc and Bernard, 2011, Thabet et al., 2014). Another approach consists in applying a transformation to put the system into a positive form before designing an interval observer (Cacace, Germani, & Manes, 2015). However, it has been proposed only for Linear Time Invariant (LTI) and Linear Time Variant (LTV) systems. A last approach has been proposed based on Müller’s theorem and interval analysis for nonlinear system (Kieffer and Walter, 2006, Meslem and Ramdani, 2011). This approach also exploits the monotone property.

The main limitation of the previous mentionedapproaches is their ability to simultaneously satisfy the conditions of the existence of the interval observer (especially cooperativity property) and a certain level of a priori given performance like the interval length. The work reported in Chebotarev et al. (2015) aims at proposing a solution to this problem within the L1L2 framework. Unfortunately, the problem of considering the Meztler property as a part of the design process, is not addressed. Furthermore, it should be pointed out that the proposed approach is based on a LTI interval observer (to deal with LPV systems) and this leads, obviously, to conservative results. The work introduced in Briat and Khammash (2016) proposes an interval observer design method based on L performance criterion, but it is limited to LTI systems.

The contribution of this paper must be thought in this context. The goal is to develop an interval observer for a general class of LPV systems, under unknown input decoupling, L2-gain and L-gain criteria (and then a mixed L2L-gain criterion), considering the cooperativity property of the observation error dynamics as a part of the design process. To this end, a new structure of an interval observer is proposed. The major advantage with regard to existing solutions is that it does not require the explicit knowledge of a (time-varying) state transformation, for the estimation error dynamics to satisfy the Meztler property.

Notations: In this paper, we adopt the notations that are commonly used in the robust control and interval communities: R and R+[0,)R are the set of real numbers and the set of nonnegative real numbers, respectively. Rn is a n-dimensional real space and Rn×m is the set of real n×m matrices. I, 0 denote respectively the identity and the null matrices of appropriate dimensions. For a matrix ARn×m, ,,<,> refer to component-wise. Denote A+=max(A,0), A=A+A, |A|=A++A and A̲,A¯=ARn×m|A̲AA¯. The same notations are adopted for vectors. For a vector vRn, the notation v refers to v=(v¯+)T(v¯)T(v̲+)T(v̲)TT, ṽ=v¯v̲, vˆ=v¯+v̲+. For a matrix A, A0 (A0) defines a strictly positive (negative) definite matrix. A refers to the Moore–Penrose inverse of a full column matrix rank A and He{A}=A+AT. In large symmetric matrix expressions, terms denoted refer to terms induced by symmetry.

Section snippets

Preliminaries

This section is devoted to definitions and lemmas that will be later used in the paper.

Definition 1

Farina & Rinaldi, 2000 Theorem 2 p. 14

A square matrix is said to be Metzler if all its off-diagonal elements are non-negative.

Definition 2

Poole & Boullion, 1974 Theorem 2.1 p. 420

Let a square matrix W=[wi,j] satisfy wij0, ij and wij>0 i=j then W is said to be a M-matrix if W1 exists and W10.

The following lemmas provide useful results for Metzler matrices and cooperative systems.

Lemma 3

Consider a (known) parameter-dependent matrixL(ρ)Rn×n whereρΨ Rs, an invertible M-matrix WRn×n and a real

Problem statement

Consider the following class of LPV systems, driven by unknown inputs w(t)Rm and d(t)Rq ẋ(t)=A(θ(t))x(t)+B(θ(t))w(t)+E(θ(t))d(t)(a)y(t)=C(θ(t))x(t)+D(θ(t))w(t)(b)Here, x(t)Rn and y(t)Rp denote state and measured output vectors, respectively. Both d(t) and w(t) are unknown input signals. However, w(t) is assumed to be bounded with known bounds w̲(t),w¯(t), whereas there is no assumption on d(t). The scheduling parameter vector θ(t)ΘRl and its time derivative θ̇(t)Γθ are assumed to be

Preliminary results

The following lemma provides a constructive method to design the lower and upper bounds of a product of a vector by a parameter dependent matrix in the set-membership context. The core element is that any matrix can always be decomposed into two (non unique) matrices whose elements are positives.

Lemma 10

Let a vector vv̲,v¯Rn and some matrices Xk(θ)Rm×n,θΘ,k{a,b}, such that X(θ)=Xa(θ)Xb(θ). Then:

(1) θΘ, assume that Xk(θ)Xk̲,Xk¯Rm×n withXk̲0. Then, with X¯=Xa¯Xa̲Xb̲Xb¯,X̲=Xb¯Xb̲Xa̲Xa

Academic example

Consider the following model with the same notations than used in the previous sections: A(θ)=θ320.50θ4θ110.20.10θ22,C(θ)=10001+θ10,B(θ)=010.2θ10.1002θ20,E(θ)=θ3+20θ1,D=010001. The parameter vector θ and its rate of variations θ̇ are unknown but assumed to be bounded by the sets θ1θ2θ3θ40.15,0.150.25,0.250.15,0.150.2,0.8,θ̇1θ̇2θ̇3θ̇40.30,0.300.25,0.250.75,0.750,0.The parameters θ1 and θ2 are respectively measured through ρ1=θ1+δ1 , ρ2=θ2+δ2 where δ10.03,0.03and δ20.03,0.03

Conclusion and future works

This paper investigated the design of an unknown input interval observer, for linear parameter varying systems, to robustly estimate the state of linear parameter varying systems, in a guaranteed way under L2 and L-gain performance. A SDP formulation is derived to design all the observer parameters. The major advantage with regard to existing theories for interval observer design is that it does not require the explicit knowledge of a (time-varying) state transformation, for the estimation

Nicolas Ellero received the Master’s degree in 2014 and the Ph.D. degree in 2018 both in automatic control from the University of Bordeaux, Bordeaux, France. He is currently an Attitude and Orbit Control System (AOCS) engineer. His activities are mainly focused on robust control and on robust stability and performance analysis.

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Nicolas Ellero received the Master’s degree in 2014 and the Ph.D. degree in 2018 both in automatic control from the University of Bordeaux, Bordeaux, France. He is currently an Attitude and Orbit Control System (AOCS) engineer. His activities are mainly focused on robust control and on robust stability and performance analysis.

David Gucik-Derigny received the Ph.D. degree in control systems from Aix-Marseille University, Marseille, France, in 2011. Since 2012, he has been an Associate Professor with Bordeaux University, Talence, France, and has integrated the IMS Laboratory. His main research interests concern model-based systems prognostic, observer based set-membership estimation, and parameter identification.

David Henry was born in Niort, France, in 1971. He received the Ph.D. degree in 1999 from the University of Bordeaux, France. He is currently a full professor at Bordeaux University, IMS laboratory (UMR CNRS n. 5218), France. His current research interests are theory and applications in model-based fault diagnosis, fault tolerant control, linear matrix inequality optimization techniques and sliding mode control with Hinf performance. His research application focuses on aeronautics and space. He has published around 40 journal papers, 6 patents with AIRBUS, 8 chapter’s book + 1 complete book. He is an expert for the European Space Agency for GNC methods, Fonds de Recherche du Québec - Nature et technologies and for Italian Ministry of Education, University and Research (MIUR). He won several international prices: best ABB application/case study paper of the Safeprocess’2015 conference, best theoretical paper of the Safecprocess’2009 conference (2nd position), best thesis ”Aerospace Valley” 2010 in the category <<aeronautic, space and embedded systems>>, best Ph.D. paper of the 2014 UKACC 10th International Conference on Control. He was involved in a wide number of aeronautical and space projects at the European level.

The material in this paper was not presented at any conference.This paper was recommended for publication in revised form by Associate Editor Tianshi Chen under the direction of Editor Torsten Söderström.

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