Automatica 72 (2016) 123–130 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Linear interval observers under delayed measurements and delay-dependent positivityI Denis Efimov a,b,c, Emilia Fridman d, Andrey Polyakov a,b,c, Wilfrid Perruquetti b,a, Jean-Pierre Richard b,a a Non-A team @ Inria, Parc Scientifique de la Haute Borne, 40 av. Halley, 59650 Villeneuve d'Ascq, France b CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France c Department of Control Systems and Informatics, ITMO University, 49 Kronverkskiy av., 197101 Saint Petersburg, Russia d School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel article info a b s t r a c t Article history: New interval observers are designed for linear systems with time-varying delays in the case of delayed Received 15 November 2014 measurements. Interval observers employ positivity and stability analysis of the estimation error system, Received in revised form which in the case of delayed measurements should be delay-dependent. New delay-dependent conditions 22 December 2015 of positivity for linear systems with time-varying delays are introduced. Efficiency of the obtained solution Accepted 10 May 2016 is demonstrated by examples. ' 2016 Elsevier Ltd. All rights reserved. Keywords: Interval observers Time delay Stability analysis 1. Introduction highly demanded in these and many others applications. Interval or set-membership estimation is a promising framework to obser- An estimation in nonlinear delayed systems is rather com- vation in uncertain systems (Gouzé, Rapaport,& Hadj-Sadok, 2000; plicated (Fridman, 2014; Sipahi, Niculescu, Abdallah, Michiels,& Jaulin, 2002; Kieffer& Walter, 2004; Mazenc& Bernard, 2011; Gu, 2011), as well as analysis of functional differential equations Moisan, Bernard,& Gouzé, 2009; Raïssi, Efimov,& Zolghadri, 2012), (Richard, 2003). Especially the observer synthesis is problemati- when all uncertainty is included in the corresponding intervals or cal for the cases when the model of a nonlinear delayed system polytopes, and as a result the set of admissible values (an interval) contains parametric and/or signal uncertainties, or when the de- for the state is provided at each instant of time. lay is time-varying and/or uncertain (Briat, Sename,& Lafay, 2011; In this work an interval observer for time-delay systems with Califano, Marquez-Martinez,& Moog, 2011; Zheng, Barbot, Boutat, delayed measurements is proposed. A peculiarity of an interval Floquet,& Richard, 2011), the frequent applications include biosys- observer is that in addition to stability conditions, some restric- tems and chemical processes. Delayed measurements usually also tions on positivity of estimation error dynamics have to be im- increase complexity of estimators, which is a case in networked posed (in order to envelop the system solutions). The existing systems. An observer solution for these more complex situations is solutions in the field (Efimov, Perruquetti,& Richard, 2013; Efimov, Polyakov,& Richard, 2015; Mazenc, Niculescu,& Bernard, 2012; Polyakov, Efimov, Perruquetti,& Richard, 2013) are based on the delay-independent conditions of positivity from Ait Rami(2009) I This work was partially supported by the Government of Russian Federation and Haddad and Chellaboina(2004). Some results on interval ob- (Grant 074-U01), the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031) and by Israel Science Foundation (Grant No 1128/14). server design for uncertain time-varying delay can be found in Efi- The material in this paper was partially presented at the 14th annual European mov et al.(2013) and Ait Rami, Schönlein, and Jordan(2013). The Control Conference, July 15–17, 2015, Linz, Austria. This paper was recommended first objective of this work is to use the delay-dependent positiv- for publication in revised form by Associate Editor Fouad Giri under the direction of ity conditions (Efimov, Polyakov, Fridman, Perruquetti,& Richard, Editor Miroslav Krstic. 2015), which are based on the theory of non-oscillatory solutions E-mail addresses: [email protected] (D. Efimov), [email protected] (E. Fridman), [email protected] (A. Polyakov), [email protected] for functional differential equations (Agarwal, Berezansky, Braver- (W. Perruquetti), [email protected] (J.-P. Richard). man,& Domoshnitsky, 2012; Domoshnitsky, 2008). Next, two http://dx.doi.org/10.1016/j.automatica.2016.05.022 0005-1098/' 2016 Elsevier Ltd. All rights reserved. 124 D. Efimov et al. / Automatica 72 (2016) 123–130 interval observers are designed for linear systems with delayed the conditions of positive invariance of polyhedral sets have been measurements (with time-varying delays) in the case of observ- similarly given in Dambrine, Richard, and Borne(1995), as well as able and detectable systems (with respect to (Efimov et al., 2015) conditions of asymptotic stability in the nonlinear case have been the present work contains new result, Theorem 12, relaxed As- considered in Borne, Dambrine, Perruquetti, and Richard(2003) sumption 1, and new examples). Efficiency of the obtained interval and Dambrine and Richard (1993, 1994). observers is demonstrated on a benchmark example from Mazenc et al.(2012) and a delayed nonlinear pendulum. 3. Delay-dependent conditions of positivity The paper is organized as follows. Some preliminaries and notation are given in Section2. The delay-dependent positivity Consider a scalar time-varying linear system with time-varying conditions are presented in Section3. The interval observer design delays (Agarwal et al., 2012): is performed for a class of linear time-delay systems (or a class of nonlinear systems in the output canonical form) with delayed xP.t/ D a0.t/xTg.t/U − a1.t/xTh.t/UC b.t/; (2) measurements in Section4. Examples of numerical simulation are x(θ/ D 0 for θ < 0; x.0/ 2 R; (3) presented in Section5. where a0 2 L1, a1 2 L1, b 2 L1, h.t/ − t 2 L1, g.t/ − t 2 L1 and h.t/ ≤ t, g.t/ ≤ t for all t ≥ 0. For the system (2) the 2. Preliminaries initial condition in (3) is, in general, not a continuous function (if x.0/ 6D 0). 2.1. Notation The following result proposes delay-independent positivity n conditions. • R is the Euclidean space (RC D fτ 2 R V τ ≥ 0g), Cτ D C.[−τ; 0U; n/ is the set of continuous maps from [−τ; 0U into R Lemma 2 (Agarwal et al., 2012, Corollary 15.7). Let h.t/ ≤ g.t/ and n for n ≥ 1; Cn D fy 2 Cn V y.s/ 2 n ; s 2 [−τ; 0Ug; R τC τ RC 0 ≤ a .t/ ≤ a .t/ for all t ≥ 0. If x.0/ ≥ 0 and b.t/ ≥ 0 for all • x is an element of Cn defined as x .s/ D x.t C s/ for all 1 0 t τ t t ≥ 0, then the corresponding solution of (2), (3) x.t/ ≥ 0 for all s 2 [−τ; 0U; t ≥ 0. •j xj denotes the absolute value of x 2 R, kxk2 is the Euclidean n n norm of a vector x 2 R , k'k D supt∈[−τ;0U k'.t/k2 for ' 2 Cτ ; Recall that in this case positivity is guaranteed for ``discontinuous'' • for a measurable and locally essentially bounded input u V initial conditions. The peculiarity of the condition 0 ≤ a1.t/ ≤ ! p k k k k D RC R the symbol u Tt0;t1/ denotes its L1 norm u Tt0;t1/ a0.t/ is that it may correspond to an unstable system (2). In p ess supfku.t/k2; t 2 Tt0; t1/g, the set of all such inputs u 2 R order to overcome this issue, delay-dependent conditions can be p with the property kukT0;C1/ < 1 will be denoted as L1; introduced. n×n • for a matrix A 2 R the vector of its eigenvalues is denoted as λ(A/; Lemma 3 (Agarwal et al., 2012, Corollary 15.9). Let h.t/ ≤ g.t/ and • ≤ 1 ≤ ≥ In and 0n×m denote the identity and zero matrices of dimensions 0 e a0.t/ a1.t/ for all t 0 with n × n and n × m, respectively; Z t • a R b corresponds to an elementwise relation R 2 f<; >; ≤; ≥ 1 1 sup a1(ξ/ − a0(ξ/ dξ < ; g (a and b are vectors or matrices): for example a < b (vectors) t2RC h.t/ e e means 8i V a < b ; for φ; ' 2 C the relation φ R ' has to be i i τ D ≥ ≥ ≥ understood elementwise for whole domain of definition of the where e exp.1/. If x.0/ 0 and b.t/ 0 for all t 0, then ≥ ≥ functions, i.e. φ.s/ R '.s/ for all s 2 [−τ; 0U; x.t/ 0 for all t 0 in (2), (3). • for a symmetric matrix Υ , the relation Υ ≺ 0 (Υ ≼ 0) means These lemmas describe positivity conditions for the system (2), that the matrix is negative (semi) definite. (3), which is more complex than (1), but scalar, they can also be extended to the n-dimensional system (1). 2.2. Delay-independent conditions of positivity Corollary 4. The system (1) with b.t/ ≥ 0 for all t ≥ 0 and initial Consider a time-invariant linear system with time-varying conditions delay: n x(θ/ D 0 for − τ ≤ θ < 0; x.0/ 2 RC; xP.t/ D A0x.t/ − A1x.t − τ.t// C b.t/; t ≥ 0; (1) n is positive if −A1 is Metzler, A0 ≥ 0, and x(θ/ D φ(θ/ for − τ ≤ θ ≤ 0; φ 2 Cτ ; −1 n n 0 ≤ .A0/i;i ≤ e.A1/i;i < .A0/i;i C τ where x.t/ 2 R , xt 2 Cτ is the state function; τ V RC ! [−τ; 0U is the time-varying delay, a Lebesgue measurable function of time, for all i D 1;:::; n.