Linear interval observers under delayed measurements and delay-dependent positivity Denis Efimov, Emilia Fridman, Andrey Polyakov, Wilfrid Perruquetti, Jean-Pierre Richard To cite this version: Denis Efimov, Emilia Fridman, Andrey Polyakov, Wilfrid Perruquetti, Jean-Pierre Richard. Linear interval observers under delayed measurements and delay-dependent positivity. Automatica, Elsevier, 2016, 72, pp.123-130. 10.1016/j.automatica.2016.05.022. hal-01312981 HAL Id: hal-01312981 https://hal.inria.fr/hal-01312981 Submitted on 9 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Linear interval observers under delayed measurements and delay-dependent positivity Denis Efimov a;b;c, Emilia Fridman d, Andrey Polyakov a;b;c, Wilfrid Perruquetti b;a, Jean-Pierre Richard b;a aNon-A team @ Inria, Parc Scientifique de la Haute Borne, 40 av. Halley, 59650 Villeneuve d'Ascq, France bCRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cit´eScientifique, 59651 Villeneuve-d'Ascq, France cDepartment of Control Systems and Informatics, ITMO University, 49 Kronverkskiy av., 197101 Saint Petersburg, Russia dSchool of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel Abstract New interval observers are designed for linear systems with time-varying delays in the case of delayed measurements. Interval observers employ positivity and stability analysis of the estimation error system, which in the case of delayed measurements should be delay-dependent. New delay-dependent conditions of positivity for linear systems with time-varying delays are introduced. Efficiency of the obtained solution is demonstrated on examples. Key words: Interval observers, Time delay, Stability analysis 1 Introduction of an interval observer is that in addition to stability conditions, some restrictions on positivity of estimation An estimation in nonlinear delayed systems is rather error dynamics have to be imposed (in order to envelop complicated [31,15], as well as analysis of functional dif- the system solutions). The existing solutions in the field ferential equations [30]. Especially the observer synthe- [23,12,14,26] are based on the delay-independent condi- sis is problematical for the cases when the model of a tions of positivity from [18,2]. Some results on interval nonlinear delayed system contains parametric and/or observer design for uncertain time-varying delay can be signal uncertainties, or when the delay is time-varying found in [28,12]. The first objective of this work is to and/or uncertain [4,5,33], the frequent applications in- use the delay-dependent positivity conditions [13], which clude biosystems and chemical processes. Delayed mea- are based on the theory of non-oscillatory solutions for surements usually also increase complexity of estima- functional differential equations [1,9]. Next, two inter- tors, which is a case in networked systems. An observer val observers are designed for linear systems with de- solution for these more complex situations is highly de- layed measurements (with time-varying delays) in the manded in these and many others applications. Interval case of observable and detectable systems (with respect or set-membership estimation is a promising framework to [13] the present work contains new result, Theorem to observation in uncertain systems [17,19,20,24,22,27], 12, relaxed Assumption 1, and new examples). Efficiency when all uncertainty is included in the corresponding in- of the obtained interval observers is demonstrated on a tervals or polytopes, and as a result the set of admissi- benchmark example from [23] and a delayed nonlinear ble values (an interval) for the state is provided at each pendulum. instant of time. The paper is organized as follows. Some preliminaries In this work an interval observer for time-delay systems and notation are given in Section 2. The delay-dependent with delayed measurements is proposed. A peculiarity positivity conditions are presented in Section 3. The in- terval observer design is performed for a class of linear ? This paper was not presented at any IFAC meeting. Cor- time-delay systems (or a class of nonlinear systems in responding author is D. Efimov. the output canonical form) with delayed measurements Preprint submitted to Automatica May 9, 2016 in Section 4. Examples of numerical simulation are pre- Under conditions of the above lemma the system has n sented in Section 5. bounded solutions for b 2 L1. Note that for linear time- invariant systems the conditions of positive invariance of polyhedral sets have been similarly given in [8], as well as 2 Preliminaries conditions of asymptotic stability in the nonlinear case have been considered in [6,7,3]. 2.1 Notation 3 Delay-dependent conditions of positivity • R is the Euclidean space (R+ = fτ 2 R : τ ≥ 0g), n n Cτ = C([−τ; 0]; R ) is the set of continuous maps from n n n [−τ; 0] into R for n ≥ 1; Cτ+ = fy 2 Cτ : y(s) 2 Consider a scalar time-varying linear system with time- n R+; s 2 [−τ; 0]g; varying delays [1]: n • xt is an element of Cτ defined as xt(s) = x(t + s) for all s 2 [−τ; 0]; x_(t) = a0(t)x[g(t)] − a1(t)x[h(t)] + b(t); (2) •j xj denotes the absolute value of x 2 R, jjxjj2 is x(θ) = 0 for θ < 0; x(0) 2 R; (3) the Euclidean norm of a vector x 2 Rn, jj'jj = n supt2[−τ;0] jj'(t)jj2 for ' 2 Cτ ; where a0 2 L1, a1 2 L1, b 2 L1, h(t) − t 2 L1, • for a measurable and locally essentially bounded in- g(t) − t 2 L1 and h(t) ≤ t, g(t) ≤ t for all t ≥ 0. For p the system (2) the initial condition in (3) is, in general, put u : R+ ! R the symbol jjujj[t0;t1) denotes its not a continuous function (if x(0) 6= 0). L1 norm jjujj[t0;t1) = ess supfjju(t)jj2; t 2 [t0; t1)g, the set of all such inputs u 2 Rp with the property p The following result proposes delay-independent posi- jjujj[0;+1) < 1 will be denoted as L1; n×n tivity conditions. • for a matrix A 2 R the vector of its eigenvalues is denoted as λ(A); Lemma 2 [1] (Corollary 15.7) Let h(t) ≤ g(t) and 0 ≤ • In and 0n×m denote the identity and zero matrices of dimensions n × n and n × m, respectively; a1(t) ≤ a0(t) for all t ≥ 0. If x(0) ≥ 0 and b(t) ≥ 0 • a R b corresponds to an elementwise relation R 2 f< for all t ≥ 0, then the corresponding solution of (2),(3) ; >; ≤; ≥} (a and b are vectors or matrices): for exam- x(t) ≥ 0 for all t ≥ 0. ple a < b (vectors) means 8i : ai < bi; for φ, ' 2 Cτ the relation φ R ' has to be understood elementwise Recall that in this case positivity is guaranteed for \dis- for whole domain of definition of the functions, i.e. continuous" initial conditions. The peculiarity of the φ(s) R '(s) for all s 2 [−τ; 0]; condition 0 ≤ a1(t) ≤ a0(t) is that it may correspond to • for a symmetric matrix Υ, the relation Υ ≺ 0 (Υ 0) an unstable system (2). In order to overcome this issue, means that the matrix is negative (semi) definite. delay-dependent conditions can be introduced. Lemma 3 [1] (Corollary 15.9) Let h(t) ≤ g(t) and 0 ≤ 2.2 Delay-independent conditions of positivity 1 e a0(t) ≤ a1(t) for all t ≥ 0 with Consider a time-invariant linear system with time- Z t 1 1 varying delay: sup [a1(ξ) − a0(ξ)]dξ < ; t2R+ h(t) e e x_(t) = A0x(t) − A1x(t − τ(t)) + b(t); t ≥ 0; (1) x(θ) = φ(θ) for − τ ≤ θ ≤ 0; φ 2 Cn; where e = exp(1). If x(0) ≥ 0 and b(t) ≥ 0 for all t ≥ 0, τ then x(t) ≥ 0 for all t ≥ 0 in (2), (3). n n where x(t) 2 R , xt 2 Cτ is the state function; τ : R+ ! [−τ; 0] is the time-varying delay, a Lebesgue measurable These lemmas describe positivity conditions for the sys- n tem (2), (3), which is more complex than (1), but scalar, function of time, τ 2 R+ is the maximum delay; b 2 L1 is the input; the constant matrices A and A have ap- they can also be extended to the n-dimensional system 0 1 (1). propriate dimensions. The matrix A0 is called Metzler if all its off-diagonal elements are nonnegative. The sys- Corollary 4 The system (1) with b(t) ≥ 0 for all t ≥ 0 tem (1) is called positive if for x ≥ 0 it has the corre- 0 and initial conditions sponding solution x(t) ≥ 0 for all t ≥ 0. n x(θ) = 0 for − τ ≤ θ < 0; x(0) 2 R+; Lemma 1 [18,2] The system (1) is positive iff A0 is Met- zler, A1 ≤ 0 and b(t) ≥ 0 for all t ≥ 0. A positive system is positive if −A1 is Metzler, A0 ≥ 0, and (1) is asymptotically stable for b(t) ≡ 0 for all τ 2 R+ n iff there are p; q 2 R+(p > 0 and q > 0) such that −1 0 ≤ (A0)i;i ≤ e(A1)i;i < (A0)i;i + τ T T p [A0 − A1] + q = 0: for all i = 1; : : : ; n.