Adaptive stabilization by delay with biased measurements Denis Efimov, Stanislav Aranovskiy, Emilia Fridman, Dmitry Sokolov, Jian Wang, Alexey Bobtsov To cite this version: Denis Efimov, Stanislav Aranovskiy, Emilia Fridman, Dmitry Sokolov, Jian Wang, et al.. Adaptive stabilization by delay with biased measurements. IFAC 2020 - 21st IFAC World Congress, Jul 2020, Berlin, Germany. hal-02634582 HAL Id: hal-02634582 https://hal.inria.fr/hal-02634582 Submitted on 27 May 2020 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. Adaptive stabilization by delay with biased measurements Denis Efimov ∗ Stanislav Aranovskiy ∗∗ Emilia Fridman ∗∗∗ Dimitry Sokolov ∗∗∗∗ Jian Wang y Alexey A. Bobtsov z ∗ Inria, Univ. Lille, CNRS, UMR 9189 - CRIStAL, F-59000 Lille, France ∗∗ CentaleSup´elec {IETR, Avenue de la Boulaie, 35576 Cesson-S´evign´e,France ∗∗∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel ∗∗∗∗ Universit´ede Lorraine, Inria, CNRS, LORIA, F-54000 Nancy, France y Hangzhou Dianzi University, Hangzhou, China z ITMO University, 197101 Saint Petersburg, Russia Abstract: The problem of output robust adaptive stabilization for a class of Lipschitz nonlinear systems is studied under assumption that the measurements are available with a constant bias. The state reconstruction is avoided by using delayed values of the output in the feedback and adaptation laws. The control and adaptation gains can be selected as a solution of the proposed linear matrix inequalities (LMIs). The efficiency of the presented approach is demonstrated on a nonlinear pendulum through simulations. 1. INTRODUCTION adaptation algorithm. The closed-loop system becomes time-delayed, then stability analysis of the regulation error Design of identification algorithms, estimators and regula- is based on the Lyapunov-Krasovskii functional proposed tors for dynamical systems are fundamental and complex in (Fridman and Shaikhet, 2016, 2017). problems studied in the control theory. In many cases, due It is important to note that there exist papers devoted to information transmission in the input-output channels, to adaptive control of time-delay systems as, for example, delays appear in the dynamics of the controlled plant. (Mirkin and Gutman, 2010; Pepe, 2004) (the uncertain Influence of a delay on the system stability is vital in parameters appear in the state equation only), or papers many cases (Gu et al., 2003; Fridman, 2014), and it usually dealt with adaptive control for systems with (multiplica- leads to degradation of the performances of regulation tive) uncertain parameters in the output equation (Zhang or estimation (Fridman, 2014). However, in some cases and Lin, 2019a,b) (without presence of time delays), but introduction of a delay may result in an improvement of to the best of our knowledge there is no theory on the the system transients (see (Fridman and Shaikhet, 2016, intersection of these approaches. This work and the con- 2017; Efimov et al., 2018) and the references therein). The sidered problem statement is motivated by a pendulum idea of these papers is that unmeasured components of control application, which is finally used for illustration. the state can be calculated using delayed values of the measured variables, which allows a design of observer to The outline of this work is as follows. The preliminaries be passed by. are given in Section 2. The problem statement and the adaptive control design are presented in sections 3 and 4, The goal of this note is to extend the results obtained respectively. A nonlinear pendulum application is consid- in (Fridman and Shaikhet, 2016, 2017) for linear systems ered in Section 5. to adaptive stabilization of a class of nonlinear systems, which contain a globally Lipschitz nonlinearity, and have a part of the measurements available with a constant bias, which is induced by a sensor error. Since for embedded con- 2. PRELIMINARIES trol and estimation solutions, the amount of computations needed for realization is a critical resource (less important Denote by R the set of real numbers and R+ = fs 2 R : than the used memory in some scenarios), in this note we s ≥ 0g. avoid to design an (reduced order) observer for the state, but introduce delayed measurements in the feedback and For a Lebesgue measurable function of time d :[a; b] ! Rm, where −∞ ≤ a < b ≤ +1, define the norm ? This work was partially supported by 111 project No. D17019 kdk[a;b) = ess supt2[a;b)jd(t)j, where j · j is the standard (China), by the Government of Russian Federation (Grant 08- Euclidean norm in m, then jjdjj = kdk and the 08), by the Ministry of Science and Higher Education of Russian R 1 [0;+1) Federation, passport of goszadanie no. 2019-0898 and by Israel space of d with kdk[a;b) < +1 (jjdjj1 < +1) we further m m Science Foundation (grant No 673/19). denote as L[a;b] (L1). n m Denote by C[a;b], a; b 2 R the Banach space of contin- and d 2 L1 there exist q ≥ 0, β 2 KL and γ 2 K such uous functions φ :[a; b] ! Rn with the uniform norm that 1;1 kφk[a;b] = supa≤s≤b jφ(s)j; and by W[a;b] the Sobolev jx(t; x0; d)j ≤ β(kx0kW; t) + γ(jjdjj1) + q 8t ≥ 0: space of absolutely continuous functions φ :[a; b] ! Rn If q = 0 then (1) is called ISS. _ with the norm kφk = kφk[a;b) + jjφjj[a;b) < +1, where W For establishment of this stability property, the Lyapunov- φ_(s) = @φ(s) is a Lebesgue measurable essentially bounded @s Krasovskii theory can be used (Pepe and Jiang, 2006; _ n function, i.e. φ 2 L[a;b]. Fridman et al., 2008; Efimov et al., 2018). Definition 2. A locally Lipschitz continuous functional V : A continuous function σ : R+ ! R+ belongs to class K if 1;1 n _ it is strictly increasing and σ(0) = 0; it belongs to class R+ × W[−τ;0] × L[−τ;0] ! R+ (i.e., V (t; φ, φ)) is called K1 if it is also radially unbounded. A continuous function simple if D+V (t; φ, d) is independent on φ¨. β : R+ × R+ ! R+ belongs to class KL if β(·; r) 2 K and β(r; ·) is decreasing to zero for any fixed r > 0. For instance, a locally Lipschitz functional V : R+ × 1;1 The symbol 1; m is used to denote a sequence of integers W[−τ;0] ! R+ is simple, another example of a simple 1; :::; m. For a symmetric matrix P 2 Rn×n, the minimum functional is given in Theorem 6 below. and the maximum eigenvalues are denoted as λmin(P ) and Definition 3. A locally Lipschitz continuous functional V : λ (P ), respectively. For a matrix A 2 n×m, jAj = 1;1 n max R R+ × W[−τ;0] × L[−τ;0] ! R+ is called practical ISS p > λmax(A A) is the induced norm. The identity matrix of Lyapunov-Krasovskii functional for the system (1) if it is dimension n × n is denoted by In. simple and there exist r ≥ 0, α1; α2 2 K1 and α; χ 2 K 1;1 m such that for all t 2 R+, φ 2 W[−τ;0] and d 2 R : 2.1 Neutral time-delay systems _ α1(jφ(0)j) ≤ V (t; φ, φ) ≤ α2(kφkW); _ + _ Consider an autonomous functional differential equation of V (t; φ, φ) ≥ maxfr; χ(jdj)g =) D V (t; φ, d) ≤ −α(V (t; φ, φ)): neutral type with inputs (Kolmanovsky and Nosov, 1986): If r = 0 then V is an ISS Lyapunov-Krasovskii functional. x_(t) = f(xt; x_ t; d(t)) (1) Theorem 4. (Fridman et al., 2008) If there exists a (prac- n 1;1 tical) ISS Lyapunov-Krasovskii functional for the system for almost all t ≥ 0, where x(t) 2 R and xt 2 W[−τ;0] −1 (1), then it is (practical) ISS with γ = α1 ◦ χ. is the state function, xt(s) = x(t + s); −τ ≤ s ≤ 0, with n m m x_ t 2 L[−τ;0]; d(t) 2 R is the external input, d 2 L1; Converse results for Theorem 4 can be found in (Pepe 1;1 n m n et al., 2017; Efimov and Fridman, 2019). f : W[−τ;0] × L[−τ;0] × R ! R is a continuous function, that is globally Lipschitz in the second variable with a constant smaller than 1, ensuring forward uniqueness 3. ROBUST OUTPUT ADAPTIVE REGULATION and existence of the system solutions (Kolmanovsky and WITH BIASED MEASUREMENTS Nosov, 1986). We assume f(0; 0; 0) = 0. For the initial 1;1 m function x0 2 W[−τ;0] and disturbance d 2 L1 denote Consider a nonlinear system for the time t ≥ 0: a unique solution of the system (1) by x(t; x0; d), which is an absolutely continuous function of time defined on x_ 1(t) = x2(t); some maximal interval [−τ; T ) for T > 0, then xt(x0; d) 2 1;1 x_ 2(t) = A21x1(t) + A22x2(t) + A23x3(t) W[−τ;0] represents the corresponding state function with +B1(u(t) + Ω(t)θ2) + L1φ(x(t)) + d1(t); xt(s; x0; d) = x(t + s; x0; d) for −τ ≤ s ≤ 0.