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1. Conventional and high order sliding mode control

Author: Yury Orlov, Andrey Polyakov, Alexander S. Poznyak, and Vadim I. Utkin
Subjects: 0209 industrial biotechnology, Computer Networks and Communications, Computer science, Applied Mathematics, Scalar (physics), Mode (statistics), Equations of motion, 02 engineering and technology, Sliding mode control, Manifold, Term (time), 020901 industrial engineering & automation, Dimension (vector space), Control and Systems Engineering, Control theory, Signal Processing, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Control (linguistics)
Abstract: Term “Conventional” sliding mode control (SMC) was introduced in the book Sliding Mode Control and Observation [1] by the authors, working in the area of high order sliding mode (HOSM) control. The term is related to all publications on n-dimensional systems with m- dimensional control and with sliding modes and state trajectories in a manifold of order n-m. Most of publications on HOSM control studied a new phenomenon for systems with a scalar control (m = 1), specifically, the existence of sliding modes in manifolds of dimension lower than n-1 with a finite reaching time. Along with implementation issues, it was natural to discuss to what extent the main principles of the conventional theory were to be revised (definitions, existence conditions, motion equations), what new properties of systems with HOSM can be expected. Partially these questions were discussed in [2] along with several international workshops on SMC and CDC in 2018. In this paper we demonstrate that main concepts are similar for both types of SMC, discuss new developments needed for HOSM control, compare the potential to suppress chattering, complexity of the both methods.
Published: 2020
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2. Sliding mode control applied to a square-back Ahmed body

Author: Franck Kerhervé, Jean-Pierre Richard, Wafik Abassi, Laurent Keirsbulck, Marc Lippert, Maxime Feingesicht, Baptiste Plumjeau, Camila Chovet, Andrey Polyakov, Jean-Marc Foucaut, Laboratoire d'Automatique, de Mécanique et d'Informatique industrielles et Humaines - UMR 8201 (LAMIH), Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-Centre National de la Recherche Scientifique (CNRS), Chalmers University of Technology [Göteborg], Turbulence Incompressible et Contrôle (TIC ), Département Fluides, Thermique et Combustion (FTC), Institut Pprime (PPRIME), Université de Poitiers-ENSMA-Centre National de la Recherche Scientifique (CNRS)-Université de Poitiers-ENSMA-Centre National de la Recherche Scientifique (CNRS)-Institut Pprime (PPRIME), Université de Poitiers-ENSMA-Centre National de la Recherche Scientifique (CNRS)-Université de Poitiers-ENSMA-Centre National de la Recherche Scientifique (CNRS), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Ecole Centrale de Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Ecole Centrale de Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mécanique des Fluides de Lille – Kampé de Fériet - UMR 9014 (LMFL), Ecole Centrale de Lille-ONERA-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Arts et Métiers Sciences et Technologies, HESAM Université (HESAM)-HESAM Université (HESAM), Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-Centre National de la Recherche Scientifique (CNRS)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France), Université de Poitiers-ENSMA-Centre National de la Recherche Scientifique (CNRS), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Centrale Lille, Laboratoire de Mécanique de Lille - FRE 3723 (LML), Université de Lille, Sciences et Technologies-Centrale Lille-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Automatique, de Mécanique et d'Informatique industrielles et Humaines - UMR 8201 [LAMIH], Turbulence Incompressible et Contrôle [TIC ], Finite-time control and estimation for distributed systems [VALSE], Laboratoire de Mécanique des Fluides de Lille – Kampé de Fériet - UMR 9014 [LMFL], Institut Pprime [PPRIME], Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL], and Laboratoire de Mécanique de Lille - FRE 3723 [LML]
Subjects: Computer science, bilinear time-delay model, General Physics and Astronomy, 02 engineering and technology, 01 natural sciences, Sliding mode control, [SPI.AUTO]Engineering Sciences [physics]/Automatic, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010305 fluids & plasmas, External flow, [SPI]Engineering Sciences [physics], flow control, sliding mode control, drag reduction unsteadiness of separated flows, 0203 mechanical engineering, Control theory, 0103 physical sciences, Aerodynamic drag, Trailing edge, ComputingMilieux_MISCELLANEOUS, Mathematical Physics, Mechanics, Lift (force), Flow control (fluid), 020303 mechanical engineering & transports, Particle image velocimetry, Drag
Abstract: International audience; We consider the first closed-loop separation control experiment on an Ahmed body using a robust, model-based strategy called "sliding mode control" (SMC). The goal is to reduce and further maintain the aerodynamic drag of the square-back Ahmed body flow at Re h = 9 × 10 4 (based on the body height). This study also investigates the practical feasibility of this approach which shows a great promise for industrial applications. The flow is manipulated by a slotted jet placed on the top trailing edge, combined with a predefined angle direction, and sensed by a drag balance. Base pressure and lift measurements are also obtained in real-time. The interaction between the air jet actuator and the mean near-wake flow are depicted by means of Particle Image Velocimetry. In order to compare this closed-loop strategy we first present two open-loop ones. Continuous blowing is initially used to directly influence the recirculation area and hence achieve a reduction in the drag. Approximately a drag reduction of 8% is accomplished making this approach the "optimal case" control strategy. However, because steady blowing mechanisms lead to the highest energy consumption scheme, this strategy will only serve as the first open-loop control reference. The second open-loop strategy involves three periodic forcing frequencies, St A = 0.0765, 0.135 & 0.405. The influence of these frequencies on the near-wake, and its further drag modification, will be further examined. The proposed sliding mode control (closed-loop strategy) is applied to the same Ahmed body configuration and compared to the open-loop cases. It will be designed on the basis of a simplified input-output model which was recently defined for another flow control application. Last, a second experiment is conducted so to show the disturbance rejection of the controller, corroborating the robustness and efficiency of this control approach. Its limitations and difficulties on an experimental setup are also discussed. SMC is able to reduce and maintain the drag to a desire set-point regardless external flow perturbations. Our control strategy was recently used in another context (flow reattachement on a wing) with the same robustness. These successes make guess it is applicable to multiple experimental and industrial contexts.
Published: 2020
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3. A Switching Controller for a Class of MIMO Bilinear Systems With Time Delay

Author: Emilia Fridman, Tonametl Sanchez, Andrey Polyakov, Laurentiu Hetel, Universidad Nacional Autónoma de México (UNAM), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Department of Electrical Engineering, Tel Aviv University [Tel Aviv], Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM), Tel Aviv University (TAU), and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: 0209 industrial biotechnology, Computer science, switched control, MIMO, 02 engineering and technology, State (functional analysis), Stability (probability), Computer Science Applications, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, bilinear systems, Trajectory, Electrical and Electronic Engineering, Constant (mathematics), Finite set, time-delay systems
Abstract: In this paper, we propose a state-dependent switching controller for multiple-input multiple-output (MIMO) bilinear systems with constant delays in both the state and the input. The control input is assumed to be restricted to take only a finite number of values. The stability analysis of the closed loop is based on a Lyapunov–Krasovskii functional, and the design is reduced to solve a system of linear matrix inequalities. The controller can be designed by considering (state) delay-dependent or delay-independent conditions.
Published: 2020
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4. State observation in microbial consortia: A case study on a synthetic producer‐cleaner consortium

Author: Alex dos Reis de Souza, Denis Efimov, Andrey Polyakov, Jean‐Luc Gouzé, Eugenio Cinquemani, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Université de Lille-Centrale Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centrale Lille-Centre National de la Recherche Scientifique (CNRS), Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Université Côte d'Azur (UCA), Analyse, ingénierie et contrôle des micro-organismes (MICROCOSME), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Grenoble Alpes (UGA), Institut National de Recherche en Informatique et en Automatique (Inria), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'océanographie de Villefranche (LOV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Modeling, simulation, measurement, and control of bacterial regulatory networks (IBIS), Laboratoire Adaptation et pathogénie des micro-organismes [Grenoble] (LAPM), Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Jean Roget, and ANR-20-CE45-0014,Ctrl-AB,Optimisation et controle de la productivité d'un écosystème algues-bactéries(2020)
Subjects: 0303 health sciences, 0209 industrial biotechnology, observability, Mechanical Engineering, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, bioprocess, 02 engineering and technology, Industrial and Manufacturing Engineering, 03 medical and health sciences, [SPI]Engineering Sciences [physics], 020901 industrial engineering & automation, Control and Systems Engineering, microbial consortia, state estimation, Electrical and Electronic Engineering, ComputingMilieux_MISCELLANEOUS, 030304 developmental biology
Abstract: International audience; In this article, the problem of state estimation of a microbial consortium is discussed. This microbial consortium is composed of two strains, called producer and cleaner, in continuous culture. The producer strain grows by consuming glucose, while the cleaner grows by consuming the acetate excreted by the producer and toxifies the environment. Considering two available measurements, the total biomass and a fluorescent reporter, an observability analysis is cast and state estimators are proposed for each state component of this system. Although the system is shown to be detectable using solely the total biomass as measurement, the advantage of measuring the fluorescent reporter on the observer design is highlighted. A numerical experiment illustrates the methodology.
Published: 2021
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5. Generalized Homogenization of Linear Observers: Theory and Experiment

Author: Gang Zheng, Siyuan Wang, Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Deformable Robots Simulation Team (DEFROST ), and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: 0209 industrial biotechnology, Mechanical Engineering, General Chemical Engineering, 010102 general mathematics, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, 01 natural sciences, Industrial and Manufacturing Engineering, 020901 industrial engineering & automation, Control and Systems Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0101 mathematics, Electrical and Electronic Engineering
Abstract: International audience; A procedure of upgrading a linear observer to homogeneous (nonlinear) one is proposed and validated by experiment. The nonlinear observer design is based on the the concept of a linear geometric homogeneity. The simple procedure developed to apply the homogeneous nonlinear observer is based on the parameters provided by the linear observer, which is potential for many applications to improve their performance. A saturation function is introduced to guaranteed the improvement of estimation quality. Finally the theoretical results are confirmed by the QDrone platform of Quanser T M .
Published: 2021
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6. Robust Stabilization of Control Affine Systems with Homogeneous Functions

Author: Denis Efimov, Andrey Polyakov, and Konstantin Zimenko
Subjects: 0209 industrial biotechnology, Class (set theory), Homogeneity (statistics), 020208 electrical & electronic engineering, Homogeneous function, 02 engineering and technology, Linear matrix, Nonlinear control, symbols.namesake, 020901 industrial engineering & automation, Control and Systems Engineering, Control system, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Affine transformation, Control (linguistics), Mathematics
Abstract: The stabilization problem of the affine control system ẋ = f0(x)+∑i=1muifi(x) with homogeneous functions f0, fi is studied. This class of systems is of interest due to the robust properties of homogeneity and the fact that many affine systems can be approximated by or transformed to the class under consideration. An advantage of the introduced design method is that the tuning rules are presented in the form of linear matrix inequalities. Performance of the approach is illustrated by a numerical example.
Published: 2020
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7. On finite-time stability of sub-homogeneous differential inclusions

Author: Andrey Polyakov, Wilfrid Perruquetti, Youness Braidiz, and Denis Efimov
Subjects: 0209 industrial biotechnology, Observer (quantum physics), 020208 electrical & electronic engineering, Mathematical analysis, Ode, 02 engineering and technology, Extension (predicate logic), Stability (probability), Section (fiber bundle), Nonlinear system, 020901 industrial engineering & automation, Differential inclusion, Control and Systems Engineering, Homogeneity (physics), 0202 electrical engineering, electronic engineering, information engineering, Mathematics
Abstract: Sub-homogeneity property is introduced and is related to a differential inclusion (DI). It is shown that a nonlinear ordinary differential equation (ODE), which may not admit a homogeneous approximation, can be transformed into a sub-homogeneous DI (which is a homogeneous extension of the original ODE). Using this homogeneous extension, one can directly recover finite-time stability property for some particular classes of nonlinear systems. In the last section, such a sub homogeneity property is used to design a nonlinear finite-time observer.
Published: 2020
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8. On fixed-time stability of a class of nonlinear time-varying systems

Author: Youness Braidiz, Andrey Polyakov, Denis Efimov, and Wilfrid Perruquetti
Subjects: Lyapunov function, 0209 industrial biotechnology, Class (set theory), Homogeneity (statistics), 020208 electrical & electronic engineering, 02 engineering and technology, Stability (probability), Section (fiber bundle), symbols.namesake, Nonlinear system, Stability conditions, 020901 industrial engineering & automation, Control and Systems Engineering, Fixed time, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Mathematics
Abstract: Fixed-time stability of a class of nonlinear time-varying systems is investigated using the introduced notions of sub- and sup-homogeneity. These concepts allow the systems (even if they do not admit homogeneous approximations) to be analyzed using the homogeneity of their extensions. Then, finite-time and fixed-time stability properties can be recovered. The proposed stability conditions are not based on Lyapunov arguments. In the last section, an example illustrates the obtained results.
Published: 2020
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9. Adaptive Discontinuous Control for Homogeneous Systems Approximated by Neural Networks

Author: Andrey Polyakov, Mariana Ballesteros, Alexander S. Poznyak, Isaac Chairez, Denis Efimov, Centro de Investigacion y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Instituto Politechnico National (IPN), and Instituto Politecnico Nacional [Mexico] (IPN)
Subjects: Lyapunov stability, 0209 industrial biotechnology, Adaptive control, Neural Networks, Computer simulation, Artificial neural network, Computer science, Homogeneity (statistics), Computer Science::Neural and Evolutionary Computation, 020208 electrical & electronic engineering, System Identification, System identification, Homogeneous function, 02 engineering and technology, Discontinuous Control, Adaptive Control, Nonlinear system, symbols.namesake, 020901 industrial engineering & automation, Homogeneous systems, Control and Systems Engineering, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols
Abstract: International audience; This study is devoted to the design of an adaptive discontinuous control based on differential neural networks (DNNs) for a class of uncertain homogeneous systems. The control is based on the universal approximation properties of artificial neural networks (ANNs) applied on a certain class of homogeneous nonlinear functions. The adaptation laws for the DNNs parameters are obtained with the application of the Lyapunov stability theory and the homogeneity properties of the approximated nonlinear system. The stability analysis of the closed loop system with the proposed controller is presented. The estimation error in the approximation of the uncertain homogeneous functions is considered in the stability analysis. The performance of the controller is illustrated by means of a numerical simulation of a homogeneous model.
Published: 2020
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10. Homogeneous Observer Design for Linear MIMO Systems

Author: Denis Efimov, Artem Kremlev, Andrey Polyakov, and Konstantin Zimenko
Subjects: 0209 industrial biotechnology, Observer (quantum physics), Degree (graph theory), Homogeneity (statistics), 020208 electrical & electronic engineering, 02 engineering and technology, State (functional analysis), Stability (probability), 020901 industrial engineering & automation, Compact space, Control and Systems Engineering, Control theory, Robustness (computer science), 0202 electrical engineering, electronic engineering, information engineering, Canonical form, Mathematics
Abstract: The paper is devoted to the problem of state observation (particularly, in finite time) of linear MIMO systems. The presented nonlinear observer does not require system transformation to a canonical form and guarantees finite-time (asymptotic with a fixed-time attraction of any compact set containing the origin) stability of observation error equation if homogeneity degree is negative (positive). The proposed observer is robust in input-to-state sense with respect to disturbances and measurement noises. Performance of the observer is illustrated by a numerical example.
Published: 2020
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11. A Consistent Discretisation method for Stable Homogeneous Systems based on Lyapunov Function

Author: Andrey Polyakov, Denis Efimov, and Tonametl Sanchez
Subjects: Lyapunov function, Surface (mathematics), 0209 industrial biotechnology, Discretization, 020208 electrical & electronic engineering, 02 engineering and technology, symbols.namesake, Nonlinear system, 020901 industrial engineering & automation, Rate of convergence, Control and Systems Engineering, Homogeneous, Stability theory, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Mathematics
Abstract: In this paper we propose a discretisation scheme for continuous and asymptotically stable homogeneous systems. This method is based on the dynamics of the system projected on a level surface of a homogeneous Lyapunov function. The discretisation method is explicit and preserves the convergence rate of the continuous-time system.
Published: 2020
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12. Observer-Based Robust Control of a Continuous Bioreactor with Heterogeneous Community

Author: Jean-Luc Gouzé, Alex dos Reis de Souza, Denis Efimov, Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)
Subjects: 0209 industrial biotechnology, Observer (quantum physics), Computer science, 020208 electrical & electronic engineering, Control (management), Robust control, Bioreactor, 02 engineering and technology, Limiting, [SPI.AUTO]Engineering Sciences [physics]/Automatic, 020901 industrial engineering & automation, Biochemical process, Control and Systems Engineering, Control theory, 0202 electrical engineering, electronic engineering, information engineering, Observer based, Observer-based control
Abstract: International audience; This work addresses the problem of robust control of the continuous bioreactor-namely the chemostat-using an observer-based approach. We design a control law aiming to regulate the growth of a certain species and to stabilize the output-a growth-associated product of interest-in a prescribed level. In addition, to avoid the single-species dogma (which takes only one species into account for control design), we add to the microbial community unknown species competing for the limiting substrate. Finally, this control architecture is able to maintain the productivity, despite this competing heterogeneity (or biodiversity) of the community.
Published: 2020
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13. METHOD OF FORMING NOMENCLATURE AND NUMBER OF SPARE PARTS FOR MAINTENANCE AND REPAIR OF SPECIAL MACHINES OF MILITARY PURPOSE

Author: Andrey Polyakov
Subjects: 020303 mechanical engineering & transports, 0203 mechanical engineering, Computer science, Spare part, 021105 building & construction, 0211 other engineering and technologies, 02 engineering and technology, Hardware_REGISTER-TRANSFER-LEVELIMPLEMENTATION, Nomenclature, Manufacturing engineering
Abstract: The system for the maintenance and repair of special military vehicles (SMV) is a complex system. The study of such systems is based primarily on system analysis and synthesis. The complexity of the study of the system of maintenance and repair of the SMV is due to the weak formalization of the processes and data on the impact of the operating time and duration of the period of time the SMV is in operation on the likelihood of their failure-free operation. To improve the efficiency of the functioning of the existing system of maintenance and repair of the SMV, a method for the formation of the nomenclature and the number of spare parts has been proposed.
Published: 2019
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14. A homogeneity property of discrete‐time systems: Stability and convergence rates

Author: Denis Efimov, Wilfrid Perruquetti, Tonametl Sanchez, Andrey Polyakov, and Jaime A. Moreno
Subjects: Lyapunov function, 0209 industrial biotechnology, Mechanical Engineering, General Chemical Engineering, Homogeneity (statistics), Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, Stability (probability), Industrial and Manufacturing Engineering, Nonlinear system, symbols.namesake, 020901 industrial engineering & automation, Discrete time and continuous time, Control and Systems Engineering, Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Dilation (morphology), 020201 artificial intelligence & image processing, Electrical and Electronic Engineering, Scaling, Mathematics
Abstract: A new definition of homogeneity for discrete-time systems is introduced. As in the continuous-time case, the property can be verified algebraically in the transition map of the system, and it implies that a dilation of the initial conditions leads to a scaling of the trajectory. Stability properties and convergence rates of the system's solutions can be established by considering only the homogeneity degree. The existence of homogeneous Lyapunov and Lyapunov-like functions is proven.
Published: 2019
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15. Homogeneous Observers for Projected Quadratic Partial Differential Equations

Author: Andrey Polyakov, Sergiy Zhuk, IBM Research - Ireland, IBM, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), and Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)
Subjects: 0209 industrial biotechnology, Partial differential equation, Discretization, Observer (quantum physics), Dynamical systems theory, Differential equation, 010102 general mathematics, 02 engineering and technology, 16. Peace & justice, 01 natural sciences, 020901 industrial engineering & automation, Transformation matrix, Quadratic equation, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Symmetric matrix, Applied mathematics, 0101 mathematics, Mathematics
Abstract: International audience; The paper proposes a new homogeneous observer for finite-dimensional projections of quadratic homogeneous hyperbolic PDEs with compact state space. The design relies upon new sufficient conditions for fixed-time convergence of observer's gain, described as a solution of a non-linear homogeneous matrix differential equations, towards an ellipsoid in the space of symmetric non-negative matrices. Convergence of the observer is analyzed, and a numerical convergence test is proposed: numerical experiments confirm the test on ODEs obtained by finite-difference discretization of Burgers-Hopf equation.
Published: 2020
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16. On finite-time stabilization of a class of nonlinear time-delay systems: Implicit Lyapunov-Razumikhin approach

Author: Wilfrid Perruquetti, Artem N. Nekhoroshikh, Denis Efimov, Andrey Polyakov, Igor B. Furtat, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), National Research University of Information Technologies, Mechanics and Optics [St. Petersburg] (ITMO), Centrale Lille, Institute of Mechanical Engineering Problems [St. Petersburg] (IPME), Russian Academy of Sciences [Moscow] (RAS), This work has been partially supported by ANR Project Finite4SoS (ANR 15-CE23-0007), by the Government of Russian Federation (Grant 08-08), by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898 and by the RFBR research project no. 19-08-00246., and ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015)
Subjects: Lyapunov function, 0209 industrial biotechnology, 020208 electrical & electronic engineering, State vector, 02 engineering and technology, Function (mathematics), 16. Peace & justice, Exponential function, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, Nonlinear system, 020901 industrial engineering & automation, Exponential stability, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, Symmetric matrix, Numerical stability, Mathematics
Abstract: International audience; Theorems on Implicit Lyapunov-Razumikhin functions (ILRF) for asymptotic, exponential, finite-time and nearly fixed-time stability analysis of nonlinear time-delay systems are presented. Based on these results, finite-time stabilization of a special class of such systems is addressed. These systems are represented by a chain of integrators with a time-delay term multiplied by a function of instantaneous state vector. Possible explicit restriction on nonlinear time-delay terms is discussed. Simple procedure of control parameters calculation is given in terms of linear matrix inequalities (LMIs). Some aspects of digital implementations of the presented nonlinear control law are touched upon. Theoretical results are illustrated by numerical simulations.
Published: 2020

17. Lyapunov-based Consistent Discretisation of Stable Homogeneous Systems

Author: Andrey Polyakov, Denis Efimov, Tonametl Sanchez, Instituto Potosino de Investigacion Cientifica y Tecnologica (IPICYT), Consejo Nacional de Ciencia y Tecnología [Mexico] (CONACYT), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), This work was supported by the Agence Nationale de la Recherche project ANR-18-CE40-0008 'DIGITSLID.', and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: Lyapunov function, 0209 industrial biotechnology, Discretization, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, homogeneous systems, 01 natural sciences, Industrial and Manufacturing Engineering, symbols.namesake, 020901 industrial engineering & automation, Exponential stability, Stability theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Nonlinear systems, Applied mathematics, 0101 mathematics, Electrical and Electronic Engineering, Mathematics, Lyapunov-based methods, Mechanical Engineering, 010102 general mathematics, Rate of convergence, Control and Systems Engineering, Homogeneous, Scheme (mathematics), discrete-time systems, symbols
Abstract: International audience; In this paper we propose a discretisation scheme for asymptotically stable homogeneous systems. This scheme exploits the information provided by a homogeneous Lyapunov function of the system. The main features of the scheme are: 1) the dis-cretisation method is explicit and; 2) the discrete-time system preserves the asymptotic stability, the convergence rate, and the Lyapunov function of the original continuous-time system.
Published: 2020
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18. Robust Adaptive Estimation in the Competitive Chemostat

Author: Andrey Polyakov, Denis Efimov, Jean-Luc Gouzé, Alex dos Reis de Souza, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Université Côte d'Azur (UCA), This work was supported by IPL COSY., and ANR-15-IDEX-0001,UCA JEDI,Idex UCA JEDI(2015)
Subjects: chemostat, Observer (quantum physics), Estimation theory, Computer science, 020209 energy, General Chemical Engineering, adaptive, State vector, 02 engineering and technology, State (functional analysis), Chemostat, Computer Science Applications, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Competition model, monitoring, 020401 chemical engineering, Coupling (computer programming), Control theory, 0202 electrical engineering, electronic engineering, information engineering, Bioreactor, state estimation, 0204 chemical engineering
Abstract: International audience; In this paper, the problem of state estimation of a bioreactor containing a single substrate and several competing species is studied. This scenario is well-known as the competition model, in which multiple species compete for a single limiting nutrient. Considering the total biomass to be the only available measurement, the challenge is to estimate the concentration of the whole state vector. To achieve this goal, the estimation scheme is built by the coupling of two estimation techniques: an asymptotic observer, which depends solely on the operating conditions of the bioreactor, and a finite-time parameter estimation technique, which drops the usual requirement of the persistence of excitation. The presented methodology achieves the estimation of each competing species and a numerical example illustrates the intended application.
Published: 2020
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19. On robustness of finite-time stability of homogeneous affine nonlinear systems and cascade interconnections

Author: Andrey Polyakov, Wilfrid Perruquetti, Youness Braidiz, Denis Efimov, Centrale Lille, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), National Research University of Information Technologies, Mechanics and Optics [St. Petersburg] (ITMO), This work was partially supported by ANR-15-CE23-0007 (Project Finite4SoS), by the Government of Russian Federation (Grant 08-08) and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898., and ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015)
Subjects: 0209 industrial biotechnology, 02 engineering and technology, robustness, Computer Science Applications, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Nonlinear dynamical systems, Stability conditions, Affine nonlinear system, 020901 industrial engineering & automation, Homogeneous systems, Control and Systems Engineering, Robustness (computer science), Cascade, Homogeneous, 0202 electrical engineering, electronic engineering, information engineering, finite-time stability, Applied mathematics, 020201 artificial intelligence & image processing, Affine transformation, Finite time, Mathematics
Abstract: International audience; This paper investigates robustness of finite-time stability property for a homogeneous nonlinear dynamical system with sufficiently small affine inputs. In addition, robust stability conditions are presented for the systems admitting homogeneous approximations at the origin or at infinity. The effects of additional stable unmodeled dynamics in the input channel on robust stability are investigated. The utility of the obtained results is illustrated via robustness analysis of homogeneous observer with time-varying gains.
Published: 2020
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20. The implicit discretization of the super-twisting sliding-mode control algorithm

Author: Denis Efimov, Andrey Polyakov, Bernard Brogliato, Modélisation, simulation et commande des systèmes dynamiques non lisses (TRIPOP), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), This work was supported by the ANR project Digitslid ANR-18-CE40-0008-01., and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: Lyapunov function, Well-posed problem, Implicit Euler discretization, 0209 industrial biotechnology, Discretization, 02 engineering and technology, Sliding mode control, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, Level set, 020901 industrial engineering & automation, Exponential stability, Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, Electrical and Electronic Engineering, Finite set, Mathematics, Lyapunov stability, 020208 electrical & electronic engineering, sliding mode control, Backward Euler method, super-twisting algorithm, Computer Science Applications, Control and Systems Engineering, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Algorithm, Numerical stability
Abstract: A preliminary version of this article was presented at the 15th Int. Workshop on Variable Structure Systems VSS’2018, Graz, Austria, July 2018.; International audience; This paper deals with the analysis of the time-discretization of the super-twisting algorithm, with an implicit Euler method. It is shown that the discretized system is well-posed. The existence of a Lyapunov function with convex level sets is proved for the continuoustime closed-loop system. Then the global asymptotic Lyapunov stability of the unperturbed discrete-time closedloop system is proved. The convergence to the origin in a finite number of steps is proved also in the unperturbed case. Numerical simulations demonstrate the superiority of the implicit method with respect to an explicit discretization with significant chattering reduction.
Published: 2020
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21. On Homogeneous Lyapunov Function Theorem for Evolution Equations

Author: Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), The author acknowledges the support of ANR DIGISLID project (ANR-18-CE40-0008), the Grant 08-08 of the Government of Russian Federation and the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898., and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: Lyapunov function, 0209 industrial biotechnology, 020208 electrical & electronic engineering, Banach space, Ode, 02 engineering and technology, Extension (predicate logic), homogeneous systems, stability of distributed parameter systems, symbols.namesake, 020901 industrial engineering & automation, Control and Systems Engineering, Homogeneous, Ordinary differential equation, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, semigroup and operator theory, Mathematics
Abstract: International audience; The existence of homogeneous Lyapunov function for a stable homogeneous ordinary differential equation (ODE) is proven by V. Zubov in 1958 and refined by L. Rosier in 1992. The present paper proposes an extension of this result to evolution equations in Banach spaces.
Published: 2020

22. On Generalized Homogenization of Linear Quadrotor Controller

Author: Andrey Polyakov, Siyuan Wang, Gang Zheng, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Deformable Robots Simulation Team (DEFROST )
Subjects: 0209 industrial biotechnology, 020901 industrial engineering & automation, Computer science, Control theory, Robustness (computer science), [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Homogeneity (physics), 0202 electrical engineering, electronic engineering, information engineering, Symmetric matrix, 020201 artificial intelligence & image processing, 02 engineering and technology, Homogenization (chemistry)
Abstract: International audience; A novel scheme for an "upgrade" of a linear control algorithm to a non-linear one is developed based on the concepts of a generalized homogeneity and an implicit homogeneous feedback design. Some tuning rules for a guaranteed improvement of a regulation quality are proposed. Theoretical results are confirmed by real experiments with the quadrotor QDrone of Quanser T M .
Published: 2020

23. Consistent Discretization of Finite-Time and Fixed-Time Stable Systems

Author: Bernard Brogliato, Denis Efimov, Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Modélisation, simulation et commande des systèmes dynamiques non lisses (TRIPOP ), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: Lyapunov stability, 0209 industrial biotechnology, Control and Optimization, Discretization, Applied Mathematics, 010102 general mathematics, 02 engineering and technology, Lipschitz continuity, 01 natural sciences, symbols.namesake, 020901 industrial engineering & automation, Exponential stability, Differential inclusion, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Ordinary differential equation, Bounded function, Euler's formula, symbols, Applied mathematics, 0101 mathematics, Mathematics
Abstract: Algorithms of implicit discretization for generalized homogeneous systems having discontinuity only at the origin are developed. They are based on the transformation of the original system to an equivalent one which admits an implicit or a semi-implicit discretization schemes preserving the stability properties of the continuous-time system. Namely, the discretized model remains finite-time stable (in the case of negative homogeneity degree), and practically fixed-time stable (in the case of positive homogeneity degree). The theoretical results are supported with numerical examples. 1. Introduction. Discretization issues are important for a digital implementation of estimation and control algorithms. Construction of a consistent stable discretization is complex for essentially non-linear ordinary differential equations (ODEs), which do not satisfy some classical regularity assumptions. For example, the sliding mode algorithms are known to be difficult in practical realization [1], [2], [3] due to discontinuous (set-valued) nature, which may invoke chattering caused by the discretization. The mentioned papers have discovered that the implicit discretization technique is useful for practical implementation of non-smooth and discontinuous control and estimation algorithms. In particular, chattering suppression in both input and output, as well as a good closed-loop performance has been confirmed experimentally in [1], [4], [5]. Finite-time stability is a desirable property for many control and estimation algorithms [6], [7], [8], [9], [10], [11]. It means that system trajectories reach a stable equilibrium (or a set) in a finite time, in contrast to asymptotic stability allowing this only for the time tending to infinity. If the settling (reaching) time is globally bounded for all initial conditions then the origin is fixed-time stable (see, e.g. [12]). The corresponding ODE models do not satisfy Lipschitz condition (at least at the origin). In the general case, an application of the conventional implicit or explicit discretization schemes does not guarantee that finite-time or fixed-time stability properties will be preserved (see, e.g. [13], [14], [15]). The latter means that the discrete-time model may be inconsistent with the continuous-time one. However, the discretized systems may remain globally finite-time stable in some cases (see [1], [2], [16], [17]). The aim of this paper is to study systematically the problem of consistent discretization of the so-called generalized homogeneous non-linear systems. The discretized model is consistent if it preserves the stability property (e.g. exponential, finite-time or fixed-time stability) of the original continuous-time system. Homogeneity is a certain form of symmetry studied in systems and control theory [9], [18], [19], [20],[21], [22], [23]. The standard homogeneity (introduced originally by L. Euler in 17th century) is the symmetry of a mathematical object f (e.g. function, vector field, operator, etc) with respect to the uniform dilation of the argument x → λx, namely, f (λx) = λ 1+ν f (x), λ > 0
Published: 2019
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24. Differential Neural Network Identification for Homogeneous Dynamical Systems

Author: Denis Efimov, Alexander S. Poznyak, Mariana Ballesteros, Isaac Chairez, and Andrey Polyakov
Subjects: Lyapunov function, 0209 industrial biotechnology, Artificial neural network, Dynamical systems theory, Computer science, 020208 electrical & electronic engineering, 02 engineering and technology, Sigmoid function, symbols.namesake, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols, Affine transformation, Algorithm
Abstract: In this paper, a non parametric identifier for homogeneous nonlinear systems affine in the input is proposed. The identification algorithm is based on the neural networks using sigmoidal activation functions. The learning algorithm is derived by means of Lyapunov function method and homogeneity theory. A numerical example demonstrates the performance of the proposed identifier.
Published: 2019
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25. Homogeneous Discrete-Time Approximation

Author: Jaime A. Moreno, Tonametl Sanchez, Denis Efimov, and Andrey Polyakov
Subjects: Imagination, 0209 industrial biotechnology, Degree (graph theory), Discretization, media_common.quotation_subject, 020208 electrical & electronic engineering, 02 engineering and technology, Stability (probability), Backward Euler method, Nonlinear system, 020901 industrial engineering & automation, Discrete time and continuous time, Control and Systems Engineering, Homogeneity (physics), 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, Mathematics, media_common
Abstract: In this paper we study some stability properties of discrete-time systems whose transition map can be approximated by a discrete-time homogeneous transition map. This allows us to identify qualitative stability properties of discrete-time systems by only knowing the discrete-time homogeneity degree of its approximation. We show how these results can be applied to the stability analysis of discrete-time systems obtained by means of the explicit and implicit Euler discretization methods.
Published: 2019
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26. Input-to-State Stability of Homogeneous Infinite Dimensional Systems with Locally Lipschitz Nonlinearities

Author: Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), The author acknowledges the support of ANR, France DIGITSLID project (ANR-18-CE40-0008) and the Ministry of Science and Higher Education of Russian Federation , passport of goszadanie no. 2019-0898., and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: 0209 industrial biotechnology, 020208 electrical & electronic engineering, Mathematical analysis, Banach space, 02 engineering and technology, State (functional analysis), Lipschitz continuity, Stability (probability), stability of distributed parameter systems, input-to-state stability, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, Homogeneous, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Evolution equation, Converse, 0202 electrical engineering, electronic engineering, information engineering, Electrical and Electronic Engineering, semigroup and operator theory, Mathematics
Abstract: International audience; The Input-to-State Stability (ISS) of homogeneous evolution equations in Banach spaces with unbounded linear operators and locally Lipschitz nonlinearities in the right-hand sides is studied. A new homogeneous converse Lyapunov theorem is presented. Similarly to finite-dimensional models, it is shown that the uniform asymptotic stability of an unperturbed homogeneous evolution equation implies its ISS with respect to homogeneously involved exogenous inputs.
Published: 2020
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27. Digital implementation of sliding-mode control via the implicit method: A tutorial

Author: Andrey Polyakov, Bernard Brogliato, Modélisation, simulation et commande des systèmes dynamiques non lisses [2020-....] (TRIPOP [2020-....]), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann [2020-....] (LJK [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....]), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189 (CRIStAL), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille, ANR-18-CE40-0008,DIGITSLID,DIGITAL SET-VALUED AND HOMOGENEOUS SLIDING MODE CONTROL AND DIFFERENTIATORS: THE IMPLICIT APPROACH(2018), Modélisation, simulation et commande des systèmes dynamiques non lisses (TRIPOP), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Work supported by the ANR project Digitslid no ANR-18-CE40-0008-01., and ANR-18-CE40-0008,DIGITSLID,Différentiateurs et commandes homogènes par modes glissants multivalués en temps discret: l'approche implicite(2018)
Subjects: 0209 industrial biotechnology, Automatic control, Discretization, Computer science, sliding-mode, General Chemical Engineering, homogenous systems, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, Sliding mode control, Industrial and Manufacturing Engineering, [SPI.AUTO]Engineering Sciences [physics]/Automatic, 020901 industrial engineering & automation, 0202 electrical engineering, electronic engineering, information engineering, Electrical and Electronic Engineering, Control (linguistics), Structure (mathematical logic), Emulation, discrete-time, Mechanical Engineering, Control engineering, Discrete time and continuous time, Control and Systems Engineering, 020201 artificial intelligence & image processing, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Focus (optics), control, super-twisting
Abstract: International audience; The objective of this article, is to provide a clear presentation of the discretization of continuous-time sliding-mode controllers, also known in the Automatic Control literature as the emulation method, when the implicit (backward) Euler scheme is used. First-order, second-order and homogeneous controllers are considered. The main theoretical results are recalled in each case, and the focus is put on the discrete-time implementation structure and on the algorithms which allow the designer to solve, at each time-step, the one-step generalized equations which are needed to compute the controllers. The article ends with some open issues.
Published: 2020
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28. A Sliding Mode Controller for a Model of Flow Separation in Boundary Layers

Author: Tonametl Sanchez, Jean-Pierre Richard, Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and This work was carried out within the framework of the CNRS Research Federation on Ground Transports and Mobility, in articulation with the ELSAT2020 project supported by the European Community, the French Ministry of Higher Education and Research, the Hauts de France Regional Council.
Subjects: Bilinear systems, 0209 industrial biotechnology, Computer science, Mechanical Engineering, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, Sliding mode control, Industrial and Manufacturing Engineering, Flow control (fluid), Flow separation, 020303 mechanical engineering & transports, 020901 industrial engineering & automation, 0203 mechanical engineering, Control and Systems Engineering, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Electrical and Electronic Engineering
Abstract: International audience; In this paper we propose a sliding mode controller for a MIMO model of flow separation in boundary layers. The model consists in a bilinear system with constant delays in both the state and the input. The main motivation to consider such a class of systems is that, it has shown to be suitable for input-output modelling and control design of some turbulent flow control systems. Stability and robustness properties of the control scheme are studied by means of Volterra equations theory, which provides easily verifiable stability conditions.
Published: 2020
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29. On finite‐time robust stabilization via nonlinear state feedback

Author: Denis Efimov, Konstantin Zimenko, Andrey Polyakov, Department of Control Systems and Informatics, Saint Petersburg State University of Information Technologies Mechanics and Optics, Saint Petersburg State University of Information Technologies Mechanics and Optics-Saint Petersburg State University of Information Technologies Mechanics and Optics, Non-Asymptotic estimation for online systems (NON-A-POST), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Contract/grant sponsor: Russian Science Foundation, and contract/grant number: 17-19-01422
Subjects: Lyapunov function, 0209 industrial biotechnology, General Chemical Engineering, Computation, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, Constructive, Industrial and Manufacturing Engineering, symbols.namesake, 020901 industrial engineering & automation, Homogeneous systems, Simple (abstract algebra), [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, Electrical and Electronic Engineering, Implicit Lyapunov Function, Mathematics, Mechanical Engineering, Homogeneity (statistics), Non-asymptotic stabilization, Linear matrix inequality, State (functional analysis), 16. Peace & justice, Control and Systems Engineering, Integrator, symbols, 020201 artificial intelligence & image processing
Abstract: International audience; A nonlinear control law is designed for finite-time stabilization of a chain of integrators. The method is based on Implicit Lyapunov Function (ILF) technique and homogeneity properties. The scheme of control parameters selection is presented by a Linear Matrix Inequality (LMI). The method is simple in implementation and does not assume on-line procedure for computation of the ILF value at the current state that is an improvement with respect to [9], [13]. The control law is presented in an explicit form and allows to find the values of all control parameters, that make the solution one of the most constructive. The theoretical results are supported by numerical example.
Published: 2018
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30. Fixed-time output stabilization and fixed-time estimation of a chain of integrators

Author: Andrey Polyakov, Francisco Lopez-Ramirez, Wilfrid Perruquetti, and Denis Efimov
Subjects: 0209 industrial biotechnology, Optimization algorithm, Computer science, Mechanical Engineering, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, Industrial and Manufacturing Engineering, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, Fixed time, Robustness (computer science), Integrator, Full state feedback, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Electrical and Electronic Engineering
Abstract: A solution to the problem of global fixed-time output stabilization and estimation of a chain of integrators is proposed. A nonlinear scheme comprising a state feedback controller and a dynamic observer are designed in order to guarantee both fixed-time estimation and fixed-time control. Robustness with respect to exogenous disturbances and measurement noises is established and a parameter optimization algorithm is provided. The performance of the obtained control and estimation algorithms are illustrated by numeric experiments.
Published: 2018
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31. Sliding mode control design using canonical homogeneous norm

Author: Andrey Polyakov
Subjects: 0209 industrial biotechnology, Mechanical Engineering, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, 02 engineering and technology, Sliding mode control, Industrial and Manufacturing Engineering, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, Homogeneous, Control theory, Norm (mathematics), 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Electrical and Electronic Engineering, Mathematics
Abstract: The problem of sliding mode control design for nonlinear plant is studied. Necessary and sufficient conditions of quadratic-like stability (stabi-lizability) for nonlinear homogeneous (control) system are obtained. Sufficient conditions of robust stability/stabilizability are deduced. The results are supported with academic examples of sliding mode control design.
Published: 2018
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32. THE COMPOSITION AND STRUCTURE OF HYDROSILICATES OBTAINED MECHANOCHEMICAL SYNTHESIS OF PORTLAND CEMENT, MODIFIED SUCROSE

Author: Andrey Polyakov and Evgeniy Shoshin
Subjects: Sucrose, Materials science, 020502 materials, Pharmaceutical Science, 02 engineering and technology, 021001 nanoscience & nanotechnology, law.invention, chemistry.chemical_compound, Portland cement, 0205 materials engineering, Complementary and alternative medicine, chemistry, Chemical engineering, law, Pharmacology (medical), Composition (visual arts), 0210 nano-technology
Published: 2018
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33. Finite-time and fixed-time observer design: Implicit Lyapunov function approach

Author: Denis Efimov, Andrey Polyakov, Wilfrid Perruquetti, Francisco Lopez-Ramirez, Non-Asymptotic estimation for online systems (NON-A-POST), Inria Lille - Nord Europe, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: Homogenous Systems, Lyapunov function, 0209 industrial biotechnology, Observer (quantum physics), 02 engineering and technology, symbols.namesake, Nonlinear system, 020901 industrial engineering & automation, Fixed-Time Stability, Control and Systems Engineering, Control theory, Robustness (computer science), [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Control system, Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, symbols, A priori and a posteriori, 020201 artificial intelligence & image processing, State observer, Electrical and Electronic Engineering, Finite-Time Stability, Implicit Lyapunov Function, Mathematics
Abstract: International audience; This work deals with the problem of finite-time and fixed-time observation of linear multiple input multiple output (MIMO) control systems. The proposed nonlinear dynamic observers guarantee convergence of the observer states to the original system state in a finite and in a fixed (defined a priori) time. Algorithms for the observers parameters tuning are also provided and a robustness analysis against input disturbances and measurement noises is carried out. The theoretical results are illustrated by numerical examples that consider both noisy and noise-free measurements and a comparison with a high-gain observer is included.
Published: 2018
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34. Acceleration of finite-time stable homogeneous systems

Author: Wilfrid Perruquetti, Arie Levant, Denis Efimov, Yotam Dvir, and Andrey Polyakov
Subjects: 0209 industrial biotechnology, Mechanical Engineering, General Chemical Engineering, MIMO, Biomedical Engineering, Zero (complex analysis), Aerospace Engineering, 02 engineering and technology, Stability (probability), Industrial and Manufacturing Engineering, Set (abstract data type), Acceleration, Matrix (mathematics), 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, Uniform boundedness, 020201 artificial intelligence & image processing, Electrical and Electronic Engineering, Mathematics
Abstract: Stabilization rates of power-integrator chains are easily regulated. It provides a framework for acceleration of uncertain multi-input multi-output (MIMO) dynamic systems of known relative degrees (RDs). The desired rate of the output stabilization (sliding-mode (SM) control) is ensured for an uncertain system, if its RD is known, and a rough approximation of the high-frequency gain matrix is available. The uniformly bounded convergence time (fixed-time stability) is obtained as a particular case. The control can be kept continuous everywhere accept the SM set, if the partial RDs are equal. Similarly uncertain smooth systems of complete MIMO RDs (i.e. lacking zero dynamics) are stabilized by continuous control at their equilibria in finite time and also accelerated. Output-feedback controllers are constructed. Computer simulation demonstrates the efficiency of the proposed approach.
Published: 2017
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35. Realization and Discretization of Asymptotically Stable Homogeneous Systems

Author: Wilfrid Perruquetti, Andrey Polyakov, Arie Levant, Denis Efimov, Non-Asymptotic estimation for online systems (NON-A-POST), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), School of Mathematical Sciences [Tel Aviv], Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University [Tel Aviv]-Tel Aviv University [Tel Aviv], Systèmes Non Linéaires et à Retards (SyNeR), Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), This work was partially supported by ANR-15-CE23-0007 (Project Finite4SoS), the Government of Russian Federation (Grant 074-U01) andthe Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031)., ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), School of Mathematical Sciences [Tel Aviv] (TAU), Raymond and Beverly Sackler Faculty of Exact Sciences [Tel Aviv] (TAU), and Tel Aviv University (TAU)-Tel Aviv University (TAU)
Subjects: 0209 industrial biotechnology, Discretization, Differential equation, 020208 electrical & electronic engineering, Mathematical analysis, Linear system, Explicit and implicit methods, 02 engineering and technology, Backward Euler method, [SPI.AUTO]Engineering Sciences [physics]/Automatic, Computer Science Applications, Euler method, symbols.namesake, 020901 industrial engineering & automation, Exponential stability, Control and Systems Engineering, Stability theory, 0202 electrical engineering, electronic engineering, information engineering, symbols, Electrical and Electronic Engineering, Mathematics
Abstract: International audience; Sufficient conditions for the existence and convergence to zero of numeric approximations to solutions of asymptotically stable homogeneous systems are obtained for the explicit and implicit Euler integration schemes. It is shown that the explicit Euler method has certain drawbacks for the global approximation of homogeneous systems with non-zero degrees, whereas the implicit Euler scheme ensures convergence of the approximating solutions to zero. Properties of absolute and relative errors of the respective discretizations are investigated.
Published: 2017
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36. On design of interval observers for parabolic PDEs

Author: Tatiana Kharkovskaia, Jean-Pierre Richard, Emilia Fridman, Andrey Polyakov, Denis Efimov, Non-Asymptotic estimation for online systems (NON-A), Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centrale Lille, Department of Electrical Engineering, Tel Aviv University [Tel Aviv], Efimov, Denis, Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Tel Aviv University (TAU)
Subjects: Distributed parameter system, 0209 industrial biotechnology, Partial differential equation, Observer (quantum physics), Heat equation, Mathematical analysis, 02 engineering and technology, Interval (mathematics), Interval observers, Space (mathematics), [SPI.AUTO]Engineering Sciences [physics]/Automatic, Noise, [SPI.AUTO] Engineering Sciences [physics]/Automatic, 020901 industrial engineering & automation, 020401 chemical engineering, Control and Systems Engineering, Boundary value problem, 0204 chemical engineering, Mathematics
Abstract: International audience; The problem of state estimation for heat non-homogeneous equations under distributed in space measurements is considered. An interval observer is designed, described by Partial Differential Equations (PDEs), for uncertain distributed parameter systems without application of finite-element approximations. Conditions of boundedness of solutions of interval observer with non-zero boundary conditions and measurement noise are proposed. The results are illustrated by numerical experiments with an academic example.
Published: 2017
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37. Fault-Tolerant Control for Physically Linked Two 2WD Mobile Robots with Actuator Faults * *This work was partially supported by the Regional project SUCRe (Surete de fonctionnement et resilience pour la gestion et le controle cooperatifs des systemes sociotechniques: Cooperation Homme(s)-Robot(s) en milieu hostile) of the Hauts de France region

Author: Maan El Badaoui El Najjar, Andrey Polyakov, Vincent Cocquempot, Gang Zheng, and Yajie Ma
Subjects: Scheme (programming language), 0209 industrial biotechnology, Engineering, business.industry, 020208 electrical & electronic engineering, SIGNAL (programming language), Mobile robot, Control engineering, Fault tolerance, 02 engineering and technology, Kinematics, Tracking (particle physics), Computer Science::Robotics, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, 0202 electrical engineering, electronic engineering, information engineering, Two-wheel drive, Actuator, business, computer, computer.programming_language
Abstract: In this paper, a fault-tolerant control scheme for two physically linked two wheel drive (2WD) mobile robots with multiple and simultaneous actuator faults is proposed. The kinematic and dynamic models are first presented. A kinematic control law is designed, based on which, a dynamic control law is investigated. By applying the proposed dynamic control signal, the desired system stability and tracking properties are ensured. Simulation results are finally presented to verify the effectiveness of the developed fault-tolerant control scheme.
Published: 2017
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38. SISO model-based control of separated flows: Sliding mode and optimal control approaches

Author: Franck Kerhervé, Maxime Feingesicht, Andrey Polyakov, and Jean-Pierre Richard
Subjects: Flow control (data), 0209 industrial biotechnology, Engineering, Variable structure control, business.industry, Mechanical Engineering, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, Bilinear interpolation, Experimental data, 02 engineering and technology, Model based control, Optimal control, 01 natural sciences, Sliding mode control, Industrial and Manufacturing Engineering, 010305 fluids & plasmas, Setpoint, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, 0103 physical sciences, Electrical and Electronic Engineering, business
Abstract: The challenging problem of active control of separated flows is tackled in the present paper using model-based design principles, and applied to data issued from a two-dimensional separated flow experiment. First, a bilinear state and input delay model of the system has been obtained from experimental data by means of a modified identification procedure. Adequacy and precision of the obtained model are demonstrated and compared with existing results. Next, two control problems (setpoint tracking and optimal control) have been formulated and studied using sliding mode control methodology and averaging analysis. The theoretical control results are supported with numerical simulations.
Published: 2017
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39. Robust Feedback Stabilization of Linear MIMO Systems Using Generalized Homogenization

Author: Denis Efimov, Wilfrid Perruquetti, Andrey Polyakov, Konstantin Zimenko, National Research University of Information Technologies, Mechanics and Optics [St. Petersburg] (ITMO), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Centre de Recherche Réseau Image SysTème Architecture et MuLtimédia (CRISTAL), École Nationale des Sciences de l'Informatique [Manouba] (ENSI), Université de la Manouba [Tunisie] (UMA)-Université de la Manouba [Tunisie] (UMA), The results on homogenization of linear MIMO systems presented in Section V were developed under support of RSF (grant 17-19-01422) in ITMO University. The other research were partially supported by ANR-15-CE23-0007 (Project Finite4SoS) and Goverment of Russian Federation (Grant 08-08)., and ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015)
Subjects: 0209 industrial biotechnology, Linear system, 02 engineering and technology, Nonlinear control, Generalized homogeneity, Homogenization (chemistry), Computer Science Applications, linear MIMO systems, finite-time stabilization, 020901 industrial engineering & automation, Compact space, Control and Systems Engineering, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Homogeneity (physics), 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, Symmetric matrix, 020201 artificial intelligence & image processing, Canonical form, Electrical and Electronic Engineering, Mimo systems, Mathematics
Abstract: A robust nonlinear control is designed for stabilizing linear multi-input–multi-output systems. The presented control law homogenizes a linear system (without its transformation to a canonical form) with a specified degree and stabilizes it in a finite time (or with a fixed-time attraction to any compact set containing the origin) if the degree of homogeneity is negative (positive). The tuning procedure is formalized in an linear matrix inequalities (LMI) form. Performance of the approach is illustrated by numerical and experimental examples.
Published: 2020
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40. On simple scheme of finite/fixed-time control design

Author: Wilfrid Perruquetti, Denis Efimov, Konstantin Zimenko, Andrey Polyakov, National Research University of Information Technologies, Mechanics and Optics [St. Petersburg] (ITMO), Non-Asymptotic estimation for online systems (NON-A-POST), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centrale Lille, and This work is supported by the Russian Science Foundation under grant 17-19-01422.
Subjects: Lyapunov function, 0209 industrial biotechnology, Homogeneity (statistics), Finite-time stabilization, 02 engineering and technology, 16. Peace & justice, Computer Science Applications, symbols.namesake, 020901 industrial engineering & automation, homogeneity, Control and Systems Engineering, Fixed time, Control theory, Integrator, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols, 020201 artificial intelligence & image processing, fixed-time stabilization, Mathematics
Abstract: International audience; Control laws are designed for stabilization of a chain of integrators of arbitrary degree in finite and fixed time. Presented control laws are obtained with use of Lyapunov function method and homogeneity concept. The present analysis is based on use of explicitly defined Lyapunov function that is the hallmark with respect to similar works. This analysis allows to get simple procedure of parameters tuning and obtain new estimates for settling-time function. The theoretical results are supported by numerical examples.
Published: 2020
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41. Minimax Sliding Mode Control Design for Linear Evolution Equations with Noisy Measurements and Uncertain Inputs

Author: Sergiy Zhuk, Orest Iftime, Andrey Polyakov, Jonathan P. Epperlein, IBM Research - Ireland, IBM, University of Groningen [Groningen], Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Research programme EEF
Subjects: 0209 industrial biotechnology, minimax, General Computer Science, evolution equations, Mechanical Engineering, 020208 electrical & electronic engineering, sliding mode, Mode (statistics), 02 engineering and technology, Minimax, Ellipsoid, Sliding mode control, Noise, 020901 industrial engineering & automation, Control and Systems Engineering, Reachability, Control theory, Bounded function, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, Electrical and Electronic Engineering, Riccati equations, Mathematics
Abstract: International audience; We extend a sliding mode control methodology to linear evolution equations with uncertain but bounded inputs and noise in observations. We first describe the reachability set of the state equation in the form of an infinite-dimensional ellipsoid, and then steer the minimax center of this ellipsoid toward a finitedimensional sliding surface in finite time by using the standard sliding mode output-feedback controller in equivalent form. We demonstrate that the designed controller is the best (in the minimax sense) in the class of all measurable functionals of the output. Our design is illustrated by two numerical examples: output-feedback stabilization of linear delay equations, and control of moments for an advection-diffusion equation in 2D.
Published: 2020
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42. Robust Stabilization of Competing Species in the Chemostat

Author: Alex dos Reis de Souza, Andrey Polyakov, Jean-Luc Gouzé, Denis Efimov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189 (CRIStAL), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille, Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and This work was supported by IPL COSY.
Subjects: 0209 industrial biotechnology, Work (thermodynamics), chemostat, sliding-mode, Feedback control, Robust control, 02 engineering and technology, Chemostat, Limiting, Industrial and Manufacturing Engineering, Computer Science Applications, [SPI.AUTO]Engineering Sciences [physics]/Automatic, 020901 industrial engineering & automation, 020401 chemical engineering, Control and Systems Engineering, Modeling and Simulation, Bounded function, Bioreactor, 0204 chemical engineering, Biological system, biological process, Parametric statistics, Mathematics
Abstract: International audience; This work addresses the problem of robust stabilization of the concentrations of two different species of living organisms, which compete for a single limiting substrate in a bioreactor. This stabilization is performed using discontinuous feedback control laws that ensure the coexistence of all species. The control laws are designed considering bounded parametric uncertainties on the kinetic rates.
Published: 2020
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43. A polytopic strategy for improved non-asymptotic robust control via implicit Lyapunov functions

Author: Alán Tapia, Denis Efimov, Andrey Polyakov, Leonid Fridman, Miguel Bernal, Instituto Tecnológico de Sonora (ITSON), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM), The authors gratefully acknowledge the financial support of CONACyT project 282013 for CVU 631139, the CONACYT sabbatical fellowship forCVU 202355, PAPIIT-UNAM IN 115419, the Government of Russian Federation (Grant 08-08), the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898, and ITSON projects PFCE 2018-19 and PROFAPI 2019-0002., and Universidad Nacional Autónoma de México (UNAM)
Subjects: Lyapunov function, 0209 industrial biotechnology, Computer science, Linear Matrix Inequalities, Perturbation (astronomy), 02 engineering and technology, Linear matrix, Computer Science Applications, [SPI.AUTO]Engineering Sciences [physics]/Automatic, symbols.namesake, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, 0202 electrical engineering, electronic engineering, information engineering, symbols, Homogeneity, 020201 artificial intelligence & image processing, Robust control, Finite-and Fixed-Time Convergence, Implicit Lyapunov Functions, Polytopic Systems, Analysis, Parametric statistics
Abstract: International audience; This paper is concerned with finite-and fixed-time robust stabilization of uncertain multi-input nonlinear systems via the implicit Lyapunov function method. Instead of splitting the system into a linear nominal model and an additive perturbation which gathers nonlinearities, parametric uncertainties , and exogenous disturbances, the methodology hereby proposed preserves some nonlinear terms in the nominal system via an exact polytopic representation which leads to design conditions in the form of linear matrix inequalities. As a result, feasible solutions are found where former approaches fail; these solutions have more accurate settling-time estimates with reduced control effort. The corresponding control law includes well-known high-order sliding modes as a particular case. Numerical simulations are provided to illustrate the advantages of the proposal.
Published: 2020
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44. Homogeneous Lyapunov Functions for Homogeneous Infinite Dimensional Systems with Unbounded Nonlinear Operators

Author: Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Ecole Centrale de Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Ecole Centrale de Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS), and Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)
Subjects: Lyapunov function, 0209 industrial biotechnology, Class (set theory), Pure mathematics, General Computer Science, Mechanical Engineering, 020208 electrical & electronic engineering, 02 engineering and technology, Lipschitz continuity, homogeneous systems, stability of distributed parameter systems, symbols.namesake, 020901 industrial engineering & automation, Control and Systems Engineering, Homogeneous, Stability theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols, Electrical and Electronic Engineering, Nonlinear operators, semigroup and operator theory, Mathematics
Abstract: International audience; The existence of a locally Lipschitz continuous homogeneous Lyapunov function is proven for a class of asymptotically stable homogeneous infinite dimensional systems with unbounded nonlinear operators.
Published: 2020
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45. Generalized Homogenization of Linear Controllers: Theory and Experiment

Author: Siyuan Wang, Andrey Polyakov, Gang Zheng, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Deformable Robots Simulation Team (DEFROST ), The authors acknowledge the support of CSC Grant, CPER DATA and ANR DIGISLID, ANR-18-CE40-0008. The second author also acknowledges the support of the Government of Russian Federation (Grant 08-08) and the Ministry of Science and Higher Education of Russian Federation, goszadanie no. 2019-0898. The third author acknowledges the support of the Project of Department of Education of Guangdong Province (No.2019KZDXM037) China, the Project Inventor (I-SITE ULNE, le programme d’Investissements d’Avenir, Métropole Européenne de Lille), and the Project ROBOCOP [ANR-19-CE19-0026]., and ANR-19-CE19-0026,ROBOCOP,robotisation de l'implant cochléaire(2019)
Subjects: 0209 industrial biotechnology, Mechanical Engineering, General Chemical Engineering, Biomedical Engineering, Aerospace Engineering, Nonlinear observer, 02 engineering and technology, Homogenization (chemistry), Industrial and Manufacturing Engineering, Nonlinear system, 020901 industrial engineering & automation, Control and Systems Engineering, Control theory, Homogeneous, Computer Science::Systems and Control, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, 020201 artificial intelligence & image processing, Electrical and Electronic Engineering, Mathematics
Abstract: A procedure of upgrading a linear observer to homogeneous (nonlinear) one is proposed and validated by experiment. The nonlinear observer design is based on the the concept of a linear geometric homogeneity. The simple procedure developed to apply the homogeneous nonlinear observer is based on the parameters provided by the linear observer, which is potential for many applications to improve their performance. A saturation function is introduced to guaranteed the improvement of estimation quality. Finally the theoretical results are confirmed by the QDrone platform of Quanser T M .
Published: 2020
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46. Integral Control Design using the Implicit Lyapunov Function Approach

Author: Angel Mercado-Uribe, Andrey Polyakov, Denis Efimov, Jaime A. Moreno, Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Universidad Nacional Autónoma de México (UNAM)
Subjects: Lyapunov function, 0209 industrial biotechnology, Perturbation (astronomy), 02 engineering and technology, 16. Peace & justice, Lipschitz continuity, symbols.namesake, 020901 industrial engineering & automation, Homogeneous, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, 020201 artificial intelligence & image processing, Mathematics
Abstract: International audience; In this paper, we design homogeneous integral controllers of arbitrary non positive homogeneity degree for a system in the normal form with matched uncer-tainty/perturbation. The controllers are able to reach finite-time convergence, rejecting matched constant (Lipschitz, in the discontinuous case) perturbations. For the design, we use the Implicit Lyapunov Function method combined with an explicit Lyapunov function for the addition of the integral term.
Published: 2019

47. Robust Control of a Competitive Environment in the Chemostat using Discontinuous Control Laws

Author: Alex dos Reis de Souza, Andrey Polyakov, Jean-Luc Gouzé, Denis Efimov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189 (CRIStAL), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Ecole Centrale de Lille, Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), and Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)
Subjects: 0209 industrial biotechnology, Work (thermodynamics), Feedback control, Control (management), 02 engineering and technology, Chemostat, Limiting, [SPI.AUTO]Engineering Sciences [physics]/Automatic, 020901 industrial engineering & automation, 020401 chemical engineering, Law, Bounded function, 0204 chemical engineering, Robust control, Mathematics
Abstract: International audience; This work addresses the problem of robust stabilizationof the concentration of two different species competingfor a single limiting substrate. This stabilization is performedby means of discontinuous feedback control laws that ensurecoexistence of all species. The control laws are designed consideringbounded uncertainties on the kinetic rates.
Published: 2019

48. On Condition for Output Finite-Time Stability and Adaptive Finite-Time Control Scheme

Author: Denis Efimov, Andrey Polyakov, Konstantin Zimenko, National Research University of Information Technologies, Mechanics and Optics [St. Petersburg] (ITMO), Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Université de Lille-Centrale Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centrale Lille-Centre National de la Recherche Scientifique (CNRS), and Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)
Subjects: Scheme (programming language), 0209 industrial biotechnology, Finite time control, Property (programming), Homogeneity (statistics), 020208 electrical & electronic engineering, 02 engineering and technology, Stability (probability), 020901 industrial engineering & automation, Control theory, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 0202 electrical engineering, electronic engineering, information engineering, Finite time, computer, computer.programming_language, Mathematics
Abstract: International audience; A sufficient condition for output finite-time stability is presented. Based on this condition a scheme of adaptive finite-time control design is provided. The presented results are obtained with the use of homogeneity property. The theoretical results are supported by numerical examples.
Published: 2019

49. Characterization of Finite/Fixed-time Stability of Evolution Inclusions

Author: Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015)
Subjects: 0209 industrial biotechnology, 020901 industrial engineering & automation, Lyapunov functional, Fixed time, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, 020208 electrical & electronic engineering, Mathematical analysis, 0202 electrical engineering, electronic engineering, information engineering, Banach space, 02 engineering and technology, Characterization (mathematics), Stability (probability), Mathematics
Abstract: International audience; Characterizations (necessary and sufficient conditions) of finite-time and fixed-time stability of evolution inclusions in Banach spaces are presented in terms of Lyapunov functionals.
Published: 2019

50. Generalized Lyapunov Exponents of Homogeneous Systems

Author: Sergiy Zhuk, Andrey Polyakov, Finite-time control and estimation for distributed systems (VALSE), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), IBM Research - Ireland, and IBM
Subjects: 0209 industrial biotechnology, Class (set theory), Zero (complex analysis), 02 engineering and technology, Lyapunov exponent, Nonlinear Sciences::Chaotic Dynamics, symbols.namesake, 020901 industrial engineering & automation, Homogeneous, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Stability theory, symbols, Applied mathematics, Mathematics
Abstract: International audience; The paper deals the method of Lyapunov exponents for a class of a generalized homogeneous systems. Homogeneous systems may have some sup-exponential and super-exponential grows. In this case, the method of Lyapunov exponents becomes non-informative, e.g. all Lyapunov exponents may equal to zero but the system is globally uniformly asymptotically stable. In this paper we propose an approach which allows us to analyze a behavior of such homogeneous systems by means of the method of Lyapunov exponents.
Published: 2019
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