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International audience; Razumikhin-like theorems on hyperexponential and fixed-time stability of time-delay systems are proposed for both explicitly and implicitly defined Lyapunov functions. While the former method is useful for stability analysis, the latter approach is more suitable for control synthesis. Examples of systems that can be stabilized hyperexponentially and in fixed time are given. The control parameters tuning algorithm is presented in the form of linear matrix inequalities. The numerical simulations illustrate the theoretical results.

International audience; The problem of finite-time stabilization of a linear plant with an optimization of both a settling time and an averaged/weighted control energy is studied using the concept of generalized homogeneity. It is shown that the optimal finite-time stabilizing control in this case can be designed by means of solving a simple linear algebraic equation. Robustness of the obtained control law is studied.

Using a new method of correlation function for amplitude and phase fluctuations (APCF), records of geomagnetic field component fluctuations (observatories Mondy and Borok) have been processed for a number of hour zones of the nightside magnetosphere. This method is meant to detect groups of equidistant frequencies inside the spectrum of source signal and also to measure a difference between two neighbor frequencies in each of these groups. The groups of equidistant frequencies in broadband spectra of these fluctuations are shown to depend on eigenfrequencies of the 2D Alfvén wave resonator, just as for dayside fluctuations [Polyakov, 2018]. An empirical relation has been found between a combination of parameters of this resonator and local time. The similarity between structural elements in the final products of processing of Alfvén fluctuation N-S and E-W components for almost all hour zones clearly indicates the reliability of the results of the APCF method in processing any broadband signals.

International audience; The methodology of the unit sliding mode control design (known since 1970s) for linear systems is revised based on the concept of the generalized homogeneity. The restriction about a consistency of the number of control inputs with the dimension of the sliding surface is eliminated. A simple procedure of control parameters tuning based on a known maximal magnitude of matched perturbations is developed. To deal with perturbations of unknown magnitude, a homogeneous sliding mode control with integral action is designed as well. The theoretical results are supported by numerical simulations.

International audience; This paper contributes to the design of a secondorder sliding-mode controller for the trajectory tracking problem in perturbed unicycle mobile robots. The proposed strategy takes into account the design of two particular sliding variables, which ensure the convergence of the tracking error to the origin in a finite time despite the effect of some external perturbations. The straightforward structure of the controller is simple to tune and implement. The global, uniform and finite-time stability of the closed-loop tracking error dynamics is demonstrated by means of Lyapunov functions. Furthermore, the performance of the proposed approach is validated through some experiments using a QBot2 unicycle mobile robot.

A generalized homogeneous control with integral action for a multiple-input plant operating under uncertainty conditions is designed. The stability analysis is essentially based on a special version of the non-smooth Lyapunov function theorem for differential equations with discontinuous right-hand sides. A Lyapunov function for analysis of the closed-loop system is presented. For negative homogeneity degree, this Lyapunov function becomes a strict Lyapunov function allowing an advanced analysis to be provided. In particular, the maximum control magnitude and the settling-time of the closed-loop system are estimated and a class of disturbances to be rejected by the control law is characterized. The control parameters are tuned by solving a system of Linear Matrix Inequalities (LMIs), whose feasibility is proved at least for small (close to zero) homogeneity degrees. The theoretical results are illustrated by numerical simulations.

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.

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.

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.

International audience; Non-overshooting stabilization is a form of safe control where the setpoint chosen by the user is at the boundary of the safe set. Exponential non-overshooting stabilization, including suitable extensions to systems with deterministic and stochastic disturbances, has been solved by the second author and his coauthors. In this paper we develop homogeneous feedback laws for fixed-time nonovershooting stabilization for nonlinear systems that are input-output linearizable with a full relative degree, i.e., for systems that are diffeomorphically equivalent to the chain of integrators. These homogeneous feedback laws can also assume the secondary role of 'fixed-time safety filters' (FxTSf filters) which keep the system within the closed safe set for all time but, in the case where the user's nominal control commands approach to the unsafe set, allow the system to reach the boundary of the safe set no later than a desired time that is independent of nominal control and independent of the value of the state at the time the nominal control begins to be overridden.

International audience; In this paper, the quadrotor stabilization under time and state constraints is studied. The objective is to design a nonlinear controller under time and state constraint for quadrotor. The nonlinear quadrotor model is built by the Euler-Lagrange approach while ignoring the Coriolis terms, hub moment and force. Based on quadrotor's dynamic model, a nonlinear feedback controller is designed for the quadrotor stabilization under time and state constraints. This feedback is an implicit PID controller where the feedback gains are obtained from LMIs (Linear matrix inequalities). LMI system characterizing the system stability and convergence properties is built based on convex embedding approach and implicit Lyapunov function method. To demonstrate the application prospects of implicit PID controller, robustness analysis is provided to show the property of implicit PID controller under external disturbance. The key novelty of this paper is that the implicit PID controller is proven feasible for applying to the quadrotor under time and state constraints, which is also the main outcome.

International audience; The problem of finite-time stabilization of linear systems under state and control input constraints in the presence of external disturbances is addressed. The proposed control is designed such that the closed-loop system is 1) superstable, when system trajectories risk to violate the state constraints, and 2) homogeneous with a negative degree, otherwise. While the former property ensures the fulfillment of the state and control constraints, the latter implies finite-time stability. The robustness of the control scheme with respect to disturbances is studied. Theoretical results are supported by numerical simulations.

International audience; The problem of finite-time stabilization of a linear plant with an optimization of both a settling time and an weighted/averaged control energy is studied using the concept of generalized homogeneity. It is shown that the optimal finite-time stabilizing control in this case can be designed solving a simple linear algebraic equation. Some issues of a practical applicability and a robustness of the obtained control law are studied.

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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.

International audience; The present paper provides a sufficient condition to ensure output finite-time and fixed-time stability. Comparing with analogous researches the proposed result is less restrictive and obtained for a wider class of systems. The presented output stability condition is used for adaptive control design, where the state vector of a plant is extended by adjustable control parameters.

International audience; This paper deals with the practical implementation of implicit homogeneous controllers (IHCs) for linearized mechanical systems. The control design includes the methodology to get gains of the IHC based on the linearized approximation of the system. If the approximation error enforced by the linearization is vanishing with the state, locally measurable and bounded, the IHC can lead the state to the origin in finite-time. This IHC control allows accelerating the convergence rate of the states. A semi-explicit algorithm is provided to exert the digital implementation of the controller. The application of the bisection method estimates the controller gain ensuring the finite-time convergence of the state to the origin. A complementary analysis provides a simplified algorithm with a reduced number of computation stages but equally efficient gain estimation. The proposed IHC is applied to a rotary inverted pendulum QUBE ™ Servo 2 platform of Quanser ®. The obtained results for the state convergence are compared with other classical feedback controllers to validate the effectiveness of the proposed scheme. The comparative analysis of the state convergence provides evidence of the faster convergence for the state trajectory to a zone centered on the origin with a smaller hypervolume than the one gotten with the classical controllers.

The rocksalt-type ferromagnetic (FM) insulator EuS (bulk TC = 17 K) grown on Bi2Se3 with well-matched (111) plane of the film and (0001) plane of the substrate is studied. The system may feature magnetic proximity effect breaking the time-reversal symmetry and opening a bandgap in the metallic topologically protected surface state of Bi2Se3. The experimental X-ray diffraction studies are combined with ab initio calculations to resolve contradictory results concerning the enhancement of the TC up to 300 K and the degree of induced magnetization in the system. It is concluded that previous studies relied on idealized and unconfirmed structure models. Herein, it is shown by surface X-ray diffraction (SXRD) with ab initio calculations that a two double layer-thick EuS film grows with a sharp interface and without chemical intermixing in a single domain state in an FCC-type stacking on the Bi2Se3(0001) surface in which the topmost layer is metallic, thereby lifting polarity. A large pz-orbital-derived top-layer sulfur magnetic moment of 0.6 μB is found, whereas for europium, μEu = 6.9 μB throughout the film is found. No magnetization within the first Bi2Se3 quintuple layer is found. The calculation of the exchange parameters Jij indicates a complex FM and antiferromagnetic ordering between europium and sulfur with a maximum Néel temperature of 226 K.

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.

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.

Spin-momentum locking in topological insulators and materials with Rashba-type interactions is an extremely attractive feature for novel spintronic devices and is therefore under intense investigation. Significant efforts are underway to identify new material systems with spin-momentum locking, but also to create heterostructures with new spintronic functionalities. In the present study we address both subjects and investigate a van der Waals-type heterostructure consisting of the topological insulator Bi

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.

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.

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.

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.

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.

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 .

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