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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

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.

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.

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.

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.

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.

This paper presents a revised version of the new signal processing method based on the analysis of a specially constructed correlation function of amplitude and phase fluctuations (APCF). This method allows us to detect the presence of a group of equidistant frequencies in the spectrum of the original signal and to measure the difference of two adjacent frequencies Δf in such a group. The end product of the processing is a histogram of a set of Δf. The effect of noise that may be present in the original signal has been examined. It has been shown that even with a very high level of noise when its component in the spectrum completely absorbs and masks spectral peaks of equidistant frequencies of the desired signal, the APCF method copes with the problem of detecting these frequencies. This method was first applied to processing of natural signals, for which recordings of geomagnetic field disturbances of an ultralow frequency (ULF) range were used. The comparison of one of the histograms with the traditional spectrum indicates that the chaotic spectrum, which has always been considered to be a noise spectrum, actually has a strictly ordered structure. It has been found that most spectral peaks belong to one of the sets (more than 10) of equidistant frequency groups. In the entire spectrum, peaks of these groups are superimposed on each other and form a complex chaotic sequence. The analysis of peaks of all histograms allows us to conclude that the equidistant frequency groups, which correspond to peaks in each histogram, are eigenfrequencies of the 2D Alfvén wave resonator. The existence of such a resonator in the magnetosphere in the vicinity of the outer edge of the plasmopause has been predicted in theoretical studies [Guglielmi, Polyakov, 1983; Leonovich, Mazur, 1987]. The processing method APCF enables us to experimentally confirm this prediction.

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.

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.

The Mutnovsky geothermal area is part of the Eastern Kamchatka active volcano belt. Mutnovsky, 80 kY old and an aging strato-volcano (a complex of 4 composite volcanic cones), acts as a magma- and water-injector into the 25-km-long North Mutnovsky extension zone. Magmatic injection events (dykes) are associated with plane-oriented MEQ (Micro Earth Quakes) clusters, most of them occurring in the NE sector of the volcano (2 × 10 km2) at elevations from −4 to −2 km, while some magmatic injections occur at elevations from −6.0 to −4.0 km below the Mutnovsky production field. Water recharge of production reservoirs is from the Mutnovsky volcano crater glacier (+1500 to +1800 masl), which was confirmed by water isotopic data (δD, δ18O) of production wells at an earlier stage of development. The Mutnovsky (Dachny) 260–310 °C high-temperature production geothermal reservoir with a volume of 16 km3 is at the junction of NNE- and NE-striking normal faults, which coincides with the current dominant dyke injection orientation. TOUGH2-modeling estimates of the reservoir properties are as follows: the reservoir permeability is 90–600 e-15 m2, the deep upflow recharge is 80 kg/s and the enthalpy is 1420 kJ/kg. Modeling was used to reproduce the history of the Mutnovsky (Dachny) reservoir exploitation since 1983 with an effective power of 48 MWe by 2016. Modeling also showed that the reservoir is capable of yielding 65–83 MWe of sustainable production until 2055, if additional production drilling in the SE part of the field is performed. Moreover, this power value may increase to 87–105 MWe if binary technologies are applied. Modeling also shows that the predicted power is sensitive to local meteoric water influx during development. Conceptual iTOUGH2-EOS1sc thermal hydrodynamic modeling of the Mutnovsky magma-hydrothermal system as a whole reasonably explains its evolution over the last 1500–5000 years in terms of heat recharge (dyke injection from the Mutnovsky-4 funnel) and mass recharge (water injection through the Mutnovsky-2 and Mutnovsky-3 funnels) conditions as previously mentioned.

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.

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.

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.

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.

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.

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.

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.

The celebrated Lyapunov function method (or the direct Lyapunov method) introduced in the Ph.D. thesis of A. M. Lyapunov in 1892 is a simple effective tool for stability analysis of differential equations. The main advantage of this method lies in the fact that a decision on stability or instability can be made by means of a certain investigation of the right-hand side of a differential equation without finding its solutions. Initially, the Lyapunov function method was limited by a regular class of ODE with continuous right-hand sides. The later evolution of the ODE theory and its applications had required extensions of this method to differential equations with discontinuous right-hand sides, functional differential equations, PDEs, and evolution systems.

The high-order sliding mode (HOSM) is discussed here as an alternative methodology to the conventional sliding mode control to design feedback control, observers, and to handle the chattering problem. Here, the principle ideas of HOSM methodology are presented. Similar to the second-order sliding mode, HOSM implies that sliding mode occurs in the manifold of a smaller order after a finite time interval.

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.

This chapter introduces linear dilations in normed vector spaces and studies their properties. The related notions (such as a generator of a dilation) are considered. Some monotonicity properties of dilations in Banach, Hilbert and Euclidean spaces are studied. Geometrical structures induced by linear dilations (such as homogeneous cones, spheres and balls) are presented. A class of linear dilations in function spaces is studied. All results are supported with examples in finite-dimensional and infinite-dimensional spaces.

A homogeneity-based approach to the stability, regularity, and robustness analysis of dynamical systems in finite-dimensional and infinite-dimensional spaces is introduced in this chapter. In particular, a dilation symmetry of solutions of homogeneous evolution systems is established; a simple way for global expansion of regularity of solutions is provided; a homogeneity-based finite-time and fixed-time stability analysis is developed; the existence of a homogeneous Lyapunov function for any stable homogeneous evolution system is proven; a homogeneous Lyapunov function for a stable homogeneous ODE is characterized by a quadratic-like functional \({\mathbb {R}}^n\rightarrow {\mathbb {R}}\); a robustness (input-to-state stability) of homogeneous evolution systems is studied.

An optimal control design is the well-known problem of mathematical control theory where a geometry of a system plays an important role. Being a kind of symmetry, the homogeneity is expected to be useful for optimization purposes. In this chapter, we study homogeneous optimal control problems in \({\mathbb {R}}^n\) and revise the conventional tools of an optimal control design (the Bellman function method and the Pontryagin maximum principle) under the assumption about a homogeneity of the system model and the cost functional.

In this chapter, the Lyapunov stability tools for sliding modes analysis are considered. The notion of finite-time stabilization is introduced, and the finite-time suitability analysis is discussed based on the homogeneity property of the considered controllable system. The realization of fixed-time stability is also presented.

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.

In this chapter, we propose a sliding-mode controller for a class of scalar bilinear systems with delay in both the input and the state. Such systems have shown to be adequate for input–output modeling and control of a class of turbulent flow systems. Since the sliding dynamics is infinite dimensional and described by an integral equation, we show that the stability and robustness analysis is simplified by using Volterra operator theory.