Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay

The observer design problem for nonlinear time-delay systems becomes more and more a subject of research in constant evolution Germani et al. (2002), Germani & Pepe (2004), Aggoune et al. (1999), Raff & Allgöwer (2006), Trinh et al. (2004), Xu et al. (2004), Zemouche et al. (2006), Zemouche et al. (2007). Indeed, time-delay is frequently encountered in various practical systems, such as chemical engineering systems, neural networks and population dynamic model. One of the recent application of time-delay is the synchronization and information recovery in chaotic communication systems Cherrier et al. (2005). In fact, the time-delay is added in a suitable way to the chaotic system in the goal to increase the complexity of the chaotic behavior and then to enhance the security of communication systems. On the other hand, contrary to nonlinear continuous-time systems, little attention has been paid toward discrete-time nonlinear systems with time-delay. We refer the readers to the few existing references Lu & Ho (2004a) and Lu & Ho (2004b), where the authors investigated the problem of robust H∞ observer design for a class of Lipschitz time-delay systems with uncertain parameters in the discrete-time case. Their method show the stability of the state of the system and the estimation error simultaneously. This chapter deals with observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main result lies in the use of a new structure of the proposed observer inspired from Fan & Arcak (2003). Using a Lyapunov-Krasovskii functional, a new nonrestrictive synthesis condition is obtained. This condition, expressed in term of LMI, contains more degree of freedom than those proposed by the approaches available in literature. Indeed, these last use a simple Luenberger observer which can be derived from the general form of the observer proposed in this paper by neglecting some observer gains. An extension of the presented result to H∞ performance analysis is given in the goal to take into account the noise which affects the considered system. A more general LMI is established. The last section is devoted to systems with differentiable nonlinearities. In this case, based on the use of the Differential Mean Value Theorem (DMVT), less restrictive synthesis conditions are proposed

Robust optimal design of FOPID controller for five bar linkage robot in a cyber-physical system: a new simulation-optimization approach

This paper aims to further increase the reliability of optimal results by setting the simulation conditions to be as close as possible to the real or actual operation to create a Cyber-Physical System (CPS) view for the installation of the Fractional-Order PID (FOPID) controller. For this purpose, we consider two different sources of variability in such a CPS control model. The first source refers to the changeability of a target of the control model (multiple setpoints) because of environmental noise factors and the second source refers to an anomaly in sensors that is raised in a feedback loop. We develop a new approach to optimize two objective functions under uncertainty including signal energy control and response error control while obtaining the robustness among the source of variability with the lowest computational cost. A new hybrid surrogate-metaheuristic approach is developed using Particle Swarm Optimization (PSO) to update the Gaussian Process (GP) surrogate for a sequential improvement of the robust optimal result. The application of efficient global optimization is extended to estimate surrogate prediction error with less computational cost using a jackknife leave-one-out estimator. This paper examines the challenges of such a robust multi-objective optimization for FOPID control of a five-bar linkage robot manipulator. The results show the applicability and effectiveness of our proposed method in obtaining robustness and reliability in a CPS control system by tackling required computational efforts

Sur l'observation de l'état des systèmes dynamiques non linéaires

The objective of this thesis was to develop observers synthesis methods providing nonrestrictive synthesis conditions. Three methods are proposed and different classes of systems are treated. The first one is the method of transformation into LPV system based on the use of the Differential Mean Value Theorem (DMVT). This technique, which provides nonrestrictive synthesis conditions, is extended to several classes of nonlinear systems such as nondifferentiable systems, systems with nonlinear outputs, systems with unknown inputs, time-delay systems and discrete-time systems. The only limitation related to this method lies in the fact that it is applicable only for nonlinearities with bounded jacobian. In order to overcome this limitation, a second method is obtained by combining the DMVT technique with a new structure of generalized Luenberger observers. Thanks to this structure, new synthesis conditions are established. These conditions are valid even if the jacobian of the nonlinearity is not bounded. In addition, a new observers synthesis method for discrete-time systems is also proposed. This method uses conjointly the Lipschitz condition with the Lyapunov standard function. Improvements, which allow to obtain nonrestrictive synthesis conditions, are proposed by using a new more general Lyapunov function (which takes account of the nonlinearity of the system) and with a generalized Luenberger observer (OLG) which allows to reduce the effect of the Lipschitz constant. Finally, the obtained results are validated by an application to synchronization and encoding/decoding in chaotic communication systemsL'objectif de cette thèse était de développer des méthodes de synthèse d'observateurs offrant des conditions de synthèse non contraignantes. Trois méthodes ont été proposées et différentes classes de systèmes ont été traitées. La première est la méthode de transformation en système LPV basée sur l'utilisation du théorème des accroissements finis (DMVT). Cette technique, qui fournit des conditions de synthèse non restrictives, est étendue à plusieurs classes de systèmes non linéaires tels que les systèmes non différentiables, les systèmes à sorties non linéaires, les systèmes à entrées inconnues, les systèmes à retard et les systèmes à temps discret. La seule limitation liée à la méthode est le fait qu'elle n'est applicable que pour des non-linéarités à jacobiennes bornées. Afin de surmonter cette limitation, une deuxième méthode est obtenue en combinant la technique du DMVT avec une nouvelle structure d'observateurs de type Luenberger généralisés. Grâce à cette structure, de nouvelles conditions de synthèse sont établies. Ces conditions sont valables même si la jacobienne de la non-linéarité n'est pas bornée. Par ailleurs, une nouvelle méthode de synthèse d'observateurs spécifique aux systèmes à temps discret est également proposée. Cette méthode utilise la condition de Lipschitz conjointement avec la fonction de Lyapunov standard. Des améliorations, qui permettent d'obtenir des conditions de synthèse non contraignantes, sont ensuite proposées en faisant appel à une nouvelle fonction de Lyapunov plus générale (qui tient compte de la non-linéarité du système) et à un observateur de Luenberger généralisé (OLG) qui permet de réduire l'effet de la constante de Lipschitz. Enfin, les résultats obtenus sont validés par une application à la synchronisation et au cryptage/décryptage dans les systèmes de communications chaotique

On the observation of the state of nonlinear dynamical systems

L'objectif de cette thèse était de développer des méthodes de synthèse d'observateurs offrant des conditions de synthèse non contraignantes. Trois méthodes ont été proposées et différentes classe de systèmes ont été traitées. La première est la méthode dePas de résum

Observer design for nonlinear systems by using high-gain and LPV/LMI-based technique

International audienceThis note deals with observer design for nonlinear systems. The main contribution of this work consists in providing a new high-gain observer design method with lower gain compared to the standard high-gain observer. This new observer, called HG/LMI observer is obtained by combining the standard high-gain methodology with the LPVLMI-based technique. We will show through analytical developments how the new observer provides a lower gain. A numerical example is given to illustrate the performance of the new HG/LMI observer

Observer design for non-globally Lipschitz nonlinear systems using Hilbert projection theorem

Presented at 61st IEEE Conference on Decision and Control, CDC 2022, Cancun, Mexique, December 6-9, 2022International audienceThis paper deals with observer design for a class of nonlinear systems. More specifically, we will propose a mathematically rigorous technique to handle systems having non-globally Lipschitz properties on the whole set $\mathbb{R}^{n}$. The unique assumption on the nonlinearity is to be globally Lipschitz on a compact convex set $\Omega \subset \mathbb{R}^{n}$, in which lives the system state. The idea consists in extending the nonlinear function to globally Lipschitz one from $\Omega$ to the whole space $\mathbb{R}^{n}$. Such an extension is performed by using the famous {\it Hilbert} projection theorem, which generalizes some existing results in the literature. The projection is then involved in the observer structure to overcome the non-satisfaction of the global property by the original nonlinear function. More importantly, to overcome the conservatism related to the boundedness of the system states, an extension to systems having only some bounded states is proposed under different but less conservative assumptions. It is shown that all the observer design methods using global Lipschitz property can be applied straightforwardly without changing the synthesis conditions

LMI-Based Observer Design for Non-Globally Lipschitz Systems Using Kirszbraun-Valentine Extension Theorem

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Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: An LMI approach

International audienceIn this paper, we address the problem of observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main contribution lies in the use of a new structure of the proposed observer with a novel Lyapunov-Krasovskii functional. Thanks to these designs, new nonrestrictive synthesis conditions, expressed in terms of linear matrix inequalities (LMIs), are obtained. Indeed, the obtained LMIs contain more degree of freedom than those established by the approaches available in the literature which consider a simple Luenberger observer with a simple Lyapunov function for the stability analysis. An extension of the presented result to H-inf performance analysis is given in order to take into account the noise (if it exists) affecting the considered system

Observer Design for Non-Globally Lipschitz Nonlinear Systems Using Hilbert Projection Theorem

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