Interval estimation of switched Takagi-Sugeno systems with unmeasurable premise variables

International audienceThis paper deals with interval observers design for nonlinear switched systems. The nonlinear modes are represented by the Takagi-Sugeno (T-S) fuzzy models with premise variables depending on unmeasurable terms, e.g. the state vector. This T-S structure can be used to represent exactly a nonlinear switched system in a compact set of the state space. The introduced method in this paper allows to compute the lower and upper bounds of the system state under the assumption that the disturbances as well as the measurement noises are unknown but bounded. First, the stability conditions of the proposed T-S interval observers are developed via Linear Matrix Inequality (LMI) formulations to ensure the convergence of the nonnegative observation error dynamics. Then, changes of coordinates are employed to relax the restrictive requirement of nonnegativity constraints. Theoretical results are applied to a numerical example to illustrate the effectiveness of the proposed method

Interval Estimation for Linear Switched System

International audienceIn this paper, the problem of state estimation is investigated for linear switched system, a subclass of hybrid systems. It will be shown that the interval observer is very often exists under moderate conditions at least in discrete time instants from continuous-time measurements. The novelty consists in proposing new conditions of cooperativity for switched systems in discrete time instants, which guarantee errors nonnegativity of interval observation. The efficiency of the interval observers is shown through simulation examples

Interval estimation for second-order delay differential equations with delayed measurements and uncertainties

International audienceThe interval estimation design is studied for a second-order delay differential equation with position delayed measurements, uncertain input and initial conditions. The proposed method contains two consecutive interval observers. The first one estimates the interval of admissible values for the position without delay for each instant of time using new delay-dependent conditions on positivity. Then derived interval estimates of the position are used to design the second observer estimating an interval of admissible values for the velocity of the considered dynamical system. The results are illustrated by numerical experiments for an example

Robust Output Feedback MPC: An Interval-Observer Approach

International audienceIn this work, we address the problem of output-feedback Model Predictive Control (MPC) of constrained, linear, discrete-time systems corrupted by additive perturbations on both state and output. The use of estimated variables in MPC is challenging and computationally expensive due to constraint satisfaction. To overcome this issue, the proposed approach incorporates interval observers on the MPC scheme to cope with uncertainty, leading to a novel, simple and very intuitive methodology providing robust constraint satisfaction with reduced computational complexity

Interval estimation for systems with time delays and algebraic constraints

International audienceThe problem of interval observer design is addressed for a class of descriptor linear systems with delays. Two sets of conditions are proposed. First, an interval observation for any input in the system is provided. Second, the control input is designed together with the observer gains in order to guarantee interval estimation and stabilization simultaneously. Efficiency of the proposed approach is illustrated by numerical experiment

On Design of Interval Observers with Sampled Measurement

International audienceNew design of interval observers for continuous-time systems with discrete-time measurements is proposed. For this purpose new conditions of positivity for linear systems with sampled feedbacks are obtained. A sampled-data stabilizing control is synthesized based on provided interval estimates. Efficiency of the obtained solution is demonstrated on examples

Interval observers design for hybrid and biological systems

International audienceThis work deals with interval observer design techniques. In the firstpart, the problem of interval observer design is studied for a classof linear hybrid systems. Several observers are proposed orientedon different conditions of positivity and stability for estimation errordynamics. Efficiency of the proposed approach is demonstrated bycomputer experiments for academic and bouncing ball systems. Notethat interval observer design techniques for linear hybrid systemshave been developed for the first time in the present work. The secondpart is devoted to the interval estimation of sequestred infected erythrocytesin plasmodium falciparum malaria patients. An advantageof the interval approaches in this case is that they give a bound ofthe errors at any time, which can be controlled in order to ensurethe positivity of the state estimates of the system. Thus, intervalestimation is very close to the reality in this case and has not beendeveloped before the present work. An interval observer in orderto estimate the sequestered parasite population is proposed in thisreport. Its efficiency is demonstrated by computer simulations

Stabilization of linear impulsive systems under dwell-time constraints: Interval observer-based framework

International audienceThe problem of interval observer design is studied for a class of linear impulsive systems. Ranged and minimum dwell-time constraints are considered under detectability assumption. The first contribution of this paper lies in designing interval observers for linear impulsive systems under ranged and minimum dwell-time constraints, and investigating positivity of the estimation error dynamics in addition to stability. Several observers are designed oriented on different conditions of positivity and stability for estimation error dynamics. The boundedness of the estimation error (input-to-state stability property) and the observer stability conditions are stated as infinite-dimensional linear programming problems. Next, an output stabilizing feedback design problem is discussed, where the stability is checked using linear matrix inequalities (LMIs). Efficiency of the proposed approach is demonstrated by computer simulations for a commercial electric vehicle equipped with a low power range extender fuel cell, a bouncing ball, an academic linear impulsive system and for Fault Detection and Isolation (FDI) and Fault-Tolerant Control (FTC) of a power split device with clutch for heavy-duty military vehicles

Invariance positive et observateur intervalles appliqués aux systèmes linéaires à retards sous contraintes

RÉSUMÉ Cette thèse porte sur l’étude du problème de la commande avec contraintes des systèmes linéaires continus à retards. Deux approches dans la littérature se sont données à développer des méthodes adéquates pour examiner la stabilité et contribuer à des procédures et outils de stabilisation. La première, considère l’effet de la saturation, tandis que la deuxième approche, basée sur la théorie d’invariance positive, repose principalement sur la conception d’une loi de contrôle non saturante ayant un comportement linéaire dans le domaine des contraintes. Des résultats concernant l’application du concept d’invariance positive à la stabilisation des systèmes à retards soumis à des contraintes ont été développés, mais restent restrictifs, de fait qu’ils sont indépendants du retard, un paramètre essentiel du système. On développe alors, dans la première partie de cette thèse des conditions nécessaires et suffisantes dépendantes du retard afin de garantir l’invariance positive de domaine des contraintes par rapport aux trajectoires de systèmes autonomes à retard. Ce résultat repose sur la transformation de premier ordre, basée sur la formule de Newton-Leibniz, de système original à retard discret, en un système à retard distribué. Une fonction de Lyapunov-Razumikhin associée au système à retard distribué garantissant la stabilité asymptotique dépendante de retard de système original est proposée. L’objectif principal visé dans la deuxième partie de cette thèse est d’appliquer le résultat du concept d’invariance positive dépendante du retard au problème de la commande sous contraintes, dissymétriques ainsi que symétriques, des systèmes à retards. Ainsi des conditions permettant la synthèse d’un régulateur par retour d’état stabilisant le système en boucle fermée en présence des contraintes, sont données. Ces conditions permettent de formuler un algorithme basé sur des schémas de Programmation Non Linéaire (NLP), ayant pour objectif la détermination du régulateur stabilisant le système en boucle fermée avec une borne maximale du retard. En effet la loi de retour d’état calculée assure, d’une part, la stabilité asymptotique de système sans retard, et d’autre part, la maintenir pour une valeur d’une borne maximale de retard, tout en respectant les contraintes : c’est la loi de commande sous contraint robuste vis à vis le retard. Les résultats obtenus sont intéressants et plus généraux que ceux développés dans la littérature. La troisième partie de cette thèse montre, pour la première fois à notre connaissance, que les observateurs intervalles, en appliquant le concept d’invariance positive, peuvent apporter des réponses intéressantes au problème de la commande sous contraintes des systèmes linéaires à retards, variable dans le temps.----------ABSTRACT In this thesis, the stabilization problem of linear continuous-time delay system with constrainted control is studied. There are two main approaches in the literature dealing with the problem of performance and stability of dynamical constrained control systems. The first one considers the effect of saturation while guaranteeing asymptotic stability. The second one, so-called positive invariance approach, is based on the design of the control law which works inside a region of linear behavior where saturations do not occur. Most of the works related to positive invariance concept have been developed for time delay systems with constrained control, but remain so restrictive, given that they are independent of delay, which is an essential parameter of the system. In the first part of this thesis, the necessary and sufficient algebraic conditions with delay dependence allowing to obtain the largest positively invariant set of delay system are given. The results can include information on the size of delay, and therefore, can be delay dependence positively invariant conditions. Based on the Newton-Leibniz formula, these results use a transformation form an original system with discrete delay to a system with distributed delay. A Lyapunov-Razumikhin function for system with distributed delay, in order to guarantee the asymptotic stability of the original system is proposed. The second part of this thesis, is to apply the concept of the delay dependent positive invariance to the robust regulator problem of continuous time delay system with symmetric and non-symmetric constraints. In fact the synthesis of state-feedback controllers is solved based on delay-dependent positively invariant set of system in closed-loop. We first obtain the necessary and sufficient algebraic conditions with delay dependence allowing to obtain the largest positively invariant set of delay systems, then we convert the constrained control problem into a Non-Linear Programming (NLP) problem with delay the objective function to be maximized. Indeed the control is firstly chosen in order to stabilize the closed loop system, free of delay, then to guarantee the asymptotic stability of the closed loop system with delay-dependence. To the best of our knowledge, it is the first time, that the output stabilization problem for time-varying delay systems with constrained control based on the interval observer technique by using the dependent delay positive invariance concept is studied. Hence, first both matrices observer gain, the lower and the upper, are obtained by solving a Sylvester’s matrix equation. Second, the interval observer is developed and guaranteed the positivity of the upper and lower observations errors

Interval estimation for uncertain systems with time-varying delays

International audienceThe estimation problem for uncertain time-delay systems is addressed. A design method of reduced-order interval observers is proposed. The observer estimates the set of admissible values (the interval) for the state at each instant of time. The cases of known fixed delays and uncertain time-varying delays are analyzed. The proposed approach can be applied to linear delay systems and nonlinear time-delay systems in the output canonical form. It involves the properties of quasi-monotone/Metzler/cooperative systems. In this framework, it is shown that if under a suitable coordinate transformation the delay-free subsystem is cooperative, then the delayed estimation error dynamics inherits this property. The conditions to find the observer gains are formulated in the form of LMI. The framework efficiency is demonstrated on examples of nonlinear systems