Benoı̂t Marx

Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi Sugeno model with unmeasurable premise variables

In this paper, a proportional integral (PI) and a proportional multiple integral observer (PMI) are proposed in order to estimate the state and the unknown inputs of nonlinear systems described by a Takagi-Sugeno model with unmeasurable premise variables. This work is an extension to nonlinear systems of the PI and PMI observers developed for linear systems. The state estimation error is written as a perturbed system. First, the convergence conditions of the state estimation errors between the system and each observer are given in LMI (Linear Matrix Inequality) formulation. Secondly, a comparison between the two observers is made through an academic example.

Fault detection, isolation and estimation for Takagi–Sugeno nonlinear systems

Abstract This article is dedicated to the problem of fault detection, isolation and estimation for nonlinear systems described by a Takagi–Sugeno (T–S) model. One of the interests of this type of models is the possibility to extend some tools and methods from the linear system case to the nonlinear one. The principle of the proposed strategy is to transform the problem of simultaneously minimizing the perturbation effect and maximizing the fault effect, on the residual vector, in a simple problem of L 2 - norm minimization. A linear system is used to define the ideal response of the residual signal to the fault. Then the aim is to synthesize a residual generator that both minimizes the difference between real and ideal responses and the influence of the disturbance. The minimization problem is formulated by the bounded real lemma (BRL) and linear matrix inequality (LMI) formalism. After studying the general framework, a special case of systems with actuator and sensor faults is considered where the fault incidence matrix is not full column rank. Simulation examples are given to illustrate the proposed method. Finally, Polyas theorem is used to reduce the conservatism of the proposed result. The obtained relaxation is also illustrated by a numerical example.

State and unknown input estimation for nonlinear systems described by Takagi-Sugeno models with unmeasurable premise variables

This paper presents a new method to synthesize observers for continuous time nonlinear systems described by Takagi-Sugeno (TS) model with unmeasurable premise variables. First, convergence conditions are established in order to guarantee the convergence of the state estimation error. These conditions are given in Linear Matrix Inequality (LMI) formulation. Secondly, a classical Proportional Integral Observer (PIO) is extended to the considered nonlinear systems in order to estimate the state and the unknown inputs (UI).

An approach for the state estimation of Takagi-Sugeno models and application to sensor fault diagnosis

In this paper, a new method to design an observer for nonlinear systems described by Takagi-Sugeno (TS) model, with unmeasurable premise variables, is proposed. Most of existing work on TS models consider models with measurable decision variables. As a consequence, these works cannot be applied when the decision variables are not available to measurement. The idea of the proposed approach is to rewrite the TS model with unmeasurable premise variable into an uncertain TS model by introducing the estimated state in the model. The convergence of the state estimation error is studied using the Lyapunov theory and the stability conditions are given in terms of Linear Matrix Inequalities (LMIs). Finally, an academic example is given to illustrate the proposed approach, with an application to sensor fault detection and isolation using an observer bank.

Observer based actuator fault tolerant control for nonlinear Takagi-Sugeno systems : an LMI approach

A new actuator fault tolerant control strategy is proposed for nonlinear Takagi-Sugeno (T-S) systems. The control law aims to compensate the actuator faults and allows the system states to track a reference corresponding to a fault free situation. The design of such a control law requires the knowledge of the faults, this task is achieved with a proportional integral observer (PIO). The robust stability of the system with the fault tolerant control law is analyzed with the Lyapunov theory and the ℒ2 optimization. Sufficient stability conditions are obtained in terms of linear matrix inequalities (LMIs). The gains of the FTC are obtained by solving these LMIs. A simulation example is finally proposed.

State estimation of nonlinear systems using multiple model approach

This paper addresses the problem of state estimation of nonlinear systems described by a Takagi-Sugeno multiple model with unmeasurable decision variables. The method is based on the reformulation of the multiple model in an equivalent form. First, the convergence conditions of the state estimation error are established using the Lyapunov method and they are expressed in LMI formulation. Secondly, performances of the observer are enhanced by pole clustering and L2 attenuation of bounded exogenous disturbances. Finally, the method is applied to estimate the state of a link flexible joint robot.

Advances in observer design for Takagi-Sugeno systems with unmeasurable premise variables

This paper proposes a new approach of observer design for nonlinear systems described by a Takagi-Sugeno model. Its main contribution concerns models with premise variables depending on the system states which are completely or partially unknown. This case is more difficult than when the premise variables are known or measured. Indeed, in this case, weighting functions of the observer depend on state estimates and the state estimation error is then governed by a Lipschitz nonlinear system. Here, two main results are established. Firstly, relaxed stability conditions are provided, using a nonquadratic Lyapunov function, to guarantee asymptotic stability of the observer. This aims to reduce the conservativeness compared to the existing works and enhance the maximal admissible Lipschitz constant for which the linear matrix inequality (LMI) conditions are feasible. Secondly, the Input-to-State Stability concept combined to a nonquadratic Lyapunov function are used to guarantee a bounded state estimation error which relaxes the conservativeness related to the Lipschitz constant. The robustness aspect is dealt with respect to some bounded modeling uncertainties and additive bounded perturbations. The stability conditions are expressed in terms of LMI.

State estimation of the three-tank system using a multiple model

This paper addresses the exact transformation of nonlinear systems into a multiple model form with unmeasurable premise variables. The multiple model structure serves to treat the observability and the state estimation problem of nonlinear systems. Using a method with no information loss, a nonlinear system is transformed into a multiple model, depending on the choice of premise variables. It is a key point, since it allows to choose, between different multiple model forms, the one that has suitable structure and properties, in order to design an observer. The convergence conditions of the state estimation error are expressed in LMI formulation using the Lyapunov method. These proposals are investigated and applied to the three-tank system.

New fault tolerant control strategy for nonlinear systems with multiple model approach

This paper addresses a new methodology to construct a fault tolerant control (FTC) in order to compensate actuator faults in nonlinear systems. This approach is based on the representation of the nonlinear model with a multiple model under Takagi-Sugenos form. The proposed control requires a simultaneous estimation of the system states and of the occurring actuator faults. The performance of the control depends on the quality of the estimations, indeed, it is important to estimate accurately and rapidly the states and the faults. This task is then performed with an Adaptive Fast State and Fault Observer (AFSFO). The stability conditions are established with Lyapunov theory and expressed in linear matrix inequality formulation to ease the design of the FTC. Furthermore, relaxed stability conditions are given with the use of the Polyas theorem.

Contribution to the constrained output feedback control

In this paper, a Takagi-Sugeno model is used to represent the nonlinear behaviour of an actuator with saturation constraint. The control design is based on an output feedback controller (static or dynamic) depending on the saturation levels. Stabilization conditions are derived with the Lyapunov method and expressed in terms of linear matrix inequalities. Stabilisation conditions are addressed using a descriptor approach for the system modelling. An academic example is also presented with a comparison between different approaches.