Original articlesState and unknown inputs estimation for Takagi–Sugeno systems with immeasurable premise variables: Proportional Multiple Integral observer design
Introduction
The multiple model approach is an elegant and a powerful way to tackle the difficulties of the control and the observation of real complex nonlinear systems [[13], [15]].
The basic idea of this technique is to apprehend the behavior of nonlinear systems by a family of local models (linear or affine) in different operating zones. The contribution of each local model is quantified by an activation function depending on measurable variables (inputs or outputs of the system) or immeasurable variables (state variables).
Various methods have been developed to obtain the multiple model of a system
the linearization of nonlinear models around chosen operating points and using adequate weighting functions [[1], [12], [13], [14]],
the identification of the parameters of local models and those of weighting functions using experimental data [[1], [4], [20]],
the Sector Nonlinearity Transformation (SNT) which gives an exact representation of nonlinear system without any loss of information [19].
Most of the existing works on multiple models are dedicated to those with measurable premise variables. In [10] and [18], quadratic and non-quadratic stability conditions for T–S fuzzy models are established using the Lyapunov theory and the Linear Matrix Inequality (LMI) formalism. In [2] and [17], the authors generalized the unknown input observer proposed by [8] to linear systems. The problem of state estimation and the diagnosis of multiple model systems have been treated in [3] and [10]. In [16], a sliding mode observer is developed for uncertain T–S models.
In the literature only few works dealt with multiple models with immeasurable premise variables in spite of their advantages
description of an important class of nonlinear systems [21],
use of one model for the diagnosis of actuator and sensors faults,
in the context of encryption using chaotic multiple models, some authors [9] have noticed an improvement of transmission security.
In [5], the Thau–Luenberger observer is generalized to T–S systems with immeasurable premise variables. The authors of [6] proposed a sliding mode observer for this class of systems. In the works of [11], two approaches for the design of multiple observers with immeasurable premise variables have been developed: The Lipschitz approach and the approach.
In this paper, we propose the design of a Proportional Multiple Integral (PMI) observer based on the Lipschitz approach for Takagi–Sugeno systems with immeasurable premise variables.
The main contribution of this paper is the introduction of an output error injection in the premise variable of the PMI observer to relax the conditions of the exponential convergence of the state estimation error.
We organize the remainder of this paper as follows. Section 2 introduces the problem and some background. In Section 3, the PMI observer is designed and the LMI conditions of exponential convergence of state estimation error are formulated. The efficacy of the designed observer is illustrated by a simulation example in Section 4. Finally, Section 5 ends the paper highlighting the main achievements and the open problems of the work.
Section snippets
Problem statement
Consider the Takagi–Sugenomodel with immeasurable premise variables subjected to unknown inputs where is the state vector, the input vector, the unknown input vector and the output vector.
, , , and are constant matrices with appropriate dimensions.
() are the activation functions depending on the state of the system. These functions verify the following convexity properties
The scalar
Proportional multiple integral observer design
In order to assess the exponential convergence of the observer (10), let us consider the state estimation error
The system (6) can be expressed as where
The dynamics of the state estimation error is where:
A. Case 1: The matrix is invertible
Theorem 1 The state estimation error between the system (1) and the PMI observer (10) converges exponentially to zero with rate if there exist two
Simulation example
To illustrate the method, we consider a chaotic nonlinear system represented by a multiple model with immeasurable premise variables. This multiple model is composed of two local models with three states and two outputs where
The activation functions are defined by
The simulation of the multiple model without the unknown input and with
Conclusion and future works
In this article, state and unknown input estimation of Takagi–Sugeno systems with immeasurable premise variables via a Proportional Multiple Integral observer is treated.
The stability conditions expressed in terms of Linear Matrix Inequalities are significantly relaxed by the introduction of an output error injection in the premise variable of the observer.
The efficiency of the PMI observer is illustrated by a simulation example of a chaotic system.
The future works will concern
– the
References (21)
- et al.
A non-iterative neuro-fuzzy based identification method for Hammerstein processes
J. Process Control
(2005) - et al.
Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems
Fuzzy Sets and Systems
(2007) H filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach
Fuzzy Set Syst.
(2009)- et al.
Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models
IEEE Trans. Syst. Man Cybern. B
(2002) - A. Akhenak, M. Chadli, J. Ragot, D. Maquin, Estimation of state and unknown inputs of a nonlinear system represented by...
- Akhenak, M. Chadli, J. Ragot, D. Maquin, Design of sliding mode unknown input observer for uncertain Takagi-Sugeno...
Fuzzy Modeling for Control
(1998)- P. Bergsten, R. Palm, Thau-Luenberger observers for T-S fuzzy systems, in: 9th IEEE International Conference on Fuzzy...
- P. Bergsten, R. Palm, D. Driankov, Fuzzy observers, in: IEEE International Fuzzy Systems Conference, Melbourne...
- et al.
Linear Matrix Inequalities in Systems and Control Theory
(1994)
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