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A new method to design a fuzzy bilinear observer (FBO) with unknown inputs is developed for a class of nonlinear systems. The nonlinear system is modeled as a fuzzy bilinear model (FBM). This kind of T-S fuzzy model is especially suitable for a nonlinear system with a bilinear term. The proposed fuzzy bilinear observer subject to unknown inputs is developed to ensure the asymptotic convergence of the error dynamic using the Lyapunov method. The proposed design conditions are given in linear matrix inequality (LMI) formulation. The paper studies also the problem of fault detection and isolation. An unknown input fuzzy bilinear fault diagnosis observer design is proposed. This work is given for both continuous and discrete cases of fuzzy bilinear models. Illustrative examples are chosen to provide the effectiveness of the given methodology.

In the recent past decades, there has been important increasing interest in the state observer design of dynamic systems subjected to unknown inputs that play an essential role in robust model-based fault detection. The case of unknown input linear system has been considered by different authors [

On the other hand, since bilinear systems present the main advantage of representing an intermediate structure between linear and nonlinear models, a considerable attention has been paid for the study of this class of process [

However, many physical systems are nonlinear in nature. For such system, the use of the well-known linear techniques may reduce in bad performance and even instability. Generally, analysis for nonlinear systems is a quite involved procedure. In these last decades, a T-S fuzzy approach to represent or approximate a large class of nonlinear systems is developed [

It is of importance to design observers for linear or nonlinear systems partially driven by unknown inputs [

Considering the advantages of bilinear systems and fuzzy control, the fuzzy bilinear system (FBS) based on the T-S fuzzy model with bilinear rule consequence has attracted the interest of researchers [

In this paper, we propose a novel approach of designing a fuzzy bilinear observer for a class of nonlinear system. The nonlinear system is modeled as a fuzzy bilinear model subject to unknown inputs. This kind of T-S fuzzy model is especially suitable for a nonlinear system with a bilinear term. The considered bilinear observer is obtained by a convex interpolation of unknown input bilinear observers. This interpolation is obtained throughout the same activation functions as the fuzzy bilinear model. Based on Lyapunov theory, the synthesis conditions of the given fuzzy observer are expressed in LMI terms. The design conditions lead to the resolution of linear constraints easy to solve with existing numerical tools. The given observer is then applied for fault detection. These results are provided for both continuous-time and discrete-time T-S bilinear models.

To the best of our knowledge, the FBO synthesis and fault diagnosis for fuzzy bilinear model subjected to unknown input seem not fully addressed in the past works. Moreover, in contrast with previous works, the proposed design is given in LMI formulation solved simultaneously.

This paper is organized as follows. In Section

In this section, fuzzy bilinear systems in the continuous and discrete-time cases are introduced. Indeed, the T-S fuzzy model is described by

Then, the overall FBS can be described as follows:

Matrices

The following section is dedicated to the state estimation of the FBS (

Considering an FBS subject to unknown inputs, the FBS (

Our objective is to design T-S fuzzy bilinear observer of the fuzzy bilinear system (

The dynamics of the state estimation error is governed by

Hence, if the following constraints are satisfied

The parameter gains

The estimation error for continuous case is given by

The following theorem gives sufficient design conditions for the unknown inputs FBS (

If there exist a symmetric definite positive matrix

In order to establish the stability of the estimation error

Using (

From (

Then, the derivative of the Lyapunov function is negative if

Taking into account (

Similarly, using the following variable change:

Classical numerical tools may be used for solving the LMI problem (

For discrete-time case, the estimation error is given by

The following result gives linear conditions to design discrete-time unknown inputs DFBS (

If there exists a symmetric definite positive matrix

To prove the asymptotic convergence of the DFBS (

Sufficient conditions for the negativity of (

Substituting (

A design procedure to design FBO for both continuous and discrete cases is summarized as follows.

Solve linear constraints (

Deduce

Knowing that

In the following, the proposed observer is used for fault detection and isolation of actuator fault.

The fault detection and isolation problem for nonlinear systems is far more complicated. In this section, an unknown input fuzzy bilinear fault diagnosis observer is considered for nonlinear model in T-S fuzzy modeling. Based on proposed unknown inputs fuzzy bilinear observer, a fuzzy bilinear system affected by an actuator fault vector

The following unknown input fuzzy bilinear fault detection observer is proposed:

The determination of gain matrices in (

If the following conditions are satisfied:

Multiplying (

Taking into account the constraint (

A suitable choice of

Then, the observer gains are obtained by the following result.

If there exist a symmetric definite positive matrix

The proof of this result is similar to the one of Theorem

To illustrate the theoretical development and the design algorithm, numerical examples are proposed in the following section.

In this section, we consider two examples: the first is an academic example in discrete-time case, and the second is a physical model of an isothermal continuous stirred tank reactor (CSTR) for the Van de Vusse reactor system.

Let us consider now the following discrete system defined by

This system can be written as

Input signal.

Solving the design conditions (

Therefore, the observer gains are computed from (

These parameters define completely the observer

To show the effectiveness of the designed observer, simulation results are presented in Figures

Trajectories of (a)

Trajectories of (a)

Trajectories of (a)

It can be deduced from Figures

In this subsection, we intend to apply the proposed design to an isothermal continuous stirred tank reactor (CSTR) (see, e.g., [

The dynamics of CSTR for the Van de Vusse reactor can be described by the following nonlinear second order system:

The system (

The observer gains are obtained by solving design conditions (

Then the observer is completely defined from (

Figures

The state

The state

In this paragraph, we will consider the same system of isothermal stirred tank reactor subject to actuator fault:

Figure

Residual signal

In this paper, a bilinear observer design is proposed for a class of unknown inputs nonlinear system. Such design is based on a T-S fuzzy bilinear model representation, particularly suitable for a nonlinear system with a bilinear term. The proposed results are developed for both continuous-time and discrete-time cases. The synthesis conditions lead to the resolution of linear constraints easy to solve with existing numerical tools. The proposed unknown input bilinear observer structure is applied for fault detection. Two illustrative examples are also given.

Based on the results in the paper, interesting future studies may be extended the proposed technique to uncertain fuzzy bilinear systems or fuzzy bilinear systems with time-delay, and can also be considered the problem of pole placements to improve the performance of the proposed fuzzy bilinear observer.

_{∞}model reduction of T-S fuzzy stochastic systems