Mathematical Problems in Engineering

1. Introduction

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 [1–4], and many types of full order and reduced order unknown input observers (UIOs) are now available.

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 [5, 6]. The observer design for bilinear systems with unknown input has been an important research topic during the last years. The works of [7, 8] have considered the design of UIO for bilinear systems in which the error estimation dynamics is linear. Under suitable transformation, the design of UIO for bilinear systems proposed in [9, 10] is equivalent to the design of UIO for linear systems in which the unknown input links with the non measurable states.

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 [11–14]. Then, in the field of stability analysis and stabilization, many works including delay and uncertainty have been developed and applied in a lot of practical situations [15–20]. For the state estimation problems and its application in fault diagnosis for uncertain T-S fuzzy models, robust of fault detection filter is developed in LMI terms [21–25].

It is of importance to design observers for linear or nonlinear systems partially driven by unknown inputs [26–29]. Such a problem arises in systems subject to disturbances and in many applications such as robust control, fault detection and isolation (FDI), system supervision, and fault-tolerant control. The design of observers for nonlinear systems is a challenging problem and has received a considerable amount of attention in the literature. In many approaches, the transform of nonlinear systems to bilinear T-S models provides a better approximation than classical T-S models [30, 31]. Motivated by this, we consider in this work bilinear T-S fuzzy models whose consequent parts are bilinear systems with 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 [30, 32–35]. For example, robust stabilization for the T-S FBS has studied in [30, 33, 34], and extension to the T-S FBS with time delay is given in [35]. An adaptive fuzzy-bilinear-observer- (FBO-) based synchronization design for generalized Lorenz system (GLS) was also examined in [36], and in [32] an observer is designed using iterative procedure.

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 2 the considered structure of the FBS is presented. In Section 3, the synthesis of fuzzy bilinear observers with unknown input in continuous and discrete cases is presented. Section 4 is devoted to the problem of fault detection by using unknown input fuzzy bilinear fault diagnosis observer. In Section 5, two examples to illustrate the proposed approach are proposed. The practical use of the theoretic study is illustrated by applying the proposed design to an isothermal continuous stirred tank reactor (CSTR).

Notation. In the rest of the paper, the following useful notation is used: denotes the set of real numbers, denotes the transpose of the matrix , denotes symmetric positive definite matrix, denotes the Moore-Penrose inverse of , and denotes symmetric matrix where . The operator denotes the time derivative for continuous-time models, that is, , and the shift operator for discrete-time models, that is, . For simplicity, in the sequel, we will simply write .

2. General Structure of a Fuzzy Bilinear Model

In this section, fuzzy bilinear systems in the continuous and discrete-time cases are introduced. Indeed, the T-S fuzzy model is described by if-then rules and used to present a fuzzy bilinear system. The th rule of the FBS for nonlinear systems is represented by the following form: then where denotes the th fuzzy rule for all , is the number of if-then rules, are the premise variables assumed to be measurable, and is the membership degree of in the fuzzy set . is the state vector, is the control input, and is the system output. The matrices , , are known matrices.

Then, the overall FBS can be described as follows: with , and verify the following properties:

Remark 2.1. Matrices , , , and can be obtained by using the polytopic transformation [37]. The advantage of this method is (i) to lead to a bilinear transformation of the nonlinear model without any approximation error and (ii) to reduce the number of local models compared to other methods [33].

The following section is dedicated to the state estimation of the FBS (2.3), subject to unknown inputs, where the vector is assumed depending on measurable variables.

3. Design of an Unknown Input Fuzzy Bilinear Observer

Considering an FBS subject to unknown inputs, the FBS (2.3) can be rewritten as follows: where is the unknown inputs vector and is a matrix with full column rank. In order to estimate state of (3.1), we propose a full-order observer of the form where is the estimated state vector and the activation functions are the same as those used in FBS (3.1). , , , , and are constant matrices with appropriate dimensions.

Our objective is to design T-S fuzzy bilinear observer of the fuzzy bilinear system (3.1) for system (3.1) subject to unknown input such that the estimation error converges towards zero when . Note that estimation error can be rewritten as follows: where .

The dynamics of the state estimation error is governed by

Hence, if the following constraints are satisfied the estimation error becomes

The parameter gains , , , , and should be determined such that the state estimate converges asymptotically to system state .

3.1. Continuous-Time Case

The estimation error for continuous case is given by

The following theorem gives sufficient design conditions for the unknown inputs FBS (3.1).

Theorem 3.1. If there exist a symmetric definite positive matrix , matrices , , , such that the following linear conditions hold for all : then the state estimation of the CFBO (3.2) converges globally asymptotically to the state of the CFBS (3.1). The observer gains are determined by

Proof. In order to establish the stability of the estimation error , let us consider the following Lyapunov function:
Using (3.12), the derivative of the Lyapunov function (3.18) is given by
From (3.6) and using (3.10), we get with
Then, the derivative of the Lyapunov function is negative if
Taking into account (3.10) and considering the variable change: we get the LMI (3.13). Taking account (3.10) and (3.23), equality (3.16) is derived from (3.9).
Similarly, using the following variable change: we get equalities (3.14) and (3.15) from (3.8) and (3.7), respectively, which ends the proof.

Remark 3.2. Classical numerical tools may be used for solving the LMI problem (3.13) subject to linear equality constraints (3.14)–(3.16). Solving this linear problem allows to deduce the observer parameters from , , , , and as mentioned by (3.17).

3.2. Discrete-Time Case

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

The following result gives linear conditions to design discrete-time unknown inputs DFBS (3.1).

Theorem 3.3. If there exists a symmetric definite positive matrix , matrices , , , such that the following linear conditions hold for all then the state estimation of the DFBO (3.2) converges globally asymptotically to the state of the DFBS (3.1). The observer gains are determined by (3.17).

Proof. To prove the asymptotic convergence of the DFBS (3.1), sufficient conditions are derived using quadratic Lyapunov function (3.18). Indeed, the variation along the solution of (3.26) is
Sufficient conditions for the negativity of (3.31) are
Substituting (3.20) in (3.32), we obtain Thus by taking account the expression of (3.10), introducing the same variables change (3.23)-(3.24), and then applying Schur complement to (3.33), we get LMI conditions (3.27). Equality constraints (3.28)–(3.30) are obtained as previously mentioned which ends the proof.

3.3. Design Algorithm

A design procedure to design FBO for both continuous and discrete cases is summarized as follows. (1)Solve linear constraints (3.13)–(3.16) for continuous-time (or (3.27)–(3.30) for discrete-time) case to get , , , and .(2)Deduce .(3)Knowing that , the observer gains are computed as follows:

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

4. Fault Detection and Isolation for Fuzzy Bilinear System

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 is considered. In this section, a residual generation is considered in order to be sensitive to fault vector and insensitive to the unknown inputs . Then, the considered system is as follows: where represents the vector of faults and the represents matrix with appropriate dimensions.

The following unknown input fuzzy bilinear fault detection observer is proposed: where represents the estimated vector and is the output signal called the residual.

The determination of gain matrices in (4.2) will be determined to ensure the convergence of the estimated errors. In order to describe the dynamic of unknown input fuzzy bilinear fault detection observer (4.2), the state estimation error is defined by . Then, from (4.1) and (4.2), we have

If the following conditions are satisfied: we get

Multiplying (4.8) by we get

Taking into account the constraint (4.7), (4.10) becomes or equivalently Then we get where is the pseudoinverse of and is an arbitrary matrix. Substituting (4.13) into (4.8) leads to

A suitable choice of and satisfying the relation (4.14) is

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

Theorem 4.1. If there exist a symmetric definite positive matrix , matrices , , such that the following linear conditions hold for all or then the state estimation of the FBO (4.2) converges globally asymptotically to the state of the FBS (4.1). The observer gains are determined by where , , and are given in (4.16), (4.15), and (4.13), respectively.

Proof. The proof of this result is similar to the one of Theorem 3.1.

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

5. Simulation Examples

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.

5.1. Example 1: Synthesis of a Discrete Fuzzy Bilinear Observer

Let us consider now the following discrete system defined by where

This system can be written as with Using the polytopic transformation [37] with and , the DFBS can be described as (2.3) where with is the input signal given in Figure 1, and is the unknown input taken as a sine wave signal of amplitude 0.1 and frequency 50 rad/s.

Figure 1: Input signal.

Solving the design conditions (3.27)–(3.30), we get

Therefore, the observer gains are computed from (3.17) as follows: