Unknown Input Observer Design for Fuzzy Bilinear System : An LMI Approach

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.


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 continuoustime 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, X T denotes the transpose of the matrix X, X 0 denotes symmetric positive definite matrix, X −1 denotes the Moore-Penrose inverse of X, and A * B C denotes symmetric matrix where * B T .The operator δ denotes the time derivative for continuous-time models, that is, δ x t ẋ t , and the shift operator for discrete-time models, that is, δ x t x t 1 .For simplicity, in the sequel, we will simply write h i ξ t h i t .

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 ith rule of the FBS for nonlinear systems is represented by the following form: where R i denotes the ith fuzzy rule for all i {1, . . ., r}, r is the number of if-then rules, ξ i t are the premise variables assumed to be measurable, and F ij ξ j t is the membership degree of ξ j t in the fuzzy set F ij .x t ∈ n is the state vector, u t ∈ is the control input, and y t ∈ p is the system output.The matrices Then, the overall FBS can be described as follows: Remark 2.1.Matrices A i , B i , N i , and C 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 ξ t is assumed depending on measurable variables.

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 d t ∈ q is the unknown inputs vector and F i ∈ n×q 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 x t ∈ n is the estimated state vector and the activation functions are the same as those used in FBS 3.1 .H i , M i , L i , J i , and E 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 e t x t − x t 3.3 converges towards zero when t → ∞.Note that estimation error can be rewritten as follows: where T I n EC.
The dynamics of the state estimation error is governed by Hence, if the following constraints are satisfied TF i 0, 3.9 T I n EC, 3.10 the estimation error becomes

3.11
The parameter gains H i , M i , L i , J i , and E should be determined such that the state estimate x t converges asymptotically to system state x t .

Continuous-Time Case
The estimation error for continuous case is given by

3.12
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 P , matrices W i , V i , S, R i such that the following linear conditions hold for all i 1, . . ., r: 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 3.17 Proof.In order to establish the stability of the estimation error e t , let us consider the following Lyapunov function: V t e T t Pe t , P P T > 0.

3.18
Using 3.12 , the derivative of the Lyapunov function 3.18 is given by

3.19
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: S PE, 3.23 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 P , W i , V i , S, and R i as mentioned by 3.17 .

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

3.26
The following result gives linear conditions to design discrete-time unknown inputs DFBS 3.1 .
Thus by taking account the expression of T 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.

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 W i , V i , S, R i and P > 0.
2 Deduce K i P −1 W i .
3 Knowing that T I EC, the observer gains are computed as follows:

3.34
In the following, the proposed observer is used for fault detection and isolation of actuator fault.

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 f t ∈ nf is considered.In this section, a residual generation is considered in order to be sensitive to fault vector f t and insensitive to the unknown inputs d t .Then, the considered system is as follows: where f t represents the vector of faults and the G i represents matrix with appropriate dimensions.
The following unknown input fuzzy bilinear fault detection observer is proposed: where z t represents the estimated vector and r t 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 e t z t − Tx t .Then, from 4.1 and 4.2 , we have

4.3
If the following conditions are satisfied: T F i 0, 4.7
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.

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.

Example 1: Synthesis of a Discrete Fuzzy Bilinear Observer
Let us consider now the following discrete system defined by This system can be written as

Mathematical Problems in Engineering
Using the polytopic transformation 37 with −0.2 < x 3 t < 0.2 and 0 < x 2 3 t < 0.33, the DFBS can be described as 2.3

5.5
where with

5.7
u t is the input signal given in Figure 1, and d t is the unknown input taken as a sine wave signal of amplitude 0. Therefore, the observer gains are computed from 3.17 as follows:

Mathematical Problems in Engineering
These parameters define completely the observer x t z t − Ey t .

5.10
To show the effectiveness of the designed observer, simulation results are presented in Figures 2, 3, and 4 for initial conditions given by x 0 0.5 0.5 0.5 T and x 0 1 1 1 T .It can be deduced from Figures 2, 3, and 4 that the proposed observer succeeds to track the system trajectories in spite of the presence of the unknown input.

Example 2: Synthesis of a Continuous Fuzzy Bilinear Observer
In this subsection, we intend to apply the proposed design to an isothermal continuous stirred tank reactor CSTR see, e.g., 38-40 .The dynamics of CSTR for the Van de Vusse reactor can be described by the following nonlinear second order system:

5.11
where the state x 1 represents the concentration of the reactant inside the reactor mol/L and the state x 2 is the concentration of the product in the CSTR output stream mol/L .The output y determines the grade of the final product.The input-feed stream to the CSTR consists of a reactant with concentration C A0 , and the controlled input is the dilution rate u F/V h −1 , where F is the input flow rate to the reactor L/h and V is the constant volume of the CSTR liters .In all the following discussions, the kinetic parameters are chosen to be k 1 50 h −1 , k 2 100 h −1 , k 3 10 L/ molh , C A0 10 mol/L, and V 1 L as in 39 .
The system 5.11 can be written as ẋ t A x t x t Bu t Nx t u t Fd t , 5.12 where 5.13   with x 1 t ∈ 1, −1 .This system can be represented using the polytopic transformation 37 as follows:

Observer Design
The observer gains are obtained by solving design conditions 3.13 -3.

5.17
Then, the fuzzy bilinear state and their estimation are given in the following figures.Figures 5 and 6 show, respectively, the evolution of the state variables x 1 and x 2 of the considered system and their corresponding observer estimation x 1 and x 2 with the input signal u t 4.5 sin 0.5 πt and the initial conditions x 0 1 1 T and x 0 0.5 0.5 T .

Residual Generation
In this paragraph, we will consider the same system of isothermal stirred tank reactor subject to actuator fault:  Figure 7 displays the convergence of the residual corresponding to the actuator fault signal.One can see that the residual is almost zero throughout the time simulation run except at time t 6 s where it appears at the actuator fault and disappears at t 8 s. Figure 7 shows that the residual r t is sensitive to f t and insensitive to d t .So the designed unknown input fuzzy bilinear fault diagnosis observer can be efficiently used to detect faults.

Conclusion
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 timedelay, and can also be considered the problem of pole placements to improve the performance of the proposed fuzzy bilinear observer.

Figure 5 :Figure 6 :
Figure 5: The state x 1 and its estimate.
Theorem 3.3.If there exists a symmetric definite positive matrix P , matrices W i , V i , S, R i such that the following linear conditions hold for all i 1, . . ., r 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 .
t B i u t N i x t u t F i d t G i f t , B i , N i , F i ,and C being the same previous matrices and