Unknown input observer for Takagi-Sugeno implicit models with unmeasurable premise variables

,


INTRODUCTION
It is well-known that equations in descriptor form offer significant flexibility in modelling nonlinear systems. Indeed, mobile robots, electrical systems, biological processes and several industrial applications are described by a system of differential and algebraic equations called implicit, singular or descriptor models, see for example [1], [2] and references therein for some applications. Furthermore, many physical phenomena, like impulses and hysteresis which are important in circuit theory, cannot be handled properly in ordinary models. Descriptor representation provides an appropriate way to deal with such problems.
This work focuses on the development of algorithms to estimate the state and unknown inputs (UIs) for implicit nonlinear models through the application of Takagi-Sugeno (T-S) formalism. In fact, many processes such as nonlinear auto regressive moving average model with eXogenous inputs (NARMAX), Hammerstein, Wiener, Hammerstein-Wiener are described by nonlinear systems [3]- [7]. Consequently, the resulting models are very complex and the analysis and the controller/observer synthesis based on the model becomes more ❒ ISSN: 2088-8708 dificult to achieve. The standard T-S approach [8], [9] is a powerful and practical modeling tool for these complex systems. It allows to obtain a model taking into account the non-linearities of the system and offering a simple and easily exploitable structure from a mathematical point of view which facilitates observer synthesis and the control design of nonlinear systems, see for example [10]- [14]. Additionally, we distinguish between a T-S fuzzy model with measurable premise variables (MPVs) and a T-S fuzzy model with unmeasurable premise variables (UPVs) depending on the nature of the variables involved in the activation functions called premise variables. Note that in [15], [16] fuzzy implicit model is defined by expanding the standard T-S model [8].
It is well known that industrial processes are often subject to disturbances. They are called UI when they affect the process input. They can spring from measurement uncertainties, sensor, actuator faults, or noises due to the environment of the process. Given their harmful effects on the normal operation of the process, their real-time estimation can be used for the synthesis of a control algorithm which is capable of minimizing these effects. Due to the increasing demand for reliability and maintainability of automatic control systems, the synthesis of observers for dynamic models subjected to UIs (called unknown inputs observers (UIO)) is one of the most important areas of research during the last two decades. Based on different approaches, this theme has been the subject of numerous theoretical developments. This is especially due to its essential role in the fault diagnosis and fault tolerant control strategy development. Several research works using different approaches have been published in [17]- [20]. Many publications have been interested in the design of UIs for the case of T-S explicit models, we may refer to [21]- [26]. In the case of T-S implicit systems, several UIOs have been developed. In [27], a design of UIO is proposed for a class of nonlinear implicit systems described by T-S structure with MPV. The basic idea of the proposed approach is based on the separation between dynamic and static relations in the T-S implicit model to estimate both the system state and the UIs concurrently. Developing an UIO for T-S fuzzy systems satisfying Lipschitz conditions is the aim of the work presented in [28]. In [29] the authors used the result obtained in [27] to develop a new approach making it possible to estimate simultaneously both non-measurable states and unknown faults in the actuators and sensors for T-S implicit model with MPV.
Most of the UIOs are synthesized to estimate the state and UI in the case of T-S implicit models with MPVs. The main contribution of this research is to introduce a novel method of fuzzy unknown inputs observer (FUIO) design for a class of continuous-time Takagi-Sugeno implicit models (CTSIMs) with UPVs permitting simultaneous estimation of the unmeasurable states and UIs. Furthermore, as compared to T-S systems with MPVs, the T-S structure with UPVs can describe a broader variety of non-linear systems. Indeed, one of the most well-known strategies for converting a nonlinear model to a T-S model is the sector nonlinearity approach [30]. This transformation usually allows for the creation of a T-S model with UPVs. It should also be noted that if the output of the system is chosen as a premise variable and this output is affected by disturbances, the resulting T-S system does not exactly describe the system. T-S models with UPVs are more harder to treat than those with UPVs. For this reason, few studies are dedicated to this class of models despite of their interest. The design method in this work is based on the singular value decomposition (SVD) approach and the use of an augmented system structure. The Lyapunov theory is used to investigate the global exponential stability of the state estimation error, and the stability conditions are expressed in terms of linear matrix inequalities (LMIs). Besides, in most works, the authors proposed observers for CTSIMs in descriptor form instead of the standard TS observer. It is important to keep in mind that designing a descriptor observer is not easy because the slow dynamic (resulting from the algebraic equations) is directly estimated, which can depend on impulsive behavior and influence the observer stability. In addition to this, the main drawback of these observers lies in their implementations. However, the solution we offer in this paper, which is presented in an explicit form, allows us to avoid this problem.
The paper is organized as: in section 2, the mathematical of the considered CTSIMs subject to UI to be studied is described. Section 3 presents the suggested method and the main result of this work. More precisely, the structure of the FUIO is proposed. Besides, the stability analysis, and the convergence conditions are demonstrated. Finally, a numerical example of this theoretical result and simulation results are given to show the interests of the proposed design framework.

Int J Elec & Comp Eng
ISSN: 2088-8708 ❒ 5037 where ζ ∈ R n , u ∈ R m and y ∈ R p represent, respectively, the state, the input and the output vectors of the sysytem. The unknown inputs are modeled by d ∈ R σ . A(ζ) ∈ R n×n , B(ζ) ∈ R n×m , Λ ∈ R n×σ are nonlinear matrices functions. C ∈ R p×n , M ∈ R n×n , D ∈ R p×σ are real known constant matrices, with rank(M ) = r < n. By using the nonlinear sector transformation [30], the system (1) can be represented by the CTSIM: collect the contribution of all sub-systems.
Along this paper, we assume that the weighing functions ρ i (β) depend on unmeasurable premise variables (state of the system), and satisfy the following constraints: Assume that d is a constant unknown control input every time interval, as (5): Let's start by defining the augmented state vector: As a result, the system (2) can be extended in the augmented form as (6): where Let us make the following assumption before giving the main result [1], [31]. Assume that: (3) are impulse observable and detectable.
According to hypothesis H 3 ), there is a non-singular matrix Ψ 1 Ψ 2 Ψ 3 Ψ 4 such that: are constant matrices that may be obtained via the singular value decomposition of Γ C .

THE PROPOSED METHOD AND MAIN RESULT
According to the transformation of CTSIM (2) into the form (6), the suggested FUIO which is not in implicit form allowing simultaneous estimation of UIs and unmeasurable states of system (2) takes the following structure: where x represents the estimated vector, the activation functions ρ i (β) are dependent on the unmeasurable premise variables andẑ is the estimated augmented state vector. The unknown matrices N i , L 1i , L 2i , G i and K must be obtained in the aim of a exponential convergence ofẑ converges to z. The matrices Ψ 2 and Ψ 4 satisfy (8).
The augmented state estimate error is defined as (10): by inserting (8) and (9) into (10) we obtain: so, error dynamics will be (12):ė from (6) and (9), we obtain: by substituting (11), (13) becomes: where provided the matrices N i , L 1i , L 2i , G i and K satisfy: and  (14) becomes: then, from (8), (16) and (17), we have (19) then using the fact that where Then, the (18) becomes: multiplying by q i=1 ρ i (β), (23) can be simplified as (24): where Let e = [e T z T ] T , we have (26) where Thus, the goal is to find the observer gains N i , L 1i , L 2i , G i (i = 1, . . . , q) and K that will ensure the stability of (26) while attenuating the effect of the input u on z. Therefore, the following theorem can be used to express the convergence condition of (9).

❒ ISSN: 2088-8708
Theorem (Under assumptions 1 and 2): Given λ > 0, the state estimation error between the CTSIM (2) and its FUIO (9) converges towards zero, if there are matrices P = P 1 0 0 P 2 , Q, W j , j = 1, . . . , q and a positive scalar α, such that the following LMIs hold: where with The observer gains N j , L 1j , L 2j , G j and K are given by (32) and the attenuation level is where Ψ 1 , Ψ 2 , Ψ 3 and Ψ 4 satisfy (8) proof: Consider the quadratic Lyapunov function: and By using (26), the time derivative of V can be written as (37): By using (26) and (35), the (37) becomes: The stability of (26) and the bounded transfer from u to z by α is assured as: The state estimation error converges exponentially if the following condition holds: From (34) and (39), inequality (42) becomes: This implicates q i,j=1 Then, from (21), (25), (27), (36) and the use of the following change of variables: it follows that (46) is equivalent to (29). Finally, according to Lyapunov stability theory, if the LMI conditions (29) are satisfied, the (24) is exponentially stable.

SIMULATION RESULTS
In this section, an academic example is used to show the validness of the proposed method. Consider a CTSIM defined by (48): where ζ = (ζ 1 , ζ 2 , ζ 3 , ζ 4 ) T ∈ R 4 , u ∈ R, d ∈ R and y ∈ R 2 . The matrices numerical values are: The weighting functions are given by: Under hypothesis H 3 ), the following matrices Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 satisfying (8) were obtained Using the MATLAB LMI toolbox, The following observer gains result from the resolution of LMIs (29) of Theorem 1 with λ = 5: The unknown input d is described as shown in Figure 1, and the following is the definition of the input signal u: The initial conditions of the state  It is obvious from this example that the proposed strategy presents better results than the approaches given by [27] and [28]. Indeed, the method developed is based on the separation between static and dynamic relations in the CTSIM. Only systems that ensure the criterion of rank described in the study allow for this separation. It is also pointed out that, to express the nonlinear system as a CTSIM, the authors took the weighing functions that are dependent on the measured variables (the output of the system). Due to the use of the same premise variables by the observer and the model, it is obvious that this class with measurable premise variables will allow a factorization of the weighting functions ρ i (β)) while analyzing the convergence of the estimate error. Noises affect the output in all practical situations. As a result, the accuracy of the model describing the system, as well as the results obtained, will be affected. In our situation, we used a CTSIM with an unmeasurable variable to simulate the rolling disc process. The activation functions used in our proposed observer are not the same as those employed in CTSIM. Furthermore, the elimination of the Lipschitz assumption of the weighting functions is another advantage of this method compared to those that use CTSIM with premise variables satisfying Lipschitz conditions (see [28]) which allowed us to to relax the LMI conditions.

CONCLUSION
This article proposes a new method to design FUIO in explicit structure for a class of CTSIMs with UPVs. Using an augmented system structure, the suggested FUIO enables to estimate, exponentially , the UIs and the system states. The Lyapunov theory is used to investigate the convergence of the observer and the the conditions that ensure the convergence of estimation errors are obtained in the LMIs formulation. A numerical example is used to verify and confirm the efficiency of the proposed method. As future works, it would be interesting to extend the present result to the design of a FUIO for a class of discrete-time T-S implicit models also for a class of time-delay T-S implicit systems and try to relax the LMI conditions and reduce its conservatism, by using different Lyapunov functions such as poly-quadratic ones.

BIOGRAPHIES OF AUTHORS
Mohamed Essabre is a professor at the department of Electrical Engineering in Faculty of Science and Technology Mohammedia (FSTM), in Hassan II University of Casablanca (Morocco). His research interests focus in nonlinear implicit model, nonlinear observer design, unknown input observer design, fault detection, Takagi-Sugeno fuzzy control. He can be contacted at email: mohamed.essabre@fstm.ac.ma.

Ilham Hmaiddouch
received the Ph.D. in Electrical Engineering from the National Higher School of Electricity and Mechanics (ENSEM), Hassan II University of Casablanca, Morocco. His research interests focus in nonlinear implicit model, nonlinear observer design, unknown input observer design, fault detection, Takagi-Sugeno fuzzy control. She can be contacted at email: ilham.hmaiddouch@yahoo.fr.