Observer Design for a Class of Nonlinear Descriptor Systems: A Takagi-Sugeno Approach with Unmeasurable Premise Variables

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Introduction
In practice, the control and the supervision of a process require the knowledge of the state of the process.One way permitting to obtain such unknown state consists in using physical sensors.However, in many cases and due to a high running cost and physical constraints, this method becomes very limited.To solve this problem, one solution is to design an observer.This method combines a priori knowledge about a physical system (nominal model) with experimental data (some on-line measurements) to provide on-line estimation of states and/or parameters.In the present work, we are concerned with the problem of the observer synthesis for nonlinear descriptor systems that can be described by dynamic models of T-S descriptor with unmeasurable premise variables.In fact, many chemical and physical processes can be described by nonlinear systems of differential and algebraic equations [1][2][3].These systems are variously called descriptor systems, singular systems, or differential algebraic equations (DAEs).This formulation includes both dynamic and static relations.Consequently, this formalism is much more general than the usual one and can model the physical constraints or impulsive behavior due to an improper part of the system.The numerical simulation of such descriptor models usually combines an ODE numerical method together with an optimization algorithm.
Recently, there has been a great deal of interest in using the approach based on the representation of the nonlinear systems by T-S models [4].This interest relies on the fact that once the T-S fuzzy models are obtained, some analysis and design tools developed in the linear system can be used, which facilitates observer or/and controller synthesis for complex nonlinear systems.Many practical problems using T-S fuzzy approach have been widely treated in the literature.The stability and stabilization problems of T-S fuzzy systems can be found in [5][6][7].In [8] a filter for nonuniformly sampled nonlinear systems represented by T-S model is proposed.For filtering problem in networked control systems where the nonlinear discrete-time system is modeled by T-S fuzzy model, we can cite [9].The T-S fuzzy observer problems for dynamic T-S fuzzy models described by ordinary differential equations (ODEs) with measurable and unmeasurable premise variables are studied in [10][11][12][13][14][15][16][17][18][19][20].Concerning nonlinear descriptor systems described by T-S descriptor models the problem of fuzzy observer design has been widely investigated; see, for instance, [21][22][23][24][25].The aim of this paper is to give a fuzzy observer design to a class of fuzzy descriptor systems permitting to estimate the unknown state without the use of an optimization algorithm.The idea of the proposed result is to separate the dynamic relations of the static relations in the descriptor model.
The outline of the paper is as follows.The main result is stated in Sections 2 and 3.It consists in showing that the state estimation problem for a class of fuzzy descriptor systems can be achieved by using a fuzzy observer having only an ODE structure.First, the method used for decomposing the differential part of the algebraic part is developed; secondly we give a fuzzy observer design permitting to estimate the unknown state.In Section 4, we illustrate the performance of the proposed observer in simulation through a model of a heat exchanger pilot process.

Fuzzy Descriptor Systems
The form of the class of Takagi-Sugeno descriptor systems with unmeasurable premise variables studied in this paper is where and  ∈ R × are real known constant matrices with where  22 constant matrices are invertible (rank( 22 ) = − ). is the number of submodels.The ℎ  () are the weighting functions that ensure the transition between the contribution of each submodel: They depend on unmeasurable premise variables (state of the system) and have the following properties: In order to design an observer for each submodel (3) ( = 1, . . ., ), we will make the following assumptions.
(H2) All submodels (3) are impulse observable; that is, rank (( (H3) All submodels (3) are detectable; that is, To design a fuzzy observer for system (1), our approach is based on the separate dynamic relations of the static relations for each submodel (3) and the global model is obtained by aggregation of the submodels.Thus, from (2), system (3) can be written as follows: Using the fact that  −1 22 exist, system (7) can be rewritten as where Then, the fuzzy descriptor system (1) can be rewritten in the following form: where

Fuzzy Observer Design
In this section, our aim is to design a fuzzy observer for descriptor system (1).Based on the separate dynamic relations of the static relations in the descriptor model (1) (see (10)) and by substituting ( 11) in (10), the proposed observer is given by the following equations: where X1 , X2 , and ŷ denote the estimated state vectors of  1 ,  2 , and output vector , respectively.The local gains   can be determined by Theorem 1.
In order to establish the conditions for the asymptotic convergence of the observer (12), we define the state estimation error: It follows from ( 10) and ( 12) that the observer error dynamic is given by the differential and algebraic equations: By adding and subtracting the term Then, ( 16) becomes where Δ  =   −   and Δ  =   −   .
Multiplying by ∑  =1 ℎ  ( 1 , ), we obtain By substituting ( 19) and ( 21) in (20), we obtain where By adding and subtracting the term Note that to prove the convergence of the estimation error  toward zero, it suffices to prove that  1 converges to zero.Thus, let X1 = (  1   1 )

𝑇
; we have where Thus, the aim is to determine the observer gains   ( = 1, . . ., ) to ensure the stability of (25) while attenuating the effect of the input  on  1 .Therefore, the convergence condition of the observer ( 12) can be formulated by the following theorem.
Theorem 1.Under the above hypotheses (H1), (H2), and (H3), the state error between the T-S descriptor model (1) and its observer (12) converges asymptotically towards zero, if there exist symmetric positive definite matrices  1 and  2 , matrices K  ,  = 1, . . ., , and a positive scalar , such that the following LMIs hold: where The gains of the observer are derived from and the attenuation level is Proof of Theorem 1.Consider the following quadratic Lyapunov function: with The time derivative of  along the trajectory of ( 25) is given by In order to ensure the stability of ( 25) and the boundedness of the transfer from  to  1 , we consider the following criterion: From ( 25) and (33), inequality (35) becomes where The inequality (36) is satisfied if Then, from ( 23), (26), and the use of the changes of variables, we establish the LMIs given by ( 27) in Theorem 1.
Thus, from (25) the LMI constraints (27) imply that X1 exponentially converges to the unknown trajectory  1 of system (10) which are identical to those of system (1).And, from system (24) and the fact that X1 converges to  1 , X2 exponentially converges to the unknown trajectory  2 of system (10) which is identical to that of system (1).

Application to a Heat Exchanger System
The aim of this section consists in applying the above fuzzy observer design (12) with unmeasurable premise variables to a descriptor model of a heat exchanger pilot process.

Physical Model.
The heat exchanger process considered is presented in Figure 1.The process is mainly built around a counterflow tubular heat exchanger.The warm water flows in a closed circuit, and the temperature in the hot water tank is fixed by an independently controlled electric heater.The cold water flows in an open circuit.The flows of either warm or cold water are controlled by two electropneumatic valves. 1 ,  3 are, respectively, the inlet temperatures of the warm and the cold water and  2 ,  4 are the correspondent outlet temperatures.The dynamics of actuators (electropneumatic valves) cannot be neglected.Indeed their time constants are equivalent to the residence time constants of the heat exchanger (0.5 s-1 s).The correspondent state variables are the displacements and the velocities of the electropneumatic valves.The temperatures are assumed to be homogeneous in the tubular heat exchanger.Under the hypotheses that the circuit of the thermal exchanger is a closed system which contains a constant mass of water, the inertia of the fluid is negligible and the flow is turbulent.
The controlled variables of our problem are the temperatures  2 and  4 , which are manipulated with the flows which are a function of electropneumatic valves current  V and  V .The current on electropneumatic valve is an actual manipulated variable of the process.Furthermore, the heatexchanger is just one part of the plant.So, the actuators should also be modeled.The electropneumatic valve is a system that exhibits inherent second-order dynamics.For the heat-exchanger, we perform the energy balance for the characterization of the temperature.A descriptor model of the process takes the form where  = ( 1 , . . .,  8 )  is the state vector,  = ( 1 ,  2 )  is the control vector, and  = ( 1 ,  4 )  = ( 2 ,  4 )  is the output measurements. 2 ,  5 are, respectively, the displacement of the warm water valve and the cold water valve. 3 ,  6 are, respectively, the velocity of the warm water valve and the cold water valve.Finally,  7 ,  8 are, respectively, the acceleration of the warm water valve and the cold water valve: ) ,  = ( ( ( ( ( 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) ; (41) , and  2 are physical constants which derive from the energy balance transfer. 0 is the static gain of the valve,  0 is the undamped natural frequency, and finally  is the damping factor.

Takagi-Sugeno Descriptor Model.
To express the model of the heat exchanger system as a Takagi-Sugeno model with the unmeasurable parameters (displacements of the valves  2 and  5 ) as decision variables, we use the procedure of fuzzy model construction given in [26].For this purpose, we rewrite (40) in the following equivalent state space form: where  =  () , =( 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ) , ) . (43) Then, we consider the sector of the nonlinearities of the terms   ∈ [  min ,   max ] of the matrix (()) with  = 1, 2: Thus, we can transform the nonlinear terms under the following shape: where Then, the global fuzzy model is inferred as where ) . (48)

Fuzzy Observer Design.
Based on the on-line measurements of the temperature of the warm water  1 and the temperature of the cold water  4 , we will show that the previous result (12) can be used to estimate the displacement, the velocity, and the acceleration of the warm water valves  2 ,  3 , and  7 and the displacement, the velocity, and the acceleration of the cold water valves  5 ,  6 , and  8 .Using Section 3, the construction of the fuzzy descriptor observer algorithm for heat exchanger system requires that the above system (47) takes the form (10).

Simulation Results
. In this section the purpose is to show by numerical simulations the good performances of the present study given in this paper.For all computer simulations results discussed in the sequel, we use the parameter values summarized in Table 1.
To simulate descriptor models (40) and (47), we use a Runge-Kutta method combined with the Newton-Raphson algorithm.
The initial conditions of the nonlinear system (40) and T-S model ( 47 (51) In order to illustrate the performances of the T-S fuzzy observer (50), we solve the LMIs given in Theorem 1.Then, note that in this practical case  2 = 0; this implies that (see ( 9))  1 = ⋅ ⋅ ⋅ =   =  1 and  1 = ⋅ ⋅ ⋅ =   = 0. Thus, the output of the system takes the following expression: Now, the LMIs (27) given in Theorem 1 become where ( Simulations results given in Figure 3 show the performances of the observer designed above with the parameters   ,  = 1, 2, 3, 4, where the dotted lines denote the state variables estimated by the fuzzy observer (50).This simulation shows that the estimation states converge to their corresponding state variables.

Conclusion
In this paper, a new method to synthesize observer for continuous-time T-S descriptor model with unmeasurable premise variables was presented.The approach is based on the separation between dynamic and static relations.The convergence conditions are obtained by using Lyapunov theory and the L 2 techniques.The existence of conditions ensuring the convergence of the state estimation error is expressed in terms of LMIs.The proposed fuzzy observer is used for the on-line estimation of unknown state in a heat exchanger model.First, the Takagi-Sugeno fuzzy model is developed to represent the descriptor nonlinear model of the heat exchanger.Next, simulation results have been given and they demonstrated the good performances of the estimator.

Table 1 :
List of parameters.