Observer Design for Simultaneous State and Fault Estimation for a Class of Discrete-time Descriptor Linear Models

. This paper investigates the problem of observer design for simultaneous states and faults estimation for a class of discrete-time descriptor linear models in presence of actuator and sensor faults. The idea of the present result is based on the second equivalent form of implicit model [1] which permits to separate the di ﬀ erential and algebraic equations in the considered singular model, and the use of an explicit augmented model structure. At that stage, an observer is built to estimate simultaneously the unknown states, the actuator faults, and the sensor faults. Next, the explicit structure of the augmented model is established. Then, an observer is built to estimate simultaneously the unknown states, the actuator faults, and the sensor faults. By using the Lyapunov approach, the convergence of the state estimation error of the augmented system is analyzed, and the observer’s gain matrix is achieved by solving only one linear matrix inequality (LMI). At long last, an illustrative model is given to show the performance and capability of the proposed strategy.


Introduction
With the emergence of the industry of the future, many industrial processes have become equipped with very sophisticated sensors in order to collect all the information relating to the discovery of any unexpected change that may occur in the process. As a result, it has become so necessary to guarantee an effective control, and corrective actions in order to maintain the best performance of the process in terms of efficiency, reliability, availability, safety and security [2][3][4]. As any technical system governed by both dynamic and static equations, it needs to be modeled in its study. Generally, the implicit model, otherwise called the descriptor or singular model [1,5,6], has become the most common in the description of the behavior of the process.
In this perspective, the aim of this work is to study the problem of observer design in order to propose actuator, and sensor faults estimation method for a class of discrete-time descriptor linear models.This problem has been widely tackled and many research works are devoted to this class of systems. They relate to explicit and singular linear models in both continuous and discrete-time cases [30][31][32][33][34][35][36][37][38][39][40]. The resolution of the design and stability problems of the observer lies in the writing of the convergence conditions in the form of LMIs solved by a dedicated software [41].
In this paper, a novel approach for designing an observer for a class of discrete-time descriptor linear models subject to actuator and sensor faults is proposed. To achieve this, we propose a new method for simultaneous state, and fault estimation that is based on the separation between differential and algebraic equations in the considered descriptor model as given in [1]. More precisely, based on an explicit augmented model structure, observer design for a class of discrete-time descriptor linear models allows the simultaneous estimation of the unknown states, the actuator faults and the sensor faults is proposed. The exponential stability of the state estimation error is studied is given in term of LMI. Besides, the proposed result is synthesized by only an explicit structure.
The rest of this article is arranged as follows. In the second section, the class of discrete-time linear implicit models with actuator and sensor faults is presented. In the third section, the main result about observer design permits to estimate simultaneously unmeasurable states, and unknown faults of actuator and sensor is stated. In the fourth section, we illustrate the performance of the developed result in simulation through an implicit model of a heat exchanger pilot process. Finally, the conclusion is given in the fifth section.
Through this paper, the notations are similar to these used in [27] with the purpose of simplifying its structure.

Model description
The class of descriptor linear models considered is presented as follows: where: The state vector is presented such as k ∈ R n−r is the vector of algebraic variables, u k ∈ R m is the control input, y k ∈ R p is the measured output vector. f ak ∈ R n a and f sk ∈ R n s are the actuator fault and sensor fault, respectively. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R p×m , D a ∈ R p×n a , E ∈ R n×n , F a ∈ R n×n a and F s ∈ R p×n s , are real known constant matrices with: where: The constant matrix A 22 is supposed invertible.
The matrix E is assumed to be of the form: and satisfies rank(E) = r < n. If it is not the case, without loss of generality, we can always find Λ and Ω such that E can be transformed into the form (4). That is: Before giving the main result, let us make the following assumption [1]: Assumption 1 : Suppose that: (1) is impulse observable and detectable.
scriptor linear model (1), the approach is based on the separation between differential and algebraic equations in model (1).
From (6) and using the fact that (A 22 ) −1 exist, the algebraic equations can be solved directly for algebraic variables, to obtain: where: Substitution of the resulting expression for Z 2 k (equation (7)) in equation (6) yields the following model: where: Which is equivalent to the following state representation: where: Assumption 2 : Suppose that the fault k is considered as a constant unknown fault signal per time interval i.e.: the system (11) under the equivalent augmented state representation given by: where:

Simultaneous State and Fault Estimation
Based on the transformation of the implicit model (1) into the equivalent form (14), the proposed observer permitting to estimate simultaneously unmeasurable state and unknown faults and their derivatives takes the following form: where: (χ 1 k ,χ 2 k ) andŷ k denote the estimated augmented state vector and the output vector respectively. G is the observer gain which is determined such that (χ 1 k ,χ 2 k ) asymptotically converges to (χ 1 k , χ 2 k ). In order to establish the condition for the asymptotic convergence of the observer (16), we define the state estimation error: It follows, from (14) and (16), the dynamics of state estimation error e k is given by the differential and algebraic equations: where Note that, to prove the convergence of the estimation error e k toward zero, it suffices to prove from (18), that e 1 k converges toward zero.
Theorem 1 There exists an observer (16) for the descriptor linear model (1) if for a scalar α > 0, there exists matrices P > 0 and W such that the following LMI holds: The gain G of the observer is derived from: quadratic Lyapunov function as follows: Its time derivative is given by: Thus, estimation error convergence is exponentially ensured if the following condition is guaranteed: Which is equivalent to the following condition: Replacing Ψ from (19) into (25), we can establish the LMI condition (20) of Theorem 1 by using the following change of variable: Thus, from the Lyapunov stability theory, if the LMI condition (20) is satisfied, the error dynamic equation (18) is exponentially asymptotically stable. This completes the proof of Theorem 1.

Application to a heat exchanger pilot process
In this section, the theory of the proposed observer (16) is applied to a heat exchanger pilot process with the aim of simultaneously estimating the unknown states and faults on-line. The following discrete-time model that we consider here is obtained by Euler discretisation of step size T=0.012s of the model given in [27]. It takes the form: where: Z k = [Z 1k , . . . , Z 8k ] T ∈ R 8 is the state vector, u k ∈ R 2 , y k ∈ R 2 , f ak ∈ R and f sk ∈ R are the control vector, the vector of output measurements, the actuator fault, and the sensor fault, respectively. The rank(E) = 6 and the numerical values of the matrices are given by: In the following simulation results, under assumption 2, faults f ak and f sk are defined as given in figure 5 and can be applied in the same time.
Therefore, to apply the proposed observer design (16) for the implicit model (27), as stated in Theorem 1, it suffices to rewrite the model (27) into its equivalent form (14) as mentioned in Section 2.
Thus, the resolution of the LMI defined in Theorem 1 with α = 0.98 lead the following numerical value:  It is easy to see that even when a fault affects the system, the proposed observer, designed by using the numerical value of the gain (28), gives a good performance which allow the estimated signals, denoted by the dashed lines, to converge to the real signals.

Conclusion
A new observer design approach for simultaneous state and fault estimation for a class of discrete-time descriptor linear models is proposed in this paper. The main idea of the present work is based on the decomposition approach in which the observer parameters are determined by using the Lyapunov theory and solving a system of LMI. It presents the extension of the work carried out in continuous case [27] to the discrete case. The good performance of the proposed observer is shown in simulation through a heat exchanger process as an illustrative application.