Article Preview
Top1. Introduction
The reconstruction of the state vector is an inevitable step for control and diagnosis of many processes (Darouach, 2005; Khadhraoui, 2014). This topic has been investigated by many researchers during the last decades (Johnson, 1975; Hostetter, 1973) aiming the synthesis of a control scheme based on an observer gain. Then, the control law is expresses mainly in function of the output measurements. For this purpose, we propose the synthesis of an unknown input functional filter for linear singular systems affected by bounded disturbances and unknown inputs. This is motivated by the importance of such system which brings together three criteria (singular, unknown input and bounded disturbance) often present in most of physical process.
Also, the singularity has, recently, become the focus of several research activities (Dai, 1989; Darouach, 1995; Darouach, 2005) due to its importance in the modeling of physical systems. In fact singular system can be viewed as a generalization of standard linear systems as they contain a dynamical and algebraic part in the systems descriptions. Then the importance of such systems comes from their ability to describe such systems despite their complex structures.
Moreover, the synthesis of observers becomes more interesting especially when we take into account defaults action on the dynamic and the output Equations (Johnson, 1975; Hostetter, 1973). This topic has been recently considered (Hou, 1992; Gaddouna, 1994; Darouach, 1996; Khadhraoui, 2014), and all related works estimate only the state vector. However, the proposed functional unknown input filter reconstructs both the functional state vector and the unknown input vector in the presence of faults. Then we can use these results in the failure detection and the control of systems in presence of disturbances.
Finally, the problem of the state observing of linear singular systems with bounded disturbances has been a great deal of investigation during the last decades (Khadhraoui, 2014; Darouach, 1996; Zasadzinski, 1998). In fact, filtering a state vector, lies on the reconstruction of a linear combination of the system states using only the input and the output measurements.
Two performance measures in filtering problem are extensively used, the 
and the 
norms (Khadhraoui, 2014, 2016; Souley, 2006). In this paper, we choose the 
filter due to its performance for minimizing the perturbation effect on the estimation error.
Motivated by these facts, we propose a functional 
filter to estimate both functional state vector and part of the unknown inputs vector based on Lyapunov-Krasovskii approach. And using the unbiasedness condition, we arrange the error expression so that the error dynamics ignores the disturbance derivative (
) (Souley, 2006). Then a Linear Matrix Inequalities (LMI) approach is developed to find the filter optimum gain.
The paper is organized as follows. Section 2 presents the difference between current work and recent published work. The third section gives assumptions used through this paper and presents the problem formulation that we propose to solve. Section 4 gives a time domain solution for the unknown input functional filter design problem with respect to the 
performance criterion. The next section summarizes the design algorithm presentation. Section 6 gives a numerical example to illustrate our approach and the last section concludes the paper.