Controllability of Singular Linear Systems by Legendre Wavelets

We propose a new method to design an observer and control the linear singular systems described by Legendre wavelets. The idea of the proposed approach is based on solving the generalized Sylvester equations. An example is also given to illustrate the procedure. 1. Introduction Singular systems, also commonly called generalized or descriptor systems in the literature, appear in many practical situations including engineering systems, economic systems, network analysis, and biological systems. In fact, many systems in the real life are singular essentially. They are usually simplified or approximated by nonsingular models because there is still lacking of efficient tools to tackle problems related to such systems. The structural analysis of linear singular systems, using either algebraic or geometric approach, has attracted considerable attention from many researchers during the last three decades (see, e.g., [1–3]). Since the introduction of Legendre wavelets method (LWM), for the resolution of variational problems, by Razzaghi and Yousefi in 2000 and 2001 [4, 5], several works applying this method were born. To mention a few, we give the resolution of differential equations [6], the study of optimal control problem with constraints [7], the resolution of linear integro-differential equations, the numerical resolution of Abel equation, and the resolution of fractional differential equations. In this paper, we propose a new method to design an observer and control the linear singular systems described by Legendre wavelets. The method is based upon expanding various time functions in the system as their truncated Legendre wavelets. The operational matrix is introduced and utilized to reduce the solution of time singular linear system to the solution of algebraic equations. Finally, we obtain the interrelations between solution problems for the linear matrix equations of Sylvester with suitable controllability and observability conditions. 2. Properties of Legendre Wavelets Wavelets are mathematical functions that are constructed using dilation and translation of a single function called the mother wavelet denoted by and satisfied certain requirements. If the dilation parameter is and translation parameter is , then we have the following family of wavelets: Restricting and to discrete values, such as , , , , and , are positive integers, we give where form a basis for . If and , then it is clear that the set forms an orthonormal basis for . Legendre wavelets have four arguments: , , is assumed to be any positive integer, is the order for Legendre