A New Fractional Order Observer Design for Fractional Order Nonlinear Systems

In this paper, a class of fractional order systems is considered and simple fractional order observers have been proposed to estimate the system’s state variables. By introducing a fractional calculus into the observer design, the developed fractional order observers guarantee the estimated states reach the original system states. Using the fractional order Lyapunov approach, the stability (zero convergence) of the error system is investigated. Finally, the simulation results demonstrate validity and effectiveness of the proposed scheme. INTRODUCTION In recent years, fractional calculus has attracted increasing interests and there has been a rapid grow in the number of applications where fractional calculus has been used [1-3]. Lately, this technique has been applied to physics and engineering science problems and fractional order systems has been widely studied in different fields of science [4, 5]. It has become apparent that a large number of physical phenomena can be modeled by fractional order models [6-8] and many real world physical systems are better characterized by fractional order differential equations. For the fractional order systems, there are many theories and criterions regarding the controllability, observability and stability, including the linear and nonlinear systems [9-15]. Besides, the design of different controllers for fractional order systems has received a great deal of attention recently and many important and significant results have been reported [16-18]. The problem of state observers that predicts the present system state is clearly central for the design of state feedback controllers. However, there are a few results regarding the state estimation for the fractional-order system [19-21]. Despite the importance of system state estimation in many practical engineering applications, to the best of the authors’ knowledge, there is no work in which the problem of designing fractional order observer for fractional order systems is investigated. In this work, we first propose a new fractional order observer for fractional order linear systems and then, an extension of the previously mentioned observer for fractional order nonlinear systems is presented. The considered class of fractional order systems has separable nonlinearity and the nonlinear part is assumed to satisfy the Lipschitz condition. Many physical systems can be expressed or transformed into this form. To prove the convergence to zero of the estimation error, we use the fractional order Lyapunov approach. In other words, by using Lyapunov approach, a sufficient condition for the asymptotic stability of the error system is given. Finally, giving numerical examples, it is shown that we extend successfully the observer design method to cope with state estimation problem for fractional order systems. The remainder of the paper is organized as follows: In Section 2, some basic concepts of fractional calculus is described. In Section 3, a class of fractional-order systems is introduced. In Section 4, two novel fractional order observers are presented and stability analysis of the fractional-order error systems when the proposed observers are applied is given. In Section 5, numerical simulations are given to confirm the effectiveness of the proposed observers. Finally, some concluding remarks are presented. FRACTIONAL-ORDER CALCULUS Fractional-order integration and differentiation is the generalization of the integer-order ones. Efforts to extend the specific definitions of the traditional integer-order to the more general arbitrary order context led to different definitions for fractional derivatives [22]. In this section, two commonly used definitions are presented. Proceedings of the ASME 2011 International Design E gi eering Technical Conferenc s & IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA