Sliding-Mode Controller Based on Fractional Order Calculus for a Class of Nonlinear Systems

Abstract

This paper presents a new approach of fractional order sliding mode controllers (FOSMC) for a class of nonlinear systems which have a single input and two outputs (SITO). Firstly, two fractional order sliding surfaces S1 and S2 were proposed with an intermediate variable z transferred from S2 to S1 in order to hierarchy the two sliding surfaces. Secondly, a control law was determined in order to control the two outputs. A sliding control stability condition was obtained by using the properties of the fractional order calculus. Finally, the effectiveness and robustness of the proposed approach were demonstrated by comparing its performance with the one of the conventional sliding mode controller (SMC), which is based on integer order derivatives. Simulation results were provided for the case of controlling an inverted pendulum system.Copyright c 2016 Institute of Advanced Engineering and Science.All rights reserved.Corresponding Author: Noureddine Bouarroudj LCP, Ecole Nationale Polytechnique, 10 av. Hassen Badi, BP. 182, El-Harrach, Alger, Algeria Email: autonour@gmail.com INTRODUCTIONSliding mode control has largely proved its effectiveness in numerous applications, see, e.g., the studies by Utkin [1] and Slotine [2]. The first step of the SMC design is to select a sliding surface that models the desired closedloop performance in state variable space. In the second step, equivalent and hitting control laws are designed such that the system state trajectories are forced towards the sliding surface and slide along it to the desired attitude.An advantage of these methods of control (SMC) is their robustness to parameters variation and bounded external disturbances. The robustness is attributed to the discontinuous term in the control input. However, this discontinuous term also causes an undesirable effect called chattering. Sometimes this discontinuous control action can even cause instability of the system. This effect can be alleviated by, for example, introducing a sat function and taking off the sgn function in the hitting control law of the SMC.In the last years, the control of single-input-two output non-linear dynamical systems has risen some interest in the control research community; in general, the controller (input) is done for the trajectory tracking of the two outputs. PID controllers were applied in [3] to the stabilization and tracking control of three types of inverted pendulum. A PID+LQR method was given in [4], in which the LQR was added negatively to the PID in order to have a resultant optimal control. A fuzzy controller with estimation of scaling factors was studied in [5]. An intelligent control system based on an interval type-2 fuzzy PD controller was studied in [6]. The SMC design for this kind of systems is a very challenging task. Such developed controllers include the decoupled SMC shown in [7], a Fuzzy Sliding Mode Control (FSMC) that was tuned using ant colony optimization [8], [9] and a decoupled SM with a fuzzy-neural network controller [10] where the fuzzy-neural network w...

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