Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 724795, 11 pages

http://dx.doi.org/10.1155/2015/724795

## Observer Based Robust Position Control of a Hydraulic Servo System Using Variable Structure Control

Research Group CEMLab, National Engineering School of Sfax, University of Sfax, 1073 Sfax, Tunisia

Received 8 May 2015; Revised 9 July 2015; Accepted 12 July 2015

Academic Editor: Rongwei Guo

Copyright © 2015 E. Kolsi-Gdoura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the position control of a hydraulic servo system rod. Our approach considers the surface design as a case of virtual controller design using the backstepping method. We first prove that a linear surface does not yield to a robust controller with respect to the unmatched uncertainty and perturbation. Next, to remedy this deficiency, a sliding controller based on the second-order sliding mode is proposed which outperforms the first controller in terms of chattering attenuation and robustness with respect to parameter uncertainty only. Next, based on backstepping a nested variable structure design method is proposed which ensures the robustness with respect to both unmatched uncertainty and perturbation. Finally, a robust sliding mode observer is appended to the closed loop control system to achieve output feedback control. The stability and convergence to reference position with zero steady state error are proven when the controller is constructed using the estimated states. To illustrate the efficiency of the proposed methods, numerical simulation results are shown.

#### 1. Introduction

Actually, the hydraulic servo systems are very popular in several industrial applications such as robotics, aerospace flight-control actuators, heavy machinery, aircrafts, automotive industry, and a variety of automated manufacturing systems. This is mainly due to their ability to produce high power and accurate and fast responses. However, these systems have a high nonlinear behavior due to the pressure flow characteristics [1] and the leakage model inside the servovalves [2]. This fact makes the control design for precise output tracking a very challenging task.

Owing to their simplicity, linear controllers of PID type [3, 4], input/output linearization controllers [5–9], and also sliding mode controllers (SMC) [10–13] have been used to control the hydraulic servo systems. However, such controllers were designed based on the plant physical model and, therefore, the plant parameters knowledge is required. Consequently, they were shown to be highly sensitive to mismatched perturbation and uncertainties, thus resulting in performance degradation.

To improve the controller performances, several strategies have been adopted such as using the self-tuned PID controller [14, 15] and nonlinear adaptive controllers [16–18]. SMC appended with some improvements have also been used. In [19, 20] SMC method has been combined with an adaptive controller, which can compensate for the system uncertain nonlinearities, for linear uncertain parameters, and especially for the nonlinear uncertain parameters to construct an asymptotically stable tracking. In [21], SMC has been used with the PID controller to achieve control of asymmetrical hydraulic cylinder trajectory tracking. To drive electrohydraulic actuators, various robust control techniques, such as and controls, were applied [22–24]. This approach enabled the compensation for the inherent nonlinearities of the actuator and rejects matched external disturbances and attenuates mismatched external disturbances. To cope with mismatched disturbances authors used the integral SMC and to remedy the slow response due to windup phenomenon a realizable reference compensation has been used to achieve fast position tracking [25, 26]. Since it has been proposed by Levant [27, 28], higher-order SMC (HOSMC) has been widely used to control electrical drives [29, 30], electropneumatic actuators [31], and electrohydraulic actuators [32, 33].

In the present paper, we are interested in controlling the position of the rod in a hydraulic servo system that consists of a four-way spool valve supplying a double effect linear cylinder with a double-rodded piston. The piston is driving a load modeled by a mass, a spring, and a sliding viscous friction. Our work aims to design a controller that may achieve the reference position in presence of mismatched parameter uncertainty and perturbation in addition to actuator saturation. To realize this objective, we start in the second section by formulating the problem and presenting the effects of using first- and second-order SMC with a linear surface. In Section 3, we present the design of a sliding surface obtained using backstepping method and variable structure controller, leading hence to a nonlinear surface that allows achieving the reference output despite the presence of uncertainties and perturbations. Numerical simulation results are presented to illustrate the efficiency of the proposed control design. In Section 4, we present a sliding observer and prove the convergence of the observer as well as the exact position tracking using the nonlinear surface SMC issued from the observer states. Finally, the conclusion and some remarks are presented in Section 5.

#### 2. Problem Statement

The electrohydraulic system that we will deal with in this paper is depicted in Figure 1 and modeled by the dynamical system (1). It has been shown in [34] that, for the symmetrical piston with equal surfaces and and assuming equal volume flow passing through (geometrically) identical ports, we can describe the system by three variable states where a differential pressure state substitutes the pressure of each chamber. This decrease in the system dimension ensures the observability when the system output is the piston position. Considerwhere denotes the difference in pressure inside the two chambers of the cylinder, and , respectively, denote the velocity and the position of the rod, and is a bounded constant or slowly varying external perturbation. In fact, from Newton’s law, a constant force perturbation leads to a constant acceleration. Thus, the velocity perturbation may be interpreted as the result of an impulsive force that acts abruptly on the system. is the total mass of the rod and the load, is the total volume of the cylinder, and is the pressure difference between the supply pressure (pressure of the pump) and the return pressure (atmospheric pressure). is the spring stiffness constant with an uncertainty and is the friction coefficient. The system parameters used for simulations are as follows: Pa, Pa, Pa, kg, m^{2}, m^{3}, kg, kg/s, N/m, m^{3} s^{−1} A^{−1} Pa^{−1/2}, m^{3} s ^{−1} Pa^{−1}, and with and being intrinsic constants modeling the leakage within the servovalve [2].