Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4505340, 10 pages

http://dx.doi.org/10.1155/2016/4505340

## Finite-Time Composite Position Control for a Disturbed Pneumatic Servo System

Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, School of Automation, Southeast University, Nanjing 210096, China

Received 5 May 2016; Accepted 19 October 2016

Academic Editor: R. Aguilar-López

Copyright © 2016 Xiaojun Wang 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 investigates the finite-time position tracking control problem of pneumatic servo systems subject to hard nonlinearities and various disturbances. A finite-time disturbance observer is firstly designed, which guarantees that the disturbances can be accurately estimated in a finite time. Then, by combining disturbances compensation and state feedback controller together, a nonsmooth composite controller is developed based on sliding mode control approach and homogeneous theory. It is proved that the tracking errors under the proposed composite control approach can be stabilized to zero in finite time. Moreover, compared with pure state feedback control, the proposed composite control scheme offers a faster convergence rate and a better disturbance rejection property. Finally, numerical simulations illustrate the effectiveness of the proposed control scheme.

#### 1. Introduction

Pneumatic servo systems play an important role and are widely used in modern industry (see pneumatic muscle systems [1, 2], pneumatic brake systems [3], pneumatic manipulators [4, 5], ball-plate pneumatic systems [6], etc.), since they have lots of advantages, such as energy saving, high power-to-weight ratio, low cost, simple structure and operation, and ease of maintenance [7, 8]. A lot of industry applications require high-precision position control of pneumatic servo systems [9–12]. Unfortunately, there exist various disturbances and hard nonlinearities due to the compressibility of gas, nonlinearity of servo valve, and so forth in pneumatic servo dynamics systems [8, 9], which brings a challenge for position control design of pneumatic servo systems. Consequently, designing a controller to deal with hard nonlinearities and disturbances is very important for improving the tracking performance of the pneumatic servo systems.

Recently, in order to attenuate the undesirable influence caused by system parameter uncertainties and external disturbances, lots of robust control approaches have been developed and widely applied in practical systems, such as sliding mode control [9, 10, 12], adaptive control [13, 14], disturbance observer based control [1, 2, 15, 16], and active disturbance rejective control [11]. Even though the aforementioned methods can attenuate the parameter uncertainties and external disturbances. However, their convergence rates are at best exponential, since most of the closed-loop pneumatic servo systems under the aforementioned control algorithms are Lipschitz continuous. As an alternative of smooth control, nonsmooth control is being more and more popular due to its many of advantages, such as faster convergence rates and better disturbance rejection properties [17, 18].

Nonsmooth control is an efficient method to achieve finite-time convergence; that is, the state of the closed-loop system under the nonsmooth control converges to zero in a finite time. Different from asymptotically stable systems, nonsmooth control systems are a kind of non-Lipschitz continuous ones, which leads to the difficulties in analysis and synthesis of nonsmooth control problems. Nonsmooth control has been widely investigated from the aspects of both theory and application, including results on finite-time stability analysis tools [18–20], lower-order systems [21, 22], cascaded system [23], and individual systems [24, 25]. In order to improve the closed-loop system performance, disturbance observer based control schemes composed of disturbance observer design and nominal feedback controller design have been proposed in [26–29]. Compared with pure feedback control methods, disturbance observer based control has several superiorities, such as faster rejection of disturbances and recovery of the nominal performances. In particular, the finite-time disturbance observer proposed in [30] can accurately estimate the disturbance in a finite time. Therefore, the undesirable influence caused by disturbances and uncertainties can be canceled in a finite time by disturbance compensation. However, to the best of our knowledge, there are no published results on finite-time disturbance observer based composite control design for pneumatic servo systems by using integral sliding mode control and homogeneous theory.

This paper studies the finite-time position control problem of a pneumatic servo system via integral sliding mode control approach and homogeneous theory [31]. Firstly, a finite-time disturbance observer is designed to estimate the disturbance. Then, by using homogeneous theory and integral sliding mode control approach, a composite finite-time controller combining disturbance compensation and state feedback is designed, which makes tracking errors globally converge to zero in a finite time. Finally, the effectiveness of the proposed control scheme is verified through numerical simulations. The main contributions of this paper are twofold. First and foremost, a finite-time disturbance observer is presented to estimate the disturbance in a finite time, such that the undesirable influence caused by the disturbance can be removed in a finite-time by using disturbance compensation. Second, compared with the smooth control schemes, the proposed finite-time composite control scheme belongs to nonsmooth control approach, which offers a faster convergence rate and a better disturbance rejection performance for the closed-loop tracking error system.

The remaining parts of this paper are organized as follows. Section 2 reviews some preliminary knowledge about some relevant basic concepts and lemmas. The pneumatic servo system model is constructed in detail in Section 3. The proposed composite controller and stability analysis are presented in Section 4. In Section 5, numerical simulations on the comparisons between the proposed composite controller and the pure state feedback controller are presented under the different conditions. Finally, conclusions are drawn in Section 6.

#### 2. Preliminaries

This section reviews some relevant basic concepts and lemmas. For , denotes the standard sign function and is denoted by .

Consider the nonlinear autonomous system, depicted bywhere is a continuous vector field on . Under this condition, the definition of finite-time stability can be represented as follows.

*Definition 1 (finite-time stability [18]). *The equilibrium of system (1) is finite-time convergent if there is an open neighbourhood of the origin and a function , such that every solution trajectory of system (1) starting from the initial point is well-defined and unique in forward time for , and . Here is called the convergence time (with respect to the initial state ). The equilibrium of system (1) is finite-time stable if it is Lyapunov stable and finite-time convergent. If , the origin is a globally finite-time stable equilibrium.

Lemma 2 (see [19]). *Let be such that the polynomial is Hurwitz, and consider the systemThere exists such that, for every , the origin is a globally finite-time stable equilibrium for system (2) under the feedback control law:where satisfywith and .*

#### 3. Modeling of Pneumatic Servo System

In this section, we firstly introduce the model of pneumatic servo system and its operating principle. Figure 1 describes the schematic of a double-acting single-rod pneumatic servo system. The pneumatic servo system is combined by a proportional valve, a double-acting single-rod cylinder, two pressure sensors, a digital computer, a linear encoder, a decoder, an analog-to-digital converter, and a digital-to-analog converter. and , respectively, represent the pressures inside the two chambers, and they are measured by pressure sensors. and are piston areas facing chamber 1 and chamber 2, respectively. In Figure 1, the pressure difference acting on the piston forces the piston/load to move, and the piston/load displacement is measured by the linear encoder. Next the displacement signal decoded by the decoder is fed back to the computer. Then, the control signal generated from computer is converted into servo valve, which determines the valve motion and thus determines the pressure inside the two chambers. Then, the dynamics of pneumatic cylinder is firstly modeled.