Mathematical Problems in Engineering

Volume 2018, Article ID 9316562, 9 pages

https://doi.org/10.1155/2018/9316562

## Finite-Time Output Feedback Control for a Rigid Hydraulic Manipulator System

^{1}School of Energy and Environment, Southeast University, Nanjing 210096, Jiangsu, China^{2}School of Automation, Southeast University & Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing 210096, Jiangsu, China

Correspondence should be addressed to Xiangyu Wang; nc.ude.ues@y.x.w

Received 12 February 2018; Accepted 25 June 2018; Published 10 July 2018

Academic Editor: Xinkai Chen

Copyright © 2018 Yong-Sheng Hao 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

The position tracking control problem of a hydraulic manipulator system is investigated. By utilizing homogeneity theory, a finite-time output feedback controller is designed. Firstly, a finite-time state feedback controller is developed based on homogeneity theory. Secondly, a nonlinear state observer is designed to estimate the manipulator’s velocity. A rigorous analysis process is presented to demonstrate the observer’s finite-time stability. Finally, the corresponding output feedback tracking controller is derived, which stabilizes the tracking error system in finite time. Simulations demonstrate the effectiveness of the designed finite-time output feedback controller.

#### 1. Introduction

In the past years, robotic system related control problem has been investigated more and more widely for robots application value [1, 2]. In addition to rapid responses and high power-to-weight ratios [3–10], a great many robots are driven by hydraulic actuators. To improve the performance for industrial hydraulic machines, it is essential to enhance the control performance of manipulators actuated by hydraulic actuators [3, 4]. However, hydraulic manipulators control problem is more challenging than their electrical counterparts, due to the nonlinear dynamics and the nonlinear mechanical linkage dynamics. On one hand, mechanical linkage dynamics are composed of strong coupling effects among various joints and significant nonlinearities in the hydraulic actuators [4]. On the other hand, with complex properties of servo valves, frictions, compressibility of hydraulic fluid and volume changes [5, 6, 11], and hydraulic actuators exhibit highly nonlinear characteristics. To realize more accurate control, the hydraulic actuators dynamics must be considered in the hydraulic manipulators control design [4, 12].

For the hydraulic manipulators position tracking control problem, researchers have proposed a great deal of meaningful results in recent years [3, 4, 12–16]. In [13], by using singular perturbation technique, a nonlinear control method was developed to solve the trajectory tracking control problem and the stabilization problem for -axis hydraulically actuated robots. Decentralized adaptive controllers were proposed for the hydraulic manipulators [14, 15]. For the Stewart-type hydraulic manipulator [16], a pressure feedback controller was presented, which allow very high proportional position error gains. However, these results have not paid much attention to stability analysis. In [3], based on Lyapunov stability theory, an adaptive robust control algorithm was proposed. With sliding mode control method, a robust control algorithm was presented in [4] to realize accurate position tracking for hydraulic manipulators. By combining pole placement technique, backstepping control, and sliding mode control together, a composite controller was designed in [12] to regulate both flexural vibrations and flexible arm motion.

In the above-mentioned literatures, the proposed methods just realize asymptotical stability for position control problems of hydraulic systems, which indicates that such systems are asymptotically stable with infinite settling time. Thus it is necessary to design more efficient controllers to offer faster convergence rates. To this end, finite-time control is a good choice. To realize finite-time control, nonsmooth control is a feasible choice. Besides faster convergence rates, nonsmooth control systems usually also possess some other superiorities, such as better disturbance rejection abilities and robustness [17]. Because of such nice features, nonsmooth control has been widely investigated from the perspectives of state feedback control [18], output feedback control [19–21], individual systems [22, 23], and multiagent systems [24–26].

Note that, in the above references, almost all the control designs are based on full-state information. However, in practice, in the case without velocity sensors, the manipulator velocity can not be measured directly. It is impractical to obtain the velocity information by differentiating the measured position, because derivation operation usually results in very noisy velocity data. To this end, an effective way is estimating the manipulator angular velocity via velocity observers.

In this paper, the position tracking control problem of the rigid hydraulic manipulator system is studied. Firstly, a finite-time state feedback controller is developed based on homogeneity theory. Secondly, in the case without angular velocity sensors, by utilizing homogeneous theory, a finite-time convergent observer is developed to obtain the angular velocity information for feedback. A rigorous stability analysis process is presented to demonstrate finite-time stability of the designed state observer. Thirdly, with the designed finite-time state feedback controller and the estimated angular velocity, a finite-time output feedback controller is given.

The remainder of this paper is arranged as follows. In Section 2, some preliminary knowledge is given. In Section 3, the model of the rigid hydraulic manipulator system is presented. The main results are presented in Section 4. Simulations on the proposed finite-time tracking controller are performed in Section 5. Finally, conclusions are drawn in Section 6.

#### 2. Preliminaries

Denote , where . is the sign function. For a vector , the notation represents the vector , where . The notation represents the vector , where . represents a diagonal matrix with the diagonal elements .

Consider the following nonlinear autonomous system:where satisfies the locally Lipschitz continuous condition. Under this condition, the definition of finite-time stability can be described as follows.

*Definition 1 (finite-time stability, [17]). *The equilibrium of system (1) is finite-time convergent if there are 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 [27]). *Consider the following system:There exists such that, for every , the origin is a globally finite-time-stable equilibrium for system (2) under the feedback control lawwhere are constants such that the polynomial is Hurwitz and satisfy , with and .*

Lemma 3 (see [19]). *For system (2), there exists proper constants , so that the observer shown below is finite-time convergent. where *

Lemma 4 (see [20]). *Consider the following system:where is a continuously homogeneous function of degree with respect to and satisfies . If system is asymptotically stable, then system (5) is locally finite-time stable, when *

#### 3. System Models and Problem Formulation

In this section, the hydraulic actuator and the model of the n-link rigid manipulator are presented firstly. Then the integrated model of them is constructed.

##### 3.1. Dynamic of the Hydraulic Actuator

The schematic of the rigid manipulator driven by single-rod hydraulic servo system is depicted in Figure 1. and represent the forward and return pressures of the cylinder, respectively. is the supply flow rate to the forward chamber and is the return flow rate to the return chamber. is the piston area facing the extending chamber and is the piston area facing the retract chamber. is the supply pump pressure and is the return pressure.