Mathematical Problems in Engineering

Volume 2016, Article ID 3839505, 13 pages

http://dx.doi.org/10.1155/2016/3839505

## Robust Observer Based Disturbance Rejection Control for Euler-Lagrange Systems

Department of Automation, Shanghai Jiao Tong University, Shanghai, China

Received 24 January 2016; Revised 11 May 2016; Accepted 5 June 2016

Academic Editor: Sergey A. Suslov

Copyright © 2016 Yanjun Zhang 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

Robust disturbance rejection control methodology is proposed for Euler-Lagrange systems, and parameters optimization strategy for the observer is explored. First, the observer based disturbance rejection methodology is analyzed, based on which the disturbance rejection paradigm is proposed. Thus, a disturbance observer (DOB) with partial feedback linearization and a low-pass filter is proposed for nonlinear dynamic model under relaxed restrictions of the generalized disturbance. Then, the outer-loop backstepping controller is designed for desired tracking performance. Considering that the parameters of DOB cannot be obtained directly based on Lyapunov stability analysis, parameter of DOB is optimized under standard control framework. By analyzing the influence of outer-loop controller on the inner-loop observer parameter, robust stability constraint is proposed to guarantee the robust stability of the closed-loop system. Experiment on attitude tracking of an aircraft is carried out to show the effectiveness of the proposed control strategy.

#### 1. Introduction

Euler-Lagrange systems widely exist in practice, such as manipulator, mobile robot, underwater vehicle, surface vessel, and aerial vehicle. Consequently, motion control of Euler-Lagrange systems has been widely explored in the past decades. Motion control systems usually work at unknown environment, and inevitably, they suffer from system uncertainties and external disturbances, which will affect the control performance or even make the system unstable [1]. To deal with this problem, numerous approaches have been proposed, such as sliding mode control [2–4], adaptive control [5–7], robust control [8–10], and intelligent control [11–13]. These control methods can more or less deal with the system uncertainties. However, facing the problems is still inevitable, such as chattering of sliding mode control, stability problem of adaptive control, conservative robust control, and convergence rate of weights in neural network and fuzzy system.

The effectiveness of disturbance observer (DOB) has been shown in many applications, such as humanoid robot control [14, 15], manipulator control [16–18], aircraft control [19, 20], optical disk control [21, 22], motor control [23, 24], and vibration control [25, 26]. Traditional DOB methodology, which is proposed based on linear system, cannot be used directly in nonlinear systems [27]. In [28], traditional linear DOB is applied for disturbance rejection of nonlinear system. However, only first-order binomial coefficient typed low-pass filter is used for DOB implementation. The performance of the closed-loop system cannot be improved effectively. Meanwhile, the optimization strategy of parameters is not investigated. Nonlinear DOB is proposed in [29, 30], which can be directly used for disturbance estimation in nonlinear systems. In this paper, we find that estimation effect of nonlinear DOB is the same as that of linear DOB with first-order low-pass filter when a constant observer gain is selected. And asymptotic stability is guaranteed simply based on the assumption that the generalized disturbances and their first-order derivatives are bounded and that the first derivatives go to zero in the steady state, which is not realistic in most conditions. Meanwhile, for a closed-loop system, the parameters of inner-loop observer depend on not only system uncertainties and measurement noise, but also the structure and parameter of outer-loop controller. However, the existing works rarely discuss parameters optimization of the observer. The influence caused by outer-loop controller is never explored in existing researches.

From the descriptions above, a robust DOB based disturbance rejection controller is proposed, and parameters optimization strategy is investigated. Nonlinear DOB and extended state observer (ESO) are first analyzed to show the essence of the disturbance estimation problem. Then, under relaxed restrictions of disturbance and system perturbation, a novel disturbance observer is proposed for nonlinear system. The observer consists of a feedback linearization compensator and a low-pass filter. The feedback linearization compensator is introduced to linearize the nonlinear dynamics into a linear part disturbed by the equivalent disturbance, whereas the low-pass filter is employed to estimate the equivalent disturbances. Then, a state feedback controller is presented for the nominal model to acquire desired performance. Stability of the overall closed-loop system is analyzed based on Lyapunov theory. At last, the influence on DOB parameters optimization caused by structure and parameter of outer-loop controller is analyzed. The robust stability constraint condition, which ensures the robust stability of the whole system, is proposed. Thus, the method can be employed to optimize the parameters of the DOB.

The main contributions of this paper are summarized as follows:(1)The disturbance rejection paradigm of the observer based disturbance rejection methodology is proposed.(2)With the proposed disturbance rejection paradigm, a novel observer, whose low-pass filter of its structure can be selected to be flexible, is proposed for nonlinear systems.(3)The parameters optimization method is investigated to make sure the designed control system can guarantee the robust stability of the closed-loop system.

The rest of this paper is organized as follows. In Section 2, a mechanical system model is established, based on which the disturbance rejection problem is formulated. In Section 3, DOB based control methodology is proposed, and parameters of DOB are optimized to guarantee the robust stability. In Section 4, attitude tracking task is carried out to show the effectiveness of the proposed strategy, followed by conclusions in Section 5.

#### 2. System Model and Problem Statement

##### 2.1. System Model

An Euler-Lagrange equation for the mechanical system is described aswhere and denote the generalized coordinates and velocities and and are the control input and external disturbance, respectively. represent the positive definite inertial matrix, represents the matrix of Coriolis and centrifugal forces, and represents the gravity term. The nonlinear functions , , and satisfy the following assumption.

*Assumption 1. *The unknown nonlinear functions , , and are continuously differentiable and locally Lipschitz.

By introducing the definitions (1) can be rewritten asAccording to the parameters perturbation, it is impossible to establish the system model accurately. By introducing the notations where subscript denotes the nominal value of the corresponding matrix and subscript denotes the part of perturbation, then, the dynamics can be described as follows:where , , and . is the perturbed term caused by the internal uncertainty, which is defined as

In practical applications, the consumption of the external disturbances is finite; that is, the external disturbance is bounded. Nevertheless, internal uncertainty usually depends on system state. Assume that the controller is defined as ; nonlinear function is continuously differentiable. Thus, from the definition of , we can also obtain that is continuously differentiable. From the above analysis, the following assumptions can be obtained.

*Assumption 2. *The external disturbance is bounded, where and represent the constant and time-varying component. The time-varying component satisfies .

*Assumption 3. *The internal uncertainties satisfy , where is classical function.

##### 2.2. Problem Formulation

For the system model described in (5), the key point of the antidisturbance control methodology is the observer configuration. The control accuracy and robustness of the overall system are largely determined by the performance of observer. Here, several widely used observers are provided for analysis. Based on the disturbance rejection paradigm, we propose a novel observer structure and parameter optimization strategy for nonlinear systems.

###### 2.2.1. Extended State Observer (ESO)

ESO is the most important part of the active disturbance rejection control (ADRC) [31]. Instead of identifying the plant dynamics off-line, ESO can estimate the combined effect of plant dynamics and external disturbance in real time. However, ESO can be only used for the standard chained systems. Here, an ESO is designed aswhere and are positive constant to be selected such that is Hurwitz.

By substituting (5) into (7) and introducing the Laplace Transformation, we finally get the following equation:where is the Laplace operator.

###### 2.2.2. Nonlinear Disturbance Observer (NDOB)

The NDOB has been widely used for nonlinear systems with uncertainties [30]. It can estimate the composite disturbances and compensate in the feedback controller. The NDOB for the dynamics of (5) is given aswhere .

From (9) we getThen, by introducing the Laplace Transformation, we finally get

In most applications, observer gain is usually selected as a positive constant.

###### 2.2.3. Disturbance Rejection Paradigm

According to the analysis above, we find that the estimation of the observer can be obtained as the real composite disturbance passing through a low-pass filter. It can be summarized that the estimation effect of the observers should fulfill the following disturbance rejection paradigm:where is the composite disturbance which contains both external disturbances and equivalent internal disturbances. is a low-pass filter such that can converge to asymptotically.

For most researches on observer based control, the structure of the low-pass filter is usually fixed by the observer structure. Meanwhile, the parameters tuning usually relies on trial and error; rarely do researches focus on the point of how to optimize the observer parameters according to the property of system uncertainties, outer-loop controller, measurement noise, and so forth. Hence, in this paper, a novel observer, whose low-pass filter can be selected to be flexible, is proposed for the nonlinear system. Particularly, the parameters optimization strategy is explored for nonlinear systems.

#### 3. Controller Design and Parameter Optimization

##### 3.1. Controller Design

The objective of controller design is that the observer is proposed to estimate the internal uncertainty and external disturbance , and thus the estimation is compensated in the closed-loop control system. Then, feedback controller is designed to stabilize the system to the equilibrium point . The control structure is shown in Figure 1.