Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6162194, 8 pages

https://doi.org/10.1155/2017/6162194

## Current-Loop Control for the Pitching Axis of Aerial Cameras via an Improved ADRC

^{1}Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei 230027, China^{2}Key Lab of Electric and Control of Anhui Province, Anhui Polytechnic University, Wuhu 241000, China

Correspondence should be addressed to BingYou Liu; nc.ude.ctsu.liam@900ybl

Received 13 July 2016; Revised 9 October 2016; Accepted 4 December 2016; Published 6 February 2017

Academic Editor: Rafael M. Herrera

Copyright © 2017 BingYou Liu 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

An improved active disturbance rejection controller (ADRC) is designed to eliminate the influences of the current-loop for the pitching axis control system of an aerial camera. The improved ADRC is composed of a tracking differentiator (TD), an improved extended state observer (ESO), an improved nonlinear state error feedback (NLSEF), and a disturbance compensation device (DCD). The TD is used to arrange transient process. The improved ESO is utilized to observe the state extended by nonlinear dynamics, model uncertainty, and external disturbances. Overtime variation of the current-loop can be predicted by the improved ESO. The improved NLSEF is adopted to restrain the residual errors of the current-loop. The DCD is used to compensate the overtime variation of the current-loop in real time. The improved ADRC is designed based on a new nonlinear function . This function exhibits enhanced continuity and smoothness compared to previously available nonlinear functions. Thus, the new nonlinear function can effectively decrease the high-frequency flutter phenomenon. The improved ADRC exhibits improved control performance, and disturbances of the current-loop can be eliminated by the improved ADRC. Finally, simulation experiments are performed. Results show that the improved ADRC displayed better performance than the proportional integral (PI) control strategy and traditional ADRC.

#### 1. Introduction

The pitching axis control system of an aerial camera adopts a servo control system with three loops, namely, current-loop, speed-loop, and position-loop. As an inner loop, the current-loop plays an important role in the pitching axis control system of an aerial camera [1, 2]. This loop should be controlled in real time to achieve good performance of the pitching axis control system. Therefore, the current-loop needs an improved control strategy. The current-loop control strategy has been extensively investigated globally, and several developments have been achieved. For example, Radwan and Mohamed [3] examined an improved vector control strategy, but the vector control strategy needs accurately known system parameters. Liu et al. [4] introduced current control strategy with an optimal variable structure that exhibits improved robustness. However, this control strategy cannot solve the jitter problem. Novak and Riener [5] examined an artificial intelligence control strategy, which does not rely on the mathematical model of the controlled object. However, this control strategy has a complex algorithm. Liu et al. [6] examined an adaptive current control technique with constant frequency based on clocked signal. This control strategy exhibits good static performance but weak restrain capacity for external disturbance. Several studies investigated a traditional active disturbance rejection control strategy, which exhibits certain anti-interference ability. In [7], Han introduced the evolution from PID to ADRC and the advantages of ADRC. In [8–10], Guo used active disturbance rejection control (ADRC) and sliding mode control (SMC) in the stabilization of the Euler-Bernoulli beam equation, one-dimensional and multidimensional antistable wave equations with boundary input disturbance. In [11], Sira-Ramírez et al. presented an ADRC scheme for the angular velocity trajectory tracking task on a substantially perturbed, uncertain, and permanent magnet synchronous motor. An ADRC was designed to solve the trajectory tracking problem of a Delta robot with uncertain dynamical model [12]. The stabilization problem of a class of nonlinear systems with actuator saturation is investigated via ADRC [13]. An adaptive extended state observer- (AESO-) based ADRC is proposed to deal with the uncertainties and applied to the air-fuel ratio (AFR) control of gasoline engine, which has large nonlinear uncertainties due to the unknown speed change, fuel film dynamics, and so on [14]. The absolute stability of nonlinear ADRC for single-input-single-output systems is analyzed by the circle criterion method [15]. In [16], an ADRC was applied to stabilization for lower triangular nonlinear systems with large uncertainties. Guo et al. [17] generalized the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. Liu et al. [18] proposed a two-layer ADRC method with the compensation of estimated equivalent input disturbances (EID) for load frequency control (LFC) of multiarea interconnected power system. However, the high-frequency flutter phenomenon is not eliminated, and the anti-interference ability needs to be enhanced. The current-loop of the pitching axis control system of an aerial camera is controlled under weightlessness. Therefore, a small interference will result in a serious error. The control system requires an improved active disturbance rejection controller (ADRC) with strong anti-interference ability.

The improved ADRC is composed of a tracking differentiator (TD), an improved extended state observer (ESO), an improved nonlinear state error feedback (NLSEF), and a disturbance compensation device (DCD). In this study, the previously available nonlinear function of a traditional ADRC is improved to a new nonlinear function with better continuity and smoothness. Thus, the improved ADRC based on the new nonlinear function exhibits better anti-interference performance. On the one hand, the total disturbances are sufficiently estimated by the improved ESO, and the state of the system does not require direct observation. Therefore, the controllability of the system can be enhanced greatly. On the other hand, control efficiency is enhanced using the improved NLSEF to conduct nonlinear calculation of the proportion signal, differential signal, and integral signal of the error [19, 20]. Extensive studies on the improved ADRC have resulted in important progress. Qi et al. [21] constructed a continuous and smooth ESO and disturbance rejection cascade decoupling control technology. An improved ADRC has been applied in a hypersonic vehicle and in the straight course error modeling of an electromagnetically controlled gyrocompass [22]. In [23], an improved ADRC based on the nonlinear arctangent function is used to reduce the observation noise in the traditional extended state observation of a system with output measurement noise. An improved ADRC is proposed to improve the tracking performance of the electromechanical actuator [24]. In [25], an improved ADRC is used to control an autonomous underwater vehicle. In [26], an enhanced ADRC is presented for a twin-rotor multi-input multioutput system with two degrees of freedom. However, few reports have focused on the use of improved ADRC in the current-loop control system of the pitching axis of an aerial camera.

The present study proposes an improved ADRC based on a new nonlinear function . Then, the controller is applied to the current-loop control system of an aerial camera pitching axis. Simulations are conducted for cases adopting the PI control strategy, traditional ADRC, and improved ADRC. Finally, the simulation results are discussed.

#### 2. Mathematical Model

The pitching axis of an aerial camera adopts a permanent magnet synchronous motor (PMSM) as an actuator. A control strategy of is used in the current-loop control system, where is the current of the direct axis. Therefore, the controlled object of the current-loop is , where is the current of the quadrature axis. The current equation of the quadrature axis in a () two-phase rotating coordinate system is given as follows:where is the stator phase resistance, is the quadrature axis inductance, is the magnetic chain produced by the rotor permanent magnet, is the equivalent voltage of the quadrature axis, and is the angular speed of the motor rotor. The mathematical model of PMSM in a () two-phase rotating coordinate system shows that the electromagnetic torque is directly controlled by . Therefore, the precise control of the electromagnetic torque can be realized by controlling the current-loop. The expression represents a coupling effect in the current equation of the quadrature axis. The current-loop is a nonlinear system under the influence of the coupling effect. Several definitions are given as follows:Then, the current equation of the quadrature axis can be expressed as follows:where and are the internal and external disturbances of the current-loop, respectively. The total disturbance of the current-loop is expressed as . Then, the current equation of quadrature axis is as follows:Equation (4) is a typical equation used to design the improved ADRC.

#### 3. Improved ADRC Design for the Current-Loop

##### 3.1. Design of the New Nonlinear Function

The nonlinear function is the core of the ADRC. The following conditions should be fully considered in designing the nonlinear function. First, the nonlinear function should exhibit good convergence around the origin. Second, the value of the nonlinear function is 0 at the origin. Third, the nonlinear function should be continuous around the origin. The traditional ADRC adopts a nonlinear function , which can be expressed as follows:The characteristics of are given as follows. The value of influences the nonlinearity degree of . The value of is usually selected between 0 and 1. The nonlinear function exhibits optimal nonlinearity when . The degree of linearity of is optimal when . The value of is the linear interval width of and related to the error range of the system. When the input is an error signal, the system can achieve rapid stability by adjusting the parameter values of . is a nonderivable function, although this function is continuous. If the value of is too small, will also cause the high-frequency flutter phenomenon. The value of is difficult to adjust, because the control performance of is sensitive to the value of . Thus, a new nonlinear function is designed. The proposed exhibits better convergence and continuity around the origin than . The expression of is calculated by the following steps.

When , is expressed as .

When , the expression of is designed as an interpolation function form by multinomial and trigonometric functions and can be expressed as follows:Part three of the above expression is selected as , but not , because exhibits better convergence than around the origin. Then, can be given as follows:where , , and are the function coefficients and can be calculated by the following steps.

In the range of , the following expression is given to meet the continuous and derivable conditions:The following expression can be obtained by (7) and (8),The values of , , and can be calculated as follows:Thus, the expression of can be obtained as follows:Simulations are performed using the two nonlinear functions under . The curves of the function are shown in Figure 1.