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

Authors: BingYou Liu; ChangAn Zhu; XingZhong Guo
Journal: Mathematical Problems in Engineering
 (2017)

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## 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 functionnewfal(·). 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.


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## Body

## 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 functionnewfal(·). 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 ofid=0 is used in the current-loop control system, where id is the current of the direct axis. Therefore, the controlled object of the current-loop is iq, where iq is the current of the quadrature axis. The current equation of the quadrature axis in a (d,q) two-phase rotating coordinate system is given as follows:(1)iq·=-RLiq-ψfLω+1Luq,where R is the stator phase resistance, L is the quadrature axis inductance, ψf is the magnetic chain produced by the rotor permanent magnet, uq is the equivalent voltage of the quadrature axis, and ω is the angular speed of the motor rotor. The mathematical model of PMSM in a (d,q) two-phase rotating coordinate system shows that the electromagnetic torque is directly controlled by iq. Therefore, the precise control of the electromagnetic torque can be realized by controlling the current-loop. The expression -ψf/Lω 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:(2)wt=-ψfLω,fiq,t=-RLiq,b=1L,u=uq.Then, the current equation of the quadrature axis can be expressed as follows:(3)iq·=fiq,t+wt+bu,where fiq,t and wt are the internal and external disturbances of the current-loop, respectively. The total disturbance of the current-loop is expressed as at=fiq,t+wt. Then, the current equation of quadrature axis is as follows:(4)iq·=at+bu.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 functionfal(·), which can be expressed as follows:(5)fale,α,δ=eαsigne,e>δeδ1-α,e≤δ,δ>0.The characteristics of fal(·) are given as follows. The value of α influences the nonlinearity degree of fal(·). The value of α is usually selected between 0 and 1. The nonlinear function exhibits optimal nonlinearity when α=0. The degree of linearity of fal(·) is optimal when α=1. The value of δ is the linear interval width of fal(·) 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 fal(·). fal(·) is a nonderivable function, although this function is continuous. If the value of δ is too small, fal(·) will also cause the high-frequency flutter phenomenon. The value of δ is difficult to adjust, because the control performance of fal(·) is sensitive to the value of δ. Thus, a new nonlinear function newfal(·) is designed. The proposed newfal(·) exhibits better convergence and continuity around the origin than fal(·). The expression of newfal(·) is calculated by the following steps.Whene>δ, newfal(·) is expressed as newfale,α,δ=eαsigne.Whene≤δ, the expression of newfal(·) is designed as an interpolation function form by multinomial and trigonometric functions and can be expressed as follows:(6)newfale,α,δ=k1e+k2e2+k3tan⁡e.Part three of the above expression is selected as tan⁡e, but not e3, because tan⁡e exhibits better convergence than e3 around the origin. Then, newfal(e,α,δ) can be given as follows:(7)newfale,α,δ=eαsigne,e>δk1e+k2e2+k3tan⁡e,e≤δ,where k1, k2, and k3 are the function coefficients and can be calculated by the following steps.In the range ofe≤δ, the following expression is given to meet the continuous and derivable conditions:(8)newfale,α,δ=δα,e=δ,newfale,α,δ=-δα,e=-δ,newfal′e,α,δ=αδα,e=±δ.The following expression can be obtained by (7) and (8),(9)k1δ+k2δ2+k3tan⁡δ=δα,-k1δ+k2δ2-k3tan⁡δ=-δα,k1+2k2δ+k3sec2⁡δ=αδα.The values of k1, k2, and k3 can be calculated as follows:(10)k1=δα+δα·tan2⁡δ-α·tan⁡δ·δαδ-tan⁡δ+δ·tan2⁡δ,k2=0,k3=α·δα+1-δαδ-tan⁡δ+δ·tan2⁡δ.Thus, the expression of newfal(e,α,δ) can be obtained as follows:(11)newfale,α,δ=eαsigne,e>δδα+δα·tan2⁡δ-α·tan⁡δ·δαδ-tan⁡δ+δ·tan2⁡δe+α·δα+1-δαδ-tan⁡δ+δ·tan2⁡δtan⁡e,e≤δ.Simulations are performed using the two nonlinear functions under α=0.05,δ=0.06. The curves of the function are shown in Figure 1.Figure 1
Curves ofnewfal(⋅) and fal(⋅) when α=0.05, δ=0.06.The simulation results demonstrate thatnewfal(e,α,δ) exhibits better continuity and derivability than fal(e,α,δ). Thus, newfal(e,α,δ) is not sensitive to the selection of the δ value and can avoid the high-frequency flutter phenomenon. In this study, the improved ADRC is designed based on newfal(e,α,δ).


### 3.2. Improved ADRC Design for the Current-Loop of the Pitching Axis of an Aerial Camera

Figure2 shows the structure of the improved ADRC of the current-loop for the pitching axis control system of an aerial camera.Figure 2
Structure of the improved ADRC of the current-loop of the pitching axis control system of an aerial camera.


#### 3.2.1. TD

TD in the improved ADRC is used to arrange the transient process. A continuous and differentiable input signal can be obtained by TD. Therefore, drastic changes in the control signal can be avoided effectively. The TD expression of the current-loop of the pitching axis control system of an aerial camera is designed as follows:(12)fh=fhanz11-iq∗,v2,r0,h0,z11·=v2,v2·=fh,where iq∗ is the current given the quadrature axis, z11 is the tracking signal for iq∗, r0 is the velocity factor, h0 is the filter factor, and r0 and h0 are used to arrange the speed of the transient process. fhanx1,x2,r0,h0 is an optimal control synthesis function and can be defined as(13)fhanv1,v2,r0,h0=-r0signα,α>δr0αδ,α≤δ,where (14)α=v2+α0-δ2signy,y>δ0v2+yh0,y≤δ0,δ=r0h0,δ0=δh0,y=v1+h0v2,α0=δ2+8r0y.v(t) is the input and v1 is the output of the TD. v1 can track v(t) as fast as possible in the case of the acceleration constraint v¨≤r. The larger the value of r0, the faster the tracking speed. v2 is the differentiation of vt when v1 is tracking vt. h0 is used to filter out the high-frequency interferences of vt. The value of h0 is parameterized as h0=0.01. The greater the value of h0, the better the filtering effect.


#### 3.2.2. Improved ESO

Improved ESO is the core of the improved ADRC, which is a nonlinear control technology that does not rely on the system model. Nonlinear dynamics, model uncertainty, and external disturbances are extended to a new state and observed by the improved ESO. The overtime variation of system can be predicted by the improved ESO. Therefore, the robustness of the system can be improved greatly. The expression of the improved ESO for the current-loop for the pitching axis control system of an aerial camera is designed as(15)e1=z21-iq,z21•=z22-β1newfale1,α0,δ1+b0ut,z22•=-β2newfale1,α1,δ1,where z21 is the tracking signal for iq, e1 is the tracking error, and z22 is the tracking signal for the total disturbance w(t). If the new nonlinear function newfal(·) is selected appropriately, the state variables z21 and z22 can indicate the state variables iq and w(t), respectively. Therefore, the improved ESO for the current-loop for the pitching axis of the aerial camera can be obtained. β1,β2 are the gain coefficients of the improved ESO. α0,α1 are the nonlinear factors and satisfy the condition 0<α1<α0≤1. δ1 is a filtering factor with a value of δ1=0.01. b0 is the estimation of the compensation factor. The internal structure of the improved ESO of the current-loop for the pitching axis of the aerial camera is shown in Figure 3.Figure 3
Internal structure of the improved ESO of the current-loop of the pitching axis of the aerial camera.The stability of the improved ESO can be proved as follows. The bandwidth of the current-loop is defined asλ. The following equation can be obtained according to previous research:(16)s+λ2=s2+β1s+β2.The nonlinear parts of the improved ESO are divided into many areas to analyze the stability of the improved ESO. All areas are replaced by piecewise linear functions. The stability of the entire area can be ensured when the stability of each area is ensured. The present study used the linear expression z21-iq to replace the nonlinear function newfal(·) to analyze the stability of the improved ESO. Therefore, (15) can be written as(17)z21•=z22-β1z21-iq+b0ut,z22•=-β2z21-iq.Laplace transform is carried out for the above equation and the following equation can be obtained:(18)z21s·s=z22s-2λz21s+2λiqs+b0us,z22s·s=-λ2z21s+λ2iqs.therefore, the following equation can be obtained:(19)z21s=λ2+2λss+λ2·iqs+b0ss+λ2·us,z22s=λ2ss+λ2·iqs+λ2b0s+λ2·us.The errors of z21(s) and z22(s) can be calculated as follows:(20)e1s=z21s-iqs,e2s=z22s-as=z22s-iqs·-b0us;then, (21) can be calculated:(21)e1s=λ2+2λss+λ2·iqs+b0ss+λ2·us-iqs,e2s=λ2ss+λ2·iqs+λ2b0s+λ2·us-iqs·s-b0us,e1s=-s2s+λ2·iqs+b0ss+λ2·us,e2s=-s2+2λss+λ2·s·iqs-s2+2λss+λ2·b0·us.This article selected iq(t) and u(t) as unit step input. Thus, iq(s)=1/s, and u(s)=1/s. Thesteady-state errors can be calculated as follows:(22)es1=lims→0⁡s·e1s=0,es2=lims→0⁡s·e2s=0;therefore, the improved ESO exhibits stability.


#### 3.2.3. Improved NLSEF

Improved NLSEF is used to combine the state variables produced by tracking the TD and improved ESO with the error of estimated value nonlinearly. The expression of the improved NLSEF for the aerial camera pitching axis current-loop is designed as(23)e2=z11-z21,u0=β3newfale2,α2,δ2,where β3 is the gain coefficient of the improved NLSEF, α2 is a nonlinear factor with a value between 0 and 1, and δ2 is a filtering factor.


#### 3.2.4. DCD

DCD is used to compensate for the total disturbance of the current-loop for the pitching axis of the aerial camera. The total disturbance is composed of internal disturbance, external disturbance, and coupling influence. The expression of the DCD for the current-loop of the pitching axis of the aerial camera is designed as(24)u=u0-z22b0,where u0 is the control signal without DCD, u is the control signal, and z22/b0 is the compensation component of total disturbance for the aerial camera pitching axis 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 functionfal(·), which can be expressed as follows:(5)fale,α,δ=eαsigne,e>δeδ1-α,e≤δ,δ>0.The characteristics of fal(·) are given as follows. The value of α influences the nonlinearity degree of fal(·). The value of α is usually selected between 0 and 1. The nonlinear function exhibits optimal nonlinearity when α=0. The degree of linearity of fal(·) is optimal when α=1. The value of δ is the linear interval width of fal(·) 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 fal(·). fal(·) is a nonderivable function, although this function is continuous. If the value of δ is too small, fal(·) will also cause the high-frequency flutter phenomenon. The value of δ is difficult to adjust, because the control performance of fal(·) is sensitive to the value of δ. Thus, a new nonlinear function newfal(·) is designed. The proposed newfal(·) exhibits better convergence and continuity around the origin than fal(·). The expression of newfal(·) is calculated by the following steps.Whene>δ, newfal(·) is expressed as newfale,α,δ=eαsigne.Whene≤δ, the expression of newfal(·) is designed as an interpolation function form by multinomial and trigonometric functions and can be expressed as follows:(6)newfale,α,δ=k1e+k2e2+k3tan⁡e.Part three of the above expression is selected as tan⁡e, but not e3, because tan⁡e exhibits better convergence than e3 around the origin. Then, newfal(e,α,δ) can be given as follows:(7)newfale,α,δ=eαsigne,e>δk1e+k2e2+k3tan⁡e,e≤δ,where k1, k2, and k3 are the function coefficients and can be calculated by the following steps.In the range ofe≤δ, the following expression is given to meet the continuous and derivable conditions:(8)newfale,α,δ=δα,e=δ,newfale,α,δ=-δα,e=-δ,newfal′e,α,δ=αδα,e=±δ.The following expression can be obtained by (7) and (8),(9)k1δ+k2δ2+k3tan⁡δ=δα,-k1δ+k2δ2-k3tan⁡δ=-δα,k1+2k2δ+k3sec2⁡δ=αδα.The values of k1, k2, and k3 can be calculated as follows:(10)k1=δα+δα·tan2⁡δ-α·tan⁡δ·δαδ-tan⁡δ+δ·tan2⁡δ,k2=0,k3=α·δα+1-δαδ-tan⁡δ+δ·tan2⁡δ.Thus, the expression of newfal(e,α,δ) can be obtained as follows:(11)newfale,α,δ=eαsigne,e>δδα+δα·tan2⁡δ-α·tan⁡δ·δαδ-tan⁡δ+δ·tan2⁡δe+α·δα+1-δαδ-tan⁡δ+δ·tan2⁡δtan⁡e,e≤δ.Simulations are performed using the two nonlinear functions under α=0.05,δ=0.06. The curves of the function are shown in Figure 1.Figure 1
Curves ofnewfal(⋅) and fal(⋅) when α=0.05, δ=0.06.The simulation results demonstrate thatnewfal(e,α,δ) exhibits better continuity and derivability than fal(e,α,δ). Thus, newfal(e,α,δ) is not sensitive to the selection of the δ value and can avoid the high-frequency flutter phenomenon. In this study, the improved ADRC is designed based on newfal(e,α,δ).


## 3.2. Improved ADRC Design for the Current-Loop of the Pitching Axis of an Aerial Camera

Figure2 shows the structure of the improved ADRC of the current-loop for the pitching axis control system of an aerial camera.Figure 2
Structure of the improved ADRC of the current-loop of the pitching axis control system of an aerial camera.


### 3.2.1. TD

TD in the improved ADRC is used to arrange the transient process. A continuous and differentiable input signal can be obtained by TD. Therefore, drastic changes in the control signal can be avoided effectively. The TD expression of the current-loop of the pitching axis control system of an aerial camera is designed as follows:(12)fh=fhanz11-iq∗,v2,r0,h0,z11·=v2,v2·=fh,where iq∗ is the current given the quadrature axis, z11 is the tracking signal for iq∗, r0 is the velocity factor, h0 is the filter factor, and r0 and h0 are used to arrange the speed of the transient process. fhanx1,x2,r0,h0 is an optimal control synthesis function and can be defined as(13)fhanv1,v2,r0,h0=-r0signα,α>δr0αδ,α≤δ,where (14)α=v2+α0-δ2signy,y>δ0v2+yh0,y≤δ0,δ=r0h0,δ0=δh0,y=v1+h0v2,α0=δ2+8r0y.v(t) is the input and v1 is the output of the TD. v1 can track v(t) as fast as possible in the case of the acceleration constraint v¨≤r. The larger the value of r0, the faster the tracking speed. v2 is the differentiation of vt when v1 is tracking vt. h0 is used to filter out the high-frequency interferences of vt. The value of h0 is parameterized as h0=0.01. The greater the value of h0, the better the filtering effect.


### 3.2.2. Improved ESO

Improved ESO is the core of the improved ADRC, which is a nonlinear control technology that does not rely on the system model. Nonlinear dynamics, model uncertainty, and external disturbances are extended to a new state and observed by the improved ESO. The overtime variation of system can be predicted by the improved ESO. Therefore, the robustness of the system can be improved greatly. The expression of the improved ESO for the current-loop for the pitching axis control system of an aerial camera is designed as(15)e1=z21-iq,z21•=z22-β1newfale1,α0,δ1+b0ut,z22•=-β2newfale1,α1,δ1,where z21 is the tracking signal for iq, e1 is the tracking error, and z22 is the tracking signal for the total disturbance w(t). If the new nonlinear function newfal(·) is selected appropriately, the state variables z21 and z22 can indicate the state variables iq and w(t), respectively. Therefore, the improved ESO for the current-loop for the pitching axis of the aerial camera can be obtained. β1,β2 are the gain coefficients of the improved ESO. α0,α1 are the nonlinear factors and satisfy the condition 0<α1<α0≤1. δ1 is a filtering factor with a value of δ1=0.01. b0 is the estimation of the compensation factor. The internal structure of the improved ESO of the current-loop for the pitching axis of the aerial camera is shown in Figure 3.Figure 3
Internal structure of the improved ESO of the current-loop of the pitching axis of the aerial camera.The stability of the improved ESO can be proved as follows. The bandwidth of the current-loop is defined asλ. The following equation can be obtained according to previous research:(16)s+λ2=s2+β1s+β2.The nonlinear parts of the improved ESO are divided into many areas to analyze the stability of the improved ESO. All areas are replaced by piecewise linear functions. The stability of the entire area can be ensured when the stability of each area is ensured. The present study used the linear expression z21-iq to replace the nonlinear function newfal(·) to analyze the stability of the improved ESO. Therefore, (15) can be written as(17)z21•=z22-β1z21-iq+b0ut,z22•=-β2z21-iq.Laplace transform is carried out for the above equation and the following equation can be obtained:(18)z21s·s=z22s-2λz21s+2λiqs+b0us,z22s·s=-λ2z21s+λ2iqs.therefore, the following equation can be obtained:(19)z21s=λ2+2λss+λ2·iqs+b0ss+λ2·us,z22s=λ2ss+λ2·iqs+λ2b0s+λ2·us.The errors of z21(s) and z22(s) can be calculated as follows:(20)e1s=z21s-iqs,e2s=z22s-as=z22s-iqs·-b0us;then, (21) can be calculated:(21)e1s=λ2+2λss+λ2·iqs+b0ss+λ2·us-iqs,e2s=λ2ss+λ2·iqs+λ2b0s+λ2·us-iqs·s-b0us,e1s=-s2s+λ2·iqs+b0ss+λ2·us,e2s=-s2+2λss+λ2·s·iqs-s2+2λss+λ2·b0·us.This article selected iq(t) and u(t) as unit step input. Thus, iq(s)=1/s, and u(s)=1/s. Thesteady-state errors can be calculated as follows:(22)es1=lims→0⁡s·e1s=0,es2=lims→0⁡s·e2s=0;therefore, the improved ESO exhibits stability.


### 3.2.3. Improved NLSEF

Improved NLSEF is used to combine the state variables produced by tracking the TD and improved ESO with the error of estimated value nonlinearly. The expression of the improved NLSEF for the aerial camera pitching axis current-loop is designed as(23)e2=z11-z21,u0=β3newfale2,α2,δ2,where β3 is the gain coefficient of the improved NLSEF, α2 is a nonlinear factor with a value between 0 and 1, and δ2 is a filtering factor.


### 3.2.4. DCD

DCD is used to compensate for the total disturbance of the current-loop for the pitching axis of the aerial camera. The total disturbance is composed of internal disturbance, external disturbance, and coupling influence. The expression of the DCD for the current-loop of the pitching axis of the aerial camera is designed as(24)u=u0-z22b0,where u0 is the control signal without DCD, u is the control signal, and z22/b0 is the compensation component of total disturbance for the aerial camera pitching axis current-loop.


## 3.2.1. TD

TD in the improved ADRC is used to arrange the transient process. A continuous and differentiable input signal can be obtained by TD. Therefore, drastic changes in the control signal can be avoided effectively. The TD expression of the current-loop of the pitching axis control system of an aerial camera is designed as follows:(12)fh=fhanz11-iq∗,v2,r0,h0,z11·=v2,v2·=fh,where iq∗ is the current given the quadrature axis, z11 is the tracking signal for iq∗, r0 is the velocity factor, h0 is the filter factor, and r0 and h0 are used to arrange the speed of the transient process. fhanx1,x2,r0,h0 is an optimal control synthesis function and can be defined as(13)fhanv1,v2,r0,h0=-r0signα,α>δr0αδ,α≤δ,where (14)α=v2+α0-δ2signy,y>δ0v2+yh0,y≤δ0,δ=r0h0,δ0=δh0,y=v1+h0v2,α0=δ2+8r0y.v(t) is the input and v1 is the output of the TD. v1 can track v(t) as fast as possible in the case of the acceleration constraint v¨≤r. The larger the value of r0, the faster the tracking speed. v2 is the differentiation of vt when v1 is tracking vt. h0 is used to filter out the high-frequency interferences of vt. The value of h0 is parameterized as h0=0.01. The greater the value of h0, the better the filtering effect.


## 3.2.2. Improved ESO

Improved ESO is the core of the improved ADRC, which is a nonlinear control technology that does not rely on the system model. Nonlinear dynamics, model uncertainty, and external disturbances are extended to a new state and observed by the improved ESO. The overtime variation of system can be predicted by the improved ESO. Therefore, the robustness of the system can be improved greatly. The expression of the improved ESO for the current-loop for the pitching axis control system of an aerial camera is designed as(15)e1=z21-iq,z21•=z22-β1newfale1,α0,δ1+b0ut,z22•=-β2newfale1,α1,δ1,where z21 is the tracking signal for iq, e1 is the tracking error, and z22 is the tracking signal for the total disturbance w(t). If the new nonlinear function newfal(·) is selected appropriately, the state variables z21 and z22 can indicate the state variables iq and w(t), respectively. Therefore, the improved ESO for the current-loop for the pitching axis of the aerial camera can be obtained. β1,β2 are the gain coefficients of the improved ESO. α0,α1 are the nonlinear factors and satisfy the condition 0<α1<α0≤1. δ1 is a filtering factor with a value of δ1=0.01. b0 is the estimation of the compensation factor. The internal structure of the improved ESO of the current-loop for the pitching axis of the aerial camera is shown in Figure 3.Figure 3
Internal structure of the improved ESO of the current-loop of the pitching axis of the aerial camera.The stability of the improved ESO can be proved as follows. The bandwidth of the current-loop is defined asλ. The following equation can be obtained according to previous research:(16)s+λ2=s2+β1s+β2.The nonlinear parts of the improved ESO are divided into many areas to analyze the stability of the improved ESO. All areas are replaced by piecewise linear functions. The stability of the entire area can be ensured when the stability of each area is ensured. The present study used the linear expression z21-iq to replace the nonlinear function newfal(·) to analyze the stability of the improved ESO. Therefore, (15) can be written as(17)z21•=z22-β1z21-iq+b0ut,z22•=-β2z21-iq.Laplace transform is carried out for the above equation and the following equation can be obtained:(18)z21s·s=z22s-2λz21s+2λiqs+b0us,z22s·s=-λ2z21s+λ2iqs.therefore, the following equation can be obtained:(19)z21s=λ2+2λss+λ2·iqs+b0ss+λ2·us,z22s=λ2ss+λ2·iqs+λ2b0s+λ2·us.The errors of z21(s) and z22(s) can be calculated as follows:(20)e1s=z21s-iqs,e2s=z22s-as=z22s-iqs·-b0us;then, (21) can be calculated:(21)e1s=λ2+2λss+λ2·iqs+b0ss+λ2·us-iqs,e2s=λ2ss+λ2·iqs+λ2b0s+λ2·us-iqs·s-b0us,e1s=-s2s+λ2·iqs+b0ss+λ2·us,e2s=-s2+2λss+λ2·s·iqs-s2+2λss+λ2·b0·us.This article selected iq(t) and u(t) as unit step input. Thus, iq(s)=1/s, and u(s)=1/s. Thesteady-state errors can be calculated as follows:(22)es1=lims→0⁡s·e1s=0,es2=lims→0⁡s·e2s=0;therefore, the improved ESO exhibits stability.


## 3.2.3. Improved NLSEF

Improved NLSEF is used to combine the state variables produced by tracking the TD and improved ESO with the error of estimated value nonlinearly. The expression of the improved NLSEF for the aerial camera pitching axis current-loop is designed as(23)e2=z11-z21,u0=β3newfale2,α2,δ2,where β3 is the gain coefficient of the improved NLSEF, α2 is a nonlinear factor with a value between 0 and 1, and δ2 is a filtering factor.


## 3.2.4. DCD

DCD is used to compensate for the total disturbance of the current-loop for the pitching axis of the aerial camera. The total disturbance is composed of internal disturbance, external disturbance, and coupling influence. The expression of the DCD for the current-loop of the pitching axis of the aerial camera is designed as(24)u=u0-z22b0,where u0 is the control signal without DCD, u is the control signal, and z22/b0 is the compensation component of total disturbance for the aerial camera pitching axis current-loop.


## 4. Simulation Experiments and Discussion

The improved ADRC is compared with the traditional ADRC and PI control strategy to accurately evaluate its performance. A PMSM is selected as the executing agency of the system. Table1 shows the PMSM parameters, Table 2 presents the traditional ADRC parameters, and Table 3 presents the improved ADRC parameters.Table 1
Parameters of the experimental permanent magnet synchronous motor.
Parameter Dimension Numerical value Power supply voltage VAC 400 Rated power W 2400 Maximum nonload speed r/min 2750 Continuous current A 5.1 Continuous torque N⋅m 6.11 Peak current A 18 Peak torque N⋅m 18 Motor pole logarithmic  16 Resistance Ω 2.9 Inductance mH 6.8Table 2
Parameters of the traditional ADRC.
Component of improved ADRC Parameter name Symbol Numerical value TD Velocity factor r 0 10 Filter factor h 0 0.02  Traditional ESO Nonlinear factor α 0 1 Nonlinear factor α 1 0.5 Linear interval width δ 1 0.01 Gain coefficient β 1 60 Gain coefficient β 2 1200  Traditional NLSEF Nonlinear factor α 2 0.5 Filter factor δ 2 0.01 Gain coefficient β 3 75  DCD Compensation coefficient b 0 1Table 3
Parameters of the improved ADRC.
Component of improved ADRC Parameter Symbol Numerical value TD Velocity factor r 0 200 Filter factor h 0 0.01  Improved ESO Nonlinear factor α 0 0.5 Nonlinear factor α 1 0.25 Linear interval width δ 1 0.01 Gain coefficient β 1 30 Gain coefficient β 2 300  Improved NLSEF Nonlinear factor α 2 0.75 Filter factor δ 2 0 Gain coefficient β 3 1000  DCD Compensation coefficient b 0 2.485Simulations are performed in the current-loop control system of the pitching axis of an aerial camera in the following three cases.First, simulations are performed to verify the response speed and steady-state performance of the improved ADRC. A current of 1 A is applied on the rotor for nonload starting. Figure4 shows the motor response curves under the PI control strategy, traditional ADRC, and improved ADRC. The control strategy based on improved ADRC exhibits shorter response time and smaller overshoot amount than the PI control strategy and the traditional ADRC control strategy. The simulation results show that the overshoot amount of the control strategy based on improved ADRC is 14.2822% and 9.4183% lower than those of the PI and traditional ADRC control strategies, respectively. The response time of the control strategy based on the improved ADRC is 1.5392 and 0.6459 s shorter than those of the PI and traditional ADRC control strategies, respectively. The current-loop parameters of the overshoot amount and the response times for the three strategies are shown in Table 4.Table 4
Parameters of the overshoot amount and the response time for nonload starting under the PI control strategy, traditional ADRC, and improved ADRC control strategy.
Control strategy Response time/s Overshoot amount PI 7.0056 16.6352% Traditional ADRC 0.8623 5.7653% Improved ADRC 0.4564 2.3470%Figure 4
Response curves of current-loop for nonload starting under the PI control strategy, traditional ADRC, and improved ADRC control strategy.Second, simulations are performed to verify the anti-interference ability of the improved ADRC. As an external disturbance, a step signal of 1 A is applied on the motor for 3 s. The current response curves are shown in Figure5. The simulation results show that the improved ADRC control strategy exhibits smaller response curve fluctuation, shorter recovery time, and smaller load disturbance influence than the PI and traditional ADRC control strategies. The simulation results show that the overshoot amount of the improved ADRC control strategy is 7.5911% and 36.4762% lower than those of the traditional ADRC and PI control strategies. The recovery time of the improved ADRC control strategy is 1.4877 and 6.9423 s shorter than those of the traditional ADRC and PI control strategies. Table 5 presents the current-loop parameters of the overshoot amount and recovery time.Table 5
Parameters of the overshoot amount and recovery time under the PI control strategy, traditional ADRC, and improved ADRC control strategy for an external disturbance applying.
Control strategy Response time/s Overshoot amount PI 6.7281 40.2103% Traditional ADRC 0.4935 7.3252% Improved ADRC 0.4758 4.7341%Figure 5
Response curves of current-loop for an external disturbance applying under the PI control strategy, traditional ADRC, and improved ADRC control strategy.Finally, simulations are performed to verify the indicator ability of the improved ADRC. A sinusoidal signal input is applied to the current-loop control system. The current output curves for the traditional ADRC and improved ADRC control strategies are shown in Figure6. According to the simulation results, the improved ADRC control strategy exhibits better indicator ability than the traditional ADRC control strategy.Figure 6
Response curves of the current-loop under the traditional ADRC and improved ADRC control strategies for a sinusoidal signal input.Simulations are also performed to verify the performance ofz22 tracking the disturbance, and the results are shown in Figure 7. Figure 7 shows that z22 can track the disturbance in real time when a unit step disturbance is applied on the current-loop at the time 3 s.Figure 7
Response curves ofz22 and the disturbance.


## 5. Conclusions

An improved ADRC of the current-loop of the pitching axis control system of an aerial camera is proposed. The new nonlinear function can effectively decrease the high-frequency flutter phenomenon, because it exhibits better continuity and smoothness than the previously available nonlinear functions. Therefore, the improved ADRC designed based on the new nonlinear function exhibits better control performances than the PI control strategy and the traditional ADRC. The nondifferentiable and discontinuous characteristics of the nonlinear function of NLSEF are overcome by the improved ADRC. Simulations are performed, and the results indicate that the system with improved ADRC control strategy exhibits better dynamic performance, static performance, and robustness than the system with PI control strategy and traditional ADRC control strategy. Extending the proposed design technique to the improved ADRC of the position-loop is an interesting topic, and this technique is the focus of our future study.



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*Source: 6162194-2017-02-06.xml*