# Fractional-Order Active Disturbance Rejection Controller Design for Large Radio Telescope Antenna

Authors: Jian-Yu Jiang; Bin Wu; Ting Zhou
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

large radio telescope antennas are required in current astronomical research and deep space communication applications. The wind disturbance becomes a serious problem with an increase in the size of the antennas. Based on the typical antenna model, this work aims to design a fractional-order active disturbance rejection controller (FOADRC) for improving the accuracy of the antenna pointing mechanism in wind disturbance. First, the fractional-order active disturbance rejection control scheme is chosen to estimate and compensate both the wind disturbance and other uncertain mechanical dynamics. Afterwards, the relationship between the parameters of fractional-order extended state observer and the performance of disturbance suppression and noise immunity is revealed. The stability of fractional-order extended state observer is proved under similar assumptions as an integer order extended state observer. In addition, a stability region boundary calculation method is presented to simplify the parameter setting procedure. Finally, we perform simulations to verify the superiority of FOADRC in the antenna control system under wind disturbance. The results demonstrate that FOADRC has a good capability of resisting wind disturbance; at the same time, it limits the response overshoot and converges faster compared to other industrial controllers.


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## Body

## 1. Introduction

Large radio telescopes are indispensable for studying various phenomena, such as planet formation, black holes, and dark energy in the universe. Green Bank Telescope is the world’s largest fully steerable telescope (100 m) that was built in America in 2001. China also plans to build a 110 m fully steerable telescope antenna called the Qitai radio telescope (QTT) [1, 2]. In the case of large antennas, the total pointing error is mainly decided by the disturbance caused by winds [3]. The methods used for reducing the effect of disturbance caused by winds have garnered a great deal of research attention in the past half-century. Some of these methods presented in the previous literature [4–7] include building an antenna in a small valley or in a radome, enhancing the stiffness of the structure, and using mesh reflector antennas. However, all of these methods degrade the performance of antennas.The control compensation is recognized as the most appropriate method for mitigating wind disturbance. In order to provide the groundwork for controller design, Gawronski developed three wind models, which are confirmed based on the experimental results [8]. The PI controllers are widely used, but are reckoned to be undesirable since they are unable to meet the stringent pointing requirements. NASA [9] investigated the relationships between the antenna performance parameters as a function of controller gains and applied the relationship to the NASA Deep Space Network antennas with dish sizes of 34 or 70 meters. However, for a 65 m antenna with a PI controller, the required pointing accuracy cannot be achieved if the speed of wind is greater than 10 m/s. Therefore, implementing the LQG algorithm yields a substantial improvement in tracking precision, as presented in Gawronski’s study regarding the impact of the weights of a closed-loop bandwidth antenna, disturbance rejection properties, and antenna acceleration. Gawronski also presented the tuning procedure of the controller based on the coordinate selection and the weight properties [10]. In addition, the author also designed and analyzed an LQG controller for NASA’s Deep Space Network antennas and executed the new control algorithms on an DSS-13 antenna. The experimental results showed an improvement in the performance of the LQG over the existing PI controller [11]. Yu-Xuan et al. proposed an LQG controller with Kalman optimal estimation to address the pointing inaccuracy of the deep space antenna due to gust disturbances [12]. For a traditional 34-m Cassegrain antenna servo system, Hu et al. designed an LQG controller with wind-gust disturbance rejection capability by virtue of state estimation and optimal feedback control. The simulation results show that the LQG controller has satisfactory results [13]. Gawronski discussed PI, LQG, and H∞ algorithms and their basic properties, tracking precision, and limitations of the NASA Deep Space Network antenna-control systems. The obstacle to applying LQG or H∞ algorithms to antenna control system is the acceleration limits on existing antenna drives, i.e., the motors and gearboxes [14].Similar to the PID algorithm, the advanced control algorithms without an accurate model are another way to improve the performance of the antenna control system. These algorithms only require modifications in certain software for making them more practical and computationally efficient. Since its inception, the active disturbance rejection control (ADRC) [15] has been used in various devices, such as motion control, DC-DC power converter, chemical processes, micro-electro-mechanical systems gyroscope, and attitude tracking of rigid spacecraft, and has achieved superior performance [16]. The ADRC offers a new and inherently robust controller building block by using an ESO to estimate both the internal plant dynamics and external disturbances. Gao proposed LADRC to avoid the repetitive controller tuning process for plants that differ mainly in terms of gain and bandwidth [17]. Qiu et al. designed LADRC for a large deep space observatory antenna by considering the wind load torque and other uncertain mechanical dynamics as total disturbances and then used ESO to estimate and compensate the total estimated disturbances. The effectiveness of the proposed approach was illustrated based on mathematical formulations and experiments [18]. Yuan et al. analyzed stability, rejection quality for external disturbance, and robustness for uncertainty, as well as noise sensitivity characteristics based on a closed-loop transfer function and frequency response [19]. Wu et al. improved the tuning efficiency with a new method to calculate the stability of regions based on D-partition, as evidenced by the digital simulations and an experiment performed using a water tank [20].The fractional-order control is another easy-to-deploy technique for improving the systems involving PID algorithms. It has been used in various applications since Podlubny proposed fractional PID controllers in 1994 [21]. It not only benefits automatic voltage regulation, permanent magnet synchronous motor speed servo systems, small fixed-wing unmanned aerial vehicle, and dc-motor with elastic shaft but also helps the designers to achieve robust performance in flexible spacecraft attitude control [22]. The stability of a certain class of fractional-order linear and nonlinear systems is investigated in [23]. Hamamci presented a method to stabilize a given fractional-order system with time delay using the D-decomposition method. This algorithm is simple and has reliable result illustrated by an example [24]. Li proposed generalized Mittag–Leffler stability and studied the stability of fractional-order nonlinear dynamic systems using the Lyapunov direct method [25]. Chen et al. presented a fractional-order active disturbance rejection control with a fractional-order extended state observer for the permanent magnet synchronous motor speed servo system. The simulations and experiments show that the proposed FOADRC performs better as compared to the integer order ADRC and FOPI controller [26]. Li et al. proposed a fractional active disturbance rejection control scheme by generalizing the traditional extended states’ observer to a fractional-order extended states’ observer. Simulation results by the fractional active disturbance rejection control and ADRC on the typical fractional-order system are compared to demonstrate the superiority and effectiveness of the proposed scheme [27]. Sheikh et al. propose a novel fractional active disturbance rejection controller for the combined operation of load frequency control and automatic voltage regulator of a hybrid power system. The robustness of the controller gains is tested under different system parameter changes from their nominal values. Results show that the developed controller operates encouragingly with reference to system stability, rapidity, and accuracy in comparison to testified control strategies available in [28].In this work, the fractional-order control integrated with active disturbance rejection control is designed for achieving high pointing accuracy in case of large radio telescope antennas under wind disturbances. The main contributions of this work are summarized:(1)
The application of fractional-order active disturbance rejection control for large radio telescope antennas under wind disturbances. The core is to consider the wind disturbance and the difference between the antenna model and the standard model as the total disturbance, so as to estimate and compensate the total disturbance using FOADRC.(2)
The stability of FOESO is proved by using the Lyapunov method. The rejection quality of external disturbance and noise sensitivity characteristics of FOADRC is studied.(3)
The stability region of FOADRC is determined by D-composition. The FOADRC controller for 25 m antenna is designed, and its effectiveness is verified based on simulations.The rest of the study is organized as follows.The models of the antenna control system and wind disturbance are presented in Section2. FOADRC for antenna is presented in Section 3. Section 4 presents the rigorous proof of stability of FOESO. In addition, the effect of parameters on FOESO performance is also analyzed in this section. Section 5 analyzes stability of FOADRC along with a detailed computation approach for the stability region as well as results. Section 6 conducts an effectiveness test of the designed FOADRC controller based on numerical simulations. Section 7 concludes this work.


## 2. Modeling of Antennas and Wind Disturbance

The modeling of antennas and wind disturbance are explained in the following sections.


### 2.1. Large Antenna Precision Servo Control System

The antenna control system includes an antenna structure, motor, reducer, amplifiers, current sensor, tachometer, encoder, and controllers. As shown in Figure1, the mechanical equations of the antenna control system consist of three parts, namely, the loop voltage equation of motor, torque balance of the motor and reducer, and torque balance of reducer and antenna structure.Figure 1
Main components of the antenna servo control system and their relationship.The mechanical equations of the antenna control system are expressed as follows:(1)Ua=e+RaIa+LadIadt,e=Keθ˙m,2KmIa=1NML+2Jmθ¨m,Jaθ¨a=2Kg1Nθm−θa,where Ua,Ra,La,Ia denote the voltage, resistance, inductance, and current of the armature, respectively, e denotes the back electromotive force, Ke denotes the back electromotive force constant, Jm,Ja represent the motor inertia and antenna inertia, respectively, θmandθa represent the angular positions of the motor and the antenna shaft, Km denotes the motor torque constant, Kg denotes the equivalent stiffness of the output shaft of a single-chain gearbox and the antenna shaft, N denotes the gearbox turn ratio, and ML denotes the mechanical load torque on the dish. The traditional three-loop control strategy is presented in Figure 2. Please note that the controllers for speed and current loops are PI controllers. It is difficult to change these controllers because they are integrated with analog circuits. The position loop controller can be easily modified as its operations are software based. The plant of the position controller is represented by the blue region in Figure 2.Figure 2
The three-loop control structure of the antennas.The position loop transfer function of the 35-m antenna presented in [29] is expressed as follows:(2)Gs=θaUg=19.712s+111.736s5+6.54s4+151s3+580s2+1370s,where θa denotes the angular position of the antenna and Ug denotes the position control voltage.


### 2.2. Wind Disturbance Models

It is well known that the wind is the main source of antenna disturbance. It disturbs the antenna structure and causes pointing errors. The velocity of the wind gust is regarded as the input. The gust component is a random process with zero mean and a spectrum called Davenport spectrum. The Davenport spectrum is mathematically expressed as follows:(3)Svω=4800vmkβω1+β2ω24/3,β=600πvm,k=12.5lnz/z02,where vm denotes the mean velocity, z denotes the distance between the antenna reflector center and the ground, and z0 denotes the height of the terrain roughness. In the simulations, a fourth-order filter named the Davenport filter is used to approximate the Davenport spectrum as(4)Hs=3.9021s3+230.1426s2−686.3151s+3.41970.331s4+38.2997s3+224.7118s2+22.7788s+0.3538.The approximate effect of the Davenport filter is presented in Figure3. It has very small errors in the target frequency band. The wind-gust torque is mathematically expressed as follows:(5)Mω=ktΔv0,Δv0=Δvαvm,kt=ctαpαπD3vm22N,α=6k,where ct denotes the torque coefficient, ap denotes the static air density, and D denotes the diameter of antenna. For a wind with a velocity of 8.94 m/s, kt=23553.4849m2/s2.Figure 3
The square root of the Davenport spectrum and magnitude-frequency curve of the filter.The torque is filtered by using a filterF that outputs a velocity, which is added to the velocity input of the antenna:(6)Fs=skd,where kd denotes the drive gain. For this antenna, the drive gain is kd=1.1×108Nms/deg. The final wind velocity is presented in Figure 4.Figure 4
The wind velocity generated using the Davenport filter.


## 2.1. Large Antenna Precision Servo Control System

The antenna control system includes an antenna structure, motor, reducer, amplifiers, current sensor, tachometer, encoder, and controllers. As shown in Figure1, the mechanical equations of the antenna control system consist of three parts, namely, the loop voltage equation of motor, torque balance of the motor and reducer, and torque balance of reducer and antenna structure.Figure 1
Main components of the antenna servo control system and their relationship.The mechanical equations of the antenna control system are expressed as follows:(1)Ua=e+RaIa+LadIadt,e=Keθ˙m,2KmIa=1NML+2Jmθ¨m,Jaθ¨a=2Kg1Nθm−θa,where Ua,Ra,La,Ia denote the voltage, resistance, inductance, and current of the armature, respectively, e denotes the back electromotive force, Ke denotes the back electromotive force constant, Jm,Ja represent the motor inertia and antenna inertia, respectively, θmandθa represent the angular positions of the motor and the antenna shaft, Km denotes the motor torque constant, Kg denotes the equivalent stiffness of the output shaft of a single-chain gearbox and the antenna shaft, N denotes the gearbox turn ratio, and ML denotes the mechanical load torque on the dish. The traditional three-loop control strategy is presented in Figure 2. Please note that the controllers for speed and current loops are PI controllers. It is difficult to change these controllers because they are integrated with analog circuits. The position loop controller can be easily modified as its operations are software based. The plant of the position controller is represented by the blue region in Figure 2.Figure 2
The three-loop control structure of the antennas.The position loop transfer function of the 35-m antenna presented in [29] is expressed as follows:(2)Gs=θaUg=19.712s+111.736s5+6.54s4+151s3+580s2+1370s,where θa denotes the angular position of the antenna and Ug denotes the position control voltage.


## 2.2. Wind Disturbance Models

It is well known that the wind is the main source of antenna disturbance. It disturbs the antenna structure and causes pointing errors. The velocity of the wind gust is regarded as the input. The gust component is a random process with zero mean and a spectrum called Davenport spectrum. The Davenport spectrum is mathematically expressed as follows:(3)Svω=4800vmkβω1+β2ω24/3,β=600πvm,k=12.5lnz/z02,where vm denotes the mean velocity, z denotes the distance between the antenna reflector center and the ground, and z0 denotes the height of the terrain roughness. In the simulations, a fourth-order filter named the Davenport filter is used to approximate the Davenport spectrum as(4)Hs=3.9021s3+230.1426s2−686.3151s+3.41970.331s4+38.2997s3+224.7118s2+22.7788s+0.3538.The approximate effect of the Davenport filter is presented in Figure3. It has very small errors in the target frequency band. The wind-gust torque is mathematically expressed as follows:(5)Mω=ktΔv0,Δv0=Δvαvm,kt=ctαpαπD3vm22N,α=6k,where ct denotes the torque coefficient, ap denotes the static air density, and D denotes the diameter of antenna. For a wind with a velocity of 8.94 m/s, kt=23553.4849m2/s2.Figure 3
The square root of the Davenport spectrum and magnitude-frequency curve of the filter.The torque is filtered by using a filterF that outputs a velocity, which is added to the velocity input of the antenna:(6)Fs=skd,where kd denotes the drive gain. For this antenna, the drive gain is kd=1.1×108Nms/deg. The final wind velocity is presented in Figure 4.Figure 4
The wind velocity generated using the Davenport filter.


## 3. Fractional-Order Active Disturbance Rejection Position Control Design for Antenna

The fractional-order active disturbance rejection position control design for antenna is explained in the following sections.


### 3.1. Preliminaries of Fractional Calculus and Fractional Control

The fractional-order calculus plays an important role in many complex scientific problems. In this section, we introduce the definitions and properties of fractional-order control used in this work.The uniform formula of a fractional integral withα∈0,1 is defined as follows [30]:(7)Dt−aaft=1Γα∫atfτt−τ1−α,where ft denotes an arbitrary integrable function, α denotes the integral order, a and t denote the lower and upper limits of the fractional-order integral, respectively, and Γ⋅ denotes the Gamma function. For an arbitrary real number p, the Riemann–Liouville and Caputo fractional derivatives are defined as follows [30]:(8)aDtpft=dp+1dtp+1aDt−p−p+1ft,(9)DtpaCft=aDt−p−p+1dp+1dtp+1ft,where p denotes the integer part of p and D and cD represent the Riemann–Liouville and Caputo fractional derivatives, respectively.The Laplace transform is a very useful tool in the analysis and design of control systems. The Laplace transform is performed by using the Riemann–Liouville’s mathematical model as follows [30]:(10)LDt−αft=s−αFs,LDtβft=sβFs−∑k=0n−1skDtβ−k−1ft|t=0.Eventually, the fractional-order system is established. For a single variable system, the fractional-order transfer function is expressed as follows [31]:(11)Gs=NsDs=bnsβn+bn−1sβn−1+.+b1s1β+b0sβ0ansαn+an−1sαn−1+.+a1sα1+a0sα0=∑i=0nbisβi∑i=0naisαi,where bi,ai,αi,βi denote the real numbers.Generally, it is not possible to obtain the numerical solution of fraction differential equations based on standard definitions. One intuitive idea to address this problem is to approximate numerical solution by using filters. We assume that the expected fitting range is (ωb, ωh). Then, the filter is mathematically expressed as follows [32]:(12)Gfs=K∏k=1Ns+ωk′s+ωk,ωk′=ωbωhωbk+N+1−γ/22N+1,ωk=ωbωhωbk+N+1+γ/2/2N+1,K=ωhγ.As a most natural generalization of the exponential function, the Mittag–Leffler function is particularly important in fractional-order system analysis. The mathematical definition of the Mittag–Leffler function with three parameters is [31](13)Eα,βγz=∑k=0∞γkΓαk+βzkk!,where α,β,γ∈C, Reα>0,Reβ>0,Reγ>0, and γk is the Pochhammer symbol.Property 1.
(see [30])(14)DtpaDtqaft=Dtp+qaft−∑j=1nDtq−jaftt=at−a−p−jΓ1−p−j.Lemma 1 (see [33]).
We consider the fractional nonautonomous systemDtαt0Cxt=ft,x, where ft,x represents the real-valued continuous function, we have(15)t0Dt−αft,xt≤t0Dt−αft,xt,where α≥0 and ⋅ denotes an arbitrary norm.Lemma 2 (see [25]).
Letx=0 be an equilibrium point of the asymptotic stability of the fractional nonautonomous system 0Dtαxt=ft,x and D⊂Rn be a domain containing the origin. Let Vt,xt:0,∞×D⟶R be a continuously differentiable function and locally Lipschitz with respect to x, such that(16)α1xa≤Vt,xt≤α2xab,DtβVt,xt≤−α3xab,where t≥0,x∈D,β∈0,1,α1,α2,α3,a, and b denotes the arbitrary positive constants. Then, x=0 is the Mittag–Leffler stable. If the assumptions hold globally on Rn, then x=0 is globally the Mittag–Leffler stable. In addition, if there exists a constant t1, the equilibrium point x=0 is generalized Mittag–Leffler stability for t≥t1.


### 3.2. Design of the Fractional-Order Active Disturbance Rejection Control

In this section, the fractional-order active disturbance rejection position control scheme of position control of antenna is presented. ADRC insightfully proposed the integrator chain of the controlled variable as the most fundamental structure for control systems, linear or nonlinear. Moreover, it generalized the concept of disturbance to the “total disturbance.” Figure5 presents the ADRC consists of extended state observer and controller. The function of the extended state observer is to expand the total disturbance into a new state variable of the system and then use the input and output of the system to observe all the states including the original state variables and disturbances in the system. The controller compensates the total disturbance to simplify the plant to a cascade integral form, which can be easily controlled by the tracking error and its derivative. In fact, we can use different linear or nonlinear gain combinations in the ESO and the feedback in the same ADRC structure.Figure 5
The state-space diagram of ADRC.First, the antenna control system is processed using the ADRC framework. The antenna servo control system is a typical form of integrator cascade. Therefore, a fractional-order active disturbance rejection controller for a large antenna is designed based on second-order integrator cascade form. The difference between the transfer function of the antenna and the second-order integrator cascade form can be regarded as the model uncertaintyw1, the wind gust, and friction, and the damping inertial effect from the dish can be considered as the external disturbance w2. The model uncertainty and unmodeled internal and external disturbances constitute the total disturbance. The total disturbance can be estimated and compensated based on a fractional-order extended state observer, which attenuates the adverse effects caused by the wind.The antenna control system is mathematically expressed as(17)y¨=bu+w1+w2.Letb0 represent the nominal value of b. Due to the inadequacy of differential in stability and response to high frequency noise, a fractional-order differential is introduced to design FOESO. Now, (17) can be rewritten as follows:(18)y2α=−y¨+y2α+w1+w2+b−b0u+b0u.We define the total disturbance asf=−y¨+w1+w2+b−b0u. Then, (18) is further simplified as follows:(19)y2α=f+b0u.We define(20)y=θa,x1=y,x2=yα,x3=f,where x1andx2 denote state variables and x3 denotes the extended state variable. Let h=fα; then, the following state-space representation is obtained:(21)x1α=x2,x2α=x3+b0u,x3α=h,y=x1.Now, FOESO is designed by combining the fractional differential and linear ESO. FOESO improves the observation ability and alleviates the contradiction between anti-interference ability and antinoise ability of LESO. The fractional-order extended state observer is constructed as follows:(22)z1α=β1y−z1+z2,z2α=β2y−z1+z3+b0u,z3α=β3y−z1,where β1,β2,β3 denote the observer gains of FOESO and α denotes the observer order of FOESO, α∈0,1.If the observer gainβi is configured by using the bandwidth-parameterization method, we have(23)β1,β2,β3=3ωo,3ωo2,ωo3,ωo>0,where ωo denotes the bandwidth of FOESO.Finally, with the state observer properly designed, the control law is designed as(24)u=kpr−z1−kdz2−z3b0.Using the bandwidth-parameterization method, we obtain(25)kd,kp=2ωc,ωc2,ωc>0,where ωc denotes the bandwidth of the controller.In summary, the FOADRC proposed in this study is mainly based on the structure of ADRC by improving the linear ESO to FOESO. The FOADRC controller for large antenna system is shown in Figure6.Figure 6
The proposed structure of FOADRC for the antenna.


## 3.1. Preliminaries of Fractional Calculus and Fractional Control

The fractional-order calculus plays an important role in many complex scientific problems. In this section, we introduce the definitions and properties of fractional-order control used in this work.The uniform formula of a fractional integral withα∈0,1 is defined as follows [30]:(7)Dt−aaft=1Γα∫atfτt−τ1−α,where ft denotes an arbitrary integrable function, α denotes the integral order, a and t denote the lower and upper limits of the fractional-order integral, respectively, and Γ⋅ denotes the Gamma function. For an arbitrary real number p, the Riemann–Liouville and Caputo fractional derivatives are defined as follows [30]:(8)aDtpft=dp+1dtp+1aDt−p−p+1ft,(9)DtpaCft=aDt−p−p+1dp+1dtp+1ft,where p denotes the integer part of p and D and cD represent the Riemann–Liouville and Caputo fractional derivatives, respectively.The Laplace transform is a very useful tool in the analysis and design of control systems. The Laplace transform is performed by using the Riemann–Liouville’s mathematical model as follows [30]:(10)LDt−αft=s−αFs,LDtβft=sβFs−∑k=0n−1skDtβ−k−1ft|t=0.Eventually, the fractional-order system is established. For a single variable system, the fractional-order transfer function is expressed as follows [31]:(11)Gs=NsDs=bnsβn+bn−1sβn−1+.+b1s1β+b0sβ0ansαn+an−1sαn−1+.+a1sα1+a0sα0=∑i=0nbisβi∑i=0naisαi,where bi,ai,αi,βi denote the real numbers.Generally, it is not possible to obtain the numerical solution of fraction differential equations based on standard definitions. One intuitive idea to address this problem is to approximate numerical solution by using filters. We assume that the expected fitting range is (ωb, ωh). Then, the filter is mathematically expressed as follows [32]:(12)Gfs=K∏k=1Ns+ωk′s+ωk,ωk′=ωbωhωbk+N+1−γ/22N+1,ωk=ωbωhωbk+N+1+γ/2/2N+1,K=ωhγ.As a most natural generalization of the exponential function, the Mittag–Leffler function is particularly important in fractional-order system analysis. The mathematical definition of the Mittag–Leffler function with three parameters is [31](13)Eα,βγz=∑k=0∞γkΓαk+βzkk!,where α,β,γ∈C, Reα>0,Reβ>0,Reγ>0, and γk is the Pochhammer symbol.Property 1.
(see [30])(14)DtpaDtqaft=Dtp+qaft−∑j=1nDtq−jaftt=at−a−p−jΓ1−p−j.Lemma 1 (see [33]).
We consider the fractional nonautonomous systemDtαt0Cxt=ft,x, where ft,x represents the real-valued continuous function, we have(15)t0Dt−αft,xt≤t0Dt−αft,xt,where α≥0 and ⋅ denotes an arbitrary norm.Lemma 2 (see [25]).
Letx=0 be an equilibrium point of the asymptotic stability of the fractional nonautonomous system 0Dtαxt=ft,x and D⊂Rn be a domain containing the origin. Let Vt,xt:0,∞×D⟶R be a continuously differentiable function and locally Lipschitz with respect to x, such that(16)α1xa≤Vt,xt≤α2xab,DtβVt,xt≤−α3xab,where t≥0,x∈D,β∈0,1,α1,α2,α3,a, and b denotes the arbitrary positive constants. Then, x=0 is the Mittag–Leffler stable. If the assumptions hold globally on Rn, then x=0 is globally the Mittag–Leffler stable. In addition, if there exists a constant t1, the equilibrium point x=0 is generalized Mittag–Leffler stability for t≥t1.


## 3.2. Design of the Fractional-Order Active Disturbance Rejection Control

In this section, the fractional-order active disturbance rejection position control scheme of position control of antenna is presented. ADRC insightfully proposed the integrator chain of the controlled variable as the most fundamental structure for control systems, linear or nonlinear. Moreover, it generalized the concept of disturbance to the “total disturbance.” Figure5 presents the ADRC consists of extended state observer and controller. The function of the extended state observer is to expand the total disturbance into a new state variable of the system and then use the input and output of the system to observe all the states including the original state variables and disturbances in the system. The controller compensates the total disturbance to simplify the plant to a cascade integral form, which can be easily controlled by the tracking error and its derivative. In fact, we can use different linear or nonlinear gain combinations in the ESO and the feedback in the same ADRC structure.Figure 5
The state-space diagram of ADRC.First, the antenna control system is processed using the ADRC framework. The antenna servo control system is a typical form of integrator cascade. Therefore, a fractional-order active disturbance rejection controller for a large antenna is designed based on second-order integrator cascade form. The difference between the transfer function of the antenna and the second-order integrator cascade form can be regarded as the model uncertaintyw1, the wind gust, and friction, and the damping inertial effect from the dish can be considered as the external disturbance w2. The model uncertainty and unmodeled internal and external disturbances constitute the total disturbance. The total disturbance can be estimated and compensated based on a fractional-order extended state observer, which attenuates the adverse effects caused by the wind.The antenna control system is mathematically expressed as(17)y¨=bu+w1+w2.Letb0 represent the nominal value of b. Due to the inadequacy of differential in stability and response to high frequency noise, a fractional-order differential is introduced to design FOESO. Now, (17) can be rewritten as follows:(18)y2α=−y¨+y2α+w1+w2+b−b0u+b0u.We define the total disturbance asf=−y¨+w1+w2+b−b0u. Then, (18) is further simplified as follows:(19)y2α=f+b0u.We define(20)y=θa,x1=y,x2=yα,x3=f,where x1andx2 denote state variables and x3 denotes the extended state variable. Let h=fα; then, the following state-space representation is obtained:(21)x1α=x2,x2α=x3+b0u,x3α=h,y=x1.Now, FOESO is designed by combining the fractional differential and linear ESO. FOESO improves the observation ability and alleviates the contradiction between anti-interference ability and antinoise ability of LESO. The fractional-order extended state observer is constructed as follows:(22)z1α=β1y−z1+z2,z2α=β2y−z1+z3+b0u,z3α=β3y−z1,where β1,β2,β3 denote the observer gains of FOESO and α denotes the observer order of FOESO, α∈0,1.If the observer gainβi is configured by using the bandwidth-parameterization method, we have(23)β1,β2,β3=3ωo,3ωo2,ωo3,ωo>0,where ωo denotes the bandwidth of FOESO.Finally, with the state observer properly designed, the control law is designed as(24)u=kpr−z1−kdz2−z3b0.Using the bandwidth-parameterization method, we obtain(25)kd,kp=2ωc,ωc2,ωc>0,where ωc denotes the bandwidth of the controller.In summary, the FOADRC proposed in this study is mainly based on the structure of ADRC by improving the linear ESO to FOESO. The FOADRC controller for large antenna system is shown in Figure6.Figure 6
The proposed structure of FOADRC for the antenna.


## 4. Convergence and Frequency-Band Characteristic Analysis of FOESO

The convergence and frequency-band characteristic analysis of FOESO is described in the following sections.


### 4.1. Convergence of FOESO

In the bandwidth of FOESO, we note that the gain factor isκi=3!/i!×3−i!,i=1,2,3, and βi=κi/εi. We obtain the dynamic observation error as follows:(26)e1t=x1t−z1t,z1αt=z2t+κ1εe1t,z2αt=z3t+κ2ε2e1t+b0ut,z3αt=κ3ε3e1t.Withgie1t=αie1ti=1,2,3, the forementioned system attains a high gain form and is expressed as follows:(27)e1t=x1t−z1t,z1αt=z2t+1εg1e1t,z2αt=z3t+1ε2g2e1t+b0ut,z3αt=1ε3g3e1t.Assumption 1.
There exists a positive constant, such that theα-order derivation of total disturbance f is bounded and satisfies the following condition:(28)dαft,x1t,x2t,wtdtα≤M.Assumption 2.
There exists a positive constantλii=1,2,3,4 and a continuous positive definite function V: R3⟶R for unknown function gi⋅i=1,2,3, and the following conditions are true as follows:(29)λ1x2≤Vx≤λ2x2,∂V∂x1x2−g1x1+∂V∂x2x3−g2x1−∂V∂x3g3x1≤−λ3x,∂V∂x3≤−λ4x..Assumption 3.
System (27) observational error function for (20) satisfies the Lipschitz condition, and the Lipschitz constant l is positive.Theorem 1.
If Assumptions1 and 2 are satisfied, FOESO achieves asymptotic stability.Proof.
We define system (27) observational error function for (20) as follows:(30)et=x1t−z1t,x2t−z2t,x3t−z3t=e1t,e2t,e3t.
We note(31)z∧it=eiεtε3−i,i=1,2,3.
Then, the observation error dynamics of system from (27) to (20) are expressed as follows:(32)z∧1αt=z∧2t−g1z∧1t,z∧2αt=z∧3t−g2z∧1t,z∧3αt=−g3z∧1t+εfαt.
Forsimplicity,wenoteDtα0Cz∧t=f∗t,z∧derderivativeoftheLyapunovfunctionVz∧ttothesystem32withrespecttotimetisexpressedas(33)Dt1−α0CVz∧t=Dt−α0V·z∧t,=Dt−α0∂V∂x1x2−g1x1+∂V∂x2x3−g2x1,−∂V∂x3g3x1+∂V∂x3εdαftdtα,≤Dt−α0−λ3x−λ4Mx,=−λ3+λ4MDt−α0x,=−λ3+λ4Ml−1Dt−α0f∗t,z∧,≤−λ3+λ4Ml−1Dt−α0f∗t,z∧,=−λ3+λ4Ml−1x.
Therefore, the conclusion can be obtained by using Lemma2, i.e., the proposed FOESO is the Mittag–Leffler stable and asymptotic stable.
In summary, based on the assumptions similar to the ESO presented in [34] and generalized Mittag–Leffler stability of fractional-order nonlinear dynamic systems, the dynamic system describing the observation error of the proposed FOESO is asymptotic stable.


### 4.2. Frequency-Band Characteristics Analysis of FOESO

As the core of FOADRC, the tracking and estimation capacity of FOESO determines the performance of FOADRC. In this section, we present the rejection quality for external disturbance, the difference between FOESO and ESO, and the relations between the dynamic characteristics and the observer parameters.Based on the Laplace transformation presented in (22) and (23), we have(34)z1=3ωos2α+3ωo2sα+ωo3sα+ωo3y+b0sαsα+ωo3u,z2=3ωo2+ωo3sαsα+ωo3y+b0sα+3ωosαsα+ωo3u,z3=ωo3s2αsα+ωo3y−b0ωo3sα+ωo3u.The tracking errors are mathematically expressed as follows:(35)e1=z1−y=−s3αωo+sα3y+b0sαωo+sα3u,e2=z2−yα=−s3α3ωo+sαωo+sα3y+b0sα3ωo+sαωo+sα3u,e3=z3−y2α−b0u=−s3α3ωosα+s2α+3ωo2ωo+sα3y+b0sα3ωosα+s2α+3ωo2ωo+sα3u.Considering the typicality of the analysis, the steady-state errors are obtained by considering the step signal with amplitudeK on both y and u as follows:(36)e1s=lims⟶0se1=lims⟶0−Ks3α+Kb0sαs3α+3ωos2α+3ωosα+ωo3=0e2s=lims⟶0se2=lims⟶0−Ks4α−3Kωos3α+Kb0s2α+3Kb0ωosαs3α+3ωos2α+3ωosα+ωo3=0e3s=lims⟶0se3=lims⟶0−Ks5α−3Kωos4α+Kb0−3Kωo2s3αs3α+3ωos2α+3ωosα+ωo3+3Kb0ωos2α+3Kb0ωosαs3α+3ωos2α+3ωosα+ωo3=0.The expression presented in (36) shows that the FOESO achieves unbiased estimation of system state variables and total disturbance.The analysis of the FOESO’s dynamic tracking process is presented. Whenb0=0, the response of z1 to step signal ys=K/s is expressed as follows:(37)z1=3ωos2α+3ωo2sα+ωo3sα+ωo3Ks=3Kωos2α−1sα+ωo3+3Kωo2sα−1sα+ωo3+Kωo31ssα+ωo3.By inverting the Laplace transformation, we obtain(38)z1t=3KωotαEα,α+13−ωotα+3Kωo2t2αEα,2α+13−ωotα+Kωo31−Eα−ωotαt2α−1Eα,2α2−ωotα.In this case, integer-order ESO’s response in time domain is as follows:(39)z1t=K−K12ωo2t2−2ωot+1e−ωot.The extreme point of (39) is expressed as follows:(40)z1t1=K3−1e3−3+K≈1.206K.(40) shows that no matter how the bandwidth parameter is chosen, there is always about 20% overshoot during the tracing process in integer-order ESO as presented in Figure 7.Figure 7
The response ofz1 corresponding to the step signal (ESO).Considering FOESO, theoretically, it is difficult to solve (38) for extreme values; however, the numerical results show that the order of FOESO is related to the overshoot during the tracing process, which can be reduced by decreasing the order of differentiation as presented in Figure 8.Figure 8
The response ofz1 corresponding to step signal (FOESO).The relationship between FOESO’s suppression of external disturbances, noise transfer characteristics, and parameters is discussed. First, the effect of noiseδo from y is analyzed. From (25), we obtain(41)z1δo=3ωos2α+3ωo2sα+ωo3sα+ωo3.Whenα=0.8 (fixed) and ωo=5,10,15,20,25 are considered for obtaining the Bode diagram as shown in Figure 9, the response speed of the system increases with an increase in ωo. However, the noise amplification becomes more obvious due to the increase in the high frequency band gain.Figure 9
The frequency-domain characteristic of FOESO observation noise (observer gainωo).Whenωo=10 (fixed) and α=0.6,0.7,0.8,0.9,1 are considered for obtaining the Bode diagram as shown in Figure 10, the response speed of the system increases, the gain of the high frequency band decreases, and the noise amplification effect decreases as α decreases.Figure 10
The frequency-domain characteristic of FOESO observation noise (observer orderα).Second, the effect of input disturbanceδc from u is analyzed. From (25), we obtain(42)z1δc=b0sαsα+ωo3.Whenα=0.8,b0=10 (fixed) and ωo=10,20,...,50 are considered for obtaining the Bode diagram as shown in Figure 11, the phase lag of the tracking signal decreases, while the gain of the high frequency band is not affected at all with an increase in ωo. Please note that the proposed FOESO has good suppressive ability to input disturbance.Figure 11
The frequency-domain characteristic of FOESO input disturbance (observer gainωo).Whenωo=10,b0=10 (fixed), and α=0.6,0.7,0.8,0.9,1 are considered for obtaining the Bode diagram as shown in Figure 12, the phase lag of the tracking signal decreases and the high frequency band gain decreases with a decrease in α.Figure 12
The frequency-domain characteristic of FOESO input disturbance (observer orderα).


## 4.1. Convergence of FOESO

In the bandwidth of FOESO, we note that the gain factor isκi=3!/i!×3−i!,i=1,2,3, and βi=κi/εi. We obtain the dynamic observation error as follows:(26)e1t=x1t−z1t,z1αt=z2t+κ1εe1t,z2αt=z3t+κ2ε2e1t+b0ut,z3αt=κ3ε3e1t.Withgie1t=αie1ti=1,2,3, the forementioned system attains a high gain form and is expressed as follows:(27)e1t=x1t−z1t,z1αt=z2t+1εg1e1t,z2αt=z3t+1ε2g2e1t+b0ut,z3αt=1ε3g3e1t.Assumption 1.
There exists a positive constant, such that theα-order derivation of total disturbance f is bounded and satisfies the following condition:(28)dαft,x1t,x2t,wtdtα≤M.Assumption 2.
There exists a positive constantλii=1,2,3,4 and a continuous positive definite function V: R3⟶R for unknown function gi⋅i=1,2,3, and the following conditions are true as follows:(29)λ1x2≤Vx≤λ2x2,∂V∂x1x2−g1x1+∂V∂x2x3−g2x1−∂V∂x3g3x1≤−λ3x,∂V∂x3≤−λ4x..Assumption 3.
System (27) observational error function for (20) satisfies the Lipschitz condition, and the Lipschitz constant l is positive.Theorem 1.
If Assumptions1 and 2 are satisfied, FOESO achieves asymptotic stability.Proof.
We define system (27) observational error function for (20) as follows:(30)et=x1t−z1t,x2t−z2t,x3t−z3t=e1t,e2t,e3t.
We note(31)z∧it=eiεtε3−i,i=1,2,3.
Then, the observation error dynamics of system from (27) to (20) are expressed as follows:(32)z∧1αt=z∧2t−g1z∧1t,z∧2αt=z∧3t−g2z∧1t,z∧3αt=−g3z∧1t+εfαt.
Forsimplicity,wenoteDtα0Cz∧t=f∗t,z∧derderivativeoftheLyapunovfunctionVz∧ttothesystem32withrespecttotimetisexpressedas(33)Dt1−α0CVz∧t=Dt−α0V·z∧t,=Dt−α0∂V∂x1x2−g1x1+∂V∂x2x3−g2x1,−∂V∂x3g3x1+∂V∂x3εdαftdtα,≤Dt−α0−λ3x−λ4Mx,=−λ3+λ4MDt−α0x,=−λ3+λ4Ml−1Dt−α0f∗t,z∧,≤−λ3+λ4Ml−1Dt−α0f∗t,z∧,=−λ3+λ4Ml−1x.
Therefore, the conclusion can be obtained by using Lemma2, i.e., the proposed FOESO is the Mittag–Leffler stable and asymptotic stable.
In summary, based on the assumptions similar to the ESO presented in [34] and generalized Mittag–Leffler stability of fractional-order nonlinear dynamic systems, the dynamic system describing the observation error of the proposed FOESO is asymptotic stable.


## 4.2. Frequency-Band Characteristics Analysis of FOESO

As the core of FOADRC, the tracking and estimation capacity of FOESO determines the performance of FOADRC. In this section, we present the rejection quality for external disturbance, the difference between FOESO and ESO, and the relations between the dynamic characteristics and the observer parameters.Based on the Laplace transformation presented in (22) and (23), we have(34)z1=3ωos2α+3ωo2sα+ωo3sα+ωo3y+b0sαsα+ωo3u,z2=3ωo2+ωo3sαsα+ωo3y+b0sα+3ωosαsα+ωo3u,z3=ωo3s2αsα+ωo3y−b0ωo3sα+ωo3u.The tracking errors are mathematically expressed as follows:(35)e1=z1−y=−s3αωo+sα3y+b0sαωo+sα3u,e2=z2−yα=−s3α3ωo+sαωo+sα3y+b0sα3ωo+sαωo+sα3u,e3=z3−y2α−b0u=−s3α3ωosα+s2α+3ωo2ωo+sα3y+b0sα3ωosα+s2α+3ωo2ωo+sα3u.Considering the typicality of the analysis, the steady-state errors are obtained by considering the step signal with amplitudeK on both y and u as follows:(36)e1s=lims⟶0se1=lims⟶0−Ks3α+Kb0sαs3α+3ωos2α+3ωosα+ωo3=0e2s=lims⟶0se2=lims⟶0−Ks4α−3Kωos3α+Kb0s2α+3Kb0ωosαs3α+3ωos2α+3ωosα+ωo3=0e3s=lims⟶0se3=lims⟶0−Ks5α−3Kωos4α+Kb0−3Kωo2s3αs3α+3ωos2α+3ωosα+ωo3+3Kb0ωos2α+3Kb0ωosαs3α+3ωos2α+3ωosα+ωo3=0.The expression presented in (36) shows that the FOESO achieves unbiased estimation of system state variables and total disturbance.The analysis of the FOESO’s dynamic tracking process is presented. Whenb0=0, the response of z1 to step signal ys=K/s is expressed as follows:(37)z1=3ωos2α+3ωo2sα+ωo3sα+ωo3Ks=3Kωos2α−1sα+ωo3+3Kωo2sα−1sα+ωo3+Kωo31ssα+ωo3.By inverting the Laplace transformation, we obtain(38)z1t=3KωotαEα,α+13−ωotα+3Kωo2t2αEα,2α+13−ωotα+Kωo31−Eα−ωotαt2α−1Eα,2α2−ωotα.In this case, integer-order ESO’s response in time domain is as follows:(39)z1t=K−K12ωo2t2−2ωot+1e−ωot.The extreme point of (39) is expressed as follows:(40)z1t1=K3−1e3−3+K≈1.206K.(40) shows that no matter how the bandwidth parameter is chosen, there is always about 20% overshoot during the tracing process in integer-order ESO as presented in Figure 7.Figure 7
The response ofz1 corresponding to the step signal (ESO).Considering FOESO, theoretically, it is difficult to solve (38) for extreme values; however, the numerical results show that the order of FOESO is related to the overshoot during the tracing process, which can be reduced by decreasing the order of differentiation as presented in Figure 8.Figure 8
The response ofz1 corresponding to step signal (FOESO).The relationship between FOESO’s suppression of external disturbances, noise transfer characteristics, and parameters is discussed. First, the effect of noiseδo from y is analyzed. From (25), we obtain(41)z1δo=3ωos2α+3ωo2sα+ωo3sα+ωo3.Whenα=0.8 (fixed) and ωo=5,10,15,20,25 are considered for obtaining the Bode diagram as shown in Figure 9, the response speed of the system increases with an increase in ωo. However, the noise amplification becomes more obvious due to the increase in the high frequency band gain.Figure 9
The frequency-domain characteristic of FOESO observation noise (observer gainωo).Whenωo=10 (fixed) and α=0.6,0.7,0.8,0.9,1 are considered for obtaining the Bode diagram as shown in Figure 10, the response speed of the system increases, the gain of the high frequency band decreases, and the noise amplification effect decreases as α decreases.Figure 10
The frequency-domain characteristic of FOESO observation noise (observer orderα).Second, the effect of input disturbanceδc from u is analyzed. From (25), we obtain(42)z1δc=b0sαsα+ωo3.Whenα=0.8,b0=10 (fixed) and ωo=10,20,...,50 are considered for obtaining the Bode diagram as shown in Figure 11, the phase lag of the tracking signal decreases, while the gain of the high frequency band is not affected at all with an increase in ωo. Please note that the proposed FOESO has good suppressive ability to input disturbance.Figure 11
The frequency-domain characteristic of FOESO input disturbance (observer gainωo).Whenωo=10,b0=10 (fixed), and α=0.6,0.7,0.8,0.9,1 are considered for obtaining the Bode diagram as shown in Figure 12, the phase lag of the tracking signal decreases and the high frequency band gain decreases with a decrease in α.Figure 12
The frequency-domain characteristic of FOESO input disturbance (observer orderα).


## 5. Stability Analysis

The stability analysis is described in the following sections.


### 5.1. Stability of the FOADRC

By substituting (24) into (22), we obtain(43)z1α=−β1z1+z2+β1y,z2α=−β2+kpz1−kdz2+β2y+kpr,z3α=−β3z1+β3y.Weconsider (43) and (24) as a double-input-single-output system with two inputs y and r an output u. Then, the state matrices of the system are defined as follows:(44)z1αz2αz3α=Az1z2z3+Bry,A=−β110−β2+kp−kd0−β300,B=0β1kpβ20β3.The output matrices of the system are expressed as follows:(45)u=Cz1z2z3+Dry,C=−kpb0−kdb0−1b0,D=kpb00.The state-space model of the system is transformed by the inverse Laplace transform, i.e.,G=CsαI−A−1B+D. The transfer functions from y and r to u are expressed as follows:(46)UsYs=−β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kpb0sαs2α+β1+kdsα+β1kd+β2+kp,C1s=kps3α+β1s2α+β2sα+β3β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kp,Thus, we obtain a block diagram of second-order FOADRC, as shown in Figure13.Figure 13
The block diagram of second-order FOADRC.The fractional-order feedback compensator is(47)Cs=β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kpb0sαs2α+β1+kdsα+β1kd+β2+kp.The fractional-order prefilter is(48)C1s=kps3α+β1s2α+β2sα+β3β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kp.The plant is(49)Ps=∑i=0mmism−i∑j=0nnjsn−j.The characteristic equation of the closed-loop system is expressed as follows:(50)1+CsPs=0.Substituting (47) and (49) in (50), we obtain(51)1+β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kpb0sαs2α+β1+kdsα+β1kd+β2+kp×∑i=0mmism−i∑j=0nnjsn−j=0.Simplifying (51), we obtain as follows:(52)b0sαs2α+β1+kdsα+β1kd+β2+kp×∑j=0nnjsn−j+β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kp×∑i=0mmism−i=0.Substituting (23) and (25) in (52), we obtain(53)b0sαs2α+3ωo+2ωcsα+6ωoωc+3ωo2+ωc2×∑j=0nnjsn−j+3ωoωc2+6ωo2ωc+ωo3s2α+3ωo2ωc2+2ωo3ωcsα+ωo3ωc2∑i=0mmism−i=0.As long as the parametersb0,ωc,ωo,α are selected in such a way that the root of (53) lies in the left half plane of the complex plane, the closed-loop system is BIBO stable.In summary, the conditions that the parameters should meet the BIBO stability of the closed-loop system are given by equivalent transformation of FOADRC and calculation of the characteristic equation of the closed-loop system.


### 5.2. The Calculation Method of Stability Region for FOADRC

In this section, in order to improve the tuning efficiency of FOADRC, a method for calculating the stability regions based on D-partition is proposed, and the effectiveness of the proposed calculation method is verified.Substitutings=jω in (53), the boundaries of the stability domain are described by ω=0,ω=±∞ and ω∈0,−∞∪0,+∞, named as real root boundary, infinite root boundary, and complex root boundary, respectively.(1)
ω=0, the real root boundary D0 is expressed as follows:(54)ωo3ωc2mm=0,wheremm≠0 and ωo>0, so D0 is ωc=0.(2)
ω=±∞: the infinite root boundary D∞ is(55)b0n0=0.Asb0 is the estimated value of the high-frequency gain of the plant, b0≠0. There are no relevant parameters in the equation, so this condition is invalid and ignored.(3)
ω∈0,−∞∪0,+∞: the complex root boundary Dω is(56)b0jωαjω2α+β1+kdjωα+β1kd+β2+kp∑j=0nnjjωn−j+β1kp+β2kd+β3jω2α+β2kp+β3kdjωα+β3kp∑i=0mmijωm−i=0.The noninteger power of a complex numberσ+jωγ is calculated as follows:(57)σ+jωγ=σ2+ω2γcosγtan−1ωσ+jsinγtan−1ωσ.Based on the above equation, we obtain(58)j2α=cosπα+jsinπα,jα=cosπ2α+jsinπ2α.Substituting (58) in (56) and simply replaced with real part and imaginary part, we can rewrite it as follows:(59)1+acω+jbcωaω+jbω=0.The real and imaginary parts are determined as follows:(60)Gpjω=rωeiϑω=aω+jbω,Gcjω=rcωeiϑω=acω+jbcω,where Gp and Gc, respectively, represent the frequency-domain response of the controlled plant and the feedback controller of FOADRC.Let the real and imaginary parts be zero, respectively, and the following conditions are obtained:(61)1+aωacω−bωbcω=0,aωbcω+acωbω=0.Based on the analysis of the above three boundaries and the preset conditions, the stability region boundary of FOADRC is expressed as follows:(62)D0:ωc=0,Dω:1+aωacω−bωbcω=0,aωbcω+acωbω=0,ωo=0.Using this computation method for antenna (2), the stability region for FOADRC controller of the antenna is as follows:(63)aω=2123200ω4−14035991ω2−23625900052500ω8−648071ω6+44886500ω4−193350000ω2+4692250000,bω=453689ω4+16997425ω2−4783697505ω2500ω8−648071ω6+44886500ω4−193350000ω2+4692250000,acω=ωo6ωcωo+ω2α2σ52−1+ωc2+3ωo2+σ3σ5ωc2ωo2+3ωαωc2ωo+3ω2ασ5ωc2+2ωαωcωo2+6ω2ασ5ωcωo+ω2ασ5ωo2σ1+ωoσ4σ2−ωc2ωo2+3ω2αωc2+6ω2αωcωo+ω2αωo2σ1,bcω=ωoσ4−ωc2ωo2+3ω2αωc2+6ω2αωcωo+ω2αωo26ωcωo+ω2α2σ52−1+ωc2+3ωo2+σ3σ1+ωoσ2σ5ωc2ωo2+3ωαωc2wo+3ω2ασ5ωc2+2ωαωcωo2+6ω2ασ5ωcωo+ω2ασ5ωo2σ1,σ1=b0ωαω2αcosπα+6ωcωo+ωc2+3ωo2+σ32+σ22,σ2=ω2αsinπα+2ωαωcσ4+3ωαωoσ4,σ3=ωασ52ωc+3ωo,σ4=sinπα2,σ5=cosπα2.1/b0=1.667 and we solve (61) and (62) for each ω to obtain the stability region as presented in Figures 14 and 15; it is evident that FOADRC has a larger stability region than ADRC (α=1).Figure 14
The stability region of the antenna (ωc,ωo).Figure 15
The stability region of the antenna (ωc,α,ωo).In order to verify the boundary of the stability region, we consider step signal as the input, andα=0.9andωc=6.053. The simulation results of wo= 7, 10, 14, and 17 cases are obtained. Figure 16 shows the accuracy of the boundary of the stability domain.Figure 16
The step response for the FOADRC.


## 5.1. Stability of the FOADRC

By substituting (24) into (22), we obtain(43)z1α=−β1z1+z2+β1y,z2α=−β2+kpz1−kdz2+β2y+kpr,z3α=−β3z1+β3y.Weconsider (43) and (24) as a double-input-single-output system with two inputs y and r an output u. Then, the state matrices of the system are defined as follows:(44)z1αz2αz3α=Az1z2z3+Bry,A=−β110−β2+kp−kd0−β300,B=0β1kpβ20β3.The output matrices of the system are expressed as follows:(45)u=Cz1z2z3+Dry,C=−kpb0−kdb0−1b0,D=kpb00.The state-space model of the system is transformed by the inverse Laplace transform, i.e.,G=CsαI−A−1B+D. The transfer functions from y and r to u are expressed as follows:(46)UsYs=−β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kpb0sαs2α+β1+kdsα+β1kd+β2+kp,C1s=kps3α+β1s2α+β2sα+β3β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kp,Thus, we obtain a block diagram of second-order FOADRC, as shown in Figure13.Figure 13
The block diagram of second-order FOADRC.The fractional-order feedback compensator is(47)Cs=β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kpb0sαs2α+β1+kdsα+β1kd+β2+kp.The fractional-order prefilter is(48)C1s=kps3α+β1s2α+β2sα+β3β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kp.The plant is(49)Ps=∑i=0mmism−i∑j=0nnjsn−j.The characteristic equation of the closed-loop system is expressed as follows:(50)1+CsPs=0.Substituting (47) and (49) in (50), we obtain(51)1+β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kpb0sαs2α+β1+kdsα+β1kd+β2+kp×∑i=0mmism−i∑j=0nnjsn−j=0.Simplifying (51), we obtain as follows:(52)b0sαs2α+β1+kdsα+β1kd+β2+kp×∑j=0nnjsn−j+β1kp+β2kd+β3s2α+β2kp+β3kdsα+β3kp×∑i=0mmism−i=0.Substituting (23) and (25) in (52), we obtain(53)b0sαs2α+3ωo+2ωcsα+6ωoωc+3ωo2+ωc2×∑j=0nnjsn−j+3ωoωc2+6ωo2ωc+ωo3s2α+3ωo2ωc2+2ωo3ωcsα+ωo3ωc2∑i=0mmism−i=0.As long as the parametersb0,ωc,ωo,α are selected in such a way that the root of (53) lies in the left half plane of the complex plane, the closed-loop system is BIBO stable.In summary, the conditions that the parameters should meet the BIBO stability of the closed-loop system are given by equivalent transformation of FOADRC and calculation of the characteristic equation of the closed-loop system.


## 5.2. The Calculation Method of Stability Region for FOADRC

In this section, in order to improve the tuning efficiency of FOADRC, a method for calculating the stability regions based on D-partition is proposed, and the effectiveness of the proposed calculation method is verified.Substitutings=jω in (53), the boundaries of the stability domain are described by ω=0,ω=±∞ and ω∈0,−∞∪0,+∞, named as real root boundary, infinite root boundary, and complex root boundary, respectively.(1)
ω=0, the real root boundary D0 is expressed as follows:(54)ωo3ωc2mm=0,wheremm≠0 and ωo>0, so D0 is ωc=0.(2)
ω=±∞: the infinite root boundary D∞ is(55)b0n0=0.Asb0 is the estimated value of the high-frequency gain of the plant, b0≠0. There are no relevant parameters in the equation, so this condition is invalid and ignored.(3)
ω∈0,−∞∪0,+∞: the complex root boundary Dω is(56)b0jωαjω2α+β1+kdjωα+β1kd+β2+kp∑j=0nnjjωn−j+β1kp+β2kd+β3jω2α+β2kp+β3kdjωα+β3kp∑i=0mmijωm−i=0.The noninteger power of a complex numberσ+jωγ is calculated as follows:(57)σ+jωγ=σ2+ω2γcosγtan−1ωσ+jsinγtan−1ωσ.Based on the above equation, we obtain(58)j2α=cosπα+jsinπα,jα=cosπ2α+jsinπ2α.Substituting (58) in (56) and simply replaced with real part and imaginary part, we can rewrite it as follows:(59)1+acω+jbcωaω+jbω=0.The real and imaginary parts are determined as follows:(60)Gpjω=rωeiϑω=aω+jbω,Gcjω=rcωeiϑω=acω+jbcω,where Gp and Gc, respectively, represent the frequency-domain response of the controlled plant and the feedback controller of FOADRC.Let the real and imaginary parts be zero, respectively, and the following conditions are obtained:(61)1+aωacω−bωbcω=0,aωbcω+acωbω=0.Based on the analysis of the above three boundaries and the preset conditions, the stability region boundary of FOADRC is expressed as follows:(62)D0:ωc=0,Dω:1+aωacω−bωbcω=0,aωbcω+acωbω=0,ωo=0.Using this computation method for antenna (2), the stability region for FOADRC controller of the antenna is as follows:(63)aω=2123200ω4−14035991ω2−23625900052500ω8−648071ω6+44886500ω4−193350000ω2+4692250000,bω=453689ω4+16997425ω2−4783697505ω2500ω8−648071ω6+44886500ω4−193350000ω2+4692250000,acω=ωo6ωcωo+ω2α2σ52−1+ωc2+3ωo2+σ3σ5ωc2ωo2+3ωαωc2ωo+3ω2ασ5ωc2+2ωαωcωo2+6ω2ασ5ωcωo+ω2ασ5ωo2σ1+ωoσ4σ2−ωc2ωo2+3ω2αωc2+6ω2αωcωo+ω2αωo2σ1,bcω=ωoσ4−ωc2ωo2+3ω2αωc2+6ω2αωcωo+ω2αωo26ωcωo+ω2α2σ52−1+ωc2+3ωo2+σ3σ1+ωoσ2σ5ωc2ωo2+3ωαωc2wo+3ω2ασ5ωc2+2ωαωcωo2+6ω2ασ5ωcωo+ω2ασ5ωo2σ1,σ1=b0ωαω2αcosπα+6ωcωo+ωc2+3ωo2+σ32+σ22,σ2=ω2αsinπα+2ωαωcσ4+3ωαωoσ4,σ3=ωασ52ωc+3ωo,σ4=sinπα2,σ5=cosπα2.1/b0=1.667 and we solve (61) and (62) for each ω to obtain the stability region as presented in Figures 14 and 15; it is evident that FOADRC has a larger stability region than ADRC (α=1).Figure 14
The stability region of the antenna (ωc,ωo).Figure 15
The stability region of the antenna (ωc,α,ωo).In order to verify the boundary of the stability region, we consider step signal as the input, andα=0.9andωc=6.053. The simulation results of wo= 7, 10, 14, and 17 cases are obtained. Figure 16 shows the accuracy of the boundary of the stability domain.Figure 16
The step response for the FOADRC.


## 6. Simulation Results

In this section, we evaluate the proposed FOADRC control scheme by performing simulations in MATLAB. The transfer function of the antenna plant and the method to simulate the wind disturbance are presented in Section2. The parameters of FOADRC are ωo=5,ωc=8.05,andα=0.9, the parameters of ADRC are ωo=5andωc=8.05, and the parameters of PI are kp=13.1071andki=8.0575. The performance of FOADRC is mainly evaluated based on the response towards a step signal, response to a sinusoidal signal, doubling of disturbance, and parameters’ perturbation. Please note that the simulation time is 20 s. In addition to the overshoot and adjustment time, the error integral criteria are also considered to measure the size of system errors. The integral square error (ISE), integral time squared error (ITSE), integral absolute error (IAE), and integral time absolute error (ITAE) are computed by using the following expressions:(64)ISE=∫0∞et2dt,ITSE=∫0∞etdt,IAE=∫0∞etdt,ITAE=∫0∞tetdt.


### 6.1. Tracking Performance Analysis

Considering that the input is a step signal with an amplitude of 3 mdeg and start time of 0, the response curves of PI, ADRC, and FOADRC are presented in Figure17. It is evident that ADRC and FOADRC have a significantly less overshoot than the PI controller, and FOADRC achieves a smaller adjustment time as compared to ADRC. The detailed results are presented in Tables 1 and 2. From Table 1, it is evident that, as compared with the PI controller, the FOADRC reduces overshoot by 38.2%, adjustment time of 2% error band calculation by 30%, and adjustment time of 5% error band calculation by 50.2%. As compared with ADRC, the FOADRC overshoots are nearly identical, but the adjustment time of 2% error band calculation is reduced by 22% and that of 5% error band calculation is reduced by 25.3%.Figure 17
The step responses for three controllers.Table 1
The dynamic performance of three controllers.
ControllersOvershoot (%)Adjustment time (s)2% error band5% error bandPI34.85.144.7308ADRC21.354.651463.1587FOADRC21.493.59902.3581Table 2
The error integral criteria of three controllers.
ControllersISEITSEΙΑΕΙΤΑΕPI6.9436.6965.15915.19ADRC3.8471.7032.85.543FOADRC3.6241.1052.3124.419Table2 presents the four error integral criteria which show that the error in the response of FOADRC to step signals is much smaller as compared to the PI controller used in antenna and also smaller as compared to ADRC. As compared with the PI controller, the ISE index decreased by 47.8%, ITSE index decreased by 83.5%, IAE index decreased by 55.19%, and ITAE index decreased by 70.91%.Figure18 shows the error curves corresponding to the three controllers when the sinusoidal signal with amplitude of 1 mdeg and frequency of 1 rad/sec is used as an input. The proposed FOADRC traces the sinusoidal signal much better as compared to the PI controller and slightly better as compared to the ADRC.Figure 18
The error curve of sine input for three controllers.


### 6.2. Robustness Analysis

In order to verify the performance of the controller in wind disturbance and model uncertainty, the simulation is performed by increasing the wind disturbance by factors of 1.5 and 2. Figure19 presents the simulation results of the PI controller under three wind disturbances. Figure 20 shows the simulation results of the proposed FOADRC under three wind disturbances. Please note that the response of the two controllers does not deteriorate in dynamic adjustment. However, when the steady state is reached, it can be seen from the local magnification diagram that the performance deterioration of FOADRC after increasing wind disturbance is significantly less as compared to the PI controller.Figure 19
The step response of the PI controller when wind disturbance is increased.Figure 20
The step response of the ADRC controller when wind disturbance is increased.The dynamic performance indicators in the case of double wind disturbance are shown in Table3. As compared with Table 1, it can be seen that the overshoot of the three controllers hardly changes when the wind disturbance is increased by 2 times, but the adjustment time of the PI controller increases by 2–3 times. The time consumed by the proposed FOADRC to enter the 2% error band increases, but the process for entering the 5% error band is still fast.Table 3
The dynamic performance of three controllers (double wind disturbance).
ControllersOvershoot (%)Adjustment time (s)2% error band5% error bandPI32.615.640710.86ADRC21.1114.90443.28FOADRC21.5414.89813.41The error integral criteria for the case where the wind disturbance is doubled are shown in Table4. The four error integral criteria for FOADRC are still less than ADRC and PI controller. As compared with the PI controller, the FOADRC decreased by 48.28% under ISE criteria, 83.96% under ITSE criteria, 55.81% under IAE criteria, and 68.73% under ITAE criteria.Table 4
The error integral criteria of three controllers (double wind disturbance).
ControllersISEITSEΙΑΕΙΤΑΕPI7.0578.1545.87723.5ADRC3.8451.8312.9858.069FOADRC3.651.3082.5977.348The step response curves of FOADRC with parameter perturbations of the antenna transfer function model are shown in Figure21. The response curves of the four parameter perturbations overlap the original response curves completely, indicating that the FOADRC performs well in response to parameter perturbations.Figure 21
The step response of the ADRC controller when parameters are perturbed (±10% and ±20%).


## 6.1. Tracking Performance Analysis

Considering that the input is a step signal with an amplitude of 3 mdeg and start time of 0, the response curves of PI, ADRC, and FOADRC are presented in Figure17. It is evident that ADRC and FOADRC have a significantly less overshoot than the PI controller, and FOADRC achieves a smaller adjustment time as compared to ADRC. The detailed results are presented in Tables 1 and 2. From Table 1, it is evident that, as compared with the PI controller, the FOADRC reduces overshoot by 38.2%, adjustment time of 2% error band calculation by 30%, and adjustment time of 5% error band calculation by 50.2%. As compared with ADRC, the FOADRC overshoots are nearly identical, but the adjustment time of 2% error band calculation is reduced by 22% and that of 5% error band calculation is reduced by 25.3%.Figure 17
The step responses for three controllers.Table 1
The dynamic performance of three controllers.
ControllersOvershoot (%)Adjustment time (s)2% error band5% error bandPI34.85.144.7308ADRC21.354.651463.1587FOADRC21.493.59902.3581Table 2
The error integral criteria of three controllers.
ControllersISEITSEΙΑΕΙΤΑΕPI6.9436.6965.15915.19ADRC3.8471.7032.85.543FOADRC3.6241.1052.3124.419Table2 presents the four error integral criteria which show that the error in the response of FOADRC to step signals is much smaller as compared to the PI controller used in antenna and also smaller as compared to ADRC. As compared with the PI controller, the ISE index decreased by 47.8%, ITSE index decreased by 83.5%, IAE index decreased by 55.19%, and ITAE index decreased by 70.91%.Figure18 shows the error curves corresponding to the three controllers when the sinusoidal signal with amplitude of 1 mdeg and frequency of 1 rad/sec is used as an input. The proposed FOADRC traces the sinusoidal signal much better as compared to the PI controller and slightly better as compared to the ADRC.Figure 18
The error curve of sine input for three controllers.


## 6.2. Robustness Analysis

In order to verify the performance of the controller in wind disturbance and model uncertainty, the simulation is performed by increasing the wind disturbance by factors of 1.5 and 2. Figure19 presents the simulation results of the PI controller under three wind disturbances. Figure 20 shows the simulation results of the proposed FOADRC under three wind disturbances. Please note that the response of the two controllers does not deteriorate in dynamic adjustment. However, when the steady state is reached, it can be seen from the local magnification diagram that the performance deterioration of FOADRC after increasing wind disturbance is significantly less as compared to the PI controller.Figure 19
The step response of the PI controller when wind disturbance is increased.Figure 20
The step response of the ADRC controller when wind disturbance is increased.The dynamic performance indicators in the case of double wind disturbance are shown in Table3. As compared with Table 1, it can be seen that the overshoot of the three controllers hardly changes when the wind disturbance is increased by 2 times, but the adjustment time of the PI controller increases by 2–3 times. The time consumed by the proposed FOADRC to enter the 2% error band increases, but the process for entering the 5% error band is still fast.Table 3
The dynamic performance of three controllers (double wind disturbance).
ControllersOvershoot (%)Adjustment time (s)2% error band5% error bandPI32.615.640710.86ADRC21.1114.90443.28FOADRC21.5414.89813.41The error integral criteria for the case where the wind disturbance is doubled are shown in Table4. The four error integral criteria for FOADRC are still less than ADRC and PI controller. As compared with the PI controller, the FOADRC decreased by 48.28% under ISE criteria, 83.96% under ITSE criteria, 55.81% under IAE criteria, and 68.73% under ITAE criteria.Table 4
The error integral criteria of three controllers (double wind disturbance).
ControllersISEITSEΙΑΕΙΤΑΕPI7.0578.1545.87723.5ADRC3.8451.8312.9858.069FOADRC3.651.3082.5977.348The step response curves of FOADRC with parameter perturbations of the antenna transfer function model are shown in Figure21. The response curves of the four parameter perturbations overlap the original response curves completely, indicating that the FOADRC performs well in response to parameter perturbations.Figure 21
The step response of the ADRC controller when parameters are perturbed (±10% and ±20%).


## 7. Conclusions

The wind disturbance is the primary factor that affects the accuracy of large radio telescope antennas. In this study, a fractional-order active disturbance rejection controller is designed and applied to an antenna based on the existing system configuration. A fractional-order extended state observer is used to estimate the parameter uncertainties, various feedback disturbances, and wind disturbances as a total disturbance. The stability, the disturbance rejection quality, and noise sensitivity characteristic are fully analyzed. The simulation results show that the FOADRC improves the control system capability of resistance to wind disturbance and limits the response overshoot (from 34.8% to 21.49%) and coverages faster compared to other industrial controller (from 5.14 to 3.599). The fractional-order integration contains an additional weight function, and the initial stage has a larger integral response value, so the FOESO can be estimated the state more quickly and accurately, and ultimately FOADRC can achieve better control performance.



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