# Speed Control of PMSM Based on an Optimized ADRC Controller

Authors: YiBo Meng; BingYou Liu; LiChao Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Permanent-magnet synchronous motor (PMSM) is a nonlinear, multivariate, strongly coupled system with uncertain external interference. A general control method cannot meet system requirements. This study proposes an optimal control strategy based on active disturbance rejection control (ADRC) to achieve high-precision control of PMSM. First, interpolation fitting is used to construct a nonlinear function with improved continuity and derivative near the origin and segment points. Second, various components of active disturbance rejection control are constructed based on the new nonlinear function, which mainly improves the extended state observer and nonlinear state error feedback. Simulation results show that under similar parameters, the improved active disturbance rejection control has smaller overshoot, faster tracking capacity, and stronger anti-interference capability compared with traditional active disturbance rejection control. Finally, the optimized active disturbance rejection control is used in the motion control system of PMSM. Experimental results show that the proposed optimal control strategy is simple and has good control performance.


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## Body

## 1. Introduction

Permanent-magnet synchronous motors are widely used in various applications due to their simple control, high efficiency, good torque characteristics, and low loss [1]. Current control methods for PMSM include proportional–integral–derivative (PID), sliding mode, adaptive, predictive, and active disturbance rejection control [2–5]. However, the application environment is generally complex and often has various disturbances because PMSM is a nonlinear, multivariable, and strongly coupled system. Hence, these control methods cannot easily perform fast and high-precision control. For example, PID control is sensitive to changes in the parameters of the controlled object model, and only the deviations of the given feedback are adjusted. PID is a single-degree-of-freedom control, and balancing the tracking performance and immunity of the system is difficult. Sliding mode control belongs to the bang-bang control, and the control volume is a noncontinuous quantity. Therefore, the problem of “flicker” inevitably arises. Adaptive control is a time-varying nonlinear system with low control accuracy, and satisfying the control requirements is difficult. Although adjusting the controller parameters in predictive control is unnecessary, accurate object model parameters, which are sensitive to changes in model parameters, are required. Moreover, the evaluation function value needs to be calculated multiple times in a control cycle, which requires a high level of computing power for the microprocessor. This will increase the cost of the control system and reduce the efficiency of industrial production. ADRC does not depend on the parameters of the object model and can estimate and compensate for internal and external disturbances in real time. ADRC also has a strong anti-interference capability and can be used in PMSM to achieve satisfactory results [6, 7].ADRC technology has been utilized for more than 20 years because of its high precision, simple structure, strong anti-interference capability, fast response, and easy parameter setting. These characteristics have elicited the attention of scholars at home and abroad. The application of ADRC in motor, flight, robot, and other industrial types of control has produced fruitful results. ADRC technology also plays an important role in various fields. For example, ADRC technology was applied to an idle controller, and the extended state observer (ESO) was used to compensate for the interference load and frictional torque of the engine. Meanwhile, a feedforward compensator was designed to improve interference suppression performance [8]. ADRC was transformed into an indirect perturbed Lurie system and then applied to a single-input single-output system. Global and local stabilities were verified with the Lyapunov theorem, and the manner in which linear dynamics was integrated into ADRC technology was described. ADRC was introduced into the rail control of a solar sail to maintain its place on the solar–lunar orbit [9, 10]. ADRC technology was also applied to a time-delay system, which is difficult to control; the parameter tuning method of ADRC technology under different time-delay systems was also summarized [11]. A linear system was used to compensate for the actual system as the nominal system to solve the aircraft attitude tracking problem accurately and efficiently under uncertain model and external disturbances [12]. Wang Ronglin applies ADRC to a position servo system of a rocket launcher driven by a PMSM, which improves the movement accuracy and robustness of the servo system [13]. A nonaffine nonlinear system with unknown external perturbation was used to estimate the unknown parts of a system using ESO, thereby solving the difficult problem of virtual control derivative acquisition [14, 15]. Zhang xinhua applied self-disturbance rejection control technology to a three-degree-of-freedom magnetic levitation permanent magnet planar motor to solve the problem of reduced electromagnetic force due to the deflection of the mover or the sharply decreased magnetic flux density with height [16]. ESO was utilized to linearize dynamically the information of a class of objects with unknown nonlinear properties and perturbations; the optimal control law was designed with LQR theory [17]. Jin Qibing used ADRC to solve the problem of model reduction error in the design process of a two-input/two-output time-delay system controller. The proposed method reduces the influence of the reduced-order error and improves the anti-interference capability and robustness of the control system [18].However, the applications of these ADRC technologies are based on the traditional ADRC. The traditional nonlinear function of ADRC has many issues in conductivity and smoothness near the origin and segment points, and these issues reduce the performance of ADRC. Many scholars have attempted to improve ADRC to achieve higher precision and stronger anti-interference capability and reduce the limitations of the traditional ADRC. For example, the nonlinear function in ADRC was replaced by a linear function, which reduces the number of tuning parameters required in ADRC to a certain extent, and was then applied to the pressurized water reactor power. However, the linear ADRC reduced the working efficiency of the system and was prone to “peaking” in the initial state [19]. Zhao Zhiliang designed a piecewise smoothing function consisting of linear and fractional power functions and applying the new function to nonlinear ESO. The new and traditional ESO were compared through a numerical simulation. The new ESO achieved smaller peaks and stronger noise immunity than traditional ESO. However, the addition of a power function increased the complexity of the linear function and the response time of the system [20]. In another study, nonlinear ESO was modified into a linear ESO and combined with the least-squares method. The linear function was applied to the nonlinear state error feedback rate to improve the capability of the system to resist interference [21]. Liu Bingyou designed a new nonlinear function that was constructed by interpolation. The new function was applied to ESO, and excellent performance was obtained. However, only the analysis of the new ADRC at the theoretical stage was performed. No actual motor verification was conducted, and the validity of the method was not confirmed. Moreover, the gain and filter performance of the new function were not analyzed [22].Parameters are added to improve the structure of ADRC, but this addition complicates the tuning and control of the entire system. If a linear ADRC is used, then the control efficiency of the entire system will not be as high as that of nonlinear ADRC, and the “peaking” phenomenon in the initial state could lead to system instability. Therefore, this study designs an optimized ADRC that acts on the PMSM based on the nonlinear function of nonlinear ADRC. First, a mathematical model of the dynamic equation of the PMSM motion equation is established. Second, a new type of nonlinear function is designed to verify the performance of the new function. Then, the design and parameter settings of the ADRC are established. Lastly, a simulation and experimental study of the entire ADRC are performed, and system simulation is conducted by combining ADRC with PMSM. The optimized ADRC exhibits better performance than the traditional ADRC in terms of tracking accuracy and anti-interference capability. The improvement of ADRC by nonlinear function is an effective way, providing a new method for the optimization of ADRC in the future. It also provides a new idea for the combination of linear and nonlinear ADRC and promotes the development of ADRC.


## 2. Mathematical Model of PMSM

A structural diagram of vector control in PMSM is shown in Figure1. The speed loop is controlled by ADRC, and the current loop is controlled by PI.Figure 1
PMSM vector control system.A mathematical model of the PMSM used in this study is established under the following conditions:(1) the distribution of the permanent magnetic rotor field in the air gap space is a sine wave; (2) the magnetic circuit is linear, and the inductance parameters are constant, in which the core saturation of the stator is ignored; (3) core eddy current and hysteresis loss are neglected; and (4) the damping of the rotor winding is ignored. Given that the d-q mathematical model of PMSM is versatile, under these conditions, the voltage equation of PMSM is(1)Ud=Rid-ωreLqiq+Lddiddt(2)Uq=Riq+ωreLdid+ωreψ+Lqdiqdtwhere id and iq are the d- and q-components of current, respectively; Ud and Uq are the d- and q-components of voltage, respectively; Ld and Lq are the inductance on AC and DC sides, respectively;R is the stator winding resistance of the motor; ωre is the angular velocity of the motor; and ψ is the flux linkage of the permanent magnet and stator.The electromagnetic torque equation of PMSM is expressed as follows:(3)Te=32pψiq+Ld-Lqidiqwhere Te is the electromagnetic torque of the permanent magnet and stator andpis the pole pairs of the motor. In general, Ld = Lq =L. Therefore, the torque equation can be simplified as follows: (4)Te=32pψiqFurthermore, the motion equation of the PMSM is as follows:(5)Te-TL-Bω=Jdωdtwhere TL is the load torque and ω is the angular speed output of the motor.BandJare the friction coefficient and rotational inertia of the motor, respectively. Figure 2 shows a structural diagram of the dynamic characteristics of the motion equation in coordinatesd-q of PMSM.Figure 2
Motion equation dynamic characteristics of the structure.


## 3. Optimized ADRC

ADRC comprises three parts. The first part, the tracking differentiator (TD), is used to track signals, including differential ones. The second part, ESO, is utilized to observe the internal state and estimate the unpredictable state of the system. The third part, the nonlinear state error feedback (NLSEF), is used to provide a stable and effective output signal.


### 3.1. Design of Nonlinear Function

A nonlinear function is used in ESO and NLSEF. Therefore, the nonlinear function is the core of the entire design of the ADRC in the present study. The function expression offal(e,α,δ), which is generally used in this study, is as follows: (6)fale,α,δ=eαsigne,e>δeδ1-α,e≤δ,δ>0The two segment points in thefal function areδ and -δ. The function is derived at the segment point to verify its derivative. We take the derivative of thefal function at pointδ to verify its derivative. (7)fal′e,α,δ=αeα-1,e>δ1δ1-α,0<e≤δ,δ>0(8)αeα-1≠1δ1-α,e=δThe derivative values are not the same at different segment points. Consequently, the primitive function is not derivable at the segment point. Although thefal function is continuous at the origin and segment points, it is not derivable and lacks continuity and smoothness. Therefore, this study improves thefal function as a new nonlinear function with well continuity and smoothness at the origin and segment points.Whenⅇ>δ the expression ofnfal(e,α,δ) isnfal(e,α,δ)=eαsignⅇ;Whenⅇ≤δ the expression ofnfal(e,α,δ) isnfal(e,α,δ)=psine+qe2+rtane.The interpolation fitting method combined with polynomial and trigonometric function is selected because theδ in thenfal(e,α,δ) function is usually less than 1. In this interval, sine has better stationarity thane, and tane has better convergence thane3. It is also continuous and derivable at the origin point.If the fitting process is derivable and continuous, then(9)nfale,α,δ=δα,e=δnfale,α,δ=-δα,e=-δnfal′e,α,δ=αδα-1,e=δ,-δnfal′e,α,δ=δα-1,e=0The solution is as follows:(10)p=δα-α·δα-1·cos2δ·tan⁡δsin3δq=0r=-δα·cos⁡δ-α·δα-1·sin⁡δsin⁡δ·tan2δThe expression of can be obtained as(11)nfale,α,δ=eαsigne,e>δδα-α·δα-1·cos2δ·tan⁡δsin3δ·sin⁡e-δα·cos⁡δ-α·δα-1·sin⁡δsin⁡δ·tan2δ·tan⁡e,e≤δBy analyzing thenfal(e,α,δ) function, the coefficient ofe2 is set as 0 when employing the interpolation fitting method, psine and rtane are two items in the equation. Thus, the new function has improved convergence after interpolation fitting.To simulate and compare thenfal(e,α,δ)andfal(e,α,δ) function, we setδ=0.01 and α=0.25. The function curve is shown in Figure 3, which indicates that the new function has improved continuity and smoothness at the origin and segment points.Figure 3
Curves ofnfal(e,α,δ) andfal(e,α,δ).The nonlinear function is regarded as a link; therefore, the input of the link ise, the output is the dynamic value of the nonlinear function, and the ratio of the output dynamic value and inpute is defined as the equivalent gain of the link.(12)λ1e,α,δ=fale,α,δe,λ2e,α,δ=nfale,α,δeWe takeδ= 0.1, α = 0.25 as an example. The equivalent gain curve of the two nonlinear functions is shown in Figure 4. The figure indicates that when e≤δ=0.1 is the input. The gain of the original nonlinear function is constant, and the gain of the new nonlinear function is a smooth curve. The original nonlinear function has an inflection point at 0.1, and the new nonlinear function image is a smooth curve, which fully meets the characteristics of “small error, large gain, large error, and small gain,” and provides considerable convenience to system parameter adjustment.Figure 4
Equivalent gain characteristic curve ofnfal(e,α,δ) andfal(e,α,δ).The filtering and anti-interference performance of thenfal(e,α,δ) function are also important. The function filter shown in Figure 5 is used to verify the filtering and antialiasing performance of thenfal(e,α,δ) function.Figure 5
Function filter.As shown in Figure5,V is selected as the sinusoidal signal with the addition of Gaussian white noise.K is the proportional coefficient and y is the filtered output. The parameters of the selected filter are as follows:K=20, α=0.5, δ=0.1. As shown in Figures 6 and 7.Figure 6
Input signal.Figure 7
Output signal.Figures6 and 7 show that the filtered image only has a small part of irregular fluctuations and basically restores the sine function of the original input perfectly, which is sufficient to demonstrate that thenfal(e,α,δ) function filter has an excellent filtering effect. This result can prove that thenfal(e,α,δ) function has good filtering effect and antijamming performance.


### 3.2. Design of the TD

TD achieves a smooth approximation of the generalized derivative of the input signal through a nonlinear function in ADRC. The effect is to arrange the transition process according to the reference input and the control object to obtain a smooth input signal. TD’s algorithm is expressed as follows:(13)V1t+h=V1t+hV2tV2t+h=V2t+hfhanV2t-Vt,V2t,r,hwhereV(t),h,r, andfhanare the input signal, integration step, factor of the tracking rate, and the nonlinear function, respectively.The expression is as follows:(14)fhan-rad,a≤d-rsgna,a>da=V2+yh0,y≤d0V2+sgnya0-d2,y>d0a0=d2+8ryd=rh0d0=h0dy=v1-v+hv2TD uses the tracking and filtering characteristics of the input signal and can arrange transitional processes for the controller. By arranging appropriate transition processes, TD can effectively alleviate the drastic changes in the control volume and overshoot of the output caused by abrupt set value changes, thereby eliminating the conflict between rapidity and overshoot. The signal-to-noise ratio is considerably improved by integrating the method to quickly trace the generalized derivative of the input signal. Moreover, the tracking signal of the input signal, which has a filtering effect, can be obtained. The transition process is arranged in the input signal step jump, which can effectively reduce the overshoot. The tracking differentiator can break through the noise mask of the differential controller and has a significant effect on the response speed and stability of the system. The parameters for TD are set tor = 10 andh =h0 = 0.01. The unit step input signal and square wave signal are selected for the simulation. Figures 8 and 9 show the response curve of TD.Figure 8
Response curves of unit step input.Figure 9
Response curves of square wave input.Figure8 shows that when the input signal is a unit step signal, the overshoot is 0% with an adjustment time of 0.628s and the differential signal has a settling time of 0.778s. Figure 9 shows that when the input signal is a unit square wave signal with a period of 3 s, the overshoot is 0% with an adjustment time of 0.635 s, and the differential signal has a settling time of 0.735 s. These results indicate that TD can quickly and accurately track input signals without overshoot providing high-quality differential signals.


### 3.3. Design of the ESO

ESO is the core of the ADRC. Utilizing ESO could help estimate varying quantities of unknown disturbances and unmodeled dynamics via compensation in the feedback. An estimation of each state variable can be obtained through ESO, which helps achieve the purpose of object reconstruction. ESO is based only on the input and output information of the control object and does not depend on the specific mathematical model that generates the disturbance. Therefore, ESO has strong robustness and versatility. The algorithm is expressed as follows:(15)ε1=Z1-ωZ1=Z2-β11nfalε1,α1,δZ2=Z3-β12nfalε1,α2,δ+b0uZ3=-β13nfalε1,α3,δwhereε1 is the observational error;Z1is the tracking signal of the input signal;Z2 is the differential signal of the input signal;Z3 is the observed signal off(t);u is the control output;b0 is the estimated value of the compensation factor; δ is the filter factor; β11, β12, and β13 are the gains of the optimized ESO; and α1, α2, and α3 are nonlinear factors. In the optimized ESO, the perturbations of the model perturbation parameters and measurement noises are processed in a unified manner. The expanded nonlinear links are used to estimate and compensate for the real-time action, thereby simplifying the objects with nonlinear uncertainties into integral series structures. Linearization and determinism of the system are then realized. The deterministic and linearized system can be controlled using existing classical control theory. The parameters for the optimized ESO are set to α1 = 0.5, α2 = 0.25, α3 = 0.125,b0 = 2.485,δ= 0.01, and β11 = 60, β12 = 240, β13 = 890. The unit step input and square wave signal are selected for the simulation. Figures 10 and 11 show the response curve of the optimized ESO.Figure 10
Response curves of unit step input.Figure 11
Response curves of square wave input.Figure10 shows that when the input signal is a unit step signal,Z1,Z2, andZ3 steadily track the unit step signal, the differential signal of the input signal, and the total perturbation signal of the system at 0.098, 0.312, and 0.825 s, respectively. Figure 11 shows that when the input signal is a unit square wave signal with a period of 3 s,Z1,Z2, andZ3 tracing the unit square wave signal, the differential signal of the square wave signal, and the total system disturbance at the rising and falling edges of 0.095, 0.616, and 1.212 s, respectively. These findings fully demonstrate that the optimized ESO can quickly and accurately estimate the amount of unknown disturbances and unmodeled dynamic changes and can compensate for them in the feedback.


### 3.4. Design of the NLSEF

NLSEF is an error in the nonlinear algorithm controller between the estimations of state variables generated from TD and ESO and forms the controlled variable with the compensation of the total disturbance. Traditional NLSEF only uses the ratio and the differential of the error signal to calculate nonlinearity. The optimized NLSEF adds the integral part of the signal, improves the control precision of the system, and enhances system immunity. The algorithm of NLSEF is expressed as follows:(16)e1=v1-z1e0=∫e1dte2=v2-z2u0=β21nfale0,α1,δ1+β22nfale1,α2,δ1+β23nfale2,α3,δ1wheree1,e2,e3 denote the integration, error signal, and differential, of the error signal, respectively; α1, α2, and α3 are nonlinear factor; and β21, β22, and β23 are the gains of the optimized NLSEF. The parameters for the optimized NLSEF are set as α1=0.5, α2=0.25, α3=0.125,δ1=1 and β11=5, β12=25, β13=75. The unit step input and sinusoidal signal are selected for the simulation. Figures 12 and 13 show the response curve of the optimized NLSEF.Figure 12
Response curves of unit step input.Figure 13
Response curves of sinusoidal input.Figure12 shows that when the input signal is a unit step signal, the output curve stably tends to approach the input curve at 0.512 s. In Figure 13, when the input signal is a sinusoidal signal with a period of 4s, the output curve tends to be stable and does not jump in 2.012 s. These findings fully demonstrate that the control signal after the optimization of NLSEF is filtered and has better stability than that before processing. The adaptability and robustness of the system are thus improved.


### 3.5. Overall Design

Various modules of ADRC are designed in the aforementioned steps. We now combine the new modules of ADRC and use the speed loop model as the controlled object. The combination of the optimized ADRC and the speed loop is shown in Figure14.Figure 14
Combination of ADRC and PMSM.


## 3.1. Design of Nonlinear Function

A nonlinear function is used in ESO and NLSEF. Therefore, the nonlinear function is the core of the entire design of the ADRC in the present study. The function expression offal(e,α,δ), which is generally used in this study, is as follows: (6)fale,α,δ=eαsigne,e>δeδ1-α,e≤δ,δ>0The two segment points in thefal function areδ and -δ. The function is derived at the segment point to verify its derivative. We take the derivative of thefal function at pointδ to verify its derivative. (7)fal′e,α,δ=αeα-1,e>δ1δ1-α,0<e≤δ,δ>0(8)αeα-1≠1δ1-α,e=δThe derivative values are not the same at different segment points. Consequently, the primitive function is not derivable at the segment point. Although thefal function is continuous at the origin and segment points, it is not derivable and lacks continuity and smoothness. Therefore, this study improves thefal function as a new nonlinear function with well continuity and smoothness at the origin and segment points.Whenⅇ>δ the expression ofnfal(e,α,δ) isnfal(e,α,δ)=eαsignⅇ;Whenⅇ≤δ the expression ofnfal(e,α,δ) isnfal(e,α,δ)=psine+qe2+rtane.The interpolation fitting method combined with polynomial and trigonometric function is selected because theδ in thenfal(e,α,δ) function is usually less than 1. In this interval, sine has better stationarity thane, and tane has better convergence thane3. It is also continuous and derivable at the origin point.If the fitting process is derivable and continuous, then(9)nfale,α,δ=δα,e=δnfale,α,δ=-δα,e=-δnfal′e,α,δ=αδα-1,e=δ,-δnfal′e,α,δ=δα-1,e=0The solution is as follows:(10)p=δα-α·δα-1·cos2δ·tan⁡δsin3δq=0r=-δα·cos⁡δ-α·δα-1·sin⁡δsin⁡δ·tan2δThe expression of can be obtained as(11)nfale,α,δ=eαsigne,e>δδα-α·δα-1·cos2δ·tan⁡δsin3δ·sin⁡e-δα·cos⁡δ-α·δα-1·sin⁡δsin⁡δ·tan2δ·tan⁡e,e≤δBy analyzing thenfal(e,α,δ) function, the coefficient ofe2 is set as 0 when employing the interpolation fitting method, psine and rtane are two items in the equation. Thus, the new function has improved convergence after interpolation fitting.To simulate and compare thenfal(e,α,δ)andfal(e,α,δ) function, we setδ=0.01 and α=0.25. The function curve is shown in Figure 3, which indicates that the new function has improved continuity and smoothness at the origin and segment points.Figure 3
Curves ofnfal(e,α,δ) andfal(e,α,δ).The nonlinear function is regarded as a link; therefore, the input of the link ise, the output is the dynamic value of the nonlinear function, and the ratio of the output dynamic value and inpute is defined as the equivalent gain of the link.(12)λ1e,α,δ=fale,α,δe,λ2e,α,δ=nfale,α,δeWe takeδ= 0.1, α = 0.25 as an example. The equivalent gain curve of the two nonlinear functions is shown in Figure 4. The figure indicates that when e≤δ=0.1 is the input. The gain of the original nonlinear function is constant, and the gain of the new nonlinear function is a smooth curve. The original nonlinear function has an inflection point at 0.1, and the new nonlinear function image is a smooth curve, which fully meets the characteristics of “small error, large gain, large error, and small gain,” and provides considerable convenience to system parameter adjustment.Figure 4
Equivalent gain characteristic curve ofnfal(e,α,δ) andfal(e,α,δ).The filtering and anti-interference performance of thenfal(e,α,δ) function are also important. The function filter shown in Figure 5 is used to verify the filtering and antialiasing performance of thenfal(e,α,δ) function.Figure 5
Function filter.As shown in Figure5,V is selected as the sinusoidal signal with the addition of Gaussian white noise.K is the proportional coefficient and y is the filtered output. The parameters of the selected filter are as follows:K=20, α=0.5, δ=0.1. As shown in Figures 6 and 7.Figure 6
Input signal.Figure 7
Output signal.Figures6 and 7 show that the filtered image only has a small part of irregular fluctuations and basically restores the sine function of the original input perfectly, which is sufficient to demonstrate that thenfal(e,α,δ) function filter has an excellent filtering effect. This result can prove that thenfal(e,α,δ) function has good filtering effect and antijamming performance.


## 3.2. Design of the TD

TD achieves a smooth approximation of the generalized derivative of the input signal through a nonlinear function in ADRC. The effect is to arrange the transition process according to the reference input and the control object to obtain a smooth input signal. TD’s algorithm is expressed as follows:(13)V1t+h=V1t+hV2tV2t+h=V2t+hfhanV2t-Vt,V2t,r,hwhereV(t),h,r, andfhanare the input signal, integration step, factor of the tracking rate, and the nonlinear function, respectively.The expression is as follows:(14)fhan-rad,a≤d-rsgna,a>da=V2+yh0,y≤d0V2+sgnya0-d2,y>d0a0=d2+8ryd=rh0d0=h0dy=v1-v+hv2TD uses the tracking and filtering characteristics of the input signal and can arrange transitional processes for the controller. By arranging appropriate transition processes, TD can effectively alleviate the drastic changes in the control volume and overshoot of the output caused by abrupt set value changes, thereby eliminating the conflict between rapidity and overshoot. The signal-to-noise ratio is considerably improved by integrating the method to quickly trace the generalized derivative of the input signal. Moreover, the tracking signal of the input signal, which has a filtering effect, can be obtained. The transition process is arranged in the input signal step jump, which can effectively reduce the overshoot. The tracking differentiator can break through the noise mask of the differential controller and has a significant effect on the response speed and stability of the system. The parameters for TD are set tor = 10 andh =h0 = 0.01. The unit step input signal and square wave signal are selected for the simulation. Figures 8 and 9 show the response curve of TD.Figure 8
Response curves of unit step input.Figure 9
Response curves of square wave input.Figure8 shows that when the input signal is a unit step signal, the overshoot is 0% with an adjustment time of 0.628s and the differential signal has a settling time of 0.778s. Figure 9 shows that when the input signal is a unit square wave signal with a period of 3 s, the overshoot is 0% with an adjustment time of 0.635 s, and the differential signal has a settling time of 0.735 s. These results indicate that TD can quickly and accurately track input signals without overshoot providing high-quality differential signals.


## 3.3. Design of the ESO

ESO is the core of the ADRC. Utilizing ESO could help estimate varying quantities of unknown disturbances and unmodeled dynamics via compensation in the feedback. An estimation of each state variable can be obtained through ESO, which helps achieve the purpose of object reconstruction. ESO is based only on the input and output information of the control object and does not depend on the specific mathematical model that generates the disturbance. Therefore, ESO has strong robustness and versatility. The algorithm is expressed as follows:(15)ε1=Z1-ωZ1=Z2-β11nfalε1,α1,δZ2=Z3-β12nfalε1,α2,δ+b0uZ3=-β13nfalε1,α3,δwhereε1 is the observational error;Z1is the tracking signal of the input signal;Z2 is the differential signal of the input signal;Z3 is the observed signal off(t);u is the control output;b0 is the estimated value of the compensation factor; δ is the filter factor; β11, β12, and β13 are the gains of the optimized ESO; and α1, α2, and α3 are nonlinear factors. In the optimized ESO, the perturbations of the model perturbation parameters and measurement noises are processed in a unified manner. The expanded nonlinear links are used to estimate and compensate for the real-time action, thereby simplifying the objects with nonlinear uncertainties into integral series structures. Linearization and determinism of the system are then realized. The deterministic and linearized system can be controlled using existing classical control theory. The parameters for the optimized ESO are set to α1 = 0.5, α2 = 0.25, α3 = 0.125,b0 = 2.485,δ= 0.01, and β11 = 60, β12 = 240, β13 = 890. The unit step input and square wave signal are selected for the simulation. Figures 10 and 11 show the response curve of the optimized ESO.Figure 10
Response curves of unit step input.Figure 11
Response curves of square wave input.Figure10 shows that when the input signal is a unit step signal,Z1,Z2, andZ3 steadily track the unit step signal, the differential signal of the input signal, and the total perturbation signal of the system at 0.098, 0.312, and 0.825 s, respectively. Figure 11 shows that when the input signal is a unit square wave signal with a period of 3 s,Z1,Z2, andZ3 tracing the unit square wave signal, the differential signal of the square wave signal, and the total system disturbance at the rising and falling edges of 0.095, 0.616, and 1.212 s, respectively. These findings fully demonstrate that the optimized ESO can quickly and accurately estimate the amount of unknown disturbances and unmodeled dynamic changes and can compensate for them in the feedback.


## 3.4. Design of the NLSEF

NLSEF is an error in the nonlinear algorithm controller between the estimations of state variables generated from TD and ESO and forms the controlled variable with the compensation of the total disturbance. Traditional NLSEF only uses the ratio and the differential of the error signal to calculate nonlinearity. The optimized NLSEF adds the integral part of the signal, improves the control precision of the system, and enhances system immunity. The algorithm of NLSEF is expressed as follows:(16)e1=v1-z1e0=∫e1dte2=v2-z2u0=β21nfale0,α1,δ1+β22nfale1,α2,δ1+β23nfale2,α3,δ1wheree1,e2,e3 denote the integration, error signal, and differential, of the error signal, respectively; α1, α2, and α3 are nonlinear factor; and β21, β22, and β23 are the gains of the optimized NLSEF. The parameters for the optimized NLSEF are set as α1=0.5, α2=0.25, α3=0.125,δ1=1 and β11=5, β12=25, β13=75. The unit step input and sinusoidal signal are selected for the simulation. Figures 12 and 13 show the response curve of the optimized NLSEF.Figure 12
Response curves of unit step input.Figure 13
Response curves of sinusoidal input.Figure12 shows that when the input signal is a unit step signal, the output curve stably tends to approach the input curve at 0.512 s. In Figure 13, when the input signal is a sinusoidal signal with a period of 4s, the output curve tends to be stable and does not jump in 2.012 s. These findings fully demonstrate that the control signal after the optimization of NLSEF is filtered and has better stability than that before processing. The adaptability and robustness of the system are thus improved.


## 3.5. Overall Design

Various modules of ADRC are designed in the aforementioned steps. We now combine the new modules of ADRC and use the speed loop model as the controlled object. The combination of the optimized ADRC and the speed loop is shown in Figure14.Figure 14
Combination of ADRC and PMSM.


## 4. Simulation Results

The optimized ADRC model is simulated by Simulink in MATLAB, and the simulation waveform of ADRC is obtained to verify and optimize ADRC performance. The same parameters are selected for simulation in the traditional and optimized ADRC as follows. The bandwidth method of ADRC is combined, and the principle of the parameters selected is proposed as follows:


### 4.1. Parameters of TD

We do not consider the filtering, so we can chooseh =h0. The parameters in the tracking differentiator that need to be set are simulation steph and tracking factorr because they affect the tracking accuracy of the output signal.r affects the tracking accuracy of the output signal and the time of the transition process. A larger results in short transition time. Therefore, the general parameters of TD are as follows:r= 10,h=h0 = 0.02.


### 4.2. Parameters of ESO

The selected bandwidth ofδ cannot be chosen too large because it will lose the advantage of the nonlinear equation. By contrast, an extremely small bandwidth will lead to instability of the integrated system. Therefore, choosing 0.01 is appropriate. The factor of nonlinear generally meets the empirical value of 0.5, 0.25, and 0.125. The gain generally satisfies the following formula:(17)β11=3ω0,β12=3ω025,β13=ω039Whenω0 is large, the ESO can track a small amplitude of error disturbance. Increasing β13 properly improves the optimized performance but easily causes an overshoot of the total disturbance estimate, leading to the shock of the control value. Therefore, we take 60, 240, and 889.


### 4.3. Parameters of NLSEF

Although the gain is small, the error attenuation speed is high, and the anti-jamming capability is strong, an extremely small gain will cause high-frequency oscillation of the output, which will adversely affect the actual object. Therefore, we take 5, 25, and 75. The speed of error decay and the output are also appropriate.The same parameters are selected for simulation in the traditional and optimized ADRC. The specific parameter selection is shown in Table1.Table 1
Parameters of the ADRC.
Component of ADRC Parameter name Symbol Numerical value  TD Velocity factor r 0 10 Simulation step h 0 0.02  ESO Nonlinear factors α 1 0.5 Nonlinear factors α 2 0.25 Nonlinear factors α 3 0.125 Liner interval width δ 0.01 Gain β 11 60 Gain β 12 240 Gain β 13 889 Compensation factor b 0 2.485  NLSEF Gain β 21 5 Gain β 22 25 Gain β 23 75 Filter factor δ 1 1According to the PMSM vector control system and the ADRC structure diagram, we construct the optimized ADRC MATLAB Model and PMSM vector control system MATLAB model as shown in Figures15 and 16.Figure 15
Optimized ADRC MATLAB model.Figure 16
PMSM vector control system MATLAB model.(1) The two response curves of ADRC when the step input is 1 rad are shown in Figure 17. The diagram shows that the traditional ADRC overshoot is 9.76% with an adjustment time of 0.39 s and the optimized ADRC overshoot is 0% with an adjustment time of 0.24s. Hence, the optimized ADRC has a faster adjustment speed and smaller overshoot than the traditional ADRC.Figure 17
Response curves of unit step input.(2) The two response curves of ADRC when the input is sin(πt) are shown in Figure 18. The diagram shows that the time at which the traditional ADRC reaches the first peak is 0.71 s and that of the optimized ADRC is 0.54 s. This finding reveals that the optimized ADRC has a higher response speed and can form a stable sinusoidal output faster than traditional ADRC can.Figure 18
Response curves of sinusoidal input.(3) First, 1-rad step is inputted and a step disturbance is added in 2 s. Figure 19 shows the response curve of the small disturbance at 1 rad, and Figure 20 shows the other response curve of the large disturbance at 20 rad. Figure 19 shows that the overshoot of the traditional ADRC affected by a small disturbance is 17.91%, and the time needed to restore the stable output is 1.89 s. Meanwhile, the overshoot of the optimized ADRC affected by a small disturbance and the time needed to restore the stable output are essentially 0 s. Figure 20 shows that the overshoot of traditional ADRC affected by a lager disturbance is 259.34%, and the time necessary to restore the stable output is 2.01 s. By contrast, the overshoot of optimized ADRC affected by the large disturbance is 9.54%, and the time necessary to restore the stable output is 0.92 s. These results indicate that the optimized ADRC has a smaller overshoot and higher restoration speed than the traditional ADRC whether the disturbance is small or large. Therefore, the optimized ADRC has better capacity of resisting disturbance than the traditional ADRC.Figure 19
Response curves of small disturbance input.Figure 20
Response curves of big disturbance input.(4) The two response curves of ADRC when the input is a square wave pulse with an amplitude of 1 rad and a cycle of 6 s are shown in Figure 21. At phase jump, the traditional ADRC takes 1.98 s to restore the stable output, whereas the optimized ADRC takes 0.43 s. This result shows that the optimized ADRC has smaller overshoot and higher restoration speed than the traditional ADRC.Figure 21
Response curves of square wave pulse input.(5) The two response curves of ADRC when the input is square wave signal and the output is affected by white noise are shown in Figure 22. The optimized ADRC has a smaller fluctuation and better anti-interference capability than the traditional ADRC.Figure 22
Response curves of white noise input.In summary, in order to more intuitively compare the performance of traditional ADRC and optimize ADRC, the performance of the two in different situations is shown in Table2.Table 2
Comparison between Traditional ADRC and Optimized ADRC.
ADRC Input signal Overshoot(%) Adjustment time(s)  Traditional ADRC Unit step 9.76 0.39 Sinusoidal / 0.71 Small disturbance 17.91 1.89 Big disturbance 259.34 2.01 Square wave pulse / 1.98  Optimized ADRC Unit step 0 0.24 Sinusoidal / 0.54 Small disturbance 0 0 Big disturbance 9.54 0.92 Square wave pulse / 0.43


## 4.1. Parameters of TD

We do not consider the filtering, so we can chooseh =h0. The parameters in the tracking differentiator that need to be set are simulation steph and tracking factorr because they affect the tracking accuracy of the output signal.r affects the tracking accuracy of the output signal and the time of the transition process. A larger results in short transition time. Therefore, the general parameters of TD are as follows:r= 10,h=h0 = 0.02.


## 4.2. Parameters of ESO

The selected bandwidth ofδ cannot be chosen too large because it will lose the advantage of the nonlinear equation. By contrast, an extremely small bandwidth will lead to instability of the integrated system. Therefore, choosing 0.01 is appropriate. The factor of nonlinear generally meets the empirical value of 0.5, 0.25, and 0.125. The gain generally satisfies the following formula:(17)β11=3ω0,β12=3ω025,β13=ω039Whenω0 is large, the ESO can track a small amplitude of error disturbance. Increasing β13 properly improves the optimized performance but easily causes an overshoot of the total disturbance estimate, leading to the shock of the control value. Therefore, we take 60, 240, and 889.


## 4.3. Parameters of NLSEF

Although the gain is small, the error attenuation speed is high, and the anti-jamming capability is strong, an extremely small gain will cause high-frequency oscillation of the output, which will adversely affect the actual object. Therefore, we take 5, 25, and 75. The speed of error decay and the output are also appropriate.The same parameters are selected for simulation in the traditional and optimized ADRC. The specific parameter selection is shown in Table1.Table 1
Parameters of the ADRC.
Component of ADRC Parameter name Symbol Numerical value  TD Velocity factor r 0 10 Simulation step h 0 0.02  ESO Nonlinear factors α 1 0.5 Nonlinear factors α 2 0.25 Nonlinear factors α 3 0.125 Liner interval width δ 0.01 Gain β 11 60 Gain β 12 240 Gain β 13 889 Compensation factor b 0 2.485  NLSEF Gain β 21 5 Gain β 22 25 Gain β 23 75 Filter factor δ 1 1According to the PMSM vector control system and the ADRC structure diagram, we construct the optimized ADRC MATLAB Model and PMSM vector control system MATLAB model as shown in Figures15 and 16.Figure 15
Optimized ADRC MATLAB model.Figure 16
PMSM vector control system MATLAB model.(1) The two response curves of ADRC when the step input is 1 rad are shown in Figure 17. The diagram shows that the traditional ADRC overshoot is 9.76% with an adjustment time of 0.39 s and the optimized ADRC overshoot is 0% with an adjustment time of 0.24s. Hence, the optimized ADRC has a faster adjustment speed and smaller overshoot than the traditional ADRC.Figure 17
Response curves of unit step input.(2) The two response curves of ADRC when the input is sin(πt) are shown in Figure 18. The diagram shows that the time at which the traditional ADRC reaches the first peak is 0.71 s and that of the optimized ADRC is 0.54 s. This finding reveals that the optimized ADRC has a higher response speed and can form a stable sinusoidal output faster than traditional ADRC can.Figure 18
Response curves of sinusoidal input.(3) First, 1-rad step is inputted and a step disturbance is added in 2 s. Figure 19 shows the response curve of the small disturbance at 1 rad, and Figure 20 shows the other response curve of the large disturbance at 20 rad. Figure 19 shows that the overshoot of the traditional ADRC affected by a small disturbance is 17.91%, and the time needed to restore the stable output is 1.89 s. Meanwhile, the overshoot of the optimized ADRC affected by a small disturbance and the time needed to restore the stable output are essentially 0 s. Figure 20 shows that the overshoot of traditional ADRC affected by a lager disturbance is 259.34%, and the time necessary to restore the stable output is 2.01 s. By contrast, the overshoot of optimized ADRC affected by the large disturbance is 9.54%, and the time necessary to restore the stable output is 0.92 s. These results indicate that the optimized ADRC has a smaller overshoot and higher restoration speed than the traditional ADRC whether the disturbance is small or large. Therefore, the optimized ADRC has better capacity of resisting disturbance than the traditional ADRC.Figure 19
Response curves of small disturbance input.Figure 20
Response curves of big disturbance input.(4) The two response curves of ADRC when the input is a square wave pulse with an amplitude of 1 rad and a cycle of 6 s are shown in Figure 21. At phase jump, the traditional ADRC takes 1.98 s to restore the stable output, whereas the optimized ADRC takes 0.43 s. This result shows that the optimized ADRC has smaller overshoot and higher restoration speed than the traditional ADRC.Figure 21
Response curves of square wave pulse input.(5) The two response curves of ADRC when the input is square wave signal and the output is affected by white noise are shown in Figure 22. The optimized ADRC has a smaller fluctuation and better anti-interference capability than the traditional ADRC.Figure 22
Response curves of white noise input.In summary, in order to more intuitively compare the performance of traditional ADRC and optimize ADRC, the performance of the two in different situations is shown in Table2.Table 2
Comparison between Traditional ADRC and Optimized ADRC.
ADRC Input signal Overshoot(%) Adjustment time(s)  Traditional ADRC Unit step 9.76 0.39 Sinusoidal / 0.71 Small disturbance 17.91 1.89 Big disturbance 259.34 2.01 Square wave pulse / 1.98  Optimized ADRC Unit step 0 0.24 Sinusoidal / 0.54 Small disturbance 0 0 Big disturbance 9.54 0.92 Square wave pulse / 0.43


## 5. Experimental Results

The experimental platform is a 400W Yaskawa servomotor. The experimental apparatus are shown in Figure23. The motor parameters are shown in Table 3. The system measures the motor speed through a 1024 photoelectric encoder and uses the same parameters as those in the simulation for traditional ADRC and optimized ADRC.Table 3
Motor parameters.
Input voltage 200V Rated current 2.5A Working frequency 250Hz Rated power 400W Torque 1.27N∗m Rated speed 3000r/min Maximum speed 6000r/min Motor resolution 1024ctsFigure 23
Experimental apparatus.Figure24 shows the traditional ADRC (a) and the optimized ADRC (b). When the input is 1000 cts/sec, the optimized ADRC tracking curve has less fluctuation and distortion than the traditional ADRC tracking curve, which is suitable for the input signal curve.Figure 24
Response curves of 1000cts/sec.
(a) (b)Figure25 shows the traditional ADRC (a) and the optimized ADRC (b). When the input is 1000 cts/sec, the optimized ADRC has less adjustment time and overshoot than the traditional ADRC while also having smaller fluctuations in speed.Figure 25
Response curves of 10000cts/sec.
(a) (b)Figures24 and 25 fully explain that the optimized ADRC has higher tracking accuracy and smaller steady-state fluctuation error than the traditional ADRC at low and high speeds.Figures26(a), 26(b), and 26(c) show the traditional ADRC when the input is sinusoidal, and Figures 26(d), 26(e), and 26(f) show the optimized ADRC. Figures (a) and (d), (b) and (e), and (c), and (f) show an amplitude of 3000 6,000, and 12,000 cts/sec, respectively, and the frequency is 1 Hz. Figures (a), (b), and (c) show that the traditional ADRC speed amplitude is larger than the fixed amplitude and phase lag, and the tracking error is large. Figures (d), (e), and (f) show that the optimized ADRC speed amplitude is consistent with the given value, phase advance or hysteresis is absent, and the tracking error is evident.Figure 26
Response curves of sinusoidal input.
(a) (b) (c)  (d) (e) (f)Therefore, the ADRC control effect is better than that of the conventional ADRC whether in low, medium, or high speed. The experimental and simulation results fully prove that the optimization of ADRC tracking accuracy and anti-jamming performance is better than that of the traditional ADRC.


## 6. Conclusion

As a continuously developing and improving new nonlinear controller, ADRC has been increasingly applied in numerous fields. This study introduces a method to optimize ADRC, constructs a new nonlinear function using the interpolation fitting method to optimize the control effect of ADRC, and elaborates the construction process of the new nonlinear function. The new nonlinear function is then applied to the various components of ADRC. A simulation is conducted to verify the feasibility of the proposed optimization method compared with the traditional ADRC under the same conditions. The simulation verifies the good performance of the optimized ADRC. The two control methods are also compared by using an actual motor. The experimental results show that the optimized ADRC has smaller overshoot, faster adjustment time, and smaller speed fluctuation than the conventional ADRC at constant low and high speeds. At low and high speeds, the optimized ADRC also has faster tuning times, smaller phase shifts, and faster speed fluctuations than conventional ADRC. The experimental and simulation results fully prove that the optimized ADRC is more robust and adaptive and has better anti-interference capability than the traditional ADRC. It also provides a new idea for the combination of linear and nonlinear ADRC and promotes the development of ADRC. Our future work will be carried out in the following aspects:(1) the optimized ADRC is applied to the position loop and current loop in the motor; (2) the ADRC parameter setting of the speed loop is analyzed to find the optimal ADRC parameters; (3) change the inherent structure of the ADRC and improve the ADRC from the structural aspect.



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*Source: 1074702-2019-05-22.xml*