Research on linear/nonlinear active disturbance rejection switching control of PMSM speed servo system

An improved linear/nonlinear active disturbance rejection switching control strategy is proposed to enhance the dynamic response and disturbance suppression capabilities of the permanent magnet synchronous motor (PMSM) speed servo system. Firstly, facilitated by a tracking differentiator, rapid setpoint tracking is achieved without speed overshoot, effectively addressing the inherent conflict between fast system response and overshooting. Secondly, weight factors are introduced into the control law, and a novel switching criterion based on the step function is designed, enabling the system to smoothly transition between operating modes in the presence of disturbances with varying amplitudes. This facilitates accurate estimation and swift compensation for disturbances, thereby enhancing control precision and robustness. Additionally, a frequency domain analysis of the extended state observer is conducted, leading to the formulation of a parameter tuning scheme for the controller. Lastly, simulation tests validate the feasibility and effectiveness of this strategy as a PMSM speed controller. The system exhibits superior dynamic tracking and disturbance suppression performance over a wide speed range.


Introduction
As an advanced and efficient propulsion system, the permanent magnet synchronous motor (PMSM) finds widespread application across diverse domains, including robotics and electric vehicles, where stringent control performance is imperative [1][2].Traditional PMSM servo systems predominantly employ PI linear control strategies.However, when faced with nonlinear factors like model uncertainty and unknown disturbances, the PI control method encounters challenges.
Active disturbance rejection control (ADRC) was first introduced in [3] and has achieved fruitful research results [4].It has been proven that it has advantages in controlling PMSM systems because it does not rely on an accurate mathematical model of the controlled system and has strong antiinterference capabilities.In [5], nonlinear ADRC (NLADRC) is applied to address issues related to speed overshoot and effectively mitigate flux linkage and torque ripple.However, NLADRC encounters challenges in tuning and performance analysis.In contrast, linear ADRC (LADRC) is introduced in [6], along with a bandwidth-based parameter tuning method characterized by straightforward theoretical analysis and ease of implementation.LADRC in [7] improves speed regulation, especially for lowfrequency disturbances.Moreover, the mechanisms of NLADRC and LADRC are explored in [8], concluding that the gain of NLESO in NLADRC is constrained by observation noise, limiting its effectiveness in suppressing large amplitude disturbances.Conversely, LADRC's tracking performance remains stable across disturbance amplitudes.Recognizing the distinct advantages of NLADRC and LADRC, the linear/nonlinear active disturbance rejection switching control (SADRC) method is introduced to capitalize on their respective strengths.SADRC is applied as the controller of PMSM in [9], achieving accurate and robust speed control.
In light of the above analysis, this paper introduces a novel SADRC strategy to improve the control performance of the PMSM speed servo system.By introducing weighting factors and employing a step function to design a novel switching criterion, SADRC ingeniously combines the advantages of both NLADRC and LADRC.This paper provides tuning directions and rules for the parameters in the SADRC controller.The subsequent sections are as follows: Section 2 outlines the PMSM's mathematical framework; Section 3 details the design scheme and parameter tuning rules for the SADRC speed controller; Section 4 conducts numerical simulations and results analysis; Section 5 concludes the findings.

Mathematical model of the PMSM
This paper investigates the surface-mounted PMSM as the research subject.The ideal conditions assume symmetrical windings, and the effects of core saturation, eddy current, hysteresis losses, and viscous friction coefficient are disregarded.The control strategy with * 0 d i is employed.The voltage and motion equations are formulated in the synchronous rotating coordinate system as follows [10]: where m Z is the mechanical angular speed; J is the moment of inertia; e T is the electromagnetic torque; L T is the load torque; T K is the torque coefficient; d u , q u , d i , and q i are the dq -axis stator voltages and currents; s R is the stator resistance; s L is the stator inductance; p n is the number of magnetic pole pairs; f \ is the permanent magnet flux linkage.

Design of the SADRC speed controller
The SADRC speed controller comprises five components: TD, linear extended state observer (LESO), NLESO, linear state error feedback (LSEF), and NLSEF, as shown in Figure 1.

Tracking differentiator
The role of TD is to mold the transition process for the specified speed signal.While the motion equation indicates the speed servo system is a first-order model, this paper opts for a second-order tracking function, denoted as fhan, for its superior tracking performance.The TD model is expressed as follows: ), ( ( ), ( 1) Z are the mechanical angular velocity given value, transition value and transition differential value, respectively; h is the integration step size; r is the speed factor.The specific expression of fhan can be found in [9].
The adjustable parameter in TD is r, impacting the response speed of *   T is the torque output by the controller; f denotes the total disturbance; d is the unmodeled disturbance in the system model.The inaccurate estimation of b is encompassed within f and will be collectively compensated.
Assuming f is continuous, differentiable, and bounded, we choose m Z as the state variable and express f as a new state variable.Discrete LESO and NLESO are designed as follows:  decreases as e increases, adhering to the characteristics of "large error with a small gain, small error with a large gain".Additionally, it is observed that the ( ) .Generally, it is more appropriate for D to be in the range (0,1).And now, NLESO can be considered as a LESO with a variable gain, represented as NL ( ) , and its tuning strategy can refer to LESO.Building upon the optimal parameter settings in [4], the tuning equation proposed in this paper is , when o 100, 200,300 Z , the frequency domain characteristic curves of disturbance estimation error and observation noise for the LESO and NLESO designed in this paper are presented in Figures 3-6.increases, the observer attenuates the low-frequency amplitude of the disturbance estimation error to a greater extent, enhancing the system's response speed.However, the amplitude of high-frequency noise also increases simultaneously, diminishing the system's capability to suppress noise.In other words, there exists a trade-off between the system's anti-disturbance performance and its ability to suppress noise.Therefore, when adjusting o Z , both factors need to be comprehensively considered.

State error feedback
Given that the ESO performance meets the system requirements, the controller output can be expressed as * .At this point, only the proportional link is required to achieve a more ideal control effect.We design LSEF and NLSEF as follows: where Lp k and NLp k are the control law gains.The parameter to be adjusted in LSEF is the controller gain Lp k , which corresponds to the controller bandwidth and determines the speed of the system's dynamic response.For the system to adequately compensate for disturbances in real-time, the observer bandwidth needs to be greater than the controller bandwidth, typically with o Z being 3-5 times Lp k [8].Consequently, during the tuning process, an appropriate controller bandwidth can be chosen after establishing the observer bandwidth o D and G are akin to those for i D and G in NLESO.

SADRC switching mechanism
The SADRC strategy aims to combine the strengths of NLADRC and LADRC, providing the PMSM speed servo system with accurate tracking and robust disturbance rejection in diverse conditions.To achieve this, a new state error feedback control law with weight factors is introduced.A novel switching criterion, considering both observation deviation and total disturbance, is designed by using a step function to minimize system jitter and instability during the switching process.The design details are provided as follows: u represents the output of the SADRC control law; K represents the weight factor of the control law; F and J are the step functions defined by the observation deviation criterion and the total disturbance criterion, respectively; 1 e , 2 e , 1 * , and 2 * represent the thresholds of the angular velocity observation deviation and total disturbance estimation in the switching transition interval, respectively.Among them, the step function curves of F and J are depicted in Figure 7-Figure 8. Based on Equation (12), it is evident that the state error feedback control law of SADRC linearly combines LSEF and NLSEF through the weight factor K .The magnitude of K dynamically adjusts based on the system state represented by F and J , determining the control proportion of LADRC and NLADRC and facilitating the smooth switching control of SADRC.As the angular velocity of the system rapidly increases, the value of * m Z rises swiftly, causing an escalation in the values of s e and 2 Z .Consequently, F and J values gradually decrease, and K noticeably diminishes, leading to an increased proportion of LADRC action.With the adjustment of the controller, s e gradually decreases, F increases, and K gradually increases.At this point, the proportion of NLADRC action increases.When both s e and 2 Z are below the lower threshold of the switching criteria, SADRC completely switches to NLADRC, providing better dynamic response and tracking accuracy.In the event of a substantial disturbance exceeding the threshold, F and J values decrease rapidly, causing an accelerated increase in the proportion of LADRC output.SADRC switches to LADRC mode, enhancing the range of disturbance suppression and providing stronger robustness and anti-interference capabilities.
To ensure effective control performance, this paper sets 1 e and 2 e to 1 and 1.2, respectively.The value of this parameter is contingent on the characteristics of the nonlinear function

System simulation and analysis
To validate the proposed SADRC method, a simulation model of the PMSM speed servo system, as depicted in Figure 9, was constructed in MATLAB/Simulink, utilizing parameters from Table 1 for the PMSM and inverter.Subsequently, the control performance of SADRC with LADRC, NLADRC, and PI control was evaluated through comparative simulations.To ensure a fair comparison of algorithm performance under identical conditions, the controller parameters for SADRC, LADRC, and NLADRC are set as follows: r=1*10

Start-up and steady-state performance
Figure 10 shows the speed tracking, A-phase current, and shaft current waveforms of SADRC at a given speed of 3000 r/min and no-load start in 0.1 s.It is evident that SADRC swiftly follows the transition speed set in the TD link without exhibiting speed overshoot or over-current conditions.This substantiates that the proposed SADRC strategy exhibits commendable startup performance as a PMSM speed controller.Figure 11 presents the speed waveforms for all four controllers, during a no-load startup at a given speed of 3000 r/min.It is apparent that, owing to the TD link in the ADRC, the startup process is smooth without any speed overshoot, while PI control exhibits a noticeable overshoot, measuring at 3.1%.Once the system reaches a steady state, all methods adeptly track the predetermined speed.The adjustment time and steady-state error are detailed in Table 2.
In summary, the ADRC method outperforms PI control during the startup phase.NLADRC proves more efficient than LADRC due to its incorporation of nonlinear functions.The proposed SADRC exhibits control performance closely aligned with NLADRC, inheriting its dynamic response advantages of rapid speed and high control accuracy.
value of r mitigates the speed overshoot of the system, while a larger value allows * m Z to track * ref Z more swiftly.Hence, we can gradually adjust from a smaller to a larger value and consider selecting a larger r.

Figure 2 . 2 .
Figure 2. Characteristic curve of ( ) i e O function.3.2.Expanded state observer Extended state observer (ESO) constitutes the central component of SADRC.The motion equations of PMSM are organized into the following form: * * * m e 0 e L 0 e , / ( ) bT f b T f f T J b b T d Z * m e bT * e

EZ. 2 E , 1 D , 2 Dfunction is illustrated in Figure 2 .
The parameters to be adjusted in NLESO are NL1 E , NL , and G .Among them, G is generally chosen as [0.01, 0.1].To facilitate the parameter setting of NL1 E and NL 2 E , the fal function is transformed as fal( , , ) [fal( , , ) / ] It is clear that as D decreases, the nonlinearity of the ( ) i e O function increases, leading to an amplification in gain amplitude.When the error e G ! , ( ) i e O
0 u represents the output of the control law.Substituting it into the motion equation of PMSM, the system can be simplified into a standard single-integral series structure, as follows m Z .The parameters to be adjusted in NLSEF are NLp k , 3 D , and G .The physical meaning of NLp k is the same as Lp k .During tuning, NLp k can be directly taken as Lp k or finely adjusted in its vicinity.The tuning rules for 3

O
, the gain is small.* is adjusted based on the system state variable requirements.Following parameter testing, this paper designates 1 * and 2 * as 0.6 times and 0.7 times rated torque.

Figure 10 .
Figure 10.Startup performance based on the SADRC.

Figure 12 .
Figure 12.Dynamic performance of four control methods at 1000 r/min.
D is the exponential power; G is the linear interval.