Integrated position tracking control method based on SMF-LADRC

Abstract

The invention discloses an integrated position tracking control method based on SMF-LADRC, and the method aims at a flexible joint permanent magnet synchronous motor FJ-PMSM servo system, and comprises the following steps: firstly, clarifying the theoretical difference of two different single-degree-of-freedom control strategies, and comparing the stability characteristics of the two different single-degree-of-freedom control strategies; then, a two-degree-of-freedom SMF-LADRC strategy is provided, sliding mode function enhanced LESO and LESFC are combined, and the two-degree-of-freedom characteristics, parameter tuning and stability of the sliding mode function enhanced LESO and LESFC are analyzed; according to the method, the disturbance suppression performance of the system is effectively improved, the theoretical accuracy is fully verified, and the dynamic response capability of the flexible joint permanent magnet synchronous motor system is remarkably enhanced.

Description

Technical Field

[0001] This invention relates to the field of flexible joint permanent magnet synchronous motor technology, and more specifically to an integrated position tracking control method based on SMF-LADRC. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) have become a core component of high-performance industrial servo systems due to their superior power density, energy efficiency, and dynamic response capabilities. However, when these motors are used in conjunction with flexible joints, their inherent elastic characteristics can trigger complex resonant dynamics and nonlinear disturbances, thereby severely reducing positioning accuracy.

[0003] Traditional PMSM servo systems employ a cascaded control structure based on a proportional-integral-derivative (PID) controller, comprising three loops: position, velocity, and current. In this structure, the dynamic performance of the inner loop controller significantly influences the response limit of the outer loop controller, leading to a relative decrease in the importance of velocity control accuracy compared to position control accuracy during short-distance point-to-point motion. To address this issue, researchers have proposed an integrated position tracking control strategy. Specifically, by integrating the velocity and position controllers into a single unit, a dual-closed-loop PMSM servo system is constructed, achieving both high-precision control and rapid dynamic response.

[0004] Proportional-Integral (PID) control is a classic control strategy widely used in industrial process control, motor drives, and robotics. Its core idea is to achieve a rapid response to system errors and eliminate steady-state errors through the synergistic action of proportional (P) and integral (I) inputs. However, this control method also has inherent drawbacks, including integral overshoot, insufficient adaptability to time-varying / nonlinear systems, and reliance on experience for parameter tuning. PID control still has significant limitations in terms of dynamic performance, adaptability to complex systems, and anti-interference capability.

[0005] To address the limitations of traditional PID control strategies, advanced control strategies based on the concept of Active Disturbance Rejection (ADRC) have emerged. The core idea of ​​this strategy is to estimate the system state variables and total internal and external disturbances in real time using Error State Optimization (ESO) technology. This method overcomes the limitations of PID control's reliance on models and disturbances, exhibiting stronger robustness and dynamic performance in the control of nonlinear and time-varying systems. However, due to the complexity of ADRC as a nonlinear control method and its difficulty in practical engineering applications, Linear Error State Optimization (LESO) and Linear Error State Feedback Control (LESFC) laws were proposed, leading to the development of the Linear Active Disturbance Rejection (LADRC) method. Although LESO is widely used in engineering practice due to its simple structure and ease of implementation, its inherent characteristics still have the following limitations:

[0006] 1) The design of LESO is based on the nominal linear model of the original system. If the original system exhibits strong nonlinearity or there is a significant mismatch between the model parameters and the actual values, the assumption of linear extended state will fail, leading to a significant increase in the disturbance estimation error or even the loss of the compensation effect.

[0007] 2) The perturbation estimation performance of LESO is determined by the observer gain, which is essentially a low-pass filtering process. For high-frequency perturbations (such as mechanical vibration and sensor noise), the estimation lag of LESO will be amplified, making it unable to effectively track rapidly changing perturbations.

[0008] 3) The gain of LESO needs to be designed through pole placement (e.g., placing the observation poles in the left half-plane). However, the gain value of LESO directly affects the estimation accuracy and response speed.

[0009] To improve the control performance of LADRC, researchers have integrated techniques such as nonlinear extension, adaptive gain adjustment, sliding mode integration, multivariable processing, delay compensation, and data-driven adaptation. However, these improved LADRC methods have not effectively enhanced the anti-interference capability of LESO. Therefore, optimizing the LESO structure to strengthen the anti-interference performance of LADRC has become a key strategy. To this end, high-precision observers such as phase-locked loop observers (PLLO), generalized proportional-integral observers (GPIO), sliding mode observers (SMO), and cascaded LESOs have been introduced into the LADRC framework, thereby significantly improving the system's control performance.

[0010] In summary, developing an improved LADRC position controller with anti-interference capabilities and two degrees of freedom is crucial for constructing a PMSM servo system with fast dynamic response and high motion accuracy. To improve the motion accuracy, dynamic performance, and anti-interference capability of the FJ-PMSM servo system, this invention proposes an improved integrated speed-position control strategy based on the LADRC method combined with a sliding mode compensation function. The dynamic performance of this strategy depends entirely on the phase-locked loop-free (LESFC) module, while the anti-interference performance depends entirely on the observer. By independently adjusting the dynamic performance and anti-interference performance, the problem of controller parameter tuning is effectively solved, thereby significantly improving the system's flexibility. Summary of the Invention

[0011] To address the aforementioned issues, this invention proposes a sliding mode compensation function (SMF)-integral linear extended state observer (LESO) active disturbance suppression (ADR) position-velocity control strategy. This strategy not only effectively improves the disturbance suppression performance of the system, but its theoretical accuracy has also been fully verified, significantly enhancing the dynamic response capability of the flexible joint permanent magnet synchronous motor system.

[0012] The specific plan is as follows:

[0013] An integrated position tracking control method based on SMF-LADRC is proposed for the FJ-PMSM servo system of a flexible joint permanent magnet synchronous motor. The method includes the following steps: First, the theoretical differences between two different single-degree-of-freedom control strategies are clarified and their stability characteristics are compared. Then, a two-degree-of-freedom SMF-LADRC strategy is proposed, which combines sliding mode function enhanced LESO and LESFC. The two-degree-of-freedom characteristics, parameter tuning and stability are analyzed in detail.

[0014] In this embodiment, a conventional closed-loop control system is analyzed, including

[0015] A. Mathematical modeling of FJ-PMSM considering periodic and non-periodic disturbances;

[0016] Using the torque reference, the electromagnetic torque represents the reference function of the FJ-PMSM system:

[0017]

[0018] Where θ m Ω is the mechanical angle, J is the mechanical angular velocity, and T is the moment of inertia. e It is electromagnetic torque, T l B is the load torque, and B is the coefficient of viscous friction.

[0019] In engineering applications, the operation of the FJ-PMSM is affected by various disturbances, such as electrical parameter disturbances, cogging torque, magnetic flux harmonics, and sampling errors. Furthermore, cross-coupling terms exist in the current loop. Considering these disturbances, the voltage equation of the FJ-PMSM yields the following expression for the total current dynamic characteristic:

[0020]

[0021] Among them I d and I q These are the stator currents along the d-axis and q-axis, respectively. d and u q The stator voltages are for the d-axis and q-axis; furthermore, the FJ-PMSM is a surface-mount motor, therefore the stator inductance is L; r s ω represents the stator resistance. e It is the electric angular velocity, u fd and u fq These represent the voltage disturbances along the d-axis and q-axis, respectively, where k is the voltage disturbance along the d-axis. q The eigencons of the shaft system, z id and z iq It is d q The total disturbance of the shaft current loop, φ f For magnetic linkage;

[0022] When analyzing periodic disturbances in the system, due to manufacturing defects and magnetic flux saturation, the sinusoidal magnetic flux distribution in the FJ-PMSM cannot remain stable under ideal operating conditions for a long period. This leads to significant periodic harmonics in the magnetic flux waveform; the magnetic flux harmonics on the dq axis are represented as:

[0023]

[0024] Where λ d0 It is the DC component of the d-axis magnetic flux, λ d and λ q It is d q Axial flux harmonics, λ d6n and λ q6n They are d q The amplitude of the 6th magnetic flux harmonic of the axis;

[0025] In the speed control loop, the disturbance torques that need to be considered include friction torque, sampling error, unmodeled calculation error, and cogging torque. Therefore, the FJ-PMSM mechanical motion equations considering concentrated disturbances are expressed as follows:

[0026]

[0027] Where p nLet g = 1.5pnφf / J be the pole pair number, which is an inherent constant of the speed control system, and f be the total disturbance in the equation of motion.

[0028] B. Traditional LADRC strategy;

[0029] A linear tracking differentiator (LTD) is introduced to balance the response rate and overshoot; its expression is:

[0030]

[0031] Where θ r It is the reference position signal, and u1 is the smoothed θ. r r2 is the derivative of r1, r LTD It is a smoothing factor;

[0032] Since it is equivalent to a second-order integral element when the observer achieves precise observation, a proportional-derivative (PD) controller is applied to achieve stable control; to avoid θ r Oscillations caused by rapid changes, -k d Used to replace k in PD controller d ;

[0033] Treating the total disturbance as an extended state variable, the tracking error dynamics can be expressed in the form of an extended state equation:

[0034]

[0035] Among them, h e Represents the total disturbance d e The differential value;

[0036] Design an ESO based on error-based single-degree-of-freedom ADRC, such as:

[0037]

[0038] Where the symbol "^" represents the estimated value of the corresponding state variable, and l1 and l2 are the ESO gains based on the error; the bandwidth parameterization method of the ESO is as follows:

[0039]

[0040] Wherein, ω0 is considered as the bandwidth of ESO;

[0041] Assume that ESO can accurately and timely estimate the total disturbance d. e Therefore, the control law for a single-degree-of-freedom ADRC system is designed as follows:

[0042]

[0043] Where k0 is the proportional gain of the control law, which can be regarded as the bandwidth of the control law.

[0044] Substituting (9) into (6), we obtain the expected error in the closed-loop dynamics.

[0045]

[0046] C. Feedback-based control principles;

[0047] Wherein, the q-axis reference current is defined as I qre The q-axis current tracking error is constructed as e r =I qre -I q ;

[0048]

[0049] According to the feedback control law, we get

[0050]

[0051] Then, substituting (12) into (11), the control law is implemented using the estimated disturbance, and its expression is:

[0052]

[0053] Accurately determine the type of interference f in the current loop under the complex operating conditions of flexible joint motors. q Significant challenges exist. It should be noted that, in most cases, both periodic and non-periodic disturbances can be considered as uncertainties. The total disturbance must satisfy the following conditions: it must be differentiable and its derivative must be bounded.

[0054] In this embodiment, the limitations of traditional control strategies include:

[0055] A. Cumulative linear observer error

[0056] State observation error equation based on LESO

[0057]

[0058] Where e p It is the position observation error, e s It is the velocity observation error, e ob It is a disturbance observation error.

[0059] Equation (14) shows that, within the LESO framework, the presence of z1 observation error leads to e p =0; when l1=0 and l2=0, l 1ep and l 2es The items are passed to e respectively s and e obThis leads to observation errors in z2 and z3; this characteristic, known as static observation error accumulation, stems from LESO's failure to follow the pure differential relationship between its observation errors. Therefore, LESO cannot be classified as a purely differential observer, and static error accumulation is the root cause of inaccurate perturbation observations.

[0060] To address the insufficient time-varying perturbation observation capability, we first focus on the transfer function Gef-L, which describes the accuracy of LESO perturbation observations. This transfer function is derived from (14):

[0061]

[0062] Four typical time-varying disturbance terms, f1(t) = t, f2(t) = t2, f3(t) = t3 and f4(t) = sin(3πt), were selected for linearization analysis; Figure 2 Bode plots of their observation errors are given.

[0063] like Figure 2 As shown, the traditional LESO method struggles to achieve error-free static observation of time-varying disturbances. Because the observation process is affected by external disturbances, the observation error gradually increases over time. Although increasing the observer bandwidth ω0 can reduce the system observation error under normal conditions, this negatively impacts the system's dynamic performance. Therefore, it can be inferred that the traditional LADRC method is ineffective in suppressing complex time-varying disturbances. Thus, an improved observer structure needs to be designed to improve disturbance observation accuracy, thereby achieving better system control performance.

[0064] B. Coupling of dynamic performance and anti-interference performance

[0065] Two single-degree-of-freedom LADRC configurations are provided: Structure 1 and Structure 2. Structure 1 is a cascaded control structure, and Structure 2 is a cascaded control structure with feedforward.

[0066] The single-degree-of-freedom LADRC configuration of structure 1 is selected as the optimal velocity control strategy;

[0067] The single-degree-of-freedom automatic adjustment feedback controller (ADRC) can effectively suppress near-zero frequency and high-frequency signals, but its attenuation effect on low-frequency and medium-frequency signals is weak. This indicates that the linear error compensation mechanism (LESO) in the controller fails to fully compensate for the uncertain periodic disturbances in the speed loop, especially when the disturbance frequency is in the low-frequency or medium-frequency range. According to formula (10), for the tracking dynamic characteristics described in formula (6), the speed tracking error cannot be kept at the zero point. In addition, in the mechanical dynamic characteristics of the permanent magnet motor described in formula (4), there are uncertain fluctuations in the speed, which will reduce the speed performance of the system in steady state.

[0068] C. SMF Scheme Structure Description

[0069] To address the problem of observation error accumulation in the system, a Luenberger observer with a pure integral structure is established; this design is very similar to the traditional linear LESO, and its expression is:

[0070]

[0071] Where the symbol “^” represents the observed value, E1 represents the position error, and E2 represents the rotational speed observation error;

[0072] To improve the observation accuracy of the system, a compensation function F is introduced, resulting in a new observer structure expression:

[0073]

[0074] Where F needs to be close to the approximate f q ;

[0075] Choose a low-pass filter to establish f q The relationship between F and F is expressed by the following expression:

[0076]

[0077] Where k c It is the cutoff frequency of the LPF;

[0078] Combining (25) and (26), the expression for CFO is:

[0079]

[0080] Although the accuracy of CFO observations has reached a satisfactory level, f q The approximate accuracy to F is still determined by k. c Decision. When k c When the value increases, f q The approximation will be more accurate. However, as the cutoff bandwidth of the filter, k... c It cannot exceed specific operational limits. Furthermore, linear filters have insufficient accuracy in approximating nonlinear functions, while nonlinear filters lack deterministic transfer functions to construct specific observation expressions. Therefore, incorporating nonlinear functions into the compensation structure is crucial for improving the observation accuracy of nonlinear perturbations.

[0081] To improve the observation accuracy of nonlinear disturbances, a sliding mode compensation function gsmo is added to the compensation structure. The design of this function is based on the hyperbolic nonlinear fast terminal sliding mode surface HNFTSMS and the fast terminal arrival law.

[0082] let

[0083]

[0084] The sliding surface is designed as follows:

[0085]

[0086] Where a is the sliding surface; β k and c k It is a constant;

[0087] To reduce system jitter and ensure rapid system convergence within a finite time, the following fast terminal arrival rule is adopted:

[0088]

[0089] Where β1>0, β2>0, and 0<β3<1;

[0090] Establish the Lyapunov function V(a) = 0.5a 2 and its derivative is

[0091]

[0092] According to (19), the system reaches the SMF in a finite amount of time and remains around it;

[0093] Therefore, the proposed SMF-LESO is

[0094]

[0095] Designing LESFC with a PD Controller

[0096]

[0097] By integrating formulas (32) and (33), an enhanced LADRC strategy is derived and combined with SMF.

[0098] In this embodiment, parameter tuning and stability analysis include:

[0099] A. Parameter adjustment of single-degree-of-freedom ADRC

[0100] The single-degree-of-freedom active disturbance suppression control structure contains two basic parameters: the proportional gain of the control law k0 and the ESO bandwidth ω0. According to ADRC structure theory, both parameters must be greater than zero simultaneously to ensure that all poles of the closed-loop transfer functions G1(s) and G2(s) are located in the left half-plane of the s-domain, while also guaranteeing the discrete transfer function G r1 (s) and G r2 All poles of (s) lie inside the unit circle; based on this principle, the parameter tuning design for k0 and ω0 is expressed as follows:

[0101] Regarding the design of k0, it controls the dynamic response performance and serves as the performance standard for selecting the k0 parameter;

[0102] Regarding the design of ω0, the parameters are set within the range of (5-10)k0. This configuration is adopted because the closed-loop transfer functions G1(s) and G2(s) have three poles: one at -k0 and the other two identical poles at -ω0. When ω0 is at least five times k0, the pole at -k0 becomes the key pole that dominates the dynamic characteristics of the system, which greatly simplifies the dynamic analysis. In this case, the expected dynamic response can be achieved by adjusting k0 alone.

[0103] Therefore, ω0 mainly controls the interference suppression performance and serves as the performance benchmark for selection;

[0104] Furthermore, k0 and ω0 cannot be too large, as this will amplify high-frequency noise in the velocity loop and affect system stability. Therefore, the selection of k0 and ω0 must strike a balance between dynamic response performance and system stability.

[0105] B. Stability analysis of SMF-LESO

[0106] To facilitate the stability analysis process, the reference pole placement method is used to design parameters L1, L2, and K. C as follows:

[0107]

[0108] Among them, k0>0 is the only parameter that needs to be adjusted;

[0109] Let e(θ) m ω m f q )=(θ m ω m f q )-(m, m, q), and let δ(θ) m ω m f q )=e(θ m ω m f q If k = 0, then the observation error expression of SMF-LESO is:

[0110]

[0111] where δ = [δ1, δ2, δ3]T, and Let H represent the differential observation of the total perturbation; since matrix H is Hurwitz stable, there exists a positive definite Hermitian matrix B that satisfies the following condition:

[0112]

[0113] The Lyapunov function is expressed as V = δTBδ; combining (35) and (36), the resulting expression is:

[0114]

[0115] Since the function is globally Lipschitz continuous, there exists a constant q such that the following conclusion holds:

[0116]

[0117] Based on (38), the following equation is established:

[0118]

[0119] Furthermore, when k0≥1, the following will be constructed:

[0120]

[0121] Further derivation shows

[0122]

[0123] Combinations (37) and (41)

[0124]

[0125] When k0>||BNq|| 2 When +1, therefore,

[0126]

[0127] According to Lyapunov's asymptotic stability theory, the proposed SMF-LESO method has achieved convergence; from formula (43), it can be seen that when SMF-LESO converges, the limit value limt→∞||Ae(θ) m ω m f q Further analysis shows that when K ≈ 0; P =K 21 k d When =2k1 and k1>0, matrix A will become a Hurwitz matrix; according to Lyapunov stability theory, the observer component and LESFC component of the SMF-LESO method remain stable; therefore, the closed-loop system constructed using the SMF-LESO method has asymptotic stability, which shows that the method has good practical stability in engineering applications.

[0128] C. Interference Observer Design

[0129] Timely and accurate disturbance observations are crucial for effectively suppressing total disturbances, directly reflecting a key performance indicator of the system's disturbance immunity. Therefore, it is necessary to analyze the observation accuracy of SMF-LESO under different orders of disturbances and compare it with LESO to evaluate its performance; the transfer function G, which characterizes the relationship between the total disturbance observed by SMF-LESO and the actual total disturbance, is also important. Z (s) is represented as:

[0130]

[0131] According to (44), the transfer function G describing the accuracy of SMF-LESO perturbation observations is... ef (s) is:

[0132]

[0133] To maintain the generality of the analysis, five typical perturbation functions are selected: f(t) = I eat f(t) = I t f(t) = I t2 and f(t) = I t3 , where t is the time variable, I represents the disturbance amplitude, and a is a positive real number;

[0134] When the observation function is f(t) = Idiet and f(t) = Iunit, both LESO and SMF-LESO can achieve accurate observations; however, when f(t) = Idiet... t At that time, the observation error of LESO remained at the level of 3I / k0; even by increasing the value of ω0 to reduce the error, it could not be completely eliminated; in addition, when the LESO observation function is I t 2 and I t 3 At that time, its observation error always exists and shows an increasing trend over time; compared with LESO, SMF-LESO has lower observation error in I. t and I t 2 It can achieve accurate measurement and there is no steady-state observation error.

[0135] The beneficial effects of this invention are:

[0136] 1. After comprehensively comparing the two newly proposed single-degree-of-freedom ADRC schemes from the perspectives of dynamic response and anti-interference performance, a single-degree-of-freedom ADRC scheme that is more suitable for speed control was selected.

[0137] 2. A direct analysis method is proposed to prove the stability and two-degree-of-freedom characteristics of the SMF-LADRC method. The perturbation observation capability of the proposed SMF-LESO method is studied and compared with the LESO method. The results show that it has a significant advantage in perturbation observation accuracy.

[0138] 3. Based on the proposed SMF-LADRC method, an integrated position control strategy was developed. This strategy significantly improves the anti-interference performance of the FJ-PMSM servo system, achieving high-precision position control and fast dynamic response. Attached Figure Description

[0139] Figure 1 This is a structural diagram of an integrated position tracking control strategy based on the second-order LADRC method.

[0140] Figure 2 This is a graph showing the observation error of LESO perturbation.

[0141] Figure 3 These are block diagrams of two different SDOF ADRC loops, where (a) is structure 1 and (b) is structure 2.

[0142] Figure 4 This is the Bode plot of the speed loop of a single-degree-of-freedom automatic adjustment feedback control system, where (a)G p (a)Bode plot of G(s), (b)Bode plot of G(s), (c)G f Bode plot of (s).

[0143] Figure 5 This is a structural diagram of a position-velocity integral controller based on the SMF-LESO method.

[0144] Figure 6 It is a block diagram of an active disturbance suppression integrated control structure that combines a sliding mode function compensator and an extended state observer.

[0145] Figure 7 The figure shows the experimental results of the step response, where (a) is the position, (b) is the velocity, (c) is the q-axis current, and (d) is the d-axis current.

[0146] Figure 8 The figure shows the results of the sinusoidal wave tracking experiment. (a) Position, (b) Velocity, (c) q-axis current, (d) d-axis current, (e) SMF-LADRC tracking error, and (f) LADRC tracking error.

[0147] Figure 9 These are load disturbance test diagrams. Among them, (a)(c) are SMF-LESO, and (b)(d) are CLESO.

[0148] Figure 10This is a bandwidth characteristic experimental diagram. Where, (a)ω c The change of (k0), (b)ω n The change of (k1). Detailed Implementation

[0149] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0150] This invention provides an integrated position tracking control method based on SMF-LADRC for a flexible joint permanent magnet synchronous motor FJ-PMSM servo system. The method includes the following steps: First, it clarifies the theoretical differences between two different single-degree-of-freedom control strategies and compares their stability characteristics. Then, it proposes a two-degree-of-freedom SMF-LADRC strategy that combines sliding mode function enhanced LESO with LESFC, and focuses on analyzing its two-degree-of-freedom characteristics, parameter tuning, and stability.

[0151] In this embodiment, a conventional closed-loop control system is analyzed, including

[0152] A. Mathematical modeling of FJ-PMSM considering periodic and non-periodic disturbances;

[0153] Using the torque reference, the electromagnetic torque represents the reference function of the FJ-PMSM system:

[0154]

[0155] Where θ m Ω is the mechanical angle, J is the mechanical angular velocity, and T is the moment of inertia. e It is electromagnetic torque, T l B is the load torque, and B is the coefficient of viscous friction.

[0156] In engineering applications, the operation of the FJ-PMSM is affected by various disturbances, such as electrical parameter disturbances, cogging torque, magnetic flux harmonics, and sampling errors. Furthermore, cross-coupling terms exist in the current loop. Considering these disturbances, the voltage equation of the FJ-PMSM yields the following expression for the total current dynamic characteristic:

[0157]

[0158] Among them I d and I q These are the stator currents along the d-axis and q-axis, respectively. d and u q The stator voltages are for the d-axis and q-axis; furthermore, the FJ-PMSM is a surface-mount motor, therefore the stator inductance is L; r s ω represents the stator resistance. eIt is the electric angular velocity, u fd and u fq These represent the voltage disturbances along the d-axis and q-axis, respectively, where k is the voltage disturbance along the d-axis. q The eigencons of the shaft system, z id and z iq It is d q The total disturbance of the shaft current loop, φ f For magnetic linkage;

[0159] When analyzing periodic disturbances in the system, due to manufacturing defects and magnetic flux saturation, the sinusoidal magnetic flux distribution in the FJ-PMSM cannot remain stable under ideal operating conditions for a long period. This leads to significant periodic harmonics in the magnetic flux waveform; the magnetic flux harmonics on the dq axis are represented as:

[0160]

[0161] Where λ d0 It is the DC component of the d-axis magnetic flux, λ d and λ q It is d q Axial flux harmonics, λ d6n and λ q6n They are d q The amplitude of the 6th magnetic flux harmonic of the axis;

[0162] In the speed control loop, the disturbance torques that need to be considered include friction torque, sampling error, unmodeled calculation error, and cogging torque. Therefore, the FJ-PMSM mechanical motion equations considering concentrated disturbances are expressed as follows:

[0163]

[0164] Where p n Let g = 1.5pnφf / J be the pole pair number, which is an inherent constant of the speed control system, and f be the total disturbance in the equation of motion.

[0165] B. Traditional LADRC strategy;

[0166] A linear tracking differentiator (LTD) is introduced to balance the response rate and overshoot; its expression is:

[0167]

[0168] Where θ r It is the reference position signal, and u1 is the smoothed θ. r r2 is the derivative of r1, r LTD It is a smoothing factor;

[0169] Since it is equivalent to a second-order integral element when the observer achieves precise observation, a proportional-derivative (PD) controller is applied to achieve stable control; to avoid θr Oscillations caused by rapid changes, -k d Used to replace k in PD controller d ;

[0170] Treating the total disturbance as an extended state variable, the tracking error dynamics can be expressed in the form of an extended state equation:

[0171]

[0172] Among them, h e Represents the total disturbance d e The differential value;

[0173] Design an ESO based on error-based single-degree-of-freedom ADRC, such as:

[0174]

[0175] Where the symbol "^" represents the estimated value of the corresponding state variable, and l1 and l2 are the ESO gains based on the error; the bandwidth parameterization method of the ESO is as follows:

[0176]

[0177] Wherein, ω0 is considered as the bandwidth of ESO;

[0178] Assume that ESO can accurately and timely estimate the total disturbance d. e Therefore, the control law for a single-degree-of-freedom ADRC system is designed as follows:

[0179]

[0180] Where k0 is the proportional gain of the control law, which can be regarded as the bandwidth of the control law.

[0181] Substituting (9) into (6), we obtain the expected error in the closed-loop dynamics.

[0182]

[0183] C. Feedback-based control principles;

[0184] Wherein, the q-axis reference current is defined as I qre The q-axis current tracking error is constructed as e r =I qre -I q ;

[0185]

[0186] According to the feedback control law, we get

[0187]

[0188] Then, substituting (12) into (11), the control law is implemented using the estimated disturbance, and its expression is:

[0189]

[0190] Accurately determine the type of interference f in the current loop under the complex operating conditions of flexible joint motors. q Significant challenges exist. It should be noted that, in most cases, both periodic and non-periodic disturbances can be considered as uncertainties. The total disturbance must satisfy the following conditions: it is differentiable and its derivative is bounded. The integral position tracking control strategy structure based on the second-order LADRC method is as follows: Figure 1 As shown.

[0191] In this embodiment, the limitations of traditional control strategies include:

[0192] A. Cumulative linear observer error

[0193] State observation error equation based on LESO

[0194]

[0195] Where e p It is the position observation error, e s It is the velocity observation error, e ob It is a disturbance observation error.

[0196] Equation (14) shows that, within the LESO framework, the presence of z1 observation error leads to e p =0; when l1=0 and l2=0, l 1ep and l 2es The items are passed to e respectively s and e ob This leads to observation errors in z2 and z3; this characteristic, known as static observation error accumulation, stems from LESO's failure to follow the pure differential relationship between its observation errors. Therefore, LESO cannot be classified as a purely differential observer, and static error accumulation is the root cause of inaccurate perturbation observations.

[0197] To address the insufficient time-varying perturbation observation capability, we first focus on the transfer function Gef-L, which describes the accuracy of LESO perturbation observations. This transfer function is derived from (14):

[0198]

[0199] Four typical time-varying disturbance terms, f1(t) = t, f2(t) = t2, f3(t) = t3 and f4(t) = sin(3πt), were selected for linearization analysis; Figure 2Bode plots of their observation errors are given.

[0200] like Figure 2 As shown, the traditional LESO method struggles to achieve error-free static observation of time-varying disturbances. Because the observation process is affected by external disturbances, the observation error gradually increases over time. Although increasing the observer bandwidth ω0 can reduce the system observation error under normal conditions, this negatively impacts the system's dynamic performance. Therefore, it can be inferred that the traditional LADRC method is ineffective in suppressing complex time-varying disturbances. Thus, an improved observer structure needs to be designed to improve disturbance observation accuracy, thereby achieving better system control performance.

[0201] B. Coupling of dynamic performance and anti-interference performance

[0202] Figure 3 The velocity loop block diagrams of different single-degree-of-freedom ADRCs are shown to intuitively illustrate the differences between Structure 1 and Structure 2.

[0203] Two single-DOF LADRC configurations are provided: Structure 1 and Structure 2.

[0204] Depend on Figure 3 (a) It can be seen that the transfer function of structure 1 is derived as follows:

[0205]

[0206] Similarly, from Figure 3 In (b), the error e of structure 2 is transmitted to the stator current I. s The transfer function is derived as follows:

[0207]

[0208] For structure 1, the speed loop extends from the speed error e to the output speed ω. m The open-loop transfer function is derived as follows:

[0209]

[0210] Similarly, for structure 2, the open-loop transfer function of the velocity loop is derived as follows:

[0211]

[0212] For structure 1, the velocity loop starts from the reference velocity ω ref To output speed ω m The derivation of the closed-loop transfer function is as follows:

[0213]

[0214] Similarly, for structure 2, the closed-loop transfer function of the velocity loop is derived as follows:

[0215]

[0216] For structure 1, the velocity loop is from the total disturbance f to the output velocity ω. m The perturbation transfer function is derived as follows:

[0217]

[0218] Similarly, for structure 2, the perturbation transfer function of the velocity loop is derived as follows:

[0219]

[0220] To compare and analyze the relative stability, dynamic response characteristics, and interference suppression performance of the two schemes, Bode plot analysis is required. Specifically, Figure 4 (a) Showing different G p Bode plot of (s), Figure 4 (b) presents Bode plots for different G(s) functions, while Figure 4 (c) explains the different G f Bode plot of (s). The proposed comparison scheme maintains the same parameter settings, where k0 = 50 and ω0 = 500 rad / s.

[0221] like Figure 4 As shown in (a), the LADRC system of structure 1 has a larger phase margin than the velocity loop of structure 2, indicating that the control stability of structure 1 is better. Figure 4 (b) further reveals two significant advantages of Structure 1 compared to Structure 2:

[0222] Lower resonance peak value: The resonance peak value of structure 1 is significantly lower than that of structure 2. This indicates that structure 1 can significantly mitigate the velocity overshoot of the control loop during the velocity step response.

[0223] Extended cutoff frequency: Structure 1 maintains a higher system cutoff frequency, enhancing its speed control bandwidth capability. This enables Structure 1's speed control to achieve superior dynamic response performance.

[0224] final, Figure 4 (c) This indicates that the LADRC of Structure 1 is superior to Structure 2 in suppressing low-frequency and mid-frequency signals, which verifies its stronger interference suppression capability. Overall, Structure 1 outperforms Structure 2 in three key dimensions: relative stability demonstrated by excellent phase margin, dynamic response characteristics exhibited by reduced overshoot and extended bandwidth, and interference suppression characteristics with better signal attenuation performance across all frequency bands. Therefore, this study ultimately selected the single-degree-of-freedom LADRC configuration of Structure 1 as the optimal speed control strategy;

[0225] In addition, such as Figure 4 As shown in (c), the single-degree-of-freedom automatic adjustment feedback controller ADRC can effectively suppress near-zero frequency and high frequency signals, but its attenuation effect on low frequency and medium frequency signals is weak. This indicates that the linear error compensation mechanism (LESO) in the controller fails to fully compensate for the uncertain periodic disturbances in the speed loop, especially when the disturbance frequency is in the low or medium frequency range. According to formula (10), for the tracking dynamic characteristics described in formula (6), the speed tracking error cannot be kept at the zero point. In addition, in the mechanical dynamic characteristics of the permanent magnet motor described in formula (4), there are uncertain fluctuations in the speed, which will reduce the speed performance of the system in steady state.

[0226] C. SMF Scheme Structure Description

[0227] To address the problem of observation error accumulation in the system, a Luenberger observer with a pure integral structure is established; this design is very similar to the traditional linear LESO, and its expression is:

[0228]

[0229] Where the symbol “^” represents the observed value, E1 represents the position error, and E2 represents the rotational speed observation error;

[0230] To improve the observation accuracy of the system, a compensation function F is introduced, resulting in a new observer structure expression:

[0231]

[0232] Where F needs to be close to the approximate f q ;

[0233] Choose a low-pass filter to establish f q The relationship between F and F is expressed by the following expression:

[0234]

[0235] Where k c It is the cutoff frequency of the LPF;

[0236] Combining (25) and (26), the expression for CFO is:

[0237]

[0238] Although the accuracy of CFO observations has reached a satisfactory level, f q The approximate accuracy to F is still determined by k. c Decision. When k c When the value increases, f q The approximation will be more accurate. However, as the cutoff bandwidth of the filter, k...c It cannot exceed specific operational limits. Furthermore, linear filters have insufficient accuracy in approximating nonlinear functions, while nonlinear filters lack deterministic transfer functions to construct specific observation expressions. Therefore, incorporating nonlinear functions into the compensation structure is crucial for improving the observation accuracy of nonlinear perturbations.

[0239] To improve the observation accuracy of nonlinear disturbances, a sliding mode compensation function gsmo is added to the compensation structure. The design of this function is based on the hyperbolic nonlinear fast terminal sliding mode surface HNFTSMS and the fast terminal arrival law.

[0240] let

[0241]

[0242] The sliding surface is designed as follows:

[0243]

[0244] Where a is the sliding surface; β k and c k It is a constant;

[0245] To reduce system jitter and ensure rapid system convergence within a finite time, the following fast terminal arrival rule is adopted:

[0246]

[0247] Where β1>0, β2>0, and 0<β3<1;

[0248] Establish the Lyapunov function V(a) = 0.5a 2 and its derivative is

[0249]

[0250] According to (19), the system reaches the SMF in a finite amount of time and remains around it;

[0251] Therefore, the proposed SMF-LESO is

[0252]

[0253] Designing LESFC with a PD Controller

[0254]

[0255] By integrating formulas (32) and (33), an enhanced LADRC strategy is derived and combined with SMF. A schematic diagram of the position-velocity control strategy constructed based on this method is shown below. Figure 5 As shown.

[0256] In this embodiment, parameter tuning and stability analysis include:

[0257] A. Parameter adjustment of single-degree-of-freedom ADRC

[0258] The single-degree-of-freedom active disturbance suppression control structure contains two basic parameters: the proportional gain of the control law k0 and the ESO bandwidth ω0. According to ADRC structure theory, both parameters must be greater than zero simultaneously to ensure that all poles of the closed-loop transfer functions G1(s) and G2(s) are located in the left half-plane of the s-domain, while also guaranteeing the discrete transfer function G r1 (s) and G r2 All poles of (s) lie inside the unit circle; based on this principle, the parameter tuning design for k0 and ω0 is expressed as follows:

[0259] Regarding the design of k0, it controls the dynamic response performance and serves as the performance standard for selecting the k0 parameter;

[0260] Regarding the design of ω0, the parameters are set within the range of (5-10)k0. This configuration is adopted because the closed-loop transfer functions G1(s) and G2(s) have three poles: one at -k0 and the other two identical poles at -ω0. When ω0 is at least five times k0, the pole at -k0 becomes the key pole that dominates the dynamic characteristics of the system, which greatly simplifies the dynamic analysis. In this case, the expected dynamic response can be achieved by adjusting k0 alone.

[0261] Therefore, ω0 mainly controls the interference suppression performance and serves as the performance benchmark for selection;

[0262] Furthermore, k0 and ω0 cannot be too large, as this will amplify high-frequency noise in the velocity loop and affect system stability. Therefore, the selection of k0 and ω0 must strike a balance between dynamic response performance and system stability.

[0263] B. Stability analysis of SMF-LESO

[0264] To facilitate the stability analysis process, the reference pole placement method is used to design parameters L1, L2, and K. C as follows:

[0265]

[0266] Among them, k0>0 is the only parameter that needs to be adjusted;

[0267] Let e(θ) m ω m f q )=(θ m ω m f q)-(m, m, q), and let δ(θ) m ω m f q )=e(θ m ω m f q If k = 0, then the observation error expression of SMF-LESO is:

[0268]

[0269] where δ = [δ1, δ2, δ3]T, and Let H represent the differential observation of the total perturbation; since matrix H is Hurwitz stable, there exists a positive definite Hermitian matrix B that satisfies the following condition:

[0270]

[0271] The Lyapunov function is expressed as V = δTBδ; combining (35) and (36), the resulting expression is:

[0272]

[0273] Since the function is globally Lipschitz continuous, there exists a constant q such that the following conclusion holds:

[0274]

[0275] Based on (38), the following equation is established:

[0276]

[0277] Furthermore, when k0≥1, the following will be constructed:

[0278]

[0279] Further derivation shows

[0280]

[0281] Combinations (37) and (41)

[0282]

[0283] When k0>||BNq|| 2 When +1, therefore,

[0284]

[0285] According to Lyapunov's asymptotic stability theory, the proposed SMF-LESO method has achieved convergence; from formula (43), it can be seen that when SMF-LESO converges, the limit value limt→∞||Ae(θ) m ω m f q Further analysis shows that when K ≈ 0; P =K 21 k d When =2k1 and k1>0, matrix A will become a Hurwitz matrix; according to Lyapunov stability theory, the observer component and LESFC component of the SMF-LESO method remain stable; therefore, the closed-loop system constructed using the SMF-LESO method has asymptotic stability, which shows that the method has good practical stability in engineering applications.

[0286] C. Interference Observer Design

[0287] Timely and accurate disturbance observations are crucial for effectively suppressing total disturbances, directly reflecting a key performance indicator of the system's disturbance immunity. Therefore, it is necessary to analyze the observation accuracy of SMF-LESO under different orders of disturbances and compare it with LESO to evaluate its performance; the transfer function G, which characterizes the relationship between the total disturbance observed by SMF-LESO and the actual total disturbance, is also important. Z (s) is represented as:

[0288]

[0289] According to (44), the transfer function G describing the accuracy of SMF-LESO perturbation observations is... ef (s) is:

[0290]

[0291] To maintain the generality of the analysis, five typical perturbation functions are selected: f(t) = I eat f(t) = I t f(t) = I t2 and f(t) = I t3 , where t is the time variable, I represents the disturbance amplitude, and a is a positive real number;

[0292] When the observation function is f(t) = Idiet and f(t) = Iunit, both LESO and SMF-LESO can achieve accurate observations; however, when f(t) = Idiet... t At that time, the observation error of LESO remained at the level of 3I / k0; even by increasing the value of ω0 to reduce the error, it could not be completely eliminated; in addition, when the LESO observation function is I t 2 and It 3 At that time, its observation error always exists and shows an increasing trend over time; compared with LESO, SMF-LESO has lower observation error in I. t and I t 2 It can achieve accurate measurement and there is no steady-state observation error.

[0293] In this embodiment, the experiments and analyses are as follows:

[0294] A. Experimental Setup

[0295] Table 1 details the parameters of the FJ-PMSM motor. The experimental platform uses an RTU-BOX-206 controller for PWM waveform modulation and ADC signal sampling. This controller boasts powerful hardware processing capabilities and flexible software configuration, making it suitable for various application scenarios. The motor driver employs a three-phase integrated half-bridge inverter topology, containing six power semiconductor switches, powered by an adjustable 310V DC main power supply and a 12-24V auxiliary DC braking voltage source. The control system utilizes SVPWM technology in conjunction with i... d =0A strategy, where the current loop is adjusted by a PI controller. d and i q The position and velocity loops are controlled by an integrated SMF-LADRC strategy. The overall control principle diagram is shown below. Figure 6 As shown. The experiment was conducted under a 10A DC bus current limit and a 10kHz sampling frequency.

[0296] It is worth noting that FJ-PMSM systems typically integrate harmonic reducers to achieve high torque output and precise motion control. This type of reducer amplifies the motor torque by reducing speed at the output shaft; the reduction ratio used in this study is 100:1 (R = 100). High-ratio harmonic reducers have three core advantages:

[0297] Torque amplification: Exponential torque enhancement (theoretical gain = R). Load capacity: Improved load-bearing capacity. Dynamic compliance: Vibration absorption and structural deformation adaptation.

[0298] Crucially, the inherent inertia of harmonic reducers necessitates a large starting torque even under no-load conditions. Therefore, stable operation—manifested as regulation of the q-axis current and balance of the three-phase currents—requires the FJ-PMSM to exceed a minimum speed threshold.

[0299] Table 1 PMSM System Parameters

[0300] parameter symbol value Extreme logarithm <![CDATA[n p ]]> 10 d-axis inductance <![CDATA[L d ]]> 0.18mH q-axis inductor <![CDATA[L q ]]> 0.18mH Stator resistance <![CDATA[R s ]]> 0.19Ω Permanent magnet flux linkage <![CDATA[Ψ f ]]> 5.3mWb DC link voltage <![CDATA[U dc ]]> 48V Rated speed N 3500rpm Rated power <![CDATA[P N ]]> 400W inertia J 3.76×10⁻⁵ kgm²

[0301] B. Step-reaction experiment

[0302] Two different methods were used to compare and analyze position and velocity control: a traditional second-order LADRC strategy and an SMF-LADR control strategy. In the current control loop, the PI parameter is set to k. p =0.79 and k i =10.88.

[0303] For both control methods, the parameter is set to r. LTD =30, gain b0=59.4, and the observer bandwidth ω0 is configured to 300 rad / s. The only difference is in the configuration of the SMF parameters, where β1=0.03, β2=0.02, β3=0.0001, c1=1.79, c2=1.27 and c3=0.39.

[0304] To verify the dynamic response performance of the proposed SMF-LADRC control strategy, a position step response experiment was conducted. Reference position θ r The signal was set to a 10-radian step signal, starting at 0.6 seconds, lasting for 0.6 seconds, with a period of 1.2 seconds, and was tested under no additional load conditions. The experimental results are as follows: Figure 7 As shown.

[0305] Comparative analysis of position, velocity, and q-axis current responses under different control methods reveals that the traditional second-order LADRC strategy exhibits significantly greater velocity jitter during step signal tracking, and its harmonic distortion in the control current is higher than that of the SMF-LADRC strategy. This indicates that the secondary adjustment process of the SMF-LADRC strategy has a negligible impact on the system's stable operation. Therefore, although both control strategies can achieve step position tracking, the SMF-LADRC strategy provides a faster and more stable control response.

[0306] C. Sine wave tracking experiment

[0307] The FJ-PMSM servo system tracks time-varying θ. r Time-varying current and speed fluctuations can occur, leading to frictional torque and cross-coupling interference. These phenomena can cause motor overheating and premature component aging. Therefore, time-varying trajectory tracking experiments are crucial for evaluating the system's effectiveness in suppressing nonlinear time-varying disturbances. Furthermore, the tracking accuracy θ... r This directly reflects the servo system's ability to suppress interference under dynamic operating conditions.

[0308] sinusoidal reference position signal θ r It possesses sufficiently smooth characteristics, thus eliminating the need for a linear tracking differentiator. In the sinusoidal trajectory tracking experiment, θ in LESFC... r It is used as a target tracking signal. Figure 8The experimental results shown demonstrate the system's response characteristics when tracking a sinusoidal reference position with an amplitude of 10.0 radians and a period of 0.76 seconds under no-load conditions.

[0309] To accurately evaluate θ in different systems mr To improve tracking accuracy, we define the tracking error as θ. err =θ mr -θ m .like Figure 8 As shown, the traditional PMSM servo system using the second-order LADRC method will θ err The amplitude is controlled within ±0.81 radians, equivalent to 8.1% of the reference position signal amplitude. In contrast, the FJ-PMSM servo system equipped with the SMCFO-LADRC method proposed in this study will reduce θ... err Controlled within ±0.41 radians, this is only 4.1% of the reference amplitude. The SMCFO-LADRC method improves the positioning and tracking accuracy of the PMSM servo system by 50.6%, fully demonstrating the significant advantages of this technology in suppressing time-varying interference.

[0310] D. Load disturbance test

[0311] Load disturbances place higher demands on the FJ-PMSM servo system. Evaluating the effectiveness of different control methods in reducing servo system position fluctuations under load disturbances effectively measures its anti-interference performance. This invention employs a scheme of releasing a load block from a fixed height during free fall to simulate load disturbances: First, the initial position of the FJ-PMSM is adjusted to ensure that the cantilever beam remains horizontal to the ground when the system moves to 10 degrees. Then, a 984-gram load block is released from 30 centimeters away from the robotic arm to impact the system, generating an irregular load that interacts with the FJ-PMSM servo system.

[0312] like Figure 9 As shown, under load disturbance conditions, replacing the traditional second-order LADRC algorithm with the SMF-LADRC method proposed in this invention significantly reduces the displacement fluctuation amplitude of the PMSM servo system from 0.56 radians to 0.26 radians, a reduction of 0.3 radians. This fully demonstrates that SMF-LADRC is superior to the traditional second-order LADRC algorithm in suppressing system disturbances. Specifically, this method improves the disturbance suppression performance of the FJ-PMSM servo system by 54%.

[0313] E. Bandwidth Characteristics Experiment

[0314] The two-dimensional degree-of-freedom characteristics of the proposed SMF-LADRC method were verified experimentally. At the start of the experiment, the cantilever beam was unloaded. First, the initial position of the FJ-PMSM servo system was adjusted to keep the cantilever beam parallel within a 12.5 radian range. After 0.6 seconds, a combined load disturbance generated by a freely falling 984g load block was applied to the robotic arm. The experimental results are as follows: Figure 10 As shown.

[0315] Furthermore, by analyzing and adjusting ω c (k0) and ω n The influence of (k1) on the system's step response rate and position fluctuations verifies that the proposed SMF-LADRC method has two-dimensional degrees of freedom.

[0316] Figure 10 (a) shows that, with ω c (k0) and ω n The influence of (k1) on the system's step response rate and position fluctuations: As the value of (k0) increases, the dynamic performance of the SMF-LADRC method is improved, while its anti-interference performance is unaffected by ω. c The effect of changes in (k0). Figure 10 (b) indicates that different ω n The step response curves at (k1) values ​​almost completely overlap, and the position fluctuation amplitude varies with ω. n The dynamic performance of the SMF-LADRC method depends only on the parameters of LESFC, while its anti-interference performance is entirely determined by the parameters of SMF-LESO.

[0317] Therefore, the dynamic and anti-interference performance of the SMF-LADRC method are independent of each other, making it a true 2DOF control method.

[0318] The results show that the PMSM servo system using SMCFO-LADRC achieves complete decoupling between dynamic performance and disturbance suppression, improving disturbance suppression capability by 54% compared to traditional second-order LADRC technology. In summary, the SMF-LADRC method constitutes a truly superior two-degree-of-freedom control scheme with excellent anti-interference capabilities, effectively overcoming the limitations of traditional second-order LADRC in suppressing nonlinear time-varying disturbances and solving the coupling problem between dynamic response and disturbance suppression performance.

[0319] The technical means disclosed in this invention are not limited to those disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications are also considered within the scope of protection of this invention.

Claims

1. An integrated position tracking control method based on SMF-LADRC, specifically for the FJ-PMSM servo system of a flexible joint permanent magnet synchronous motor, characterized by: The process includes the following steps: first, clarifying the theoretical differences between two different single-degree-of-freedom control strategies, and then comparing their stability characteristics; Subsequently, a two-degree-of-freedom SMF-LADRC strategy is proposed, which combines the sliding mode function enhanced LESO with LESFC, and its two-degree-of-freedom characteristics, parameter tuning and stability are analyzed.

2. The integrated position tracking control method based on SMF-LADRC according to claim 1, characterized in that: First, we analyze conventional closed-loop control systems, including A. Mathematical modeling of FJ-PMSM considering periodic and non-periodic disturbances; Using the torque reference, the electromagnetic torque represents the reference function of the FJ-PMSM system: Where θ m Ω is the mechanical angle, J is the mechanical angular velocity, and T is the moment of inertia. e It is electromagnetic torque, T l B is the load torque, and B is the coefficient of viscous friction. The voltage equations of the FJ-PMSM derive the following expression for the dynamic characteristics of the total current: Where I d and I q These are the stator currents along the d-axis and q-axis, respectively. d and u q The stator voltages are for the d-axis and q-axis; furthermore, the FJ-PMSM is a surface-mount motor, therefore the stator inductance is L; r s ω represents the stator resistance. e It is the electric angular velocity, u fd and u fq These represent the voltage disturbances along the d-axis and q-axis, respectively, where k is the voltage disturbance along the d-axis. q The eigencons of the shaft system, z id and z iq It is d q The total disturbance of the shaft current loop, φ f For magnetic linkage; Due to manufacturing defects and magnetic flux saturation, the sinusoidal magnetic flux distribution in the FJ-PMSM cannot remain stable under ideal operating conditions for a long period, which leads to significant periodic harmonics in the magnetic flux waveform; the magnetic flux harmonics on the dq axis are represented as: Where λ d0 It is the DC component of the d-axis magnetic flux, λ d and λ q It is d q Axial flux harmonics, λ d6n and λ q6n They are d q The amplitude of the 6th magnetic flux harmonic of the axis; In the speed control loop, the FJ-PMSM mechanical motion equations considering concentrated disturbances are expressed as: Where p n Let g = 1.5pnφf / J be the pole pair number, which is an inherent constant of the speed control system, and f be the total disturbance in the equation of motion. B. Traditional LADRC strategy; A linear tracking differentiator (LTD) is introduced to balance the response rate and overshoot; its expression is: Where θ r It is the reference position signal, and u1 is the smoothed θ. r r2 is the derivative of r1, r LTD It is a smoothing factor; Since it is equivalent to a second-order integral element when the observer achieves precise observation, a proportional-derivative (PD) controller is applied to achieve stable control; to avoid θ r Oscillations caused by rapid changes, -k d Used to replace k in PD controller d ; Treating the total disturbance as an extended state variable, the tracking error dynamics can be expressed in the form of an extended state equation: Among them, h e Represents the total disturbance d e The differential value; Design an error-based single-degree-of-freedom ADRC ESO as follows: Where the symbol "^" represents the estimated value of the corresponding state variable, l1 and l2 are the error-based ESO gains; the bandwidth parameterization method of ESO is as follows: Where ω0 is the bandwidth of ESO; Assume that ESO can accurately and timely estimate the total disturbance d. e A control law for a single-degree-of-freedom ADRC was designed: Where k0 is the proportional gain of the control law, which can be regarded as the bandwidth of the control law; Substituting (9) into (6), we obtain the expected error in the closed-loop dynamics: C. Feedback-based control principles; The q-axis reference current is defined as I qre The q-axis current tracking error is constructed as e r =I qre -I q ; According to the feedback control law, we get: Then, substituting (12) into (11), the control law is implemented using the estimated disturbance, and its expression is: The total disturbance must satisfy the following conditions: it is differentiable and its derivative is bounded.

3. The integrated position tracking control method based on SMF-LADRC according to claim 2, characterized in that: Secondly, the limitations of traditional control strategies include: A. Cumulative linear observer error State observation error equation based on LESO Where e p It is the position observation error, e s It is the velocity observation error, e ob It is a disturbance observation error; Equation (14) shows that, within the LESO framework, the presence of z1 observation error leads to e p =0; when l1=0 and l2=0, l 1ep and l 2es The items are passed to e respectively s and e ob This leads to observation errors in z2 and z3; To address the insufficient time-varying perturbation observation capability, we first focus on the transfer function Gef-L, which describes the accuracy of LESO perturbation observations. This transfer function is derived from (14): Four typical time-varying disturbance terms, f1(t) = t, f2(t) = t2, f3(t) = t3 and f4(t) = sin(3πt), were selected for linearization analysis; Design an improved observer structure to enhance disturbance observation accuracy, thereby achieving better system control performance; B. Coupling of dynamic performance and anti-interference performance Two single-degree-of-freedom LADRC configurations are provided: Structure 1 and Structure 2. Structure 1 is a cascaded control structure, and Structure 2 is a cascaded control structure with feedforward. The single-degree-of-freedom LADRC configuration of structure 1 was selected as the optimal velocity control strategy. The single-degree-of-freedom automatic adjustment feedback controller ADRC can effectively suppress near-zero frequency and high frequency signals, but its attenuation effect on low frequency and medium frequency signals is weak. This indicates that the linear error compensation mechanism LESO in the controller fails to fully compensate for the uncertain periodic disturbances in the speed loop, especially when the disturbance frequency is in the low or medium frequency range. According to formula (10), for the tracking dynamic characteristics described in formula (6), the speed tracking error cannot be kept at the zero point. In addition, in the mechanical dynamic characteristics of the permanent magnet motor described in formula (4), there are uncertain fluctuations in the speed, which will reduce the speed performance of the system in steady state. C. SMF Scheme Structure Description To address the problem of observation error accumulation in the system, a Luenberger observer with a pure integral structure is established; its expression is: Where the symbol "^" represents the observed value, E1 represents the position error, and E2 represents the rotational speed observation error; To improve the system's observation accuracy, a compensation function F is introduced, resulting in a new observer structure expression: Where F needs to be close to the approximate f q ; Choose a low-pass filter to establish f q The relationship between F and F is expressed by the following expression: Where k c It is the cutoff frequency of the LPF; Combining (25) and (26), the expression for CFO is: To improve the observation accuracy of nonlinear disturbances, a sliding mode compensation function gsmo is added to the compensation structure. The design of this function is based on the hyperbolic nonlinear fast terminal sliding mode surface HNFTSMS and the fast terminal arrival law. let The sliding surface is designed as follows: Where a is the sliding surface; β k and c k It is a constant; To reduce system jitter and ensure rapid system convergence within a finite time, the following fast terminal arrival rule is adopted: Where β1>0, β2>0, and 0<β3<1; Establish the Lyapunov function V(a) = 0.5a 2 Its derivative is: According to (19), the system reaches the SMF in a finite amount of time and remains around it; Therefore, the proposed SMF-LESO is Designing LESFC with a PD Controller By integrating formulas (32) and (33), an enhanced LADRC strategy is derived and combined with SMF.

4. The integrated position tracking control method based on SMF-LADRC according to claim 3, characterized in that: Finally, parameter tuning and stability analysis include: A. Parameter adjustment of single-degree-of-freedom ADRC The single-degree-of-freedom active disturbance suppression control structure contains two basic parameters: the proportional gain of the control law k0 and the ESO bandwidth ω0. According to ADRC structure theory, both parameters must be greater than zero simultaneously to ensure that all poles of the closed-loop transfer functions G1(s) and G2(s) are located in the left half-plane of the s-domain, while also guaranteeing the discrete transfer function G r1 (s) and G r2 All poles of (s) lie inside the unit circle; based on this principle, the parameter tuning design for k0 and ω0 is expressed as follows: Regarding the design of k0, it controls the dynamic response performance and serves as the performance standard for selecting the k0 parameter; Regarding the design of ω0, the parameter is set within the range of k0. This configuration is adopted because the closed-loop transfer functions G1(s) and G2(s) have three poles: one located at -k0 and the other two identical poles located at -ω0. When ω0 reaches at least five times k0, the pole located at -k0 will become the key pole that dominates the dynamic characteristics of the system. In this case, only k0 needs to be adjusted to achieve the expected dynamic response. Therefore, ω0 mainly controls the interference suppression performance and serves as the performance benchmark for selection; Furthermore, the selection of k0 and ω0 must strike a balance between dynamic response performance and system stability; B. Stability analysis of SMF-LESO To facilitate the stability analysis process, the reference pole placement method is used to design parameters L1, L2, and K. C as follows: Among them, k0>0 is the only parameter that needs to be adjusted; Let e(θ) m ω m f q )=(θ m ω m f q )-(m, m, q), and let δ(θ) m ω m f q )=e(θ m ω m f q If k = 0, then the observation error expression of SMF-LESO is: where δ = [δ1, δ2, δ3]T, and Let H represent the differential observation of the total perturbation; since matrix H is Hurwitz stable, there exists a positive definite Hermitian matrix B that satisfies the following condition: The Lyapunov function is expressed as V = δTBδ; combining (35) and (36), the expression obtained is: Since the function is globally Lipschitz continuous, there exists a constant q such that the following conclusion holds: Based on (38), the following equation is established: Furthermore, when k0≥1, the following will be constructed: Further derivation shows that: Combinations (37) and (41) When k0>||BNq|| 2 When +1, therefore, According to Lyapunov's asymptotic stability theory, the proposed SMF-LESO method has achieved convergence; from formula (43), it can be seen that when SMF-LESO converges, the limit value limt→∞||Ae(θ) m ω m f q Further analysis shows that when K ≈ 0; P =K 21 k d When =2k1 and k1>0, matrix A will become a Hurwitz matrix; according to Lyapunov stability theory, the observer component and LESFC component of the SMF-LESO method remain stable; therefore, the closed-loop system constructed using the SMF-LESO method has asymptotic stability. C. Interference Observer Design The observation accuracy of SMF-LESO under different orders of perturbation is analyzed and compared with LESO to evaluate its performance; the transfer function G, which characterizes the relationship between the total observed perturbation and the actual total perturbation of SMF-LESO, is also analyzed. Z (s) is represented as: According to (44), the transfer function G describing the accuracy of SMF-LESO perturbation observations is... ef (s) is: To maintain the generality of the analysis, five typical perturbation functions are selected: f(t) = I eat f(t) = I t f(t) = I t2 and f(t) = I t3 , where t is the time variable, I represents the disturbance amplitude, and a is a positive real number; When the observation function is f(t) = Idiet and f(t) = Iunit, both LESO and SMF-LESO can achieve accurate observations; however, when f(t) = Idiet... t At that time, the observation error of LESO remained at the level of 3I / k0; even by increasing the value of ω0 to reduce the error, it could not be completely eliminated; in addition, when the LESO observation function is I t 2 and I t 3 At that time, its observation error always exists and shows an increasing trend over time; compared with LESO, SMF-LESO has lower observation error in I. t and I t 2 It can achieve accurate measurement and there is no steady-state observation error.

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