A Fixed-Time Integral Sliding Mode Control Method for Permanent Magnet Synchronous Motors Based on Extended State Observer

Abstract

This invention belongs to the field of permanent magnet synchronous motor control technology, specifically relating to a fixed-time integral sliding mode control method for permanent magnet synchronous motors based on an extended state observer. The method includes the following steps: observing the total disturbance of the system through an extended state observer and embedding the observed total disturbance into the designed controller as a feedforward term to compensate for unknown disturbances; designing an improved integral sliding surface and using it to design a fixed-time sliding mode controller so that the system state converges within a fixed time, the integral term effectively suppresses chattering, and enhances the controller's performance; proving the closed-loop stability of the system based on Lyapunov theory; simulation experiments demonstrate that this method can rapidly compensate for disturbances, effectively suppress chattering, improve the system's tracking performance, effectively enhance the control accuracy and anti-disturbance capability of the permanent magnet synchronous motor servo system, and simultaneously ensure system robustness.

Description

Technical fields:

[0001] This invention belongs to the field of permanent magnet synchronous motor control technology, specifically relating to a fixed-time integral sliding mode control method for permanent magnet synchronous motors based on an extended state observer. Background technology:

[0002] As science and technology continue to advance to higher standards, the functions and effects that permanent magnet synchronous motors (PMSMs) need to achieve in applications are becoming increasingly complex and diverse. To achieve a finite settling time, terminal sliding mode control (TSMC) has been proposed; however, its control law carries the risk of singularities. To avoid singularity issues, non-singular terminal sliding mode control (NTSMC) has been proposed for designing speed controllers for PMSM systems.

[0003] Since traditional control methods struggle to achieve ideal control performance, ensuring fixed-time convergence of the system state while compensating for disturbances, and maximizing the response speed and control accuracy of PMSM (Permanent Magnet Synchronous Motor) systems, has become a pressing issue, leading to the development of various control methods and controller designs. For example, to address issues such as attitude tracking errors, relative position, and chattering in six-degree-of-freedom tracking systems for spacecraft, a continuous adaptive non-singular fixed-time fast terminal sliding mode control method without mass, inertia matrix, or disturbance information is proposed using fixed-time adaptive technology. To address uncertainties such as load variations, friction, and external disturbances in nonholonomic mobile robot systems, a multivariable fixed-time torque controller based on a disturbance observer is designed. To improve the tracking accuracy and optimize convergence time of robot manipulators, a fixed-time adaptive neural network controller is proposed. For uncertain, non-strict feedback nonlinear systems, a fixed-time adaptive fuzzy backstepping control scheme is designed to achieve excellent tracking performance and maintain the state within a predefined time-varying compact region during operation.

[0004] Existing technologies still have some problems, such as the difficulty in obtaining initial values ​​of state variables, chattering, and the inability to guarantee convergence time in PMSM applications. Summary of the Invention:

[0005] The purpose of this invention is to overcome the above-mentioned shortcomings of the prior art and propose a fixed-time integral sliding mode control method for permanent magnet synchronous motors based on an extended state observer. The integral term of the fixed-time sliding surface selected by the extended state observer can effectively suppress chattering, while ensuring that the system state can converge in a fixed time, thereby accelerating the system response speed and improving the system's anti-disturbance capability.

[0006] To achieve the above objectives, the present invention provides a fixed-time integral sliding mode control method for permanent magnet synchronous motors based on an extended state observer, comprising the following steps:

[0007] S1. Establish the mathematical model of the permanent magnet synchronous motor:

[0008]

[0009] Among them, i d i q These represent the currents along the d-axis and q-axis, respectively; u d u q R represents the voltage along the d-axis and q-axis, respectively; s L is the stator winding resistance; d L q The inductances along the d-axis and q-axis are respectively; ψ f The magnetic flux generated by a permanent magnet; n p ω is the number of pole pairs of the permanent magnet synchronous motor; T is the angular velocity of the motor's mechanical rotor; L Where J is the load torque; J is the moment of inertia; and B is the friction damping coefficient.

[0010] To facilitate the design of the extended state observer, equation (2) is transformed as follows:

[0011]

[0012] in, u = bi q * ;u is a virtual control input, This represents the total system disturbance.

[0013] S2. Design an extended state observer

[0014] As can be seen from equation (3), d is the lumped disturbance composed of load torque, disturbance inertia, and friction damping. After the disturbance is observed and compensated by the observer, the system can be approximated as a first-order integral system, simplifying the model and reducing the amount of computation. The total disturbance of the system is estimated as follows:

[0015] S2.1 Establish the dynamic model of the extended state observer:

[0016]

[0017] Where p1 = 2p, p2 = -p 2 p is the pole of the extended state observer; For estimating the motor speed; This is an estimate of the total system disturbance d;

[0018] S2.2 Defines the observation error of the extended state observer:

[0019]

[0020] Where η1 is the observation error of velocity and η2 is the observation error of disturbance;

[0021] Taking the derivative of equation (5), we can obtain the dynamic equation of the observation error as follows:

[0022]

[0023] S3. Design of a fixed-time integral sliding mode controller based on an extended state observer.

[0024] The total disturbance of the system is observed by an extended state observer, and the observed total disturbance is embedded into the designed controller as a feedforward term to compensate for the total disturbance, thereby eliminating the impact of the disturbance on the system; the specific process is as follows:

[0025] S3.1 Establish the velocity error state equation:

[0026] The speed tracking error e is:

[0027] e = ω * -ω (7)

[0028] Where, ω * ω represents the desired speed, and ω represents the actual output angular velocity of the motor's mechanical rotor.

[0029] Differentiating equation (7), we obtain the velocity error state equation as follows:

[0030]

[0031] in,

[0032] S3.2 Establish a fixed-time integral sliding mode controller model:

[0033] The fixed-time integral sliding surface is selected as follows:

[0034]

[0035] Where k1 and k2 are positive real numbers, 0 < α < 1, β > 1;

[0036] Taking the derivative of equation (9), we obtain the derivative of the sliding surface as follows:

[0037]

[0038] To maintain the sliding mode state on the sliding surface, a fixed-time integral sliding mode control law is designed as follows:

[0039]

[0040] Among them, k0, k3, and k4 are all positive real numbers, and k0 > 0.5 and α2 > 1;

[0041] Substituting formula (11) into formula (10), we get

[0042]

[0043] in, And |n|≤n * n * It is a positive real number;

[0044] S4. Choose the Lyapunov function and prove the system stability.

[0045] Consider a closed-loop control system consisting of a permanent magnet synchronous motor servo system, a fixed-time integral sliding mode controller, and an extended state observer. All signals are stable, and the tracking error converges to a sufficiently small region within a fixed time. Three Lyapunov functions are selected to prove that the system state approaches the sliding surface, the system state "slides" from the sliding surface to the origin, and the tracking error converges to a sufficiently small region. The smaller the region, the closer the motor speed is to the desired value, and the stronger the control performance. The specific process is as follows:

[0046] Proof: Select the first Lyapunov function

[0047]

[0048] Differentiating formula (13) yields

[0049]

[0050] According to Young's inequality, we get

[0051]

[0052]

[0053] Substituting (15) and (16) into (14) yields

[0054]

[0055] Where L1 = min{2k0-1,2k3,2k4,-1}, L1 and L2 are bounded constants because the estimation error η2 is also bounded, and the system is stable in finite time.

[0056] Next, we will further demonstrate that the error system can stabilize on the sliding surface over a fixed time:

[0057] The second Lyapunov function is selected as

[0058]

[0059] Differentiating formula (18) yields

[0060]

[0061] Where, v = 1 / 2n 2 , |n|≤n * Bounded, according to Lemma 2 of fixed time, the error system will reach the sliding surface in a fixed time, and its convergence time is: T≤1 / (αθ(1-p))+1 / (βθ(q-1)), α=k3, β=k4, p=(α1+1) / 2, q=(α2+1) / 2, 0<θ<1;

[0062] When the tracking error e stabilizes at the sliding surface, formula (9) can be written as follows:

[0063]

[0064] The third Lyapunov function is selected as

[0065]

[0066] Differentiating formula (21) yields

[0067]

[0068] Calculation shows According to Lemma 3 of the fixed time, the tracking error e will converge to zero in a fixed time.

[0069] Assuming that the tracking error dynamic equation (8) satisfies that the total disturbance is bounded and the rate of change of the disturbance is bounded, then under the proposed control strategy, the sliding mode manifold converges to the desired state in finite time. The tracking error e converges within a fixed time to |e|≤Φ=min{(Λ / k1)} 1 / α ,(Λ / k2) 1 / β},

[0070] Proof: From formula (19), we can obtain

[0071]

[0072] According to Lemma 3 of the fixed-time principle, the system state tends to the sliding surface within a fixed time interval. To ensure the fixed-time stability of the system, the following condition must be satisfied: Right now Therefore, as long as |s|>Θ1, the sliding mode variable s can reach |s|≤Θ1 in a fixed time.

[0073] Furthermore, formula (19) can also be written as

[0074]

[0075] Based on the same principle as formula (23), it is only necessary to satisfy the condition. Right now Therefore, as long as |s|>Θ2, the sliding mode variable s can reach |s|<Θ2 in a fixed time.

[0076] From formula (10) we get

[0077]

[0078] Since |s|≤Θ, formula (12) can be changed to

[0079]

[0080] Substituting (26) into (12), we get

[0081]

[0082] Therefore, as long as |e|>(Λ / k1) 1 / α If |e| ≤ Φ1, then the tracking error e can converge to |e| ≤ Φ1 within a fixed time.

[0083] Formula (27) can also be written as

[0084]

[0085] Therefore, as long as |e|>(Λ / k2) 1 / β If |e| ≤ Φ2, then the tracking error e can converge to |e| ≤ Φ2 within a fixed time.

[0086] Compared with existing technologies, this invention, targeting permanent magnet synchronous motor (PMSM) systems, utilizes the sliding mode concept, combining an extended state observer and a fixed-time integral sliding surface to design a fixed-time integral sliding mode control method for PMSMs based on an extended state observer. The extended state observer estimates the total disturbance of the system and is embedded in the sliding controller to compensate for the disturbance. The integral term contained in the fixed-time integral sliding surface can effectively suppress chattering and enhance the performance of the controller. The composite controller ensures that the system state can accurately and quickly track the reference signal, achieving rapid convergence. This method can quickly compensate for disturbances, improve the robustness of the system, effectively suppress chattering, improve the tracking performance of the system, and ensure the control accuracy of the system. Attached image description:

[0087] Figure 1 This is a schematic diagram illustrating the flow principle of the fixed-time integral sliding mode method based on the extended state observer involved in this invention.

[0088] Figure 2 This is a simulation image of a permanent magnet synchronous motor observed under the desired sinusoidal signal, as per the present invention.

[0089] Figure 3 This is a simulation image of a permanent magnet synchronous motor observed under the expected step signal, as per the present invention.

[0090] Figure 4 This is a simulation image of a permanent magnet synchronous motor observed under the desired composite time-varying signal, as per the present invention.

[0091] Figure 5 This is a simulation image of the speed tracking of a permanent magnet synchronous motor under the desired sinusoidal signal, as per the present invention.

[0092] Figure 6 This is a simulation image of the speed tracking error of a permanent magnet synchronous motor under the desired sinusoidal signal, as per the present invention.

[0093] Figure 7 This is a simulation image of the control input of a permanent magnet synchronous motor under the desired sinusoidal signal, as per the present invention.

[0094] Figure 8 The image shows an experimental waveform of the starting speed of the permanent magnet synchronous motor at 200 r / min, as per the present invention.

[0095] Figure 9 The image shown is a simulated experimental image of the speed waveform of the permanent magnet synchronous motor under load and unload conditions at a speed of 200 r / min, as per the present invention.

[0096] Figure 10 These are experimental images of the starting speed waveform and the speed waveform under load and unload conditions of the permanent magnet synchronous motor at a speed of 600 r / min, as per the present invention.

[0097] Figure 11 The image shows the experimental waveform of the permanent magnet synchronous motor under load and unload at a speed of 600 r / min, as per the present invention.

[0098] Figure 12 These are experimental images of the starting speed waveform and the speed waveform under load and unload conditions of the permanent magnet synchronous motor at a speed of 1000 r / min, as per the present invention.

[0099] Figure 13 The image shows the experimental waveform of the permanent magnet synchronous motor under load and unload conditions at a speed of 1000 r / min, as per the present invention.

[0100] Specific implementation formula:

[0101] The present invention will be further described below with reference to specific embodiments and accompanying drawings.

[0102] Example 1:

[0103] The fixed-time integral sliding mode control method for permanent magnet synchronous motors based on an extended state observer, as described in this embodiment, includes the following steps:

[0104] S1. Establish the mathematical model of the permanent magnet synchronous motor:

[0105]

[0106] Among them, i d i q U represents the current along the d-axis and q-axis, respectively; d U q R represents the voltage along the d-axis and q-axis, respectively; s L is the stator winding resistance; d L q The inductances along the d-axis and q-axis are respectively; ψ f The magnetic flux generated by a permanent magnet; n p ω is the number of pole pairs of the permanent magnet synchronous motor; T is the angular velocity of the motor's mechanical rotor; L Where J is the load torque; J is the moment of inertia; and B is the friction damping coefficient.

[0107] To facilitate the design of the extended state observer, equation (2) is transformed as follows:

[0108]

[0109] in, u = bi q * ;u is a virtual control input, This represents the total system disturbance.

[0110] S2. Design an extended state observer

[0111] As can be seen from equation (3), d is the lumped disturbance composed of load torque, disturbance inertia, and friction damping. After the disturbance is observed and compensated by the observer, the system can be approximated as a first-order integral system, simplifying the model and reducing the amount of calculation. The specific process of estimating the total disturbance of the system is as follows:

[0112] S2.1 Establish the dynamic model of the extended state observer:

[0113]

[0114] Where p1 = 2p, p2 = -p2 p is the pole of the extended state observer; For estimating the motor speed; This is an estimate of the total system disturbance d;

[0115] S2.2 Defines the observation error of the extended state observer:

[0116]

[0117] Where η1 is the observation error of velocity and η2 is the observation error of disturbance;

[0118] Taking the derivative of equation (5), we can obtain the dynamic equation of the observation error as follows:

[0119]

[0120] S3. Design of a fixed-time integral sliding mode controller based on an extended state observer.

[0121] The total disturbance of the system is observed by an extended state observer, and the observed total disturbance is embedded into the designed controller as a feedforward term to compensate for the total disturbance, thereby eliminating the impact of the disturbance on the system. The specific process is as follows:

[0122] S3.1 Establish the velocity error state equation:

[0123] The speed tracking error is:

[0124] e = ω * -ω (7)

[0125] Where, ω * ω represents the desired speed, and ω represents the actual output speed.

[0126] Differentiating equation (7), we obtain the velocity error state equation as follows:

[0127]

[0128] in,

[0129] S3.2 Establish a fixed-time integral sliding mode controller model:

[0130] The fixed-time integral sliding surface is selected as follows:

[0131]

[0132] Where k1 and k2 are positive real numbers, 0 < α < 1, β > 1;

[0133] Taking the derivative of equation (9), we obtain the derivative of the sliding surface as follows:

[0134]

[0135] To maintain the sliding mode state on the sliding surface, a fixed-time integral sliding mode control law is designed as follows:

[0136]

[0137] Where k0, k3, and k4 are positive real numbers, and k0 > 0.5, α2 > 1;

[0138] Substituting (11) into (10) gives

[0139]

[0140] in, And |n|≤n * n * It is a positive real number;

[0141] S4. Choose the Lyapunov function and prove the system stability.

[0142] Consider a closed-loop control system consisting of a permanent magnet synchronous motor servo system, a fixed-time integral sliding mode controller, and an extended state observer. All signals are stable, and the tracking error converges to a sufficiently small region within a fixed time. Three Lyapunov functions are selected to prove that the system state approaches the sliding surface, the system state "slides" from the sliding surface to the origin, and the tracking error converges to a sufficiently small region. The smaller the region, the closer the motor speed is to the desired value, and the stronger the control performance. The specific process is as follows:

[0143] Proof: Select the first Lyapunov function

[0144]

[0145] Differentiating formula (13) yields

[0146]

[0147] According to Young's inequality, we get

[0148]

[0149]

[0150] Substituting (15) and (16) into (14) yields

[0151]

[0152] Where L1 = min{2k0-1,2k3,2k4,-1}, L1 and L2 are bounded constants because the estimation error η2 is also bounded, and the system is stable in finite time.

[0153] Next, we will further demonstrate that the error system can stabilize on the sliding surface within a fixed time.

[0154] The second Lyapunov function is selected as

[0155]

[0156] Differentiating formula (18) yields

[0157]

[0158] Where, v = 1 / 2n 2 , |n|≤n * Bounded, according to Lemma 2 of fixed time, the error system will reach the sliding surface in a fixed time, and its convergence time is: T≤1 / (αθ(1-p))+1 / (βθ(q-1)), α=k3, β=k4, p=(α1+1) / 2, q=(α2+1) / 2, 0<θ<1;

[0159] When the tracking error e stabilizes at the sliding surface, formula (9) can be written as follows:

[0160]

[0161] The third Lyapunov function is selected as

[0162]

[0163] Differentiating formula (21) yields

[0164]

[0165] Calculation shows According to Lemma 3 of the fixed time, the tracking error e will converge to zero in a fixed time.

[0166] Assuming that the tracking error dynamic equation (8) satisfies that the total disturbance is bounded and the rate of change of the disturbance is bounded, then under the proposed control strategy, the sliding mode manifold converges to the desired value in finite time. The tracking error e converges within a fixed time to |e|≤Φ=min{(Λ / k1)} 1 / α ,(Λ / k2) 1 / β},Λ=-k0Θ-k3Θ α1 -k4Θ α2 +n;

[0167] Proof: From formula (19), we can obtain

[0168]

[0169] According to Lemma 3 of the fixed-time condition, the system state can approach the sliding surface within a fixed time interval. To ensure the fixed-time stability of the system, the following condition must be satisfied: Right now Therefore, as long as |s|>Θ1, the sliding mode variable s can reach |s|≤Θ1 in a fixed time.

[0170] Formula (19) can be rewritten as

[0171]

[0172] Based on the same principle as (23), it is only necessary to satisfy the condition. Right now Therefore, as long as |s|>Θ2, the sliding mode variable s can reach |s|<Θ2 in a fixed time.

[0173] From formula (10) we get

[0174]

[0175] Since |s|≤Θ, formula (12) can be rewritten as

[0176]

[0177] Substituting formula (26) into formula (12), we get

[0178]

[0179] Therefore, as long as |e|>(Λ / k1) 1 / α If |e| ≤ Φ1, then the tracking error e can converge to |e| ≤ Φ1 within a fixed time.

[0180] Furthermore, formula (27) can also be written as

[0181]

[0182] Therefore, as long as |e|>(Λ / k2) 1 / β If |e| ≤ Φ2, then the tracking error e can converge to |e| ≤ Φ2 within a fixed time.

[0183] The fixed-time lemmas involved in this embodiment are all existing technologies, namely:

[0184] Lemma 1: Consider the following system

[0185] x = f(x), f(0) = 0, x ∈ R n (1)

[0186] Where, f(·):R n →R nIf the equilibrium x = 0 of system (1) is a continuous function, then it is a finite-time stable equilibrium if the equilibrium x = 0 is Lyapunov stable and finite-time convergent. That is, there exists a function T(x0) such that... And for all t ≥ T(x0), x(t,x0) = 0, where x(t,x0) = 0 is the solution of system (1) starting from x(0) = x0, if the equilibrium x = 0 of system (1) is a finite-time stable equilibrium, and the previous finite convergence time T(x0) is independent of the initial state, that is, T(x0) for all x0 ∈ R n It is a constant;

[0187] Lemma 2: For system (1), if there exists a continuous positive definite function V(x) such that x∈R n If α > 0, β > 0, 0 < p < 1, and q > 1, then the system is practically fixed-time stable, and the fixed settling time is expressed as... Where 0 < θ < 1;

[0188] Lemma 3: For system (1), if there exists a continuous positive definite function V(x) such that Where α>0, β>0, 0 <p<1,q> 1. If the system is globally stable at a fixed time, the fixed setup time is expressed as... Where 0 < θ < 1.

[0189] Example 2:

[0190] To verify the feasibility of the method described in Example 1, this example first performs simulation analysis on MATLAB, and then conducts experimental verification using the LINKS-RT system, a hardware-in-the-loop simulation platform developed by Beijing Lingsichuangqi Co., Ltd., comparing it with a linear sliding mode controller based on an extended state observer and a PI controller. The specific process is as follows:

[0191] (I) Simulation Analysis: The simulation parameters of the motor model are shown in Table 1. The simulation parameters of the three controllers are shown below:

[0192] (1) The parameters of the controller (ESO+FTSMC) using the fixed-time integral sliding mode control method based on the extended state observer described in Example 1 are: k0=5, k1=6, k2=8, k3=0.9, k4=0.9, p=500, α=0.7, β=1.3, α1=0.88, α2=1.55;

[0193] (2) The parameters of the linear sliding mode controller (ESO+LSMC) based on the extended state observer are: k0=10, k1=6, α=1, p=500;

[0194] (3) The PI controller parameters are: k p =15,k i =800.

[0195] Table 1 Simulation parameters of permanent magnet synchronous motor system

[0196] System parameters numerical values unit Moment of inertia p 0.003 <![CDATA[kg·m 2 ]]> <![CDATA[Number of pole pairs n p > 4 — Stator resistance R 2.875 Ω <![CDATA[Stator inductance L d > 0.0085 H <![CDATA[Rotor inductance L q > 0.0085 H Damping coefficient B 0.008 — <![CDATA[Rotor flux linkage ψ f > 0.32 Wb

[0197] The simulation performance comparison results of the three controllers are shown in Table 2.

[0198] Table 2 Comparison of Simulation Performance of Three Control Methods

[0199]

[0200] Substituting the above motor model simulation parameters and the three controller parameters into the control law and sliding mode controller simulation model of Example 1, the simulation results are as follows: The simulation image of the permanent magnet synchronous motor under the expected values ​​of sinusoidal signal d = 0.2sin(t), step signal d = 2.5, and composite time-varying signal (see equation (29)) is as follows. Figure 2-4 As shown; simulation images of speed tracking, speed tracking error, and control input of a permanent magnet synchronous motor under a sinusoidal signal expectation are as follows. Figure 5-7 As shown;

[0201]

[0202] from Figure 2-4 It can be seen that the designed extended state observer can effectively and quickly observe and track.

[0203] from Figure 5-7 It can be seen that the designed control method has no obvious chattering and has good robustness, control performance and disturbance rejection capability.

[0204] (II) Experimental Verification: This mainly includes two parts: first, comparing the control performance of the three controllers during motor startup; second, comparing the motor's anti-interference capability, i.e., the control performance of the three controllers when the motor is subjected to external load interference; the specific steps are as follows:

[0205] The experimental controller parameters are as follows:

[0206] (1) The parameters of the controller (ESO+FTSMC) based on the extended state observer are: k0=1.5,k1=1.2,k2=8,k3=0.9,k4=0.9,p=1000,α=0.9,β=0.5,α1=0.001,α2=1.2;

[0207] (2) The parameters of the linear sliding mode controller (ESO+LSMC) based on the extended state observer are: k0=20, k1=60, α=1, p=1000;

[0208] (3) The PI controller parameters are: k p =0.4, k i =0.05.

[0209] The experimental performance comparison results of the three controllers are shown in Table 3.

[0210] Table 3 Comparison of experimental performance of the three controllers

[0211]

[0212] The above-mentioned experimental controller parameters were loaded into the LINKS-RT system for experimental verification. Schematic diagrams of the motor starting speed waveform and load-reducing speed waveforms at 200 r / min, 600 r / min, and 1000 r / min are shown below. Figure 8-13 As shown.

[0213] from Figure 8-13 It can be seen that the designed control method has small error, high precision, fast response, can effectively suppress chattering, and has good tracking performance and control precision.

[0214] In summary, the fixed-time integral sliding mode control method for permanent magnet synchronous motors based on extended state observers can quickly compensate for disturbances, effectively suppress chattering, improve the tracking performance and control accuracy of the system, and ensure the robustness of the system.

Claims

1. A fixed-time integral sliding mode control method for a permanent magnet synchronous motor based on an extended state observer, characterized in that, Includes the following steps: S1. Establish the mathematical model of the permanent magnet synchronous motor: Among them, i d i q These represent the currents along the d-axis and q-axis, respectively; u d u q R represents the voltage along the d-axis and q-axis, respectively; s L is the stator winding resistance; d L q The inductances along the d-axis and q-axis are respectively; ψ f The magnetic flux generated by a permanent magnet; n p ω is the number of pole pairs of the permanent magnet synchronous motor; T is the angular velocity of the motor's mechanical rotor; L Where J is the load torque; J is the moment of inertia; and B is the friction damping coefficient. To facilitate the design of the extended state observer, equation (2) is transformed as follows: in, u = bi q * ;u is a virtual control input, This represents the total system disturbance. S2. Design an extended state observer As can be seen from equation (3), d is the lumped disturbance composed of load torque, disturbance inertia, and friction damping. After the disturbance is observed and compensated by the observer, the system can be approximated as a first-order integral system, simplifying the model, reducing the amount of calculation, and estimating the total disturbance of the system. The specific process is as follows: S2.1 Establish the dynamic model of the extended state observer: Where p1 = 2p, p2 = -p 2 p is the pole of the extended state observer; For estimating the motor speed; This is an estimate of the total system disturbance d; S2.2 Defines the observation error of the extended state observer: Where η1 is the observation error of velocity and η2 is the observation error of disturbance; Taking the derivative of equation (5), we can obtain the dynamic equation of the observation error as follows: S3. Design of a fixed-time integral sliding mode controller based on an extended state observer. The total disturbance of the system is observed by an extended state observer, and the observed total disturbance is embedded into the designed controller as a feedforward term to compensate for the total disturbance, thereby eliminating the impact of the disturbance on the system; the specific process is as follows: S3.1 Establish the velocity error state equation: The speed tracking error e is: e=ω * -oh (7) Where, ω * ω represents the desired speed, and ω represents the actual output angular velocity of the motor's mechanical rotor. Differentiating equation (7), we obtain the velocity error state equation as follows: in, S3.2 Establish a fixed-time integral sliding mode controller model: The fixed-time integral sliding surface is selected as follows: Where k1 and k2 are positive real numbers, 0 < α < 1, β > 1; Taking the derivative of equation (9), we obtain the derivative of the sliding surface as follows: To maintain the sliding mode state on the sliding surface, a fixed-time integral sliding mode control law is designed as follows: Among them, k0, k3, and k4 are all positive real numbers, and k0 > 0.5 and α2 > 1; Substituting formula (11) into formula (10), we get in, And |n|≤n * n * It is a positive real number; S4. Choose the Lyapunov function and prove the system stability. Consider a closed-loop control system consisting of a permanent magnet synchronous motor servo system, a fixed-time integral sliding mode controller, and an extended state observer. All signals are stable and the tracking error can converge to a sufficiently small region within a fixed time. Three Lyapunov functions are selected to prove that the system state approaches the sliding surface, the system state "slides" from the sliding surface to the origin, and the tracking error converges to a sufficiently small region. The smaller the region, the closer the motor speed is to the desired value, which also represents a stronger control performance.

2. The fixed-time integral sliding mode control method for permanent magnet synchronous motors based on an extended state observer according to claim 1, characterized in that, The specific process of step S4 is as follows: Select the first Lyapunov function Differentiating formula (13) yields According to Young's inequality, we get Substituting (15) and (16) into (14) yields Where L1 = min{2k0-1,2k3,2k4,-1}, L1 and L2 are bounded constants because the estimation error η2 is also bounded, and the system is stable in finite time. Next, we will further demonstrate that the error system can stabilize on the sliding surface over a fixed time: The second Lyapunov function is selected as Differentiating formula (18) yields Where, v = 1 / 2n 2 , |n|≤n * Bounded, according to the fixed-time lemma, the error system will reach the sliding surface in a fixed time, and its convergence time is: T≤1 / (αθ(1-p))+1 / (βθ(q-1)), α=k3, β=k4, p=(α1+1) / 2, q=(α2+1) / 2, 0<θ<1; When the tracking error e stabilizes at the sliding surface, formula (9) can be written as follows: The third Lyapunov function is selected as Differentiating formula (21) yields Calculation shows According to the fixed-time lemma, the tracking error e will converge to zero within a fixed time. Assuming that the tracking error dynamic equation (8) satisfies that the total disturbance is bounded and the rate of change of the disturbance is bounded, then under the proposed control strategy, the sliding mode manifold converges to the desired state in finite time. The tracking error e converges within a fixed time to |e|≤Φ=min{(Λ / k1)} 1 / α ,(Λ / k2) 1 / β }, Proof: From formula (19), we can obtain According to the fixed-time lemma, the system state can tend to the sliding surface within a fixed time interval; to ensure the fixed-time stability of the system, the following condition must be satisfied. Right now Therefore, as long as |s|>Θ1, the sliding mode variable s can reach |s|≤Θ1 in a fixed time. Furthermore, formula (19) can also be written as Based on the same principle as formula (23), it is only necessary to satisfy the condition. Right now Therefore, as long as |s|>Θ2, the sliding mode variable s can reach |s|<Θ2 in a fixed time. From formula (10) we get Since |s|≤Θ, formula (12) can be changed to Substituting (26) into (12), we get Therefore, as long as |e|>(Λ / k1) 1 / α If |e| ≤ Φ1, then the tracking error e can converge to |e| ≤ Φ1 within a fixed time. Formula (27) can also be written as Therefore, as long as |e|>(Λ / k2) 1 / β If |e| ≤ Φ2, then the tracking error e can converge to |e| ≤ Φ2 within a fixed time.

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