SPMSM non-cascade sensorless control method based on cascade expansion state observer

Abstract

The invention discloses an SPMSM non-cascade sensorless control method based on a cascade expansion state observer, and the method comprises the steps: rewriting a mechanical motion equation and a stator voltage equation of a motor into a second-order equation, constructing a cascade expansion observer according to the second-order equation, and carrying out the self-adaptive nonlinear fast terminal sliding mode control. And the stability and convergence of the observer and the controller are analyzed through the Lyapunov stability theorem, and the SPMSM non-cascade sensorless control based on the cascade expansion state observer is constructed, so that the system reliability and the control precision are effectively improved. Therefore, the SPMSM vector control method of the improved cascade linear expansion observer is simple, convenient to operate, simple, practical, safe and reliable.

Description

Technical Field

[0001] The present invention belongs to the technical field of motor control, and in particular relates to a SPMSM non-cascade sensorless control method based on a cascade extended state observer. Background Art

[0002] SPMSM (Surface-mounted Permanent Magnet Synchronous Motor) has been widely used in aerospace, electric vehicles and industrial automation due to its advantages such as high power density, large torque output and wide speed range. SPMSM can not only achieve efficient energy conversion, but also maintain stable performance under high speed and high load conditions, meeting the needs of modern high-tech applications.

[0003] In SPMSM drives, integrated sensorless control technology is gaining more and more attention. This control strategy eliminates the need for traditional position sensors by utilizing the motor's own electrical parameters and operating status, thereby significantly reducing system costs. In addition, sensorless control also improves system reliability because it removes vulnerable sensor components and reduces potential failure points; therefore, SPMSM and its sensorless control technology play an increasingly important role in promoting the popularization of electric vehicles and improving the level of industrial automation.

[0004] In the related prior art, the literature [J.Mao, H.Li, L.Yang, H.Zhang, L.Liu, X.Wang, and J.Tao, “Non-cascaded model-free predictive speed control of SPMSM drive system, IEEE Trans.Energy Convers., vol.37, no.1, pp.153-162, Mar.2022] proposes a non-cascaded model-free predictive speed control method using composite variables to establish a super-local model, which uses Lyapunov function to ensure global convergence. The literature [X.Zhang, Z.Wang, X.Li, and J.Bian, “Non-cascaded predictive speed synchronous control of dual permanent magnet motor system without weight coefficient tuning,” in Proc.33rd Chin.Control Decis.Conf(CCDC), Kunming, China, May 2021,pp.5701-5706] introduces a finite control set strategy for dual permanent magnet motor systems, adopts a non-cascade structure, and predicts the reference voltage vector based on current and speed information. The literature [H.Li,S.Wang,Y.Xie,S.Zheng,and P.Shi,“Virtual reference-based fuzzy non cascade speed controlfor PMSM systems with unmatched disturbances and current constraints,lEEETrans.Fuzzy Syst.,vol.3l,no.12pp.4249-4261,Dec.2023] proposes a fuzzy non-cascade speed control scheme based on virtual reference. Although these non-cascade controls show good application prospects in permanent magnet synchronous motor systems, further improvements are still needed to improve the speed, dynamic response and anti-perturbation capability of the system. Summary of the invention

[0005] In view of the above, the present invention provides a SPMSM non-cascade sensorless control method based on a cascade extended state observer, which combines the extended state observer with the optimization control strategy to improve the dynamic response and anti-interference ability of the system, and can solve the problem of parameter mismatch in the SPMSM cascade control structure leading to decreased dynamic performance and increased static deviation, thereby optimizing the dynamic performance, steady-state accuracy and system robustness of the SPMSM.

[0006] A non-cascade sensorless control method of SPMSM based on cascade extended state observer comprises the following steps:

[0007] (1) Based on the basic structure and working principle of SPMSM, the mechanical motion equation of SPMSM and the stator current mathematical model of SPMSM in dq axis coordinates are established;

[0008] (2) Based on the above mechanical motion equations and stator current mathematical model, a second-order extended observer mathematical model is constructed;

[0009] (3) Optimizing the mathematical model of the second-order extended observer to obtain a linear extended observer equation;

[0010] (4) Design an adaptive nonlinear fast terminal sliding mode controller, and then calculate the d-axis stator voltage reference value u according to the linear expansion observer equation d * and q-axis stator voltage reference value u q * , thereby achieving high-precision control of the SPMSM.

[0011] Furthermore, the mechanical motion equation is expressed as follows:

[0012]

[0013] Where: e is the rotation speed of SPMSM, ψ f is the magnetic flux of SPMSM, T e is the electromagnetic torque of SPMSM, T L is the load torque of SPMSM, B is the viscosity coefficient of SPMSM, J is the moment of inertia of SPMSM, n p is the number of pole pairs of SPMSM, i q is the q-axis stator current of SPMSM, and t represents time.

[0014] Furthermore, the expression of the stator current mathematical model is as follows:

[0015]

[0016] Where: R sis the stator resistance of the SPMSM, L d and L q are the d-axis stator inductance and q-axis stator inductance of SPMSM, u d and u q are the d-axis stator voltage and q-axis stator voltage of the SPMSM, respectively, i d is the d-axis stator current of the SPMSM.

[0017] Furthermore, the expression of the second-order extended observer mathematical model is as follows:

[0018]

[0019] in: is the speed ω e The first derivative of is the q-axis stator current i q The first derivative of , α = 3n p ψ f / 2J, β=1 / L q ,i q * is the reference value of the q-axis stator current, f1 is the observed unknown disturbance caused by the unmodeled dynamics, unknown perturbations and uncertainties in the mechanical part of the SPMSM drive system, and f2 is the observed unknown disturbance caused by the unmodeled dynamics, unknown perturbations and uncertainties in the q-axis related part of the SPMSM drive system.

[0020] Furthermore, in step (3), two cascaded second-order extended observers ESO1 and ESO2 are designed to construct and observe unknown disturbances f1 and f2, thereby obtaining the following linear extended observer equation;

[0021]

[0022] in: is the speed ω e The second-order derivative of is the first-order derivative of the observed unknown disturbance f1.

[0023] Furthermore, the expression of the observed unknown disturbance f1 is as follows:

[0024]

[0025] Where: z1 = s1 and is the reference value of the motor speed, and are the motor speed observation values ​​estimated by ESO1 and ESO2, e1 and e2 correspond to the speed errors of ESO1 and ESO2 respectively. and are the disturbance observation values ​​estimated by ESO1 and ESO2, l1~l4 are the gain coefficients of the observer, and They are and The first derivative of and They are and The first derivative of .

[0026] Furthermore, the expression of the observed unknown disturbance f2 is as follows:

[0027]

[0028] Where: x1 = η1 and is the reference value of the q-axis stator current, and are the q-axis stator current observations estimated by ESO1 and ESO2, c1 and c2 correspond to the current errors of ESO1 and ESO2, respectively. and are the disturbance observation values ​​estimated by ESO1 and ESO2 respectively, m1~m4 are the gain coefficients of the observer, and They are and The first derivative of and They are and The first derivative of .

[0029] Furthermore, the expressions of the gain coefficients l1-l4 and m1-m4 are as follows:

[0030]

[0031] Where: c11 and ω c12 is the bandwidth of ESO1, ω c21 and ω c22 is the bandwidth of ESO2.

[0032] Furthermore, in step (4), the d-axis stator voltage reference value u d * The error between the d-axis stator current reference value and the measured value is obtained after PI (proportional integral) control, and the q-axis stator voltage reference value u q * It is calculated by the following expression;

[0033]

[0034] Where: uc is the output voltage of the adaptive nonlinear fast terminal sliding mode controller.

[0035] Furthermore, the output voltage u of the adaptive nonlinear fast terminal sliding mode controller is c The expression is as follows:

[0036] u c =u e +u sw

[0037]

[0038] u sw =-k1σ-k2sig g (σ)

[0039]

[0040] Where: u e is the equivalent control law, u sw is the switching control law, k1 and k2 are positive gain constants, σ ​​is the sliding surface, sig g (σ)=sign(σ)|σ| g , sig b1 (e)=sign(e)|e| b1 , sign() is a sign function, a1, a2, b1, b2, g are all constants and satisfy a1>0, a2>0, 0<g<1, 1<b1<2, b1>b2, e represents the speed error, is the first-order derivative of e.

[0041] The present invention rewrites the mechanical motion equation and stator voltage equation of the motor into second-order equations through a permanent magnet synchronous motor current sensorless vector control strategy based on an improved extended observer, and constructs a cascaded extended observer and an adaptive nonlinear fast terminal sliding mode control based on this. The stability and convergence of the observer and the controller are analyzed by the Lyapunov stability theorem, and an SPMSM non-cascade sensorless control based on a cascaded extended state observer is constructed, which effectively improves the system reliability and control accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 This is a block diagram of a non-cascade sensorless control system based on the SPMSM improved extended state observer of the present invention.

[0043] Figure 2 The structural block diagram of the second-order extended observer ESO1 designed for the present invention.

[0044] Figure 3The structural block diagram of the second-order extended observer ESO2 designed for the present invention.

[0045] Figure 4 Schematic diagram of the comparison curve of different k1 values ​​for the improved expanded observer without sensor control.

[0046] Figure 5 Schematic diagram of the comparison curve of different k2 values ​​for the improved expanded observer sensorless control.

[0047] Figure 6 For the improved extended observer sensorless control different ω c11 Schematic diagram of the comparison curve of values.

[0048] Figure 7 For the improved extended observer sensorless control different ω c21 Schematic diagram of the comparison curve of values.

[0049] Figure 8 Schematic diagram of the comparison curves of different controllers of unimproved and improved expanded observer sensorless control during the startup phase.

[0050] Fig. 9 It is a schematic diagram of the speed response comparison curve of different controllers of unimproved and improved expanded observer sensorless control under the condition of sudden speed increase.

[0051] Fig.10 It is a schematic diagram of the speed response comparison curve of different controllers of unimproved and improved expanded observer sensorless control under the condition of sudden speed drop.

[0052] Fig.11 It is a schematic diagram of the speed response comparison curve of different controllers of unimproved and improved expanded observer sensorless control under load mutation conditions. DETAILED DESCRIPTION

[0054] In order to describe the present invention more specifically, the technical solution of the present invention is described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0055] The present invention is based on the SPMSM non-cascade sensorless control method of the cascade extended state observer, which is specifically as follows:

[0056] First, based on the basic structure and working principle of SPMSM, its stator current equation and mechanical motion equation can be established in the dq axis coordinate system. Assuming that the d axis and q axis are aligned with the stator flux of the motor respectively, and ignoring the influence of core saturation and current harmonics, the mechanical motion equation is:

[0057]

[0058] Where: eis the motor speed, ψ f is the magnetic flux, T e is the electromagnetic torque, T L is the load torque, B is the viscosity coefficient, J is the moment of inertia, n p is the pole pair number, i q is the q-axis stator current.

[0059] The stator current equation is:

[0060]

[0061] Where: R s is the stator resistance, L d and L q are the d-axis stator inductance and the q-axis stator inductance, u d and u q are the d-axis stator voltage and the q-axis stator voltage respectively, i d is the d-axis stator current.

[0062] Then, the mechanical motion equation and stator voltage equation are rewritten;

[0063]

[0064] Where: α = 3n p ψ f / 2J, a=1 / J, a1=B / J, β=b=1 / L q , b1=R s / L q , b2=-L q / L d .

[0065] When considering the disturbance of motor parameter changes, the above formula can be expressed as:

[0066]

[0067] Where: f1 and f2 represent the unmodeled dynamics, unknown perturbations, and uncertain disturbances associated with the mechanical part and q-axis of the PMSM drive system, respectively.

[0068] According to the above formula, we get:

[0069]

[0070] Theorem 1: The system disturbance and its derivatives estimated by the cascaded extended state observer are ultimately uniformly bounded.

[0071] Proof: We will now prove the stability of the chosen cascaded extended state observer by the following proof.

[0072] Let eij =e j , where i or j = 1, 2. The error equation for the second-order ESO1 used to observe f2 can be derived as follows:

[0073]

[0074] Where: e 12 is the estimation error associated with the disturbance x2 and can be expressed as:

[0075]

[0076] let From the above formula, we can get:

[0077]

[0078] Where:

[0079] Since both eigenvalues ​​of H are -1, we can conclude that H is Hurwitz stable. Therefore, there exists a unique positive definite matrix P such that:

[0080] H T P+PH=-I

[0081] Where:

[0082] Select the Lyapunov function as V(γ)=γ T Pγ, therefore:

[0083]

[0084] Since h is a global Lipschitz with respect to x, there exists a constant ζ such that h ≤ ζ||xz 1i ||. For all x, z 1i , we can derive the following relationship:

[0085]

[0086] When ω0 ≥ 1, we can observe that Therefore, we have the inequality:

[0087]

[0088] In the formula: ρ=||PKξ||+1, substituting the formula into the equation, we get:

[0089]

[0090] When ω0>ρ, we can get The same applies to e2i . Therefore, we can conclude that:

[0091]

[0092] Where: i,j=1,2.

[0093] Based on the above analysis, it can be concluded that Lyapunov's asymptotic stability theorem holds, indicating that the cascade extended state observer (CESO) is convergent. In addition, when the system reaches a steady state, the right side of the above equation converges to 0, so it can be inferred that:

[0094]

[0095] The observation results of the above formula show that when the value of m2 is significantly larger than h, the steady-state error is small enough.

[0096] The present invention uses pole placement technology for the gain design of the second-order LESO, and achieves ideal closed-loop system performance based on the pole placement technology:

[0097]

[0098] Where: c21 and ω c22 They represent the bandwidths of the second-order linear extended state observer (LESO) for estimating x2 and η2, respectively. It is worth noting that in order to estimate signals contaminated by high-frequency noise, ESO1 selects a lower observer bandwidth to prevent the noise from being over-amplified. At the same time, in order to estimate signals that are less affected by noise, ESO2 selects a higher observer bandwidth to maintain a higher tracking capability. At the same time, after multiple tests in the experiment, it is in line with 2ω c21 =ω c22 In this case, using cascaded ESOs has several advantages. It can not only estimate x2 and η2 and aggregate disturbance f2 based on the measured signal, but also enhance the observation of disturbances and suppress the amplification of noisy measurements compared with conventional ESOs.

[0099] In summary, the above sections cover the observer design and stability analysis of the collective perturbation f2. The same process can be applied to the collective perturbation f1, and the corresponding observer gain is:

[0100]

[0101] Where: c11 and ω c12 denote the bandwidths of the second-order LESO used to estimate z2 and s2, respectively, and these bandwidths satisfy 2ω c11 =ω c12 conditions.

[0102] Taking into account the influence of parameter intake and unknown perturbations, in order to achieve high-performance control of SPMSM, accelerate the speed transient response, and improve the steady-state control accuracy, the present invention designs an ANFTSMC (adaptive non-singular fast terminal sliding mode control) speed controller by adaptively changing the influence coefficient of the NFTSMC (non-singular fast terminal sliding mode control) jitter term. It can effectively reduce jitter and achieve rapid convergence.

[0103]

[0104] Where: u c is the controller output.

[0105] Order sig b (e) = sign(x) | x b , the sliding surface of the composite single-loop controller can be designed as:

[0106]

[0107] In the formula: a1>0, a2>0, 1<b1<2, b1>b2.

[0108] The derivation of the above formula is as follows:

[0109]

[0110] After selecting the NFTSM (non-singular fast terminal sliding mode) surface described in the above formula, in order to ensure that the state variables remain on the sliding mode surface σ = 0, the equivalent control law is derived from the condition σ = 0:

[0111]

[0112] In order to facilitate the rapid convergence of the state variables to the NFTSM surface, the switching control law u sw Designed for:

[0113] u sw =-k1σ-k2sig g (σ)

[0114] Where: k1 and k2 are positive gain constants, which will be discussed further later. By selecting the value of g so that 0 < g < 1, the jitter effect is suppressed and the system state is smoothly approximated to the dynamic sliding surface.

[0115] It is worth noting that larger g-values ​​result in smoother switching forces, but at the expense of reduced system robustness.

[0116] In order to solve the problem of k2 for u q To reduce the impact of jitter, we introduce an adaptive parameter k2' for k2, whose value is:

[0117]

[0118] Where: k 2max is the maximum gain, k s is the gain, k2 is the minimum gain.

[0119] The new model-free sliding mode speed loop control law is designed using the state equation, equivalent control law equation, and switching control law equation, which leads to the following expression:

[0120] u c =u e +u sw

[0121] In order to evaluate the stability of the proposed control law, we now perform a stability analysis. The proof of the asymptotic stability of the ANFTSMC controller is given as follows. Proof Consider the following Lyapunov function for the analysis:

[0122]

[0123] From the above formula, the first-order derivative of the Lyapunov function V(t) is:

[0124]

[0125] For the above equation, if and only if σ(t) = 0, Established.

[0126] In summary, the above formula is constant and non-positive. Therefore, the designed convergence law satisfies the sliding mode reachability condition, indicating that the sliding mode controller has asymptotic stability.

[0127] The proof of local finite-time stability is as follows:

[0128] Theorem 2: If the Lyapunov function satisfies the given inequality:

[0129]

[0130] Among them: δ>0, ξ>0, and 0<λ<1.

[0131] When the system converges globally, the time it takes to converge to 0 is:

[0132]

[0133] Using Theorem 2 and the above formula, we can prove the convergence in finite time:

[0134]

[0135] So we can write:

[0136]

[0137] According to Theorem 2, the initial value V(0) = V0, and the convergence time can be limited by the following inequality:

[0138]

[0139] In summary, using the control rate u q * , the permanent magnet synchronous motor can asymptotically converge to the desired reference position in a finite time.

[0140] Finally, the resulting linear extended observer is used to achieve high-precision control of the SPMSM.

[0141] In this embodiment, the simulation motor parameters are shown in Table 1:

[0142] Table 1: Motor parameters

[0143]

[0144] Figure 1 The basic framework of non-cascade sensorless control based on cascaded linear extended state observer is demonstrated, in which the inputs of the two observers are the speed and q-axis current respectively. By observing and suppressing the disturbance, the robustness of the system can be qualitatively improved, and a certain filtering effect can be achieved. Figure 2 and Figure 3 The connection structure of two cascaded second-order extended observers is further refined.

[0145] By systematically varying one parameter while keeping the others constant, we can experimentally study the individual effects of each parameter on the behavior of the entire system:

[0146] The influence of k1: Figure 4 They are the comparison curves of speed, d-axis current, q-axis current, and q-axis voltage, respectively. Figure 4 The green dotted line in the middle represents the experimental results when k1=60, the blue dotted line represents the experimental results when k1=130, and the orange solid line represents the experimental results when k1=200.

[0147] The impact of k2: Figure 5 The speed, d-axis current, q-axis current and q-axis voltage are given respectively. Figure 5 The green dotted line in the middle represents the experimental results when k2=0.1, the blue dotted line represents the experimental results when k2=0.01, and the orange solid line represents the experimental results when k2=0.001.

[0148] ω c11 Impact: Figure 6 The comparison curves of speed, d-axis current, q-axis current and q-axis voltage are depicted respectively. Figure 6 The green dashed line in the middle represents ω c11 =2, the blue dotted line represents ω c11 =5, the orange solid line represents ω c11 The experimental results when =15.

[0149] ω c21 Impact: Figure 7 They are the comparison curves of speed, d-axis current, q-axis current, and q-axis voltage, respectively. Figure 7 The green dashed line in the middle represents ω c21 =200, the blue dotted line indicates ω c21 =500, the orange solid line represents ω c21 = Experimental results when 800.

[0150] Based on the above results, we can draw the following conclusions about the impact of different parameter values ​​on system performance: smaller k1 values ​​lead to faster voltage and speed tracking speeds, but may cause overshoot and increase of transient q-axis current at the initial moment; on the other hand, larger k2 values ​​lead to faster tracking speeds, which helps to suppress transient q-axis current at the initial moment. c11 A smaller value of ω results in a faster response, but also results in a relatively large transient q-axis current when the speed changes suddenly; in addition, a smaller value of c21 Values ​​allow the system to reach steady state faster, but they may also cause overshoot. In summary, these parameter values ​​significantly affect the performance of the system, affecting aspects such as speed tracking, transient current behavior and response speed.

[0151] In this structure, the system uses a cascaded approach to construct two second-order extended observers. The first second-order extended observer uses the input speed signal to estimate the primary disturbances related to the moment of inertia, torque, and speed changes. The role of this observer is to identify the rapidly changing dynamic disturbances in the system and generate preliminary disturbance information. The second second-order extended observer uses the disturbance estimate of the first observer as one of the inputs, while continuing to process the speed signal to further correct and compensate for the residual disturbances in the system. Through this cascade design, the system's anti-interference ability is significantly enhanced, allowing the control system to maintain a high degree of stability and precise control performance under complex working conditions.

[0152] Below we compare the accuracy of the proposed method with other sensorless observers, taking PI control and SMC (Sliding Mode Control) as examples for comparison:

[0153] from Figure 8It can be clearly seen that compared with PI and SMC, the control method of the present invention (CO-ANFTSMC) achieves faster speed tracking. In addition, compared with PI and SMC control, the fluctuations of d-axis current, q-axis current and q-axis voltage under CO-ANFTSMC control are significantly smaller, and CO-ANFTSMC effectively reduces the transient current at the initial moment. Figure 9 to Figure 10 It can be seen that compared with PI and SMC, CO-ANFTSMC responds faster when the speed increases and decreases suddenly, and the speed fluctuation is relatively small. The speed response curve when the load suddenly increases and decreases is as follows: Fig.11 As shown in Figure 1, when loading, the torque changes from 0 torque to 1 torque, and when unloading, the torque changes from 1 torque to 0 torque. Compared with PI and SMC, the use of CO-ANFTSMC can restore the permanent magnet synchronous motor system to steady state faster.

[0154] To solve the problem of parameter mismatch leading to decreased dynamic performance and increased static deviation of SPMSM, we proposed a CESO-based SPMSM speed-current single-loop control method. By comparing the experiments of PI and SMC under different working conditions and analyzing the simulation results, we draw the following conclusions:

[0155] (1) A CESO-based ANFTSMC speed controller was designed, which can restore the motor speed, current and torque to the given values ​​in a very short time under the condition of motor parameter input and unknown disturbance, thereby improving the response speed and robustness of the system. Compared with PI and SMC control, the fluctuations of d-axis current, q-axis current and q-axis voltage are significantly reduced, effectively suppressing the transient current at the initial moment. The design purpose of CESO is to feed back the unknown part of the external disturbance to the controller for compensation and to suppress the amplification of noise measurement.

[0156] (2) Through comprehensive comparison with the PI and SMC control algorithms, it is verified that the algorithm of the present invention still has high precision control performance under different working conditions such as motor parameters and load time variation. While ensuring the reliability and safety of the system, it reduces the system jitter and enhances the robustness of the system.

[0157] (3) The non-serial structure control method is adopted to improve the bandwidth of the controller. In addition, due to the large number of parameters, how to accurately design the parameters in theory and smoothly promote them to engineering is a work that needs to be done in the future.

[0158] The above description of the embodiments is to facilitate the understanding and application of the present invention by those skilled in the art. It is obvious that those skilled in the art can easily make various modifications to the above embodiments and apply the general principles described herein to other embodiments without creative work. Therefore, the present invention is not limited to the above embodiments. Improvements and modifications made by those skilled in the art to the present invention based on the disclosure of the present invention should be within the protection scope of the present invention.

Claims

1. A non-cascade sensorless control method of SPMSM based on a cascade extended state observer, comprising the following steps: (1) Based on the basic structure and working principle of SPMSM, the mechanical motion equation of SPMSM and the stator current mathematical model of SPMSM in dq axis coordinates are established; (2) Based on the above mechanical motion equations and stator current mathematical model, a second-order extended observer mathematical model is constructed; (3) Optimizing the mathematical model of the second-order extended observer to obtain a linear extended observer equation; (4) Design an adaptive nonlinear fast terminal sliding mode controller, and then calculate the d-axis stator voltage reference value u according to the linear expansion observer equation d * and q-axis stator voltage reference value u q * , thereby achieving high-precision control of the SPMSM.

2. The SPMSM non-cascade sensorless control method according to claim 1, characterized in that: The mechanical motion equation is expressed as follows: Where: e is the rotation speed of SPMSM, ψ f is the magnetic flux of SPMSM, T e is the electromagnetic torque of SPMSM, T L is the load torque of SPMSM, B is the viscosity coefficient of SPMSM, J is the moment of inertia of SPMSM, n p is the number of pole pairs of SPMSM, i q is the q-axis stator current of SPMSM, and t represents time.

3. The SPMSM non-cascade sensorless control method according to claim 2, characterized in that: The expression of the stator current mathematical model is as follows: Where: R s is the stator resistance of the SPMSM, L d and L q are the d-axis stator inductance and q-axis stator inductance of SPMSM, u d and u q are the d-axis stator voltage and q-axis stator voltage of the SPMSM, respectively, i d is the d-axis stator current of the SPMSM.

4. The SPMSM non-cascade sensorless control method according to claim 3, characterized in that: The expression of the mathematical model of the second-order extended observer is as follows: in: is the speed ω e The first derivative of is the q-axis stator current i q The first derivative of , α = 3n p ψ f / 2J, β=1 / L q ,i q * is the reference value of the q-axis stator current, f1 is the observed unknown disturbance caused by the unmodeled dynamics, unknown perturbations and uncertainties in the mechanical part of the SPMSM drive system, and f2 is the observed unknown disturbance caused by the unmodeled dynamics, unknown perturbations and uncertainties in the q-axis related part of the SPMSM drive system.

5. The SPMSM non-cascade sensorless control method according to claim 4, characterized in that: In the step (3), two cascaded second-order extended observers ESO1 and ESO2 are designed to construct and observe unknown disturbances f1 and f2, thereby obtaining the following linear extended observer equation; in: is the speed ω e The second-order derivative of is the first-order derivative of the observed unknown disturbance f1.

6. The SPMSM non-cascade sensorless control method according to claim 4, characterized in that: The expression of the observed unknown disturbance f1 is as follows: Where: z1 = s1 and is the reference value of the motor speed, and are the motor speed observation values ​​estimated by ESO1 and ESO2, e1 and e2 correspond to the speed errors of ESO1 and ESO2 respectively. and are the disturbance observation values ​​estimated by ESO1 and ESO2, l1~l4 are the gain coefficients of the observer, and They are and The first derivative of and They are and The first derivative of .

7. The SPMSM non-cascade sensorless control method according to claim 6, characterized in that: The expression of the observed unknown disturbance f2 is as follows: Where: x1 = η1 and is the reference value of the q-axis stator current, and are the q-axis stator current observations estimated by ESO1 and ESO2, c1 and c2 correspond to the current errors of ESO1 and ESO2, respectively. and are the disturbance observation values ​​estimated by ESO1 and ESO2 respectively, m1~m4 are the gain coefficients of the observer, and They are and The first derivative of and They are and The first derivative of .

8. The SPMSM non-cascade sensorless control method according to claim 7, characterized in that: The expressions of the gain coefficients l1~l4 and m1~m4 are as follows: Where: c11 and ω c12 is the bandwidth of ESO1, ω c21 and ω c22 is the bandwidth of ESO2.

9. The SPMSM non-cascade sensorless control method according to claim 5, characterized in that: In step (4), the d-axis stator voltage reference value u d * The error between the d-axis stator current reference value and the measured value is obtained after PI control, and the q-axis stator voltage reference value u q * It is calculated by the following expression; Where: u c is the output voltage of the adaptive nonlinear fast terminal sliding mode controller.

10. The SPMSM non-cascade sensorless control method according to claim 9, characterized in that: The adaptive nonlinear fast terminal sliding mode controller outputs the voltage u c The expression is as follows: in c =in e +in sw Where: u e is the equivalent control law, u sw is the switching control law, k1 and k2 are positive gain constants, σ ​​is the sliding surface, sig g (σ)=sign(σ)|σ| g , sig b1 (e)=sign(e)|e| b1 , sign() is a sign function, a1, a2, b1, b2, g are all constants and satisfy a1>0, a2>0, 0<g<1, 1<b1<2, b1>b2, e represents the speed error, is the first-order derivative of e.

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