A method for field weakening control of a permanent magnet synchronous motor by layered projection

Abstract

This invention discloses a field weakening control method for a permanent magnet synchronous motor (PMSM) based on layered projection, comprising the following steps: S1, based on the mathematical model of the PMSM and the field weakening control principle, constructing constraint conditions including a current limit circle and a voltage limit ellipse, and introducing the traditional gradient descent method for dividing the field weakening region and correcting the current command; S2, proposing a current reference value generation method based on layered projection. This method takes the initial current command generated by the Levenberg-Marquardt algorithm as input, and successively corrects the command through voltage constraint projection and current constraint projection, finally generating a feasible reference command that satisfies the voltage and current constraints, which is then input to the current controller to achieve optimal torque tracking within the field weakening region. This invention effectively alleviates the problem of the traditional gradient descent method easily getting trapped in local optima, exhibits higher computational efficiency in real-time control, adapts to parameter changes in the PMSM, and improves the overall stability of the system in the field weakening region.

Description

Technical Field

[0001] This invention relates to the field of permanent magnet synchronous motor (PMSM) technology, and specifically to a field weakening control method for layered projection of permanent magnet synchronous motor. Background Technology

[0002] With its high power density, excellent torque performance, and superior operating efficiency, the PMSM (Motor-Mounted Module Motor) has become a core actuator in high-precision motion control scenarios such as electric drive systems for new energy vehicles, industrial robots, and precision CNC machine tools. To meet the wide speed range requirements of these applications, especially when DC bus voltage is limited, field weakening control technology is essential. By applying a negative d-axis current, the air gap magnetic field generated by the permanent magnet is weakened, allowing the motor speed to exceed the base speed without exceeding the inverter's maximum output voltage. However, the dynamic performance and robustness of field weakening control systems have always been key challenges in engineering practice: parameter disturbances, flux linkage decay caused by temperature changes, and smooth switching between different field weakening regions can all lead to current oscillations, torque fluctuations, and even instability. Therefore, researching advanced field weakening control strategies that can achieve rapid dynamic response and strong robustness across the entire speed range has significant theoretical and engineering value for promoting the development of high-performance electric drive systems.

[0003] However, the realization of the aforementioned theoretical advantages is largely limited by the implementation methods of traditional field weakening control strategies. Within the field-oriented control (FOC) framework, traditional field weakening strategies typically utilize proportional-integral (PI) regulators to adjust the flux linkage level when the inverter voltage reaches saturation. While these feedback-based methods can handle parameter variations, the auxiliary PI loop increases control complexity and complicates the parameter tuning process. Furthermore, the dynamic performance of such PI-based schemes often fails to meet the requirements of high-bandwidth applications.

[0004] Due to the diverse and nonlinear parameters of motors and their drivers, quickly and accurately identifying the optimal operating point is one of the major challenges in the field of permanent magnet synchronous motor (PMSM) control. Besides traditional lookup table (LUT)-based methods and minimum current search-based methods, field weakening control algorithms reported in the literature generally fall into two categories: signal injection-based methods and model-based techniques. The basic principle of signal injection-based methods is to track the zero rate of change of torque with respect to current angle by injecting a disturbance signal into the current. Recent literature reports various maximum torque-to-current ratio (MTPA) operating parameters, such as operating speed, mechanical power, stator flux linkage, stator current amplitude, and reconfigured DC link current. These signal injection-based methods are insensitive to motor parameters; however, they introduce disturbances into the system, leading to additional losses and degraded control performance. Virtual signal injection-based methods avoid the disturbance problems caused by the actual injected signal, but introduce tracking errors because only nominal inductance is used. An enhanced virtual signal injection technique considers nonlinear motor parameters to eliminate these tracking errors; however, compared to other signal injection-based methods, this method loses the advantage of parameter insensitivity.

[0005] Model-based methods derive the optimal current reference value from the mathematical model of the PMSM. The maximum torque / voltage (MTPV) condition is derived using the Lagrange multiplier method. However, in the derivation, the d-axis and q-axis inductances are treated as constant values. Some researchers have proposed improved MTPA and MTPV conditions by considering the derivatives of the inductance with respect to the d-axis and q-axis currents to achieve a more accurate optimal operating point. Other methods implicitly reformulate the optimization problem, incorporating stator resistance and mutual inductance in the calculations. Traditional model-based methods typically employ optimality criteria and related constraints, such as current and voltage boundaries, to form constrained optimization problems in different operating regions. For given torque and speed reference values, the intersections of the torque hyperbola, voltage ellipse, current circle, and field weakening trajectory need to be calculated. Therefore, when determining the operating region, a logical sequence needs to be established first, followed by the selection of a suitable optimization problem. Numerical algorithms, such as Newton's method, Ferrari squares, and polynomial approximation, are also used. However, these methods involve complex equations, increasing the computational burden on practical targets. As the synthesis process progresses, real-time implementation may require longer execution times, leading to a degraded dynamic performance. Other methods in the weak magnetic region are based on voltage feedback. The current reference value is automatically adjusted according to the voltage tracking error. These methods are unaffected by parameters. However, their transient performance is limited.

[0006] These advanced control strategies share the commonality of solving constrained optimization problems in real time to ensure optimal performance. Gradient descent (GD) provides a computationally efficient solution for minimizing the cost function. This technique has been used to detect the maximum torque-to-current ratio (MTPA) angle and determine the direction of voltage drop. However, standard gradient descent solvers are inherently only applicable to unconstrained problems, which is clearly insufficient for scenarios with strict current and voltage constraints in PMSM control. To address this issue, gradient projection algorithms have emerged as a powerful extension. By projecting unconstrained updates onto a defined set of constraints, they can handle the inherent operating boundaries in motor drives. Summary of the Invention

[0007] To address the aforementioned problems, this invention proposes a novel field weakening control method for permanent magnet synchronous motors (PMSMs) based on the Levenberg-Marquardt algorithm (LMA) combined with gradient projection. First, its key contribution lies in the efficient integration of LMA to quickly and reliably determine the optimal current reference value. This method effectively overcomes the inherent weakness of traditional gradient methods, which are prone to getting trapped in local optima, by leveraging the robustness of LMA to initial conditions. Furthermore, the proposed strategy significantly improves computational efficiency, making it highly suitable for real-time control applications. The application of this algorithm enhances the robustness and dynamic response performance of the field weakening control system. Second, to ensure safe and constrained operation, gradient projection is strictly applied in each iteration to rigorously limit current and voltage. This constraint treatment prevents problems such as inverter overload and motor instability. Finally, experimental results validate the effectiveness of the proposed field weakening method.

[0008] The specific plan is as follows:

[0009] A field weakening control method for a permanent magnet synchronous motor based on layered projection includes the following steps:

[0010] S1. Based on the mathematical model of permanent magnet synchronous motor and the field weakening control principle, a constraint condition including the current limit circle and the voltage limit ellipse is constructed, and the gradient descent method is used to preliminarily divide the field weakening region and correct the current command.

[0011] S2. Based on step S1, a current reference value generation method based on layered projection is proposed. This method takes the initial current command generated by the Levenberg-Marquardt algorithm as input, and corrects the command successively through voltage constraint projection and current constraint projection. Finally, a feasible reference command that meets the voltage and current constraints is generated and input to the current controller to achieve optimal torque tracking in the field weakening region.

[0012] Further, step S1 includes:

[0013] S11, Mathematical Model of Permanent Magnet Synchronous Motor

[0014] The PMSM electrodynamic equations in a synchronous rotor coordinate system with reference to the d-axis aligned with the rotor flux, and by implementing the Park transformation while keeping the current and voltage moduli constant:

[0015]

[0016] Among them, u d u q i d and i q These represent the stator voltage and current of the PMSM on the dq axis, respectively; ω e L represents the electric angular velocity of the motor. s It is the stator inductance; R s It is the stator resistance, and ψ f It is a permanent magnet flux chain;

[0017] The mechanical motion equations and electromagnetic torque T of PMSM e Represented as:

[0018]

[0019] Where, ω m T represents mechanical angular velocity; l B represents the load torque; P represents the damping coefficient; n Indicates the number of pole pairs of the motor;

[0020] S12, Field weakening principle of permanent magnet synchronous motor

[0021] In the weak magnetic field region, motor operation is limited by both the inverter and the motor's operating limits; the current and voltage limits are as follows:

[0022]

[0023] Among them, U s max For the maximum output voltage, I s max For maximum output current; when using space vector pulse width modulation, U s max Set as ;

[0024] Under steady-state operation and with stator resistance voltage drop negligible, the stator voltage equation simplifies to:

[0025]

[0026] Substituting equation (4) into equation (3), the voltage limit is re-expressed in terms of current as follows:

[0027]

[0028] Using equations (4) and (5), the limiting trajectories of voltage and current can be obtained; both trajectories are circular. Within the rated speed range, the MTPA strategy is optimal, maximizing torque per ampere within the current limit. For SPMSM, since there is no reluctance in its design, i d =0 control strategy is equivalent to MTPA control strategy. Above base speed, the motor operates in a constant power field-weakening region limited by voltage and current. The intersection of the speed-related voltage-limiting ellipse and the current-limiting circle defines the optimal operating trajectory segment AB. In the low-power field-weakening sub-region, the MTPV strategy maximizes torque under voltage saturation by adjusting the dq-axis current along segment BC.

[0029] S13, Traditional field weakening control based on gradient descent

[0030] When θ is less than 90°, the motor operates in the weak magnetic field region; when θ is greater than 90°, the motor operates in the MTPV region.

[0031] The direction of constant torque is along -i d Direction; calculate the standard vector (V) using gradient descent. d V q The direction of voltage drop; to obtain the direction of output voltage drop, the cost function is set as:

[0032]

[0033] When the constant speed motor operates at high speed, the voltage drop of the stator winding is negligible, and the output voltage equation in the dq synchronous rotating coordinate system becomes formula (4). The expression for the direction of voltage drop is:

[0034]

[0035] θ is determined based on the calculated cosine value, thereby identifying the field weakening region where the motor is currently operating.

[0036]

[0037] The direction of MTPV is the tangent direction of its trajectory, therefore we can obtain

[0038]

[0039] The expression for the obtained current setpoint is:

[0040]

[0041] Where α and β are the gain coefficients used to set the current correction value. The current setpoint is corrected. During the field weakening process, the input current of the current regulator can be expressed as...

[0042]

[0043] in, and These are the reference current values ​​for the d-axis and q-axis, respectively. and This is the correction value for the current; and This is the corrected current value.

[0044] Furthermore, in step S2,

[0045] The current reference value generation method based on layered projection takes the initial current command generated by LMA as input. This intermediate current command is then further processed by a voltage constraint and current constraint projection module, which considers the voltage limit loop constraint at the current speed and corrects the command. Finally, a feasible reference command that simultaneously satisfies both current and voltage constraints is input to the current controller, thereby optimizing control performance while ensuring stable operation of the field weakening control.

[0046] Below base speed, current limitation is the primary constraint, and MTPA control is employed to minimize copper losses and provide the required torque. However, in the high-speed, field-weakening region, the primary constraint shifts to the inverter DC-link voltage limitation. Under these conditions, the control objective is to ensure accurate torque tracking within voltage constraints while minimizing current amplitude to reduce copper losses.

[0047] The goal of field weakening control is to achieve maximum torque while satisfying voltage constraints; the constant torque trajectory defined by equation (12) and the voltage ellipse defined by equation (13) are denoted as i d and i q The function, i.e., f(i) d i q ) and g(i d i q ):

[0048]

[0049]

[0050] Solving the simultaneous equations redefines the problem as an unconstrained optimization problem, namely minimizing the objective function E(x), which represents the sum of squared errors; E(x) is defined as:

[0051]

[0052] Where x is a current vector called a parameter, defined as i dq =[i d i q ] T And F(x)=[f (i d i q ) g(i d i q )] T ;

[0053] To find the minimum point of E(x), i.e., x * It applies a numerical optimization method for solving nonlinear least squares problems; it is a method starting from the operation point x. 0 The initial iterative process converges to x. * .

[0054] Further, step S2 includes:

[0055] S21. Applications of gradient descent algorithm and Gauss-Newton iteration algorithm

[0056] To find the minimum of the objective function, the gradient descent algorithm (GD) is the simplest and most intuitive technique. Its parameters are updated at each step by subtracting a scaled gradient, as follows:

[0057]

[0058] Where, α k It is the step size; ▽E(x) k ) represents E(x) k The gradient of ), ▽E(x) k ) is defined as

[0059]

[0060] J(x) k ) is F(x k The Jacobian matrix of ) is defined as

[0061]

[0062] In x *Near α, the convergence rate is linear and usually very slow because when α... k When fixed, ▽E(x) k ) in x * The convergence rate is close to 0; by utilizing the curvature information of E(x), the convergence speed can be improved. However, this method does not use the curvature of E(x). This makes it difficult to set α. k Instead, it should be modified in each iteration to improve speed.

[0063] The Gauss-Newton Iterative Algorithm (GNA) utilizes curvature and gradient information to improve convergence speed; it is an improvement on Newton's method, deriving the same iterative function.

[0064]

[0065] Where H(x) k ) is F(x k The Hessian matrix of ) is defined as J(x) k ) T J(x k The curvature of E(x) is proportional to the curvature of E(x).

[0066] However, the convergence rate depends on the initial point x. 0 Sensitive, especially x 0 Stay away from x * When; when J(x) k When the condition is not only singular but also pathological, the risk of poor convergence increases, which can lead to oscillatory responses under high-saturation operating conditions of the PMSM.

[0067] Applications of S22 and Levenberg-Marquardt algorithms

[0068] LMA (Latent Motion Analysis) is one of the most widely used nonlinear least squares solvers in optimization algorithms. This algorithm is based on gradient search at extreme points and incorporates a damping parameter μ to adaptively adjust between gradient descent and Gauss-Newton iteration algorithms, thus combining the advantages of both. Specifically, when μ is small, the iteration step size is close to that of Newton's method, resulting in fast convergence; while when μ is large, the iteration step size approximates that of gradient descent, leading to strong stability. This mechanism allows LMA to achieve a good balance between convergence speed and numerical robustness when solving complex nonlinear least squares problems.

[0069] LMA combines the advantages of GD and GNA. Parameters are updated according to the following update rules:

[0070]

[0071] Where μ kIt is the damping coefficient, μ k Multiply by the diagonal of the Hessian matrix to scale each component of the gradient;

[0072] When μ k When the value is small, use the GNA step size; when μ k When the value is large, the GD method is followed; typically, the initial value of μk is large, so the first step is in the direction of gradient descent. The logic behind this is that GNA is more efficient in the final iterations, while the GD method is useful at the beginning of the process because the ideal solution is still far away at this point.

[0073] LMA through the damping factor μ k ·diag(H(x k Enhance the robustness of GNA, even from a suboptimal initial guess x. 0 It can also converge to the optimal solution x * This mechanism can be interpreted as a trust region method that mitigates poor convergence by limiting the step size. LMA dynamically adjusts the step size in x. k Stay away from x * Gradient descent behavior and x k Approaching x * Interpolation is performed between the GNA behavior at different times, making it particularly suitable for highly saturated conditions where fast and robust calculations of accurate current references are required.

[0074] Numerical algorithms are compared in terms of condition number, which shows their sensitivity to small perturbations in J(x). The condition number K(J) of LMA is defined as follows:

[0075]

[0076] Damping coefficient μ k The error is dynamically adjusted based on its evolution during the iteration process, and it fluctuates due to parameter updates or saturation levels; μ k Adjustment is achieved through the gain ratio, which is defined as the difference between the actual cost function reduction and the predicted reduction. However, this incurs additional computational burden. Alternatively, a hybrid framework combining these two methods can be implemented to minimize computation time; for example, GNA can be used at low torque reference values, while LMA can be used under high torque conditions.

[0077] S23. Method for proposing weak magnetic regions considering constraints

[0078] The transformation expression for voltage constraint projection is shown in (23), and the transformation expression for torque constraint projection is shown in (24); in geometry, for a given point H(i d i q P1 is the point on the ellipse projected from point H along the line HO, where O is the center point of the ellipse (-ψ). f / L s P2 is the point H projected onto the torque curve along the q-axis;

[0079] Treating projection onto the current circle as a special case, we derive a closed-form expression:

[0080]

[0081] The proposed expression for the oblique projection operation on the voltage boundary is as follows:

[0082]

[0083] The proposed oblique projection operation is represented on the torque curve as follows:

[0084]

[0085] in and These represent the voltage amplitude and maximum voltage limit at point H, respectively.

[0086] When the reference values ​​for torque and speed are T and ω respectively, point S is obtained by implementing the GD algorithm. It can be seen that both current and voltage constraints have been met; therefore, the optimal solution is found by projecting onto the voltage and current boundaries in sequence; for example, the MTPA point is projected onto the voltage boundary through equation (22) to obtain S1, and then S1 is projected onto the current circle through equation (21) to obtain S2. By repeating these two projection operations, such as S1, S2, S3..., the optimal solution can finally be obtained.

[0087] When the torque and speed reference values ​​are T and ω respectively, point R can be obtained by implementing the GD algorithm. At this time, the voltage constraint has been reached. However, it is impossible to determine whether the final optimal solution is inside or outside the current constraint circle. Therefore, the proposed method first finds the optimal solution by projecting onto the voltage and torque boundaries, and then, once the current boundary is reached, replaces the oblique projection onto the torque curve in equation (23) with the orthogonal projection onto the current constraint in equation (21). For example, the maximum torque-to-current ratio (MTPA) point is projected onto the voltage boundary using equation (22), denoted as R1, and then R is projected onto the current constraint using equation (23). e1 Projecting onto the torque curve, still denoted as R1. Next, project R1 onto the voltage boundary again to obtain R2. At this point, the current limit has been reached. Therefore, using equation (21), project R2 onto the current constraint to obtain R2. By repeating these projection operations, such as R2, R3, R4..., the optimal solution is obtained;

[0088] If the constraints are not met, the point will be iteratively projected onto the current boundary using equation (21), or onto the voltage boundary using equation (22), or onto the torque curve using equation (23); based on these projection operations, the current reference value is ensured to be within the voltage and current constraints.

[0089] S24. Model-free predictive current control based on ESO

[0090] To mitigate the impact of motor parameters on control performance, a hyperlocal mathematical model is employed in the current loop. This method decomposes the current model into input and lumped disturbance state variables; therefore, the mathematical expression of equation (1) is represented by the following hyperlocal model independent of motor parameters:

[0091]

[0092] Among them, i dq =[i d i q ] T ;u s =[u d u q ] T b is considered as the gain of the input matrix; f dq =[f d f q ] T It is an unknown current disturbance that is related to the system parameters and the nonlinearity in (1);

[0093] MFPC based on ESO was designed, using i dq A hyperlocal model with F as the state variable and current error as feedback:

[0094]

[0095] Where, z1=[ ] T It is the observed dq-axis current i dq The value of z2; ] T It is the observed value of the permanent magnet synchronous motor current disturbance F, e1=[e d e q ] T β1 and β2 are the current observation errors; β1 and β2 are the coefficients of ESO.

[0096] Where, z1=[ ] T It is the observed dq-axis current idq The value of z2; ] T It is the observed value of the permanent magnet synchronous motor current disturbance F, e1=[e d e q ] T β1 and β2 are the current observation errors; β1 and β2 are the coefficients of ESO.

[0097] Considering digital delay, the reference voltage formula is:

[0098]

[0099] The beneficial effects of this invention are:

[0100] 1. The Levenberg-Marquardt algorithm (LMA) is innovatively introduced. This is a highly efficient nonlinear optimization method that can quickly converge to the optimal solution. The proposed method effectively alleviates the problem of traditional gradient descent methods easily getting trapped in local optima, thanks to its robustness in initial point selection. The algorithm exhibits higher computational efficiency in real-time control, adapts to parameter variations in PMSM, and improves the overall stability of the system in the weak magnetic region.

[0101] 2. Introducing gradient projection ensures that current and voltage constraints are strictly met in each iteration, thereby guaranteeing the safe operation of the system. This avoids inverter overload or motor instability caused by improper constraint handling in traditional methods. Attached Figure Description

[0102] Figure 1 This is a voltage and current limit trajectory diagram for the PMSM.

[0103] Figure 2 This is a flowchart of the proposed algorithm.

[0104] Figure 3 This is a projection operation diagram of the proposed method in the weak magnetic region.

[0105] Figure 4 This is a projection calculation diagram of the method of the present invention in the weak magnetic region, considering voltage and current constraints. (a) MTPA(S) point is outside the current circle, and (b) MTPA(R) point is inside the current circle.

[0106] Figure 5 This is the control block diagram of the proposed method.

[0107] Figure 6The results are experimental findings using different μ values ​​under the proposed control method. (a) Rotation speed increased from 3000 to 5000 r / min, μ = 0.1. (b) Rotation speed increased from 3000 to 5000 r / min, μ = 1. (c) Rotation speed increased from 3000 to 5000 r / min, μ = 2.

[0108] Figure 7 These are experimental results under varying torque conditions. (a) Load torque increases from 30 to 50 Nm. (b) Load torque decreases from 50 to 30 Nm.

[0109] Figure 8 This is the experimental result of parameter mismatch. (a) 100%L s Current tracking during parameter changes. (b) 150%L s Current tracking during parameter changes. (c) 200%L s Current tracking when parameters change. Detailed Implementation

[0110] 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.

[0111] like Figure 5 As shown, this invention provides a field weakening control method for a permanent magnet synchronous motor using layered projection, comprising the following steps:

[0112] S1. Based on the mathematical model of permanent magnet synchronous motor and the field weakening control principle, a constraint condition including the current limit circle and the voltage limit ellipse is constructed, and the gradient descent method is used to preliminarily divide the field weakening region and correct the current command.

[0113] S2. Based on step S1, a current reference value generation method based on layered projection is proposed. This method takes the initial current command generated by the Levenberg-Marquardt algorithm as input, and corrects the command successively through voltage constraint projection and current constraint projection. Finally, a feasible reference command that meets the voltage and current constraints is generated and input to the current controller to achieve optimal torque tracking in the field weakening region.

[0114] Further, step S1 includes:

[0115] S11, Mathematical Model of Permanent Magnet Synchronous Motor

[0116] The PMSM electrodynamic equations in a synchronous rotor coordinate system with reference to the d-axis aligned with the rotor flux, and by implementing the Park transformation while keeping the current and voltage moduli constant:

[0117]

[0118] Among them, u d u q i d and i q These represent the stator voltage and current of the PMSM on the dq axis, respectively; ω e L represents the electric angular velocity of the motor. s It is the stator inductance; R s It is the stator resistance, and ψ f It is a permanent magnet flux chain;

[0119] The mechanical motion equations and electromagnetic torque T of PMSM e Represented as:

[0120]

[0121] Where, ω m T represents mechanical angular velocity; l B represents the load torque; P represents the damping coefficient; n Indicates the number of pole pairs of the motor;

[0122] S12, Field weakening principle of permanent magnet synchronous motor

[0123] In the weak magnetic field region, motor operation is limited by both the inverter and the motor's operating limits; the current and voltage limits are as follows:

[0124]

[0125] Among them, U s max For the maximum output voltage, I s max For maximum output current; when using space vector pulse width modulation, U s max Set as ;

[0126] Under steady-state operation and with stator resistance voltage drop negligible, the stator voltage equation simplifies to:

[0127]

[0128] Substituting equation (4) into equation (3), the voltage limit is re-expressed in terms of current as follows:

[0129]

[0130] like Figure 4As shown, the limiting trajectories of voltage and current can be obtained using equations (4) and (5); both trajectories are circular. Within the rated speed range, the MTPA strategy is optimal, maximizing torque per ampere within the current limit. For SPMSM, since there is no reluctance in its design, i d =0 control strategy is equivalent to MTPA control strategy. Above base speed, the motor operates in a constant power field-weakening region limited by voltage and current. The intersection of the speed-related voltage-limiting ellipse and the current-limiting circle defines the optimal operating trajectory segment AB. In the low-power field-weakening sub-region, the MTPV strategy maximizes torque under voltage saturation by adjusting the dq-axis current along segment BC.

[0131] S13, Traditional field weakening control based on gradient descent

[0132] like Figure 1 As shown, this is the torque and voltage direction diagram of the PMSM. According to... Figure 1 The value of θ determines the field weakening region. When θ is less than 90°, the motor operates in the field weakening region; when θ is greater than 90°, the motor operates in the MTPV region.

[0133] The direction of constant torque is along -i d Direction; calculate the standard vector (V) using gradient descent. d V q The direction of voltage drop; to obtain the direction of output voltage drop, the cost function is set as:

[0134]

[0135] When the constant speed motor operates at high speed, the voltage drop of the stator winding is negligible, and the output voltage equation in the dq synchronous rotating coordinate system becomes formula (4). The expression for the direction of voltage drop is:

[0136]

[0137] θ is determined based on the calculated cosine value, thereby identifying the field weakening region where the motor is currently operating.

[0138]

[0139] The direction of MTPV is the tangent direction of its trajectory, therefore we can obtain

[0140]

[0141] The expression for the obtained current setpoint is:

[0142]

[0143] Where α and β are the gain coefficients used to set the current correction value. The current setpoint is corrected. During the field weakening process, the input current of the current regulator can be expressed as...

[0144]

[0145] in, and These are the reference current values ​​for the d-axis and q-axis, respectively. and This is the correction value for the current; and This is the corrected current value.

[0146] Furthermore, in step S2,

[0147] The current reference value generation method based on layered projection takes the initial current command generated by LMA as input. This intermediate current command is then further processed by a voltage constraint and current constraint projection module, which considers the voltage limit loop constraint at the current speed and corrects the command. Finally, a feasible reference command that simultaneously satisfies both current and voltage constraints is input to the current controller, thereby optimizing control performance while ensuring stable operation of the field weakening control.

[0148] Below base speed, current limitation is the primary constraint, and MTPA control is employed to minimize copper losses and provide the required torque. However, in the high-speed, field-weakening region, the primary constraint shifts to the inverter DC-link voltage limitation. Under these conditions, the control objective is to ensure accurate torque tracking within voltage constraints while minimizing current amplitude to reduce copper losses.

[0149] The goal of field weakening control is to achieve maximum torque while satisfying voltage constraints; the constant torque trajectory defined by equation (12) and the voltage ellipse defined by equation (13) are denoted as i d and i q The function, i.e., f(i) d i q ) and g(i d i q ):

[0150]

[0151]

[0152] Solving the simultaneous equations redefines the problem as an unconstrained optimization problem, namely minimizing the objective function E(x), which represents the sum of squared errors; E(x) is defined as:

[0153]

[0154] Where x is a current vector called a parameter, defined as i dq =[i d i q ] T And F(x)=[f (i d i q ) g(i d i q )] T ;

[0155] To find the minimum point of E(x), i.e., x * It applies a numerical optimization method for solving nonlinear least squares problems; it is a method starting from the operation point x. 0 The initial iterative process converges to x. * .

[0156] Further, step S2 includes:

[0157] S21. Applications of gradient descent algorithm and Gauss-Newton iteration algorithm

[0158] To find the minimum of the objective function, the gradient descent algorithm (GD) is the simplest and most intuitive technique. Its parameters are updated at each step by subtracting a scaled gradient, as follows:

[0159]

[0160] Where, α k It is the step size; ▽E(x) k ) represents E(x) k The gradient of ), ▽E(x) k ) is defined as

[0161]

[0162] J(x) k ) is F(x k The Jacobian matrix of ) is defined as

[0163]

[0164] In x * Near α, the convergence rate is linear and usually very slow because when α...k When fixed, ▽E(x) k ) in x * The convergence rate is close to 0; by utilizing the curvature information of E(x), the convergence speed can be improved. However, this method does not use the curvature of E(x). This makes it difficult to set α. k Instead, it should be modified in each iteration to improve speed.

[0165] The Gauss-Newton Iterative Algorithm (GNA) utilizes curvature and gradient information to improve convergence speed; it is an improvement on Newton's method, deriving the same iterative function.

[0166]

[0167] Where H(x) k ) is F(x k The Hessian matrix of ) is defined as J(x) k ) T J(x k The curvature of E(x) is proportional to the curvature of E(x).

[0168] However, the convergence rate depends on the initial point x. 0 Sensitive, especially x 0 Stay away from x * When; when J(x) k When the condition is not only singular but also pathological, the risk of poor convergence increases, which can lead to oscillatory responses under high-saturation operating conditions of the PMSM.

[0169] Applications of S22 and Levenberg-Marquardt algorithms

[0170] LMA (Latent Motion Analysis) is one of the most widely used nonlinear least squares solvers in optimization algorithms. This algorithm is based on gradient search at extreme points and incorporates a damping parameter μ to adaptively adjust between gradient descent and Gauss-Newton iteration algorithms, thus combining the advantages of both. Specifically, when μ is small, the iteration step size is close to that of Newton's method, resulting in fast convergence; while when μ is large, the iteration step size approximates that of gradient descent, leading to strong stability. This mechanism allows LMA to achieve a good balance between convergence speed and numerical robustness when solving complex nonlinear least squares problems.

[0171] LMA combines the advantages of GD and GNA. Parameters are updated according to the following update rules:

[0172]

[0173] Where μ k It is the damping coefficient, μ kMultiply by the diagonal of the Hessian matrix to scale each component of the gradient;

[0174] When μ k When the value is small, use the GNA step size; when μ k When the value is large, the GD method is followed; typically, the initial value of μk is large, so the first step is in the direction of gradient descent. The logic behind this is that GNA is more efficient in the final iterations, while the GD method is useful at the beginning of the process because the ideal solution is still far away at this point.

[0175] LMA through the damping factor μ k ·diag(H(x k Enhance the robustness of GNA, even from a suboptimal initial guess x. 0 It can also converge to the optimal solution x * This mechanism can be interpreted as a trust region method that mitigates poor convergence by limiting the step size. LMA dynamically adjusts the step size in x. k Stay away from x * Gradient descent behavior and x k Approaching x * Interpolation is performed between the GNA behavior at different times, making it particularly suitable for highly saturated conditions where fast and robust calculations of accurate current references are required.

[0176] Numerical algorithms are compared in terms of condition number, which shows their sensitivity to small perturbations in J(x). The condition number K(J) of LMA is defined as follows:

[0177]

[0178] Damping coefficient μ k The error is dynamically adjusted based on its evolution during the iteration process, and it fluctuates due to parameter updates or saturation levels; μ k Adjustment is achieved through the gain ratio, which is defined as the difference between the actual cost function reduction and the predicted reduction. However, this incurs additional computational burden. Alternatively, a hybrid framework combining these two methods can be implemented to minimize computation time; for example, GNA can be used at low torque reference values, while LMA can be used under high torque conditions.

[0179] S23. Method for proposing weak magnetic regions considering constraints

[0180] Figure 3 Schematic diagrams of the classical projection operation and the proposed projection operation are shown. The transformation expression for voltage-constrained projection is shown in (23), and the transformation expression for torque-constrained projection is shown in (24); in geometry, for a given point H(i d i q ),like Figure 3As shown, P1 is the point on the ellipse projected from point H along the line HO, where O is the center point of the ellipse (-ψ). f / L s P2 is the point H projected onto the torque curve along the q-axis;

[0181] Treating projection onto the current circle as a special case, we derive a closed-form expression:

[0182]

[0183] The proposed expression for the oblique projection operation on the voltage boundary is as follows:

[0184]

[0185] The proposed oblique projection operation is represented on the torque curve as follows:

[0186]

[0187] in and These represent the voltage amplitude and maximum voltage limit at point H, respectively.

[0188] When the reference values ​​for torque and speed are T and ω respectively, point S is obtained by implementing the GD algorithm, as shown below. Figure 3 As shown. It can be seen that both the current and voltage constraints have been met; therefore, the optimal solution is found by projecting successively onto the voltage and current boundaries; for example, in Figure 3 In the equation (22), point MTPA is projected onto the voltage boundary to obtain S1. Then, S1 is projected onto the current circle through equation (21) to obtain S2. By repeating these two projection operations, such as S1, S2, S3, etc., the optimal solution can be obtained.

[0189] When the reference values ​​for torque and speed are T and ω respectively, point R can be obtained by implementing the GD algorithm, as shown in the figure. At this time, the voltage constraint has been reached. However, it is impossible to determine whether the final optimal solution is inside or outside the current constraint circle. Therefore, the proposed method first finds the optimal solution by projecting onto the voltage and torque boundaries, and then, once the current boundary is reached, replaces the oblique projection onto the torque curve in equation (23) with the orthogonal projection onto the current constraint in equation (21). For example, in Figure 3 In the middle, the maximum torque-to-current ratio (MTPA) point is projected onto the voltage boundary using equation (22), denoted as R1. Then, R is calculated using equation (23). e1Projecting onto the torque curve, still denoted as R1. Next, project R1 onto the voltage boundary again to obtain R2. At this point, the current limit has been reached. Therefore, using equation (21), project R2 onto the current constraint to obtain R2. By repeating these projection operations, such as R2, R3, R4..., the optimal solution is obtained;

[0190] Figure 2 The process of the proposed method in the weak magnetic region is demonstrated. Figure 3 The projection operation in the dq coordinate system is shown when only voltage constraints are considered.

[0191] If the constraints are not met, the point will be iteratively projected onto the current boundary using equation (21), or onto the voltage boundary using equation (22), or onto the torque curve using equation (23); based on these projection operations, the current reference value is ensured to be within the voltage and current constraints.

[0192] S24. Model-free predictive current control based on ESO

[0193] To mitigate the impact of motor parameters on control performance, a hyperlocal mathematical model is employed in the current loop. This method decomposes the current model into input and lumped disturbance state variables; therefore, the mathematical expression of equation (1) is represented by the following hyperlocal model independent of motor parameters:

[0194]

[0195] Among them, i dq =[i d i q ] T ;u s =[u d u q ] T b is considered as the gain of the input matrix; f dq =[f d f q ] T It is an unknown current disturbance that is related to the system parameters and the nonlinearity in (1);

[0196] MFPC based on ESO was designed, using i dq A hyperlocal model with F as the state variable and current error as feedback:

[0197]

[0198] Where, z1=[ ] T It is the observed dq-axis current idq The value of z2; ] T It is the observed value of the permanent magnet synchronous motor current disturbance F, e1=[e d e q ] T β1 and β2 are the current observation errors; β1 and β2 are the coefficients of ESO.

[0199] Considering digital delay, the reference voltage formula is:

[0200]

[0201] Experimental results

[0202] To verify the effectiveness of the proposed scheme, experimental verification was conducted on a three-phase permanent magnet synchronous motor drive system. This experiment comprehensively studied the proposed control strategy. This study aims to evaluate the dynamic performance and robustness of the proposed strategy, with particular focus on its performance under weak magnetic field conditions.

[0203] Figure 6 Experimental results are presented under the proposed control strategy (μ=1). Clearly, not only is speed overshoot suppressed, but the settling time is also significantly reduced. When the target speed is 5000 r / min, the settling time is shorter than that of the traditional method. Furthermore, due to the elimination of speed fluctuations, i d and i q The fluctuations disappeared, and the waveform became smoother. Figure 6 It should be noted that a relative error still exists when the rotor speed is 5000 r / min. Furthermore, in order to increase the rotor speed to follow the target value, i q Greater than the steady-state value, i d It will continue to increase negatively.

[0204] Figure 6 Experimental results using different μ values ​​under the proposed control strategy are presented. It can be concluded that μ independently affects the settling time without affecting the suppression of speed overshoot. When μ is set to 1, compared to μ values ​​of 0.1 and 2, the current response and trajectory exhibit superior dynamic performance, and the current stability is also better. Therefore, it offers high flexibility in parameter setting and allows for a large adjustment range of μ.

[0205] Figure 7 This demonstrates the situation where the torque command changes from 30 to 50 Nm at 5000 r / min. The permanent magnet synchronous motor operates at a speed of 5000 r / min. Figure 7The torque step response characteristics shown demonstrate the relative performance of the proposed control strategy compared to the conventional control strategy under sudden load torque changes. After introducing a step load disturbance at 0.4 seconds, all control methods achieve stable current tracking and fast dynamic response, with steady-state accuracy remaining within acceptable tolerances. Although the conventional strategy effectively expands the motor's operating speed range, experimental results confirm that the proposed method outperforms it in terms of dynamic performance and disturbance rejection characteristics. Figure 7 During this process, the load torque decreases from 50 Nm to 30 Nm, the speed increases to 5040 r / min after the torque change, and then quickly recovers to 5000 r / min. Under these conditions, i d and i q The absolute values ​​of all decrease. Therefore, in the proposed control system, the speed drop and rise caused by torque changes are small, and steady state can be quickly reconstructed.

[0206] To verify the robustness of the system under parameter mismatch conditions, experiments were conducted by setting the inductance parameter in the controller to 150% and 200% of the actual value of the motor. Figure 8 The paper presents a comparison of steady-state experimental results under operating conditions of 5000 r / min speed and 30 Nm load torque, with respect to inductor parameter mismatch. It can be seen that the proposed method exhibits increased current ripple under parameter mismatch, but still demonstrates extremely low sensitivity to parameter deviations while remaining stable, indicating stronger parameter robustness. In contrast, the control performance of traditional methods is highly dependent on precise motor parameters, is very sensitive to parameter changes, and shows a significant decrease in control performance under parameter mismatch.

[0207] 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. A field weakening control method for a permanent magnet synchronous motor using layered projection, characterized in that, Includes the following steps: S1. Based on the mathematical model of permanent magnet synchronous motor and the field weakening control principle, a constraint condition including the current limit circle and the voltage limit ellipse is constructed. The traditional gradient descent method is introduced to divide the field weakening region and correct the current command. S2. Based on step S1, a current reference value generation method based on layered projection is proposed. This method takes the initial current command generated by the Levenberg-Marquardt algorithm as input, and corrects the command successively through voltage constraint projection and current constraint projection. Finally, a feasible reference command that meets the voltage and current constraints is generated and input to the current controller to achieve optimal torque tracking in the field weakening region.

2. The field weakening control method for layered projection of a permanent magnet synchronous motor according to claim 1, characterized in that: Step S1 includes: S11, Mathematical Model of Permanent Magnet Synchronous Motor The PMSM electrodynamic equations in a synchronous rotor coordinate system with reference to the d-axis aligned with the rotor flux, and by implementing the Park transformation while keeping the current and voltage moduli constant: where u d , u q , i d and i q denote the stator voltages and currents of the PMSM in the dq axes, respectively; ω e denotes the electrical angular speed of the machine; L s is the stator inductance; R s is the stator resistance, and ψ f is the permanent magnet flux linkage. The mechanical motion equations and electromagnetic torque T of PMSM e Represented as: Where, ω m T represents mechanical angular velocity; l B represents the load torque; P represents the damping coefficient; n Indicates the number of pole pairs of the motor; S12, Field weakening principle of permanent magnet synchronous motor In the weak magnetic field region, motor operation is limited by both the inverter and the motor's operating limits; the current and voltage limits are as follows: Among them, U s max For the maximum output voltage, I s max For maximum output current; when using space vector pulse width modulation, U s max Set as ; Under steady-state operation and with stator resistance voltage drop negligible, the stator voltage equation simplifies to: Substituting equation (4) into equation (3), the voltage limit is re-expressed in terms of current as follows: S13, Traditional field weakening control based on gradient descent The direction of constant torque is along -i d Direction; calculate the standard vector (V) using gradient descent. d V q The direction of voltage drop; to obtain the direction of output voltage drop, the cost function is set as: When the constant speed motor operates at high speed, the voltage drop of the stator winding is negligible, and the output voltage equation in the dq synchronous rotating coordinate system becomes formula (4); the expression for the direction of voltage drop is... θ is determined based on the calculated cosine value, thereby identifying the field weakening region where the motor is currently operating. The direction of MTPV is the tangent direction of its trajectory, therefore we obtain The expression for the obtained current setpoint is: Where α and β are the gain coefficients used to set the current correction value; the current setpoint is corrected; during the field weakening process, the input current of the current regulator is expressed as... in, and These are the reference current values ​​for the d-axis and q-axis, respectively. and This is the correction value for the current; and This is the corrected current value.

3. The field weakening control method for layered projection of a permanent magnet synchronous motor according to claim 1, characterized in that: In step S2 The goal of field weakening control is to achieve maximum torque while satisfying voltage constraints; the constant torque trajectory defined by equation (12) and the voltage ellipse defined by equation (13) are denoted as i d and i q The function, i.e., f(i) d i q ) and g(i d i q ): Solving the simultaneous equations redefines the problem as an unconstrained optimization problem, namely minimizing the objective function E(x), which represents the sum of squared errors; E(x) is defined as: Where x is a current vector called a parameter, defined as i dq =[i d i q ] T And F(x)=[f (i d i q ) g (i d i q )] T ; To find the minimum point of E(x), i.e., x * It applies a numerical optimization method for solving nonlinear least squares problems; it is a method starting from the operation point x. 0 The initial iterative process converges to x. * .

4. The field weakening control method for layered projection of a permanent magnet synchronous motor according to claim 1, characterized in that: Step S2 includes: S21. Applications of gradient descent algorithm and Gauss-Newton iteration algorithm The parameters of the gradient descent (GD) algorithm are updated at each step by subtracting a scaled gradient, as follows: Where, α k It is the step size; ▽E(x) k ) represents E(x) k The gradient of ), ▽E(x) k ) is defined as J(x) k ) is F(x k The Jacobian matrix of ) is defined as The Gauss-Newton iterative algorithm GNA utilizes curvature and gradient information to improve convergence speed; the same iterative function is derived: Where H(x) k ) is F(x k The Hessian matrix of ) is defined as J(x) k ) T J(x k The curvature of E(x) is proportional to the curvature of E(x). However, the convergence rate depends on the initial point x. 0 Sensitive, especially x 0 Stay away from x * When; when J(x) k When the condition is not only singular but also pathological, the risk of poor convergence increases, which can lead to oscillatory responses under high-saturation operating conditions of the PMSM. Applications of S22 and Levenberg-Marquardt algorithms LMA parameters are updated according to the following update rules: Where μ k It is the damping coefficient, μ k Multiply by the diagonal of the Hessian matrix to scale each component of the gradient; When μ k Hours, using GNA step size; when μ k When large, follow the GD method; LMA through the damping factor μ k ·diag(H(x k Enhance the robustness of GNA, even from a suboptimal initial guess x. 0 It can also converge to the optimal solution x * LMA dynamically in x k Stay away from x * Gradient descent behavior and x k Approaching x * Interpolate between the GNA behaviors at different times; Numerical algorithms are compared in terms of condition number, which shows their sensitivity to small perturbations in J(x). The condition number K(J) of LMA is defined as follows: Damping coefficient μ k The error is dynamically adjusted based on its evolution during the iteration process, and it fluctuates due to parameter updates or saturation levels; μ k The adjustment is made by the gain ratio, which is defined as the difference between the actual cost function reduction and the predicted reduction. S23. Method for proposing weak magnetic regions considering constraints The transformation expression for voltage constraint projection is shown in (22), and the transformation expression for torque constraint projection is shown in (23); in geometry, for a given point H(i d i q P1 is the point on the ellipse projected from point H along the line HO, where O is the center point of the ellipse (-ψ). f / L s P2 is the point H projected onto the torque curve along the q-axis; Treating projection onto the current circle as a special case, we derive a closed-form expression: The proposed expression for the oblique projection operation on the voltage boundary is as follows: The proposed oblique projection operation is represented on the torque curve as follows: in and These represent the voltage amplitude and maximum voltage limit at point H, respectively. First, the optimal solution is found by projecting onto the voltage and torque boundaries. Then, once the current boundary is reached, the oblique projection onto the torque curve in equation (22) is replaced by the orthogonal projection onto the current constraint in equation (21). By repeating these projection operations, the optimal solution is obtained. If the constraints are not met, the point will be iteratively projected onto the current boundary using equation (21), or onto the voltage boundary using equation (22), or onto the torque curve using equation (23); based on these projection operations, the current reference value is ensured to be within the voltage and current constraints. S24. Model-free predictive current control based on ESO The current model is decomposed into input and lumped disturbance state variables; therefore, the mathematical expression of equation (1) is represented by the following hyperlocal model independent of the motor parameters: Among them, i dq =[i d i q ] T ;u s =[u d u q ] T b is considered as the gain of the input matrix; f dq =[f d f q ] T It is an unknown current disturbance that is related to the system parameters and the nonlinearity in (1); MFPC based on ESO was designed, using i dq A hyperlocal model with F as the state variable and current error as feedback: Where, z1=[ ] T It is the observed dq-axis current i dq The value of z2; ] T It is the observed value of the permanent magnet synchronous motor current disturbance F, e1=[e d e q ] T β1 and β2 are the current observation errors; β1 and β2 are the coefficients of ESO. Considering digital delay, the reference voltage formula is:

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