Error correction method for resolver based on phase-locked loop and fused least square method

Abstract

The application relates to a resolver error correction method based on a phase-locked loop and a fusion least square method, which comprises the following steps: firstly, an ellipse model is constructed by using a general quadratic curve equation, and initial values of ellipse coefficients are obtained by taking the square sum of algebraic distances from sampling points to the ellipse model as an objective function; secondly, the ellipse coefficients are converted into geometric parameters, and iterative optimization is carried out by taking the square sum of geometric distances from the sampling points to the ellipse boundary as an objective function, so that scale ambiguity and curvature deviation inherent in algebraic distance fitting are eliminated; finally, a Huber loss function is introduced to calculate the weight of each sampling point, and iterative optimization is carried out by taking the product sum of the square of orthogonal geometric distances of each sampling point and the weight as a cost function, so that the interference of outliers and non-Gaussian noise is inhibited; then, physical error parameters are inversely solved from the robust ellipse coefficients, an inverse transformation model is constructed, real-time I / Q signals are online corrected, a distorted ellipse track is reconstructed into a standard circular track, and the angle and angular velocity of a rotor are output by a phase-locked loop closed loop tracking.

Description

Technical Field

[0001] This invention relates to the field of motor control and position sensor signal processing technology, and more specifically, to a rotary transformer error correction method based on phase-locked loop and fused least squares method. Background Technology

[0002] Permanent magnet synchronous motors (PMSMs) are widely used in electric vehicle drives and other fields due to their high power density and efficiency. Achieving high-performance field-oriented control of PMSMs requires real-time, high-precision acquisition of the rotor's absolute position and speed. A rotary transformer, an angular position sensor based on the principle of electromagnetic induction, has an electrical structure equivalent to a transformer with a variable coupling coefficient. It does not rely on magnetic materials, is typically integrated into the motor shaft, and offers advantages such as strong resistance to electromagnetic interference, tolerance to dust and moisture, resistance to high and low temperatures, and high reliability, making it a preferred solution for automotive-grade position sensors.

[0003] The two analog signals output by the resolver need to be demodulated to obtain the rotor angle. Existing demodulation methods can be divided into two categories: hardware methods and software methods. Hardware methods mainly rely on dedicated resolver-to-digital converter chips (such as AD2S1210). Although such hardware solutions can provide reliable angle data, they significantly increase material costs, occupy printed circuit board space, and their tracking bandwidth is limited by external components, resulting in poor flexibility. In addition, the supply of such chips is highly concentrated in specific regions, posing supply chain security risks and the potential for import disruptions, making it difficult to meet the industry's demand for self-sufficiency, controllability, and low cost.

[0004] In software demodulation, the direct arctangent calculation method is extremely sensitive to high-frequency noise, requiring heavy filtering and inevitably introducing phase lag and limiting control bandwidth. The phase-locked loop (PLL) method, as a closed-loop control system, possesses excellent low-pass filtering characteristics, directly obtaining smooth speed signals and theoretically achieving zero-lag tracking of ramp inputs, thus offering advantages in software demodulation. However, in practical applications, due to the non-ideal characteristics of the resolver and sampling circuit, the output signal generally suffers from amplitude imbalance, DC bias, and quadrature phase mismatch. These errors distort the I / Q signal trajectory, which should be a standard circle, into an eccentric rotating ellipse, directly causing periodic errors in rotor angle demodulation, leading to torque pulsation, decreased control accuracy, and even affecting motor starting and load-carrying capacity. Most existing phase-locked loop (PLL) software demodulation methods do not effectively correct the aforementioned I / Q signal distortions online. Furthermore, single least-squares fitting algorithms struggle to simultaneously balance computational speed, estimation accuracy, and robustness to outlier noise: algebraic least-squares is fast but suffers from scale ambiguity and curvature bias, while geometric least-squares is accurate but requires good initial values ​​and is computationally intensive and slow. Both are sensitive to outliers and non-Gaussian noise introduced by electromagnetic interference and vibration in actual sampled data. Therefore, achieving high-precision, low-latency, and robust online correction of rotary transformer demodulation errors without adding dedicated hardware is a pressing technical problem in this field. Summary of the Invention

[0005] The technical problem this invention aims to solve is how to overcome the problems of high hardware chip cost, poor flexibility, and supply chain risks in existing rotary transformer demodulation schemes, as well as the lack of effective online correction for I / Q signal distortion in software demodulation and the difficulty of balancing speed, accuracy, and robustness with a single least squares algorithm.

[0006] This invention provides a method for error correction of a rotary transformer based on a phase-locked loop and fused least squares method. The rotary transformer includes an excitation winding on the rotor and sine and cosine windings on the stator. The method includes the following steps: S1, apply high-frequency excitation to the excitation winding, collect the output signals of the sine winding and cosine winding to calculate the in-phase component I and the quadrature component Q, and accumulate N sampling points on the IQ plane to form a point cloud; S2, construct an elliptic model using the general quadratic curve equation, and use minimizing the sum of squared algebraic distances from the sampling points to the elliptic model as the objective function to solve for the initial values ​​of the elliptic coefficients to construct the algebraic elliptic model. S3, convert the elliptic coefficients into geometric parameters and use them as the current geometric parameters. The ellipse boundary is determined by the current geometric parameters. The current geometric parameters are iteratively updated with the objective function of minimizing the sum of squared geometric distances from the sampling points to the ellipse boundary. After the iteration converges, the geometric parameters are reversed to optimize the elliptic coefficients to construct the geometric ellipse model. S4 uses the optimized elliptic coefficients as the current parameters, determines the elliptic boundary based on the current parameters, calculates the orthogonal geometric distance from the sampling point to the elliptic boundary, and calculates the weight of each sampling point using the Huber loss function; iteratively updates the current parameters using the cost function of minimizing the sum of the products of the squares of the orthogonal geometric distances of each sampling point and the weights, and outputs the robust elliptic coefficients after the iteration converges. S5, based on the robust elliptic coefficients, the physical error parameters are solved by inverse solution; based on this, an inverse transformation model is constructed to correct the real-time I / Q signal, and the corrected I / Q signal is input into the phase-locked loop for closed-loop tracking, and the rotor angle and angular velocity are output.

[0007] Compared with existing technologies, this application has the following advantages: First, it constructs an elliptical model using a general quadratic curve equation, and uses minimizing the sum of squared algebraic distances from sampling points to the elliptical model as the objective function to analytically solve for the initial values ​​of the elliptic coefficients. Then, it converts the elliptic coefficients into geometric parameters, and uses minimizing the sum of squared geometric distances from sampling points to the elliptical boundary as the objective function for iterative optimization, eliminating the scale ambiguity and curvature deviation inherent in algebraic distance fitting. Finally, it introduces the Huber loss function to calculate the weights of each sampling point, and uses minimizing the sum of the products of the squared orthogonal geometric distances of each sampling point and the weights as the cost function for iterative optimization, suppressing the interference of outliers and non-Gaussian noise. Furthermore, it uses robust elliptic coefficients to inversely solve for physical error parameters and constructs an inverse transformation model to perform online correction of real-time I / Q signals, reconstructing the distorted elliptical trajectory into a standard circular trajectory, which is then fed into a phase-locked loop to track and output the rotor angle and angular velocity. This application uses software algorithms instead of dedicated hardware chips, eliminating the need for additional hardware costs and avoiding supply chain risks.

[0008] In one possible implementation, the sine winding and the cosine winding are spatially orthogonal to each other; The carrier frequency of the high-frequency excitation is greater than the rotor electrical angular frequency. The output signals of the sine winding and cosine winding are amplitude modulation signals modulated by the rotor position amplitude.

[0009] Compared with existing technologies, by spatially orthogonalizing the sine and cosine windings, the two output signals have a fixed 90° phase difference, providing a correct orthogonal basis for subsequent angle demodulation. By setting the carrier frequency of the high-frequency excitation to be greater than the rotor's electrical angular frequency, the modulation signal and the rotation frequency are fully separated in the spectrum, facilitating the effective extraction of baseband components without introducing spectral aliasing. By performing envelope demodulation on the two amplitude-modulated signals, the amplitude of the I / Q components is directly proportional to the sine and cosine values ​​of the rotor angle, reliably encoding the angle information into the amplitude information of the baseband signal. This provides a physical basis for subsequent I / Q point cloud construction and ellipse fitting, ensuring the accuracy and reliability of the input data.

[0010] In one possible implementation, S1 specifically includes: S11, apply a high-frequency sinusoidal excitation signal to the excitation winding, and collect the output signals of the sinusoidal winding and the cosine winding; S12, after enveloping demodulation and analog-to-digital conversion of the output signal, we obtain the in-phase component I and the quadrature component Q, which are proportional to the sine and cosine values ​​of the rotor angle, respectively. S13, with the in-phase component I as the horizontal axis and the orthogonal component Q as the vertical axis, forms a two-dimensional IQ plane; N sampling points are accumulated on the IQ plane to form a point cloud.

[0011] Compared with existing technologies, this method converts analog amplitude-modulated signals into digital baseband signals by performing envelope demodulation and analog-to-digital conversion on the two output signals. By constructing a two-dimensional IQ plane with the in-phase component I as the horizontal axis and the quadrature component Q as the vertical axis and accumulating N sampling points to form a point cloud, the discrete sampled values ​​output by the rotary transformer are mapped to a point set distribution on a two-dimensional plane. This transforms the three types of errors originally implicit in the time-domain signal—channel gain imbalance, quadrature phase mismatch, and DC bias—into geometric distortions (elliptical eccentricity, rotation, and center translation) of the point cloud trajectory. Thus, the problem of identifying error parameters is transformed into a geometric fitting problem of a planar ellipse, making the error correction problem have clear geometric intuition and mathematical operability.

[0012] In one possible implementation, the expression for constructing the elliptic model in S2 using the general quadratic curve equation is as follows: ; In the formula, Indicates in-phase components, Indicates orthogonal components, Let be the elliptic coefficients, and satisfy ? .

[0013] Compared with existing technologies, by constructing an elliptic model using a general quadratic curve equation, a mathematical description of the trajectory of I / Q signals with errors is established, transforming the problem of identifying error parameters at the physical level into the problem of estimating elliptic coefficients at the mathematical level.

[0014] In one possible implementation, the expression in S2, with the objective function being the minimization of the sum of squared algebraic distances from the sampling points to the elliptical model, is as follows: ; In the formula, This represents the total number of sampling points. Indicates the first The in-phase components of each sampling point Indicates the first Orthogonal components of each sampling point; Solving for the initial values ​​of the elliptic coefficients in S2 to construct the algebraic elliptic model specifically includes: By introducing Lagrange multipliers And add elliptical scale constraints For the objective function with respect to the elliptic coefficient vector Taking the partial derivatives and setting them to zero, we obtain the generalized eigenvalue equations. ,in, It is a second-order moment matrix composed of the coordinates of the sampling points. For coefficients only For a sparse matrix with elliptic scaling constraints, solving the generalized eigenvalue equation yields multiple eigenvalues. The elliptic coefficient vector corresponding to the smallest positive value is then selected. , elliptic coefficient vector The six components are assigned to the elliptic coefficients in sequence. That is, the initial values ​​of the elliptic coefficients are obtained analytically; the initial values ​​of the elliptic coefficients are substituted into the elliptic model to obtain the algebraic elliptic model.

[0015] Compared with existing technologies, by introducing Lagrange multipliers and adding elliptic scale constraints, the problem of minimizing the sum of squared algebraic distances under constraints is transformed into solving a generalized eigenvalue equation. The eigenvector corresponding to the smallest positive eigenvalue is taken as the initial value of the elliptic coefficients. Thus, the analytical solution is obtained directly through a single matrix operation, which has significant advantages such as fast computation speed, no iteration required, and no convergence problem. It overcomes the defects of nonlinear optimization methods, such as sensitivity to initial values ​​and easy getting trapped in local optima. At the same time, although the analytical solution has scale ambiguity and curvature deviation, it can provide a rough outline containing the approximate position, shape and orientation of the ellipse, providing a high-quality initial value for subsequent iterative optimization with the goal of minimizing the sum of squared geometric distances. This solves the core problems of difficult cold start and excessive dependence on initial values ​​in iterative optimization with geometric distance as the objective function.

[0016] In one possible implementation, S3 specifically includes: S31, the algebraic ellipse model is converted into an algebraic ellipse matrix formed by matrices. The initial values ​​of the ellipse coefficients are converted into geometric parameters including the coordinates of the ellipse center, the major semi-axis, the minor semi-axis, and the rotation angle by solving the matrix eigenvalue decomposition and the linear equation system. The geometric parameters are used as the initial values ​​of the current geometric parameters. The current geometric parameters are substituted into the algebraic ellipse matrix to obtain the ellipse boundary. S32, construct the objective function to minimize the elliptical boundary at each sampling point under the current geometric parameters, with the expression: ; In the formula, Indicates the first Position vector of each sampling point To the current geometric parameters Lower Elliptical Boundary orthogonal geometric distance, , Represents the boundary of an ellipse any point on, , Represents Euclidean distance. ; S33 uses the Gauss-Newton method to solve the objective function and outputs the increment of the geometric parameters; based on the increment, the current geometric parameters are updated and returned to S32 for iteration until the relative change of two adjacent objective function values ​​is less than a preset threshold, the iteration converges and the converged geometric parameters are output; S34. The converged geometric parameters are expanded into a general quadratic curve equation through coordinate rotation and translation transformations to obtain the optimized elliptic coefficients; and the optimized elliptic coefficients are substituted into the elliptic model to obtain the geometric elliptic model.

[0017] Compared with existing technologies, this method transforms the elliptic coefficients into geometric parameters consisting of the ellipse center coordinates, semi-major axis, semi-minor axis, and rotation angle through eigenvalue decomposition and solving linear equations. This reduces the iteration variables from six algebraic coefficients to five independent parameters with clear geometric meaning, lowering the dimensionality of the optimization space. Iterative updates are performed using the minimum sum of squared geometric distances from each sampling point to the ellipse boundary determined by the current geometric parameters as the objective function. The optimized distances have clear Euclidean geometric meaning and are statistically unbiased, overcoming the scale ambiguity and curvature bias caused by the lack of physical meaning in minimizing the sum of squared algebraic distances. The Gauss-Newton method is used for iterative solving, providing high-quality initial values ​​from minimizing the sum of squared algebraic distances, resulting in rapid convergence with only a few iterations. The converged geometric parameters are then inverted back to optimize the elliptic coefficients through coordinate rotation and translation transformations, achieving a complete closed loop from algebraic initial values ​​to geometric refinement. This improves fitting accuracy while maintaining a consistent elliptic coefficient expression with the preceding and following steps, effectively reducing systematic bias in elliptic coefficient calculation.

[0018] In one possible implementation, S4 specifically includes: S41, using the optimized elliptic coefficients as the initial values ​​for the current parameters; S42, Calculate the elliptical boundary from each sampling point to the current parameters. The orthogonal geometric distance is used as the residual, and the calculation formula is: ; In the formula, Indicates the current parameter. Indicates the elliptical boundary under the current parameters any point on, Indicates the first The coordinates of each sampling point; The residuals of all sampling points are sorted to form a residual vector. ; S43, a robust metric for calculating residuals using absolute median: ; In the formula, Represents the absolute median difference function; And the standardized residuals for each sampling point are calculated based on robust scalar calculations: ; S44, based on the standardized residuals of each sampling point, the Huber loss function is used to calculate the weight of each sampling point: ; In the formula, Represents the Huber threshold constant; S45, Construct a cost function that minimizes the sum of the products of the squared orthogonal geometric distances at each sampling point and the weights: ; In the formula, Indicates the first Each sampling point in the current parameter The weights below, Indicates the first Each sampling point to the ellipse boundary under the current parameters The orthogonal geometric distance; S46 uses the Gauss-Newton method to solve the cost function and outputs the parameter increment; based on the parameter increment, the current parameter is updated and returned to S42 for iteration until the relative change of two adjacent cost function values ​​is less than a preset threshold, the iteration converges and the robust elliptic coefficient is output.

[0019] Compared with existing technologies, a robust mechanism is constructed by introducing the Huber loss function in the iterative optimization: In each iteration, the orthogonal geometric distance from each sampling point to the current ellipse boundary is first calculated as the residual. The robust scale of the residual is estimated by using the absolute median difference and the residual is standardized to make the residual scale-independent. The weight of each sampling point is determined by the standardized residual based on the Huber function. When the residual of the sampling point falls within the normal range, the weight is 1. When the residual exceeds the threshold, the weight automatically decays as the residual increases. Iterative optimization is performed with the minimum sum of the products of the squared orthogonal geometric distance and the weight as the cost function. This overcomes the inherent defect of sensitivity to outliers when using the sum of squared algebraic distances or the sum of squared geometric distances as the objective function. Even under harsh working conditions where electromagnetic interference and vibration introduce outliers, the ellipse fitting results can still remain stable and will not produce significant deviations due to the presence of individual abnormal sampling points. This significantly enhances the anti-interference capability and engineering applicability of the system in real industrial environments.

[0020] In one possible implementation, the physical error parameters in step S5 are solved by inverse solving using robust elliptic coefficients, specifically including: First, the robust elliptic coefficients are solved by matrix eigenvalue decomposition and a system of linear equations to obtain the coordinates of the elliptic center. Rotation angle Long half shaft and short half shaft Among them, in-phase DC bias Orthogonal DC bias ; Next, by , Combined with rotation angle Determine the gain allocation to obtain the in-phase channel gain. and orthogonal channel gain ; Then, based on the rotation angle Long half shaft and short half shaft Orthogonal phase error calculated using trigonometric functions : ; Thus, the physical error parameters are obtained: in-phase DC bias. Orthogonal DC bias In-phase channel gain Gain of positive traffic lane and orthogonal phase error .

[0021] Compared with existing technologies, this method establishes an analytical inverse relationship between robust elliptic coefficients and physical error parameters: the in-phase DC bias and quadrature DC bias are directly obtained from the coordinates of the ellipse center; the in-phase channel gain, quadrature channel gain, and quadrature phase error are jointly solved from the major and minor semi-axes and the rotation angle, mapping the abstract mathematical coefficients back to five error parameters with clear physical meanings. This inverse process is entirely based on analytical formulas, requiring no iteration and involving minimal computation. Furthermore, the five physical parameters obtained correspond one-to-one with the actual operating state of the rotary transformer, making the physical meaning of the error parameters clear and explicit. This facilitates engineers in determining the specific fault type and severity of the sensor or sampling circuit based on the inverse solution results, combining parameter identification and fault diagnosis functions, and providing accurate parameter basis for subsequent inverse transformation correction.

[0022] In one possible implementation, S5 constructs an inverse transform model to correct the real-time I / Q signal, inputs the corrected I / Q signal into the phase-locked loop for closed-loop tracking, and outputs the rotor's angle and angular velocity, specifically including: First, based on DC bias Orthogonal DC bias In-phase channel gain Gain of positive traffic lane and orthogonal phase error Construct the inverse transformation matrix: ; ; In the formula, This represents the corrected in-phase component. This represents the corrected orthogonal components. This represents the in-phase component sampled in real time. This represents the actual orthogonal components collected; Next, the original I / Q signals sampled in real time are subjected to inverse transform matrix operations to obtain the corrected in-phase and quadrature components; Finally, the corrected in-phase and quadrature components are fed into the phase-locked loop, and then sequentially tracked by the phase detector, proportional-integral controller, and integrator in a closed loop to output the rotor's angle and angular velocity.

[0023] Compared with existing technologies, this method constructs an inverse transform model based on the DC bias, channel gain, and quadrature phase error obtained from the inverse transformation. For the real-time sampled raw I / Q signals, the in-phase and quadrature DC biases are subtracted separately, and then the channel gain imbalance and quadrature phase error are simultaneously eliminated via the inverse transform matrix. The corrected I / Q signals are restored to a standard circular trajectory centered on the origin, strictly corresponding to the sine / cosine values ​​of the actual rotor angle. The corrected clean signal is then fed into a phase-locked loop (PLL), and through a phase detector, proportional-integral controller, and integrator closed-loop tracking, the estimated electrical angle and estimated angular velocity of the rotor are output. This combined strategy of "inverse transform online correction + PLL closed-loop tracking" fundamentally eliminates the periodic demodulation error of the rotor angle caused by I / Q signal distortion. Even under low signal-to-noise ratio and impulse noise interference conditions, the PLL can still achieve high-precision tracking with zero phase lag, ultimately ensuring the accuracy and stability of the motor's field-oriented control. Attached Figure Description

[0024] Figure 1 This is a diagram of the internal structure of the rotary transformer of this application; Figure 2 This is a flowchart illustrating the process of this application; Figure 3 This is a schematic diagram illustrating the impact of I / Q channel errors on the demodulated signal constellation diagram. Figure 4 This is a performance comparison chart between the present application and existing algorithms under non-Gaussian noise and outlier interference; Figure 5 This is a time-domain simulation waveform of the online error correction process in this application; Figure 6 This is a comparison chart of the computational efficiency and convergence speed between this application and existing algorithms. Detailed Implementation

[0025] First, those skilled in the art should understand that these embodiments are merely used to explain the technical principles of the embodiments of this application and are not intended to limit the scope of protection of the embodiments of this application. Those skilled in the art can make adjustments as needed to adapt to specific application scenarios.

[0026] In the description of the embodiments of this application, it should be noted that, unless otherwise explicitly specified and limited, the terms "connected" and "linked" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium. Those skilled in the art can understand the specific meaning of the above terms in the embodiments of this application based on the specific circumstances.

[0027] In the embodiments of this application, unless otherwise expressly specified and limited, "above" or "below" the second feature can mean that the first feature is in direct contact with the second feature, or that the first feature is in indirect contact with the second feature through an intermediate medium. Furthermore, "above," "on top of," and "over" the second feature can mean that the first feature is directly above or diagonally above the second feature, or simply that the first feature is at a higher horizontal level than the second feature. "Below," "below," and "under" the second feature can mean that the first feature is directly below or diagonally below the second feature, or simply that the first feature is at a lower horizontal level than the second feature.

[0028] The present application will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0029] See Figures 1-6 As shown, this application discloses an error correction method for a rotary transformer based on a phase-locked loop and fused least squares method, such as... Figure 1 As shown, the rotary transformer includes an excitation winding on the rotor and sine and cosine windings on the stator. The sine and cosine windings are spatially orthogonal, meaning the central axis of the sine windings (S2, S4) and the central axis of the cosine windings (S1, S3) are physically 90 degrees apart on the inner circumference of the stator core. The excitation winding is excited by a high-frequency sine signal. In this embodiment, the excitation signal is a sine wave with a carrier frequency of 10kHz and an effective value of 7V, which is much higher than the rotor's electrical angular frequency. As the rotor rotates, the magnetic coupling coefficient between the rotor and stator windings changes according to the sine and cosine of the rotor angle. The excitation signal is modulated by the rotor position amplitude, inducing two orthogonal analog envelope signals carrying angle information on the stator. The rotary transformer of this application is installed on a built-in permanent magnet synchronous motor, a surface-mounted permanent magnet synchronous motor, or other AC motors equipped with a rotary transformer.

[0030] The method of this application includes the following steps: S1, apply high-frequency excitation to the excitation winding, collect the output signals of the sine and cosine windings to calculate the in-phase component I and the quadrature component Q, and accumulate N sampling points on the IQ plane to form a point cloud; specifically including: S11, apply a high-frequency sinusoidal excitation signal to the excitation winding, and collect the output signals of the sinusoidal winding and cosine winding; S12, the two output signals are amplitude modulation signals modulated by the rotor position amplitude. After envelope demodulation and analog-to-digital conversion, in-phase component I and quadrature component Q, which are proportional to the sine and cosine values ​​of the rotor angle, are obtained respectively. At this time, both the in-phase component and the quadrature component contain errors. S13, with the in-phase component I as the horizontal axis and the orthogonal component Q as the vertical axis, forms a two-dimensional IQ plane; N sampling points are accumulated on the IQ plane to form a point cloud.

[0031] Ideally, the trajectory of the I / Q components is a standard circle centered at the origin, satisfying: ; ; In the formula, This is the actual angle of the rotor. Let be the radius of the ideal circular trajectory.

[0032] However, in practical applications, due to the non-ideal characteristics of the rotary transformer sensor and sampling circuit, the trajectory is distorted into an eccentric rotating ellipse under the combined effects of the gain imbalance between the in-phase and positive-phase channels, the quadrature phase error, and the DC bias.

[0033] like Figure 3 As shown, the constellation trajectory of the I / Q signal undergoes different distortions under different error conditions: like Figure 3 As shown in (a), in the ideal case: no amplitude imbalance, no phase error, no DC bias, the I / Q trajectory is a standard circle with the origin as the center, satisfying that the in-phase component I0 is equal to A multiplied by the rotor angle cosine and the quadrature component Q0 is equal to A multiplied by the rotor angle sine.

[0034] like Figure 3 As shown in (b), in the case of amplitude imbalance: when the in-phase channel gain Gain with positive traffic flow When they are not equal, the trajectory distortion is an ellipse with its principal axis aligned with the coordinate axes, and its major semi-axis a and minor semi-axis b satisfy the following: ; ; like Figure 3As shown in (c), in the case of phase mismatch: when the local oscillator of the positive traffic lane deviates from the angle by 90°. At that time, the orthogonal component becomes A multiplied by the sine. The trajectory is distorted into a rotating ellipse.

[0035] like Figure 3 As shown in (d), under DC bias: when there is in-phase DC bias Orthogonal DC bias At that time, the trajectory is an eccentric circle.

[0036] When the above three types of errors act together, as follows: Figure 3 As shown in (e), it can be represented as a linear transformation of the ideal components followed by superposition and translation. The distorted I / Q components satisfy: ; ; In the formula, , For I / Q channel gain, This is the quadrature phase error. , DC bias, Let be the radius of the ideal circular trajectory.

[0037] Therefore, the general quadratic curve equation is used to describe the eccentric rotating ellipse.

[0038] S2, construct an elliptic model using the general quadratic curve equation, and use minimizing the sum of squared algebraic distances from the sampling points to the elliptic model as the objective function to solve for the initial values ​​of the elliptic coefficients to construct the algebraic elliptic model. The expression for constructing the ellipse model using the general quadratic curve equation is as follows: ; In the formula, This represents the total number of sampling points. Indicates the first The in-phase components of each sampling point Indicates the first The orthogonal components of each sampling point.

[0039] To find the initial values ​​of the elliptic coefficients, Lagrange multipliers are introduced. And add elliptical scale constraints. For the objective function with respect to the elliptic coefficient vector Taking the partial derivatives and setting them to zero, we obtain the generalized eigenvalue equations. ,in, It is a second-order moment matrix composed of the coordinates of the sampling points. For coefficients only For a sparse matrix with elliptic scaling constraints, solving the generalized eigenvalue equation yields multiple eigenvalues. The elliptic coefficient vector corresponding to the smallest positive value is then selected. , elliptic coefficient vector The six components are assigned to the elliptic coefficients in sequence. That is, the initial values ​​of the elliptic coefficients are obtained analytically; the initial values ​​of the elliptic coefficients are substituted into the elliptic model to obtain the algebraic elliptic model.

[0040] S3 converts the elliptic coefficients into geometric parameters and uses them as the current geometric parameters. The ellipse boundary is determined by the current geometric parameters. The current geometric parameters are iteratively updated with the objective function of minimizing the sum of squared geometric distances from the sampling points to the ellipse boundary. After convergence, the geometric parameters are inverted to optimize the elliptic coefficients to construct the geometric ellipse model. Specifically, this includes: S31, convert the algebraic elliptic model into an algebraic elliptic matrix formed by matrices; the expression is: ; The initial values ​​of the elliptic coefficients are transformed into geometric parameters, including the coordinates of the ellipse center, the major semi-axis, the minor semi-axis, and the rotation angle, through matrix eigenvalue decomposition and solving a system of linear equations. Specifically: Calculating the center of an ellipse using a matrix-based ellipse model , , ; After translating the center of the ellipse, construct the quadratic term matrix. Calculate the quadratic term matrix eigenvalues and , The semi-major axis of the ellipse is obtained. ellipse semi-major axis , ; Depend on Obtain the rotation angle of the ellipse To form complete geometric parameters .

[0041] With geometric parameters Initial values ​​for the current geometric parameters Substituting the current geometric parameters into the algebraic elliptic matrix yields the elliptic boundary.

[0042] S32, construct the objective function to minimize the elliptical boundary at each sampling point under the current geometric parameters, with the expression: ; In the formula, Indicates the first Position vector of each sampling point To the current geometric parameters Lower Elliptical Boundary orthogonal geometric distance, , Represents the boundary of an ellipse any point on, , Represents Euclidean distance. .

[0043] S33 uses the Gauss-Newton method to solve the objective function and outputs the increments of the geometric parameters; based on the increments, it updates the current geometric parameters and returns to S32 for iteration until the relative change between two adjacent objective function values ​​is less than a preset threshold, at which point the iteration converges and the converged geometric parameters are output; specifically including: Calculate the distance from each sampling point to the ellipse orthogonal geometric distance of the boundary , Indicates the first Geometric parameters for the next iteration; Constructing residual vectors Calculate the residual vector For geometric parameters Jacobian matrix , its first Line number The column is represented as: That is, the partial derivative of the orthogonal geometric distance of each sampling point with respect to the five components of the geometric parameters; Solve the normal equation Obtain the increment of geometric parameters ,in, Represents the identity matrix; Incremental based on geometric parameters Update geometric parameters until the objective function values ​​of two consecutive intervals are reached. Output the geometric parameters of iterative convergence after the relative change is less than a preset threshold. .

[0044] S34, the converged geometric parameters The optimized elliptic coefficients are obtained by expanding the equation into a general quadratic curve through coordinate rotation and translation transformations; specifically including: According to geometric parameters Defined with the center of the ellipse A local coordinate system is established with the origin as the origin, the direction of the major semi-axis of the ellipse as the horizontal axis, and the direction of the minor semi-axis of the ellipse as the vertical axis. The coordinates in this local coordinate system are... ; In this local coordinate system, the standard ellipse equation satisfies: ; Transform the standard ellipse equation in the local coordinate system to the IQ plane coordinate system through rotation and translation, and let... The optimized elliptic coefficients are obtained by solving the following equation: ; ; ; ; ; ; And optimize the elliptic coefficient Substituting the ellipse model yields the geometric ellipse model.

[0045] S4 uses the optimized elliptic coefficients as the current parameters, determines the ellipse boundary based on the current parameters, calculates the orthogonal geometric distance from the sampling points to the ellipse boundary, and calculates the weight of each sampling point using the Huber loss function; iteratively updates the current parameters using the cost function of minimizing the sum of the products of the squared orthogonal geometric distances of each sampling point and the weights, and outputs the robust elliptic coefficients after the iteration converges; specifically including: S41, using the optimized elliptic coefficients as the initial values ​​for the current parameters. .

[0046] S42, Calculate the elliptical boundary from each sampling point to the current parameters. The orthogonal geometric distance is used as the residual, and the calculation formula is: ; In the formula, Indicates the current parameter. Indicates the elliptical boundary under the current parameters any point on, Indicates the first The coordinates of each sampling point; The residuals of all sampling points are sorted to form a residual vector. .

[0047] S43, a robust metric for calculating residuals using absolute median: ; In the formula, Represents the absolute median difference function; And the standardized residuals for each sampling point are calculated based on robust scalar calculations: .

[0048] S44, based on the standardized residuals of each sampling point, the Huber loss function is used to calculate the weight of each sampling point: ; In the formula, This represents the Huber threshold constant, as described in the embodiments of this application. When the residual of the sampling point falls within the normal range, the weight is 1. When the residual exceeds the threshold, the weight automatically decreases as the residual increases. The principle is to assign low weight to large noise and high weight to small noise. Threshold The selection of is a statistical conclusion: assuming the noise follows a Gaussian distribution, we take . This allows Huber regression to achieve approximately 95% of the efficiency of ordinary least squares under pure Gaussian noise; that is, the weight is 1 when the data point falls within the normal noise range to maintain high accuracy; when the data point deviates too much, the weight is automatically reduced, thereby significantly enhancing the robustness of the system while maintaining the accuracy of the algorithm.

[0049] S45, Construct a cost function that minimizes the sum of the products of the squared orthogonal geometric distances at each sampling point and the weights: ; In the formula, Indicates the first Each sampling point in the current parameter The weights below, Indicates the first Each sampling point to the elliptical boundary under the current parameters The orthogonal geometric distance.

[0050] S46 uses the Gauss-Newton method to solve the cost function and outputs the parameter increment; based on the parameter increment, the current parameters are updated and the process returns to S42 for iteration until the relative change between two adjacent cost function values ​​is less than a preset threshold. The iteration converges, and the robust elliptic coefficients are output, specifically including: First, regarding the current parameters Construct residual vectors and weight diagonal matrix ; Secondly, calculate the residual vector. For the current parameter Jacobian matrix : ; The Jacobian matrix in the first Line number Column elements In this embodiment, the partial derivatives of the elliptic coefficients are approximated using the forward difference method: ; In the embodiments of this application, The partial derivatives of the other coefficients of the elliptic coefficients are similar; Next, the weighted normal equation is constructed and solved: The coefficient increment is obtained. ; Update the current parameters by coefficient increment. Then, update the current parameters. coefficients in Apply elliptic constraints Then return to S42, continuing until the relative change in parameters between two consecutive iterations is less than a preset threshold, and output the updated current parameters as robust elliptic coefficients. .

[0051] S5, based on the robust elliptic coefficients, the physical error parameters are solved by inverse solution; based on this, an inverse transformation model is constructed to correct the real-time I / Q signal, and the corrected I / Q signal is input into the phase-locked loop for closed-loop tracking, and the rotor angle and angular velocity are output.

[0052] Specifically, the physical error parameters derived from the robust elliptic coefficients include: First, based on robust elliptic coefficients Convert to geometric parameters, including the center of the ellipse. Rotation angle Long half shaft and short half shaft The calculation formula is: ; ; ; Next, by , Combined with rotation angle Determine the gain allocation to obtain the in-phase channel gain. and orthogonal channel gain ; Then, based on the rotation angle Long half shaft and short half shaft Orthogonal phase error calculated using trigonometric functions : ; Thus, the physical error parameters are obtained: in-phase DC bias. Orthogonal DC bias In-phase channel gain Gain of positive traffic lane and orthogonal phase error .

[0053] The correspondence between ellipticity coefficients and physical error parameters is shown in Table 1.

[0054] Table 1. Physical meaning and geometric influence of ellipticity coefficients ; An inverse transform model is constructed based on the physical error parameters obtained from the inverse solution, wherein the I / Q signal containing the error satisfies the forward distortion model with respect to the ideal I / Q signal: ; Inverting the above forward distortion model yields the inverse transformation matrix: ; ; In the formula, This represents the corrected in-phase component. This represents the corrected orthogonal components. This represents the in-phase component sampled in real time. This represents the actual orthogonal components collected; Next, the original I / Q signals sampled in real time are subjected to inverse transform matrix operations to obtain the corrected in-phase and quadrature components; the corrected output... satisfy: ; ; Finally, the corrected in-phase and quadrature components are fed into the phase-locked loop, and then sequentially tracked by the phase detector, proportional-integral controller, and integrator in a closed loop to output the rotor's angle and angular velocity.

[0055] Specifically, the phase-locked loop consists of a phase detector, a proportional-integral controller, and an integrator, and its input is the demodulated in-phase component. and orthogonal components Output rotor position; The phase detector output is: ; In the formula, Indicates angular error. The electrical angle output by the phase-locked loop. This is the actual angle of the rotor. ,in, This represents the proportionality coefficient. Represents the integral coefficient. This represents the Laplace operator.

[0056] The angle is obtained by inputting the signal to the integrator after adjustment by the proportional-integral controller. The closed-loop transfer function is: ; Through design and The closed-loop transfer function is stable and converges; after the closed-loop transfer function is stable, the angle... Converging to the actual angle Simultaneously, the integrator outputs the angular velocity. .

[0057] Simulation verification To verify the validity of this application, an elliptical trajectory model incorporating amplitude imbalance, phase mismatch, and DC bias was constructed and simulated in the experiment.

[0058] Simulation parameter settings include: channel gain , Phase error (Approximately 0.1396 radians), DC bias , 200 sampling points were randomly selected from the generated elliptical trajectory, and the error parameters were estimated and compared using three methods: algebraic least squares, geometric least squares, and the fusion least squares method of this invention.

[0059] Simulation results show that algebraic least squares can calculate error parameters at an extremely fast speed, and the estimated phase error and DC bias are highly consistent with the true values, but there is a significant scale ambiguity problem—gain ratio. / While approximately 0.833 was accurately identified, the absolute gain deviated from the set value and was sensitive to outliers. Geometric least squares successfully eliminated algebraic bias by minimizing the orthogonal geometric distance. The identified error parameters (including absolute gain) , The original method perfectly matches the true value and has extremely high numerical accuracy, but the calculation speed is slow and it is still sensitive to outliers under non-Gaussian noise. The fusion least squares method of this invention effectively combines the initialization advantages of algebraic least squares with the geometric accuracy of geometric least squares, and introduces the Huber weight function. It not only achieves high numerical accuracy (the in-phase channel gain is about 0.999997 and the positive channel gain is about 1.200003), but also effectively suppresses the influence of outliers through the reweighting mechanism, realizing fast, stable and accurate parameter estimation.

[0060] like Figure 4 As shown, Figure 4 For the comparison of constellation trajectory fitting, the green dotted line indicates that algebraic least squares has a scale bias, the blue dotted line indicates that geometric least squares is biased due to outliers, while the present invention (red dashed line) accurately tracks the true value; the estimation errors in gain ratio, phase offset and DC bias are significantly smaller than those of algebraic least squares and geometric least squares.

[0061] like Figure 5As shown, the simulation introduces physical sensor errors consistent with the experimental setup, with the gray solid line representing the original distorted signal. After the fusion least squares estimator is started at t=0.25 seconds, the corrected signal (thick solid line) quickly and smoothly converges from the distorted state to perfectly match the ideal reference waveform (black dashed line), which intuitively verifies the superior online parameter identification capability and fast convergence characteristics of this invention.

[0062] like Figure 6 As shown, the computational overhead of the three algorithms is compared: Algebraic least squares has the shortest average execution time due to its analytical characteristics, approximately 0.041 milliseconds (1 iteration), but suffers from scale bias and limited accuracy; Traditional geometric least squares requires an average of 5 iterations to converge during cold start, with the highest latency of approximately 1.146 milliseconds; This invention accelerates the nonlinear optimization process by using the output of algebraic least squares as the precise initial value, reducing the required number of iterations to 2 and the average execution time to 0.525 milliseconds, which is approximately 54% lower than the computational overhead of traditional geometric least squares. Table 2 shows a comparison of the execution time and number of iterations for the three algorithms.

[0063] Table 2 Comparison of execution time and number of iterations for the three algorithms ; The simulation results above verify that the present invention can meet the requirements of embedded microcontrollers for low-latency operation while ensuring high estimation accuracy.

[0064] In summary, without adding dedicated hardware chips, this invention effectively corrects the amplitude imbalance, quadrature phase mismatch, and DC bias distortion of the rotary transformer I / Q signals by using step-by-step fusion least squares error parameter identification and inverse model online correction, combined with phase-locked loop closed-loop tracking. Even under low signal-to-noise ratio and impulse noise conditions, it can still reconstruct an ideal circular trajectory from the distorted input, maintaining high estimation accuracy and zero-phase lag tracking.

[0065] In the description of the embodiments of this application, it should be noted that the terms "inner" and "outer" and other terms indicating direction or positional relationship are based on the direction or positional relationship shown in the drawings. This is only for the convenience of description and does not indicate or imply that the device or component must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, it should not be construed as a limitation of this application.

[0066] In the description of this application, the references to terms such as "an embodiment," "some embodiments," "in this embodiment," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in a suitable manner in any one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0067] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for correcting errors in a rotary transformer based on a phase-locked loop and fused least squares method, characterized in that, The rotary transformer includes an excitation winding on the rotor and sine and cosine windings on the stator, and includes the following steps: S1, apply high-frequency excitation to the excitation winding, collect the output signals of the sine winding and cosine winding to calculate the in-phase component I and the quadrature component Q, and accumulate N sampling points on the IQ plane to form a point cloud; S2, construct an elliptic model using the general quadratic curve equation, and use minimizing the sum of squared algebraic distances from the sampling points to the elliptic model as the objective function to solve for the initial values ​​of the elliptic coefficients to construct the algebraic elliptic model. S3, convert the elliptic coefficients into geometric parameters and use them as the current geometric parameters. The ellipse boundary is determined by the current geometric parameters. The current geometric parameters are iteratively updated with the objective function of minimizing the sum of squared geometric distances from the sampling points to the ellipse boundary. After the iteration converges, the geometric parameters are reversed to optimize the elliptic coefficients to construct the geometric ellipse model. S4 uses the optimized elliptic coefficients as the current parameters, determines the elliptic boundary based on the current parameters, calculates the orthogonal geometric distance from the sampling point to the elliptic boundary, and calculates the weight of each sampling point using the Huber loss function; iteratively updates the current parameters using the cost function of minimizing the sum of the products of the squares of the orthogonal geometric distances of each sampling point and the weights, and outputs the robust elliptic coefficients after the iteration converges. S5, based on the robust elliptic coefficients, the physical error parameters are solved by inverse solution; based on this, an inverse transformation model is constructed to correct the real-time I / Q signal, and the corrected I / Q signal is input into the phase-locked loop for closed-loop tracking, and the rotor angle and angular velocity are output.

2. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 1, characterized in that, The sine winding and the cosine winding are orthogonal to each other in space; The carrier frequency of the high-frequency excitation is greater than the rotor electrical angular frequency. The output signals of the sine winding and cosine winding are amplitude modulation signals modulated by the rotor position amplitude.

3. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 1, characterized in that, S1 specifically includes: S11, apply a high-frequency sinusoidal excitation signal to the excitation winding, and collect the output signals of the sinusoidal winding and the cosine winding; S12, after enveloping demodulation and analog-to-digital conversion of the output signal, we obtain the in-phase component I and the quadrature component Q, which are proportional to the sine and cosine values ​​of the rotor angle, respectively. S13, with the in-phase component I as the horizontal axis and the orthogonal component Q as the vertical axis, forms a two-dimensional IQ plane; N sampling points are accumulated on the IQ plane to form a point cloud.

4. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 1, characterized in that, The expression for constructing the ellipse model using the general quadratic curve equation in S2 is as follows: ； In the formula, Indicates in-phase components, Indicates orthogonal components, Let be the elliptic coefficients, and satisfy ? .

5. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 4, characterized in that, The expression for S2, with the objective function being the minimization of the sum of squared algebraic distances from the sampling points to the elliptic model, is as follows: ； In the formula, This represents the total number of sampling points. Indicates the first The in-phase components of each sampling point Indicates the first Orthogonal components of each sampling point; Solving for the initial values ​​of the elliptic coefficients in S2 to construct the algebraic elliptic model specifically includes: By introducing Lagrange multipliers And add elliptical scale constraints. For the objective function with respect to the elliptic coefficient vector Taking the partial derivatives and setting them to zero, we obtain the generalized eigenvalue equations. ,in, It is a second-order moment matrix composed of the coordinates of the sampling points. For coefficients only For a sparse matrix with elliptic scale constraints, solving the generalized eigenvalue equation yields multiple eigenvalues. The elliptic coefficient vector corresponding to the smallest positive value is then selected. , elliptic coefficient vector The six components are assigned to the elliptic coefficients in sequence. That is, the initial values ​​of the elliptic coefficients are obtained analytically; the initial values ​​of the elliptic coefficients are substituted into the elliptic model to obtain the algebraic elliptic model.

6. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 5, characterized in that, S3 specifically includes: S31, the algebraic ellipse model is converted into an algebraic ellipse matrix formed by matrices. The initial values ​​of the ellipse coefficients are converted into geometric parameters including the coordinates of the ellipse center, the major semi-axis, the minor semi-axis, and the rotation angle by solving the matrix eigenvalue decomposition and the linear equation system. The geometric parameters are used as the initial values ​​of the current geometric parameters. The current geometric parameters are substituted into the algebraic ellipse matrix to obtain the ellipse boundary. S32, construct the objective function to minimize the elliptical boundary at each sampling point under the current geometric parameters, with the expression: ； In the formula, Indicates the first Position vector of each sampling point To the current geometric parameters Lower Elliptical Boundary orthogonal geometric distance, , Represents the boundary of an ellipse any point on, , Represents Euclidean distance. ; S33 uses the Gauss-Newton method to solve the objective function and outputs the increment of the geometric parameters; based on the increment, the current geometric parameters are updated and returned to S32 for iteration until the relative change of two adjacent objective function values ​​is less than a preset threshold, the iteration converges and the converged geometric parameters are output; S34. The converged geometric parameters are expanded into a general quadratic curve equation through coordinate rotation and translation transformations to obtain the optimized elliptic coefficients; and the optimized elliptic coefficients are substituted into the elliptic model to obtain the geometric elliptic model.

7. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 6, characterized in that, S4 specifically includes: S41, using the optimized elliptic coefficients as the initial values ​​for the current parameters; S42, Calculate the elliptical boundary from each sampling point to the current parameters. The orthogonal geometric distance is used as the residual, and the calculation formula is: ； In the formula, Indicates the current parameter. Indicates the elliptical boundary under the current parameters any point on, Indicates the first The coordinates of each sampling point; The residuals of all sampling points are sorted to form a residual vector. ; S43, a robust metric for calculating residuals using absolute median: ； In the formula, Represents the absolute median difference function; And the standardized residuals for each sampling point are calculated based on robust scalar calculations: ； S44, based on the standardized residuals of each sampling point, the Huber loss function is used to calculate the weight of each sampling point: ； In the formula, Represents the Huber threshold constant; S45, Construct a cost function that minimizes the sum of the products of the squared orthogonal geometric distances at each sampling point and the weights: ； In the formula, Indicates the first Each sampling point in the current parameter The weights below, Indicates the first Each sampling point to the elliptical boundary under the current parameters The orthogonal geometric distance; S46 uses the Gauss-Newton method to solve the cost function and outputs the parameter increment; based on the parameter increment, the current parameter is updated and returned to S42 for iteration until the relative change of two adjacent cost function values ​​is less than a preset threshold, the iteration converges and the robust elliptic coefficient is output.

8. The method for error correction of a rotary transformer based on phase-locked loop and fused least squares method according to claim 7, characterized in that, In S5, the physical error parameters are derived by inverse solving using robust elliptic coefficients, specifically including: First, the robust elliptic coefficients are solved by matrix eigenvalue decomposition and a system of linear equations to obtain the coordinates of the elliptic center. Rotation angle Long half shaft and short half shaft Among them, in-phase DC bias Orthogonal DC bias ; Next, by , Combined with rotation angle Determine the gain allocation to obtain the in-phase channel gain. and orthogonal channel gain ; Then, based on the rotation angle Long half shaft and short half shaft Orthogonal phase error calculated using trigonometric functions : ； Thus, the physical error parameters are obtained: in-phase DC bias. Orthogonal DC bias In-phase channel gain Gain of positive traffic lane and orthogonal phase error .

9. The error correction method for a rotary transformer based on a phase-locked loop and fused least squares method according to claim 8, characterized in that, Based on this, S5 constructs an inverse transform model to correct the real-time I / Q signal. The corrected I / Q signal is then input into the phase-locked loop for closed-loop tracking, outputting the rotor's angle and angular velocity, specifically including: First, based on DC bias Orthogonal DC bias In-phase channel gain Gain of positive traffic lane and orthogonal phase error Construct the inverse transformation matrix: ； ； In the formula, This represents the corrected in-phase component. This represents the corrected orthogonal components. This represents the in-phase component sampled in real time. This represents the actual orthogonal components collected; Next, the original I / Q signals sampled in real time are subjected to inverse transform matrix operations to obtain the corrected in-phase and quadrature components; Finally, the corrected in-phase and quadrature components are fed into the phase-locked loop, and then sequentially tracked by the phase detector, proportional-integral controller, and integrator in a closed loop to output the rotor's angle and angular velocity.

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