Robust complex coefficient repetitive control method for permanent magnet synchronous motor based on position domain internal model

Abstract

The present invention discloses a method for robust complex coefficient repetitive control of a permanent magnet synchronous motor based on a position domain internal model, which relates to the technical field of permanent magnet synchronous motor control. First, the time domain periodic characteristics of the spindle speed are used to construct an improved time domain complex coefficient repetitive controller. Second, based on the position domain periodic characteristics of the cutting force, a position domain repetitive extended state observer is constructed to online estimate non-periodic disturbances such as parameter perturbations and position domain periodic disturbances caused by the cutting force. Finally, the separation design principle and the small gain theorem are applied to derive the system stability conditions, and an optimization algorithm is used to synchronously optimize the controller parameters. The system design method based on time domain complex coefficient repetitive control and position domain repetitive extended state observer of the present invention designs an improved complex coefficient repetitive control system based on the position domain repetitive extended state observer to achieve feedforward compensation for position domain periodic disturbances and non-periodic disturbances and fast and high-precision tracking of time domain periodic reference speed signals.

Description

Technical Field

[0001] The present invention relates to the technical field of permanent magnet synchronous motor control, and in particular to a method for robust complex coefficient repetitive control of a permanent magnet synchronous motor based on a position domain internal model. Background Art

[0002] During the CNC turning of complex non-circular parts, if constant speed turning is used, the cutting force will fluctuate significantly with position due to the varying arc lengths per unit angle of the non-circular workpiece, easily causing chatter. This not only directly reduces machining accuracy but, in severe cases, can even affect the stability of the machining process. Variable spindle speed machining is an effective approach to addressing this issue. By varying the spindle rotational speed, the relative instantaneous speed of the tool at the cutting point can be kept constant, thereby maintaining relatively stable cutting forces and effectively suppressing chatter. Because spindle speed tracking accuracy significantly impacts workpiece machining precision, it is crucial to develop a control algorithm to ensure high-precision spindle speed tracking. Variable spindle speed machining involves varying the spindle speed based on a sinusoidal excitation signal at the base speed (i.e., the speed used in constant speed turning), ensuring that the reference speed signal exhibits a well-defined temporal periodicity.

[0003] During variable-speed machining, the position-periodic disturbances caused by the cutting force on the spindle system can reduce the tracking accuracy of the spindle speed. This means that the cutting force is a position-dependent periodic disturbance on the spindle drive motor. Therefore, it is necessary to achieve rapid and high-precision tracking of the time-domain periodic spindle speed while suppressing the position-domain periodic disturbances caused by the cutting force on the spindle motor, thereby further improving the tracking accuracy of the spindle speed. Summary of the Invention

[0004] In view of the above-mentioned problems existing in the prior art, the purpose of the present invention is to provide a robust complex coefficient repetitive control method for a permanent magnet synchronous motor based on a position domain internal model. The problems of suppressing the position domain periodic disturbance caused by the cutting force of the non-circular turning spindle system during variable speed processing and tracking the time domain periodic spindle speed are studied, and a complex coefficient repetitive control method based on a position domain repetitive extended state observer is proposed. This method constructs a position domain repetitive extended state observer based on the position domain internal model of the cutting force to realize online estimation and active compensation of non-periodic disturbances such as position periodic disturbances and uncertainties caused by the cutting force. The cutting force model and related available position information are used to design a position domain periodic signal internal model repetitive controller to estimate the position periodic disturbance caused by the cutting force. An improved complex coefficient repetitive controller is designed based on the time domain periodic spindle speed characteristics to realize fast and high-precision tracking of the spindle time domain periodic speed. In addition, an optimization algorithm is used to synchronously optimize the controller parameters.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is:

[0006] A method for robust complex coefficient repetitive control of a permanent magnet synchronous motor based on a position domain internal model comprises the following steps:

[0007] Step S1: constructing a mechanical motion equation of a speed loop of a drive motor, wherein the drive motor drives a non-circular turning spindle, wherein the mechanical motion equation takes into account parameter perturbations, disturbances caused by position periodic cutting forces, and external disturbances;

[0008] Step S2: using the time-domain periodic characteristics of the spindle speed, construct a time-domain improved complex coefficient repetitive controller;

[0009] Step S3: Based on the position-domain periodic characteristics of the cutting force, a position-domain repeated expansion state observer is constructed to online estimate the non-periodic disturbance and the position-domain periodic disturbance caused by the cutting force;

[0010] Step S4: constructing a composite control law to achieve feedforward compensation for periodic and non-periodic disturbances in the position domain and fast and high-precision tracking of the periodic reference speed signal in the time domain;

[0011] Step S5: Provide system stability conditions and parameter optimization.

[0012] As a further improvement of the above technical solution:

[0013] Step S1 includes the following steps:

[0014] Step S11: constructing a cutting force model based on the cutting force empirical formula and the spindle rotation angle position;

[0015] Step S12: Determine the type of the drive motor and construct a speed loop mechanical motion equation of the drive motor.

[0016] The mechanical motion equation of the speed loop constructed in step S1 is:

[0017] ;

[0018] in, is the spindle rotation angular acceleration, is the coefficient of the q-axis current of the permanent magnet synchronous motor, , p is the number of rotor pole pairs, 、 is the nominal value, is the q-axis stator current, represents the total disturbance including parameter perturbation, disturbance caused by position periodic cutting force and external disturbance, is a non-periodic disturbance including parameter perturbation and external disturbance, is the position periodic disturbance caused by cutting force, 、 are the perturbations of the permanent magnet flux linkage and moment of inertia, is the viscous damping coefficient, is the spindle rotation angular velocity, is an external disturbance, is the disturbance caused by the position periodic cutting force.

[0019] In step S2, the transfer function of the time-domain improved complex coefficient repetitive controller is:

[0020] ;

[0021] in, is the internal model of the spindle signal, s is the Laplace operator, is a complex coefficient filter, is the cutoff frequency, j is the imaginary unit, n is the order of the suppressed harmonic signal, is the imaginary part of the complex coefficient.

[0022] In step S2, arrive The transfer function is:

[0023] ;

[0024] in, 、 are the input signal and output signal of the complex coefficient filter, respectively. For adjustment Phase, yes The imaginary signal obtained by the first-order inertia link is For adjustment Amplitude.

[0025] Step S3 includes the following steps:

[0026] Step S31: designing an extended state observer;

[0027] Step S32: construct an improved position domain repetitive controller based on the position periodic characteristics of the disturbance caused by the cutting force.

[0028] Step S33: Construct a position domain repeated expansion state observer.

[0029] The expanded state space model of the mechanical motion equation is:

[0030] ;

[0031] ;

[0032] in, is the first-order derivative of the state vector, is the state vector of the permanent magnet synchronous motor speed loop system, is the control input, is the first-order derivative of the total disturbance, is the system output;

[0033] In step S31, an extended state observer is designed according to the extended state space model as follows:

[0034] ;

[0035] in, represents the derivative of the observer state vector, is the state vector of the observer, and They are and Estimates of , T represents the transpose of the matrix, is the observer gain, 、 are the different gains of the extended state observer;

[0036] In step S32, the transfer function of the improved position domain repetitive controller is:

[0037] ;

[0038] in, is a first-order low-pass filter, e is a natural constant, is the position period, is the position domain Laplace transform operator, is a positive number;

[0039] In step S33, the transfer function in step S32 is embedded in the extended state observer, and the mutual conversion between the time domain signal and the position domain signal is realized through the domain conversion operator, and the estimated value of the unestimated partial disturbance is obtained: ;

[0040] in, is the repetitive controller feedforward gain, Repeat the controller output for the position domain, and denote the position domain Laplace transform and the position domain inverse Laplace transform, respectively. is the gain of the extended state observer, , , the estimated value of the total disturbance is:

[0041] ;

[0042] in, for The estimated value of , combined with the estimated value of the extended state space model, the extended state observer and the total disturbance, is obtained as the disturbance estimation error: .

[0043] In step S4, a composite control law is constructed based on the improved complex coefficient repetitive control feedforward, reconstructed state feedback and disturbance compensation as a control input.

[0044] In step S5, the composite control law is substituted into the extended state space model as a control input, and combined with the estimation error, the structural equivalent transformation is performed on the permanent magnet synchronous motor control system block diagram based on the position domain repeated extended state observer and the time domain improved complex coefficient repetitive controller, and the system is decomposed into two series subsystems, namely subsystem 1 and subsystem 2, wherein subsystem 1 is a tracking control subsystem based on the complex coefficient repetitive controller and reconstructed state feedback, and subsystem 2 is a disturbance estimation and compensation subsystem based on the position domain repeated extended state observer;

[0045] Assumption 1: Perturbation in the mechanical motion equation of the speed loop 、 and its first-order derivative , Bounded;

[0046] For subsystem 1, if Assumption 1 holds, and the following conditions also hold: (a) There is no unstable zero-pole cancellation; (b) is stable; (c) ; Then the closed-loop subsystem 1 based on repetitive controller and state feedback is asymptotically stable;

[0047] in, For arrive The transfer function of

[0048] For subsystem 2, if Assumption 1 is true, and the following conditions are also true: (1) No unstable zero-pole cancellation; (2) is stable; (3) ; Then the disturbance estimation and compensation subsystem 2 based on the position domain repeated extended state observer is exponentially stable;

[0049] in, For transfer function to the unestimated disturbance;

[0050] When optimizing parameters, the particle swarm optimization algorithm is used to repeatedly control the feedforward gain of the complex coefficients , integral control feedforward gain , reconstructed state feedback gain , position domain repetitive control feedforward gain and the extended state observer gain Perform synchronization optimization.

[0051] In order to comprehensively evaluate the tracking and anti-interference performance of the system, the following performance indicators are used:

[0052] ;

[0053] in, , and are the weight constants, is the spindle speed time cycle, t represents time, is the number of repetition cycles, 、 and Respectively Particle No. The fitness value, system tracking error and total disturbance estimation error corresponding to the iteration.

[0054] The beneficial effects of the present invention are:

[0055] (1) The position domain repetitive extended state observer achieves real-time estimation of both periodic and non-periodic disturbances in the position domain. Compared with traditional repetitive control and extended state observer methods, it improves the disturbance estimation capability of the system, thereby improving the anti-disturbance performance of the system. The introduction of phase compensation improves the steady-state performance of the system. It achieves high-precision estimation of the disturbance caused by the position periodic cutting force, thereby further improving the overall disturbance suppression performance. The two controllers can be designed separately, making the control system design more flexible.

[0056] (2) The time-domain complex coefficient repetitive controller realizes the fast and high-precision tracking of the time-domain periodic spindle speed, solves the trade-off problem between stability and control performance caused by the low-pass filter, and realizes the high-precision tracking of the time-domain periodic reference speed and dynamic compensation of the position-domain periodic disturbance by designing the composite control law, thus improving the transient and steady-state tracking performance and disturbance suppression performance of the system. Compared with the traditional repetitive control method, the introduction of the integral controller (A R The improved complex coefficient repetitive controller based on the complex coefficient filter not only improves the transient performance of the system tracking, but also reduces the steady-state tracking error.

[0057] (3) Traditional repetitive controllers and extended state observers use fixed parameters, resulting in a trade-off between tracking control, disturbance rejection, and dynamic performance. This invention uses the system stability condition as the target constraint and utilizes a particle swarm optimization algorithm to search for the optimal controller parameter combination, resulting in a closed-loop system with good steady-state tracking, disturbance rejection, and dynamic performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 This is a structural block diagram of a permanent magnet synchronous motor control system based on a position domain repetitive extended state observer and a time domain complex coefficient repetitive controller according to the present invention.

[0059] Figure 2 yes Figure 1 Equivalent block diagram.

[0060] Figure 3 It is an equivalent block diagram of subsystem 1 of the present invention.

[0061] Figure 4 It is an equivalent block diagram of subsystem 2 of the present invention.

[0062] Figure 5 It is a reference input speed signal of a complex coefficient repetitive control method based on a position domain extended state observer in one embodiment of the present invention. and output speed Comparison curve simulation results.

[0063] Figure 6 It is the control input of the complex coefficient repetitive control method based on the position domain extended state observer in one embodiment of the present invention. Simulation results.

[0064] Figure 7 The speed tracking error of the complex coefficient repetitive control method based on the position domain extended state observer in one embodiment of the present invention is Simulation results.

[0065] Figure 8 This is a time domain control input curve diagram of an embodiment of the present invention.

[0066] Figure 9 This is a position domain control input curve diagram of an embodiment of the present invention.

[0067] Figure 10 This is a time domain total disturbance estimation diagram according to an embodiment of the present invention.

[0068] Figure 11 This is a total disturbance estimation diagram in the position domain according to an embodiment of the present invention.

[0069] Figure 12 This is a time domain tool cutting trajectory tracking diagram of an embodiment of the present invention.

[0070] Figure 13 This is a position domain tool cutting trajectory tracking diagram of an embodiment of the present invention.

[0071] Figure 14 This is a time domain tool cutting trajectory tracking error diagram of an embodiment of the present invention.

[0072] Figure 15 This is a tool cutting trajectory tracking error diagram in the position domain according to an embodiment of the present invention.

[0073] Figure 16 This is a comparison of the speed tracking error between the ICR_ESO method according to an embodiment of the present invention and the method according to the present invention.

[0074] Figure 17 This is an embodiment of the present invention, and a comparison of the speed tracking error between the method of the present invention and the IRC_SRCESO method. DETAILED DESCRIPTION

[0075] The following describes the specific embodiments of the present invention in detail with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only used to illustrate and explain the present invention and are not intended to limit the present invention.

[0076] For ease of description, spatially relative terms such as "above," "above," "on the upper surface of," and "upper" may be used herein to describe the spatial positional relationship of a device or feature to other devices or features as shown in the figures. It should be understood that spatially relative terms are intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if a device in a drawing is inverted, a device described as "above" or "on top of" another device or structure would then be positioned as "below" or "below" the other device or structure. Thus, the exemplary term "above" can include both the "above" and "below" orientations. The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatially relative descriptions used herein should be interpreted accordingly.

[0077] A robust complex coefficient repetitive control method for a permanent magnet synchronous motor based on a position domain internal model is proposed. This method targets a spindle drive servo system with non-periodic disturbances in the time domain and periodic disturbances in the position domain caused by cutting forces. An improved time domain complex coefficient repetitive controller and a position domain repetitive extended state observer are constructed to construct a composite control law as the control input. , ensuring a closed-loop system ( Figure 1 ) while achieving a stable time domain periodic reference input signal Fast and high-precision tracking and active compensation of non-periodic disturbances and position periodic disturbances.

[0078] In this scheme, the spindle drive motor control system structure based on the position domain repetitive extended state observer and the time domain improved complex coefficient repetitive controller is as follows: Figure 1As shown in the figure, it includes four parts: a control object, a position domain repetitive extended state observer, a time domain improved complex coefficient repetitive controller, and a composite control law based on disturbance compensation, reconstructed state feedback and feedforward control.

[0079] The control method comprises the following steps:

[0080] Step S1: Constructing a mechanical motion equation of a speed loop of a driving motor, wherein the driving motor drives a non-circular turning spindle. The mechanical motion equation takes into account parameter perturbations, disturbances caused by periodic cutting forces in the position domain, and external disturbances.

[0081] This step includes the following steps:

[0082] Step S11: Construct a cutting force model based on the empirical formula of cutting force and the angular position of the spindle. Assume that the angular displacement of the workpiece as the spindle rotates is:

[0083] (1)

[0084] in, is the initial angular displacement, is the instantaneous angular velocity of the main shaft, The angular displacement time variable representing the spindle rotation motion is due to the spindle speed:

[0085] (2)

[0086] This means There exists an inverse function: . is the main axis rotation angular velocity.

[0087] Define a domain conversion operator as:

[0088] (3)

[0089] is the space of Lebesgue square integrable functions on the position domain, is the weighted function space in the time domain, is the time domain signal, is the signal in the position domain.

[0090] Take the elliptical cross-section machining part as an example, let the major axis be a, the minor axis be b, and θ be the rotation angle position. When the workpiece rotates, the tool displacement z (mm) and the cutting speed are (m / s) is calculated as follows:

[0091] (4)

[0092] The tangential cutting force of the tool on the workpiece Usually calculated by the exponential empirical formula:

[0093] (5)

[0094] in, is the influence coefficient of workpiece, tool material, etc. on the tangential force component, and f is the feed rate, that is, the distance the tool moves along the workpiece surface during the machining process. c 、f c 、n c They are the exponential influence coefficients of back cutting depth, feed rate and cutting speed on the tangential force component, which are all positive numbers and can be calculated by looking up the table.

[0095] According to the torque formula, the disturbance caused by the cutting force can be obtained as:

[0096] (6)

[0097] From (4)-(6), we can see that when the spindle is running at variable speed, It is a trigonometric function with a known position period of π and a time period caused by the cutting force, that is, The position period is (see the introduction to formula (9) below).

[0098] Step S12: Constructing the mechanical motion equation of the speed loop of the permanent magnet synchronous drive motor.

[0099] According to the spindle drive characteristics, the surface-mounted permanent magnet synchronous motor is considered as the non-circular turning spindle drive motor. In the dq axis reference coordinate system, its speed loop mechanical motion equation is: for:

[0100] (7)

[0101] in, is the q-axis stator current, J is the moment of inertia, p is the number of rotor poles, is the magnetic link, is the viscous damping coefficient, is the motor mechanical angular velocity, is the disturbance caused by the position periodic cutting force, For external disturbances.

[0102] The influence of load changes, thermal effects, machine tool vibration and other factors during the turning process will cause perturbations in electrical and mechanical parameters. Assume:

[0103] (8)

[0104] in, 、 is the nominal value, 、 are the perturbations of the permanent magnet magnetic flux and moment of inertia respectively.

[0105] Taking parameter perturbation, position domain period (disturbance caused by cutting force and external disturbance) as total disturbance, Equation (7) can be rewritten as:

[0106] (9)

[0107] in, , represents the total disturbance, is a non-periodic disturbance including parameter perturbation and external disturbance, It is the position periodic disturbance caused by cutting force, that is, the disturbance caused by periodic cutting force in position domain.

[0108] Assumption 1: (9) where the disturbance 、 and its first-order derivative , Bounded.

[0109] Step S2: Using the time-domain periodic characteristics of the spindle speed, a time-domain improved complex coefficient repetitive controller is constructed.

[0110] In this step, in order to achieve high-precision tracking of the spindle speed and solve the trade-off between stability and control performance caused by the filter, the spindle speed time period is used. Design a complex coefficient repetitive controller. Since the basic speed of the spindle is a constant, introduce its signal internal model , thereby improving the transient performance of the control system. To this end, an improved complex coefficient repetitive controller (ICR) is proposed, whose transfer function is:

[0111] (10)

[0112] Where s is the Laplace operator, represents the amplitude of the Laplace transform of the basic speed signal, is a complex coefficient filter:

[0113] (11)

[0114] represents the fundamental frequency of the periodic signal, n is the order of the harmonic signal suppression, is the cutoff frequency, j is the imaginary unit. Let the input signal of the complex coefficient filter be , the output is ,satisfy:

[0115] (12)

[0116] in, It represents the output signal of the input signal after the time delay link, T is the time domain period of the spindle speed, is the spindle speed tracking error in the time domain.

[0117] The filtering process of the complex coefficient filter is Indicates that the complex signal The expression is:

[0118] (13)

[0119] Complex signal It is constructed through the first-order inertia link. The transfer function of the first-order inertia link is for:

[0120] (14)

[0121] in, yes The imaginary signal obtained by the first-order inertia link is Its derivative, a is used to adjust Amplitude, β is used to adjust Phase. The inverse Laplace transform yields the time domain differential equation for extracting the imaginary part of the signal:

[0122] (15)

[0123] Filtered complex signal The expression is:

[0124] (16)

[0125] in, represents the real part of the complex signal after filtering, represents the imaginary part of the complex signal after filtering. and They are and The Laplace transform of .

[0126] According to (11) to (16), the differential equations for extracting the real and imaginary parts of the complex signal are obtained:

[0127] (17)

[0128] in, , represents the real derivative of the complex signal after filtering, represents the imaginary derivative of the filtered complex signal.

[0129] from arrive The transfer function for:

[0130] (18)

[0131] Step S3: Based on the position-domain periodic characteristics of the cutting force, a position-domain repeated expansion state observer is constructed to online estimate non-periodic disturbances such as parameter perturbations and position-domain periodic disturbances caused by the cutting force.

[0132] Step S31: Design an extended state observer.

[0133] Assume that the state vector of the permanent magnet synchronous motor speed loop system is , the system output is , the control input is , total disturbance The first-order derivative of is , the expanded state space model of formula (9) is:

[0134] (19)

[0135] in, is the first-order derivative of the state vector, and the system matrices are:

[0136]

[0137] According to formula (19), the extended state observer is designed as:

[0138] (20)

[0139] in, is the state vector of the observer, and They are and Estimates, is the state vector matrix, is the control input matrix, is the observer gain, represents the output variable of the observer, represents the derivative of the observer state vector.

[0140] Since the traditional extended state observer cannot realize the position periodic disturbance caused by cutting force In order to further improve the estimation accuracy of the total disturbance, the present invention proposes a position domain repeated extended state observer.

[0141] Step S32: Construct an improved position domain repetitive controller based on the position domain periodic characteristics of the disturbance caused by the cutting force, and define the signal The position domain Laplace transform of is:

[0142] (twenty one)

[0143] in, is the Laplace transform operator in the position domain. Then, the periodic disturbance in the position domain caused by the cutting force is The position domain Laplace transform of is:

[0144] (twenty two)

[0145] because The dynamic behavior of is dominated by the denominator, so Considered as a disturbance The model within the position domain.

[0146] Therefore, the transfer function of the improved position domain repetitive controller is:

[0147] (twenty three)

[0148] Among them, the first-order low-pass filter Used to ensure system stability.

[0149] A first-order low-pass filter using zero initial conditions in the position domain :

[0150] (twenty four)

[0151] in, and are the state and input of the low-pass filter, is a time constant, and the position domain first-order low-pass filter satisfies the following amplitude characteristics:

[0152] (25)

[0153] in, is the current frequency, To suppress the maximum angular frequency of the disturbance signal caused by the cutting force. In addition, the embedding of the filter will cause phase lag, and the lag value is:

[0154] (26)

[0155] In formula (23) For phase correction, compensation exist The phase lag produced at the fundamental and harmonics of the frequency. To this end, the correction factor is selected , such that:

[0156] (27)

[0157] This means that at a certain frequency, The phase lag caused by this can be corrected by the periodic correction unit Precise compensation.

[0158] Step S33: Construct a position domain repeated expansion state observer. is embedded into the extended state observer, such as Figure 1 As shown, through the domain conversion operator and The mutual conversion between time domain signal and position domain signal is realized. The estimated value of the unestimated part of the disturbance is:

[0159] (28)

[0160] in, is the repetitive controller feedforward gain, Repeat the controller output for the position domain, and denote the position domain Laplace transform and the position domain inverse Laplace transform, respectively. , is the estimated error of the reconstructed state.

[0161] Therefore, the estimate of the total disturbance becomes:

[0162] (29)

[0163] in, for The estimated value of , combined with (19) (20) and (29), the perturbation estimation error is .

[0164] Step S4: Construct a composite control law to realize the system's feedforward compensation for periodic and non-periodic disturbances in the position domain and fast and high-precision tracking of the periodic reference speed signal in the time domain.

[0165] Based on the improved complex coefficient repetitive control feedforward, reconstructed state feedback and disturbance compensation, a composite control law is constructed as the control input:

[0166] (30)

[0167] in, (31)

[0168] is the integral output signal, , is the system tracking error, is the reconstructed state feedback gain, is the complex coefficient repetitive control feedforward gain; is the integral feedforward gain, is the disturbance compensation gain, and its value is , Permanent magnet synchronous motor q Coefficient of shaft current.

[0169] Step S5: Provide system stability conditions and parameter optimization methods.

[0170] Step S51: deriving system stability conditions.

[0171] Substitute (30) into (19) and combine the estimated error to obtain the attached Figure 1 Perform structural equivalent transformation to obtain Figure 2 The original system is decomposed into two subsystems connected in series, where subsystem 1 above the dotted line is a tracking control subsystem based on a complex coefficient repetitive controller and reconstructed state feedback, and its dynamic equation is:

[0172] (32)

[0173] Below the dotted line is the disturbance compensation subsystem 2, whose output is:

[0174] (33)

[0175] Figure 2 In total disturbance To system output The transfer function is:

[0176] (34)

[0177] in, is the total disturbance The Laplace transform of , From the total disturbance to the disturbance estimation error The transfer function, , ;

[0178] (35)

[0179] For To output The transfer function of Stable, then subsystem 1 is stable. is the error from disturbance estimation to The transfer function of and If both are stable, then subsystem 2 is stable. Subsystem 1 and subsystem 2 are connected in series. According to the separation design principle, if both subsystems are stable, the stability of the entire system can be guaranteed.

[0180] For subsystem 1, the embedding of repeated controller is not considered. arrive The transfer function is:

[0181] (36)

[0182] Will Figure 2 Neutron system 1 undergoes structural equivalent transformation to obtain Figure 3 .set up:

[0183] (37)

[0184] in Represents the largest singular value.

[0185] If subsystem 1 ( Figure 3 ) satisfies assumption 1, and the following conditions are also true:

[0186] (a) There is no unstable zero-pole cancellation;

[0187] (b) is stable;

[0188] (c) ;

[0189] Then the closed-loop subsystem 1 based on repetitive controller and state feedback is asymptotically stable. (See formula (18)).

[0190] prove:

[0191] The structural equivalent transformation of subsystem 1 is as follows Figure 3 As shown, conditions (a) and (b) ensure that the original closed-loop subsystem 1 without the repetitive controller is asymptotically stable. From the small gain theorem, we can see that conditions (b) and (c) ensure that the control system of the closed-loop subsystem 1 is stable, so subsystem 1 is stable.

[0192] For subsystem 2, let , without considering the embedding of the position domain repeat controller, The transfer function to the unestimated disturbance is:

[0193] (38)

[0194] in, is the position domain repetitive control feedforward gain.

[0195] In the present invention, are the different gains of the extended state observer.

[0196] Will Figure 2 Neutron system 2 undergoes structural equivalent transformation to obtain Figure 4 .

[0197] For subsystem 2 ( Figure 4 ), if assumption 1 is established and the following conditions are also established:

[0198] (1) There is no unstable zero-pole cancellation;

[0199] (2) is stable;

[0200] (3) ;

[0201] but Figure 4 The disturbance estimation and compensation subsystem 2 based on the position domain repeated extended state observer is shown to be exponentially stable.

[0202] prove:

[0203] Perform structural equivalent transformation on subsystem 2 to obtain the equivalent system structure diagram ( Figure 4 ),in is a phase compensation factor, Denotes the position domain period of the disturbance, which is π. Conditions (1) and (2) ensure that the original closed-loop system without the repetitive controller is asymptotically stable.

[0204] ,and , by the small gain theorem, conditions (1) to (3) ensure that the closed-loop subsystem 2 is exponentially stable.

[0205] Step S52: Particle swarm optimization algorithm design.

[0206] The present invention adopts particle swarm optimization algorithm to repeatedly control the feedforward gain of complex coefficients , integral control feedforward gain , reconstructed state feedback gain , position domain repetitive control feedforward gain and the extended state observer gain Perform synchronous optimization. During iteration, the particle speed and position updates follow the following rules:

[0207] (39)

[0208] Where k is the current iteration number, , represent the velocity and position of the i-th particle at the k-th iteration, and are the best positions of the i-th particle and the entire particle swarm in the k-th iteration, 、 is the learning factor, and both are greater than zero. and is a random number in the interval (0,1). is the inertia weight, whose value decreases linearly with the number of iterations k:

[0209] (40)

[0210] Where K is the total number of iterations, and are the maximum and minimum values of the inertia weight respectively. It is larger in the early stage of iteration, which helps to enhance the global search capability, and smaller in the later stage, which helps to improve the local search capability.

[0211] In order to comprehensively evaluate the tracking and anti-interference performance of the system, the following performance indicators are used:

[0212] (41)

[0213] in, , and is the weight constant of each item, N is the number of repeated cycles, 、 and .

[0214] are the fitness value, system tracking error, and total disturbance estimation error corresponding to the i-th particle, respectively. The first two items are used to evaluate the tracking performance of the system, and the third item is used to evaluate the anti-disturbance performance of the system.

[0215] A specific numerical example is given below.

[0216] The method proposed in this invention is applied to the speed control of the permanent magnet synchronous drive motor of the non-circular turning spindle. The parameters and variables are listed in Table 1.

[0217] The experiment was set up using aluminum alloy material and diamond cutting tools to machine an elliptical part with a long axis a = 37 mm and a short axis b = 36.75 mm. According to the existing reference cutting force influence coefficients are: =504.2, =0.7, =0.72, =0.05. When the spindle speed n=25r / s and the feed rate is 0.16667mm / r. Assume that the spindle angular velocity and speed are:

[0218] (42)

[0219] (43)

[0220] Table 1 Parameters and variables of permanent magnet synchronous drive motor

[0221]

[0222] From the above formula, we can see that the basic spindle speed is =1500r / min, speed variation ±300r / min, the period of variable reference input signal is =0.2s, the fundamental frequency is =10πrad / s, maximum angular velocity =60πrad / s. Select the cutoff frequency =5 =50πrad / s, a=10π, β=6.283. According to formula (6), the cutting force position period is =π, =2, take =3π.

[0223] Select the relevant parameters of the particle swarm optimization algorithm: the number of particles is 40, the number of optimization parameters is 5, the maximum number of iterations K=40, the number of repeated control cycles N=15, the learning factor = =1.45, inertia weight =0.9, =0.4, the weights of the objective function =0.6, =1, =0.3. The controller parameter search range is as follows:

[0224] ;

[0225] The parameters obtained by solving are:

[0226] (44)

[0227] correspond: ;

[0228] Therefore, all system stability conditions are met and the control system based on the proposed method is stable.

[0229] Figure 5 、 6 Figures 7 and 8 show that the closed-loop control system of a permanent magnet synchronous motor based on a position-domain repetitive extended state observer and a complex coefficient repetitive controller is stable. The reference input signal tracked is in the acceleration phase from 0 to 0.2 seconds, and in the speed cycle phase from t > 0.2 seconds, with a time period of 0.2 seconds. The control input curve shows non-periodic fluctuations. This shows that the proposed method not only tracks the time-periodic reference input signal but also suppresses position-periodic disturbances. During the acceleration phase from 0 to 0.2 seconds, the steady-state error is 0.00238 r / min. During the speed cycle phase after 0.2 seconds, the maximum steady-state tracking error is 0.002994 r / min, which is 0.0001663% of the reference speed.

[0230] from Figure 8 、 9 It can be seen that the control input curve fluctuation varies with respect to the time period, while the position period is fixed at π. Figure 10 、 11 It can be found that the total disturbance The total disturbance curve estimated by this method almost completely coincides with the total disturbance curve, including position periodic disturbances. Therefore, the proposed method not only tracks the spindle speed in the time domain, but also achieves high-precision estimation of position periodic disturbances and non-periodic disturbances.

[0231] According to the tool displacement formula, the tool trajectory tracking effect is as follows Figure 12 、 13 and Figure 14 、 15 When the spindle is speed-varied for non-circular turning, the tool cutting displacement changes in time domain period while the position period is fixed at π. Figure 12 、 13 As can be seen from the figure, the actual trajectory is highly consistent with the expected trajectory, indicating that the accuracy of non-circular turning can be effectively improved by improving the tracking accuracy of the spindle speed control. Figure 14 、 15 The maximum trajectory steady-state tracking error was observed to be 0.0000148 mm, which is 0.00592% of the desired tool displacement.

[0232] In order to verify the effectiveness of the position domain repeated extended state observer introduced in this paper, the proposed method is compared with the complex coefficient repeated method based on the extended state observer (ICR_ESO). To ensure the fairness of the comparison, the same parameters are selected. The simulation comparison results are shown in Figure 2. Figure 16As shown in the figure, the tracking error curve of the complex coefficient repetitive method based on the extended state observer has large non-periodic fluctuations, with a maximum tracking error of 0.4646 r / min, which is 0.0258% of the reference input signal, seriously affecting the machining accuracy of non-circular turning. Clearly, the introduction of the position domain repetitive internal model significantly suppresses the position periodic disturbance caused by the cutting force and improves the steady-state tracking accuracy of the reference speed.

[0233] In order to verify the superiority of introducing complex coefficient repetitive controller to control the system, the proposed method is compared with the control method based on position domain repetitive extended state observer and improved repetitive controller (IRC_SRCESO), and the parameters are exactly the same. Figure 17 It was observed that the tracking error of the method of the present invention is smaller than that of the latter in both transient and steady states. The control system based on the position domain repetitive extended state observer and the improved repetitive controller has a steady-state error of 0.238 r / min in the acceleration stage of 0 to 0.2s; in the speed cycle operation stage after 0.2s, the maximum steady-state tracking error is 0.3018 r / min, which is 0.01677% of the reference input. The tool cutting estimated tracking error based on this method is 0.001488 mm, which is 0.5952% of the expected trajectory. Therefore, the use of a complex coefficient repetitive controller in the present invention can further improve the tracking accuracy of the spindle speed, which also illustrates the superiority of the method proposed in the present invention.

[0234] Results show that while the improved repetitive control method can improve the transient performance of spindle acceleration, the improved complex coefficient repetitive controller can not only improve transient performance but also further enhance steady-state tracking accuracy, thereby improving non-circular turning accuracy. Furthermore, the proposed position-domain repetitive extended state observer can achieve high-precision estimation of non-periodic disturbances such as parameter perturbations, as well as periodic position disturbances caused by cutting forces, further improving non-circular turning accuracy.

[0235] Finally, it is necessary to explain here that the above embodiments are only used to further illustrate the technical solution of the present invention in detail and cannot be understood as limiting the scope of protection of the present invention. Some non-essential improvements and adjustments made by technicians in this field based on the above content of the present invention all fall within the scope of protection of the present invention.

Claims

1. A robust complex coefficient repetitive control method for a permanent magnet synchronous motor based on a position domain internal model, characterized in that: The steps include: Step S1: constructing a mechanical motion equation of a speed loop of a drive motor, wherein the drive motor drives a non-circular turning spindle, wherein the mechanical motion equation takes into account parameter perturbations, disturbances caused by position periodic cutting forces, and external disturbances; Step S2: Using the time-domain periodic characteristics of the spindle speed, a time-domain improved complex coefficient repetitive controller is constructed; wherein the transfer function of the time-domain improved complex coefficient repetitive controller is: ； in, is the internal model of the spindle signal, s is the Laplace operator, is a complex coefficient filter, is the cutoff frequency, j is the imaginary unit, n is the order of the suppressed harmonic signal, is the imaginary part of the complex coefficient, is the spindle speed time cycle; Step S3: Based on the position-domain periodic characteristics of the cutting force, a position-domain repeated expansion state observer is constructed to online estimate the non-periodic disturbance and the position-domain periodic disturbance caused by the cutting force; Step S4: constructing a composite control law to achieve feedforward compensation for periodic and non-periodic disturbances in the position domain and fast and high-precision tracking of the periodic reference speed signal in the time domain; Step S5: Provide system stability conditions and parameter optimization.

2. The control method according to claim 1, wherein: Step S1 includes the following steps: Step S11: constructing a cutting force model based on the cutting force empirical formula and the spindle rotation angle position; Step S12: Determine the type of the drive motor and construct a speed loop mechanical motion equation of the drive motor.

3. The control method according to claim 1 or 2, characterized in that: The mechanical motion equation of the speed loop constructed in step S1 is: ； in, is the spindle rotation angular acceleration, Permanent magnet synchronous motor q The coefficient of shaft current, , p is the number of rotor pole pairs, 、 is the nominal value, is the q-axis stator current, represents the total disturbance including parameter perturbation, disturbance caused by position periodic cutting force and external disturbance, is a non-periodic disturbance including parameter perturbation and external disturbance, is the position periodic disturbance caused by cutting force, 、 are the perturbations of the permanent magnet flux linkage and moment of inertia, is the viscous damping coefficient, is the main axis rotation angular velocity, is an external disturbance, is the disturbance caused by the position periodic cutting force.

4. The control method according to claim 3, wherein: In step S2, the transfer function from the input signal of the complex coefficient filter to the output signal of the complex coefficient filter is: ； in, For adjustment Phase, yes The imaginary signal obtained by the first-order inertia link is For adjustment Amplitude.

5. The control method according to claim 3 or 4, characterized in that: Step S3 includes the following steps: Step S31: designing an extended state observer; Step S32: construct an improved position domain repetitive controller based on the position periodic characteristics of the disturbance caused by the cutting force. Step S33: Construct a position domain repeated expansion state observer.

6. The control method according to claim 5, characterized in that: The expanded state space model of the mechanical motion equation is: ； ； in, is the first-order derivative of the state vector, is the state vector of the permanent magnet synchronous motor speed loop system, is the control input, is the first-order derivative of the total disturbance, is the system output; In step S31, an extended state observer is designed according to the extended state space model as follows: ； in, represents the derivative of the observer state vector, is the state vector of the observer, and They are and Estimates of , T represents the transpose of the matrix, is the observer gain, 、 are the different gains of the extended state observer; In step S32, the transfer function of the improved position domain repetitive controller is: ； in, is a first-order low-pass filter, is the time-delay link in the position domain, is the position period, is the position domain Laplace transform operator, is the correction factor; In step S33, the transfer function in step S32 is embedded in the extended state observer, and the mutual conversion between the time domain signal and the position domain signal is realized through the domain conversion operator, and the estimated value of the unestimated partial disturbance is obtained: , in, is the repetitive controller feedforward gain, Repeat the controller output for the position domain, and denote the position domain Laplace transform and the position domain inverse Laplace transform, respectively. is the gain of the extended state observer, , , The total disturbance is estimated to be: ； in, for The estimated value of , combined with the estimated value of the extended state space model, the extended state observer and the total disturbance, is obtained as the disturbance estimation error: .

7. The control method according to claim 1, wherein: In step S4, a composite control law is constructed based on the improved complex coefficient repetitive control feedforward, reconstructed state feedback and disturbance compensation as a control input.

8. The control method according to claim 6, wherein: In step S5, the composite control law is substituted into the extended state space model as a control input, and combined with the estimation error, the structural equivalent transformation is performed on the permanent magnet synchronous motor control system block diagram based on the position domain repeated extended state observer and the time domain improved complex coefficient repetitive controller, and the system is decomposed into two series subsystems, namely subsystem 1 and subsystem 2, wherein subsystem 1 is a tracking control subsystem based on the complex coefficient repetitive controller and reconstructed state feedback, and subsystem 2 is a disturbance estimation and compensation subsystem based on the position domain repeated extended state observer; Assumption 1: Perturbation in the mechanical motion equation of the speed loop 、 and its first-order derivative , Bounded, For subsystem 1, if Assumption 1 holds, and the following conditions also hold: (a) There is no unstable zero-pole cancellation; (b) is stable; (c) ; Then the closed-loop subsystem 1 based on repetitive controller and state feedback is asymptotically stable; in, For arrive The transfer function, is the output signal of the complex coefficient filter, For subsystem 2, if Assumption 1 is true, and the following conditions are also true: (1) No unstable zero-pole cancellation; (2) is stable; (3) ; Then the disturbance estimation and compensation subsystem 2 based on the position domain repeated extended state observer is exponentially stable; in, For to the unestimated disturbance; where, When optimizing parameters, the particle swarm optimization algorithm is used to repeatedly control the feedforward gain of the complex coefficients. , integral control feedforward gain , reconstructed state feedback gain , position domain repetitive control feedforward gain and the extended state observer gain Perform synchronization optimization.

9. The control method according to claim 8, characterized in that: In order to comprehensively evaluate the tracking and anti-interference performance of the system, the following performance indicators are used: ； in, , and are the weight constants, is the spindle speed time cycle, t represents time, N is the number of repeated cycles, 、 and are the fitness value, system tracking error and total disturbance estimation error corresponding to the i-th particle at the k-th iteration, respectively.

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