Improved Complex Coefficient Repetitive Control Method for Permanent Magnet Synchronous Motor Based on Disturbance Compensation

Abstract

The present invention discloses an improved complex coefficient repetitive control method for a permanent magnet synchronous motor based on disturbance compensation, which relates to the technical field of permanent magnet synchronous motor control. First, according to the speed loop of the permanent magnet synchronous drive motor for the non-circular cutting spindle, an extended state observer is established to estimate and actively compensate the aperiodic disturbance in real time; then, by using the periodic characteristics of the spindle speed change, a complex coefficient repetitive controller is constructed, and combined with the internal model of the basic speed of the spindle, an improved complex coefficient repetitive controller is built; then, based on the Lyapunov stability theory, the stability condition of the closed-loop system is obtained, and the controller parameters are solved based on linear matrix inequalities. The improved complex coefficient repetitive control method for a permanent magnet synchronous motor based on disturbance compensation of the present invention designs an improved complex coefficient repetitive control system based on an extended state observer, which not only realizes fast and high-precision tracking of aperiodic speed signals and periodic speed signals, but also realizes high-precision suppression of aperiodic disturbances and different harmonic periodic disturbances.

Description

Technical Field

[0001] The present invention relates to the technical field of permanent magnet synchronous motor control, and particularly to an improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation. Background Art

[0002] During the process of non-circular NC turning of complex parts, if a constant rotational speed turning method is adopted, due to the difference in the surface arc length per unit angle of non-circular workpieces, the cutting force will fluctuate greatly with the change of position, which is extremely likely to cause chatter. This will not only directly reduce the machining accuracy, but seriously affect the stability of the machining process in severe cases. Spindle speed variation machining is an effective way to solve this problem. By changing the spindle rotation speed, the relative instantaneous speed of the tool at the cutting point can be kept constant, thereby maintaining the relative stability of the cutting force and effectively suppressing the chatter phenomenon. Since the spindle speed tracking accuracy has an important impact on the machining accuracy of workpieces, it is particularly necessary to propose a corresponding control algorithm to ensure the high-precision tracking of the spindle speed. Spindle speed variation machining means that the rotational speed changes according to a sine excitation signal on the basic rotational speed (i.e., the rotational speed of constant-speed turning), so that the reference rotational speed signal has a clear time periodicity. Although the repetitive control method based on the internal model principle can track and suppress periodic signals with high precision, there is no learning effect in the first cycle and the response starts from the second cycle. Therefore, making full use of its self-learning characteristics and optimizing the response speed is extremely crucial for improving the machining accuracy and transient performance.

[0003] In addition, during the machining process, there will inevitably be non-periodic disturbances such as parameter perturbations and periodic disturbances such as current harmonics. Therefore, while ensuring the fast and high-precision tracking of the spindle speed, it is also necessary to achieve the suppression of non-periodic disturbances and different harmonic periodic disturbances in order to achieve high-precision machining of non-circular turning. Summary of the Invention

[0004] In view of the above technical problems to be solved, the present invention provides an improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation.

[0005] The technical solution proposed by the present invention to solve the above technical problems is as follows:

[0006] An improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation. First, according to the speed loop of the permanent magnet synchronous drive motor of the non-circular turning spindle, an extended state observer is established to estimate and actively compensate the disturbance in real time; then, using the periodic characteristics of the spindle speed variation, a complex coefficient repetitive controller is constructed, and combined with the internal model of the spindle basic speed, an improved complex coefficient repetitive controller is built; then, based on the Lyapunov stability theory, the stability conditions of the closed-loop system are obtained, and the controller parameters are solved based on linear matrix inequalities.

[0007] As a further improvement of the above technical solution:

[0008] Preferably, the transfer function of the improved complex coefficient repetitive controller is:

[0009] ;

[0010] Wherein, is the internal model of the base speed , represents the magnitude of the Laplace transform of the base speed signal, represents the time delay link of the periodic signal, s represents the Laplace transform operator, T represents the time period of the periodic reference speed and the periodic disturbance, is a complex coefficient filter, and the transfer function of the complex coefficient filter is:

[0011] ;

[0012] Wherein, represents the fundamental frequency of the periodic signal, is the order for suppressing harmonic signals;

[0013] The frequency characteristic of the improved complex coefficient repetitive controller is:

[0014] ;

[0015] Wherein, j represents the imaginary unit, k represents the harmonic order, is the cut-off angular frequency of the low-pass filter.

[0016] Preferably, let the output signal of the integrator in the improved complex coefficient repetitive controller be , and obtain . The composite control law constructed based on the improved complex coefficient repetitive control, state feedback, and disturbance compensation is:

[0017] ;

[0018] Wherein, ;

[0019] Wherein, is the system tracking error, is the state feedback gain, is the integral feedforward gain, is the repetitive control feedforward gain; is the disturbance compensation gain, and its value is , is the input coefficient of the d-axis current loop of the permanent magnet synchronous motor . is the estimated mechanical angular velocity of the motor ; is the estimated total disturbance ; represents the output of the improved complex coefficient repetitive controller represents the input signal of the integrator in the improved complex coefficient repetitive controller represents the real part of the filtered complex signal.

[0020] Preferably, the filtering process of the complex coefficient filter is represented by where the complex signal has the expression:

[0021] ;

[0022] In the formula, is the real part of the complex signal, is the imaginary part of the complex signal, is the complex coefficient filter; the filtered complex signal has the expression:

[0023] ;

[0024] wherein, represents the signal after being filtered by the complex coefficient filter, represents the real part of the filtered complex signal, represents the imaginary part of the filtered complex signal, and are respectively and the Laplace transforms of;

[0025] The differential equations for extracting the real and imaginary parts of the complex signal are obtained:

[0026] ;

[0027] wherein, represents the derivative of the real part of the filtered complex signal, represents the derivative of the imaginary part of the filtered complex signal.

[0028] Preferably, in the complex coefficient filter, the transfer function from to is:

[0029] ;

[0030] wherein, and are adjustment parameters, adjusts The amplitude of Adjust The phase of It shows the actual filter transfer function when implementing a complex coefficient filter When, by adjusting And Change the transfer function of the filter.

[0031] Preferably, the dynamic equation of the extended state observer is:

[0032] ;

[0033] Where Is the state vector of the observer, Is the state observer gain, Is the composite control law, Is the system output, Represents the output variable of the state observer, Represents the derivative of the state vector of the state observer, A Represents the extended state matrix, B Represents the extended state input matrix, C Represents the extended state output matrix;

[0034] Define the state estimation error as:

[0035] ;

[0036] Where Is the system state vector, Represents the state estimation error, Is the rotational speed estimation error, Is the disturbance estimation error; The estimated error state equation is obtained as:

[0037] ;

[0038] Where Represents the derivative of the state estimation error, E Represents the extended state derivative matrix, Represents the derivative of the total disturbance.

[0039] Preferably, the stability condition is obtained in the following manner:

[0040] For a given positive scalar And , if there exists a positive definite matrix , And a matrix of appropriate dimension And Such that the linear matrix inequality holds:

[0041] ;

[0042] Among them, ;

[0043] ;

[0044] Among them, 、 、 、 、 、 、 、 、 、 、 、 are intermediate variables, used to adjust the state feedback gain , used to adjust the state observer gain , and used to adjust the repetitive control feedforward gain , and used to adjust the integral feedforward gain .

[0045] Preferably, the controller parameters include the state feedback gain , the repetitive control feedforward gain , the integral feedforward gain and the state observer gain , designed as:

[0046] ;

[0047] Among them, C represents the extended state output matrix.

[0048] The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation provided by the present invention has the following advantages compared with the prior art:

[0049] (1) The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation of the present invention. The improved complex coefficient repetitive controller breaks through the stability limitation of the original architecture by introducing a complex coefficient filter into the traditional repetitive controller, effectively solving the trade-off contradiction between stability and control performance of traditional filters. For the composite speed tracking of the drive motor under the spindle variable speed machining condition (i.e., superimposing a sinusoidal fluctuation on the basic speed signal (constant speed)), this controller forms a dual internal model structure by paralleling the integral link (the internal model corresponding to the basic speed) and the complex coefficient repetitive controller, which not only overcomes the defect that the traditional repetitive controller cannot track in the first cycle, but also significantly improves the transient response performance. The improved controller has both multi-modal tracking and disturbance suppression capabilities: on the one hand, it can achieve high-precision and fast tracking of aperiodic signals (such as acceleration commands, constant value settings) and periodic signals (such as sinusoidal fluctuations); on the other hand, by using the frequency domain characteristics of the complex coefficient filter, it can accurately suppress periodic disturbances with different harmonic frequencies, and at the same time enhance the system robustness through the synergistic effect of the dual internal models, reducing overshoot and oscillation in the initial adjustment stage while ensuring the steady-state accuracy.

[0050] (2) The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation of the present invention changes the first-order low-pass filter in the traditional repetitive controller to a complex coefficient filter, solves the trade-off problem between stability and control performance, and improves the tracking accuracy of the system; when implementing the complex coefficient filter, by adjusting two parameters to change the amplitude and phase of the imaginary part signal, the flexibility of system design is improved.

[0051] (3) The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation of the present invention introduces the internal model of the spindle basic speed, improves the transient performance of the system response, and effectively solves the problem that the inability to track in the first cycle of the repetitive controller has no impact on the system performance. <�

[0052] (4) The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation of the present invention, compared with the complex coefficient repetitive controller, combines with an extended state observer to achieve real-time estimation and active compensation estimation of parametric uncertainties, and the two can be designed separately, improving the flexibility of control system design; according to the system stability conditions, the controller parameters are calculated using linear matrix inequalities, making the system have satisfactory uncertainty suppression ability and periodic reference signal tracking performance. Description of the Drawings

[0053] Figure 1 is the block diagram of the permanent magnet synchronous motor control system of the present invention.

[0054] Figure 2 is the block diagram of the complex coefficient filtering process of the present invention.

[0055] Figure 3It is a specific implementation block diagram of the complex coefficient filtering process of the present invention.

[0056] Figure 4 is the reference input speed signal in the experimental verification of the present invention and the output speed comparison curve.

[0057] Figure 5 is the control input in the experimental verification of the present invention curve.

[0058] Figure 6 is the speed tracking error in the experimental verification of the present invention curve.

[0059] Figure 7 is the time-domain tracking curve of the tool cutting trajectory in the experimental verification of the present invention.

[0060] Figure 8 is the position-domain tracking curve of the tool cutting trajectory in the experimental verification of the present invention.

[0061] Figure 9 is the time-domain tracking error curve of the tool cutting trajectory in the experimental verification of the present invention.

[0062] Figure 10 is the position-domain tracking error curve of the tool cutting trajectory in the experimental verification of the present invention.

[0063] Figure 11 is the speed tracking error of the present invention in the experimental verification without introducing an integrator (complex coefficient repetitive controller CFR) and with introducing an integrator (improved complex coefficient repetitive controller ICR) comparison curve.

[0064] Figure 12 is the speed tracking error of the present invention in the experimental verification using a traditional filter (improved repetitive controller IRC) and a complex coefficient filter (improved complex coefficient repetitive controller ICR) comparison curve. Specific Embodiments

[0065] The following details the specific embodiments of the present invention. It should be understood that the specific embodiments described herein are only for explaining and illustrating the present invention and are not used to limit the present invention.

[0066] Based on the disturbance compensation improved complex coefficient repetitive control method for permanent magnet synchronous motors of the present invention, for the spindle drive servo system with parametric uncertainties and external disturbances, the control input is designed , while ensuring the robust stability of the entire closed-loop system, realizing the periodic reference input signal High-precision tracking. In the present invention, an extended state observer is designed in the speed loop of the permanent magnet synchronous drive motor to achieve real-time estimation and dynamic compensation of the total disturbance, and an improved complex coefficient repetitive controller is designed to achieve high-precision tracking of the spindle speed. As Figure 1 shown, the permanent magnet synchronous motor control system structure based on the complex coefficient repetitive controller and the extended state observer includes five parts: the controlled object, the improved complex coefficient repetitive controller, the extended state observer, the disturbance compensator, and the state feedback controller.

[0067] The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation in the present invention includes the following steps:

[0068] Step S1, determination of spindle motion parameters.

[0069] Taking the surface-mounted permanent magnet synchronous motor as the non-circular turning spindle drive motor, in the d-q axis reference coordinate system, its mechanical motion equation is:

[0070] (1)

[0071] Among them, is the q-axis stator current, J is the moment of inertia, p is the number of pole pairs of the rotor, is the magnetic flux, is the viscous damping coefficient, is the mechanical angular velocity of the motor, is the load disturbance, is the external disturbance.

[0072] The influence of factors such as load changes, thermal effects, and machine tool vibrations during the turning process will cause electrical and mechanical parameter perturbations. Let:

[0073] (2)

[0074] Among them, , are the nominal values, , represent the perturbation amounts of the permanent magnet flux and moment of inertia parameters.

[0075] Regarding the parameter perturbations, load disturbances, and external disturbances as the total disturbance, the mechanical motion equation can be rewritten as:

[0076] (3)

[0077] Among them, represents the input coefficient of the q-axis current, , is the total disturbance including parameter perturbation and external disturbance.

[0078] Spindle speed change processing, that is, the speed changes according to the sinusoidal excitation signal at the basic speed, and the angular velocity of the drive motor is used as the periodic reference input signal. Expressed as:

[0079] (4)

[0080] in, is the basic rotational angular velocity of the main axis, is the amplitude of the spindle angular velocity change, is the frequency of change, is the relative amplitude of velocity change, , is the relative frequency of speed change, , describes the amplitude and frequency of the spindle relative to the basic speed, and is the characteristic parameter of the spindle speed change. The basic speed is the constant speed when the spindle is constantly processing non-circular turning.

[0081] Step S2: Design an extended state observer.

[0082] Assume the total disturbance and its first-order derivative Bounded.

[0083] set up is the system state vector, is the system output, As the control input, the expanded state space model of the mechanical motion equation is:

[0084] (5)

[0085] in, A represents the expanded state matrix, B represents the expanded state input matrix, C represents the expanded state output matrix, E represents the identity matrix, represents the derivative of the expansion state; the system matrices are:

[0086] ;

[0087] The extended state observer is designed as:

[0088] (6)

[0089] in, is the state vector of the observer, represents the transpose of the matrix, yes The estimation of is the total disturbance The estimation of is the state observer gain l 1 represents the angular velocity observation gain l 2 represents the disturbance observation gain represents the output variable of the observer represents the derivative of the observer state vector.

[0090] Define the state estimation error as:

[0091] (7)

[0092] where represents the state estimation error is the rotational speed estimation error is the disturbance estimation error is The estimation of is the total disturbance The estimation of.

[0093] From equations (5) and (6), the estimated error state equation is:

[0094] (8)

[0095] where represents the derivative of the state estimation error.

[0096] Step S3, construct an improved complex coefficient repetitive controller.

[0097] S3-1, traditional repetitive controller.

[0098] The angular velocity of the motor rotation The period of is . The traditional repetitive controller The transfer function of is:

[0099] (9)

[0100] where s represents the Laplace transform operator represents the time-delay link is a first-order low-pass filter; The transfer function of is:

[0101] (10)

[0102] where Represents the cut-off angular frequency of the low-pass filter.

[0103] The filter satisfies the frequency characteristic:

[0104] (11)

[0105] Where, Is the highest frequency of the reference input signal, Is the cut-off angular frequency of the low-pass filter, satisfying ; Represents the amplitude-frequency characteristic of the low-pass filter, j Represents the imaginary unit. Substituting the low-pass filter into the repetitive controller gives its frequency-domain characteristic As:

[0106] (12)

[0107] Assume that the current harmonics include the fundamental wave and the Nth harmonic. For the entire linear control system, the output and tracking error also include the fundamental wave and the Nth harmonic. Therefore, for the repetitive controller There is:

[0108] (13)

[0109] Where, Represents the fundamental frequency of the periodic signal and the current harmonics, k Represents the harmonic order, N Represents the maximum harmonic order.

[0110] Can be simplified to:

[0111] (14)

[0112] Is the output signal of the repetitive controller. Assume To the output The transfer function is , From the above equation, when Is much larger than , It can make Large enough, so that the frequency characteristic of the transfer function from the reference input signal To the output Satisfies:

[0113] (15)

[0114] Where, Represents the repetitive control feedforward gain, Represents The frequency characteristic of.

[0115] Obviously, the traditional repetitive control system can track the periodic reference input speed when the cut-off angular frequency of the filter is large enough. However, a large means that the low-pass filter has a high bandwidth, which may amplify the influence of measurement noise and thus affect the system performance.

[0116] S3-2, complex coefficient repetitive controller.

[0117] To solve the problems of the above traditional repetitive controller, the present invention constructs a complex coefficient repetitive controller. The complex coefficient repetitive controller has the following expression:

[0118] (16)

[0119] where is a complex coefficient filter, expressed as:

[0120] (17)

[0121] where is the imaginary part of the complex coefficient. Let the input signal of the complex coefficient filter be , and the output be , satisfying the following conditions:

[0122] (18)

[0123] where represents the output signal of the input signal passing through the time-delay link, represents the speed tracking error.

[0124] Substitute into to obtain the filtering characteristics of the complex coefficient repetitive controller:

[0125] (19)

[0126] Since , then:

[0127] (20)

[0128] Therefore, when , .

[0129] This shows that by designing a complex coefficient filter of the corresponding order , the complex coefficient repetitive controller enables the system to completely track harmonic signals. Comparing equations (14) and (20) gives:

[0130] (21)

[0131] That is . Therefore, when the cut-off frequency is the same, in the low-frequency range ( is small), the steady-state tracking performance of the system based on complex coefficient repetitive control for periodic signals is better than that of the system based on traditional repetitive control.

[0132] S3-3, Improved complex coefficient repetitive controller.

[0133] Since the speed change is a sinusoidal change on the basic speed when the main shaft runs with variable speed, there is a constant signal. The gains of both the traditional repetitive controller and the complex coefficient repetitive controller are infinite at , and both can completely track this constant signal at steady state. However, since repetitive control cannot track periodic signals in the first cycle and has no impact on the system, the transient performance in the first cycle is poor. To improve the transient performance of the control system, the present invention proposes an improved complex coefficient repetitive controller (ICR), and the transfer function is as follows:[[]]

[0134] (22)

[0135] wherein is the basic rotational speed (constant speed machining rotational speed) is the internal model of represents the amplitude of the Laplace transform of the basic rotational speed signal represents the time delay link of the periodic signal s represents the Laplace transform operator T represents the time period of the periodic signal is a complex coefficient filter, and the transfer function of the complex coefficient filter is as follows:[[]]

[0136] ;

[0137] wherein represents the fundamental frequency of the periodic signal is the order for suppressing harmonic signals

[0138] Substituting into , the frequency characteristic of the improved complex coefficient repetitive controller is obtained as follows:[[]]

[0139] (23)

[0140] Substituting into , the above formula is simplified to:[[]]

[0141] (24)

[0142] Obviously, when , , represents the amplitude-frequency characteristic of the improved complex coefficient repetitive controller. The designed complex coefficient filter optimizes the improved complex coefficient repetitive controller by directionally selecting the target harmonic components (such as harmonics of a specific order), enabling it to track the corresponding harmonic components in the reference input signal without steady-state error. Further, when all target harmonic disturbances (such as harmonic currents of each order) exist simultaneously, the controller can achieve global suppression based on the frequency-domain selectivity of the complex coefficient filter, completely eliminating the influence of multi-frequency harmonic disturbances on the system.

[0143] Let the integral output signal in the improved complex coefficient repetitive controller be , , and construct the composite control law based on the improved complex coefficient repetitive control, state feedback, and disturbance compensation as:

[0144] (25)

[0145] where (26)

[0146] In the formula, is the system tracking error, is the state feedback gain, is the integral feedforward gain is the repetitive control feedforward gain; is the disturbance compensation gain, and its value is ; represents the output of the improved complex coefficient repetitive controller, represents the output signal of the integrator in the improved complex coefficient repetitive controller, represents the input signal of the integrator in the improved complex coefficient repetitive controller.

[0147] Step S4, implementation of the complex coefficient filter.

[0148] The filtering process of the complex coefficient filter is represented by , as Figure 2 shown, it is necessary to construct a complex signal with orthogonal components. Taking the original input signal as the real part of the complex signal, its imaginary part needs to satisfy a 90-degree phase shift relationship. Although traditional integral or differential operations can generate orthogonal components, they may cause system stability problems. The present invention uses a first-order inertial link instead, that is:

[0149] (27)

[0150] where Control the amplitude characteristic, Control the phase characteristic. The time-domain dynamic equation obtained by inverse Laplace transform is:

[0151] (28)

[0152] Wherein, Represents the signal processed by the inertia link, and the parameter Is used for amplitude calibration, Is responsible for phase adjustment. By setting a smaller Value, when the operating frequency Is, Relative to Will exhibit an approximate 90-degree phase lag, satisfying the imaginary part orthogonality condition. Thus, the complex signal is defined as:

[0153] (29)

[0154] After the complex signal is processed by the complex coefficient filter, the output signal is decomposed into:

[0155] (30)

[0156] Wherein, Is the signal filtered by the complex coefficient filter, Represents the real part of the filtered complex signal, Represents the imaginary part of the filtered complex signal.

[0157] Combining equations (17), (28), and (30) to derive the extraction differential equations for the real and imaginary parts of the filtered complex signal:

[0158] (31)

[0159] Wherein, the dot variable represents the time-domain derivative of the corresponding signal, Is the cut-off frequency, Is the order for suppressing harmonic signals. From the above equation, the real and imaginary parts of the filtered complex signal can be extracted.

[0160] The implementation block diagram of the complex coefficient filter is as shown in Figure 2 Indicates that from equations (27)-(31), the specific implementation of the filtering process of the complex coefficient filter in Figure 3 Can be obtained, and the transfer function from To Can be obtained (As shown in Figure 1 )Is:

[0161] (32)

[0162] and are adjustment parameters, adjust the amplitude of, adjust the phase of, indicating the actual filter transfer function when implementing the complex coefficient filter When, by adjusting and change the transfer function of the filter, so as to obtain a satisfactory filtering effect.

[0163] The improved complex coefficient repetitive controller of the present invention breaks through the stability limitation of the original architecture by introducing a complex coefficient filter into the traditional repetitive controller, and effectively solves the trade-off contradiction between the stability and control performance of the traditional filter. For the composite speed tracking of the drive motor under the spindle variable speed machining condition (i.e., superimposing a sinusoidal fluctuation on the basic speed signal (constant speed)), the controller forms a double internal model structure by connecting the integral link (the internal model corresponding to the basic speed) in parallel with the complex coefficient repetitive controller, which not only overcomes the defect that the traditional repetitive controller cannot track in the first cycle, but also significantly improves the transient response performance. The improved controller has both multi-modal tracking and disturbance rejection capabilities: on the one hand, it can achieve high-precision and fast tracking of aperiodic signals (such as acceleration commands, constant value settings) and periodic signals (such as sinusoidal fluctuations); on the other hand, by using the frequency domain characteristics of the complex coefficient filter, it can accurately suppress periodic disturbances of different harmonic frequencies, and at the same time enhance the system robustness through the synergistic effect of the double internal model, reducing overshoot and oscillation in the initial adjustment stage while ensuring the steady-state accuracy.

[0164] Step S5, stability conditions and parameter design of the permanent magnet synchronous motor control system.

[0165] Since the stability of a linear system is independent of the input signal, therefore, in order to determine the stability conditions of the system, let . Based on equations (3), (8), (18), (25) and (26), the permanent magnet synchronous motor speed system based on the improved complex coefficient repetitive controller and the extended state observer is expressed as:

[0166] (33)

[0167] Define the state vector , the above equation can be transformed into: (34)

[0168] where the coefficient matrices , , are respectively:

[0169]

[0170] The above equation shows that the system is a time-delay system. The stability condition is given by:

[0171] For a given positive scalar and , if there exists a positive definite matrix , and the dimension matrix and So that the following linear matrix inequality holds:

[0172] (35)

[0173] The permanent magnet synchronous motor speed system (Equation (33)) based on the improved complex coefficient repetitive controller and extended state observer is asymptotically stable. 、 、 They are:

[0174] (36)

[0175] (37)

[0176] in, 、 、 、 、 、 、 、 、 、 、 、 is an intermediate variable; among them, Used to adjust the state feedback gain , Used to adjust the state observer gain , and Used to adjust the repetitive control feedforward gain , and Used to adjust the integral feedforward gain These tuning parameters enhance the design flexibility and reduce the conservatism of the linear matrix inequality solutions.

[0177] By solving the linear matrix inequality, the state feedback gain , repeated control feedforward gain , integral feedforward gain and the state observer gain It can be obtained by the following design:

[0178] (38)

[0179] Proof of stability condition:

[0180] Consider the following Lyapunov function:

[0181] (39)

[0182] where (40)

[0183] and and (both ) are the adjustment parameters of the linear matrix inequality.

[0184] The derivative of this Lyapunov function along the system (Equation (34)) is:

[0185] (41)

[0186] where represents the system matrix, represents the system vector, which are respectively:

[0187] (42)

[0188] For Equation (41), if then the control system is asymptotically stable. According to the Schur complement lemma, is equivalent to:

[0189] (43)

[0190] Substitute , and into (43), and let:

[0191] , ;

[0192] We can obtain (35), (36), (37) and (38), where is a sufficiently small positive number used to ensure that the linear matrix inequality can be solved smoothly.

[0193] Experimental verification:

[0194] Apply the method described in the present invention to the speed control of the spindle permanent magnet synchronous drive motor servo system, and its parameters and variables are shown in Table 1.

[0195] Table 1 Parameters and variables

[0196]

[0197] Set the test to machine a long shaft and a short shaft of an elliptical part using an aluminum alloy material and a diamond tool. and a short shaft Let the angular velocity be:

[0198] (44)

[0199] The spindle speed is:

[0200] (45)

[0201] Therefore, the basic rotational angular velocity of the spindle is , and the characteristic parameter of the speed change is taken as: , at this time the basic spindle speed is , the speed change , the change frequency is , that is, the period of the reference input signal is , the fundamental frequency is . Select the cut-off angular frequency , and select the following parameters to solve the LMI formula (35):

[0202] ;

[0203] Get and .

[0204] The system simulation results based on the method proposed in the present invention are as Figures 4 to 6 shown. The actual output change curve of the spindle speed coincides highly with the given variable speed curve, and can achieve fast and high-precision tracking of the spindle speed. In the working stage of 0 - 0.2 s, the spindle makes an accelerated rotational motion, and the system can quickly track. Within 0.2 - 2 s, the spindle speed changes periodically, and the system quickly enters the steady state in the first cycle. The maximum steady-state error is 0.1906 r / min, which is about 0.0106% of the reference signal.

[0205] Assume that the initial position of the tool is at the long axis. When the spindle processes the elliptical part at a constant speed with the basic speed, the radial movement displacement of the tool over time can be expressed as:

[0206] <> (46)

[0207] where represents the displacement amount of the tool cutting, a represents the length of the long axis of the machined part, and b represents the length of the short axis of the machined part.

[0208] In variable-speed non-circular turning, when the spindle speed changes according to Equation (44), the radial movement displacement of the tool over time becomes:

[0209] (47)

[0210] According to Equation (47), the comparison curves of the expected and actual tool paths during variable-speed machining of an elliptical part in non-circular turning and the tool cutting path tracking error are as Figures 7 to 10 shown. From Figure 7 and Figure 9 it can be seen that at this time, the machining path changes periodically with respect to time, while from Figure 8 and Figure 10 it can be seen that it is fixed periodically with respect to position, with a period of π, a maximum error of 0.000704 mm, which is approximately 0.2816% of the reference signal.

[0211] To verify the superiority of the improved complex coefficient repetitive controller (ICR) of the present invention in the transient and steady-state performance of the control system, the method of the present invention is simulated and compared with the complex coefficient repetitive control system (CFR) based on the extended state observer. To ensure fairness in the comparison, the controller parameters are selected to be the same, and the repetitive control feedforward gain of the complex coefficient repetitive control system is taken as , and the comparison results of the spindle drive motor speed tracking error are as Figure 11 shown.

[0212] It can be clearly observed from the figure that the transient performance of the improved complex coefficient repetitive control system of the present invention is superior to that of the complex coefficient repetitive control system, and the steady-state error is also smaller. The spindle drive motor system based on the complex coefficient repetitive controller has a large tracking error during the acceleration rotation process from 0 to 0.2 seconds. During the variable-speed process from 0.2 to 2 seconds, the first change cycle is not fully tracked, with a maximum tracking error of 1.321 r / min. The second cycle enters the steady state, with a maximum steady-state tracking error of 0.533 r / min, which is approximately 0.0296% of the reference signal. The steady-state error of the method of the present invention is only 35.7% of that of the unimproved system. Therefore, the improved complex coefficient repetitive control system designed based on the extended state observer of the present invention ensures the given speed tracking accuracy during spindle variable-speed machining and improves the machining accuracy of non-circular turning to a certain extent.

[0213] To verify the superiority of using a complex coefficient filter in the repetitive controller, the proposed method is compared with the traditional repetitive control system based on the extended state observer. To ensure fairness in the comparison, an improved repetitive controller (IRC) with an internal model of the reference speed is connected in parallel to the traditional repetitive controller, and the controller parameters are selected to be the same as those of the proposed method. The comparison results of the spindle drive motor speed tracking error based on the two control methods are as Figure 12 shown.

[0214] From Figure 7 It can be observed that the introduction of the rotational speed internal model significantly reduces the tracking error during the acceleration rotation stage of the spindle motor from 0 to 0.2 seconds, and the two curves are highly similar. However, the control system error based on the complex coefficient repetitive controller is smaller. During the variable speed machining stage from 0.2 to 2 seconds, the steady-state tracking error of the system based on the traditional repetitive controller is 0.731 r / min, which is about 0.0406% of the given rotational speed. The steady-state tracking error of the system of the method of the present invention is 72.9% of this method, effectively improving the tracking accuracy of the given rotational speed during the steady state of spindle variable speed machining and improving the machining accuracy of non-circular turning.

[0215] The above-mentioned embodiments are only preferred embodiments of the present invention and do not impose any form of limitation on the present invention. Although the present invention has been disclosed as above with preferred embodiments, it is not intended to limit the present invention. Therefore, any simple modifications, equivalent changes, and decorations made to the above embodiments based on the technical essence of the present invention without departing from the technical solution of the present invention shall fall within the scope of protection of the technical solution of the present invention.

Claims

1. An improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation, characterized in that, First, according to the speed loop of the permanent magnet synchronous drive motor for non-circular turning spindle, an extended state observer is established to estimate and actively compensate for disturbances in real time; then, using the periodic characteristics of spindle speed change, a complex coefficient repetitive controller is constructed, and combined with the internal model of the basic spindle speed, an improved complex coefficient repetitive controller is built; then, based on the Lyapunov stability theory, the stability conditions of the closed-loop system are obtained, and the controller parameters are solved based on linear matrix inequalities. Transfer function of the improved complex coefficient repetitive controller is as follows: ； Among them, is the base speed of the inner model, represents the amplitude of the Laplace transform of the base speed signal, represents the time delay link of the periodic signal, s represents the Laplace transform operator, T represents the time period of the periodic reference speed and the periodic disturbance, is a complex coefficient filter, and the transfer function of the complex coefficient filter is: ； Among them, represents the fundamental frequency of the periodic signal, is the order for suppressing the harmonic signal; Frequency Characteristics of Improved Complex-Coefficient Repetitive Controller are as follows: ； Among them, j represents the imaginary unit, k represents the harmonic order, is the cut-off angular frequency of the low-pass filter; Let the output signal of the integrator in the improved complex coefficient repetitive controller be , and we obtain . The composite control law constructed based on the improved complex coefficient repetitive control, state feedback, and disturbance compensation is as follows: ； Among them, ； wherein, is the system tracking error, is the state feedback gain, is the integral feedforward gain, is the repetitive control feedforward gain; is the disturbance compensation gain, and its value is , is the permanent magnet synchronous motor input coefficient of the d-axis current loop; is the estimation of the mechanical angular velocity of the motor, is the estimation of the total disturbance ; represents the output of the improved complex coefficient repetitive controller, represents the input signal of the integrator in the improved complex coefficient repetitive controller, represents the real part of the filtered complex signal.

2. The improved complex coefficient repetitive control method for permanent magnet synchronous motor based on disturbance compensation according to claim 1, characterized in that The filtering process of the complex coefficient filter is represented by where the complex signal has the following expression: ； wherein, is the real part of the complex signal, is the imaginary part of the complex signal, is a complex coefficient filter; the filtered complex signal has the following expression: ； Among them, denotes the signal after being filtered by a complex coefficient filter, denotes the real part of the complex signal after filtering, denotes the imaginary part of the complex signal after filtering, and are respectively and the Laplace transforms of; The differential equations for extracting the real and imaginary parts of the complex signal are obtained: ； Among them, represents the real part derivative of the filtered complex signal, represents the imaginary part derivative of the filtered complex signal.

3. The improved complex coefficient repetitive control method for permanent magnet synchronous motors based on disturbance compensation according to claim 2, wherein In the complex coefficient filter, from to the transfer function is: ； Among them, and are adjustment parameters, adjust the amplitude of, adjust the phase of, indicates the actual filter transfer function when implementing the complex coefficient filter By adjusting and the transfer function of the filter is changed.

4. The improved complex coefficient repetitive control method for permanent magnet synchronous motor based on disturbance compensation according to claim 3, characterized in that The dynamic equation of the extended state observer is: ； Among them, is the state vector of the observer, is the state observer gain, is the composite control law, is the system output, represents the output variable of the state observer, represents the derivative of the state vector of the state observer, A represents the extended state matrix, B represents the extended state input matrix, C represents the extended state output matrix; The state estimation error is defined as: ； Among them, is the system state vector, represents the state estimation error, is the rotational speed estimation error, is the disturbance estimation error; the corresponding state equation of the estimation error is: ； wherein, represents the derivative of the state estimation error, E represents the expanded state derivative matrix, represents the derivative of the total disturbance.

5. The improved complex coefficient repetitive control method for permanent magnet synchronous motor based on disturbance compensation according to claim 4, characterized in that, The stability conditions are obtained in the following way: For a given positive scalar , if there exists a positive definite matrix , and a matrix with appropriate dimensions such that the linear matrix inequality holds: ； Among them, ； ； Among them, , , , , , , , , , , , are intermediate variables, used to adjust the state feedback gain , used to adjust the state observer gain , and used to adjust the repetitive control feedforward gain , and used to adjust the integral feedforward gain .

6. The improved complex coefficient repetitive control method for a permanent magnet synchronous motor based on disturbance compensation according to claim 5, wherein The controller parameters include a state feedback gain , a repetitive control feedforward gain , an integral feedforward gain and a state observer gain , which are designed as follows: ； Among them, C represents the expansion state output matrix.

Images