Permanent magnet synchronous motor robust complex coefficient repetitive control method based on position domain internal model

Abstract

The invention discloses a permanent magnet synchronous motor robust complex coefficient repetitive control method based on a position domain internal model, and relates to the technical field of permanent magnet synchronous motor control, and the method comprises the steps: constructing a time domain improved complex coefficient repetitive controller through employing the time domain periodic characteristics of a main shaft rotating speed; secondly, on the basis of the position domain periodic characteristics of the cutting force, a position domain repetitive expansion state observer is constructed, and on-line estimation is carried out on aperiodic disturbance such as parameter perturbation and position domain periodic disturbance caused by the cutting force; and finally, deducing a system stability condition by applying a separation design principle and a small gain theorem, and synchronously optimizing controller parameters by adopting an optimization algorithm. The invention discloses a system design method based on time domain complex coefficient repetitive control and a position domain repetitive expansion state observer. An improved complex coefficient repetitive control system based on a position domain repetitive expansion state observer is designed to realize feed-forward compensation of periodic disturbance and non-periodic disturbance of a position domain and fast and high-precision tracking of a time domain periodic reference rotating speed signal.

Description

Technical Field

[0001] The 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] In the process of non-circular CNC turning of complex parts, if the constant speed turning method is used, due to the difference in the surface arc length per unit angle of the non-circular workpiece, the cutting force will fluctuate greatly with the change of position, which is very easy to cause chatter. This will not only directly reduce the machining accuracy, but also affect the stability of the machining process in serious cases. Spindle speed change 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 influence on the workpiece machining accuracy, it is particularly necessary to propose a corresponding control algorithm to ensure high-precision tracking of the spindle speed. Spindle speed change refers to the speed changing according to the sinusoidal excitation signal on the basic speed (i.e., the speed of constant speed turning), so that the reference speed signal has a clear time periodicity.

[0003] During variable speed machining, the position periodic disturbance caused by the cutting force on the spindle system will reduce the tracking accuracy of the spindle speed, that is, the cutting force is a position-related periodic disturbance for the spindle drive motor. Therefore, it is necessary to achieve fast and high-precision tracking of the time-domain periodic spindle speed while suppressing the disturbance caused by the position-domain periodic cutting force on the spindle motor, so as to further improve the tracking accuracy of the spindle speed. Summary of the invention

[0004] In view of the above problems existing in the prior art, the purpose of the present invention is to provide a method for robust complex coefficient repetitive control of a permanent magnet synchronous motor based on a position domain internal model. The problem of suppressing the position domain periodic disturbance caused by the cutting force of a non-circular turning spindle system during variable speed processing and tracking and controlling the time domain periodic spindle speed is 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: 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: 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, and 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, construct a time-domain improved complex coefficient repetitive controller; 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 disturbances and non-periodic disturbances in the position domain and fast and high-precision tracking of periodic reference speed signals in the time domain; Step S5: Provide system stability conditions and parameter optimization.

[0006] As a further improvement of the above technical solution: Step S1 includes the following steps: Step S11: constructing a cutting force model according to the empirical formula of cutting force and the rotation angle position of the spindle; Step S12: Determine the type of the drive motor and construct the speed loop mechanical motion equation of the drive motor.

[0007] The mechanical motion equation of the speed loop constructed in step S1 is: ; in, is the angular acceleration of the spindle, 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, perturbation caused by position periodic cutting force and external perturbation, is a non-periodic disturbance including parameter perturbations and external disturbances, is the position periodic disturbance caused by cutting force, , are the perturbations of the permanent magnet flux linkage and the moment of inertia, is the viscous damping coefficient, is the main axis rotation angular velocity, is the external disturbance, is the disturbance caused by the position periodic cutting force.

[0008] In step S2, the transfer function of the time-domain improved complex coefficient repetitive controller is: ; in, is the internal model of the main axis signal, s is the Laplace operator, is a complex coefficient filter, is the cut-off frequency, j is the imaginary unit, n is the order of the suppressed harmonic signal, is the imaginary part of the complex coefficient.

[0009] In step S2, from arrive The transfer function is: ; 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.

[0010] Step S3 includes the following steps: Step S31: designing an extended state observer; Step S32: construct an improved position domain repetitive controller according to the position periodic characteristics of the disturbance caused by the cutting force, Step S33: construct a position domain repeated expansion state observer.

[0011] 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, Output for the system; 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 An estimate 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, e is a natural constant, is the position period, is the position domain Laplace transform operator, is a positive number; 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 estimated value of the total disturbance is: ; in, for The estimated value of is combined with the estimated value of the extended state space model, the extended state observer and the total disturbance to obtain the disturbance estimation error: .

[0012] In step S4, a composite control law is constructed as a control input based on improved complex coefficient repetitive control feedforward, reconstructed state feedback and disturbance compensation.

[0013] In step S5, the composite control law is substituted into the extended state space model as the control input, and combined with the estimation error, the permanent magnet synchronous motor control system structure block diagram based on the position domain repeated extended state observer and the time domain improved complex coefficient repetitive controller is structurally equivalently transformed and decomposed into two series-connected subsystems, namely subsystem 1 and subsystem 2, wherein subsystem 1 is a tracking control subsystem based on the complex coefficient repetitive controller and the reconstructed state feedback, and subsystem 2 is a disturbance estimation 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 of For subsystem 2, if Assumption 1 holds, and the following conditions also hold: (1) There is no unstable zero-pole cancellation; (2) is stable; (3) ; Then the disturbance estimation compensation subsystem 2 based on the position domain repeated extended state observer is exponentially stable; in, For The transfer function to the unestimated disturbance; 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.

[0014] In order to comprehensively evaluate the tracking and anti-disturbance performance of the system, the following performance indicators are used: ; in, , and is the weight constant of each item, 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.

[0015] The beneficial effects of the present invention are: (1) The position domain repeated extended state observer achieves real-time estimation of periodic disturbances and non-periodic disturbances in the position domain. Compared with the traditional repeated 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. (2) The time-domain complex coefficient repetitive controller realizes fast and high-precision tracking of the time-domain periodic spindle speed, solves the trade-off 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 a 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.

[0016] (3) Traditional repetitive controllers and extended state observers use fixed parameters, and there is a trade-off between tracking control, disturbance suppression and dynamic performance. The present invention uses the system stability condition as the target constraint and uses the particle swarm optimization algorithm to search for the optimal controller parameter combination, so that the closed-loop system has good steady-state tracking, disturbance rejection and dynamic performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 It 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.

[0018] Figure 2 yes Figure 1 Equivalent block diagram of .

[0019] Figure 3 It is an equivalent block diagram of subsystem 1 of the present invention.

[0020] Figure 4 It is an equivalent block diagram of subsystem 2 of the present invention.

[0021] 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 Compare the simulation results of the curve.

[0022] Figure 6 is a control input of a complex coefficient repetitive control method based on a position domain extended state observer according to an embodiment of the present invention. Simulation results.

[0023] 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.

[0024] Figure 8 4 is a time domain control input curve diagram of an embodiment of the present invention.

[0025] Fig. 9 It is a position domain control input curve diagram of an embodiment of the present invention.

[0026] Fig.10 It is a time domain total disturbance estimation diagram of an embodiment of the present invention.

[0027] Fig.11 It is a total disturbance estimation diagram in the position domain according to an embodiment of the present invention.

[0028] Fig.12 This is a time domain tool cutting trajectory tracking diagram of an embodiment of the present invention.

[0029] Fig.13 It is a position domain tool cutting trajectory tracking diagram of an embodiment of the present invention.

[0030] Fig.14 It is a time domain tool cutting trajectory tracking error diagram of an embodiment of the present invention.

[0031] Fig.15 It is a tool cutting trajectory tracking error diagram in the position domain of an embodiment of the present invention.

[0032] Fig.16 It is a comparison of the speed tracking error between the ICR_ESO method and the method of the present invention based on an embodiment of the present invention.

[0033] Fig.17 This is an embodiment of the present invention, in which the speed tracking error of the method of the present invention is compared with that of the IRC_SRCESO method. DETAILED DESCRIPTION

[0034] The specific implementation of the present invention is described in detail below in conjunction with the accompanying drawings. It should be understood that the specific implementation described here is only used to illustrate and explain the present invention, and is not used to limit the present invention.

[0035] For ease of description, spatially relative terms such as "above", "above", "on the upper surface of", "above", etc. may be used here to describe the spatial positional relationship between a device or feature and other devices or features as shown in the figure. It should be understood that spatially relative terms are intended to include different orientations of the device in use or operation in addition to the orientation described in the figure. For example, if the device in the accompanying drawings is inverted, the device described as "above other devices or structures" or "above other devices or structures" will be positioned as "below other devices or structures" or "below other devices or structures". Thus, the exemplary term "above" can include both "above" and "below". The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatially relative descriptions used here are interpreted accordingly.

[0036] A robust complex coefficient repetitive control method for a permanent magnet synchronous motor based on an internal model in the position domain is proposed. The method is aimed at 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 complex coefficient repetitive controller in the time domain and a repetitive extended state observer in the position domain are constructed to construct a composite control law as the control input. , ensuring a closed-loop system ( Figure 1 ) while achieving stability in the time domain periodic reference input signal Fast and high-precision tracking and active compensation of non-periodic disturbances and position periodic disturbances.

[0037] In this scheme, the control system structure of the spindle drive motor based on the position domain repetitive extended state observer and the time domain improved complex coefficient repetitive controller is as follows: Figure 1 As shown, 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.

[0038] The control method comprises the following steps: 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, and the mechanical motion equation takes into account parameter perturbations, disturbances caused by periodic cutting forces in the position domain, and external disturbances.

[0039] This step includes the following steps: 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 with the rotation of the spindle is: (1) in, is the initial angular displacement, is the instantaneous angular velocity of the main axis, Represents the angular displacement time variable of the spindle rotation motion, due to the spindle speed: (2) This means There is an inverse function: . is the main axis rotation angular velocity.

[0040] Define a domain conversion operator as: (3) is the space of Lebesgue square integrable functions on the position domain, is the weighting function space in the time domain, is the time domain signal, is a signal in the position domain.

[0041] 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 machining displacement z (mm), cutting speed (m / s) is calculated as follows: (4) The tangential cutting force of the tool on the workpiece Usually calculated by the exponential empirical formula: (5) in, is the influence coefficient of the workpiece, tool material, etc. on the tangential force, and f is the feed rate, that is, the distance the tool moves along the workpiece surface during processing. 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, all of which are normal numbers and can be calculated by looking up the table.

[0042] According to the torque formula, the disturbance caused by the cutting force can be obtained as: (6) From equations (4)-(6), we can see that when the spindle is running at variable speed, is a trigonometric function of the cutting force with a known position period of π and a time period of π, that is, The position period is is a periodic disturbance of (see the introduction to formula (9) below).

[0043] Step S12: constructing the speed loop mechanical motion equation of the permanent magnet synchronous drive motor.

[0044] 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: (7) in, is the q-axis stator current, J is the moment of inertia, p is the number of rotor pole pairs, 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 disturbance.

[0045] The influence of load changes, thermal effects, machine tool vibration and other factors during turning processing will cause electrical and mechanical parameter perturbations. Assume: (8) in, , is the nominal value, , They are the perturbations of the permanent magnet magnetic flux and the moment of inertia respectively.

[0046] Taking parameter perturbation, position domain period (perturbation caused by cutting force and external perturbation) as the total perturbation, equation (7) is rewritten as: (9) in, , represents the total disturbance, is a non-periodic disturbance including parameter perturbations and external disturbances, It is the position periodic disturbance caused by the cutting force, that is, the disturbance caused by the periodic cutting force in the position domain.

[0047] Assumption 1: (9) In formula , and its first-order derivative , Bounded.

[0048] Step S2: Using the time-domain periodic characteristics of the spindle speed, a time-domain improved complex coefficient repetitive controller is constructed.

[0049] 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, and its transfer function is: (10) Where s is the Laplace operator, represents the amplitude of the Laplace transform of the basic speed signal, is a complex coefficient filter: (11) represents the fundamental frequency of the periodic signal, n is the order of the suppressed harmonic signal, is the cutoff frequency, j is the imaginary unit. Let the input signal of the complex coefficient filter be , the output is ,satisfy: (12) 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 time domain.

[0050] The filtering process of the complex coefficient filter is Indicates that the complex signal The expression is: (13) Complex signal It is constructed through the first-order inertia link. The transfer function of the first-order inertia link is for: (14) 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: (15) The filtered complex signal The expression is: (16) 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 .

[0051] According to (11) to (16), the differential equations for extracting the real and imaginary parts of the complex signal are obtained: (17) in, , represents the real derivative of the complex signal after filtering, Represents the imaginary derivative of the complex signal after filtering.

[0052] from arrive The transfer function for: (18) Step S3: Based on the position-domain periodic characteristics of the cutting force, a position-domain repeated extended state observer is constructed to online estimate the non-periodic disturbances such as parameter perturbations and the position-domain periodic disturbances caused by the cutting force.

[0053] Step S31: Design an extended state observer.

[0054] Assume that the state vector of the permanent magnet synchronous motor speed loop system is , the system output is , the control input is , the total disturbance The first-order derivative of is , the expanded state space model of formula (9) is: (19) in, is the first-order derivative of the state vector, and the system matrices are:

[0055] According to formula (19), the extended state observer is designed as: (20) in, is the state vector of the observer, and They are and The estimate, 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.

[0056] 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 expanded state observer.

[0057] 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: (twenty one) 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: (twenty two) because The dynamic behavior of is dominated by the denominator, so Considered as a disturbance The model is located in the position domain.

[0058] Therefore, the transfer function of the improved position domain repetitive controller is: (twenty three) Among them, the first-order low-pass filter Used to ensure the stability of the system.

[0059] First-order low-pass filter using zero initial condition in position domain : (twenty four) 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: (25) 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 a phase lag, and the lag value is: (26) In formula (23), For phase correction, compensation exist The phase lag produced at the fundamental and harmonic frequencies of the , so that: (27) This means that at a certain frequency, The phase lag caused can be corrected by the period correction unit Precise compensation.

[0060] 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: (28) 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 reconstructed state estimation error.

[0061] Therefore, the estimate of the total disturbance becomes: (29) in, for The estimated value of , combined with (19), (20) and (29), the perturbation estimation error is .

[0062] Step S4: construct a composite control law to realize the system's feedforward compensation for periodic disturbances and non-periodic disturbances in the position domain and fast and high-precision tracking of the periodic reference speed signal in the time domain.

[0063] 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: (30) in, (31) 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, whose value is , Permanent magnet synchronous motor q Coefficient of shaft current.

[0064] Step S5: Provide system stability conditions and parameter optimization methods.

[0065] Step S51: deriving system stability conditions.

[0066] Substituting (30) into (19) and combining the estimated error, we can get 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: (32) Below the dotted line is the disturbance compensation subsystem 2, whose output is: (33) Figure 2 The total disturbance To system output The transfer function is: (34) in, The total disturbance The Laplace transform of , From the total disturbance to the disturbance estimation error The transfer function of , ; (35) For To output The transfer function of is stable, then subsystem 1 is stable. is the error from disturbance estimation to The transfer function of and If both subsystems 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.

[0067] For subsystem 1, the embedding of repeated controllers is not considered. arrive The transfer function is: (36) Will Figure 2 The neutron system 1 undergoes structural equivalent transformation to obtain Figure 3 .set up: (37) in Represents the largest singular value.

[0068] If subsystem 1 ( Figure 3 ) satisfies assumption 1, and the following conditions are also true: (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. (See formula (18)).

[0069] prove:

[0070] 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, it can be seen that conditions (b) and (c) ensure that the control system of the closed-loop subsystem 1 is stable, so subsystem 1 is stable.

[0071] For subsystem 2, let , without considering the embedding of the position domain repeated controller, from The transfer function to the unestimated disturbance is: (38) in, is the position domain repetitive control feedforward gain.

[0072] In the present invention, are the different gains of the extended state observer.

[0073] Will Figure 2 The neutron system 2 undergoes structural equivalent transformation to obtain Figure 4 .

[0074] For subsystem 2 ( Figure 4 ), if assumption 1 is established and the following conditions are also established: (1) There is no unstable zero-pole cancellation; (2) is stable; (3) ; but Figure 4 The disturbance estimation compensation subsystem 2 shown based on the position domain repeated extended state observer is exponentially stable.

[0075] prove: Perform structural equivalent transformation on subsystem 2 to obtain an equivalent system structure diagram ( Figure 4 ),in is a phase compensation factor, represents 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.

[0076] ,and , by the small gain theorem, conditions (1)~(3) ensure that the closed-loop subsystem 2 is exponentially stable.

[0077] Step S52: particle swarm optimization algorithm design.

[0078] 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: (39) Where k is the current iteration number, , denote the velocity and position of the i-th particle at the k-th iteration, respectively. and are the best positions of the i-th particle and the entire particle swarm in the k-th iteration, respectively. , are learning factors, and are all 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: (40) 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.

[0079] In order to comprehensively evaluate the tracking and anti-disturbance performance of the system, the following performance indicators are used: (41) in, , and is the weight constant of each item, N is the number of repeated cycles, , and .

[0080] are the fitness value, system tracking error and total disturbance estimation error corresponding to the i-th particle. 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.

[0081] A specific numerical example is given below.

[0082] The method proposed in the present invention is applied to the speed control of the permanent magnet synchronous drive motor of the non-circular turning spindle, and the parameters and variables are listed in Table 1.

[0083] The experiment was set up using aluminum alloy material and diamond tool to machine an elliptical part with a long axis of 37 mm and a short axis of 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: (42) (43) Table 1 Parameters and variables of permanent magnet synchronous drive motor

[0084] From the above formula, it can be seen that the basic spindle speed is =1500r / min, speed variation ±300r / min, the period of variable reference input signal is =0.2s, 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.

[0085] 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, weights of each objective function =0.6, =1, =0.3. The controller parameter search range is as follows: ; The parameters obtained by solving are: (44) correspond: ;

[0086] Therefore, all system stability conditions are met and the control system based on the proposed method is stable.

[0087] Figure 5 , 6 , 7 show that the closed-loop control system of the permanent magnet synchronous motor based on the position domain repeated extended state observer and the complex coefficient repetitive controller is stable. The reference input signal tracked is in the acceleration stage from 0 to 0.2s, t>0.2s is the speed cycle operation stage, and the time period is 0.2s. From the control input curve, it can be seen that there are non-periodic fluctuations. This shows that the proposed method not only realizes the tracking of the time period reference input signal, but also realizes the suppression of the position period disturbance. In the acceleration stage from 0 to 0.2s, the steady-state error is 0.00238r / min; in the speed cycle operation stage after 0.2s, the maximum steady-state tracking error is 0.002994r / min, which is 0.0001663% of the reference speed.

[0088] 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. Fig.10 , 11 It can be found that the total disturbance The position periodic disturbance is included, and the total disturbance curve estimated by this method almost completely coincides with the total disturbance curve. Therefore, the proposed method not only realizes the tracking of the time-domain periodic spindle speed, but also realizes the high-precision estimation of the position periodic disturbance and non-periodic disturbance.

[0089] According to the tool displacement formula, the tool trajectory tracking effect is as follows Fig.12 , 13 and Fig.14 , 15When the spindle speed is changed during non-circular turning, the tool cutting displacement time domain period changes while the position period is fixed at π. Fig.12 , 13 It can be seen that 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. Fig.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.

[0090] In order to verify the effectiveness of the position domain repeated extended state observer introduced in this invention, 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. Fig.16 As 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, and the maximum tracking error is 0.4646r / min, which is 0.0258% of the reference input signal, seriously affecting the machining accuracy of non-circular turning. Obviously, after introducing the position domain repetitive internal model, the position periodic disturbance caused by the cutting force is significantly suppressed, and the steady-state tracking accuracy of the reference speed is improved.

[0091] 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. Fig.17 It is observed that the tracking errors of the method of the present invention are smaller than those of the latter in both transient and steady states. The control system based on the position domain repetitive expanded state observer and the improved repetitive controller has a steady-state error of 0.238r / min in the acceleration stage of 0~0.2s; in the speed cycle operation stage after 0.2s, the maximum steady-state tracking error is 0.3018r / min, which is 0.01677% of the reference input. The tool cutting estimated tracking error based on this method is 0.001488mm, 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.

[0092] The results show that although the improved repetitive control method can improve the transient performance of spindle acceleration rotation, the improved complex coefficient repetitive controller can not only improve the transient performance, but also further improve the steady-state tracking accuracy, thereby improving the non-circular turning accuracy. On the other hand, the proposed position domain repetitive extended state observer can achieve high-precision estimation of non-periodic disturbances such as parameter perturbations and position periodic disturbances caused by cutting forces, further improving the non-circular turning accuracy.

[0093] 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 method for robust complex coefficient repetitive control of a permanent magnet synchronous motor based on position domain internal model, characterized in that: The steps include: 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, and 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, construct a time-domain improved complex coefficient repetitive controller; 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 disturbances and non-periodic disturbances in the position domain and fast and high-precision tracking of periodic reference speed signals in the time domain; Step S5: Provide system stability conditions and parameter optimization.

2. The control method according to claim 1, characterized in that: Step S1 includes the following steps: Step S11: constructing a cutting force model according to the empirical formula of cutting force and the rotation angle position of the spindle; Step S12: Determine the type of the drive motor and construct the 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 angular acceleration of the spindle, 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, perturbation caused by position periodic cutting force and external perturbation, is a non-periodic disturbance including parameter perturbations and external disturbances, is the position periodic disturbance caused by cutting force, , are the perturbations of the permanent magnet flux linkage and the moment of inertia, is the viscous damping coefficient, is the main axis rotation angular velocity, is the external disturbance, is the disturbance caused by the position periodic cutting force.

4. The control method according to claim 3, characterized in that: In step S2, the transfer function of the time-domain improved complex coefficient repetitive controller is: ； in, is the internal model of the main axis signal, s is the Laplace operator, is a complex coefficient filter, is the cut-off frequency, j is the imaginary unit, n is the order of the suppressed harmonic signal, is the imaginary part of the complex coefficient.

5. The control method according to claim 4, characterized in that: 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 through the first-order inertia link is For adjustment Amplitude.

6. The control method according to claim 4 or 5, 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 according to the position periodic characteristics of the disturbance caused by the cutting force, Step S33: construct a position domain repeated expansion state observer.

7. The control method according to claim 6, 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, Output for the system; 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 An estimate 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 location 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 estimated value of the total disturbance is: ； in, for The estimated value of is combined with the estimated value of the extended state space model, the extended state observer and the total disturbance to obtain the disturbance estimation error: .

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

9. The control method according to claim 7, characterized in that: In step S5, the composite control law is substituted into the extended state space model as the control input, and combined with the estimation error, the permanent magnet synchronous motor control system structure block diagram based on the position domain repeated extended state observer and the time domain improved complex coefficient repetitive controller is structurally equivalently transformed and decomposed into two series-connected subsystems, namely subsystem 1 and subsystem 2, wherein subsystem 1 is a tracking control subsystem based on the complex coefficient repetitive controller and the reconstructed state feedback, and subsystem 2 is a disturbance estimation 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 of For subsystem 2, if Assumption 1 holds, and the following conditions also hold: (1) There is no unstable zero-pole cancellation; (2) is stable; (3) ; Then the disturbance estimation compensation subsystem 2 based on the position domain repeated extended state observer is exponentially stable; in, For The transfer function to the unestimated disturbance; 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.

10. The control method according to claim 9, characterized in that: In order to comprehensively evaluate the tracking and anti-disturbance performance of the system, the following performance indicators are used: ； in, , and is the weight constant of each item, 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.

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