CN112787551A - Method for improving robustness and contour performance of double-shaft or three-shaft feed driving system - Google Patents

Method for improving robustness and contour performance of double-shaft or three-shaft feed driving system Download PDF

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CN112787551A
CN112787551A CN201911062233.3A CN201911062233A CN112787551A CN 112787551 A CN112787551 A CN 112787551A CN 201911062233 A CN201911062233 A CN 201911062233A CN 112787551 A CN112787551 A CN 112787551A
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moment
matrix
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shaft
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张秀云
王志强
方沂
徐征
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Tianjin University of Technology and Education China Vocational Training Instructor Training Center
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P5/00—Arrangements specially adapted for regulating or controlling the speed or torque of two or more electric motors
    • H02P5/46—Arrangements specially adapted for regulating or controlling the speed or torque of two or more electric motors for speed regulation of two or more dynamo-electric motors in relation to one another
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/0003—Control strategies in general, e.g. linear type, e.g. P, PI, PID, using robust control
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/22—Current control, e.g. using a current control loop
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P25/00—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details
    • H02P25/02—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details characterised by the kind of motor
    • H02P25/022—Synchronous motors
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P6/00—Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/04—Arrangements for controlling or regulating the speed or torque of more than one motor
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P6/00—Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/04—Arrangements for controlling or regulating the speed or torque of more than one motor
    • H02P2006/045—Control of current
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P2207/00—Indexing scheme relating to controlling arrangements characterised by the type of motor
    • H02P2207/05—Synchronous machines, e.g. with permanent magnets or DC excitation

Abstract

The invention discloses a method for improving the robustness and the profile performance of a double-shaft or three-shaft feed driving system. A double-shaft or three-shaft position-current prediction controller is designed through a model prediction control method based on a state space, a contour error and a tracking error are added into a cost function as internal variables of a double-shaft or three-shaft feed driving system, and then a control input signal enabling tracking precision and contour precision to be coordinated and optimal is searched by using the ideas of multi-step prediction, rolling optimization and feedback correction in model prediction control. Meanwhile, according to actual requirements, constraint conditions are considered, and a posterior constraint limiting network of current increment is established; finally, the purpose of improving the tracking and contour control performance of the double-shaft or three-shaft feeding driving system is achieved.

Description

Method for improving robustness and contour performance of double-shaft or three-shaft feed driving system
Technical Field
The invention relates to the field of servo drive control, in particular to a method for improving the robustness and the profile performance of a double-shaft or three-shaft feed drive system.
Background
The permanent magnet synchronous motor driven multi-axis motion system has wide application prospect in the technical field of high-speed and high-precision modern numerical control due to the advantages of the multi-axis motion system in the aspects of power density, operation efficiency, response speed and the like. The contour machining precision is an important performance index of a multi-axis motion control technology, and in the tracking control of a multi-axis contour track, the contour precision of multi-axis contour tracking is generally evaluated by contour errors. The profile error is ultimately reflected in the product, which can cause a reduction in the precision of the product. Therefore, reducing the profile error to improve the profile accuracy is of great significance to multi-axis motion control.
In the actual production and processing process, besides tracking a linear motion track, a complex contour motion track with more turns needs to be tracked, the problems of large contour error and low dynamic response speed at the turning point of the track can be caused by the problems of motion inertia of each axis, untimely response and the like, the contour tracking precision of a multi-axis motion system can be influenced by the factors of nonlinear friction, slow time-varying disturbance and the like, and the research of a high-precision and high-performance contour control strategy is particularly important.
Disclosure of Invention
The invention aims to provide a method for improving the robustness and the profile performance of a double-shaft or three-shaft feed driving system aiming at the technical defects of large profile error and low dynamic response speed at a track turning point in the prior art.
The technical scheme adopted for realizing the purpose of the invention is as follows:
a method of improving robustness and profile performance of a dual-axis feed drive system, comprising the steps of:
step 1, establishing a state equation of a double-shaft feed driving system;
step 2, multi-step prediction: performing multiple iterations on the state equation of the double-shaft feed driving system to obtain a matrix of multi-step prediction output;
step 3, establishing a cost function, substituting the matrix predicted and output in the step 2 into the cost function, carrying out minimum solution, further obtaining a model prediction control increment vector matrix, obtaining a control increment vector at the current moment and a control quantity vector at the current moment, and acting the control increment vector at the current moment on the double-shaft feed driving system;
and 4, carrying out posterior check by using a limiting module: and (3) applying the control quantity corresponding to the control quantity vector passing through the limiting module to the double-shaft feed driving system as current loop input, and applying the control increment vector passing through the limiting module to the state equation in the step (1) to form a closed-loop control network.
Preferably, the state equation in step 1 is
Figure BDA0002258281510000021
Figure BDA0002258281510000022
Δiq(k)＝[Δiqx T(k) Δiqy T(k)]T；y(k)＝[θx(k) θy(k)]T；
x(k)＝[θx(k) θx(k-1) θx(k-2) θy(k) θy(k-1) θy(k-2)]T；
Figure BDA0002258281510000023
Δiqi(k)＝[Δiqi(k) Δiqi(k-1)]T；ci＝[1 0 0]；x(k)＝[θi(k) θi(k-1) θi(k-2)]T；
i∈{x，y}；
In the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000024
ai1＝-1-ai2,
Figure BDA0002258281510000025
Figure BDA0002258281510000026
Tsk, k-1 and k-2 are respectively the k moment, the k-1 moment and the k-2 moment;
Jeqiis equivalent moment of inertia, and Jeqi＝Ji+Miri 2/4π2；BeqiIs an equivalent viscous friction coefficient, and Beqi＝Bi+Ciri 2/4π2
JiIs the rotational inertia of the motor; thetaiIs the rotor mechanical angle; b isiThe viscous friction coefficient of the motor; ciIs the viscous friction coefficient of the motion mechanism; miIs the moving part mass; tau isiIs the load torque; ktiThe torque coefficient of the permanent magnet synchronous motor is; i.e. iqiIs the q-axis component of the stator current; Δ iqi(k) Is the control increment of the motor.
Preferably, the matrix of the multi-step prediction output in step 2 is:
Y＝Fx(k)+GΔIq (2)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000027
Y＝[yT(k+1) yT(k+2) … yT(k+N)]T；
Figure BDA00022582815100000210
Figure BDA0002258281510000029
in the formula, T represents transposition, N is preDomain measurement, NcIs the control domain.
Preferably, the cost function in step 3 is
Figure BDA0002258281510000031
In the formula (3), Y is represented by the formula (2) theta*Gives a sequence for the motor rotor angle, and Θ*＝[θ*T(k+1)θ*T(k+2)…θ*T(k+N)]T，θ*(k)＝[θx *(k)θy *(k)]T；QcIs a matrix of contour error weight coefficients, and Qc＝ωcIN×N；QaIs a tracking error weight coefficient matrix, and Qa＝ωaI(3N)×(3N)；QuTo control the incremental weight coefficient matrix, and Qu＝(diag[ωux，ωuy]，...，diag[ωux，ωuy])(2Nc)×(2Nc)(ii) a Wherein ω iscAnd ωaRespectively profile error and tracking error coefficients, omegauxAnd ωuyControl incremental weight coefficients for the x and y axes, respectively, and ωc、ωa、ωuxAnd ωuyAre all more than or equal to 0, and I is an identity matrix;
l is a matrix of the relationship between profile error and tracking error, is
Figure BDA0002258281510000032
By applying to the merit function (3)
Figure BDA0002258281510000033
Solving to obtain a control increment vector matrix, which is as follows:
ΔIq＝(GTLTQcLG+GTQaG+Qu)-1×GT(LTQcL+Qa)(Θ*-Y) (4)
the model predictive control quantity vector at the current moment is as follows:
iq(k)＝iq(k-1)+Δiq(k) (5)
in the formula (5), the reaction mixture is,
Figure BDA0002258281510000034
preferably, the control increment vector and the control quantity vector passing through the limiting module at the current moment are respectively delta iq *(k) And iq *(k) The limiting module in the step 4 is:
Figure BDA0002258281510000035
as is clear from the formulae (1) and (6), Δ iq *(k) Is expressed as Δ iq *(k)＝[Δiqx *(k) Δiqx *(k-1) Δiqy *(k) Δiqy *(k-1)]T，Δiqx *(k) And Δ iqx *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqy *(k) And Δ iqy *(k-1) control increments of the y motor at the time k and the time k-1 respectively; i.e. iq *(k) Is represented by the form iq *(k)＝[iqx *(k) iqx *(k-1) iqy *(k) iqy *(k-1)]T，iqx *(k) And iqx *(k-1) control quantities of x motors at the time k and the time k-1, iqy *(k) And iqy *(k-1) respectively representing the control quantity of the y motor at the k moment and the k-1 moment; will iqx *(k) And iqy *(k) The double-shaft feed driving system is used as an input of a double-current closed-loop control system and acts on the double-shaft feed driving system.
A method for improving the robustness and the profile performance of a triaxial feed drive system comprises the following steps:
step 1, establishing a state equation of a three-axis feeding driving system;
step 2, multi-step prediction: performing multiple iterations on the state equation of the three-axis feed driving system to obtain a matrix of multi-step prediction output;
step 3, establishing a cost function, substituting the matrix predicted and output in the step 2 into the cost function, carrying out minimum solution, further obtaining a model prediction control increment vector matrix, obtaining a control increment vector at the current moment and a control quantity vector at the current moment, and acting the control increment vector at the current moment on the triaxial feed driving system;
and 4, carrying out posterior check by using a limiting module: and (3) acting the control quantity corresponding to the control quantity vector passing through the limiting module on the triaxial feed driving system in the step (1) as a current loop input, and acting the control increment vector passing through the limiting module on the state equation in the step (1) to form a closed-loop control network.
Preferably, the state equation of the three-axis feed driving system in step 1 is as follows:
Figure BDA0002258281510000041
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000042
Δiq(k)＝[Δiqx T(k) Δiqy T(k) Δiqz T(k)]T；y(k)＝[θx(k) θy(k) θz(k)]T；x(k)＝[θx(k) θx(k-1) θx(k-2) θy(k) θy(k-1) θy(k-2) θz(k) θz(k-1) θz(k-2)]T；
Figure BDA0002258281510000043
Δiqi(k)＝[Δiqi(k) Δiqi(k-1)]T；ci＝[1 0 0]i∈{x，y，z}；
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000044
ai1＝-1-ai2,
Figure BDA0002258281510000045
Figure BDA0002258281510000046
Tsk, k-1 and k-2 are respectively the k moment, the k-1 moment and the k-2 moment;
Jeqiis equivalent moment of inertia, and Jeqi＝Ji+Miri 2/4π2；BeqiIs an equivalent viscous friction coefficient, and Beqi＝Bi+Ciri 2/4π2
JiIs the rotational inertia of the motor; thetaiIs the rotor mechanical angle; b isiThe viscous friction coefficient of the motor; ciIs the viscous friction coefficient of the motion mechanism; miIs the moving part mass; tau isiIs the load torque; ktiThe torque coefficient of the permanent magnet synchronous motor is; i.e. iqiBeing the q-axis component of the stator current, Δ iqi(k) Is the control increment of the motor;
preferably, the matrix of the multi-step prediction output in step 2 is:
Y＝Fx(k)+GΔIq (8)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000047
Y＝[yT(k+1) yT(k+2) … yT(k+N)]T；
Figure BDA0002258281510000048
Figure BDA0002258281510000051
where T denotes transpose, N is the prediction field, NcIs the control domain.
Preferably, the cost function in step 3 is:
Figure BDA0002258281510000052
in the formula, theta*Gives a sequence for the motor rotor angle, and Θ*＝[θ*T(k+1) θ*T(k+2) … θ*T(k+N)]T，θ*(k)＝[θx *(k) θy *(k) θz *(k)]T；QcIs a matrix of contour error weight coefficients, and Qc＝ωcIN×N；QaIs a tracking error weight coefficient matrix, and Qa＝ωaI(3N)×(3N)；QuTo control the incremental weight coefficient matrix, and Qu＝(diag[ωux，ωuy，ωuz]，...，diag[ωux，ωuy，ωuz])(3Nc)×(3Nc)(ii) a Wherein ω iscAnd ωaRespectively profile error and tracking error coefficients, omegaux、ωuy、ωuzControl incremental weight coefficients for the x, y, and z axes, respectively, and ωc、ωa、ωux、ωuyAnd ωuzAre all more than or equal to 0, and I is an identity matrix;
l is a matrix of the relationship between profile error and tracking error, is
Figure BDA0002258281510000053
By applying to the merit function (9)
Figure BDA0002258281510000054
Solving to obtain a control increment vector matrix, which is as follows:
ΔIq＝(GTLTQcLG+GTQaG+Qu)-1×GT(LTQcL+Qa)(Θ*-Y) (10)
the model predictive control quantity vector at the current moment is as follows:
iq(k)＝iq(k-1)+Δiq(k) (11)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000055
preferably, the limiting module in step 4 is:
Figure BDA0002258281510000061
as is clear from the formulae (7) and (12), Δ iq *(k) Is expressed as Δ iq *(k)＝[Δiqx *(k)Δiqx *(k-1)Δiqy *(k)Δiqy *(k-1)Δiqz *(k)Δiqz *(k-1)]T，Δiqx *(k) And Δ iqx *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqy *(k) And Δ iqy *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqz *(k) And Δ iqz *(k-1) respectively controlling increment of the z motor at the k moment and the k-1 moment; i.e. iq *(k) Is represented by the form iq *(k)＝[iqx *(k)iqx *(k-1)iqy *(k)iqy *(k-1)iqz *(k)iqz *(k-1)]T，iqx *(k) And iqx *(k-1) control quantities of x motors at the time k and the time k-1, iqy *(k) And iqy *(k-1) control quantities of the motor at the time k and the time k-1, iqz *(k) And iqz *When (k-1) is each kThe control quantity of the motor at the moment of time k-1; will iqx *(k)、iqy *(k) And iqz *(k) The three-current closed-loop control system is used as an input of the three-current closed-loop control system and acts on the three-shaft feed driving system.
Compared with the prior art, the invention has the beneficial effects that:
1. the model predictive control of the invention has the advantages of intuitive modeling, simple control structure, fast dynamic response, easy processing of system constraint and the like when processing multivariable and multi-target control, and the research on the predicted contour control strategy of the invention is favorable for improving the motion stability and contour accuracy of a control system, is favorable for further meeting the requirements of advanced manufacturing technology, and provides powerful technical support for high-speed and high-precision processing technology necessary for the current manufacturing industry.
2. The method is different from the traditional cascade control structure, is based on the idea of unified modeling, takes a motion mechanism of a double-shaft or three-shaft feed driving system and a permanent magnet synchronous motor as a whole, and simultaneously establishes a unified model by combining an internal model principle to compensate the non-modeled nonlinear friction and the slowly time-varying disturbance.
3. The method can improve the transient profile tracking performance of the system on the premise of ensuring the steady-state profile tracking performance, particularly when a large turn occurs, the profile error can be reduced, the adjusting speed can be accelerated, and meanwhile, the method has better robustness to constant and slow time-varying disturbance.
Drawings
FIG. 1 is a general schematic diagram for improving the robustness and profile performance of a dual axis feed drive system.
FIG. 2 is a schematic diagram of the error of arbitrary trajectory profile in plane and space.
Detailed Description
The present invention will be described in further detail with reference to specific examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1
The invention takes the motion mechanism and the motors as a whole to carry out unified modeling, and combines the internal model principle to inhibit the non-modeled nonlinear friction and the slowly time-varying disturbance. A double-shaft or three-shaft position-current prediction controller is designed through a model prediction control method based on a state space, a contour error and a tracking error are added into a cost function as internal variables of a double-shaft or three-shaft feed driving system, and then a control input signal enabling tracking precision and contour precision to be coordinated and optimal is searched by using the ideas of multi-step prediction, rolling optimization and feedback correction in model prediction control. Meanwhile, according to actual requirements, constraint conditions are considered, and a posterior constraint limiting network of current increment is established; finally, the purpose of improving the tracking and contour control performance of the double-shaft or three-shaft feeding driving system is achieved.
For the model predictive profile control strategy of the dual-axis feed drive system, the structural block diagram of the model predictive profile control system of the invention is shown in fig. 1.
In the figure, θ is a rotor mechanical angle vector, and θ ═ θx θy]T，θxFor x-axis rotor mechanical angle, thetayIs the mechanical angle of the y-axis rotor; theta*Is a desired rotor mechanical angle vector, and θ*＝[θx * θy *]T，θx *Desired rotor mechanical angle for x-axis, θy *The rotor mechanical angle is desired for the y-axis.
For the model prediction contour control strategy of the biaxial or triaxial feed driving system, a corresponding plane or space arbitrary trajectory contour error schematic diagram is shown in fig. 2.
In the figure, P*And P is an expected position point, P is an actual position point, S is a point of the actual position closest to the expected contour, and N is the center of the circular arc track. The tracking error e is defined as the distance between the actual position point and the desired position point, ex、eyRespectively tracking errors of an x axis and a y axis, the contour error epsilon is defined as the shortest distance between an actual position point and an expected contour, beta is the included angle between the tangent line of the expected position point of the contour track and the positive direction of the x axis, R is the radius of a track osculating circle,
Figure BDA0002258281510000071
to approximate the contour error, t is the unit tangent vector at the desired location point, L is the relationship vector of the contour error and the tracking error, and for a planar arbitrary trajectory, L ═ Lx Ly]T；LxIs the x-axis component of L, and Lx＝sinβ-ex/2R；LyIs the y-axis component of L, and Ly＝-cosβ-ey/2R。
Example 2
For the model predictive profile control method of the dual-axis feed driving system, the scheme of embodiment 1 is further described by combining the calculation formula and examples, and the following description refers to:
(1) establishment of unified model of double-shaft feed driving system
The mathematical model of the motion mechanism in the double-shaft feed driving system is as follows:
Figure BDA0002258281510000072
where k is the current time, MiIs the moving part mass; p is a radical ofiThe actual position of the track;
Figure BDA0002258281510000073
is piThe first derivative of the first time of the first,
Figure BDA0002258281510000074
is piSecond derivative of, CiIs the coefficient of viscous friction of the moving parts; f. ofiThe driving force is used for driving the sliding block to move; ffiIs considered to be a non-linear friction of unknown constant perturbations.
The motion equation of the permanent magnet synchronous motor for driving the double-shaft feeding driving system is as follows:
Figure BDA0002258281510000081
in the formula, JiIs the rotational inertia of the motor; thetaiIs the rotor mechanical angle; b isiThe viscous friction coefficient of the motor; tau isiIs the load torque; ktiThe torque coefficient of the permanent magnet synchronous motor is; i.e. iqiIs the q-axis component of the stator current;
Figure BDA0002258281510000082
is thetaiThe second derivative of (a), which can be used to represent the rotor mechanical angular acceleration;
Figure BDA0002258281510000083
is thetaiThe first derivative of (a) may be used to represent the mechanical angular speed of the rotor.
The relationship between the driving force for driving the slide block to move and the load torque, the actual position of the track and the mechanical angle of the motor rotor is as follows:
fi＝2πτi/ri,pi＝riθi/2π (3)
in the formula, riThe pitch of the ball screw.
Combining the formulas (1) to (3), the ideal model for establishing the double-shaft feed driving system is as follows:
Figure BDA0002258281510000084
in the formula, JeqiIs equivalent moment of inertia, and Jeqi＝Ji+Miri 2/4π2；BeqiIs an equivalent viscous friction coefficient, and Beqi＝Bi+Ciri 2/4π2。
Converting equation (4) to a transfer function form as:
Figure BDA0002258281510000085
to suppress constant and slowly time-varying disturbances, a zeroth order keeper is first added, then the z-transform is applied and the transformed transfer function is multiplied by z-1/(z-1), so the modified transfer function can be written as:
Figure BDA0002258281510000086
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000087
ai1＝-1-ai2,
Figure BDA0002258281510000088
Figure BDA0002258281510000089
Tsis the sampling period, i ∈ { x, y }.
Writing the transfer function of the formula (6) into a form of a state equation, obtaining a state equation of an x-axis motor when the formula (6) i is equal to x, obtaining a state equation of a y-axis motor when the formula (6) i is equal to y, and combining the state equations of the two motors to obtain a state equation of the two-axis feed driving system simultaneously, wherein the state equation can be expressed as:
Figure BDA0002258281510000091
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000092
Δiq(k)＝[Δiqx T(k) Δiqy T(k)]T；y(k)＝[θx(k) θy(k)]T；x(k)＝[θx(k) θx(k-1) θx(k-2) θy(k) θy(k-1) θy(k-2)]T；
Figure BDA0002258281510000093
Δiqi(k)＝[Δiqi(k) Δiqi(k-1)]T；ci＝[1 0 0]；x(k)＝[θi(k) θi(k-1) θi(k-2)]T；i∈{x，y}，Δiqi(k) is the control increment of the motor.
(2) Multi-step prediction
By iterating the equation of state (7) of the dual-axis feed drive system a plurality of times, a matrix of multi-step prediction outputs is obtained, which is:
Y＝Fx(k)+GΔIq (8)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000094
Y＝[yT(k+1) yT(k+2) … yT(k+N)]T；
Figure BDA0002258281510000098
Figure BDA0002258281510000096
where T denotes transpose, N is the prediction field, NcIs the control domain.
(3) Value function solving
To control both tracking error and contour error, the cost function may be chosen as:
Figure BDA0002258281510000097
wherein Y is represented by the formula (8) (. theta.)*Gives a sequence for the motor rotor angle, and Θ*＝[θ*T(k+1) θ*T(k+2) … θ*T(k+N)]T，θ*(k)＝[θx *(k) θy *(k)]T；QcIs a matrix of contour error weight coefficients, and Qc＝ωcIN×N；QaIs a tracking error weight coefficient matrix, and Qa＝ωaI(3N)×(3N)；QuTo control the incremental weight coefficient matrix, and Qu＝(diag[ωux，ωuy]，...，diag[ωux，ωuy])(2Nc)×(2Nc)(ii) a Wherein ω iscAnd ωaRespectively profile error and tracking error coefficients, omegauxAnd ωuyControl incremental weight coefficients for the x and y axes, respectively, and ωc、ωa、ωuxAnd ωuyAre all greater than or equal to 0, and I is an identity matrix.
L is a matrix of the relationship between profile error and tracking error, is
Figure BDA0002258281510000101
By applying to the merit function (9)
Figure BDA0002258281510000102
Solving to obtain a control increment vector matrix, which is as follows:
ΔIq＝(GTLTQcLG+GTQaG+Qu)-1×GT(LTQcL+Qa)(Θ*-Y) (10)
the model predictive control quantity vector at the current moment is as follows:
iq(k)＝iq(k-1)+Δiq(k) (11)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000103
obtaining an N-step prediction output matrix through multi-step prediction in the second step, substituting the prediction output matrix Y into the value function selected in the third step, and carrying out minimum solving to further obtain a model prediction control incremental vector matrix delta IqAccording to the basic principle of model predictive control, only the control increment vector delta i at the current moment is usedq(k) Acting on the dual axis feed drive system.
(4) Restraint module
However, in the actual dual-shaft feeding driving system, the stator voltage and the stator current of the driving motor should satisfy the following constraint conditions (taking the x-axis as an example) under the limitation of the maximum value of the voltage and the current of the three-phase inverter on the direct current side, the rated output current and the voltage and the current of the motor itself
Figure BDA0002258281510000104
In the formula usxIs the phase voltage amplitude; u. ofdx、uqxD and q axis components of the stator voltage, respectively; u. oflimxIs the maximum value of the phase voltage; i.e. isxIs the current amplitude; i.e. idi、iqiD-axis and q-axis components of the stator current, respectively; i.e. ilimxIs the current maximum.
In order to simplify the implementation and reduce the on-line calculation time, a posterior implementation method is adopted, a posterior constraint limiting network considering current increment is provided, for example, a limiting module in fig. 1, a current control increment vector delta i acting on the current moment is solved and determined through a cost function in the previous stepq(k) Then making the current control quantity vector iq(k) After passing through the limiting module, the result delta i is obtainedq *(k) The feedback is used for model predictive control modeling to form a closed-loop control network. The concrete implementation formula of the limiting module is
Figure BDA0002258281510000111
As is clear from the formulae (7) and (13), Δ iq *(k) Is expressed as Δ iq *(k)＝[Δiqx *(k) Δiqx *(k-1) Δiqy *(k) Δiqy *(k-1)]T，Δiqx *(k) And Δ iqx *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqy *(k) And Δ iqy *(k-1) control increments of the y motor at the time k and the time k-1 respectively; i.e. iq *(k) Is represented by the form iq *(k)＝[iqx *(k) iqx *(k-1) iqy *(k) iqy *(k-1)]T，iqx *(k) And iqx *(k-1) control quantities of x motors at the time k and the time k-1, iqy *(k) And iqy *(k-1) respectively representing the control quantity of the y motor at the k moment and the k-1 moment; will iqx *(k) And iqy *(k) The double-shaft feed driving system is used as an input of a double-current closed-loop control system and acts on the double-shaft feed driving system.
Example 3
For the model predictive contour control method of the three-axis feed driving system, the following further introduces the scheme in embodiment 1 by combining the calculation formula and the example, see the following description for details, this embodiment emphasizes the different parts from embodiment 2, and the same parts are not repeated:
(1) establishment of unified model of triaxial feed driving system
The equation of state for a three-axis feed drive system can be expressed as:
Figure BDA0002258281510000112
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000113
Δiq(k)＝[Δiqx T(k) Δiqy T(k) Δiqz T(k)]T；y(k)＝[θx(k) θy(k) θz(k)]T；x(k)＝[θx(k) θx(k-1) θx(k-2) θy(k) θy(k-1) θy(k-2) θz(k) θz(k-1) θz(k-2)]T；
Figure BDA0002258281510000114
Δiqi(k)＝[Δiqi(k) Δiqi(k-1)]T；ci＝[1 0 0]。
(2) multi-step prediction
The matrix of the multi-step prediction output is:
Y＝Fx(k)+GΔIq (15)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000115
Y＝[yT(k+1) yT(k+2) … yT(k+N)]T；
Figure BDA0002258281510000116
Figure BDA0002258281510000121
where T denotes transpose, N is the prediction field, NcIs the control domain.
(3) Value function solving
By solving the cost function, we can obtain:
ΔIq＝(GTLTQcLG+GTQaG+Qu)-1×GT(LTQcL+Qa)(Θ*-Y) (16)
in the formula, theta*Gives a sequence for the motor rotor angle, and Θ*＝[θ*T(k+1) θ*T(k+2) … θ*T(k+N)]T，θ*(k)＝[θx *(k) θy *(k) θz *(k)]T；QcIs a matrix of contour error weight coefficients, and Qc＝ωcIN×N；QaIs a tracking error weight coefficient matrix, and Qa＝ωaI(3N)×(3N)；QuTo control the incremental weight coefficient matrix, and Qu＝(diag[ωux，ωuy，ωuz]，...，diag[ωux，ωuy，ωuz])(3Nc)×(3Nc)(ii) a Wherein ω iscAnd ωaRespectively profile error and tracking error coefficients, omegaux、ωuy、ωuzControl incremental weight coefficients for the x, y, and z axes, respectively, and ωc、ωa、ωux、ωuyAnd ωuzAre all greater than or equal to 0, and I is an identity matrix.
L is a matrix of the relationship between profile error and tracking error, is
Figure BDA0002258281510000122
The model predictive control quantity vector at the current moment is as follows:
iq(k)＝iq(k-1)+Δiq(k) (17)
in the formula (I), the compound is shown in the specification,
Figure BDA0002258281510000123
(4) restraint module
The concrete implementation formula of the limiting module is
Figure BDA0002258281510000124
As is clear from the formulae (14) and (18), Δ iq *(k) Is expressed as Δ iq *(k)＝[Δiqx *(k)Δiqx *(k-1)Δiqy *(k)Δiqy *(k-1)Δiqz *(k)Δiqz *(k-1)]T，Δiqx *(k) And Δ iqx *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqy *(k) And Δ iqy *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqz *(k) And Δ iqz *(k-1) respectively controlling increment of the z motor at the k moment and the k-1 moment; i.e. iq *(k) Is represented by the form iq *(k)＝[iqx *(k)iqx *(k-1)iqy *(k)iqy *(k-1)iqz *(k)iqz *(k-1)]T，iqx *(k)And iqx *(k-1) control quantities of x motors at the time k and the time k-1, iqy *(k) And iqy *(k-1) control quantities of the motor at the time k and the time k-1, iqz *(k) And iqz *(k-1) respectively representing the control quantity of the z motor at the k moment and the k-1 moment; will iqx *(k)、iqy *(k) And iqz *(k) The three-current closed-loop control system is used as an input of the three-current closed-loop control system and acts on the three-shaft feed driving system.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (10)

1. A method of improving robustness and profile performance of a dual-axis feed drive system, comprising the steps of:
step 1, establishing a state equation of a double-shaft feed driving system;
step 2, multi-step prediction: performing multiple iterations on the state equation of the double-shaft feed driving system to obtain a matrix of multi-step prediction output;
step 3, establishing a cost function, substituting the matrix predicted and output in the step 2 into the cost function, carrying out minimum solution, further obtaining a model prediction control increment vector matrix, obtaining a control increment vector at the current moment and a control quantity vector at the current moment, and acting the control increment vector at the current moment on the double-shaft feed driving system;
and 4, carrying out posterior check by using a limiting module: and (3) applying the control quantity corresponding to the control quantity vector passing through the limiting module to the double-shaft feed driving system as current loop input, and applying the control increment vector passing through the limiting module to the state equation in the step (1) to form a closed-loop control network.
2. The method for improving robustness and profile performance of a dual axis feed drive system as claimed in claim 1 wherein the state equations of step 1 are
Figure FDA0002258281500000011
Figure FDA0002258281500000012
Δiq(k)＝[Δiqx T(k) Δiqy T(k)]T；y(k)＝[θx(k) θy(k)]T；x(k)＝[θx(k) θx(k-1) θx(k-2) θy(k) θy(k-1) θy(k-2)]T；
Figure FDA0002258281500000013
Δiqi(k)＝[Δiqi(k) Δiqi(k-1)]T；ci＝[1 0 0]；x(k)＝[θi(k) θi(k-1) θi(k-2)]T；
i∈{x，y}；
In the formula (I), the compound is shown in the specification,
Figure FDA0002258281500000014
Figure FDA0002258281500000015
Tsk, k-1 and k-2 are respectively the k moment, the k-1 moment and the k-2 moment;
Jeqiis equivalent moment of inertia, and Jeqi＝Ji+Miri 2/4π2；BeqiIs an equivalent viscous friction coefficient, and Beqi＝Bi+Ciri 2/4π2
JiIs the rotational inertia of the motor; thetaiIs the rotor mechanical angle; b isiFor viscous friction of electric machinesA coefficient; ciIs the viscous friction coefficient of the motion mechanism; miIs the moving part mass; tau isiIs the load torque; ktiThe torque coefficient of the permanent magnet synchronous motor is; i.e. iqiIs the q-axis component of the stator current; Δ iqi(k) Is the control increment of the motor.
3. The method of improving robustness and profile performance of a dual axis feed drive system as claimed in claim 1, wherein said matrix of multi-step prediction outputs in step 2:
Y＝Fx(k)+GΔIq (2)
in the formula (I), the compound is shown in the specification,
Figure FDA0002258281500000021
Y＝[yT(k+1) yT(k+2)…yT(k+N)]T；
Figure FDA0002258281500000022
Figure FDA0002258281500000023
Figure FDA0002258281500000024
where T denotes transpose, N is the prediction field, NcIs the control domain.
4. The method for improving robustness and profile performance of a dual axis feed drive system as claimed in claim 1 wherein said cost function of step 3 is
Figure FDA0002258281500000025
In the formula (3), Y is represented by the formula (2) theta*Gives a sequence for the motor rotor angle, and Θ*＝[θ*T(k+1)θ*T(k+2)…θ*T(k+N)]T，θ*(k)＝[θx *(k)θy *(k)]T；QcIs a matrix of contour error weight coefficients, and Qc＝ωcIN×N；QaIs a tracking error weight coefficient matrix, and Qa＝ωaI(3N)×(3N)；QuTo control the incremental weight coefficient matrix, and Qu＝(diag[ωux，ωuy]，...，diag[ωux，ωuy])(2Nc)×(2Nc)(ii) a Wherein ω iscAnd ωaRespectively profile error and tracking error coefficients, omegauxAnd ωuyControl incremental weight coefficients for the x and y axes, respectively, and ωc、ωa、ωuxAnd ωuyAre all more than or equal to 0, and I is an identity matrix;
l is a matrix of the relationship between profile error and tracking error, is
Figure FDA0002258281500000026
By applying to the merit function (3)
Figure FDA0002258281500000027
Solving to obtain a control increment vector matrix, which is as follows:
ΔIq＝(GTLTQcLG+GTQaG+Qu)-1×GT(LTQcL+Qa)(Θ*-Y) (4)
the model predictive control quantity vector at the current moment is as follows:
iq(k)＝iq(k-1)+Δiq(k) (5)
in the formula (5), the reaction mixture is,
Figure FDA0002258281500000028
5. the improved double-shaft feed drive system robustness of claim 1The method for the performance of the stick and the contour is characterized in that a control increment vector and a control quantity vector of the current moment passing through a limiting module are respectively made to be delta iq *(k) And iq *(k) The limiting module in the step 4 is:
Figure FDA0002258281500000031
as is clear from the formulae (1) and (6), Δ iq *(k) Is expressed as Δ iq *(k)＝[Δiqx *(k)Δiqx *(k-1)Δiqy *(k)Δiqy *(k-1)]T，Δiqx *(k) And Δ iqx *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqy *(k) And Δ iqy *(k-1) control increments of the y motor at the time k and the time k-1 respectively; i.e. iq *(k) Is represented by the form iq *(k)＝[iqx *(k)iqx *(k-1)iqy *(k)iqy *(k-1)]T，iqx *(k) And iqx *(k-1) control quantities of x motors at the time k and the time k-1, iqy *(k) And iqy *(k-1) respectively representing the control quantity of the y motor at the k moment and the k-1 moment; will iqx *(k) And iqy *(k) The double-shaft feed driving system is used as an input of a double-current closed-loop control system and acts on the double-shaft feed driving system.
6. A method for improving the robustness and profile performance of a three-axis feed drive system, comprising the steps of:
step 1, establishing a state equation of a three-axis feeding driving system;
step 2, multi-step prediction: performing multiple iterations on the state equation of the three-axis feed driving system to obtain a matrix of multi-step prediction output;
step 3, establishing a cost function, substituting the matrix predicted and output in the step 2 into the cost function, carrying out minimum solution, further obtaining a model prediction control increment vector matrix, obtaining a control increment vector at the current moment and a control quantity vector at the current moment, and acting the control increment vector at the current moment on the triaxial feed driving system;
and 4, carrying out posterior check by using a limiting module: and (3) acting the control quantity corresponding to the control quantity vector passing through the limiting module on the triaxial feed driving system in the step (1) as a current loop input, and acting the control increment vector passing through the limiting module on the state equation in the step (1) to form a closed-loop control network.
7. The method for improving the robustness and profile performance of a three-axis feed drive system of claim 6, wherein the state equation of the three-axis feed drive system in step 1:
Figure FDA0002258281500000032
in the formula (I), the compound is shown in the specification,
Figure FDA0002258281500000033
Δiq(k)＝[Δiqx T(k) Δiqy T(k) Δiqz T(k)]T；y(k)＝[θx(k) θy(k) θz(k)]T；x(k)＝[θx(k) θx(k-1) θx(k-2) θy(k) θy(k-1) θy(k-2) θz(k) θz(k-1) θz(k-2)]T；
Figure FDA0002258281500000041
Δiqi(k)＝[Δiqi(k) Δiqi(k-1)]T；ci＝[1 0 0]
i∈{x，y，z}；
in the formula (I), the compound is shown in the specification,
Figure FDA0002258281500000042
ai1＝-1-ai2,
Figure FDA0002258281500000043
Figure FDA0002258281500000044
Tsk, k-1 and k-2 are respectively the k moment, the k-1 moment and the k-2 moment;
Jeqiis equivalent moment of inertia, and Jeqi＝Ji+Miri 2/4π2；BeqiIs an equivalent viscous friction coefficient, and Beqi＝Bi+Ciri 2/4π2
JiIs the rotational inertia of the motor; thetaiIs the rotor mechanical angle; b isiThe viscous friction coefficient of the motor; ciIs the viscous friction coefficient of the motion mechanism; miIs the moving part mass; tau isiIs the load torque; ktiThe torque coefficient of the permanent magnet synchronous motor is; i.e. iqiBeing the q-axis component of the stator current, Δ iqi(k) Is the control increment of the motor.
8. The method for improving the robustness and profile performance of a three-axis feed drive system of claim 6, wherein the matrix of the multi-step prediction output in step 2 is:
Y＝Fx(k)+GΔIq (8)
in the formula (I), the compound is shown in the specification,
Figure FDA0002258281500000045
Y＝[yT(k+1) yT(k+2)…yT(k+N)]T；
Figure FDA0002258281500000046
Figure FDA0002258281500000047
Figure FDA0002258281500000048
where T denotes transpose, N is the prediction field, NcIs the control domain.
9. The method for improving the robustness and profile performance of a three-axis feed drive system as claimed in claim 6, wherein the cost function in step 3 is:
Figure FDA0002258281500000049
in the formula, theta*Gives a sequence for the motor rotor angle, and Θ*＝[θ*T(k+1)θ*T(k+2)…θ*T(k+N)]T，θ*(k)＝[θx *(k)θy *(k)θz *(k)]T；QcIs a matrix of contour error weight coefficients, and Qc＝ωcIN×N；QaIs a tracking error weight coefficient matrix, and Qa＝ωaI(3N)×(3N)；QuTo control the incremental weight coefficient matrix, and Qu＝(diag[ωux，ωuy，ωuz]，...，diag[ωux，ωuy，ωuz])(3Nc)×(3Nc)(ii) a Wherein ω iscAnd ωaRespectively profile error and tracking error coefficients, omegaux、ωuy、ωuzControl incremental weight coefficients for the x, y, and z axes, respectively, and ωc、ωa、ωux、ωuyAnd ωuzAre all more than or equal to 0, and I is an identity matrix;
l is a matrix of the relationship between profile error and tracking error, is
Figure FDA0002258281500000051
By applying to the merit function (9)
Figure FDA0002258281500000052
Solving to obtain a control increment vector matrix, which is as follows:
ΔIq＝(GTLTQcLG+GTQaG+Qu)-1×GT(LTQcL+Qa)(Θ*-Y) (10)
the model predictive control quantity vector at the current moment is as follows:
iq(k)＝iq(k-1)+Δiq(k) (11)
in the formula (I), the compound is shown in the specification,
Figure FDA0002258281500000053
10. the method for improving the robustness and profile performance of a three-axis feed drive system of claim 6, wherein the limiting module of step 4 is:
Figure FDA0002258281500000054
as is clear from the formulae (7) and (12), Δ iq *(k) Is expressed as Δ iq *(k)＝[Δiqx *(k)Δiqx *(k-1)Δiqy *(k)Δiqy *(k-1)Δiqz *(k)Δiqz *(k-1)]T，Δiqx *(k) And Δ iqx *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqy *(k) And Δ iqy *(k-1) control increments of the motor at time k and time k-1, respectively, Δ iqz *(k) And Δ iqz *(k-1) respectively controlling increment of the z motor at the k moment and the k-1 moment; i.e. iq *(k) Is represented by the form iq *(k)＝[iqx *(k) iqx *(k-1) iqy *(k) iqy *(k-1) iqz *(k) iqz *(k-1)]T，iqx *(k) And iqx *(k-1) control quantities of x motors at the time k and the time k-1, iqy *(k) And iqy *(k-1) control quantities of the motor at the time k and the time k-1, iqz *(k) And iqz *(k-1) respectively representing the control quantity of the z motor at the k moment and the k-1 moment; will iqx *(k)、iqy *(k) And iqz *(k) The three-current closed-loop control system is used as an input of the three-current closed-loop control system and acts on the three-shaft feed driving system.
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