CN112947083B - Nonlinear model predictive control optimization method based on magnetic suspension control system - Google Patents
Nonlinear model predictive control optimization method based on magnetic suspension control system Download PDFInfo
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Abstract
The invention provides a nonlinear model predictive control optimization method based on a magnetic suspension control system. According to the model prediction control principle, a nonlinear motion model combined with a disturbance estimation value is taken as a prediction model, the motion range of a magnetic suspension workbench and the maximum output of a current power amplifier are taken as constraint conditions, the total cost of tracking errors and current loads in a prediction step length is taken as a target function, and the problem of multi-step rolling optimization control of the magnetic suspension workbench is established; finally, the optimization control problem is converted into a nonlinear Eulerian Langerian equation set, and an optimization control strategy is solved by using a parallelization Newton method capable of being rapidly executed on line to form closed-loop control; the method solves the problem of real-time application of nonlinear model predictive control on a high-frequency sampling magnetic suspension system, and has universality, higher accuracy and convergence rate; the control method provided by the invention improves the accuracy of the magnetic suspension workbench tracking the expected track and improves the dynamic characteristic of the system.
Description
Technical Field
The invention relates to the technical field of magnetic suspension workbench motion control, in particular to a nonlinear model predictive control optimization method based on a magnetic suspension control system.
Background
The magnetic suspension technology is an effective means for providing multi-degree-of-freedom motion in ultra-precision application in industries such as semiconductor processing, bioengineering, aerospace part manufacturing and the like, and because of the full suspension characteristic of no mechanical and electrical connection, the multi-degree-of-freedom magnetic suspension workbench needs to perform complex translation and rotation control on all six axes even if the multi-degree-of-freedom magnetic suspension workbench is simple in set point adjustment. In addition, the inherent characteristics of an actual system, such as nonlinear coupling effects between multiple degrees of freedom motions, modeling inaccuracies, unknown disturbances, etc., will significantly degrade the stage performance and present difficulties to the motion controller design compared to a nominal system. The model predictive control is a mainstream advanced control method developed in recent years, depends on the solution of an optimization control problem under economic excitation, and has the potential of being applied to a nonlinear multiple-input multiple-output magnetic levitation system under the condition of considering state constraint and control constraint.
The development and implementation of advanced control schemes such as fuzzy control, iterative learning control, neural network control, adaptive robust control and the like are of great significance for improving the positioning and track tracking performance of the multi-degree-of-freedom magnetic suspension workbench. These controllers calculate the dynamic forces or moments required for accurate motion of the system based on a decoupled linear model or a taylor-extended linearized model. However, when the drive currents are bounded, they cannot handle explicit nonlinear process constraints from force and torque to current, and thus cannot provide the desired control performance and dynamic response. Although the nonlinear model predictive control is successfully applied to the single-degree-of-freedom magnetic suspension system, the computational burden of exponential explosion makes the nonlinear model predictive control difficult to apply to the multi-degree-of-freedom magnetic suspension workbench, and real-time closed-loop control cannot be realized.
An integrator is a practical method for model predictive control to attenuate the effects of disturbances and uncertainties in industrial applications. However, due to the coupling of the integration behavior with other control performance (e.g., strict constraint satisfaction, transient response, and stability), it is difficult to select model predictive control parameters that achieve the desired control performance for a closed-loop system. Alternatively, the disturbance observer can be used as a feed forward compensation for the controller without sacrificing nominal performance to enhance robustness against disturbances and uncertainties. In general, a typical application of a disturbance observer in model predictive control is based on lumped disturbance estimation of a nominal system as a correction term to feed forward directly to a control input, and since the disturbance estimation is not included in a backward horizontal optimization process, these model predictive controllers are relatively difficult to obtain active control performance and always cause static errors.
The traditional nonlinear model predictive controller is difficult to ensure high-precision tracking when the magnetic suspension workbench has disturbance and uncertainty, and the optimization calculation under the fast sampling frequency is difficult to meet the real-time requirement. Therefore, the nonlinear model predictive control of the magnetic suspension workbench and the online optimization method thereof are needed.
Disclosure of Invention
In order to solve the above technical problems, the present invention aims to provide a nonlinear model predictive control optimization method based on a magnetic levitation control system, so as to solve the problem that the traditional nonlinear model predictive controller is difficult to realize closed-loop real-time and accurate track tracking control on a magnetic levitation table.
The technical scheme of the system is that the magnetic suspension control system comprises: the device comprises a magnetic suspension workbench, a laser sensor, an analog-to-digital converter, a microprocessor, a digital-to-analog converter and a current power amplifier; the magnetic suspension workbench consists of a stator and a rotor; the stator is composed of a plurality of track coils; the rotor consists of a plurality of groups of Halbach magnetic arrays and a back plate; the stator and the rotor are wirelessly connected;
the laser sensor, the analog-to-digital converter, the microprocessor and the digital-to-analog converter are sequentially connected in series;
the laser sensor is arranged at the periphery of a rotor of the magnetic suspension workbench and used for measuring six-degree-of-freedom motion information of the magnetic suspension workbench and transmitting the six-degree-of-freedom motion information to the analog-to-digital converter;
the analog-to-digital converter performs analog-to-digital conversion on the six-degree-of-freedom motion information of the magnetic suspension workbench to obtain a six-degree-of-freedom motion digital signal of the magnetic suspension workbench, and transmits the six-degree-of-freedom motion digital signal to the microprocessor;
the microprocessor obtains a current control rate through the nonlinear model predictive control optimization method according to the six-degree-of-freedom motion digital signal of the magnetic suspension workbench, and transmits the current control rate to the digital-to-analog converter;
the digital-to-analog converter performs digital-to-analog conversion on the current control rate to obtain an analog current control rate, and transmits the analog current control rate to the current power amplifier for power amplification to obtain a driving current;
the current power amplifier transmits driving current to the stator of the magnetic suspension workbench, and when a plurality of track coils in the stator of the magnetic suspension workbench are energized with current, magnetic fields around the Halbach magnetic array on the rotor generate magnetic force under the excitation of the current so as to drive the magnetic suspension workbench to move in six degrees of freedom, so that closed-loop control is formed.
The technical scheme of the method is a nonlinear model predictive control optimization method, which comprises the following steps:
step 1: constructing a six-degree-of-freedom motion coordinate system by a microprocessor;
step 2: the microprocessor establishes a six-degree-of-freedom magnetic model in a suspension state;
and step 3: equating uncertainty which is not considered in modeling and external disturbance to disturbance lumped on an independent coil, establishing a nonlinear motion model of the magnetic suspension workbench by combining the magnetic force model in the step 2, and designing the lumped disturbance of the nonlinear disturbance observer estimation system according to the model;
and 4, step 4: according to a model prediction control principle, a discrete nonlinear motion model combined with a disturbance estimation value is used as a prediction model, the motion range of the magnetic suspension workbench and the maximum output of a current power amplifier are used as constraint conditions, the tracking error in a prediction step length and the total cost of current load are used as objective functions, and the problem of nonlinear multistep rolling optimization control of the magnetic suspension workbench is established.
And 5: and (4) converting the nonlinear multistep rolling optimization control problem model of the magnetic suspension workbench in the step (4) into a nonlinear Euler Langerian equation set, solving an optimization control strategy by using a parallelization Newton method capable of being rapidly executed on line, and realizing closed-loop control of the system.
Preferably, the six-degree-of-freedom motion coordinate system in step 1 is as follows:
the center of the upper surface of the stator is taken as the origin of a fixed coordinate system(s)so, taking the center of the lower surface of the mover as the origin of the moving coordinate system { t }to, relative displacement amount P of the tablex、Py、PzAt the originto is characterized by a vector difference under a fixed coordinate system, and the relative rotation amounts alpha, beta and gamma are characterized by the rotation amounts of the movable coordinate axis relative to the fixed coordinate axis;
preferably, the six-degree-of-freedom magnetic force model in the suspension state in step 2 is:
combining the mechanical structure and the coordinate system definition of the magnetic suspension workbench in the step 1, according to a magnetic field calculation method based on harmonic analysis and a Lorenz integral rule, the force and the moment acting on the magnetic suspension workbench are as follows:
wherein the content of the first and second substances,
wherein, BrIs the remanent magnetization of the permanent magnet, NcoilIs the number of turns of the track coil, wcAnd hcTrack width and height, respectively, of the track coilmAnd hmLength and height of the individual permanent magnets, K being the thrust coefficient determined by the system structure, λ 2 pi/lmIs the spatial harmonic number, L is the distance from the center of the single Halbach magnetic array to the center of the whole magnetic array, and u ═ i1,i2,i3,i4,i5,i6,i7,i8]TIs the control input of the system under the drive of 8 independent coils;
preferably, the step 3 is a model of nonlinear motion of the magnetic suspension workbench considering lumped current disturbance:
the influence caused by unmodeled dynamics of the system, inaccurate magnetization of the magnet, non-uniform winding of the coil and electromagnetic disturbance of the current amplifier is assumed to be in a lumped form equivalent to that of 8 coils
Giving out;
because of the complete suspension characteristic without mechanical connection, the magnetic force model is further expanded into a form of a nonlinear motion model:
wherein the content of the first and second substances,
the position and the speed of the six degrees of freedom of the system form state variables, namely the six degrees of freedom motion information of the magnetic suspension workbench is obtained by the measurement of the laser sensor;
M＝diag(m,m,m,Iα,Iβ,Iγ) Is a diagonal matrix formed by the rotor mass and the rotational inertia on 3 rotating shafts,
is the gravitational acceleration vector;
the nonlinear disturbance observer in the step 3 is as follows:
aiming at the nonlinear motion model, a disturbance observer in the following form is designed:
wherein the content of the first and second substances,
is an estimate of the disturbance to the system,
is the internal state vector of the disturbance observer, h-r＝hT·(h·hT)-1And
are respectively h and
right inverse of (d), gain of observer to ensure fast convergence of disturbance estimation deviation
A positive definite and symmetrical matrix should be selected;
preferably, the discrete nonlinear motion model combined with the disturbance estimation value in step 4 is taken as a prediction model:
to improve the accuracy of the predicted state, the nonlinear motion model and the estimated value of the disturbance described in step 3 are combined
To estimate the state of the system at a future time:
at system sampling time TsAnd a sampling time tk＝kTsDiscretizing the state of the estimated system at the future time under the condition that k is {0,1, 2. }, and obtaining a prediction model of the system under N prediction steps as follows:
wherein the content of the first and second substances,
is an inverse time function;
and 4, taking the motion range of the magnetic suspension workbench and the maximum output of the current power amplifier as constraint conditions:
considering that the motion range of the actuator and the maximum output of the current power amplifier are constrained as follows:
converting the constraints into mixed inequality constraints:
and 4, taking the total cost of the tracking error and the current load in the prediction step as an objective function:
with xrefConsidering the positively determined tracking error weight for the reference trajectory
And current load weight
Defining the objective function under N prediction steps as:
4, the nonlinear multi-step rolling optimization control problem of the magnetic suspension workbench is as follows:
according to a model prediction control principle, a prediction model, mixed inequality constraints and an objective function of a system under N prediction step lengths are considered, and a magnetic suspension workbench nonlinear multi-step rolling optimization control problem model to be solved is as follows:
wherein the content of the first and second substances,
and
respectively is a state sequence and a control sequence under N prediction step lengths;
preferably, the non-linear euler langery equation set in step 5 is:
converting the mixed inequality constraint of step 4 into a logarithmic barrier function:
followed by the definition of a Lagrange multiplier sequence
And Hamiltonian function of the problem to be optimized:
optimal sequence of states according to the KKT condition
Control sequence
And lagrange multiplier sequence
The following non-linear euler langery equation set should be satisfied:
wherein the initial optimum state
And tail optimal Lagrange multiplier
Is a boundary condition of the system of equations.
And
are respectively H about
And
a gradient of (a);
consider integrationOptimizing variables
Correction matrix
In combination with
Referring to the left side of the equation of the nonlinear eulerian langery system of equations, the parallelized newton method described in step 5 can be performed quickly on-line.
The parallel Newton method capable of being rapidly executed on line in the step 5 mainly comprises the following steps:
step 5.1, preprocessing variables to be optimized according to boundary conditions;
step 5.2, introduction
Approximate estimate of (2)
Roughly updating the integration optimization variables by a gradient descent method in combination with the correction matrix theta;
step 5.3, due to the introduction of
Based on approximate deviation
Optimizing variables for integration in a backward manner
And (6) carrying out correction.
Step 5.4, due to the introduction of
Based on approximate deviation
Optimizing variables for integration in a forward manner
Carrying out correction;
step 5.5, calculating KKT residual epsilon under current iterationjIf the residual error is less than the preset termination residual error epsilon, exiting the while loop and outputting zj+1And thetajOtherwise, returning to the starting point of the cycle;
solving the nonlinear multistep rolling optimization control problem of the magnetic suspension workbench by using a parallelization Newton method capable of being rapidly executed on line at each sampling moment, and outputting an integrated optimization variable zj+1In (1)
Applied to a coil to drive the worktable to move, and updating the state variable x (t) through the six-freedom-degree motion information of the magnetic suspension worktable measured by the laser sensor at the next sampling momentk+1) Solving for an estimate of the disturbance
And let z1＝zj+1，Θ0＝ΘjAnd repeatedly calculating the optimized control rate to form a closed-loop system.
The invention has the following beneficial effects:
a simplified magnetic force model of the magnetic suspension workbench is established, and the model structure is refined, so that the online calculated amount can be reduced;
a disturbance observer is designed aiming at a nonlinear motion model of a magnetic suspension workbench disturbed by a system, and an estimated value of disturbance is combined into a prediction model so as to improve the disturbance resistance and tracking precision of a track tracker;
under the rapid sampling frequency, the provided parallel Newton method can calculate the nonlinear multistep rolling optimization control problem of the magnetic suspension workbench in real time on a processor with a parallel framework, has better universality, can be widely used for solving the optimization problem in a nonlinear model prediction controller, and is favorable for popularization and application.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings of the embodiments will be briefly described below.
FIG. 1: a flow chart provided for an embodiment of the invention.
FIG. 2: a top view of a magnetic levitation table is provided for one embodiment of the present invention.
FIG. 3: a left view of a magnetic levitation table provided in an embodiment of the present invention.
FIG. 4: a magnetic levitation table back view is provided for one embodiment of the present invention.
FIG. 5: the method of the present invention provided for one embodiment of the present invention is outlined in comparison with other methods.
FIG. 6: the invention provides a method and a trajectory tracking comparison chart for other methods.
FIG. 7: the invention provides a profile error comparison graph for the method provided by one embodiment of the invention and other methods.
FIG. 8: the proposed method of the present invention provides a comparison of the frequency response of one embodiment of the present invention with other methods.
Detailed Description
In order to facilitate the understanding and implementation of the present invention for those of ordinary skill in the art, the present invention is further described in detail with reference to the accompanying drawings and examples, it is to be understood that the embodiments described herein are merely illustrative and explanatory of the present invention and are not restrictive thereof.
The following describes an embodiment of the present invention with reference to fig. 1 to 8, specifically:
the technical scheme of the system in the embodiment of the invention is that the magnetic suspension control system comprises: the device comprises a magnetic suspension workbench, a laser sensor, an analog-to-digital converter, a microprocessor, a digital-to-analog converter and a current power amplifier; the magnetic suspension workbench consists of a stator and a rotor; the stator is composed of a plurality of track coils; the rotor consists of a plurality of groups of Halbach magnetic arrays and a back plate; the stator and the rotor are wirelessly connected;
the laser sensor, the analog-to-digital converter, the microprocessor and the digital-to-analog converter are sequentially connected in series;
the laser sensor is arranged at the periphery of a rotor of the magnetic suspension workbench and used for measuring six-degree-of-freedom motion information of the magnetic suspension workbench and transmitting the six-degree-of-freedom motion information to the analog-to-digital converter;
the analog-to-digital converter performs analog-to-digital conversion on the six-degree-of-freedom motion information of the magnetic suspension workbench to obtain a six-degree-of-freedom motion digital signal of the magnetic suspension workbench, and transmits the six-degree-of-freedom motion digital signal to the microprocessor;
the microprocessor obtains a current control rate through the nonlinear model predictive control optimization method according to the six-degree-of-freedom motion digital signal of the magnetic suspension workbench, and transmits the current control rate to the digital-to-analog converter;
the digital-to-analog converter performs digital-to-analog conversion on the current control rate to obtain an analog current control rate, and transmits the analog current control rate to the current power amplifier for power amplification to obtain a driving current;
the current power amplifier transmits driving current to the stator of the magnetic suspension workbench, and when a plurality of track coils in the stator of the magnetic suspension workbench are energized with current, magnetic fields around the Halbach magnetic array on the rotor generate magnetic force under the excitation of the current so as to drive the magnetic suspension workbench to move in six degrees of freedom, so that closed-loop control is formed.
The model of the laser sensor is ZX2-LD 50L;
the model of the analog-to-digital converter is PXIe-7856R;
the model of the microprocessor is PXIe-8880;
the model of the digital-to-analog converter is PXIe-7856R;
the model of the current power amplifier is PA 12A;
the magnetic suspension workbench consists of a stator and a rotor;
the stator is composed of 8 track coils;
the rotor consists of 4 groups of Halbach magnetic arrays and a back plate;
FIG. 1 is an overall process flow diagram of the present invention; the top view of the magnetic levitation table of fig. 2, the left view of the magnetic levitation table of fig. 3 and the back view of the magnetic levitation table of fig. 4 provide the mechanical structure of the magnetic levitation table and the related structural parameters, coordinate system definition. The magnetic suspension workbench comprises a Halbach magnetic array 1, track coils 2 and a back plate 3, a stator consists of 8 track coils, and a rotor consists of 4 groups of Halbach arrays and back plates.
the six-degree-of-freedom motion coordinate system in the step 1 is as follows:
the center of the upper surface of the stator is taken as the origin of a fixed coordinate system(s)so, taking the center of the lower surface of the mover as the origin of the moving coordinate system { t }to, relative displacement amount P of the tablex、Py、PzAt the originto is characterized by a vector difference under a fixed coordinate system, and the relative rotation amounts alpha, beta and gamma are characterized by the rotation amounts of the movable coordinate axis relative to the fixed coordinate axis;
step 2: on the basis of the mechanical structure and coordinate definition of the magnetic suspension workbench, a magnetic force model with six degrees of freedom in a suspension state is established;
combining the mechanical structure and the coordinate system definition of the magnetic suspension workbench in the step 1, according to a magnetic field calculation method based on harmonic analysis and a Lorenz integral rule, the force and the moment acting on the magnetic suspension workbench are as follows:
wherein the content of the first and second substances,
in the above formulas (1) and (2), Br1.2T is the remanent magnetization of the permanent magnet, Ncoil300 is the number of turns of the track coil, wc10mm and hc10mm is the track width and height, respectively, of the track coil,/m40mm and hm10mm is the length and height of the individual permanent magnets, K is the thrust coefficient determined by the system structure, λ 2 pi/lmIs the space harmonic number, L is 80mm, is the distance from the center of a single Halbach magnetic array to the center of the whole magnetic array, and u is [ i ═1,i2,i3,i4,i5,i6,i7,i8]TIs the control input of the system under the drive of 8 independent coils;
and step 3: equating uncertainty which is not considered in modeling and external disturbance to disturbance lumped on an independent coil, establishing a nonlinear motion model of the magnetic suspension workbench by combining the magnetic force model in the step 2, and designing the lumped disturbance of the nonlinear disturbance observer estimation system according to the model;
the nonlinear motion model of the magnetic suspension workbench considering the lumped current disturbance in the step 3 is as follows:
the influence brought by unmodeled dynamics of a system, inaccurate magnetization of a magnet, uneven winding of a coil, electromagnetic disturbance of a current amplifier and the like is assumed to be equivalent to a lumped form on 8 coils
It is given. Because of the complete suspension characteristic without mechanical connection, the magnetic force model is further expanded into a form of a nonlinear motion model:
in the above-mentioned formula (3),
is a system sixThe position and the speed of the degree form a state variable, namely the six-degree-of-freedom motion information of the magnetic suspension workbench, and the state variable is obtained by measuring through the laser sensor;
m-diag (2.37,2.37,2.37,9376.16,9376.16,18665.97) is a diagonal matrix consisting of mover mass and moment of inertia on 3 rotational axes, and g-0, 0,9.8,0,0,0]TIs the gravitational acceleration vector.
The nonlinear disturbance observer in the step 3 is as follows:
for the nonlinear motion model (3), a disturbance observer of the form:
wherein
Is an estimate of the disturbance to the system,
is the internal state vector of the disturbance observer, h-r＝hT·(h·hT)-1And
are respectively h and
right inverse of (c). In order to ensure fast convergence of the disturbance estimation deviation, the gain E of the observer is diag (100,100,100,100,100,100,100,100);
and 4, step 4: according to a model prediction control principle, a discrete nonlinear motion model combined with a disturbance estimation value is used as a prediction model, the motion range of a magnetic suspension workbench and the maximum output of a current power amplifier are used as constraint conditions, the total cost of tracking errors and current loads in a prediction step is used as a target function, and the nonlinear multistep rolling optimization control problem of the magnetic suspension workbench is established;
to improve the accuracy of the predicted state, a non-linear motion model (3) is combined with the estimated value of the disturbance
To estimate the state of the system at a future time:
at system sampling time Ts1ms and sampling time tk＝kTsIn the following description, the formula (5) is discretized by the backward euler method, and the prediction model of the system under N-10 prediction steps is obtained as follows:
in the above-mentioned formula (6),
is an inverse time function;
and 4, taking the motion range of the magnetic suspension workbench and the maximum output of the current power amplifier as constraint conditions:
considering that the motion range of the actuator and the maximum output of the current power amplifier are constrained as follows:
wherein the content of the first and second substances,
xmin＝-[10mm,10mm,5mm,10-2rad,10-2rad,10-2rad,500mm/s,500mm/s,50mm/s,10rad/s,10rad/s,10rad/s]T
xmax＝[10mm,10mm,5mm,10-2rad,10-2rad,10-2rad,500mm/s,500mm/s,50mm/s,10rad/s,10rad/s,10rad/s]T
umin＝-[3A,3A,3A,3A,3A,3A,3A,3A]T
umax＝[3A,3A,3A,3A,3A,3A,3A,3A]T
converting the constraints into mixed inequality constraints:
and 4, taking the total cost of the tracking error and the current load in the prediction step as an objective function:
with xrefFor the reference trajectory, considering the positive tracking error weight Q ═ diag (100,100,100,10000,10000,10000,0.001,0.001,0.001,0.001,0.001,0.001, 0.001) and the current load weight R ═ diag (0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001), the objective function for N prediction steps is defined as:
4, the nonlinear multi-step rolling optimization control problem of the magnetic suspension workbench is as follows:
according to a model prediction control principle, a prediction model (6), a mixed inequality constraint (8) and an objective function (9) are considered, and the nonlinear multi-step rolling optimization control problem of the magnetic suspension workbench to be solved is as follows:
in the above-mentioned formula (10),
and
respectively is a state sequence and a control sequence under N prediction step lengths;
and 5: converting the nonlinear multistep rolling optimization control problem of the magnetic suspension workbench in the step 4 into a nonlinear Eulerian Langerian equation set, solving an optimization control strategy by using a parallelization Newton method capable of being rapidly executed on line, and realizing closed-loop control of the system;
the nonlinear Eulerian Langerian equation set in the step 5 is as follows:
in order to solve the nonlinear multistep rolling optimization control problem (10) of the magnetic suspension workbench, a mixed inequality constraint (8) is firstly converted into a logarithmic barrier function:
followed by the definition of a Lagrange multiplier sequence
And Hamiltonian function of the problem to be optimized:
optimal sequence of states according to the KKT condition
Control sequence
And lagrange multiplier sequence
The following non-linear euler langery equation set should be satisfied:
in the above formula (13), the initial optimum state
And tail optimal Lagrange multiplier
Is a boundary condition of the system of equations.
And
are respectively H about
And
of the gradient of (c).
Considering integrated optimization variables
Correction matrix
In combination with
Refers to the left side of the equation of the nonlinear eulerian langery system of equations;
the fast online executable parallelization newton method described in step 5 is shown in table 1:
TABLE 1 parallelized Newton method for fast online execution
In table 1, M ═ 4 indicates the maximum number of iterations allowed, and κ ═ 10-3，ε＝10-5"j" refers to the jth iteration in the optimization processGeneration;
referring to table 1, the parallel newton method capable of being performed on-line quickly in the present invention mainly includes the following steps:
firstly, preprocessing variables to be optimized according to boundary conditions;
second step of introduction
Approximate estimate of (2)
Roughly updating the integration optimization variables by a gradient descent method in combination with the correction matrix theta;
third step, due to introduction of
Based on approximate deviation
Optimizing variables for integration in a backward manner
And (6) carrying out correction.
The fourth step, due to the introduction of
Based on approximate deviation
Optimizing variables for integration in a forward manner
Carrying out correction;
fifthly, calculating KKT residual error epsilon under current iterationjIf the residual error is less than the preset termination residual error epsilon, exiting the while loop and outputting zj+1And thetajOtherwise, returning to the starting point of the loop.
Using the table at each sampling instant1, solving the nonlinear multi-step rolling optimization control problem of the magnetic suspension workbench by the parallelization Newton method capable of being rapidly executed on line, and outputting an integrated optimization variable zj+1In (1)
Applied to a coil to drive the worktable to move, and updating the state variable x (t) through the six-freedom-degree motion information of the magnetic suspension worktable measured by the laser sensor at the next sampling momentk+1) Solving for an estimate of the disturbance
And let z1＝zj+1，Θ0＝ΘjAnd repeatedly calculating the optimized control rate to form a closed-loop system.
The control effect of the present invention will be described by performing a trajectory tracking experiment using the magnetic levitation table shown in fig. 4.
1. Contour tracing and trajectory tracking control experiment: in order to verify the effectiveness of the nonlinear model prediction trajectory tracker of the designed magnetic suspension workbench, xrefThe reference track is designed as a three-dimensional hexagonal profile:
fig. 5, fig. 6 and fig. 7 show the profile tracing comparison, the trajectory tracking comparison and the profile error comparison of the method proposed by the present invention and the PID control method, respectively. Looking at the local magnification curve plotted in fig. 6, this curve shows that the actual position of the proposed method of the invention is closer to the reference trajectory than to the PID. In contrast to a PID controller, overshoot in the regulation results in a larger tracking error when there is a larger change in the reference profile. In contrast, the method provided by the invention utilizes the prediction capability to advance the turning of the response curve, thereby realizing excellent track following capability. The profile error depicted in fig. 7 is defined as the shortest distance from the actual position to the reference profile, and the root mean square of the profile error of the proposed method and PID controller is 10.0um and 27.5um, respectively. It can be seen that the proposed non-linear model predictive trajectory tracker has better control performance.
2. Frequency characteristic test experiment: the magnetic suspension working table is tested under the method and the PID controller, the frequency range of 1-150Hz,sx、sy、sfrequency characteristic in the z-axis direction. As can be seen from the amplitude and phase information shown in fig. 8, the proposed method has better low frequency tracking characteristics (1-20Hz) than the PID controller. At the same time, the phase lag situation is mitigated by taking into account future reference trajectories in the objective function. In addition, the invention provides that a higher bandwidth can be provided for the system, but because the determining factors of the frequency characteristic are the mechanical structure and the maximum output of the current power amplifier, the improvement is limited. The comparison result further shows the effectiveness of the nonlinear model prediction trajectory tracker provided by the invention on improving the control and performance of the magnetic suspension workbench and the dynamic response.
Through comparison experiments, the magnetic suspension workbench trajectory tracking controller based on nonlinear model predictive control and the online optimization method thereof are proved to be suitable for motion control of the magnetic suspension workbench, and have real-time performance, universality and better tracking accuracy.
It should be understood that the above description of the preferred embodiments is given for clarity and not for any purpose of limitation, and that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (1)
1. A nonlinear model predictive control optimization method based on a magnetic suspension control system is characterized by comprising the following steps:
the magnetic suspension control system comprises: the device comprises a magnetic suspension workbench, a laser sensor, an analog-to-digital converter, a microprocessor, a digital-to-analog converter and a current power amplifier; the magnetic suspension workbench consists of a stator and a rotor; the stator is composed of a plurality of track coils; the rotor consists of a plurality of groups of Halbach magnetic arrays and a back plate; the stator and the rotor are wirelessly connected;
the laser sensor, the analog-to-digital converter, the microprocessor and the digital-to-analog converter are sequentially connected in series;
the laser sensor is arranged at the periphery of a rotor of the magnetic suspension workbench and used for measuring six-degree-of-freedom motion information of the magnetic suspension workbench and transmitting the six-degree-of-freedom motion information to the analog-to-digital converter;
the analog-to-digital converter performs analog-to-digital conversion on the six-degree-of-freedom motion information of the magnetic suspension workbench to obtain a six-degree-of-freedom motion digital signal of the magnetic suspension workbench, and transmits the six-degree-of-freedom motion digital signal to the microprocessor;
the microprocessor obtains a current control rate through the nonlinear model predictive control optimization method according to the six-degree-of-freedom motion digital signal of the magnetic suspension workbench, and transmits the current control rate to the digital-to-analog converter;
the digital-to-analog converter performs digital-to-analog conversion on the current control rate to obtain an analog current control rate, and transmits the analog current control rate to the current power amplifier for power amplification to obtain a driving current;
the current power amplifier transmits driving current to a stator of the magnetic suspension workbench, and when a plurality of track coils in the stator of the magnetic suspension workbench are energized with current, magnetic fields around the Halbach magnetic array on the rotor generate magnetic force under the excitation of the current so as to drive the magnetic suspension workbench to move in six degrees of freedom, so that closed-loop control is formed;
step 1: constructing a six-degree-of-freedom motion coordinate system by a microprocessor;
step 2: the microprocessor establishes a six-degree-of-freedom magnetic model in a suspension state;
and step 3: equating uncertainty which is not considered in modeling and external disturbance to disturbance lumped on an independent coil, establishing a nonlinear motion model of the magnetic suspension workbench by combining the magnetic force model in the step 2, and designing the lumped disturbance of the nonlinear disturbance observer estimation system according to the model;
and 4, step 4: according to a model prediction control principle, a discrete nonlinear motion model combined with a disturbance estimation value is used as a prediction model, the motion range of a magnetic suspension workbench and the maximum output of a current power amplifier are used as constraint conditions, the total cost of tracking errors and current loads in a prediction step is used as a target function, and the nonlinear multistep rolling optimization control problem of the magnetic suspension workbench is established;
and 5: converting the nonlinear multistep rolling optimization control problem model of the magnetic suspension workbench in the step 4 into a nonlinear Euler Langerian equation set, solving an optimization control strategy by using a parallelization Newton method capable of being rapidly executed on line, and realizing closed-loop control of the system;
the six-degree-of-freedom motion coordinate system in the step 1 is as follows:
the center of the upper surface of the stator is taken as the origin of a fixed coordinate system(s)so, taking the center of the lower surface of the mover as the origin of the moving coordinate system { t }to, relative displacement amount P of the tablex、Py、PzAt the originto is characterized by a vector difference under a fixed coordinate system, and the relative rotation amounts alpha, beta and gamma are characterized by the rotation amounts of the movable coordinate axis relative to the fixed coordinate axis;
the magnetic force model with six degrees of freedom in the suspension state in the step 2 is as follows:
combining the mechanical structure and the coordinate system definition of the magnetic suspension workbench in the step 1, according to a magnetic field calculation method based on harmonic analysis and a Lorenz integral rule, the force and the moment acting on the magnetic suspension workbench are as follows:
wherein the content of the first and second substances,
wherein, BrIs the remanent magnetization of the permanent magnet, NcoilIs the number of turns of the track coil, wcAnd hcTrack width and height, respectively, of the track coilmAnd hmLength and height of the individual permanent magnets, K being the thrust coefficient determined by the system structure, λ 2 pi/lmIs the spatial harmonic number, L is the distance from the center of the single Halbach magnetic array to the center of the whole magnetic array, and u ═ i1,i2,i3,i4,i5,i6,i7,i8]TIs the control input of the system under the drive of 8 independent coils;
step 3, the magnetic suspension workbench nonlinear motion model considering the lumped current disturbance:
the influence caused by unmodeled dynamics of the system, inaccurate magnetization of the magnet, non-uniform winding of the coil and electromagnetic disturbance of the current amplifier is assumed to be in a lumped form equivalent to that of 8 coils
Giving out;
because of the complete suspension characteristic without mechanical connection, the magnetic force model is further expanded into a form of a nonlinear motion model:
wherein the content of the first and second substances,
the position and the speed of the six degrees of freedom of the system form state variables, namely the six degrees of freedom motion information of the magnetic suspension workbench is obtained by the measurement of the laser sensor;
M＝diag(m,m,m,Iα,Iβ,Iγ) Is composed of rotor mass and 3 rotary inertiaThe diagonal matrix is formed by the diagonal matrix,
is the gravitational acceleration vector;
the nonlinear disturbance observer in the step 3 is as follows:
aiming at the nonlinear motion model, a disturbance observer in the following form is designed:
wherein the content of the first and second substances,
is an estimate of the disturbance to the system,
is the internal state vector of the disturbance observer, h-r＝hT·(h·hT)-1And
are respectively h and
right inverse of (d), gain of observer to ensure fast convergence of disturbance estimation deviation
A positive definite and symmetrical matrix should be selected;
step 4, taking the discrete nonlinear motion model combined with the disturbance estimation value as a prediction model:
combining the nonlinear motion model and the estimated value of the disturbance described in step 3
To estimate the state of the system at a future time:
at system sampling time TsAnd a sampling time tk＝kTsDiscretizing the state of the estimated system at the future time under the condition that k is {0,1, 2. }, and obtaining a prediction model of the system under N prediction steps as follows:
wherein the content of the first and second substances,
is an inverse time function;
and 4, taking the motion range of the magnetic suspension workbench and the maximum output of the current power amplifier as constraint conditions:
considering that the motion range of the actuator and the maximum output of the current power amplifier are constrained as follows:
converting the constraints into mixed inequality constraints:
and 4, taking the total cost of the tracking error and the current load in the prediction step as an objective function:
with xrefConsidering the positively determined tracking error weight for the reference trajectory
And current load weight
Defining the objective function under N prediction steps as:
4, the nonlinear multi-step rolling optimization control problem of the magnetic suspension workbench is as follows:
according to a model prediction control principle, a prediction model, mixed inequality constraints and an objective function of a system under N prediction step lengths are considered, and a magnetic suspension workbench nonlinear multi-step rolling optimization control problem model to be solved is as follows:
wherein the content of the first and second substances,
and
respectively is a state sequence and a control sequence under N prediction step lengths;
the nonlinear Eulerian Langerian equation set in the step 5 is as follows:
converting the mixed inequality constraint of step 4 into a logarithmic barrier function:
followed by the definition of a Lagrange multiplier sequence
And Hamiltonian function of the problem to be optimized:
optimal sequence of states according to the KKT condition
Control sequence
And lagrange multiplier sequence
The following non-linear euler langery equation set should be satisfied:
wherein the initial optimum state
And tail optimal Lagrange multiplier
Is a boundary condition of the equation set;
and
are respectively H about
And
a gradient of (a);
considering integrated optimization variables
Correction matrix
In combination with
Referring to the left side of the equation of the nonlinear Eulerian Langerian equation set, the parallelization Newton method which can be rapidly executed on line in the step 5;
the parallel Newton method capable of being rapidly executed on line in the step 5 mainly comprises the following steps:
step 5.1, preprocessing variables to be optimized according to boundary conditions;
step 5.2, introduction
Approximate estimate of (2)
Roughly updating the integration optimization variables by a gradient descent method in combination with the correction matrix theta;
step 5.3, due to the introduction of
Based on approximate deviation
Optimizing variables for integration in a backward manner
Carrying out correction;
step 5.4, due to the introduction of
Based on approximate deviation
Optimizing variables for integration in a forward manner
Carrying out correction;
step 5.5, calculating KKT residual epsilon under current iterationjIf the residual error is less than the preset termination residual error epsilon, exiting the while loop and outputting zj+1And thetajOtherwise, returning to the starting point of the cycle;
solving the nonlinear multistep rolling optimization control problem of the magnetic suspension workbench by using a parallelization Newton method capable of being rapidly executed on line at each sampling moment, and outputting an integrated optimization variable zj+1In (1)
Is applied to the coil to drive the stage to move, and is switched on at the next sampling timeUpdating state variable x (t) of six-freedom-degree motion information of the magnetic suspension workbench measured by the laser sensork+1) Solving for an estimate of the disturbance
And let z1＝zj+1，Θ0＝ΘjAnd repeatedly calculating the optimized control rate to form a closed-loop system.
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