Robust iterative learning model predictive controller for magnetically levitated planar motor and its application

Abstract

Disclosed are a robust iterative learning model predictive controller for a magnetic levitation planar motor and its application. For the repeated motion of the X-axis of the magnetic levitation planar motor, a system dynamics model and a reference trajectory model with bounded disturbances that satisfy the Lipschitz continuity condition are established, and a tracking error model is obtained based on the system dynamics model and the reference trajectory model; according to the model predictive control principle, a tracking error model combined with the disturbance of the last iteration cycle is used as a learning prediction model, the upper bound of the second norm of the tracking error based on the Lipschitz continuity characteristic of the disturbance is used as a cost function, and a robust positive invariant set based on the state feedback control rate and bounded disturbance is used as a constraint condition to construct a magnetic levitation planar motor learning optimization control problem. The present invention solves the problem that model predictive control is easily affected by disturbances in actual magnetic levitation systems, and the control method provided by the present invention is used to improve the tracking performance of the magnetic levitation planar motor in the presence of external disturbances.

Description

Technical Field

[0001] The present invention relates to the technical field of trajectory tracking control of constrained uncertain systems, and in particular to a robust iterative learning model predictive controller for a magnetically suspended planar motor and its application. Background Art

[0002] In the field of precision manufacturing, positioning mechanisms are crucial to machining accuracy. Nowadays, due to the advantages of no mechanical contact, no friction, and multiple freedoms, magnetic levitation planar motors have become positioning devices that can provide high-performance motion in the field of precision manufacturing. In relevant research centers, tracking control is one of the main control issues studied in magnetic levitation motion systems, and tracking error is also the key to judging the motion performance of magnetic levitation systems.

[0003] Advanced control methods such as adaptive robust control, neural network control, and fuzzy control are of great significance to the improvement of the motion performance of magnetic levitation systems. However, these advanced control methods cannot take into account system constraints. System constraints, whether they are state constraints determined by the safe operating area or the maximum control output limited by the power amplifier, are crucial to the safe and robust operation of the magnetic levitation system. Since constraints can be explicitly handled in a systematic manner, model predictive control has become the preferred solution for the motion control of magnetic levitation planar motor systems. For magnetic levitation planar motors, system uncertainties, including external disturbances, motion coupling between multiple degrees of freedom, and modeling errors, are inevitable in practice. Since the control performance of model predictive control is directly related to the accurate system model, these uncertainties and disturbances will affect the tracking accuracy of magnetic levitation planar motors based on model predictive control.

[0004] Considering that magnetically levitated planar motors always perform repetitive trajectory tracking tasks in practical applications, repetitive motion control methods can be introduced to improve the tracking performance of magnetically levitated planar motors. Iterative learning control stands out from many repetitive control methods because it can accurately extract hidden repetitive information from historical information. However, due to the lack of ability to handle constraints, the application of iterative learning control in magnetically levitated planar motor systems is also limited.

[0005] Traditional model predictive control cannot maintain high-precision tracking when the magnetic levitation planar motor is subjected to external disturbances, and iterative learning control cannot handle the system constraints existing in the magnetic levitation planar motor system. Therefore, a robust iterative learning model predictive control method for magnetic levitation planar motor is needed. Summary of the invention

[0006] The purpose of the present invention is to provide a robust iterative learning model predictive controller for a magnetic levitation planar motor and its application, so as to solve the problem that the traditional model predictive controller is difficult to achieve accurate tracking of the reference trajectory of the magnetic levitation planar motor and the problem that iterative learning control cannot handle the system constraints of the magnetic levitation planar motor.

[0007] According to a first aspect of an embodiment of the present invention, a method for constructing a robust iterative learning model predictive controller for a magnetic levitation planar motor is provided, comprising: for the repeated motion of the X-axis of the magnetic levitation planar motor, constructing a system dynamics model and a bounded disturbance that satisfies the Lipschitz continuity condition, and at the same time establishing a reference trajectory model, and obtaining a tracking error model through the system dynamics model and the reference trajectory model; using the disturbance of the previous iteration cycle and combining it with the tracking error model to construct a learning prediction model; defining a state feedback control rate, combining it with the bounded disturbance, solving the robust positive invariant set of the tracking error model through a geometric iteration method, and constructing constraints based on the robust positive invariant set; based on the Lipschitz continuity condition of the disturbance, establishing a cost function for the upper bound of the second norm of the tracking error; according to the model predictive control principle, combining the learning prediction model, the constraints and the cost function to construct a learning optimization control problem for the magnetic levitation planar motor, and obtaining a robust iterative learning model predictive controller.

[0008] According to a second aspect of an embodiment of the present invention, a magnetic levitation planar motor control method is provided to solve the magnetic levitation planar motor learning optimization control problem obtained by the magnetic levitation planar motor robust iterative learning model predictive controller construction method, and obtain the optimal control signal Will serve as the control input of the magnetically levitated planar motor

[0009] According to a third aspect of an embodiment of the present invention, a magnetic levitation planar motor is provided, comprising: a processor; and a memory for storing instructions executable by the processor; wherein the processor is configured to execute all or part of the steps of the magnetic levitation planar motor control method.

[0010] According to a fourth aspect of an embodiment of the present invention, there is provided a non-temporary computer-readable storage medium having a computer program stored thereon, wherein the computer program implements all or part of the steps of the magnetic levitation planar motor control method when executed by a processor.

[0011] Compared with the existing technology, the beneficial effects of the present invention are as follows: the control strategy of the present invention combines iterative learning control with model predictive control, while retaining the excellent constraint processing capability of model predictive control and the ability of iterative learning to identify repeated information, and improving the trajectory tracking accuracy of the magnetic levitation planar motor without relying on an accurate system model. According to the principles of model predictive control and iterative learning control, the disturbance information of the previous iteration cycle is introduced into the prediction model to compensate for the influence of repetitive disturbances in the system, so that the tracking accuracy in the iteration process is gradually improved. The robust positive invariant set introduced in the optimization control problem can ensure that the tracking error is always kept within the range defined by the invariant set while enhancing the anti-interference ability of the system. In addition, in order to further improve the tracking accuracy, based on the Lipschitz continuous characteristics of the disturbance, a cost function for the upper bound of the second norm of the tracking error is derived. The present invention solves the problem that the magnetic levitation planar motor is susceptible to uncertainty and disturbance when performing trajectory tracking tasks, improves the trajectory tracking accuracy of the magnetic levitation planar motor, and the proposed method has the advantages of strong robustness and good control performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] In order to more clearly illustrate the technical solution of the embodiment of the present invention, the drawings of the embodiment are briefly introduced below.

[0013] Figure 1 A flow chart of a method for constructing a robust iterative learning model predictive controller for a magnetically levitated planar motor provided in one embodiment of the present invention.

[0014] Figure 2 A schematic diagram of a magnetically levitated planar motor provided in one embodiment of the present invention.

[0015] Figure 3 A schematic diagram of a robust positive invariant set provided by an embodiment of the present invention.

[0016] Figure 4 A reference trajectory signal diagram provided by an embodiment of the present invention.

[0017] Figure 5 A framework diagram of a robust iterative learning model predictive control algorithm provided in one embodiment of the present invention.

[0018] Figure 6 A tracking error process diagram of a magnetically suspended planar motor during 15 times of trajectory tracking controlled by the method of the present invention provided in one embodiment of the present invention. DETAILED DESCRIPTION

[0019] Figure 1 The flowchart of the method for constructing a robust iterative learning model predictive controller for a magnetically suspended planar motor is shown. The following is a detailed description of each step in the method.

[0020] Step 1: According to Figure 2 , for the repeated motion of the X-axis of the magnetic levitation planar motor, a system dynamics model and a bounded disturbance that satisfies the Lipschitz continuity condition are constructed, and a reference trajectory model is established at the same time. The tracking error model is obtained through the system dynamics model and the reference trajectory model;

[0021] The system dynamics model of the X-axis repeated motion of the magnetic levitation planar motor is:

[0022]

[0023] in, is the number of iterations, is the sampling time, are the state, control input, disturbance and output of the system respectively. The parameter matrix of the system for:

[0024]

[0025] In some examples, t s =1ms is the sampling time of the system, and m=2.37kg is the mass of the motor.

[0026] The perturbation that is bounded and satisfies the Lipschitz continuity condition is:

[0027] The disturbance w in the system dynamics model k (t)Satisfy the constraints:

[0028]

[0029] in, is the feasible space of perturbations. In one example

[0030] The disturbance w of the system k (t) satisfies the Lipschitz continuity condition:

[0031] ||w k (t)-w k-1 (t)||2≤L x ·||x k (t)-x k-1 (t)||2+L u ·||u k (t)-u k-1 (t)||2 (4)

[0032] in, and L u ∈(0,1) is the Lipschitz coefficient. In one example, L x =1×10 4 , L u=0.1;||w k (t)-w k-1 (t)||2,||x k (t)-x k-1 (t)||2,||u k (t)-u k-1 (t)||2 is the binary norm of the system disturbance, state, and control input deviation between two adjacent iterations k and k-1, respectively.

[0033] The reference trajectory model is:

[0034]

[0035] in, are the state, control input, and output of the reference trajectory, respectively, and the reference trajectory model has the same parameter matrix as the system dynamics model

[0036] The tracking error model obtained by the system dynamics model and the reference trajectory model is:

[0037]

[0038] in, are the tracking error state and the tracking error respectively. The parameter matrix of the tracking error model is the same as that of the system dynamics model.

[0039] Step 2: Use the disturbance of the previous iteration cycle and combine it with the tracking error model to build a learning prediction model.

[0040] Step 2 is as follows: Combine the tracking error model of step 1 and the disturbance w of the previous iteration period k-1 k-1 (t), to estimate the tracking error state and tracking error of the system at the next moment of the current iteration cycle k, that is, to learn the prediction model:

[0041]

[0042] in, are the predicted tracking error state, control input, and tracking error, respectively.

[0043] Step 3: Define the state feedback control rate, combine the bounded disturbance, solve the robust positive invariant set of the tracking error model through the geometric iteration method, and construct constraints based on the robust positive invariant set.

[0044] The state feedback control rate is:

[0045]

[0046] in, is a feedback matrix such that the eigenvalues ​​of the matrix A+B·K are all located within the unit circle. In one example, K=[179.12 938.69].

[0047] Solving the robust positive invariant set of the tracking error model by geometric iteration mainly includes the following steps:

[0048] The first step is to substitute the state feedback control rate into the tracking error model to obtain the autonomous tracking error model:

[0049] f k (t+1)=(A+B·K)·f k (t)+B·d k (t) (9)

[0050] Among them, d k (t) = w k (t)-w k-1 (t) is the disturbance of the autonomous tracking error model.

[0051] The second step is to define the initial iteration set Where i = 0, and construct the autonomous tracking error model disturbance Feasible space in Addition to the Minkowski set, the set Defined as In one example,

[0052] The third step is to update the iteration set based on the autonomous tracking error model:

[0053]

[0054] Among them, the collection satisfy gather for

[0055] The fourth step is to compare the inclusion relationship between iteration sets. If the iteration sets between two adjacent iterations i+1 and i satisfy Exit iteration and output robust positive invariant set Otherwise, set i=i+1 and return to the third step to continue the loop update.

[0056] Robust positive invariant sets such as Figure 3 As shown, specifically:

[0057]

[0058] Construct constraints based on the robust positive invariant set: constrain the predicted tracking error state described in step 2 to the robust positive invariant set:

[0059]

[0060] Step 4: Based on the perturbed Lipschitz continuity condition, establish a cost function for the upper bound of the second norm of the tracking error.

[0061] According to the Lipschitz continuity condition of the disturbance described in step 1, the binary norm of the deviation between the predicted tracking error state and the actual tracking error state satisfies:

[0062]

[0063] Since the tracking error state and the tracking error satisfy ||e k (t+1)||2=||C·f k (t+1)||2≤||f k (t+1)||2, based on the triangle inequality The tracking error second norm ||e is obtained by the above second norm deviation. k (t+1)||2The cost function of the upper bound

[0064]

[0065] Step 5: According to the model predictive control principle, combined with the learning prediction model described in step 2, the constraints described in step 3, and the cost function described in step 4, the learning optimization control problem of the magnetic levitation planar motor is constructed, the control algorithm is applied to the actual magnetic levitation planar motor, and repeated trajectory tracking experiments are carried out in the presence of external disturbances to verify the tracking performance of the control algorithm.

[0066] The learning optimization control problem of the magnetic levitation planar motor to be solved is:

[0067]

[0068] Solve the optimal control signal obtained by the above magnetic levitation planar motor learning optimization control algorithm Will serve as the control input of the system

[0069] The specific method of applying the control algorithm to the actual magnetic levitation planar motor system is: convert the control algorithm into code form through LabVIEW-Realtime software and download it to the NI controller. The NI controller provides registers to store reference trajectory signals and the status, control input and disturbance of the historical iteration cycle system, and then realizes the control of the magnetic levitation planar motor through real-time communication between the NI controller and the magnetic levitation planar motor system.

[0070] The repeated trajectory tracking experiment described in step 5 is carried out in the presence of external disturbances. The specific method is as follows: first, the trajectory signal of each iteration cycle is generated by the reference trajectory model, including the state x of the reference trajectory r (t), control input u r (t) and the output y r (t), and store the trajectory signal in the register of the NI controller. The generated trajectory signal is as follows: Figure 4 As shown. Secondly, based on the system parameter matrix (2), the feasible space of disturbance (4) and the Lipschitz continuity condition of disturbance (3), the learning prediction model (7), the constraint conditions on the robust positive invariant set (12) and the cost function (14) are designed respectively, and then integrated into (15) to form a robust iterative learning model predictive controller. The structure of the robust iterative learning model predictive controller is as follows: Figure 5 Then, at time t of the kth iteration, the historical data x is taken from the register of the NI controller. k-1 (t),u k-1 (t), w k-1 (t) and the corresponding trajectory control input u r (t) and trajectory state x r (t), and the actual state x of the current magnetic levitation planar motor k (t) is compared to obtain the tracking error state f k (t) = x k (t)-x r (t). Finally, the robust iterative learning model predictive controller uses historical data x k-1 (t),u k-1 (t), w k-1 (t) and the current trajectory control input u r (t), system status x k (t) and the tracking error state f k (t) Solve the optimal control signal As the control input of the system To drive the moving position of the magnetic levitation planar motor in the X-axis direction, the optimal control rate is repeatedly calculated at the next sampling moment to achieve closed-loop trajectory tracking.

[0071] In an exemplary embodiment, a magnetic levitation planar motor control method is also provided to solve the magnetic levitation planar motor learning optimization control problem obtained by the magnetic levitation planar motor robust iterative learning model predictive controller construction method, and obtain the optimal control signal Will serve as the control input of the magnetically levitated planar motor

[0072] In an exemplary embodiment, a magnetically levitated planar motor is also provided, comprising: a processor; and a memory for storing instructions executable by the processor; wherein the processor is configured to execute all or part of the steps of the method.

[0073] In an exemplary embodiment, a non-transitory computer-readable storage medium is also provided, on which a computer program is stored, and when the computer program is executed by a processor, all or part of the steps of the above method are implemented. For example, the non-transitory computer-readable storage medium can be a ROM, RAM, CD-ROM, magnetic tape, floppy disk, optical data storage device, etc.

[0074] The actual tracking effect of the magnetic suspension planar motor under the control method of the present invention is as follows: Figure 6 As shown in the figure, the curve of the tracking error change in multiple iteration cycles shows that due to the inaccurate disturbance identification at the initial moment, the actual position of the magnetic levitation planar motor has a large deviation from the reference trajectory, and the deviation is Figure 3 As the iterations proceed, the tracking error becomes smaller and smaller, and the actual position of the magnetic levitation planar motor becomes closer and closer to the reference trajectory. In the 15th iteration cycle, a relatively ideal tracking effect is achieved, and the root mean square and maximum absolute value of the tracking error are 0.0047 mm and 0.0162 mm respectively.

[0075] In the presence of external interference in the magnetic levitation planar motor system, the tracking error is always limited to the range defined by the robust positive invariant set, and as the iteration proceeds, the tracking error gradually decreases and achieves an ideal tracking effect, which proves the effectiveness of the control method of the present invention.

Claims

1. A method for constructing a robust iterative learning model predictive controller for a magnetically levitated planar motor, characterized in that: include: For the repeated motion of the X-axis of the magnetic levitation planar motor, a system dynamics model and a bounded disturbance that satisfies the Lipschitz continuity condition are constructed, and a reference trajectory model is established at the same time, and a tracking error model is obtained through the system dynamics model and the reference trajectory model; Using the disturbance of the last iteration cycle and combining it with the tracking error model, a learning prediction model is constructed; Define a state feedback control rate, combine bounded disturbances, solve the robust positive invariant set of the tracking error model through a geometric iteration method, and construct constraint conditions based on the robust positive invariant set; Based on the perturbed Lipschitz continuity condition, a cost function for the upper bound of the second norm of the tracking error is established. According to the model predictive control principle, the learning prediction model, the constraint conditions and the cost function are combined to construct the learning optimization control problem of the magnetic levitation planar motor, and a robust iterative learning model predictive controller is obtained; The system dynamics model is: x k (t+1)=A·x k (t)+B·u k (t)+B·w k (t) y k (t+1)=C·x k (t+1) in, is the number of iterations, is the sampling time, are the state, control input, disturbance and output of the system respectively, and the parameter matrix of the system for: C＝[1 0] Among them, t s is the sampling time of the system, m is the mass of the motor; Perturbation w k (t)Satisfy the constraints: in, is the feasible space of perturbation; Perturbation w k (t) satisfies the Lipschitz continuity condition: ||w k (t)-w k-1 (t)||2≤L x ·‖x k (t)-x k-1 (t)||2+L u ·‖u k (t)-u k-1 (t)||2 in, and L u ∈(0,1) is the Lipschitz coefficient, ||w k (t)-w k-1 (t)||2,||x k (t)-x k-1 (t)||2,||u k (t)-u k-1 (t)||2 is the second norm of the deviation of the system disturbance, state, and control input between two adjacent iterations k and k-1 respectively; The reference trajectory model is: x r (t+1)=A·x r (t)+B·u r (t) y r (t+1)=C·x r (t+1) in, are the state, control input and output of the reference trajectory, respectively, and the reference trajectory model has the same parameter matrix as the system dynamics model The tracking error model is: f k (t+1)=A·f k (t)+B·u k (t)+B·w k (t)-B·u r (t) e k (t+1)=C·f k (t+1) in, are the tracking error state and the tracking error respectively. The parameter matrix of the tracking error model is the same as that of the system dynamics model.

2. The method according to claim 1, characterized in that Using the tracking error model and the disturbance w of the last iteration period k-1 k-1 (t) is used to estimate the tracking error state and tracking error of the system at the next moment of the current iteration cycle k, that is, the learning prediction model: in, are the predicted tracking error state, control input, and tracking error, respectively.

3. The method according to claim 2, characterized in that The state feedback control rate is: in, is a feedback matrix such that the eigenvalues ​​of the matrix A+B·K are all inside the unit circle.

4. The method according to claim 3, characterized in that The specific method of solving the robust positive invariant set of the tracking error model by the geometric iteration method is: Substituting the state feedback control rate into the tracking error model, the autonomous tracking error model is obtained: f k (t+1)=(A+B·K)·f k (t)+B·d k (t) Among them, d k (t) = w k (t)-w k-1 (t) is the disturbance of the autonomous tracking error model; Define the initial iteration set Where i = 0, and construct the autonomous tracking error model disturbance Feasible space Where ⊕ is Minkowski set addition, the set Defined as Update the iteration set based on the autonomous tracking error model: Among them, the collection satisfy gather for Compare the inclusion relationship between iteration sets. If the iteration sets between two adjacent iterations i+1 and i satisfy Exit the iteration and output the robust positive invariant set Otherwise, set i=i+1 and continue to execute the loop update.

5. The method according to claim 4, characterized in that The constraint condition constructed based on the robust positive invariant set is: constraining the predicted tracking error state to be within the robust positive invariant set 6. The method according to claim 5, characterized in that The specific method of establishing the cost function of the upper bound of the second norm of the tracking error based on the perturbation Lipschitz continuity condition is: According to the Lipschitz continuity condition of the disturbance, the binary norm of the deviation between the predicted tracking error state and the actual tracking error state satisfies: Since the tracking error state and the tracking error satisfy ||e k (t+1)||2=||C·f k (t+1)||2≤||f k (t+1)||2, based on the triangle inequality The tracking error second norm ||e is obtained by the above second norm deviation. k (t+1)||2The cost function of the upper bound 7. The method according to claim 6, characterized in that The learning optimization control problem of the magnetic levitation planar motor constructed by combining the learning prediction model, the constraint conditions and the cost function is:

8. A method for controlling a magnetically suspended planar motor, characterized in that: The robust iterative learning model predictive controller obtained by the method for constructing the robust iterative learning model predictive controller of the magnetic levitation planar motor according to claim 7 performs repeated trajectory tracking in the presence of external disturbances, and the specific process is as follows: The reference trajectory model is used to generate a trajectory signal for each iteration cycle, including the state x of the reference trajectory. r (t), control input u r (t) and the output y r (t); At time t of the kth iteration, take out the historical data x k-1 (t),u k-1 (t), w k-1 (t) and the corresponding trajectory control input u r (t) and the reference trajectory state x r (t), and the actual state x of the current magnetic levitation planar motor k (t) is compared to obtain the tracking error state f k (t) = x k (t)-x r (t); Robust iterative learning model predictive controller using historical data x k-1 (t),u k-1 (t), w k-1 (t) and the current trajectory control input u r (t), actual state x k (t) and the tracking error state f k (t) Solve the optimal control signal As the control input of the system To drive the moving position of the magnetic levitation planar motor in the X-axis direction, the optimal control rate is repeatedly calculated at the next sampling moment to achieve closed-loop trajectory tracking.

9. A magnetically suspended planar motor, characterized in that: include: processor; a memory for storing instructions executable by the processor; Wherein, the processor is configured to execute the steps of the method of claim 8.

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