Characteristic polynomial-based precision positioning platform zero-error model prediction control method

Abstract

The invention provides a precise positioning platform zero-error model prediction control method based on a characteristic polynomial, and belongs to the field of precise positioning platform ultra-precise tracking control, and the method comprises the following steps: S1, recognizing an autoregression model with external input according to the input and output data of a precise positioning platform; s2, establishing an augmentation system containing a tracking error according to the characteristic polynomial of the reference trajectory; s3, designing an optimization problem with constraints based on the augmented system containing the tracking error; and S4, solving an optimization problem according to a coordinate descent method to obtain a model prediction controller, and realizing zero-error tracking of the multiple-input multiple-output system. In order to solve the problem that a precision positioning platform is low in tracking precision when tracking an unstable reference trajectory or control input is limited, reference trajectory signals with characteristic polynomials can be tracked in a zero-error mode when constraint conditions are met, and the solving efficiency of a constraint optimization problem is guaranteed through application of a coordinate descent method. And the provided model prediction controller can be applied to high-frequency sampling hardware.

Description

Technical Field

[0001] The present invention belongs to the field of ultra-precision tracking control of precision positioning platforms, and particularly relates to a zero-error model predictive control method for precision positioning platforms based on characteristic polynomials, which is mainly used for realizing multi-degree-of-freedom zero-error tracking of a reference trajectory only according to historical output data under the constraint of limited hardware conditions of the precision positioning platform. Background Art

[0002] Due to its excellent positioning accuracy, precision positioning platforms are widely used in high-precision machining fields such as chip manufacturing and laser processing. However, in theoretical derivations, the control systems of precision positioning platforms often face the problem of being unable to fully achieve zero-error tracking. This is because the feedforward controller in existing control algorithms is difficult to accurately compensate for the dynamics of precision positioning platforms, resulting in a deviation between the actual output and the reference trajectory. In addition, the control input and displacement output of precision positioning platforms are usually physically constrained. For example, the control input is limited by the maximum power and voltage of the actuator, and the displacement output is affected by the structural rigidity, friction, and other mechanical limitations of the platform itself. Although some existing control algorithms can achieve zero-error tracking without any constraints, in practical applications, the existence of constraints makes most control algorithms without considering constraints difficult to be implemented in real systems, thus resulting in the inability to achieve the desired tracking performance. Therefore, how to design a control algorithm that can effectively achieve accurate tracking under these physical limitations has become a major challenge in the current research on precision positioning platform control technology.

[0003] Optimal control theory designs a controller by designing a cost function related to the tracking error and the control input and solving the optimization problem. In general optimal control methods, the control input itself is directly used in the cost function, and the obtained controller does not make full use of the reference trajectory, and can only limit the tracking error within a certain range, and cannot perform zero-error tracking. In addition, if the cost function is designed based on an infinite time domain, the handling of actual constraints will not be flexible enough to effectively cope with the performance problems caused by constraint limitations in real systems. Summary of the Invention

[0004] Aiming at the control constraints and tracking accuracy problems existing when the precision positioning platform tracks the reference trajectory, the present invention proposes a zero-error model predictive control method for the precision positioning platform based on the characteristic polynomial. First, an autoregressive model with external input for each degree of freedom is identified according to the input and output data of the precision positioning platform; subsequently, a non-minimum state space of the precision positioning platform is constructed through the uniform processing of the autoregressive models of each degree of freedom, and an augmented system containing the tracking error is further constructed based on the characteristic polynomial of the reference trajectory; thereafter, a new cost function is constructed based on the above augmented system and the characteristic polynomial of the reference trajectory, and a constrained optimization problem is designed through the parameter matrix; finally, the relatively complex constrained optimization problem is transformed into a dual problem, and the dual problem is solved by the coordinate descent strategy to further obtain the model predictive controller, thereby realizing the zero-error tracking control of the multi-input multi-output system.

[0005] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial, comprising the following steps:

[0007] S1. Identify an autoregressive model with external input according to the input and output data of the precision positioning platform;

[0008] S2. Establish an augmented system containing the tracking error according to the characteristic polynomial of the reference trajectory;

[0009] S3. Design a constrained optimization problem based on the augmented system containing the tracking error;

[0010] S4. Solve the optimization problem according to the coordinate descent method to obtain the model predictive controller, and realize the zero-error tracking of the multi-input multi-output system.

[0011] The beneficial effects of the present invention are as follows:

[0012] 1. The present invention first constructs an autoregressive model with external input for each degree of freedom through the historical input and output data of the precision positioning platform. On this basis, a state space of the tracking error is further constructed according to the characteristic polynomial of the reference trajectory signal. This state space only depends on the historical control input, historical output and historical tracking error of the precision positioning platform, ensuring that the state estimation process is not involved in the subsequent controller design process.

[0013] 2. In the process of designing the cost function, on the one hand, ordinary optimal control methods directly introduce control inputs into the cost function, resulting in the controller not being able to achieve precise zero-error tracking at the theoretical level; on the other hand, the infinite-horizon cost function cannot flexibly handle the control constraints brought about by actual situations. The present invention constructs a finite-horizon cost function, in which the control inputs are introduced based on the preprocessing of the reference trajectory polynomial, and through the definition of multiple parameter matrices, the original problem is transformed into a constrained optimization problem only for the control input sequence.

[0014] 3. Since the control frequency of the precision positioning platform is extremely high in practical applications, the solution of the constrained optimization problem may be restricted. The present invention transforms the relatively complex original constrained optimization problem into a dual problem under simple constraints, and gives a solution approach to this dual problem through the coordinate descent method, enabling this precision tracking algorithm to be directly applied to the hardware platform with high-speed sampling. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 is a flowchart of the zero-error model predictive control method for the precision positioning platform based on the characteristic polynomial of the present invention;

[0016] Figure 2 is the output displacement and reference trajectory diagram of the precision positioning platform when inputting a sinusoidal reference trajectory;

[0017] Figure 3 is the tracking error diagram of the precision positioning platform when inputting a sinusoidal reference trajectory;

[0018] Figure 4 is the control input diagram of the precision positioning platform when inputting a sinusoidal reference trajectory;

[0019] Figure 5 is the output displacement and reference trajectory diagram of the precision positioning platform when inputting a composite reference trajectory;

[0020] Figure 6 is the tracking error diagram of the precision positioning platform when inputting a composite reference trajectory;

[0021] Figure 7 is the control input diagram of the precision positioning platform when inputting a composite reference trajectory. DETAILED DESCRIPTION OF THE INVENTION

[0022] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific implementation examples described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0023] AsFigure 1 As shown in the figure, the present invention proposes a zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial, including the following steps:

[0024] Step S1: Identify an autoregressive model with external input according to the input and output data of the precision positioning platform, including:

[0025] The precision positioning platform is a multi-input multi-output system. Affected by processing and assembly accuracy, there is coupling between the displacement outputs of different degrees of freedom. Therefore, the displacement outputs of different degrees of freedom are all affected by multiple control inputs. Based on this, an autoregressive model with external input of the precision positioning platform is established as follows:

[0026] ,

[0027] where the subscript represents the variable related to the th degree of freedom of the precision positioning platform, , is the output displacement of the th degree of freedom of the precision positioning platform at time , is the number of historical outputs contained in the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the model parameter related to the historical output of the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the control input of the precision positioning platform at time , is the number of control inputs of the precision positioning platform, is the number of historical inputs contained in the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the model parameter related to the historical input of the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the set composed of all real numbers, and the superscript represents the transpose of the matrix;

[0028] Swept-frequency signals with rich frequency spectrum information are sequentially input into different input channels of the precision positioning platform, and all output displacements of the th degree of freedom of the precision positioning platform under this input are recorded. Different and are selected, and the system is identified according to the above autoregressive model structure. A series of and that minimize the difference between the identification result and the actual data are selected.

[0029] Step S2: Establish an augmented system containing the tracking error based on the characteristic polynomial of the reference trajectory, including:

[0030] Define the output displacement vector of the precision positioning platform at time as , where denotes stacking all the internal vectors into a column vector in sequence, used to define variables, and establish the system state vector at time

[0031] as:

[0032] , , where

[0033] ,

[0034] Among them, the specific form of the parameter intermediate matrix is:

[0035] ,

[0036] ,

[0037] ;

[0038] The elements in are , where is the number of matrix parameters related to the output vector, When = 0, The elements in are , where is the number of matrix parameters related to the input vector, When = 0, is the dimensional identity matrix, is the dimensional identity matrix; diag() represents a diagonal matrix.

[0039] Express the characteristic polynomial of the reference trajectory as:

[0040] ,

[0041] Among them, is the hysteresis operator, is the intermediate variable, is the degree of the reference trajectory characteristic polynomial, is the coefficient of this polynomial, satisfying , is the reference trajectory vector that each degree of freedom needs to track;

[0042] Based on the above non-minimal state space model, the augmented system is constructed as:

[0043] ,

[0044] Among them, is the state variable of the augmented system, the state variable constructed based on the reference trajectory characteristic polynomial, the control input constructed based on the reference trajectory characteristic polynomial, is the precision positioning platform is the tracking error vector at time , and the total error vector formed by the tracking error vectors at multiple historical times

[0045] , , ,

[0046] , .

[0047] Step S3: Design a constrained optimization problem based on the augmented system containing the tracking error, including:

[0048] To enable the outputs of each degree of freedom of the precision positioning platform to accurately track the corresponding reference trajectory, design the following finite-time cost function :

[0049] ,

[0050] Among them, and are the weight parameters related to the error and control input in the cost function respectively, is the prediction step length in the cost function, and define multiple parameter intermediate matrices:

[0051] Tracking error matrix at multiple future times ,

[0052] Control input matrix at multiple future times ,

[0053] Control input matrices constructed based on the reference trajectory characteristic polynomial at multiple future moments ,

[0054] Control input matrices at multiple known historical moments ,

[0055] Intermediate matrices , intermediate matrix ,

[0056] ,

[0057] ,

[0058] wherein, the intermediate matrix is order square matrix, and the intermediate matrix is rows, column matrix;

[0059] Then, based on the augmented system, the above cost function can be expressed in the following compact form:

[0060] ,

[0061] Furthermore, the following relationship can be constructed:

[0062] ;

[0063] According to the actual situation, the state variable constraints and output constraints of the system are uniformly transformed into control input constraints and represented by matrices as:

[0064] ,

[0065] wherein, is row column constant matrix parameter, is a constant vector parameter containing elements, is for the number of inequality constraints. Then, the trajectory tracking problem of the precision positioning platform can be transformed into solving the following constrained optimization problem:

[0066] ,

[0067] Subject to: ,

[0068] ,

[0069] wherein, is the optimal solution to the above optimization problem.

[0070] Step S4: Solve the optimization problem according to the coordinate descent method to obtain a model predictive controller, and achieve zero-error tracking of the multi-input multi-output system, including:

[0071] Transform the above constrained optimization problem into the following Lagrangian polynomial :

[0072] ,

[0073] where is the Lagrange multiplier, the intermediate matrix , the intermediate matrix , find the partial derivative of the above formula with respect to and set it to 0, we can get that when the cost function is minimized, the control input sequence satisfies ;

[0074] Substitute the above formula into the Lagrangian polynomial, and the original constrained optimization problem can be transformed into the following dual problem:

[0075] ,

[0076] satisfies: ,

[0077] where is the optimal solution to the above dual problem, the intermediate matrix , the intermediate vector , then solve the above dual problem by the coordinate descent method. Specifically, it can be solved by performing the following assignment operations multiple times:

[0078] The -th element in ,

[0079] where is the -th element on the diagonal of the square matrix , is the -th element in , is the -th row of the square matrix . To further simplify the computational complexity, the above assignment operation can be replaced with the following form:

[0080] Intermediate variable ,

[0081] ,

[0082] where Represents an assignment operation, which means assigning the operation result of the expression after the symbol to the variable before the symbol. Assign values to all elements in and stop assigning values after times, and obtain the Lagrange multiplier value at the current moment , is used to determine and the proximity parameter at the current moment . The larger is, the closer is to

[0083] ,

[0084] Select the first elements in

[0085] as the control input at the current moment, and zero-error tracking of the reference trajectory can be achieved under the constraint conditions.

[0086] Embodiment

[0087] In the present invention, the simulation step size is set to 0.0001 s (i.e., the sampling frequency is 10 kHz), and the discrete system model of the single-degree-of-freedom precision positioning platform is , the prediction step size is 10, the initial control input is 0, and all initial values of the system state are 0. is , is , the constraint matrix , , the number of iterations ;

[0088] (2) Sine signal tracking test:

[0089] First, input a sine wave signal with an amplitude of 1 μm and a frequency of 200 Hz as the reference trajectory signal to be tracked, and the simulation duration is 0.04 s; Figure 2 is the output displacement and reference trajectory diagram of the precision positioning platform when inputting the sine reference trajectory; Figure 3 is the tracking error diagram of the precision positioning platform when inputting the sine reference trajectory. After stabilization, the tracking error almost reaches the upper limit of the calculation accuracy; Figure 4 is the control input diagram of the precision positioning platform when inputting the sine reference trajectory;

[0090] (3) Composite signal tracking test:

[0091] A composite signal formed by superimposing sine signals with an amplitude of 1 μm and a frequency of 200 Hz, an amplitude of 2 μm and a frequency of 50 Hz, and an amplitude of 3 μm and a frequency of 70 Hz is used as the reference trajectory signal, and the simulation duration is 0.1 s; Figure 5 It is the output displacement and reference trajectory diagram of the precision positioning platform when inputting the composite reference trajectory; Figure 6 It is the tracking error diagram of the precision positioning platform when inputting the composite reference trajectory. After stabilization, the tracking error almost reaches the upper limit of the calculation accuracy; Figure 7 It is the control input diagram of the precision positioning platform when inputting the composite reference trajectory.

[0092] In the specific embodiments described above, the purpose, technical solutions, and beneficial effects of the present invention have been further described in detail. It should be understood that the above are only specific embodiments of the present invention and are not used to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial, characterized in that, It includes the following steps: S1. Identify an autoregressive model with external input according to the input-output data of the precision positioning platform; S2. Establish an augmented system containing tracking error according to the characteristic polynomial of the reference trajectory; S3. Design an optimization problem with constraints based on the augmented system containing tracking error; S4. Solve the optimization problem according to the coordinate descent method to obtain a model predictive controller, and achieve zero-error tracking of the multi-input multi-output system.

2. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 1, characterized in that The S1 includes: The displacement outputs of different degrees of freedom are affected by multiple control inputs. Based on this, an autoregressive model with external input of the following precision positioning platform is established: ， wherein, is the th degree of freedom of the precision positioning platform, is the output displacement of the th degree of freedom of the precision positioning platform at moment, is the number of historical outputs contained in the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the model parameter related to the historical outputs of the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the control input of the precision positioning platform at moment, is the number of control inputs of the precision positioning platform, is the number of historical inputs contained in the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the model parameter related to the historical inputs of the autoregressive model of the output displacement of the th degree of freedom of the precision positioning platform, is the set composed of all real numbers, and the superscript represents the transpose of a matrix.

3. The zero-error model predictive control method for the precision positioning platform based on the characteristic polynomial according to claim 2, characterized in that, The S1 further includes: Input sweep signals with rich spectral information into different input channels of the precision positioning platform in sequence, and record all the output displacements of the th degree of freedom of the precision positioning platform under this input. Select different and . Identify the system according to the above autoregressive model, and select a series of and that minimize the difference between the identification result and the actual data.

4. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 3, wherein The S2 includes: Define the precision positioning platform The output displacement vector at a moment is , and establish it according to the historical input and output data of the precision positioning platform The system state vector at a moment ; Among them, means stacking all the internal vectors in sequence into a column vector, is used to define a variable, , is the maximum value of the relevant elements.

5. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 4, wherein The S2 further includes: According to the identification result, obtain the non-minimal state space model of the precision positioning platform, and construct an augmented system based on this non-minimal state space model as: ， Among them, is the state variable of the augmented system, the state variable constructed based on the characteristic polynomial of the reference trajectory , the control input constructed based on the characteristic polynomial of the reference trajectory , is the characteristic polynomial of the reference trajectory, is the precision positioning platform is the tracking error vector at time , and the total error vector composed of the tracking error vectors at multiple historical times ; is the parameter intermediate matrix.

6. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 5, characterized in that The S3 includes: In order to enable the outputs of all degrees of freedom of the precision positioning platform to accurately track the corresponding reference trajectories, a finite-time cost function as shown below is designed : ， Among them, and are the weight parameters related to the error and control input in the cost function respectively, is the prediction step length in the cost function, and the sum matrix of tracking errors at multiple future moments is defined as , the control input matrix at multiple future moments , the control input matrix constructed based on the characteristic polynomial of the reference trajectory at multiple future moments , the control input matrix at multiple known historical moments , construct the matrix relationship , , is the intermediate matrix.

7. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 6, characterized in that, The S3 further includes: According to the actual situation, the state variable constraints and output constraints of the system are uniformly transformed into control input constraints, and are represented by a matrix as , where is a constant matrix parameter with rows and columns, is a constant vector parameter with elements, is the number of inequality constraints for denotes the time. Then, the trajectory tracking problem of the precision positioning stage is transformed into solving the following constrained optimization problem: ， Satisfy: , ； Among them, represents the finite-time cost function and the control input matrix at multiple future moments that obtains the minimum value value.

8. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 7, wherein The S4 includes: Transform the constrained optimization problem into the following Lagrange polynomial : ， wherein, is the Lagrange multiplier, and the above formula takes the partial derivative with respect to the control input matrix at multiple future times and sets it to 0, so that when the finite-time cost function is minimized, the control input matrix at multiple future times satisfies ; , are all intermediate matrices.

9. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 8, characterized in that, The S4 further includes: After substituting into the Lagrange polynomial, the original constrained optimization problem is transformed into the following dual problem: ， Satisfy: ; Among them, is the optimal solution of the above dual problem, is the intermediate matrix, is the intermediate vector, means making the smallest value.

10. The zero-error model predictive control method for a precision positioning platform based on a characteristic polynomial according to claim 9, wherein The S4 further includes: Solve the dual problem by the coordinate descent method, and solve it by performing the following assignment operations multiple times: Intermediate variable , , ， Among them, represents an assignment operation, which means assigning the operation result of the expression after the symbol to the variable before the symbol. is a square matrix the th element on the diagonal, is the th element in is a square matrix the th row; Assign values to all elements in After times, stop the assignment and obtain the Lagrange multiplier value at the current moment , and further approximately obtain the optimal control input sequence at the current moment as . Select The first elements in are used as the control input at the current moment to achieve zero-error tracking of the reference trajectory under the constraint conditions.

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