Two-dimensional curved surface workpiece machining system control method based on frequency domain data driving

Abstract

The invention relates to a two-dimensional curved surface workpiece machining system control method based on frequency domain data driving. The method comprises the steps that a two-dimensional curved surface workpiece machining system model is established; performing equal-interval frequency domain sampling on the established two-dimensional curved surface workpiece processing system model to obtain frequency response of the two-dimensional curved surface workpiece processing system; calculating a frequency domain estimation function of the system based on the set frequency domain decomposition order; initializing system input and output and ILC tracking errors, and setting error precision of the ILC tracking errors; calculating a frequency domain approximation error; selecting an ILC control gain; calculating a time-frequency control law based on a frequency domain estimation function of the system and ILC control gain; performing two-dimensional inverse Z transformation on the time-frequency control law to obtain system output; and when the error precision of the ILC tracking error is smaller than the preset error precision, iteration is stopped. According to the method, the problem of excessive dependence on a system model in the prior art is solved, the modeling process of a complex system is greatly simplified, and meanwhile, the control robustness and precision of a two-dimensional curved surface workpiece machining system are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of iterative learning control, and more specifically to a control method for a two-dimensional curved surface workpiece machining system based on frequency domain data drive. Background Art

[0002] Iterative Learning Control (ILC) is an advanced control strategy. Its core is to capture the feedback information generated by the system when it repeatedly performs the same task, and use this information to continuously optimize the control input through an iterative process to gradually reduce the tracking error of the system and achieve accurate tracking of the predetermined trajectory. Unlike traditional feedback control methods that rely on precise mathematical models, ILC significantly improves the dynamic performance and accuracy of the system by continuously learning and adjusting from historical operating data. This method is particularly suitable for systems with repetitive operation characteristics, such as robotics, precision machining, production line automation, and chemical process control.

[0003] In the field of precision machining, machining systems for two-dimensional curved workpieces (such as CNC machining control systems) often show obvious repeatability characteristics. With the continuous improvement of machining accuracy requirements, the difficulty of machining also increases. In the actual machining process, factors such as the geometric deviation of the machine tool and the deformation of the workpiece itself may lead to the accumulation of form and position errors, thereby affecting the machining quality. Therefore, in these application scenarios, it is very important to design a special control system to achieve effective control, and the application of two-dimensional ILC is particularly critical. However, for traditional two-dimensional discrete systems, the positive solution of the Lyapunov equation is not always a necessary and sufficient condition for the asymptotic stability of the system. This essential difference means that the ILC control method on the one-dimensional system cannot be simply applied to the two-dimensional system.

[0004] Although rich research results have been achieved in the field of one-dimensional ILC, the research results of two-dimensional ILC are very limited. At present, the research of two-dimensional ILC mainly focuses on time domain analysis and design, which usually requires high accuracy and complexity of the system model. Compared with the time domain method, the frequency domain design method can convert the convolution operation into a multiplication operation in the frequency domain, thereby significantly reducing the computational complexity and improving the efficiency. In addition, through the loop shaping technology in the frequency domain, the performance and robustness goals can be achieved more intuitively and efficiently, optimizing the trade-off between performance and robustness under system uncertainty conditions. Summary of the invention

[0005] The present invention aims to overcome the problem of excessive reliance on the model of the two-dimensional curved surface workpiece processing system in the prior art, and provide a two-dimensional curved surface workpiece processing system control method based on frequency domain data drive, which greatly simplifies the modeling process of the complex system and provides a Improves the control robustness and accuracy of two-dimensional curved surface workpiece machining systems.

[0006] The present invention adopts the following technical solution:

[0007] A control method for a two-dimensional curved surface workpiece processing system based on frequency domain data drive, comprising:

[0008] S1, system model building step;

[0009] Establish a two-dimensional curved surface workpiece processing system model as follows:

[0010]

[0011] Among them, χ,γ∈{0,1,2,…,N-1} respectively represent the running time index of the two-dimensional surface workpiece machining system in two dimensions, N represents the maximum running time of the system; k represents the number of iterations; u k (χ,γ) and y k (χ,γ) represent the input and output of the kth iteration of the two-dimensional surface workpiece machining system; x 1,k (χ,γ) represents the position of the tool at the time index χ and γ in the kth iteration of the two-dimensional surface workpiece machining system; 2,k (χ,γ) represents the tool velocity at the time index χ and γ in the kth iteration of the two-dimensional surface workpiece machining system; 1,k (χ+1,γ) represents the position of the tool at the time index χ+1 and γ in the kth iteration of the two-dimensional surface workpiece machining system; 2,k (χ,γ+1) represents the speed of the tool of the two-dimensional curved surface workpiece machining system at the time index of χ and γ+1; and is a completely unknown real matrix of the system with suitable dimensions;

[0012] S2, frequency response acquisition step;

[0013] The established two-dimensional curved surface workpiece machining system model is sampled in the frequency domain at equal intervals to obtain the frequency response of the two-dimensional curved surface workpiece machining system. as follows:

[0014]

[0015] Among them, z -1 =e jθT and is a complex variable, j is an imaginary unit, θ and are the frequencies of the horizontal and vertical signals respectively, and T is the sampling period; is the number of sampling points in the frequency domain;

[0016] θr and Respectively expressed in When the number of sampling points is , the two dimensions of the two-dimensional surface correspond to the angular frequencies of the rth and sth points;

[0017] S3, frequency domain estimation function calculation step;

[0018] Set the frequency domain decomposition order to l and calculate the frequency domain estimation function of the two-dimensional surface workpiece machining system as follows:

[0019]

[0020] in, represents the p-order rational basis function; represents the q-order rational basis function; a l 、a ν , b l and b ν represents a complex number in the unit disk, and They are respectively a l 、a ν , b l and b ν The complex conjugate of

[0021] and Obtained by the following formula:

[0022] make l≤m, then in, represents the unit disk, represents the number of frequency domain decomposition factors, m represents the maximum order of frequency domain decomposition; <·,·> represents inner product operation;

[0023] S4, initialization step;

[0024] When k=1, the initial input u of the two-dimensional curved surface workpiece machining system 1 (χ,γ), the system output is y 1 (χ,γ); at the same time, the ideal tracking trajectory y of the two-dimensional curved surface workpiece processing system r (χ,γ) and y 1 (χ,γ) is subtracted to obtain the ILC tracking error e when k = 1 1 (χ,γ); Set the error accuracy of the two-dimensional curved surface workpiece processing system output to δ;

[0025] S5, frequency domain approximation error calculation step;

[0026] Calculate the frequency domain approximation error Φ as follows:

[0027]

[0028] Among them, sup represents the supremum of the function value; Represent the transfer function of the two-dimensional curved surface workpiece machining system; M represents the standard remainder factor of frequency domain decomposition;

[0029] S6, control gain selection step;

[0030] According to the convergence condition of ILC |ρ|<1 / Φ, select the ILC control gain ρ;

[0031] S7, time-frequency control law calculation step;

[0032] The Z-domain expression of the time-frequency control law is calculated as follows:

[0033]

[0034] Among them, U k+1 (z -1 ,ξ -1 ) and U k (z -1 ,ξ -1 ) represent the Z-domain expressions of the time-frequency control law of the k+1th and kth iterations respectively; E k (z -1 ,ξ -1 ) represents the ILC tracking error in the Z domain at the kth iteration; e k (χ,γ) represents the ILC tracking error of the two-dimensional curved surface workpiece machining system at the kth iteration; is the two-dimensional Z transform;

[0035] S8, system output calculation steps;

[0036] Calculate the input of the k+1th iteration of the 2D surface workpiece machining system Substitute it into the two-dimensional curved surface workpiece machining system model to obtain the output y of the k+1th iteration of the two-dimensional curved surface workpiece machining system k+1 (χ,γ), where represents the two-dimensional inverse Z transform;

[0037] S9, iteration stop judgment step;

[0038] Calculate the ILC tracking error e of the two-dimensional curved surface workpiece machining system at the k+1th iteration k+1 (χ,γ)=y r (χ,γ)-yk+1 (χ,γ), and judge ||e k+1 (χ,γ)|| 2 <δ is true, if true, stop the iteration, otherwise execute k=k+1 and turn to execute S7~S9, where ||·|| 2 represents the 2-norm.

[0039] It can be seen from the above description of the present invention that, compared with the prior art, the present invention has the following beneficial effects:

[0040] The present invention discloses a control method for a two-dimensional curved surface workpiece machining system based on frequency domain data drive, which integrates the concepts of data drive and frequency domain design. The method proposes a time-frequency control law, which contains both time domain and frequency domain components, providing greater flexibility for adjusting learning performance. In addition, according to the proposed control convergence condition, it becomes easy to select a suitable control gain, thereby simplifying the design of the two-dimensional curved surface workpiece machining system, overcoming the excessive reliance of the prior art on the two-dimensional curved surface workpiece machining control system model, greatly simplifying the modeling process of complex systems, and improving the control robustness and accuracy of the two-dimensional curved surface workpiece machining system. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a flow chart of a two-dimensional curved surface workpiece processing system control method based on frequency domain data drive according to an embodiment of the present invention;

[0042] Figure 2 The ideal surface tracking trajectory of the embodiment of the present invention;

[0043] Figure 3 is the ILC tracking error index TE when both the input and output of the system of the embodiment of the present invention are affected by noises with different signal-to-noise ratios (k) Schematic diagram of

[0044] Figure 4 The ILC tracking error index TE of the proposed time-frequency control law is shown in the embodiment of the present invention when the iteration index k=8. (k) Change relationship diagram;

[0045] Figure 5 The ILC tracking error index TE of the proposed time-frequency control law is shown in the embodiment of the present invention when the iteration index k=50. (k) Change relationship diagram. DETAILED DESCRIPTION

[0046] The present invention is further described below through specific implementation modes.

[0047] See also Figure 1As shown, the present invention provides a control method for a two-dimensional curved surface workpiece processing system based on frequency domain data drive, which includes the following steps.

[0048] S1, system model building step.

[0049] During the precision machining of two-dimensional curved workpieces, the control system must accurately control the movement of the print head to ensure that the workpiece material strictly follows the predetermined trajectory and is accurately deposited layer by layer with the appropriate thickness. The core goal of iterative learning control is to continuously optimize the control strategy and adjust the system input so that the system output can accurately track the preset ideal trajectory. The two-dimensional curved workpiece machining system model is as follows:

[0050]

[0051] Among them, χ,γ∈{0,1,2,…,N-1} respectively represent the running time index of the two-dimensional surface workpiece machining system in two dimensions, N represents the maximum running time of the system; k represents the number of iterations; u k (χ,γ) and y k (χ,γ) represent the input and output of the kth iteration of the two-dimensional surface workpiece machining system; x 1,k (χ,γ) represents the position of the tool at the time index χ and γ in the kth iteration of the two-dimensional surface workpiece machining system; 2,k (χ,γ) represents the tool velocity at the time index χ and γ in the kth iteration of the two-dimensional surface workpiece machining system; 1,k (χ+1,γ) represents

[0052] The position of the tool at the time index χ+1 and γ in the kth iteration of the two-dimensional surface machining system; x 2,k (χ,γ+1) represents the speed of the tool on the two-dimensional surface at the time index χ and γ+1;

[0053] and is a completely unknown real matrix of the system with suitable dimensions.

[0054] Assume that the ideal tracking trajectory of the two-dimensional curved surface workpiece machining system model (1) is y r (χ,γ), then the ILC tracking error of the k-th system operation is:

[0055] e k (χ,γ)=y r (χ,γ)-y k (χ,γ) (2)

[0056] The initial system state of the 2-D discrete-time system (1) should satisfy x1,k (0,γ)=x 2,k (χ,0)=0, the condition is χ≥0,γ≥0, k=0,1,2,….

[0057] e k The Z-domain expression of (χ,γ) is:

[0058]

[0059] Among them, E k (z -1 ,ξ -1 ) is the ILC tracking error in the Z domain at the kth iteration, Y r (z -1 ,ξ -1 ) is the ideal tracking trajectory of the system in the Z domain, Y k (z -1 ,ξ -1 ) is the Z-domain expression of the system output at the kth iteration, z- 1 =e jθT and is a complex variable (j is an imaginary unit, θ and is the frequency, T is the sampling period), is the two-dimensional Z transform.

[0060] By performing two-dimensional Z transformation on both sides of the two-dimensional discrete system equation (1), the two-dimensional curved surface workpiece processing system (1) can be expressed in the Z domain as:

[0061]

[0062] Therefore, mathematical transformation of formula (4) yields

[0063]

[0064] Among them, J(z -1 ,ξ -1 ) represents the two-dimensional transfer function model of the two-dimensional curved surface workpiece machining system (1), and I n is the n-th order identity matrix.

[0065] The goal of frequency domain 2D ILC design is to design an ILC control law to adjust the system input U k (z -1 ,ξ -1 ), so that when the iteration index k approaches infinity, That is to say, curved surface workpiece processing can produce ideal workpieces.

[0066] S2, frequency response acquisition step.

[0067] The established two-dimensional curved surface workpiece machining system model (1) is sampled in the frequency domain at equal intervals to obtain the frequency response of the two-dimensional curved surface workpiece machining system. as follows:

[0068]

[0069] Among them, z -1 =e jθT and is a complex variable, j is an imaginary unit, θ and are the frequencies of the horizontal and vertical signals respectively, and T is the sampling period; is the number of sampling points in the frequency domain; θ r and Respectively expressed in When the number of sampling points is , the two dimensions of the two-dimensional surface correspond to the angular frequencies of the r-th and s-th points; r and s represent the measured values ​​of the frequency domain sampling of the two dimensions of the two-dimensional surface respectively.

[0070] S3, frequency domain estimation function calculation step.

[0071] Set the frequency domain decomposition order to l and calculate the frequency domain estimation function of the two-dimensional surface workpiece machining system as follows:

[0072]

[0073] in, represents the p-order rational basis function; represents the q-order rational basis function; p and q represent the order of the rational basis function, a l 、a ν , b l and b ν represents a complex number in the unit disk, and They are respectively a l 、a ν , b l and b ν The complex conjugate of

[0074] and Obtained by the following formula:

[0075] make but in, represents the unit disk, represents the number of frequency domain decomposition factors, m represents the maximum order of frequency domain decomposition; <·,·> represents inner product operation;

[0076] S4, initialization step.

[0077] When k=1, the initial input u of the two-dimensional curved surface workpiece machining system 1 (χ,γ), the system output is y 1 (χ,γ); at the same time, the ideal tracking trajectory y of the two-dimensional curved surface workpiece processing system r (χ,γ) and y 1 (χ,γ) is subtracted to obtain the ILC tracking error e when k = 1 1 (χ,γ); Set the error accuracy of the two-dimensional surface workpiece processing system output to δ.

[0078] S5, frequency domain approximation error calculation step.

[0079] Calculate the frequency domain approximation error Φ as follows:

[0080]

[0081] Among them, sup represents the supremum of the function value; Represent the transfer function of the two-dimensional curved surface workpiece machining system; M represents the standard remainder factor of frequency domain decomposition.

[0082] S6, control gain selection step;

[0083] According to the convergence condition of ILC |ρ|<1 / Φ, the ILC control gain ρ is selected.

[0084] S7, time-frequency control law calculation step.

[0085] The Z-domain expression of the time-frequency control law is calculated as follows:

[0086]

[0087] Among them, U k+1 (z -1 ,ξ -1 ) and U k (z -1 ,ξ -1 ) represent the Z-domain expressions of the time-frequency control law of the k+1th and kth iterations respectively; E k (z -1 ,ξ -1 ) represents the ILC tracking error in the Z domain at the kth iteration; e k (χ,γ) represents the ILC tracking error of the two-dimensional curved surface workpiece machining system at the kth iteration; is the two-dimensional Z transform.

[0088] In this embodiment, the time-frequency control law expression of the kth iteration is as follows:

[0089]

[0090] Where χ,γ∈{0,1,2,…,N-1} is the system running time index, z -1 =e jθT and is a complex variable (j is an imaginary unit, θ and is the frequency, T is the sampling period), is the two-dimensional inverse Z transform, k is the number of iterations, u k+1 (χ,γ) and u k (χ,γ) are the inputs of the k+1th and kth system iterations respectively, e k (χ+1,γ+1) is the ILC tracking error of the kth system iteration, is the transfer function of the two-dimensional system J(z -1 ,ξ -1 ), and ρ is the ILC control gain.

[0091] S8, the system outputs the calculation steps.

[0092] Calculate the input of the k+1th iteration of the 2D surface workpiece machining system Substitute it into the two-dimensional curved surface workpiece machining system model to obtain the output y of the k+1th iteration of the two-dimensional curved surface workpiece machining system k+1 (χ,γ), where Represents the two-dimensional inverse Z transform.

[0093] S9, iteration stop judgment step;

[0094] Calculate the ILC tracking error e of the two-dimensional curved surface workpiece machining system at the k+1th iteration k+1 (χ,γ)=y r (χ,γ)-y k+1 (χ,γ), and judge ||e k+1 (χ,γ)|| 2 <δ is true, if true, stop the iteration, otherwise execute k=k+1 and turn to execute S7~S9, where ||·|| 2 represents the 2-norm.

[0095] The frequency domain data driven two-dimensional curved surface workpiece processing system control method of the present invention will be verified through an actual controlled system as follows.

[0096] In order to verify the effectiveness of the method proposed in this invention, the following two-dimensional curved surface workpiece machining system model is considered as the controlled system for verification:

[0097]

[0098] Assume that the two-dimensional curved workpiece machining system (8) works repeatedly in a finite time period {0,1,2,Λ,N}, N=30. The required ideal curved surface tracking trajectory is Its changing relationship is Figure 2 shown.

[0099] In order to evaluate the proposed frequency domain data-driven ILC design, the following evaluation indicators TE (k) To measure the accuracy of ILC tracking, that is, the average error in the frequency domain, the calculation formula is in and

[0100] Considering that there is additive random noise in both the input and output frequency domain measurements of the two-dimensional curved workpiece machining system, a noise with a signal-to-noise ratio (SNR) of 30 dB is added to the input and a noise with an SNR of 25 dB is added to the output. First, according to S3 above, the corresponding estimated transfer function is obtained Then, the initial control input u 1 (χ,γ)=90cos(0.1πχγ) is applied to the two-dimensional curved surface workpiece processing system (8). Next, according to the above S6, the control gain value of the time-frequency control law is selected as ρ=0.21. Figure 3 The green solid line in the figure shows the ILC tracking error index TE when a noise with an SNR of 25dB is added to both the input and output of the system. (k) In order to more intuitively show the impact of noise on the time-frequency control law, Figure 3 The ILC tracking error index TE is also shown under noise conditions of SNR 18dB, 20dB and 25dB. (k) The change relationship. Figure 3 It can be observed that a higher signal-to-noise ratio means less noise in the signal, a higher quality of the frequency domain signal, and therefore a higher execution efficiency of the proposed control method. Figure 4 and Figure 5 The running results of the tracking trajectory and the ideal surface tracking trajectory under different iteration indexes are shown. Obviously, as the iteration index increases, the ILC tracking error will converge to a bounded value, with good robust convergence performance.

[0101] The above embodiments are only used to illustrate the present invention, and are not used to limit the present invention. As long as they are based on the technical essence of the present invention, changes and modifications to the above embodiments will fall within the scope of the claims of the present invention.

Claims

1. A control method for a two-dimensional curved surface workpiece processing system based on frequency domain data drive, characterized in that: include: S1, system model building step; Establish a two-dimensional curved surface workpiece processing system model as follows: Among them, χ,γ∈{0,1,2,…,N-1} respectively represent the running time index of the two-dimensional surface workpiece machining system in two dimensions, N represents the maximum running time of the system; k represents the number of iterations; u k (χ,γ) and y k (χ,γ) represent the input and output of the kth iteration of the two-dimensional surface workpiece machining system; x 1,k (χ,γ) represents the position of the tool at the time index χ and γ in the kth iteration of the two-dimensional surface workpiece machining system; 2,k (χ,γ) represents the tool velocity at the time index χ and γ in the kth iteration of the two-dimensional surface workpiece machining system; 1,k (χ+1,γ) represents the position of the tool at the time index χ+1 and γ in the kth iteration of the two-dimensional surface workpiece machining system; 2,k (χ,γ+1) represents the speed of the tool of the two-dimensional curved surface workpiece machining system at the time index of χ and γ+1; and is a completely unknown real matrix of the system with suitable dimensions; S2, frequency response acquisition step; The established two-dimensional curved surface workpiece machining system model is sampled in the frequency domain at equal intervals to obtain the frequency response of the two-dimensional curved surface workpiece machining system. as follows: Among them, z -1 =e jθT and is a complex variable, j is an imaginary unit, θ and are the frequencies of the horizontal and vertical signals respectively, and T is the sampling period; is the number of sampling points in the frequency domain; θ r and Respectively expressed in When the number of sampling points is , the two dimensions of the two-dimensional surface correspond to the angular frequencies of the rth and sth points; S3, frequency domain estimation function calculation step; Set the frequency domain decomposition order to l and calculate the frequency domain estimation function of the two-dimensional surface workpiece machining system as follows: in, represents the p-order rational basis function; represents the q-order rational basis function; a l 、a ν , b l and b ν represents a complex number in the unit disk, and They are respectively l 、a ν , b l and b ν The complex conjugate of ; and Obtained by the following formula: make l≤m, then in, represents the unit disk, represents the number of frequency domain decomposition factors, m represents the maximum order of frequency domain decomposition; <·,·> represents inner product operation; S4, initialization step; When k=1, the initial input u1(χ,γ) of the two-dimensional curved surface workpiece machining system is set to y1(χ,γ), and the system output is y1(χ,γ). At the same time, the ideal tracking trajectory y r The ILC tracking error e1(χ,γ) is obtained by subtracting y1(χ,γ) from y1(χ,γ) when k=1; the error accuracy of the output of the two-dimensional curved surface workpiece machining system is set to δ; S5, frequency domain approximation error calculation step; Calculate the frequency domain approximation error Φ as follows: Among them, sup represents the supremum of the function value; Represent the transfer function of the two-dimensional curved surface workpiece machining system; M represents the standard remainder factor of frequency domain decomposition; S6, control gain selection step; According to the convergence condition of ILC |ρ|<1 / Φ, select the ILC control gain ρ; S7, time-frequency control law calculation step; The Z-domain expression of the time-frequency control law is calculated as follows: Among them, U k+1 (z -1 ,ξ -1 ) and U k (z -1 ,ξ -1 ) represent the Z-domain expressions of the time-frequency control law of the k+1th and kth iterations respectively; E k (z -1 ,ξ -1 ) represents the ILC tracking error in the Z domain at the kth iteration; e k (χ,γ) represents the ILC tracking error of the two-dimensional curved surface workpiece machining system at the kth iteration; is the two-dimensional Z transform; S8, system output calculation steps; Calculate the input of the k+1th iteration of the 2D surface workpiece machining system Substitute it into the two-dimensional curved surface workpiece machining system model to obtain the output y of the k+1th iteration of the two-dimensional curved surface workpiece machining system k+1 (χ,γ), where represents the two-dimensional inverse Z transform; S9, iteration stop judgment step; Calculate the ILC tracking error e of the two-dimensional curved surface workpiece machining system at the k+1th iteration k+1 (χ,γ)=y r (χ,γ)-y k+1 (χ,γ), and judge ||e k+1 (χ,γ)||2<δ holds true. If so, stop the iteration. Otherwise, execute k=k+1 and turn to execute S7 to S9, where ||·||2 represents the 2-norm.

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