Parameter search space calculation method

Abstract

To solve the problem of the prior art, in that attainable control performance may be limited, in addition, in order to calculate a large search space, the determination of the solvency of an optimization problem has to be made many times, and a time taken to calculate the search space becomes long.SOLUTION: A parameter search space calculation method comprises: Step 1 of taking an initial FB parameter ρini and stability H (ρini) calculated for the ρini, as initial data DH=(ρini;H(ρini)), and initializing an estimated iterator as i=1; Step 2 of calculating an estimated stability H^(ρ) and its uncertainty σ2(ρ) under an assumption that the stability follows a statistical distribution; Step 3 of calculating an acquisition function and taking an FB parameter ρnext at which the acquisition function is minimum, as a next search point; and Step 4 of adding the ρnext and stability H(ρnext) to DH. In the case of i<n (an upper limit number of iterations), i=i+1 is set and the process returns to Step 2, and in the case of i=n, the process ends and a space of H^(ρ)≥0 is set as a search space T.SELECTED DRAWING: Figure 4

Description

[Technical Field]

[0001] The present invention relates to a method for calculating a parameter search space in the automatic adjustment of feedback control parameters. [Background technology]

[0002] In industrial servo control, where high-speed and high-precision performance is required, automatic adjustment technology, which enables automatic and rapid control adjustment, is one of the important control design technologies for achieving and maintaining desired servo performance and improving productivity. Parameter search in automatic adjustment is generally performed based on optimization algorithms, and various automatic adjustment methods have been proposed. When adjusting feedback (FB) control parameters, parameter search involves changes in the stability of the control system, thus requiring automatic adjustment that takes stability into account. Most prior studies on automatic adjustment of FB control parameters do not explicitly specify how to set the parameter search space, and in practice, trial-and-error methods such as restricting the search space to a narrow space are necessary. However, restricting the search space to a narrow space is undesirable because it leads to limitations in the achievable control performance.

[0003] Focusing on the aforementioned problem, Non-Patent Literature 1 proposes an automatic tuning method for FB control parameters, which calculates a parameter space (hereinafter referred to as the stability space) in which the FB control system has predetermined stability (gain margin, phase margin) as a search space, and performs an active parameter search within that space. There, as a method for calculating the parameter search space, a method is proposed that utilizes the solvability of a constrained optimization problem to calculate a parameter space that satisfies predetermined stability, and it is possible to calculate the search space in a shorter time than the method of calculating the stability space by exhaustive search.

[0004] Patent Document 1 discloses a parameter search method in which an order is assigned to a plurality of parameter search points included in a parameter search space, and the values ​​indicating the order are used as search targets to be manipulated in order to search for parameters, and an optimization algorithm (genetic algorithm) is executed for this purpose. Patent Document 2 discloses a parameter search method that uses Bayesian optimization (Non-Patent Document 2) as the optimization algorithm to determine the confidence limit of the constraint function for unexplored parameters, and sets the parameter that optimizes the objective function within the parameter range in which the constraint function is estimated to satisfy predetermined conditions as the next search point. [Prior art documents] [Patent Documents]

[0005] [Patent Document 1] Japanese Patent Publication No. 2020-140345 [Patent Document 2] Japanese Patent Publication No. 2019-113985 [Non-patent literature]

[0006] [Non-Patent Document 1] Y. Maeda, N. Gou, and M. Iwasaki, “A Stable Parameter Area Calculation Method for Advanced Autotuning of Feedback Controller,” in Proceedings of the 17th IEEE International Conference on Industrial Informatics, pp.541-546, 2019. [Non-Patent Document 2] B. Shahriari, K. Swersky, Z. Wang, RP Adams, and ND Freitas, “Taking the human out of the loop: A review of Bayesian optimization,” in Proceedings of the IEEE, vol.104, no.1, pp.148-175, 2016. [Non-Patent Document 3] Y. Maeda and M. Iwasaki, “Circle Condition-Based Feedback Controller Design for Fast and Precise Positioning,” IEEE Transactions on Industrial Electronics, vol.61, no.2, pp.1113-1122, 2014. [Non-Patent Document 4] Mochihashi, Ohba: "Gaussian Processes and Machine Learning," Kodansha (2019) [Overview of the Initiative] [Problems that the invention aims to solve]

[0007] However, the method for calculating the parameter search space in Non-Patent Document 1 includes: • Since the search space that can be calculated is a subspace of the stable space, the achievable control performance may be limited. • To calculate a vast search space, it is necessary to perform the solvability check of the optimization problem many times, resulting in a very long time to calculate the search space. This problem has been identified. Patent Document 2 focuses on optimizing the objective function related to control performance, and therefore does not guarantee that a vast stable space can be estimated with high accuracy as a search space.

[0008] This invention proposes a Bayesian optimization-based search space estimation method as a way to efficiently calculate a vast stability space, which is used as the parameter search space in automatic tuning. The proposed method assumes that stability follows a statistical distribution and efficiently estimates the stability space using Bayesian optimization, taking into account the estimation of stability and its uncertainty. This enables the calculation of a vast search space in a shorter time than described in Non-Patent Document 1. [Means for solving the problem]

[0009] The present invention, which solves the above problems, is as follows. (1) A method for calculating a parameter search space in the automatic adjustment of feedback control parameters, the initial feedback parameter ρ ini and the stability H(ρ ini ) calculated for ρ ini are used to form the initial data D H =(ρ ini ; H(ρ ini )). Step 1 initializes the iteration variable i with i := 1. Step 2 calculates the estimated stability H^(ρ) and its uncertainty σ 2 (ρ) under the assumption that the stability follows a statistical distribution. Step 3 calculates the acquisition function and sets the feedback parameter ρ next at which the acquisition function is minimized as the next search point. Step 4 adds ρ next and the stability H(ρ next ) to D H . If i is less than the maximum number of iterations n, i := i + 1 and the process returns to Step 2. When i = n, the process ends, and the space where H^(ρ) ≥ 0 is defined as the search space T. This is a method for calculating a parameter search space characterized by the above steps. Here, "H^" is, for example, the ^ that should be written above H as described in equation (8). For the sake of convenience, ^ is written following H. (2) The method for calculating a parameter search space according to (1), characterized in that the statistical distribution assumed for the estimated stability H^(ρ) in Step 2 is a Gaussian distribution. (3) The method for calculating a control parameter search space according to (1), characterized in that when H(ρ1), H(ρ2),... H(ρ N ) are known, H(ρ1), H(ρ2),... H(ρ N ) follow a Gaussian process. (4) The method for calculating a parameter search space according to any one of (1) to (3), characterized in that in Step 3, the acquisition function is a function for estimating the boundary of the parameter space that satisfies a predetermined stability. (5) The method for calculating a parameter search space according to any one of (1) to (3), characterized in that in Step 3, the minimization of the acquisition function is performed by Bayesian optimization. [Advantages of the Invention]

[0010] According to the present invention, a vast parameter search space can be calculated in a shorter time than with conventional technology, and automatic tuning can be performed more effectively than with currently mainstream technology. Improved efficiency in automatic tuning leads to increased productivity and enhanced product value. [Brief explanation of the drawing]

[0011] [Figure 1] This figure shows an example configuration of a positioning control system having an automatic adjustment function for FB control parameters, which is one embodiment of the present invention. [Figure 2] This is a diagram illustrating the concept of a circular condition constraint. [Figure 3] This figure shows a conceptual diagram (D=3) of the calculation of the search space using the conventional method. [Figure 4] This is a flowchart of the proposed search space calculation method. [Figure 5] (a) This figure shows the stability estimation model before updating the Gaussian process, and (b) This figure shows a conceptual diagram of the acquisition function. [Figure 6] This figure shows a conceptual diagram of the stability estimation model after updating the Gaussian process. [Figure 7] This figure shows the frequency characteristics of a galvanometer scanner. [Figure 8] These figures show (a) RG∩TG, (b) RG\TG, and (c) TG\RG for the stable space estimated by the proposed method. [Figure 9] This figure shows the Nyquist trajectory based on parameters within the TG (Training Graph). [Figure 10] This diagram shows a comparison between SG and TG. [Figure 11] This figure shows a comparison of the open-loop frequency characteristics of SG and TG. [Modes for carrying out the invention]

[0012] Embodiments of the present invention will be described below with reference to the drawings. The present invention is not limited to the following embodiments, and modifications, alterations, and improvements may be made without departing from the scope of the invention.

[0013] (FB control parameter automatic adjustment problem) Figure 1 shows a block diagram of a 2-degree-of-freedom positioning control system with an automatic FB control parameter adjustment function. In the figure, r: position command, y: position response, e: trajectory tracking error, u FF : Feedforward (FF) control input, u: control input, P: controlled object, C(ρ): FB controller, ρ: D FB control parameters to be adjusted. The automatic adjustment system selects ρ from the search space using a search algorithm (optimization algorithm), and searches for ρ that has the desired positioning performance by repeatedly executing positioning operations and evaluating control performance. The characteristic of this automatic adjustment method is that by using a parameter space with a predetermined stability as the search space, parameter search can be performed without worrying about instability.

[0014] (A method for calculating the search space that considers the stability of the FB control system) Conventional method: This method, which utilizes the solvability of a constrained optimization problem, calculates a parameter space that satisfies predetermined stability (gain margin, phase margin) before parameter search, using the solvability of an optimization problem constrained by stability. Below, we will describe the definition of the stability constraint and the method for calculating the search space.

[0015] (Constraints on stability (Non-patent document 3)) Figure 2 shows a conceptual diagram of the Nyquist trajectory L(jω) and the circular condition for the open-loop characteristic L(s)=P(s)C(s) of the FB control system. In the figure, L(jω): Nyquist trajectory, Cs: predetermined stability margin (gain margin g) m [dB] and phase margin φ m [deg]) defined by the center (-c s , j0), radius r s This is the designated circle. Circle C s The point where this intersects with the real axis is the reciprocal of the gain margin, 1 / G. m (g m=20log 10 G m [dB]) Circle C s The intersection of the unit circle centered at the origin is the phase margin Φ m ( m =180Φ m G m and Φ m Using c s and r s It can be expressed by the following equation.

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[0016] Stability margin G m , Φ m The conditions for guaranteeing this are discrete angular frequency points Ω α (α=1,…,N α ) Nyquist trajectory L(jΩ α ) is circle C s This means passing outside the boundary, and can be expressed by the following equation.

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[0017] (Method for calculating the search space) Figure 3 shows a conceptual diagram of the search space S calculated by the conventional method when the number of FB control parameters D=3. In the figure, region 1 is the stable space R, and region 3 is the stable space R. D A 3-dimensional polyhedron D, composed of 27 lattice points, is calculated in the search space S, ρ β (β=0,1,…3 D -1) is a lattice point, and let ρ0 be the lattice point at the center of the polyhedron. S is calculated by solving the following optimization problem.

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[0018] Here, J(ρ0) is the objective function for having a predetermined servo bandwidth. When the size of a certain polyhedron is defined, if we can find a solution for ρ0 that satisfies equation (3) (solvency), then 3 D The parameter space inside a D-dimensional polyhedron represented by a lattice of points is determined to be a space that satisfies a predetermined stability. On the other hand, if ρ0 satisfying equation (3) cannot be found, it is determined that the polyhedron of that size does not fit in the stable space R. Therefore, equation (3) is solved while gradually decreasing the size of the polyhedron, and the interior space of the set of polyhedra when it becomes solvable becomes the search space S calculated by the conventional method. However, if the stable space R can be approximated by a set of polyhedra, then S ≈ R, but from the inventors' experience, a set of polyhedra often cannot accurately represent S, and S becomes a subset of R.

[0019] (Proposed method: A method based on Bayesian optimization) Figure 4 shows a flowchart of the proposed search space calculation method, and the processing content at each step is explained below. Step 1: Initialize the estimation iterator i to i:=1, and set the initial FB parameter ρ ini Stability H(ρ ini ) is calculated. This is the initial data D H =(ρ ini ,H(ρ ini )) Step 2: Estimate the stability H^(ρ) and its uncertainty σ based on the Gaussian process. 2 Calculate (ρ). Step 3: Calculate the acquisition function and find the FB parameter ρ that minimizes the acquisition function. next Let this be the next search point. Step 4 ρ next Its stability H(ρ next ) to data D H Add to it. Step 5: If i is less than the upper limit of iterations n, set i:=i+1 and repeat Steps 2 to 4. When i becomes n, the space where H^(ρ)≧0 is the search space T. In the following sections, we will describe the stability estimation based on the Gaussian process in Step 2, and the acquisition function used in Step 3.

[0020] (Estimation of stability based on Gaussian processes) Stability H(ρ) is given by H=[H(ρ1)H(ρ2)...H(ρ N )] T ∈R N×1 Assume that is known. Assume that H follows a Gaussian process, and define a Gaussian process regression model H^(ρ) for H. Here, T represents the transpose symbol. Let the average value of H, m, be given by the following equation.

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[0021] In this case, H follows the following Gaussian process.

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[0022] From this Gaussian process, a certain input ρ * Output H(ρ) * Let's consider predicting ). Given H, ρ * H(ρ) * ) similarly follows a Gaussian process in equation (5). Therefore, H(ρ * Adding ) to H * =[H(ρ1)...H(ρ N ) H(ρ * )] T For each input {ρ1,…,ρ N ρ * The (N+1)×(N+1) dimension covariance matrix calculated from} is K * Therefore, its Gaussian distribution is expressed by the following equation.

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[0023] Therefore, input ρ * H(ρ) * ) can be calculated as a conditional probability. From equation (6), H(ρ * The conditional probability of ) is expressed by the following formula:

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[0024] (An acquisition function for boundary search in stable spaces) In Bayesian optimization, the acquisition function determines the next search point. In this invention, R is efficiently estimated by using an acquisition function that estimates the boundary of the stable space R. Since the boundary of R can be expressed as H(ρ)=0, the acquisition function is the absolute value of the stability estimate H^(ρ) and its uncertainty σ. 2 It is defined as a linear sum of (ρ) by the following equation.

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[0025] In the upper part of the figure (Figure 5(a)), the dashed line 11 represents the true stability H(ρ), the solid line 12 represents the estimated stability H^(ρ), the dotted line 13 represents H^(ρ)±κσ(ρ), and point 14 represents the known data D. HRegion 2 represents the region where H^(ρ)≧0 is estimated to satisfy a predetermined stability. Meanwhile, in the lower part of the figure (Figure 5(b)), the dashed line 21 represents |H^(ρ)| and the solid line 22 represents the acquisition function. From the figure, the acquisition function takes a small value in the region estimated to be the stability boundary (H^(ρ)=0) by the estimated model H^(ρ), and the acquisition function has a minimum value in the region where the uncertainty κσ(ρ) is large (ρ=-5.2 in this embodiment). Therefore, the next search point ρ is the ρ at which the acquisition function is minimized. next By actively exploring stability boundaries with high uncertainty, it is possible to efficiently estimate stability boundaries.

[0026] Figure 6 shows the estimated stability when the Gaussian process is updated after obtaining the stability H(ρ=-5.2) where the acquisition function is minimized in Figure 5 (ρ=-5.2). It can be seen that the estimation accuracy of H^(ρ) has improved and the estimated stable space is approaching the true stable space. In the proposed method, the parameter space H^(ρ)≧0 at the end of the flowchart in Figure 4 is used as the search space T. By using an appropriate kernel function and κ, T can be obtained as a good approximation of R by repeatedly performing optimization calculations.

[0027] <Simulation> (Overview of the test device and positioning control system) In this embodiment, as an example of an industrial servo mechanism, a galvanometer scanner for a printed circuit board laser drilling machine is used as the control target, and a parameter space with a predetermined stability is estimated (see Non-Patent Literature 1 for details of the galvanometer scanner used). Figure 7 shows the frequency characteristics from the control input u to the position response y of the galvanometer scanner measured by the sinusoidal sweep method. From the figure, it can be seen that the galvanometer scanner has a first resonance at 2.9 kHz, a second resonance at 5.8 kHz, and multiple resonance modes in the high-frequency region above 10 kHz. In this embodiment, the search space is calculated using the non-parametric model in Figure 7, and the predetermined stability margin is set as shown in Table 1, considering the robust stability when the characteristics of the controlled object fluctuate. Also, the c at that time s and r s This will be listed in the same table. [Table 1]

[0028] The positioning control system considered in the search space calculation is a one-degree-of-freedom control system with u FF =0 in FIG. 1, and the FB controller C is configured by cascading a PID controller and two resonance compensation filters as follows.

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[0029] Note that the initial parameters are the values shown in Table 2.

Table 2

[0030] (Estimation result of search space) In the comparative evaluation of the search spaces S and T calculated by the conventional method and the proposed method, a grid shown in Table 3 is created, and a parameter set G on the grid is introduced. Then, the parameter set of the stable space R calculated by the full search method of Non-Patent Document 1 is R G , the parameter set of the search space S calculated by the conventional method is S G , and the parameter set of the search space T calculated by the proposed method is T G , and they are defined as follows. Hereinafter, the point sets R G , S G , T G are respectively represented as the true stable space, the search space calculated by the conventional method, and the search space T calculated by the proposed method.

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Table 3

[0031] The conventional method considered the calculation time and the size of the search space calculated, changed the size of the polyhedron 17 times, and calculated the search space S by solving Equation (3). The proposed method uses the initial parameters shown in Table 2 to construct the initial data D H (ρ ini ; H(ρ ini )) and set n = 200 and κ = 0.01 to estimate the search space T so as to obtain sufficient estimation accuracy for the true stable space R. Note that in this embodiment, the minimization of the acquisition function in Equation (9) calculates the acquisition function for all elements of the parameter set G on the above-mentioned grid and determines the minimum point.

[0032] R G and T G are shown in comparison in Fig. 8. Here, X\Y represents the difference set obtained by subtracting a set Y from a set X. The points in Fig. (a) are R G ∩T G , the points in Fig. (b) are R G \T G , and the points in Fig. (c) are T G \R G . From the figure, the search space calculated by the proposed method occupies most of the points in Fig. (a) and is in good agreement with the true stable space R G . The quantitative evaluation of the reproducibility of T G with respect to R G is shown in Table 4. n(X) represents the number of elements in a set X, and R G and T G had a coincidence rate of 97.45%. Fig. 9 shows the Nyquist trajectories for the case of the parameters within T G . The left figure (a) shows the Nyquist trajectories randomly extracted from 50 points from R G ∩T G , and the right figure (b) shows the Nyquist trajectories randomly extracted from 50 points from T G \R G . All the non-circular lines such as C s in the left figure (a) satisfy a predetermined stability constraint, whereas Cs Non-circular lines do not satisfy the stability constraint, but the impact of the constraint violation on stability is negligible, so they can be considered parameters that do not pose a practical problem. [Table 4]

[0033] Next, as a comparison between the conventional method and the proposed method, S G and T G Figure 10 shows this. In the figure, point 31 is S G Point 32 is T G Therefore, the proposed method obtains a much larger search space than conventional methods. Table 5 shows S G and T G This shows a quantitative comparison of the number of parameter points, and it can be seen that the proposed method can calculate a vast search space with about three times the number of parameter points as the conventional method. Figure 11 shows a comparison of the open-loop characteristics in the search spaces calculated by the proposed method and the conventional method. In the figure, dashed lines 41 and 42 represent S, respectively. G and T G This is an open-loop frequency response plotted by randomly selecting 100 parameters from the given set. The proposed method exhibits a greater change in open-loop characteristics, particularly in the low-frequency range, compared to conventional methods, suggesting that it is possible to explore a wide range of parameters while satisfying the required stability. [Table 5]

[0034] Finally, Table 6 shows the time required for the exhaustive search method, the conventional method, and the proposed method to compare the time required to calculate the search space. The exhaustive search method takes the longest time because it must evaluate equation (2) for all parameters in the set G. The conventional method can calculate the search space in about 17% of the time of the exhaustive search method, but as shown in Figure 10, the calculated search space is a true stable space R. G It is clearly a smaller subspace. In contrast, the proposed method takes approximately 57% of the time of the conventional method, as shown in Figure 8, R G This allows us to acquire a vast exploration space equivalent to that of [another game]. [Table 6]

[0035] Based on the above, numerical simulations using a galvanometer scanner as the controlled object have demonstrated that the parameter search space calculation method based on Bayesian optimization proposed in this invention can calculate a larger parameter search space in a shorter time than the method utilizing the solvability of a constrained optimization problem proposed in Non-Patent Document 1. In this embodiment, the results of calculating the parameter search space independently of automatic tuning are shown, but it is also possible to calculate a parameter search space that satisfies a predetermined stability in parallel with automatic tuning. In that case, this can be achieved by combining the objective function related to control performance and equation (9) related to stability to form the acquisition function. [Industrial applicability]

[0036] The present invention makes it possible to adjust feedback control parameters in, for example, machine tools and conveying equipment for logistics. [Explanation of symbols]

[0037] 1: Stable space R 2: Region where H^(ρ)≧0 is estimated to satisfy a given stability. 11: True stability H(ρ) 12: Estimation of stability H^(ρ) 13: H^(ρ)±κσ(ρ) 14: Known Data D H 21:|H^(ρ)| 22: Acquisition Function 31:S G (point) 32:T G (point) 41:T G Open-loop frequency characteristics of the internal parameters 42:S G Open-loop frequency characteristics of the internal parameters

Claims

1. A method for calculating a parameter search space in the automatic adjustment of feedback control parameters, the initial feedback parameter ρ ｉｎｉ and the stability H(ρ ｉｎｉ ) calculated for the ρ ｉｎｉ are used to form the initial data D Ｈ =(ρ ｉｎｉ ; H(ρ ｉｎｉ )), and step 1 of initializing the iteration counter i of the estimation to i:=1; step 2 of calculating the estimated stability H^(ρ) and its uncertainty σ ２ (ρ) assuming that the stability follows a statistical distribution; step 3 of calculating an acquisition function and setting the feedback parameter ρ ｎｅｘｔ at which the acquisition function is minimized as the next search point; step 4 of adding the ρ ｎｅｘｔ and the stability H(ρ ｎｅｘｔ ) to the D Ｈ ; if the i is less than the maximum number of iterations n, then return to step 2 with i:=i + 1, and end when i = n, and a method for calculating a parameter search space characterized in that the space where H^(ρ)≥0 is set as the search space T.

2. The parameter search space calculation method according to claim 1, characterized in that, in step 2, the statistical distribution assumed to follow the stability estimate H^(ρ) is a Gaussian distribution.

3. H(ρ １ ), H(ρ ２ ), ...H(ρ Ｎ When H(ρ) is known, １ ), H(ρ ２ ), ...H(ρ Ｎ The method for calculating the control parameter search space according to claim 1, characterized in that the process follows a Gaussian process.

4. The parameter search space calculation method according to any one of claims 1 to 3, characterized in that, in step 3, the acquisition function is a function that estimates the boundary of a parameter space that satisfies a predetermined stability.

5. The method for calculating the parameter search space according to any one of claims 1 to 3, characterized in that, in step 3, the minimization of the acquisition function is performed by Bayesian optimization.

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