Data processing apparatus, control system, data processing method, and program

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The data processing apparatus and method address the challenge of estimating the controllability Gramian for systems with unknown models by employing a data-driven, continuous-time approach, overcoming the limitations of discrete-time models and enhancing control system analysis accuracy.

JP7688947B2Active Publication Date: 2025-06-05THE JAPAN SCI & TECH AGENCY

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Patent Information

Application Number: JP2023545687
Authority / Receiving Office: JP · JP
Patent Type: Patents
Current Assignee / Owner
Priority Date: 2021-09-03
Filing Date: 2022-09-02
Publication Date: 2025-06-05
Estimated Expiration: 2042-09-02

AI Technical Summary

Technical Problem

Existing methods struggle to estimate the controllability Gramian for systems with unknown mathematical models, especially in continuous-time systems, as they rely on discrete-time models that lose physical information and fail to utilize prior knowledge about continuous-time systems.

Method used

A data processing apparatus and method that estimates the controllability Gramian using a data-driven approach for continuous-time systems by acquiring time-series state data, defining matrices based on this data, and numerically solving linear equations to estimate the controllability Gramian, even in the presence of noise.

Benefits of technology

Enables the estimation of the controllability Gramian for systems with unknown mathematical models, effectively addressing the limitations of discrete-time models by leveraging continuous-time data and prior knowledge, thus improving the accuracy and effectiveness of control system analysis.

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Abstract

This data processing device 1 estimates extrema [formula 140] G(∞) in [formula 144] t=∞ for a controllability grammian [formula 138] G(t) defined by [formula 2] (BB) when [formula 1] (AA) is established, where [formula 142] X(t) is an n-dimensional vector representing the state of an item being controlled, [formula 139] u(t) is an m-dimensional vector representing a control input, A is an unknown n×n matrix, and B is a known nXm matrix, the data processing device comprising: a data acquisition unit 10 for acquiring a set of chronological state data [formula 5] x([t 11, t 12], x11), x([t 2 1, t 2 2], x11), etc., x([t q 1, t q2], xq1) in q time sections [formula 4] [t i 1, t i 2] (i=1, 2, etc., q) when [formula 39] u(t)Ξ0 holds; a controllability grammian calculation unit 12 for estimating [formula 146] G(∞)=X by deriving, through numerical calculation, a solution X to a linear equation [formula 6] (CC) in relation to [formula 157] E(t):=[x(t+t 11, t 11, x 11)x(t+t 2 1, t 2 1, x 2 1), etc., x(t+t n 1, t n 1, x n 1)] and [formula 158] E 0:=[x 11 x 21, etc., x n 1)]; and an output unit 15 for outputting an input matrix when the controllability grammian is maximum, on the basis of an estimated maximization condition.

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Description

Technical Field

[0001] The present invention relates to a data processing apparatus, a control system, a data processing method, and a program.

Background Art

[0002] As an index indicating the controllability of a system, the controllability Gramian is known (see, for example, Non-Patent Documents 1 to 6). Whether a system is controllable can be determined by whether the controllability Gramian is regular. Furthermore, the magnitude of the eigenvalue of the controllability Gramian quantitatively indicates the degree of influence of the input on the state.

Prior Art Documents

Non-Patent Documents

[0003]

Non-Patent Document 1

Non-Patent Document 2

Non-Patent Document 3

Non-Patent Document 8

Non-Patent Document 9

Non-Patent Document 10

Summary of the Invention

Problems to be Solved by the Invention

[0004] The controllability Gramian can be easily calculated if the mathematical model of the system is known (hereinafter, the calculation method based on the mathematical model of the system is called the “model-based method”). However, the required amount of data for system modeling is not always available. In such cases, the mathematical model of the system cannot be obtained.

[0005] In contrast to the model-based method, a method that calculates based on the data of the state trajectory of the system without using the mathematical model is called the “data-driven method”. Since the data-driven method does not require identifying the mathematical model of the system, it has the advantage of requiring fewer decision variables than the model-based method.

[0006] A method of estimating a controllability Gramian using a data-driven approach (for example, Non-Patent Document 7) or a method of maximizing a controllability Gramian (for example, Non-Patent Document 8) is known. All of these methods are discrete-time models formulated based on discrete time. However, physical information contained in data is well represented in a continuous-time model formulated based on continuous time, whereas there is a problem that the visibility deteriorates in a discrete-time model. That is, there is a problem that it is difficult to utilize prior knowledge regarding the characteristics of a continuous-time system in a conventional discrete-time model.

[0007] The present invention has been made in view of such problems, and an object thereof is to estimate a controllability Gramian for a system with an unknown mathematical model using a data-driven approach for a continuous-time system.

Means for Solving the Problems

[0008] In order to solve the above problems, a data processing apparatus according to an aspect of the present invention

Number

[0009] In one embodiment, a set of time-series state data may include noise. The controllability Gramian calculation unit solves the linear equation

Number

[0010] In one embodiment, the controllability Gramian calculation unit performs numerical calculation using prior knowledge about the signs of some or all of the matrix components of the solution X.

[0011] Another aspect of the present invention is also a data processing device. This device

Number

[0012] In a certain embodiment,

Number

[0013] In one embodiment,

Number

[0014] Another aspect of the present invention is a control system. This control system is for controlling a controlled object, and includes a sensor that detects a set of time-series state data from the controlled object, the aforementioned data processing device, and a control unit that controls the controlled object. The sensor transmits the detected set of time-series state data to the data acquisition unit of the data processing device, and the data processing device transmits the estimated

Number

[0015] Another aspect of the present invention is a data processing method. This method

Number

[0016] Yet another aspect of the present invention is also a data processing method. This method

Number

[0017] Another aspect of the present invention is a program. This program

Number

[0018] Yet another aspect of the present invention is also a program. This program

Number

[0019] Yet another aspect of the present invention is a data processing device. This data processing device

Number

[0020] In addition, any combination of the above components, and those obtained by converting the expression of the present invention among an apparatus, a method, a system, a recording medium, a computer program, etc. are also effective as aspects of the present invention.

Effects of the Invention

[0021] According to the present invention, for a system with an unknown mathematical model, the controllability Gramian can be estimated using a data-driven method for a continuous-time system.

Brief Description of the Drawings

[0022]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

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Figure 8

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Figure 9

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Figure 10

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Figure 11

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Figure 12

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Figure 13

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Figure 14

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Figure 15

Number

Figure 16

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Figure 17

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Figure 18

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Figure 19

Mode for Carrying Out the Invention

[0023] Hereinafter, the present invention will be described with reference to the drawings based on preferred embodiments. The embodiments are illustrative and not restrictive of the invention. Not all features and combinations thereof described in the embodiments are necessarily essential to the invention. The same or equivalent components, members, and processes shown in each drawing are denoted by the same reference numerals, and repeated explanations are omitted as appropriate. Also, the scales and shapes of the respective parts shown in each figure are set for convenience in order to facilitate the explanation, and are not to be construed restrictively unless otherwise specified. Further, when terms such as "first", "second", etc. are used in this specification or claims, unless otherwise specified, these terms do not represent any order or importance, but are only for distinguishing one configuration from another. Also, in each drawing, some members that are not important for explaining the embodiments are omitted from the display.

[0024] Before describing specific embodiments, first, the basic findings will be described. [Definition of Symbols] Hereinafter, the definitions of mathematical symbols used in this specification will be described. R: Real number

Math

Mathematics

[0025] [Linear System and Controllability Gramian] Consider the following linear system.

Mathematics

Number

[0026] The following lemma is fundamental to the calculation of the controllability Gramian (see, for example, Non-Patent Document 9).

[0027] (Lemma 1) For the system shown in (1), consider the Lyapunov equation

Number

[0028] [Data-Driven Estimation and Maximization Problems] If the mathematical model of (1) is obtained, the controllability Gramian

Number

[0029]

Number

[0030] Based on the above, the problem of estimating and maximizing [Number theory] is organized as follows. (Problem 1) Regarding the system shown in (1), assume that the matrix A is an unknown full-rank matrix. Given a set of data [Number theory] consisting of q segments of state trajectories. However [Number theory] it is (i) When a known input matrix B is given, [Mathematics] is estimated. (ii) [Mathematics] Under the condition of [Mathematics] the input matrix [Mathematics] is obtained. Here [Mathematics] is a predetermined set of possible input matrices.

[0031] Pay attention to the following four points. First, in Problem 1(i), it is assumed that the input matrix B is known. On the other hand, the main purpose of Problem 1(ii) is to find the optimal input channel for controlling the system. From this perspective, Problem 1(i) corresponds to the performance analysis of the input matrix B. That is, in this case, [Mathematics] the input matrix B will be evaluated for the performance measurement of

[0032] Second, generally [Mathematics] increases with the norm of B. Therefore, for considering the maximization of [Mathematics] in Problem 1(ii), [Mathematics] the condition of

[0033] Thirdly, B can be flexibly selected, but the following two settings are fundamental.

Number

[0034] Fourthly, the obtained data set is a continuous-time signal and is assumed to be noise-free. However, even when this assumption does not hold, the method described in this specification is applicable to the above problems. This will be described later.

[0035] [Solution of the data-driven Lyapunov equation in stability analysis]

[0036] Consider the system of (1) when B = 0.

Number

[0037] Here, although A is unknown, consider solving (5) when a set of state trajectory data is given. This problem is formulated as follows. (Problem 2) Regarding the system shown in (4), assume that the matrix A is a full-rank matrix and is unknown. The symmetric matrix

Number

[0038] The solution to Problem 2 is given as follows. Consider the state trajectory x from the initial state x(0) of (4). By creating a quadratic form of x(t) with respect to both sides of (5),

Number

[0039] By applying the data set of Problem 2 to (9), the following q equations are obtained.

Number

[0040] As described above, the solution to Problem 2 is obtained as follows.

[0041] (Lemma 2) Consider Problem 2. [Number] If so, there exists a unique solution Y in (10), which is equal to the solution to Problem 2. (End of Lemma 2)

[0042] Lemma 2 shows that the solution to Problem 2 can be obtained by solving the linear equation (10) (or equivalently (11)) using the set of data of the state trajectory of the system shown in (4).

[0043] [Estimation of Controllability Gramian] Hereinafter, the solution method for Problem 1(i) will be described.

[0044] [Data Conversion] According to the aforementioned Lemma 1, Problem 1(i) is reduced to solving the Lyapunov equation (3) by a data-driven approach. Hereinafter, the solution method for (3) in the database is obtained using the same method as that for obtaining (10). Similar to (6), a quadratic form of the left side of (3) and the state x(t) when B = 0 in (1) is created. [Number] However, this time, unlike the case of (8), generally, [Number] (and [Number] are) [Number] cannot be replaced. This is because x(t) is the state of (1), and generally [Number] Therefore,

[0045] First, introduce the following system defined by the transpose matrix of matrix A.

Number

[0046] is used.

Number

[0047] However, here, the set of data of the state trajectory in (13) in Problem 1 has not been obtained. This difficulty can be solved by the following result.

[0048] (Lemma 3) In Problem 1

Number

[0049] (Proof) Given

Number

Mathematics

[0050] By this lemma, the data set of Problem 1 can be converted into the data set of the state trajectory of (13). Here

Mathematics

[0051] [Data-driven estimation] (16), (19) and [Number] yields a linear equation [Number] is obtained. This is [Number] and vector [Number] with respect to [Number] is expressed as.

[0052] As a result, the solution to Problem 1 is obtained as follows.

[0053] (Theorem 1) Consider Problem 1(i). [Number] is assumed. [Number] If so, a unique solution X exists in (20), which is equal to the solution to Problem 1 (end of Theorem 1).

[0054] According to Theorem 1, the solution to Problem 1(i) is given as the solution to the linear equation (20).

[0055] In the above, it was assumed that the set of data obtained was a continuous-time signal and contained no noise. However, this is not the only case, and as shown below, the above results can also be applied to data containing noise.

[0056] Data

Number

[0057] Next, assume that a set of data is given as non-periodic sample data of the state trajectory. Since the left side of (20) consists of the endpoints of each state trajectory, it can be calculated from the sample data of the state trajectory. On the other hand, the right side of (20) is the integral of the quadratic form of the state trajectory. This can be approximately calculated from the sample data, for example, by using the trapezoidal rule. Using these principles, an approximate solution to the above problem can be obtained.

[0058] [Maximization of the controllability Gramian] The solution method for Problem 1 (ii) will be described below.

[0059] [Characterization of the input matrix] The controllability Gramian with respect to the input matrix B [Number] shows the maximization of

[0060] (Lemma 4) Regarding the system shown in (1), assume that the matrix A is a full-rank matrix. Consider the following maximization problem. [Number] Here [Number] is a given set of possible input matrices. [Number] Let (since matrix A is a full-rank matrix, this is a finite positive definite value). At this time, the following proposition holds. (i) [Number] Let be the unit eigenvector corresponding to the largest eigenvalue of

Number

[0061] (Proof) (i) (See, for example, Non-Patent Document 10). (ii) Let the i-th diagonal component of matrix B be [Number] Set it as. From (2) and the properties of the trace, [Number] is obtained. For any [Number] with respect to, [Number] is a diagonal matrix. Thus, from (28), [Number] (That is [Number] ) For any [Number] with respect to, [Number] holds. On the other hand, from (29) and [Number] the definition of, [Number] (27) holds with respect to. Furthermore [Number] is (Proof completed).

[0062] [Data-driven maximization problem] Lemma 4 states that [Math.] the input matrix that maximizes is, in (25), [Math.] is characterized by. On the other hand [Math.] is [Math.] equal to the unique solution of the Lyapunov equation (5) with respect to. Thus, from Lemma 2 and Lemma 4, the following holds.

[0063] (Theorem 2) Consider Problem 1 (ii). Assume that (12) holds for the data set given in Problem 1. [Math.] be [Math.] with respect to [Math.] be the solution of. Further, in Lemma 4 [Math.] with respect to [Math.] and [Math.] define in the same way as, [Math.] and

Number

[0064] Theorem 2 shows that the solution of Problem 1 (ii) can be obtained by solving the linear equation (10) and creating the input matrix by the method of Lemma 4.

[0065] [First Embodiment] FIG. 1 is a functional block diagram of a data processing apparatus 1 according to the first embodiment. The data processing apparatus 1 includes a data acquisition unit 10, a controllability Gramian calculation unit 12, and an output unit 15. The data processing apparatus 1

Number

Mathematics

[0066] The data acquisition unit 10 acquires a set of time-series state data in a plurality of time intervals. In the following description, the data acquisition unit 10 acquires a set of time-series state data in q time intervals

Mathematics

[0067] The controllability Gramian calculation unit 12, based on the set of state data acquired by the data acquisition unit 10

Mathematics

[0068] The output unit 15 outputs the controllability Gramian estimated by the controllability Gramian calculation unit 12.

[0069] Here, the controlled object is, for example, as follows. (Example 1) Power system. The time series data is the power generation amount, power consumption amount, temperature, humidity, etc. of each power plant. (Example 2) Robot. The time series data is the load, posture, external force, motion state, etc. (Example 3) Human body. The time series data is blood pressure, pulse, body temperature, blood components, etc. (Example 4) Chemical plant. The time series data is temperature, humidity, atmospheric pressure, dust amount in the air, etc. (Example 5) Device to be maintained. The time series data is the failure alarm frequency, electrical resistance, metal fatigue degree, heat generation amount, etc. These controlled objects are configured as a network system. The mathematical model describing the time series data output from here is complex, and generally the mathematical model itself is unknown. In contrast, by estimating and outputting the controllability Gramian using this embodiment, it is possible to determine which node of the controlled object configured as a network system should be controlled as an input channel while keeping the mathematical model unknown. The above features are common to all the following embodiments.

[0070] According to this embodiment, for a system with an unknown mathematical model, the controllability Gramian can be estimated using a data-driven approach based on the acquired set of continuous-time state data.

[0071] [Second Embodiment] The set of time-series state data acquired by the data acquisition unit may include noise. In this case, instead of (20), the controllability Gramian calculation unit 12 [Number] calculates an approximate solution to Problem 1(i) from [Number] to estimate [Number] and [Number] are defined with respect to p pieces of data and [Number] [Number]

[0072] According to this embodiment, even when the state data includes noise, the controllability Gramian can be estimated using a data-driven approach.

[0073] [Third Embodiment]

[0074] When the state data includes noise, knowledge about the signs of some or all of the matrix components of the solution X of (23) may be given as prior knowledge. In this case, the controllability Gramian calculation unit 12 performs numerical calculations using the prior knowledge.

[0075] According to this embodiment, even when the state data includes noise, the controllability Gramian can be estimated more accurately using a data-driven approach.

[0076] [Fourth Embodiment] FIG. 2 is a functional block diagram of a data processing apparatus 1 according to the fourth embodiment. The data processing apparatus 2 includes a data acquisition unit 10, a maximization condition calculation unit 14, and an output unit 15. The data processing apparatus 2 [Number] is an n-dimensional vector representing the state of the control target, [Number] is an m-dimensional vector representing the control input, A is an unknown n×n matrix, and B is a known n×m matrix. [Number] When it holds, [Number] the controllability Gramian defined by [Number] of [Number] the limit in [Number] the trace of [Number] estimates the matrix B when it reaches the maximum.

[0077] The data acquisition unit 10 acquires a set of time-series state data in a plurality of time intervals. In the following description, the data acquisition unit 10 has q time intervals [Mathematics] A set of time-series state data in [Mathematics] is to be obtained.

[0078] Based on the set of state data acquired by the data acquisition unit 10, the maximization condition calculation unit 14 [Mathematics] Based on, the solution of the linear equation [Mathematics] of [Mathematics] By numerically calculating [Mathematics] calculate the matrix B when is maximized.

[0079] Based on the maximization condition estimated by the maximization condition calculation unit 14, the output unit 15 outputs the input matrix when the controllability Gramian is maximized.

[0080] According to this embodiment, for a system with an unknown mathematical model, based on the acquired set of continuous-time state data, the input matrix when the controllability Gramian is maximized can be estimated using a data-driven approach.

[0081] [Fifth Embodiment] In the fifth embodiment, [Mathematics] When, the maximization condition calculation unit 14 [Mathematics] By calculating the unit eigenvector corresponding to the maximum eigenvalue of the first input matrix [Number] is obtained.

[0082] According to this embodiment, [Number] when it is, an effective input matrix can be obtained.

[0083] [Sixth Embodiment] In the sixth embodiment, [Number] when it is, [Number] when the maximum among the diagonal components of is the (k, k) component, the maximization condition calculation unit calculates an n×n matrix in which the (k, k) component is 1 and the other components are 0 to the second input matrix [Number] is obtained.

[0084] According to this embodiment, [Number] when it is, an effective input matrix can be obtained.

[0085] [Seventh Embodiment (1)] Figure 3 is a functional block diagram of a control system 3 according to the seventh embodiment. The control system 3 includes a sensor 16, a data processing device 1, and a control unit 18. The data processing device 1 includes a data acquisition unit 10 and a controllability Gramian calculation unit 12. That is, the data processing device 1 is configured and operates in the same manner as the data processing device 1 in FIG. 1. The control system 3 controls an external controlled object 100 by inputting a control signal to the controlled object 100.

[0086] The sensor 16 detects a set of time-series state data from the controlled object 100. Then, the sensor 16 transmits the detected set of state data to the data acquisition unit 10 of the data processing device 1. The data processing device 1

Number

[0087] According to this embodiment, time-series state data from an external controlled object is detected, the controllability Gramian is estimated using a data-driven approach based on the detected set of state data, and the controlled object can be appropriately controlled based on the estimated controllability Gramian.

[0088] [Seventh Embodiment (2)] FIG. 19 is a functional block diagram of a control system 5 according to a modification of the seventh embodiment. The control system 5 includes a sensor 16, a data processing device 1, and a control unit 19. The data processing device 1 includes a data acquisition unit 10 and a controllability Gramian calculation unit 12. The control unit 19 includes a control input value determination unit 191 and an input channel determination unit 192. That is, the control system 5 is different from the control system 3 in FIG. 3 in that the control unit 19 is configured to include the control input value determination unit 191 and the input channel determination unit 192. Other configurations of the control system 5 are common to those of the control system 3. Hereinafter, the description will focus on the parts different from the control system 3, and redundant descriptions will be omitted.

[0089] A set of state data detected by the sensor 16 is transmitted to the control input value determination unit 191. Based on this set of state data, the control input value determination unit 191 determines what kind of control should be performed on the control target determined by the input channel determination unit 192. The control input value determination unit 191 transmits the determined control content to the input channel determination unit 192.

[0090] The estimated controllability Gramian is transmitted to the input channel determination unit 192. Based on this controllability Gramian, the input channel determination unit 192 determines which control target among the control targets 100 should be controlled. The input channel determination unit 192 selects the determined control target and executes the control determined by the control input value determination unit 191 for the selected control target.

[0091] According to the present embodiment, time-series state data from an external control target is detected, the controllability Gramian is estimated using a data-driven method based on the detected set of state data, an appropriate control target is selected based on the estimated controllability Gramian, and the selected control target can be appropriately controlled.

[0092] [Eighth Embodiment] FIG. 4 is a functional block diagram of a control system 4 according to the eighth embodiment. The control system 4 includes a sensor 16, a data processing device 2, and a control unit 18. The data processing device 2 includes a data acquisition unit 10 and a maximization condition calculation unit 14. That is, the control system 2 is configured in the same manner as the data processing device 2 in FIG. 2 and operates in the same manner. The control system 4 controls the external controlled object 100 by inputting a control signal to the external controlled object 100.

[0093] The sensor 16 detects a set of time-series state data from the controlled object 100. Then, the sensor 16 transmits the detected set of state data to the data acquisition unit 10 of the data processing device 1. The data processing device 2

Number

[0094] According to the present embodiment, time-series state data from an external controlled object is detected, and based on the detected set of state data, the input matrix when the controllability Gramian becomes maximum is estimated using a data-driven approach, and the controlled object can be appropriately controlled based on the estimated input matrix.

[0095] [Ninth Embodiment] FIG. 5 is a flowchart showing the processing procedure of a data processing method according to the ninth embodiment. This method

Number

[0096] In step S1, this method

Number

[0097] In step S2, this method

Number

Mathematics

[0098] In step S3, this method outputs the controllability Gramian estimated in step S2.

[0099] According to this embodiment, for a system with an unknown mathematical model, based on the acquired set of continuous-time state data, the controllability Gramian can be estimated using a data-driven approach with a computer.

[0100] [Tenth Embodiment] FIG. 6 is a flowchart showing the processing procedure of the data processing method according to the tenth embodiment. This method

Mathematics

[0101] In step S1, this method

Mathematics

[0102] In step S4, this method

Mathematics

[0103] In step S5, this method outputs the input matrix when the controllability Gramian is maximized based on the maximization condition estimated in step S4.

[0104] According to this embodiment, for a system with an unknown mathematical model, based on the acquired set of continuous-time state data, the input matrix when the controllability Gramian is maximized can be estimated using a computer by means of a data-driven approach.

[0105] [11th Embodiment] The 11th embodiment is a program. This program

Number

[0106] According to this embodiment, for a system with an unknown mathematical model, software for estimating the controllability Gramian using a data-driven method can be implemented as a program based on the acquired set of continuous-time state data.

[0107] [Embodiment 12] Embodiment 12 is a program. This program

Number

Mathematics

[0108] is maximized, and a step of outputting an input matrix when the controllability Gramian is maximized based on the estimated maximization condition are executed by a computer.

[0109] [Verification 1] Hereinafter, results are shown according to the first embodiment for a set of time-series state data without noise. In the system shown in (1) [Number] Find the solution to Problem 1(i) when [Number] [Number] Let

[0110] For comparison, the controllability Gramian [Number] obtained using (3) and Matlab (registered trademark)'s "lyap" has the following true values. [Number]

[0111] Figures 7 to 12 show time-series data without noise. Specifically, Figure 7 shows the time-series data in the time interval

Number

[0112] The data acquisition unit 10 of the data processing apparatus 1 in FIG. 1 acquires a set of time-series state data in six time intervals shown in FIGS. 7 to 12

Number

[0113] The controllability Gramian calculation unit 12 obtains the following values by numerically solving (22).

Number

[0114] [Verification 2] Next, the results obtained according to the second embodiment are shown for a set of time-series state data including noise. FIGS. 13 to 18 show time-series data without noise. Specifically, FIG. 13 shows the time-series data in the time interval

Number

[0115] The controllability Gramian calculation unit 12 obtains the following values by numerically solving (23). [Number] Although this result is worse compared to (31), it generally agrees with the true value (30).

[0116] [Verification 3] Regarding the set of time-series state data including noise, the results actually obtained using the third embodiment are described. The set of time-series state data in this case is also shown in Figures 13 to 18 as described above.

[0117] Here [Number] Suppose that prior knowledge is obtained that has the following pattern of signs. [Number] However, * indicates that the sign is unknown. Then [Number] The problem of estimating is reduced to the following optimization problem. [Number] Here [Number] and [Number] relates to (21) and is used in (23).

Number

[0118] By numerically calculating (33),

Number

[0119] [Verification 4] Hereinafter, using the data of FIGS. 7 to 12, the results obtained according to the fourth embodiment will be described. In this case,

Number

[0120] [Verification 5] The results obtained according to the fifth embodiment are described below using the data in FIGS. 7 to 12. [Number] In the case of, by numerical calculation according to the embodiment, [Number] Regarding, the first input matrix [Number] is obtained. With this first input matrix, [Number] takes the maximum value [Number] This For the result of, as a comparative example, an input matrix where all components are 1 / 3 [Number] is used, [Number] is [Number] This is significantly smaller than [Number] obtained in the embodiment.

[0121] [Verification 6] The results obtained according to the sixth embodiment are described below using the data in FIGS. 7 to 12. [Number] In the case of, by numerical calculation according to the embodiment,

Number

[0122] In Table 1,

Number

Table 1

[0123] [Modes when the action changes] In the modes described above, one of the goals was to estimate the index of "ease of control" when the input channels to each node of the network were fixed (Problem 1 (i)). That is, the following problem was discussed. (Problem 1) Regarding the system represented by Equation (1), assume that the matrix A is an unknown full-rank matrix. A set of data consisting of q segments of state trajectories

Number

[0124] That is, in the embodiment described above, it is assumed that the amount of action (an index indicating the connection strength between nodes or the amount of distributed data) between each node constituting the network is all fixed values. In this case, it was possible to know "how to select the input channel in order to improve the ease of control". On the other hand, hereinafter, the estimation of the ease of control when the amount of action is adjusted (the estimation of the controllability Gramian when the amount of action is changed) will be described. The goal is to know "how to adjust the amount of action in order to improve the ease of control".

[0125] Hereinafter, consider the estimation of the controllability Gramian when a change amount Δ is added to an unknown matrix A. This problem can be formulated as follows. (Problem 3) Consider the system of Equation (1). However, [Number] is known, but [Number] is assumed to be unknown. Also, assume that data of N state trajectories are given to this system as follows. [Number] However, [Number] is. Also [Number] is such that [Number] is arbitrarily given. At this time [Number] Find.

[0126] Problem 3 is to estimate the value of the controllability Gramian from data when the A matrix is changed by Δ for the system of equation (1). This problem appears, for example, when applying state feedback to a system or adjusting the connection strength between nodes in a network system.

[0127] [Data-driven solution of Lyapunov equation] Below, as a preparation for Problem 13, a method for solving the Lyapunov equation using the state trajectory of the system will be described. Consider the following system. [Number] Here [Number] is a stable matrix. Also, the state trajectory of the system of equation (35) is, as described above, [Number] is represented by. For this system, consider the Lyapunov equation [Number] where [Number] is [Number] is a matrix for which becomes stable, and [Number] is a symmetric matrix. At this time, it is known that equation (36) has a unique solution.

[0128] At this time, consider the following problem. (Problem 4) Consider the system of equation (35). However

Number

[0129] The solution to Problem 4 is given as follows. First, consider the quadratic form with respect to the state orbit

Number

[0130] Next, consider rewriting Equation (40) in a matrix form. First, Equation (39) can be written as, using lvec and rvec, [Number] Considering this, Equation (40) can be equivalently rewritten as [Number] where [Number] is a matrix determined by the state trajectory data of the system in Equation (35), and its i-th row [Number] is respectively [Number] [Number] is given by

[0131] At this time, the solution to Problem 4 is obtained as follows. (Theorem 3) Consider Problem 4. If [Number] then the solution to Equation (40) is unique within the range of symmetric matrices and is equal to the solution to Problem 4. (End of Theorem 3)

[0132] [Data-driven estimation of the controllability Gramian] Next, consider the solution to Problem 3.

[0133] [Data conversion] For the dual system of equation (35) [Eqn.] If state trajectory data is available, using Theorem 3 [Eqn.] can be calculated. This fact is shown as follows. From Lemma 1, the controllability g ramian [Eqn.] is [Eqn.] obtained as the unique solution of. Here, when equation (43) is for equation (36) for [Eqn.] [Eqn.] [Eqn.] Note that corresponding to the case where, following the above discussion [Eqn.] is obtained. Therefore, by applying the state trajectory of the system of equation (13) to equation (44), [Eqn.] a linear equation with a solution can be constructed from only the state trajectory data.

[0134] However, in Problem 3, the state trajectory of the system of Equation (13) is

Number

[0135] (Lemma 5) The state trajectory of the system of Equation (13) is expressed as

Number

[0136] [Data-driven estimation] Using the above results, give the solution to Problem 3. First, the function

Number

[0137] Therefore, the solution to Problem 3 is obtained as follows. (Theorem 4) Consider Problem 3. Assume that N ≥ n. The E in equation (45) 0 is regular, and

Number

[0138] [Embodiment 13] Similar to the first embodiment, the data processing apparatus according to the thirteenth embodiment will be described with reference to FIG. 1. FIG. 1 is a functional block diagram of a data processing apparatus 1 according to the thirteenth embodiment. The data processing apparatus 1 includes a data acquisition unit 10, a controllability Gramian calculation unit 12, and an output unit 15. The data processing apparatus 1 [Number] is an n-dimensional vector representing the state of the control target, [Number] is an m-dimensional vector representing the control input, A is an unknown n×n matrix, and B is a known n×m matrix. [Number] When it holds, [Number] the limit of the controllability Gramian of matrix A defined by [Number] is [Number] estimated.

[0139] The data acquisition unit 10 acquires a set of time-series state data in a plurality of time intervals. In the following description, the data acquisition unit 10 acquires a set of time-series state data in q time intervals [Number] in [Number] shall be obtained.

[0140] Based on the set of state data acquired by the data acquisition unit 10, the controllability Gramian calculation unit 12 [Number] defines [Number] represented by [Number] and calculates [Number] [Number] and solves the linear equation Linear equation [Number] by numerically solving the solution X of [Number] and estimates

[0141] The output unit 15 outputs the controllability Gramian estimated by the controllability Gramian calculation unit 12.

[0142] According to this embodiment, the controllability Gramian when the action amount between each node constituting the network changes can be estimated.

[0143] As described above, the present invention has been described based on embodiments. It is understood by those skilled in the art that these embodiments are examples, and various modifications are possible for each combination of their components and each processing process, and such modifications are also within the scope of the present invention.

[0144] Any combination of the above-described embodiments and variations is also useful as an embodiment of the present invention. The new embodiments resulting from the combination have the effects of each of the combined embodiments and variations.

Industrial Applicability

[0145] The principle of the present invention can be applied to the control of systems in various fields as follows. (Application Example 1) When the control target is a power system, optimal power supply control can be performed from time-series data such as the power generation amount, power consumption amount, temperature, and humidity of each power plant. (Application Example 2) When the control target is a robot, optimal posture control can be performed from time-series data such as load, posture, external force, and motion state. (Application Example 3) When the control target is a human body, optimal dosing control can be performed from time-series data such as blood pressure, pulse, body temperature, and blood components. (Application Example 4) When the control target is a chemical plant, optimal operation control can be performed from time-series data such as temperature, humidity, atmospheric pressure, and the amount of dust in the air. (Application Example 5) When the control target is the maintenance of a device, optimal maintenance control can be performed from time-series data such as the frequency of failure alarms, electrical resistance, metal fatigue degree, and calorific value.

Explanation of Reference Numerals

[0146] 1 ··· Data processing device. 2 ··· Data processing device. 3 ··· Control system. 4 ··· Control system. 5 ··· Control system. 10 ··· Data acquisition unit. 12 ··· Controllability Gramian calculation unit. 14 ··· Maximization condition calculation unit. 15 ··· Output unit. 16 ··· Sensor. 18 ··· Control unit. 19 ··· Control unit. 100 ··· Control target. 191 ··· Control input value determination unit. 192 ··· Input channel determination unit. S1 ··· Step of acquiring a set of time-series state data. S2 ··· Step of estimating the controllability Gramian. S3 ··· Step of outputting the controllability Gramian. S4 ··· Step of estimating the condition when the controllability Gramian is maximized. S5 ··· Step of outputting the input matrix when the controllability Gramian is maximized.

Claims

1. 【No. 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, and B be a known n × m matrix. [0010] When holds, [0025] The controllability gramian defined by [Number 138] of [Number 144] Limit in [Number 140] A data processing device for estimating [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition unit for acquiring [74] Represented by [Number 75] Define [Number 81] [Number 89] Calculate [Number 157] [Number 158] Regarding linear equation [006] By calculating the solution X of [Number 146] A controllability Gramian calculator that estimates and an output unit that outputs the estimated controllability gramian.

2. the set of time-series state data includes noise; The controllability Gramian calculation unit calculates a linear equation [006] Instead, the linear equation [Number 108] By calculating the solution X of [Number 146] 2. The data processing apparatus according to claim 1, further comprising:

3. The controllability gramian calculation unit 3. The data processing apparatus according to claim 2, wherein the numerical calculation is performed using prior knowledge regarding the signs of some or all of the matrix elements of the solution X.

4. 【Number 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, and B be a known n × m matrix. [0010] When holds, [0025] The controllability gramian defined by [Number 138] of [Number 144] Limit in [Number 140] Trace [Number 141] A data processing device for estimating a matrix B when is maximized, comprising: [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition unit for acquiring the data; [0070] The linear equation [0080] Solution [0097] By calculating numerically, [Number 141] and an output unit that outputs an input matrix when the controllability Gramian is maximized based on the estimated maximization condition.

5. 【Fig. 47】 When this is the case, the maximization condition calculation unit [0090] By computing the unit eigenvector corresponding to the largest eigenvalue of First Input Matrix ##EQU00011## 5. The data processing apparatus according to claim 4, wherein the data processing apparatus calculates:

6. 【Fig. 49】 When [0097] If the maximum diagonal element of is the (k, k) element, then The maximization condition calculation unit calculates an n×n matrix in which the (k, k) component is 1 and the other components are 0, Second Input Matrix ##EQU00012## 5. The data processing apparatus according to claim 4, wherein the data processing apparatus calculates:

7. A control system for controlling a control target, comprising: A sensor for detecting a set of time-series state data from the controlled object; A data processing device according to any one of claims 1 to 3, A control unit that controls the control target, The sensor transmits a set of detected time-series status data to a data acquisition unit of the data processing device; The data processing device estimates [Number 146] to the control unit, The control unit is [Number 146] A control system comprising: a control unit that controls the controlled object based on the control unit;

8. A control system for controlling a control target, comprising: A sensor for detecting a set of time-series state data from the controlled object; A data processing device according to any one of claims 4 to 6, A control unit that controls the control target, The sensor transmits a set of detected time-series status data to a data acquisition unit of the data processing device; The data processing device estimates [Number 141] transmits the matrix B when is maximized to the control unit; The control unit is [Number 146] and controlling the controlled object using the control signal.

9. 【Fig. 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, and B be a known n × m matrix. [0010] When holds, [0025] The controllability gramian defined by [Number 138] of [Number 144] Limit in [Number 140] A data processing method for estimating Using the data acquisition unit, [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition step for acquiring Using the controllability Gramian calculator, [74] Represented by [Number 75] Define [Number 81] [Number 89] Calculate [Number 157] [Number 158] With respect to the linear equation [006] By calculating the solution X of [Number 146] A computational step of estimating Using the output section, and outputting the estimated controllability gramian.

10. 【Number 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, and B be a known n × m matrix. [0010] When holds, [0025] The controllability gramian defined by [Number 138] of [Number 144] Limit in [Number 140] Trace [Number 141] A data processing method for estimating a matrix B when is maximized, comprising the steps of: Using the data acquisition unit, [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition step for acquiring Using the maximization condition calculation unit, [0070] The linear equation [0080] Solution [0090] By calculating numerically, [Number 141] A calculation step of estimating the matrix B when and outputting, using an output unit, an input matrix when the controllability Gramian is maximized based on the estimated maximization condition.

11. 【Number 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, and B be a known n × m matrix. [0010] When holds, [0025] The controllability gramian defined by [Number 138] of [Number 144] Limit in [Number 140] A program for estimating [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition step for acquiring [74] Represented by [Number 75] Define [Number 81] [Number 89] Calculate [Number 157] [Number 158] With respect to the linear equation [006] By calculating the solution X of [Number 146] A computational step of estimating and outputting the estimated controllability gramian.

12. 【Number 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, and B be a known n × m matrix. [0010] When holds, [0025] The controllability gramian defined by [Number 138] of [Number 144] Limit in [Number 140] Trace [Number 141] A program for estimating a matrix B when is maximized, [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition step for acquiring [0070] The linear equation [0080] Solution [0097] By calculating numerically, [Number 141] is maximized; and based on the estimated maximization condition, outputting an input matrix when the controllability Gramian is maximized.

13. 【Number 142】 is an n-dimensional vector representing the state of the controlled object, [Number 139] Let be an m-dimensional vector representing the control input, A be an unknown n × n matrix, B be a known n × m matrix, and Δ be the change in A. [0010] When holds, [0025] The limit of the controllability Gramian of matrix A defined by [Number 231] When [Number 232] A data processing device for estimating [0039] q time intervals when [0045] A set of time-series state data in [0050] A data acquisition unit for acquiring [74] Represented by [Number 75] Define [Number 81] [Number 89] Calculate [Number 157] [Number 158] Regarding linear equation [Number 225] By calculating the solution X of [Number 233] A controllability Gramian calculator that estimates and an output unit that outputs the estimated controllability gramian.

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Control system design method of electric-hydraulic shaking table
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Control system design method of electric-hydraulic shaking table