# US 11,443,219 B2 · Model estimation system, method, and program

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- Number: US 11,443,219 B2
- Document ID: US11443219B2
- Status: Granted
- Kind: B2
- Application number: 16/481,715
- Filed: 2018-01-18
- Published: 2020-01-23
- Granted: 2022-09-13
- Earliest claimed: 2017-02-17
- Assignees: NEC Corporation
- Applicants: NEC Corporation
- Inventors: Riki Eto; Yoshio Kameda
- Web page: https://patentgrep.com/patent/US11443219B2

## CPC

- G06N 7/00: Computing arrangements based on specific mathematical models
- G05B 13/04: involving the use of models or simulators
- G06F 30/20: Design optimisation, verification or simulation (optimisation, verification or simulation of circuit designs G06F30/30)
- G06N 20/00: Machine learning
- G06F 17/13: Differential equations (using digital differential analysers G06F7/64)
- G06F 17/16: Matrix or vector computation {, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization (matrix transposition G06F7/78)}
- G06F 2111/10: Numerical modelling
- G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
- G05B: CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS (systems for controlling or regulating non-electric variables G05D; systems for regulating electric or magnetic variables G05F; control devices or systems insofar as characterised by mechanical features only G05G)
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

A model estimation system estimates a model of a system represented by an ordinary differential equation with all coefficients being non-zero, and with which input data and a state at each time can be obtained. When an order of the ordinary differential equation and input data and a state at multiple past times in the system are inputted, a model expression construction unit constructs an expression representing a model by using a first matrix that is a matrix according to the order and has only some elements as unknown elements and a second matrix that is a matrix according to the order and has only some one element as an unknown element. A model estimation unit uses input data and a state at multiple past times, to estimate the model by learning unknown elements of the first matrix and the unknown element of the second matrix.

## Claims

### Claim 1

A model estimation system for estimating a model of a system that is represented by an ordinary differential equation with all coefficients being non-zero and allows input data and a state at each time to be obtained, the model estimation system comprising:
at least one memory configured to store program instructions;
at least one processor configured to execute the program instructions to implement:
a model expression construction unit which constructs an expression representing a model for predicting a state of the system at a next time from input data and a state of the system at one time, when an order of the ordinary differential equation and input data and a state at multiple past times in the system are inputted, by using a first matrix that is a matrix according to the order and has only some elements as unknown elements and a second matrix that is a matrix according to the order and has only one element as an unknown element;
a model estimation unit which estimates the model by machine learning unknown elements of the first matrix and an unknown element of the second matrix in the expression by using input data and a state at the multiple past times; and
a model implementing unit which implements controllability of a target system based on transition from an initial state to a target state of the estimated model.

### Claim 2 (depends on 1)

The model estimation system according to claim 1, wherein
the model expression construction unit
constructs an expression represented by Expression (16) having b0 to bn−1 and cn as unknown elements, with: an order of an ordinary differential equation as n; input data to the system at a time k as uk; a state of the system at a time k as [yk, yk(1), . . . , yk(n)]T, wherein notation in superscript parenthesis indicates a differential order; a state of the system at a time k+1 following a time k as [yk+1, yk+1 (1), . . . , yk+1 (n)]T; and a time period from one time to a next time as Δt, and
the model estimation unit
estimates b0 to bn−1 and cn in the expression by using input data and a state at multiple past times in the system.
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### Claim 3

A model estimation method for estimating a model of a system that is represented by an ordinary differential equation with all coefficients being non-zero and allows input data and a state at each time to be obtained, wherein
a computer
constructs an expression representing a model for predicting a state of the system at a next time from input data and a state of the system at one time, when an order of the ordinary differential equation and input data and a state at multiple past times in the system are inputted, by using a first matrix that is a matrix according to the order and has only some elements as unknown elements and a second matrix that is a matrix according to the order and has only one element as an unknown element; and
estimates the model by machine learning unknown elements of the first matrix and the unknown element of the second matrix in the expression by using input data and a state at the multiple past times; and
implements controllability of a target system based on transition from an initial state to a target state of the estimated model.

### Claim 4 (depends on 3)

The model estimation method according to claim 3, wherein
the computer
constructs an expression represented by Expression (17) having b0 to bn−1 and cn as unknown elements, with: an order of an ordinary differential equation as n; input data to the system at a time k as uk; a state of the system at a time k as [yk, yk(1), . . . , yk(n)]T, wherein notation in superscript parenthesis indicates a differential order; a state of the system at a time k+1 following a time k as [yk+1, yk+1 (1), . . . , yk+1 (n)]T; and a time period from one time to a next time as Δt, and
estimates b0 to bn−1 and cn in the expression by using input data and a state at multiple past times in the system.
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### Claim 5

A non-transitory computer-readable recording medium in which a model estimation program is recorded, the model estimation program causing a computer to estimate a model of a system that is represented by an ordinary differential equation with all coefficients being non-zero and allows input data and a state at each time to be obtained, the model estimation program causes
the computer to execute:
a model expression construction process of constructing an expression representing a model for predicting a state of the system at a next time from input data and a state of the system at one time, when an order of the ordinary differential equation and input data and a state at multiple past times in the system are inputted, by using a first matrix that is a matrix according to the order and has only some elements as unknown elements and a second matrix that is a matrix according to the order and has only one element as an unknown element;
a model estimation process of estimating the model by learning unknown elements of the first matrix and the unknown element of the second matrix in the expression by using input data and a state at the multiple past times; and
an implementation process of implementing controllability of a target system based on transition from an initial state to a target state of the estimated model.

### Claim 6 (depends on 5)

The non-transitory computer-readable recording medium according to claim 5, wherein
the model estimation program causes the computer,
in the model expression construction process,
to construct an expression represented by Expression (18) having b0 to bn−1 and cn as unknown elements, with: an order of an ordinary differential equation as n; input data to the system at a time k as uk; a state of the system at a time k as [yk, yk(1), . . . , yk(n)]T, wherein notation in superscript parenthesis indicates a differential order; a state of the system at a time k+1 following a time k as [yk+1, yk+1 (1), . . . , yk+1 (n)]T; and a time period from one time to a next time as Δt, and,
in the model estimation process,
to estimate b0 to bn−1 and cn in the expression by using input data and a state at multiple past times in the system
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