Nonlinear dynamics model structure and model parameter estimation method

Abstract

To provide: a structure of a nonlinear dynamic model which can be constructed without repeatedly solving a complicated nonlinear optimization problem, and which allows for visually grasping a non-linearity as a shape and which can enhance accuracy of system identification; and a model parameter estimation method.SOLUTION: A structure of a nonlinear dynamic model is expressed by a non-linearity notation function representing a non-linearity and a dynamic characteristic notation function representing a dynamic characteristic. Structural parameters thereof are constructed of non-linearity notation parameters of the non-linearity notation function and dynamic notation parameters of a dynamic notation function. The structural parameters are estimated by optimization calculations that minimize an evaluation function regarding an error between predicted values obtained from the nonlinear dynamic model and actually measured values obtained from a target.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a structure of a nonlinear dynamic characteristics model and a model parameter estimation method, and more particularly to a structure of a nonlinear dynamic characteristics model and a model parameter estimation method that can be constructed without repeatedly solving complex nonlinear optimization problems, that allows nonlinearity to be visually grasped as a shape, and that can improve the accuracy of system identification. [Background technology]

[0002] It is necessary to know the dynamic characteristics of a system in question in order to analyze, predict, control, simulate, diagnose abnormalities, etc. The dynamic characteristics of a system can be expressed as a dynamic characteristics model, which can be used for system analysis, prediction, control, simulation, diagnoses abnormalities, etc. The process of determining a mathematical model (dynamic characteristics model) that best describes a system under a certain evaluation function based on the time series signals of the observed values ​​(actual measured values) of the system is called system identification. Systems generally exhibit nonlinear behavior, so a nonlinear dynamic characteristics model is required as the dynamic characteristics model, and it is desirable that the system identification be highly accurate and that the constituent parameters be easy to estimate.

[0003] Patent Document 1 describes a nonlinear dynamic characteristics model that approximately simulates the nonlinear responsiveness of a controlled object that changes sharply depending on operating conditions (hereinafter referred to as operating point) and improves prediction accuracy. This nonlinear dynamic characteristics model is a dynamic characteristics model for a controlled object that exhibits nonlinear behavior, and has a structure having linear part parameters and nonlinear part parameters as constituent parameters in order to approximately simulate the nonlinear behavior of the controlled object, and the nonlinear parameter part is made up of basis functions with a center parameter and a spread parameter as adjustment parameters.

[0004] In the process of estimating the configuration parameters, the parameter space is divided into a linear parameter space and a nonlinear parameter space, and a search of the nonlinear parameter space is performed once a search of the linear parameter space is completed, or alternatively, a search of the linear parameter space is performed once a search of the nonlinear parameter space is completed. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] JP 3571330 B2 Summary of the Invention [Problem to be solved by the invention]

[0006] The dynamic characteristics model disclosed in Patent Document 1 has a structure having linear part parameters and nonlinear part parameters as constituent parameters, and the nonlinear part parameters are composed of basis functions with a center parameter and a spread parameter as adjustment parameters. A plurality of normal distribution functions are used as basis functions, the mean and standard deviation of which are the center parameter and the spread parameter, and the coefficient of multiplying and adding up the plurality of normal distribution functions is the linear part parameter.

[0007] Here, in the process of estimating the constituent parameters, the parameters are estimated by dividing the space into linear and nonlinear parameter spaces. However, the estimation formulas and the estimation process are complex, and as described below, depending on the data, it may be difficult to estimate the parameters.

[0008] First, the linear part parameters, which are coefficients that are product-sum functions of the normal distribution function, represent both the nonlinearity expressed by the normal distribution function and the dynamic characteristics that are the coefficients of the time series, making it difficult to grasp each of the nonlinearity and dynamic characteristics.

[0009] Secondly, since the mean and standard deviation of the normal distribution function are used as the adjustment parameters, it is necessary to solve a complex nonlinear optimization problem (for example, "Optimization of Nonlinear Systems" by Masatoshi Sakawa, Morikita Publishing) through repeated calculations using the descent method or the like in a single search of the nonlinear parameter space, which requires repeated calculations for each search.

[0010] Thirdly, in the search for nonlinear parameter space to solve a nonlinear optimization problem, depending on the data used, it is possible to search for a local optimum rather than a global optimum (for example, Shimizu Kiyotaka, "Mathematical Programming", Shokodo, p.60), which can make it difficult to estimate parameters. In addition, when considering how to deal with this, the parameter estimation process becomes more complicated.

[0011] An object of the present invention is to provide a nonlinear dynamic characteristics model structure and a model parameter estimation method that can be constructed without repeatedly solving complex nonlinear optimization problems, that allows nonlinearity to be visually grasped as a shape, and that can improve the accuracy of system identification. [Means for solving the problem]

[0012] In order to solve the above problems, the present invention provides a structure of a nonlinear dynamic characteristic model for simulating a system exhibiting nonlinear behavior, the structure of the nonlinear dynamic characteristic model including a nonlinearity representation function representing nonlinearity and a dynamic characteristic representation function representing dynamic characteristics. Combined operations And the behavior is Distinguished between nonlinearity and dynamic characteristics The structure is constructed so as to be represented by the above-mentioned formula, and the configuration parameters are a nonlinearity representation parameter of the nonlinearity representation function and a dynamic characteristic representation parameter of the dynamic characteristic representation function. It is divided into The present invention is characterized in that:

[0013] Here, the structure of the nonlinear dynamics model is such that its behavior is expressed in the form of a product of a nonlinearity representation function and a dynamics representation function. Distinguished between nonlinearity and dynamic characteristics It is preferred to have the structure constructed as shown.

[0014] The nonlinearity representation function is preferably an orthogonal function, and the orthogonal function is preferably expressed by Fourier calculation.

[0015] The present invention also provides a method for estimating model parameters of a structure of a nonlinear dynamic characteristics model, characterized in that the constituent parameters are adjusted and the constituent parameters are estimated by an optimization calculation that minimizes an evaluation function relating to an error between a predicted value obtained from the nonlinear dynamic characteristics model and an actual measurement value corresponding to the predicted value obtained from an object. Effect of the Invention

[0016] The structure of the nonlinear dynamic characteristic model of the present invention that simulates an object exhibiting nonlinear behavior is a nonlinearity representation function that represents nonlinearity and a dynamic characteristic representation function that represents dynamic characteristics. Combined operations And the behavior is Distinguished between nonlinearity and dynamic characteristics Since the structure is constructed as shown in FIG. 1 and the constituent parameters consist of nonlinearity representation parameters of the nonlinearity representation function and dynamic characteristic representation parameters of the dynamic characteristic representation function, the nonlinearity and dynamic characteristics can be easily grasped as individual functions.

[0017] In addition, the structure of the nonlinear dynamic characteristic model is expressed as a product of the nonlinearity representation function and the dynamic characteristic representation function. Distinguished between nonlinearity and dynamic characteristics By having a structure constructed as shown, the function representing the nonlinearity and the function representing the dynamic characteristics can be separated, and the effect of the nonlinearity on the dynamic characteristics can be confirmed.

[0018] In addition, by using orthogonal functions as the nonlinearity representation functions, the components of the orthogonal function system are independent, uncorrelated, and linearly independent (for example, Arima Satoshi, "Linear Algebra," Tokyo Tosho, p. 49), so the nonlinearity representation function parameters have a linear relationship with the components of the orthogonal functions. This eliminates the need for repeated calculations in one search, and allows the application of the usual least squares method. In addition, since there is no need to solve a nonlinear optimization problem, there is no problem with searching for a locally optimal solution.

[0019] Furthermore, by expressing orthogonal functions using Fourier calculations (for example, "Spectral Analysis" by Hino Mikio, Asakura Publishing), the nonlinearity representation function can be understood as a waveform consisting of a fundamental frequency and its integer multiples, so that it is possible to adjust from rough changes in nonlinearity to fine changes by adjusting the frequency multiple used.

[0020] In addition, the model parameter estimation method for the structure of a nonlinear dynamic characteristics model of the present invention estimates the constituent parameters by adjusting the constituent parameters and performing optimization calculations to minimize an evaluation function related to the error between a predicted value obtained from the nonlinear dynamic characteristics model and an actual value corresponding to the predicted value obtained from the target. In other words, instead of searching for parameters by dividing into a linear parameter space and a nonlinear parameter space as in Patent Document 1, the parameter space is divided into a nonlinearity representation parameter space constituting a nonlinearity representation function representing nonlinearity and a dynamic characteristic representation parameter space constituting a dynamic characteristic representation function representing dynamic characteristics, and the nonlinearity and dynamic characteristics are separated and optimized in separate parameter spaces, so that the nonlinearity representation parameter space and the dynamic characteristic representation parameter space do not interfere with each other, and parameters can be estimated efficiently. [Brief description of the drawings]

[0021] [Figure 1] FIG. 2 is a diagram showing a target (system) for constructing a nonlinear dynamic characteristic model and its inputs and outputs. [Diagram 2] An example of input variables, nonlinear variables, and output variables input to and output from the target (system) is shown below. [Diagram 3] FIG. 13 is a diagram showing an example of a change in a nonlinearity representation function value with respect to a nonlinear variable. [Figure 4] FIG. 11 is a diagram showing a comparison between actual measured values ​​of output variables in a target (system) and specific examples of predicted values ​​of output variables based on the nonlinear dynamic characteristics model of the present invention, in which (a) shows the time series signals of the actual measured values ​​of output variables and the predicted values ​​of output variables, (b) shows an enlarged portion of the signals, and (c) shows a histogram of output variable errors (distribution of errors between the actual measured values ​​of output variables and the predicted values ​​of output variables). [Diagram 5]FIG. 1 is a diagram showing an example of the control performance of a control using a nonlinear dynamic characteristic model of the present invention and a control using an existing linear model in comparison. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0022] The present invention will now be described with reference to the drawings.

[0023] Fig. 1 shows a system for which a nonlinear dynamics model is constructed and its inputs and outputs. Hereinafter, the system for which a nonlinear dynamics model is constructed is simply referred to as the system. System 1 has input variables, nonlinear variables, and output variables as inputs and outputs, and exhibits nonlinear behavior.

[0024] 2 is a diagram showing an example of input variables, nonlinear variables, and output variables input and output to and from system 1, where each variable is a time-series signal related to system 1. Note that, in the following, an example will be described in which the input and output to system 1 are input variables 1 and 2, a nonlinear variable, and an output variable, but the number of each variable input to system 1 is not limited to this.

[0025] The nonlinear dynamic characteristics model is constructed based on the measured values ​​of input variables, nonlinear variables, and output variables so as to simulate the nonlinear behavior of System 1, and outputs predicted values ​​of output variables for the input variables and nonlinear variables. Here, the model is based on a linear system model based on the auto-regressive AR (Auto-Regressive) model of system identification (Takayoshi Nakamizo, Signal Analysis and System Identification, Corona Publishing, p.123).

[0026] (Structure of nonlinear dynamics model) First, the structure of the nonlinear dynamic characteristics model of the present invention will be described. In the present invention, nonlinear behavior among the behaviors exhibited by the system is distinguished from dynamic characteristics, and the structure of the nonlinear dynamic characteristics model is a nonlinearity representation function that represents nonlinearity and a dynamic characteristics representation function that represents dynamic characteristics. Combined operations And the behavior is Distinguished between nonlinearity and dynamic characteristics The structure is constructed as shown in Equation (1). Distinguishing between nonlinear and dynamic characteristics,A nonlinearity representation function that represents nonlinearity and a dynamic characteristic representation function that represents dynamic characteristics. Combined operations This is a basic equation expressed as: and is used for estimating parameters of the nonlinear dynamic characteristics model of the present invention. In the following, construction of the structure of the nonlinear dynamic characteristics model will be explained based on the equation expressing the behavior of the system. @@@@@@end

[0027]

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[0028] The meanings of the symbols in formula (1) are as follows: u: Input variable 1 (hereinafter referred to as input variable 1u) d: Input variable 2 (hereinafter referred to as input variable 2d) X: Nonlinear variable (hereinafter referred to as nonlinear variable X) y: Output variable (hereinafter referred to as output variable y) y t : Actual measured value of output variable at time t Φ0(X): The bias component of the nonlinear expression function with X as the nonlinear variable Φ y (X): A nonlinearity representation function for the output variable y with X as a nonlinear variable. Φ u (X): A nonlinearity representation function for the input variable 1u with X as a nonlinear variable. Φ d (X): A nonlinearity representation function for input variables 2d with X as a nonlinear variable Γ0: DC component of dynamic characteristic function Gamma y : Dynamic characteristic expression function for output variable y Gamma u : Dynamic characteristic expression function for input variable 1u Gamma d : Dynamic characteristic expression function for input variable 2d e t : Actual value of output variable y t and the error (residual) between the output variable prediction value by the nonlinear dynamics model Here, the nonlinear variable X may be different for each nonlinearity representation function. In addition, Φ described below is the nonlinearity representation function Φ0(X), Φ y (X),Φ u (X),Φ d (X), and Γ is the dynamic characteristic representation function Γ0, Γ y ,Γ u ,Γ d is a general term for.

[0029] Error e on the right side of equation (1) t The terms other than the output variable y t That is, the nonlinear dynamics model is expressed by equation (2).

[0030]

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[0031] Here, 〈y t 〉 is the output variable predicted value at time t by the nonlinear dynamics model. Explanation of symbols already explained will be omitted. The same will be omitted below.

[0032] The dynamic characteristic representation function Γ is expressed by equation (3).

[0033]

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[0034] The meanings of the symbols in equation (3) are as follows: α0: Bias of dynamic characteristic parameter α y,i (i=1, ,p): Dynamic characteristic representation parameter for output variable y α u,i (i=1,...,q u ): Dynamic characteristic representation parameters for input variable 1u α d,i (i=1,...,q d ): Dynamic characteristic representation parameters for input variable 2d τ: Dead time between output variable y and input variables 1u, 2d p: degree of output variable y q u :The degree of the input variable 1u q d : the degree of the input variable 2d y t-i ,u t-τ-i ,d t-τ-i : Actual measured values ​​of output variable y, input variable 1u, and input variable 2d at time ti In addition, α described below is the dynamic characteristic representation parameter α0, α y,i (i=1,...,p),α u,i (i=1,...,q u ),α d,i (i=1,...,q d ) is a general term for

[0035] The nonlinearity representation function Φ is expressed by equation (4).

[0036]

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[0037] The meanings of the symbols in equation (4) are as follows: c 0,0 ,c 01,k ,c 02,k (k=1,...,m): Nonlinearity parameters for DC, cosine, and sine components that indicate the bias of nonlinear properties c y,0 ,c y1,k ,c y2,k (k=1,...,m): Nonlinearity parameters for the DC, cosine, and sine components of the output variable y c u,0 ,c u1,k ,c u2,k (k=1,...,m): Nonlinearity parameters for the DC, cosine, and sine components of the input variable 1u c d,0 ,c d1,k ,c d2,k(k=1,...,m): Nonlinearity parameters for the DC, cosine, and sine components that indicate the nonlinearity of the input variable 2d m: the order of the cosine and sine waves of the Fourier calculation, which are orthogonal functions, and 2m+1: the order of the nonlinearity representation function Φ X min : The minimum value of the nonlinear variable X as a range of change T: The range of change of the nonlinear variable X (the maximum value Xmax of the nonlinear variable X - the minimum value X min ) and the fundamental period of the Fourier calculation Here, the order m may vary depending on the variable. Note that c in the following description is the nonlinearity representation parameter c 0,0 ,c 01,k ,c 02,k (k=1,···,m),c y,0 ,c y1,k ,c y2,k (k=1,···,m),c u,0 ,c u1,k ,c u2,k (k=1,···,m),c d,0 ,c d1,k ,c d2,k (k=1, ,m).

[0038] The nonlinearity representation function Φ is, for example, a Fourier calculation of an orthogonal function, and is expressed here by the DC component of the Fourier calculation, the fundamental wave of a sine wave and a cosine wave, and harmonics indicated by their orders.

[0039] The nonlinear dynamic characteristic model (1) is expressed by a nonlinearity representation function Φ and a dynamic characteristic representation function Γ, more specifically, by the product of the nonlinearity representation function Φ, which is a Fourier operation of orthogonal functions, and the dynamic characteristic representation function Γ, and its constituent parameters are the nonlinearity representation parameter c of the nonlinearity representation function Φ and the dynamic characteristic representation parameter α of the dynamic characteristic representation function Γ.

[0040] Here, the product term of the bias component Φ0(X) of the nonlinearity representation function Φ and the DC component Γ0 of the dynamic characteristic representation function Γ may be unnecessary as a dynamic characteristic model. A linear dynamic characteristic representation function without a nonlinearity representation function is called an AR model, but in the field of control, there are many cases where there is no bias component or DC component term, and in such cases, the product term of the bias component Φ0(X) of the nonlinearity representation function Φ and the DC component Γ0 of the dynamic characteristic representation function Γ is unnecessary. Even if there is a bias component or DC component term, if a dynamic characteristic model is constructed from an equilibrium point, which is the steady state of the controlled object, the product term of the bias component Φ0(X) of the nonlinearity representation function Φ and the DC component Γ0 of the dynamic characteristic representation function Γ is unnecessary. In this way, the product term of the bias component Φ0(X) of the nonlinearity representation function Φ and the DC component Γ0 of the dynamic characteristic representation function Γ can be flexibly handled in consideration of the application of the model, the control in the system, etc.

[0041] (Model parameter estimation) Next, the estimation of the model parameters will be described. In order to construct the nonlinear dynamic characteristic model (1), the nonlinearity representation parameter c and the dynamic characteristic representation parameter α are estimated. Note that the nonlinearity representation parameter c and the dynamic characteristic representation parameter α are both linear parameters.

[0042] The nonlinearity parameter c and the dynamic characteristic parameter α are the output variable prediction value 〈y t 〉 and the actual output variable y obtained from the target system t Error e t (The error e in the nonlinear dynamics model (1) t ) square error criterion evaluation function J(E{e t 2}), where E{·} denotes the expected value.

[0043] The nonlinearity representation parameter c and the dynamic characteristic representation parameter α are both linear parameters, but since the nonlinear dynamic characteristic model equation (1) is expressed in the form of a product of the nonlinearity representation function Φ and the dynamic characteristic representation function Γ, the nonlinear dynamic characteristic model equation (1) is nonlinear.

[0044] Therefore, the nonlinearity representation parameter c constituting the nonlinearity representation function Φ and the dynamic characteristic representation parameter α constituting the dynamic characteristic representation function Γ are estimated alternately. This allows the estimation to be handled as a linear parameter estimation, and allows the nonlinearity and dynamic characteristics to be understood separately. For example, the nonlinearity representation parameter c is fixed, the nonlinearity representation function Φ is given as a calculated value, and the dynamic characteristic representation parameter α is estimated by applying the linear least squares method (Takayoshi Nakamizo, Signal Analysis and System Identification, Corona Publishing, p. 123).

[0045] When estimating the nonlinearity representation parameter c, the constraint in equation (5) can be introduced. Since the nonlinear dynamic characteristics model is in the form of the product of the nonlinearity representation function Φ and the dynamic characteristics representation function Γ, as shown in equation (1), it is effective to impose a magnitude constraint on the nonlinearity representation function Φ and use only shape information.

[0046] Since the nonlinearity representation function Φ only needs to construct a relative relationship in the operating range of the nonlinear variable X, relative shape information indicating nonlinearity can be extracted by restricting the magnitude in the estimation of the nonlinearity representation parameter c. In other words, the nonlinearity representation parameter c can be estimated using the linear least squares method with constraints of equation (5) (for the linear least squares method with constraints, see, for example, MATLAB (registered trademark) (HP homepage) "Solving Constrained Linear Least Squares Problems" lsqlin (https: / / jp.mathworks.com / help / optim / ug / lsqlin.html)).

[0047]

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[0048] The constituent parameters c and α of the nonlinear dynamics model (1) are the actual measured value of the output variable y t and the output variable prediction value 〈y t 〉 (e in the nonlinear dynamics model (1) t ) and the dynamic characteristic representation parameter α of the dynamic characteristic representation function Γ and the nonlinearity representation parameter c of the nonlinearity representation function Φ, which form the product, can be estimated alternately by linear least squares method.

[0049] In the estimation process, the dynamic characteristic parameter α or the nonlinear characteristic parameter c is fixed, and the other is estimated using the least squares method in an alternating manner, and the evaluation function J(E{e t 2}) (The error e t The iterations are performed by minimizing the evaluation function J(E{e t 2}) is equal to or smaller than a set threshold, or the set number of repetitions is reached.

[0050] When the repeated calculation is started, the nonlinearity representation parameter c is fixed to an initial value, for example, according to equation (6).

[0051]

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[0052] As shown in equation (6), the initial value of the nonlinearity representation parameter c is set to 1 for the DC component only and to zero for the sine wave and cosine wave coefficients. This gives the minimum value X of the nonlinear variable X in equation (4). min Therefore, the value of the nonlinearity representation function Φ becomes 1 within the change range T, and as a condition of no nonlinearity, the nonlinear dynamic characteristic model (1) becomes (7). By substituting the dynamic characteristic representation function Γ of (3) into (7), we derive (8).

[0053]

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[0054]

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[0055] Equation (8) is a general linear model expression consisting of the time series signals of the output variable y, the input variables 1u and 2d, and the dynamic characteristic representation parameter α. When the time series signals and the dynamic characteristic representation parameter are expressed as vectors, it becomes equation (9).

[0056]

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[0057] where [·] is a vector or matrix, [·] T is the transpose of the vector or matrix [·]. Also, y t is the actual output variable value at time t, z t is the vector of time series signals at time t, θ α is a vector of dynamics representation parameters.

[0058] N times t1, t2, t N Considering the vectors of the output variable actual values ​​and time series signals in, the vector of the output variable actual values ​​Y is the vector of the time series signals Z, and the vector of the dynamic characteristic representation parameters θ α , the error vector e α Using this, it is expressed as equation (10).

[0059]

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[0060] The evaluation function J(E{e t 2}) is minimized by δJ / (δθ α )=0, and the dynamics representation parameter vector θ α The least squares estimate of 〈θ α〉 can be derived using equation (11).

[0061]

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[0062] In the above, when starting to estimate the parameters c and α of the nonlinear dynamic characteristics model (1), the value of equation (6) was given as the initial value of the nonlinearity representation parameter c to obtain the nonlinear dynamic characteristics model (7) and (8). However, even if a value other than the initial value, for example an estimated value of the nonlinearity representation parameter c, is given to the nonlinearity representation parameter c, the nonlinearity representation function Φ can be quantified in equation (4) using the time series signal of the nonlinear variable X, and can be expressed as a linear model. This makes it possible to estimate the dynamic characteristics representation parameter α in the same way as above.

[0063] By substituting the dynamic characteristic representation parameter α estimated in the above manner, and the time series signals of the output variable y, input variable 1u, and input variable 2d into the prediction model (12) of (8), the output variable prediction value 〈y t 〉 can be obtained. t 〉 and the measured output variable y t Error e t From the evaluation function J(E{e t 2}) (Note that the first equation of (11) may be used here), the validity of the estimated parameters can be confirmed. t 2}) with a preset threshold value, the end of the repeated calculation can be determined.

[0064]

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[0065] Next, the nonlinearity representation parameter c is estimated. The dynamic characteristics representation function Γ can be quantified by substituting the dynamic characteristics representation parameter α calculated as above and the time series signals of the output variable y, input variable 1u, and input variable 2d into equation (3). By substituting the quantified dynamic characteristics representation function Γ of equation (3) and equation (4) of the nonlinearity representation function Φ to which the time series signal of the nonlinear variable X is applied into equation (1) of the nonlinear dynamic characteristics model, equation (13) can be obtained, which shows the linear relationship between the DC component, cosine wave, and sine wave of the Fourier operation (nonlinearity representation function Φ) and the nonlinearity representation parameter c.

[0066]

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[0067] The DC, cosine, and sine components of the Fourier operation (nonlinear function Φ) multiplied by the dynamic characteristic function Γ are expressed as vector p t and the nonlinearity representation parameters are expressed as a vector θ c Then, equation (13) becomes equation (14), and the nonlinearity representation parameter c (or the vector of nonlinearity representation parameters θ c ) is the evaluation function J(E{e t 2}) can be estimated by the constrained linear least squares method.

[0068]

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[0069] By applying the estimated nonlinear representation parameter c and the time series signal of the nonlinear variable X to equation (4) and substituting it into the nonlinear dynamic characteristic model of equation (2) together with the quantified dynamic characteristic representation function Γ of equation (3), the output variable prediction value 〈y t 〉 can be obtained. The output variable prediction value 〈y t 〉 and the measured output variable y t Error e t From the evaluation function J(E{e t 2}), the validity of the estimated parameters can be confirmed. The iterative calculation ends when the evaluation function J(E{e t 2}) with a preset threshold value, the determination can be made.

[0070] The nonlinearity representation function and the dynamic characteristic representation function do not form a product. For example, as shown in equation (15), the dynamic characteristic representation function Γ d is not in the form of a product with a nonlinear representation function, and the dynamic characteristic representation function Γ d In the case of only the dynamic characteristic representation function Γ, the dynamic characteristic representation parameter α can be estimated in the same manner as above, but the nonlinearity representation parameter c cannot be estimated in its original form. d does not have a parameter to be estimated (nonlinearity representation parameter c) that has been quantified, so by moving it to the left side of equation (16) and configuring the right side of equation (16) with the nonlinearity representation parameter c, the nonlinearity representation parameter c can be estimated.

[0071]

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[0072]

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[0073] Fig. 3 is a diagram showing an example of the nonlinear characteristics of a nonlinearity representing function whose parameters have been estimated. As shown in Fig. 3, the nonlinearity representing function value changes depending on the nonlinear variable. Note that Fig. 3 shows an example in which the nonlinearity representing function is an orthogonal function system (Fourier calculation), and shows the dependency of the function value on the nonlinear variable.

[0074] 4 is a diagram showing a comparison between actual measured values ​​of output variables in a system and specific examples of predicted values ​​of output variables by the nonlinear dynamic characteristic model of the present invention. The system in question here is a denitration system that reduces NOx generated by combustion in a thermal power plant by controlling the injection of NH3, and the input variable 1u, input variable 2d, nonlinear variable X, and output variable y in this specific example are the NH3 flow rate, the fuel flow control valve opening (variable related to NOx generation), the blade path temperature (temperature of gas flowing into the denitration process), and the NOx concentration at the outlet of the denitration process, respectively.

[0075] FIG. 4(a) is a diagram showing a comparison of the time series signals of the NOx concentration actual measurement value at the outlet of the denitration process in the system and the NOx concentration predicted value at the outlet of the denitration process by the nonlinear dynamics model (2) of the present invention, and FIG. 4(b) is a diagram showing an enlarged part of the figure. FIG. 4(c) is a diagram showing a histogram of NOx concentration errors (distribution of errors between the actual NOx concentration values ​​and the predicted NOx concentration values). This is the case where a DC component, a fundamental period, and up to twice the period are used as a nonlinearity representation function, and the number of repeated calculations is 500. As shown in FIG. 4, the NOx concentration predicted value by the nonlinear dynamics model of the present invention is almost the same as the actual NOx concentration value in the system, which shows that the nonlinear dynamics model of the present invention is effective.

[0076] FIG. 5 is a diagram showing an example of the control performance of the control using the nonlinear dynamics model of the present invention and the control using the existing linear model, and shows the time series signals of the output variables (controlled variables) controlled by each control. Note that the control parameters and data used other than the model are all under the same conditions. When constant set value control is performed in a system, the variance (variance) of the output variables is an index of control performance. In FIG. 5(a), the case of the control using the nonlinear dynamics model of the present invention is shown by a solid line, and the case of the control using the existing linear model is shown by a dotted line, and in FIG. 5(b), a part of the control is shown in an enlarged manner. As shown in FIG. 5, the variance of the output variables in the case of the control using the nonlinear dynamics model of the present invention is smaller than the variance of the output variables in the case of the control using the existing linear model, which shows that the nonlinear dynamics model of the present invention is effective.

[0077] The Akaike Information Criterion (AIC) is an index for evaluating and selecting models in system identification ("Methods of Time Series Analysis" edited by Akaike Hiroji, Asakura Publishing). According to this, the smaller the value, the more suitable the model is. The values ​​obtained by constructing models for the object and time series signal in Figures 1 and 2 were -1.756e+3 for the linear model, -7.79e+3 for the model in Patent Document 1, and -8.24e+3 for the nonlinear dynamic characteristics model of the present invention. This shows that the nonlinear dynamic characteristics model of the present invention highly simulates the system and is suitable as a model.

[0078] As described above, the structure of the nonlinear dynamics model of the present invention can be constructed without repeatedly solving complex nonlinear optimization problems, the nonlinearity can be visually grasped as a shape, and the accuracy of system identification is improved. Therefore, the structure of the nonlinear dynamics model of the present invention is effective for system analysis, prediction, control, simulation, abnormality diagnosis, etc.

[0079] Although the embodiments have been described above, the present invention is not limited to the above embodiments, and can be modified in various ways based on the technical viewpoints of the matters described in the claims, and includes those within the scope of the claims. For example, the number of input variables, the initial values ​​of parameter estimation, the order of the dynamic characteristic representation function and the nonlinearity representation function at the start of parameter estimation, etc. are not limited to the above embodiments, and can be modified as appropriate. In addition, the Fourier calculation of the nonlinearity representation function (orthogonal function) includes Fourier series, complex Fourier series, etc. [Explanation of symbols]

[0080] 1. System for which nonlinear dynamics model is to be constructed

Claims

1. A structure of a nonlinear dynamics model for simulating a system exhibiting nonlinear behavior, The structure of the nonlinear dynamic characteristic model is constructed so that the behavior of the model is represented by being distinguished between nonlinearity and dynamic characteristics through a calculation in which a nonlinearity representation function representing nonlinearity and a dynamic characteristic representation function representing dynamic characteristics are combined; A structure of a nonlinear dynamic characteristic model, characterized in that constituent parameters are divided into nonlinearity representation parameters of the nonlinearity representation function and dynamic characteristic representation parameters of the dynamic characteristic representation function.

2. The structure of the nonlinear dynamic characteristic model according to claim 1, characterized in that the structure of the nonlinear dynamic characteristic model is constructed so that its behavior is represented by being distinguished into nonlinearity and dynamic characteristics in the form of a product of the nonlinearity representing function and the dynamic characteristic representing function.

3. 3. The structure of the nonlinear dynamics model according to claim 1, wherein the nonlinearity representation function is an orthogonal function.

4. 4. The structure of the nonlinear dynamic characteristic model according to claim 3, wherein the orthogonal functions are expressed by Fourier calculations.

5. A method for estimating model parameters of a structure of a nonlinear dynamic characteristic model according to any one of claims 1 to 4, comprising the steps of: A model parameter estimation method, comprising: estimating the constituent parameters by an optimization calculation that minimizes an evaluation function relating to an error between a predicted value obtained from the nonlinear dynamic characteristics model and an actual measurement value corresponding to the predicted value obtained from an object.

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