JP2008287343A - Model parameter estimating arithmetic unit and method, model parameter estimating arithmetic processing program, and recording medium recording the same - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To construct a model by estimating and computing a model parameter of high accuracy and reliability even from input/output waveform data insufficient in quality and quantity in a one-input multiple-output type first-order lag system. <P>SOLUTION: This model parameter estimating arithmetic unit 10 estimating and computing a model parameter of the one-input multiple-output type first-order lag system based on input/output data, estimates and computes the model parameter considering transcendental information including at least one of a time constant and steady gain of the one-input multiple-output type first-order lag system, as at least one of a constraint condition and an evaluation function based on the input/output data, wherein the transcendental information is at least one of the upper limit values and lower limit values of the time constant and steady gain. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a model parameter estimation calculation apparatus and method for estimating and calculating a model parameter of a dynamic model using a system identification method utilizing theoretical and empirical knowledge for a system having a dynamic model, and model parameter estimation. The present invention relates to an arithmetic program and a computer-readable recording medium that records the program.

  When designing a control system of a system or simulating its behavior, a model that can simulate the behavior of the system is required. For example, when designing a model-based controller, the model is indispensable, and even when a PID (Proportional Integral Derivative) compensator is used, the model is necessary when simulating the behavior of the control system in advance.

  Generally, the method of building a model is roughly divided into the following two (or three).

  One is modeling based on the so-called first principle (referred to as white box modeling or physical modeling), and the physicochemical natural laws governing the system (kinetic equation, electromagnetic field equation, material balance, heat balance, Modeling based on chemical reaction equations). In the case of a large-scale system, this is an approach in which modeling is performed by dividing the system into subsystems and integrating them to build a model of the entire system.

  The other is called black box modeling, an approach that does not use physicochemical laws or a priori knowledge about the system. This is a method for estimating the parameters of a model from observed (finite number) of input / output waveform data by regarding the system as a black box. In the field of control engineering, this is called system identification. This is effective when the system is relatively large or complex and its dynamics cannot be described only by the laws of physicochemistry.

  Furthermore, gray box modeling exists as the third classification (when the modeling method is roughly divided into three). This is positioned between white box modeling and black box modeling. For example, it is a modeling technique that partially uses physicochemical natural laws in addition to input / output waveform data.

  Some of the systems to be modeled can express basic behaviors with equations of motion, such as robots (manipulators) with multi-link structures. However, plants with heat and chemical reactions (for example, chemical plants and Some dynamic characteristics, such as combustion furnaces, are difficult to derive from the laws of nature. In this case, black box modeling that builds a model based on input and output waveform data will be adopted, but if the system is proper and stable and there is no vibration or reverse response that can not be ignored in its step response, In many cases, the model structure can be approximated by “dead time + first-order lag system” (no problem when considered in light of the purpose of control or simulation).

  For example, according to Non-Patent Document 2, unknown parameters of a model must be estimated from input / output waveform data by some method. In this case, the order of the model is increased unnecessarily in an attempt to bring the model closer to the real system. Doing so does not necessarily improve the accuracy of the model. This is because the number of parameters to be estimated increases as the order increases, and the parameter estimation accuracy deteriorates due to noise and neglected dynamic characteristics / nonlinear characteristics.

  When the model structure is "dead time + first order lag system", the dead time value can be calculated by observing the output waveform delay with respect to the input waveform such as step response, adding the actuator operation delay and sensor detection delay time. It can be estimated by combining or analyzing input / output waveform data (for example, a cross-correlation function). There are also many systems that have no dead time or can ignore the dead time. Therefore, it is important to estimate the model parameter of the first-order lag system with high accuracy from the input / output waveform data (when there is a dead time, data obtained by shifting the time of the input or output waveform data by the dead time). .

  Here, the first-order lag system will be described. The transfer function of the first-order lag system is expressed by the following equation.

Here, K is a steady gain, and T is called a time constant. FIG. 2 is a waveform diagram showing input / output waveform data showing a step response waveform of a first-order lag system according to the prior art. Here, Y _c (s) and U _c (s) are the time series of the output waveform data y _c (t) and the input waveform data u _c (t) of the system, respectively, expressed in the frequency domain by Laplace transform. is there. As the nature of the step response waveform, it is known that the output value becomes about 63.2% of the steady gain after T seconds, and the rising slope of the response becomes K / T.

  Now, when the continuous time system of equation (1) is approximated by a discrete time system of sampling time Δt (<< T), it becomes as follows. When the denominator of the right side of equation (1) is brought to the left side, the following equation is obtained.

  Next, when the differential element s in the equation (2) is replaced with a forward difference and approximated, the following equation is obtained.

  Further, the above equation (3) is arranged to obtain the following equation.

Here, it is expressed as follows.

Here, n is an integer. And

In other words, the above equation (4) is expressed by the following equation.

  The actual input / output waveform data is affected by disturbance, noise, and the like, and there is a modeling error, so there is some error in equation (6). Therefore, strictly speaking, the error term e (k) is introduced into the equation (6) to obtain the following equation.

  This is a model format known as an ARX (Auto-Regressive model with eXogenious input) model. When the dead time is L seconds (L = τΔt), the expression (1) is expressed by the following expression.

  Equation (7) is expressed by the following equation.

  If the input waveform data is shifted by the dead time, that is,

とすれば、一般性を失うことなく（７）式で表現できるので、以降、（７）式のモデルに基づいて説明する。

Then, since it can be expressed by Expression (7) without losing generality, the following description will be based on the model of Expression (7).

  When obtaining the model parameters a and b of the equation (7) from the input / output waveform data (black box modeling), the related art (pp. 55 to 58 of Non-Patent Document 1, p. 71 to p. 77 of Reference Document 2). Then, a method using the least square method is known. The model parameter to be obtained is the vector θ

とし、

age,

と定義すると、（７）式は

(7) is defined as

と変形できる。いま、誤差項ｅ（ｋ）の時系列が平均値０、有限な分散σ^２を有する白色雑音であると仮定すると、（１０）式の１段先予測値は、

And can be transformed. Assuming that the time series of the error term e (k) is white noise having an average value of 0 and a finite variance σ ² , the one-stage prediction value of the equation (10) is

となることが知られており、このときの予測誤差は次式で表される。

The prediction error at this time is expressed by the following equation.

を最小化するようなモデルパラメータθを求める。同定入力が２次以上のＰＥ性（Persistently Exciting：持続励振性）であり、システムが安定かつ可観測であれば、（１３）式を最小化するモデルパラメータθは最小二乗計算によって求めることができることが知られている。 In order to obtain a model parameter θ that fits the input / output waveform data as much as possible, a sum of squares of prediction errors is considered as an evaluation criterion. That is, when input / output waveform data from time 0 to N is obtained,

A model parameter θ that minimizes is obtained. If the identification input is second- or higher-order PE (Persistently Exciting) and the system is stable and observable, the model parameter θ that minimizes Equation (13) can be obtained by least-squares calculation. It has been known.

  In the black box modeling as described above, it is assumed that the modeling error is white noise, or there is a condition to be satisfied such as PE property for an input signal. However, in an actual process, there are often limited experimental opportunities for collecting input / output waveform data for system identification, and limited identification input waveforms and sizes. The input / output waveform data also includes disturbances and noises. In some cases, there may be disturbances that cannot be ignored. For example, when building a model with a process that burns a wide variety of unspecified items such as a waste incinerator, the type of waste that is actually put into the furnace with respect to the speed input of the waste feeder And the amount will vary. Therefore, when a model is constructed in which the speed of the waste supply machine is input and the temperature in the furnace is output, there is a large disturbance in the input / output waveform data. In addition, since the data collection opportunities are limited, it is not possible to acquire a sufficient amount of data to counteract the influence of disturbance.

  In this way, the input / output waveform data has non-negligible disturbances and noise, the waveform and size of the identification input that can be applied are limited, and the experimental opportunities to collect input / output waveform data are limited. Since it is not possible to prepare qualitatively and quantitatively sufficient data for modeling with high accuracy, modeling work has to rely on repeated trial and error, and it takes an enormous amount of time to build the model. It was. In other words, it is necessary to repeat data acquisition, data section selection, sampling time adjustment, data preprocessing (including outlier removal, filtering, etc.) and model order selection over and over again. It was. In addition, in such a limited quality and quantity of input / output waveform data, it has been very difficult to obtain a highly accurate and reliable model.

Japanese Patent Application Laid-Open No. 2004-272916. Shuichi Adachi, "System identification for control by MATLAB", Tokyo Denki University Press, pp. 1-5, issued November 30, 1996. Toru Katayama, “Introduction to System Identification (System Control Information Library 9)”, Asakura Shoten, pp. 4-7, May 25, 1994. Herbert J. A. F. Tulleken, "Grey-box Modeling and Identification Using Physical Knowledge and Bayesian Techniques", Automatica, Vol. 29, No. 2, pp.285-308, 1993. Edited by Mikio Kubo et al., "Applied Mathematical Planning Handbook", Asakura Shoten, pp. 654-663, April 30, 2002. Tamaki Hisashi, “EEExt System Optimization”, Ohmsha, pp. 2-5, pp. 73-80, November 25, 2005.

  In order to solve the above problems, there is the following prior art as an attempt to improve the accuracy of a model by supplementing a priori knowledge.

  As a first prior art, Non-Patent Document 3 discloses a modeling technique in which a priori knowledge regarding the stability of a model and the sign of steady gain is considered as a constraint condition. The problems of this modeling technique are as follows. The a priori information to consider for the process is not just the sign or stability of the steady gain. In the case of a first-order lag system, the steady-state gain and the approximate value of the time constant (range of possible values) are often known theoretically and empirically, but this method takes these a priori knowledge into account. I can't put it in. Therefore, in this method, a model that matches a priori knowledge about the steady-state gain and the time constant is not always obtained, and there is a problem that the accuracy and reliability of the model is insufficient.

  In addition, as a second prior art, Patent Document 1 discloses the following method. First, an initial model is obtained from input / output waveform data using an existing modeling method. The model parameter is changed so that the error (weighted error) with the model parameter of the initial model becomes small, and the constraint condition regarding the steady-state gain, stability, and system response is satisfied. Is a new model. However, in this method, just because the model parameter after the change is close to the model parameter of the initial model, the dynamic characteristics of the model are not always close. For example, a state space model of order 1 with 1 input and 1 output of the following equation is used.

Here, for the output waveform data y (k), the input waveform data u (k), the state parameter x (k), and the model parameters a ₀ , b ₀ , c ₀ , d ₀ (respectively scalars), the initial model is It is assumed that (a ₀ , b ₀ , c ₀ , d ₀ ) = (0.1, _1.0 , _1.0 , 0.0). For this initial model, two models with different parameters b ₀ and c ₀ , model 1 (a ₀ , b ₀ , c ₀ , d ₀ ) = (0.1, 0.9, 0.9, ₀ . 0), model 2 (a ₀ , b ₀ , c ₀ , d ₀ ) = (0.1, 0.2, _5.0 , _0.0 ). The value closer to the initial model in the parameter space is the model 1, but the model 2 is closer in dynamic characteristics to the initial model (in this case, the initial model and the model 2 are identical models). Represent).

  In Patent Document 1, in the first place, in the case of state space expression, the model parameters are not uniquely determined (can take an infinite number of values), so it is meaningless to use the proximity of the model parameters as an evaluation function.

  In addition, input / output waveform data is used only when obtaining an initial model, and when performing optimization calculation of model parameters so as to satisfy the constraints based on a priori information as much as possible, only the initial model is used as a norm. There was a problem that no consideration was given to conformity to waveform data. Compatibility with input / output waveform data is only indirectly considered through the initial model, and whether the input / output waveform data is reflected in the modeling depends on the accuracy of the initial model.

  The object of the present invention is to solve the above problems, and to estimate and calculate model parameters with high accuracy and reliability even from input / output waveform data with insufficient quality and quantity in a first-order multiple delay system with one input and multiple outputs. Another object of the present invention is to provide a model parameter estimation calculation device and method, a model parameter estimation calculation program, and a computer-readable recording medium that records the program.

A model parameter estimation calculation device according to a first aspect of the invention is a model parameter estimation calculation device that estimates and calculates a model parameter of a first-order lag system of a one-input multiple-output system based on input / output data.
Based on the input / output data, a priori information including at least one of a time constant and a steady-state gain of the first-order lag system of the one-input / multiple-output system is considered as at least one of the constraint condition and the evaluation function. And calculating means for estimating and calculating the model parameter.

  In the model parameter estimation calculation device, the a priori information is at least one of an upper limit value and a lower limit value of a time constant and a steady gain.

  Further, in the model parameter estimation calculation device, the calculation means uses the a priori information as linear constraint information based on the input / output data, and includes output data included in the input / output data, a model A model parameter is estimated and calculated so that an evaluation function indicating an error from output data when the parameter model is estimated and calculated is minimized.

  Further, in the model parameter estimation calculation device, the evaluation function is a square error between the output data included in the input / output data and the output data when the estimation calculation is performed with the model of the model parameter, and the calculation means includes: A quadratic programming problem method is used to estimate and calculate model parameters that satisfy the linear constraints and minimize the evaluation function.

A model parameter estimation calculation method according to a second invention is a model parameter estimation calculation method for estimating and calculating a model parameter of a first order lag system of a one-input / multiple-output system based on input / output data.
Based on the input / output data, a priori information including at least one of a time constant and a steady-state gain of the first-order lag system of the one-input / multiple-output system is considered as at least one of the constraint condition and the evaluation function. And a calculation step for estimating the model parameter.

  In the model parameter estimation calculation method, the a priori information is at least one of an upper limit value and a lower limit value of a time constant and a steady gain.

  In the model parameter estimation calculation method, the calculation step uses the a priori information as linear constraint information based on the input / output data, and includes output data included in the input / output data, a model A model parameter is estimated and calculated so that an evaluation function indicating an error from output data when the parameter model is estimated and calculated is minimized.

  Further, in the model parameter estimation calculation method, the evaluation function is a square error between the output data included in the input / output data and the output data when the model parameter model is estimated and calculated, and the calculation step includes: A quadratic programming problem method is used to estimate and calculate model parameters that satisfy the linear constraints and minimize the evaluation function.

  A model parameter estimation calculation processing program according to a third invention includes a calculation step of the model parameter estimation calculation method.

  A computer-readable recording medium according to a fourth aspect of the present invention records the model parameter estimation calculation processing program.

Therefore, the model parameter estimation calculation apparatus and method according to the present invention have the following specific effects.
(1) A priori information on time constant (upper and lower limits of time constant) Modeling in consideration of a priori information (upper and lower limits of steady gain) on at least one of steady gains, the quality is low and the amount is low A model with high accuracy and reliability can be obtained from the output waveform data.
(2) Since a control system can be designed and a simulation can be performed using a model with high accuracy and reliability, the accuracy and reliability of the control system and simulation are also improved.
(3) When a priori information on time constants or steady gains is used as at least one of constraints and evaluation functions, the obtained model necessarily matches these a priori information. . In the conventional method, after obtaining a model from input / output waveform data, it is verified whether the model matches a priori information, and if it does not match, the modeling process is restarted (from the beginning or from the middle). There was a need. In particular, it is difficult to build models that match a priori information in processes with large disturbances and noise, processes with limited input waveforms that can be applied, and processes with a small number of data that can be collected. Sampling, data section selection, sampling time adjustment, data preprocessing, etc. must be repeated through trial and error, and the modeling process required a lot of time and effort. By taking into account a priori information in advance, the number of trial and error / redo in the modeling process can be greatly reduced, and a model with high accuracy and reliability can be constructed in a short period of time. As a result, the lead time for study by control system construction and simulation is shortened, and the effects can be enjoyed from an early stage.

  Hereinafter, embodiments according to the present invention will be described with reference to the drawings. In addition, in each following embodiment, the same code | symbol is attached | subjected about the same component.

FIG. 1 is a block diagram showing a configuration of a model parameter estimation calculation device 10 according to an embodiment of the present invention. As shown in FIG. 1, the model parameter estimation calculation device 10 of this embodiment is configured to include a digital computer. For example, by executing the model parameter estimation calculation process of FIG. 2, the model parameter estimation calculation result is displayed. And output. The model parameter estimation calculation device 10 is a model parameter estimation calculation device that estimates and calculates model parameters of a first order lag system of a one-input / multiple-output system based on input / output data,
(A) a CPU (central processing unit) 20 of a computer that calculates and controls the operation and processing of the model parameter estimation calculation device 10;
(B) a ROM (read only memory) 21 for storing a basic program such as an operation program and data necessary for executing the basic program;
(C) A RAM (random access memory) 22 that operates as a working memory of the CPU 20 and temporarily stores parameters and data necessary for the model parameter estimation calculation processing of FIG.
(D) a data memory 23 configured by, for example, a hard disk memory and storing data such as input parameter data and simulation result data;
(E) a program memory 24 configured by, for example, a hard disk memory and storing the model parameter estimation calculation processing program of FIG. 3 read using the CD-ROM drive device 45;
(F) A keyboard that is connected to a keyboard 41 for inputting predetermined data and instruction commands, receives data and instruction commands input from the keyboard 41, performs predetermined signal conversion and other interface processing, and transmits the data to the CPU 20 Interface 31;
(G) Connected to a mouse 42 for inputting instruction commands on the CRT display 43, receives data and instruction commands input from the mouse 42, performs predetermined signal conversion and other interface processing, and transmits them to the CPU 20. A mouse interface 32;
(H) Connected to a CRT display 43 that displays data processed by the CPU 20, a setting instruction screen, and the like, converts image data to be displayed into an image signal for the CRT display 43, outputs it to the CRT display 43, and displays it. A display interface 33;
(I) A printer connected to a printer 44 that prints data processed by the CPU 20 and a predetermined model parameter estimation calculation result, etc., performs predetermined signal conversion of print data to be printed, and outputs it to the printer 44 for printing. An interface 34;
(J) Connected to the CD-ROM drive device 45 that reads the program data of the model parameter estimation calculation processing program from the CD-ROM 45a in which the model parameter estimation calculation processing program of FIG. A drive device interface 35 for transferring program data of the program to the program memory 24 by performing predetermined signal conversion, etc .;
(K) An A / D converter 25a for A / D converting an analog input signal from the input signal generator 51, and an analog output signal output from the modeling target system 50 in response to the analog input signal And an input interface 25 including an A / D converter 25b for A / D converting an analog output signal from the output signal detector 52 to be output.
Here, these circuits 20-25 and 31-34 are connected via the bus 30. Note that the input data and output data A / D converted by the A / D converters 25 a and 25 b are stored in the data memory 23 via the bus 30.

  Next, an estimation calculation method of model parameter estimation calculation processing used in the model parameter estimation calculation apparatus 10 according to the present embodiment will be described in detail below.

（Ｂ）入出力波形データ：

に対して、モデリングの際に最小化すべき評価関数を次式のように表す。 I-th output from a certain input (i = 1, 2,..., M; where the natural number M is the number of outputs, and the subscript i of each symbol hereinafter refers to the i-th output. (A) Model parameters up to

(B) Input / output waveform data:

On the other hand, the evaluation function to be minimized in modeling is expressed as follows.

と、定常利得Ｋ_ｉの上下限

が与えられるとする。また、（６）式の離散システムの安定性の条件は、よく知られているように、次式で表される。 Now, as a priori information, the upper and lower limits of the time constant T _i (> 0)

When upper and lower limits of the steady gain _{K i}

Is given. Further, the stability condition of the discrete system of equation (6) is expressed by the following equation, as is well known.

に関して、制約条件が線形でないので、次式のようにおく。 Consider minimizing the evaluation function of equation (13) under the constraints of equations (15), (16), and (17). As it is, model parameters

Since the constraint is not linear, the following equation is used.

  At this time, the equation (13) is expressed by the following equation.

here,

（時定数は正のためである。）、サンプリングタイム

であるから、（１８）式を用いると（１５）式の制約条件は次式で表される。 here,

(The time constant is positive.) Sampling time

Therefore, when the equation (18) is used, the constraint condition of the equation (15) is expressed by the following equation.

Further, when X _i (> 0) is applied to each side of the equation (16) and the equation (19) is substituted, the constraint condition of the equation (16) is expressed by the following equation.

を（１７）式の制約条件に代入して整理すると、次式を得る。 further,

Is substituted into the constraint condition of the equation (17) and rearranged, the following equation is obtained.

  Further, when the formula (18) is substituted and arranged, the following formula is obtained.

  Here, when the formulas (21), (22), and (24) are put together, the following formula is obtained.

は、（２１）式の左側の制約条件

を満たせば自然に成り立つので、（２５）式では考慮していない。パラメータ

に対して、（２５）式の線形制約条件を満たし、（２０）式の評価関数を最小化するものを求める問題は、例えば非特許文献４−５にあるように、二次計画問題として広く知られ、当該二次計画問題法に係る既知の数値計算アルゴリズムを用いて、すなわち、市販の数値計算ツールを用いて、パラメータ

を求めることができる。そして、（１８）式及び（１９）式から、

によって、モデルパラメータである時定数Ｔ_ｉと定常利得Ｋ_ｉの値を求めることができる。 In addition, the constraint condition on the left side of equation (24)

Is the constraint on the left side of equation (21)

Since it is natural if it satisfies the above, it is not considered in the equation (25). Parameters

On the other hand, the problem of obtaining the one that satisfies the linear constraint condition of the expression (25) and minimizes the evaluation function of the expression (20) is widely used as a quadratic programming problem as described in Non-Patent Document 4-5, for example. Using known numerical algorithms related to the quadratic programming problem, i.e. using commercially available numerical tools, parameters

Can be requested. From the equations (18) and (19),

Thus, the values of the time constant T _i and the steady gain K _i that are model parameters can be obtained.

  FIG. 3 is a flowchart showing model parameter estimation calculation processing executed by the model parameter estimation calculation apparatus 10 of FIG. The algorithm of the model parameter estimation calculation processing program according to the present embodiment will be described below using the flowchart of FIG.

[Step S1]
An analog input signal is applied from the input signal generator 51 to the modeling target system, and an analog output signal output from the modeling target system 50 in response thereto is detected by the output signal detector 52. The analog input signal is input to an A / D converter 25 a in the input interface 25 and A / D converted, and then stored in the data memory 23. The analog output signal is input to the A / D converter 25b in the input interface 25 and A / D converted, and then stored in the data memory 23. At this time, what is used as input waveform data and what is used as output waveform data is determined in accordance with the purpose of control or simulation. For example, when designing a control system, it is necessary to decide what to control (control amount: output waveform data) and what to operate (operation input: input waveform data). When there are a plurality of candidates as input / output waveform data, all the data may be collected.

  Specifically, for example, in the control of a waste treatment plant, the dust supply speed of the dust feeder is used as the model input, the furnace temperature and the boiler steam generation amount are used as the model output, and the supply air amount is used as the model output. It can be considered that the input is the sand layer temperature and the output of the model. In the converter pressure control system of the converter, it is conceivable that the damper opening of the exhaust system called RSE (ring / slit / element) is used as the model input, and the pressure at the converter furnace port is used as the model output. In the temperature prediction simulation of the steel sheet heating furnace, the fuel supply amount of the heating burner can be used as a model input, and the temperature of the steel sheet can be used as a model output.

  At the time of modeling, step input, sine wave superposition, pseudo white binary signal (M series, etc.), random input signal, etc. are used as input waveforms applied to the process. The input data to be applied has conditions (PE property, etc.) that should be theoretically satisfied, and ideally these conditions are desirably satisfied. However, the process to be modeled is often limited in the size and waveform of the input that can be applied on the device or in operation, and it is not always possible to apply ideal input data. .

[Step S2]
A data section to be used for modeling is selected from input / output waveform data including input data and output data stored in the data memory 23. Input / output waveform data may contain disturbances, noise, and abnormal values (sensor detection errors and abnormalities during data recording), and the effects of these disturbances, noise, and abnormal values during modeling. It is better to extract and use data with a smaller part. On the other hand, even if the length of the data used for modeling is too short, the estimation accuracy of the model parameters deteriorates due to the influence of disturbance and noise. The data section used for modeling is selected in consideration of these balances, and the selection result is input using the keyboard 41 or the mouse 42, for example.

[Step S3]
The sampling time Δt of the model is set and input using the keyboard 41 or the mouse 42, for example. The sampling time Δt is set according to the response speed of the system to be modeled, the frequency band to be modeled, and the like.

[Step S4]
Decimation (referred to as data filtering and thinning process) based on the sampling time Δt set in step S3, removing abnormal values, filtering to reduce the influence of disturbance and noise, etc. Pre-process input / output waveform data.

[Step S5]
Based on a priori information (theoretical / empirical detection), the upper and lower limits of the steady gain and the upper and lower limits of the time constant are set and input using the keyboard 41 or the mouse 42, for example. At this time, it is not always necessary to set both the upper limit value and the lower limit value, and only the upper limit value or the lower limit value may be set. Further, a restriction may be provided only on the steady gain, or a restriction may be provided only on the time constant. However, when the steady gain and the approximate value of the time constant are known, it is desirable to set upper and lower limits for each. This is to improve the accuracy and reliability of the model.

を０と置き換えればよい。これは、（２４）式の安定性の制約条件を考慮するためである。定常利得の下限値を制約として考慮しない場合には、（２５）式の不等式の３行目を取り除けばよい。同様に、定常利得の上限値を制約として考慮しない場合には、（２５）式の不等式の４行目を取り除けばよい。そして、 [Step S6]
Under the constraint condition of equation (25), a quadratic programming problem that minimizes the evaluation function of equation (20) is solved. Since the solving method of the quadratic programming problem is widely known, when solving these problems, a commercially available optimization calculation tool may be used, or a program for solving the problem yourself may be constructed. If the lower limit value of the time constant is not considered as a constraint, the first line of the inequality of equation (25) may be removed. When the upper limit value of the time constant is not considered as a constraint, the right side of the second line of equation (25)

May be replaced with 0. This is to take into account the stability constraint of equation (24). If the lower limit value of the steady gain is not considered as a constraint, the third line of the inequality of equation (25) may be removed. Similarly, when the upper limit value of the steady gain is not considered as a constraint, the fourth line of the inequality (25) may be removed. And

から、（２６）式と（２７）式によって、時定数Ｔと定常利得Ｋの値を計算する。 [Step S7]
Parameters obtained in the previous step

Thus, the values of the time constant T and the steady gain K are calculated by the equations (26) and (27).

[Step S8]
Verify that the model with the calculated model parameters is valid. For example, using data of a portion not used for modeling, the output waveform simulated using the model is compared with the output waveform data collected from the actual system for verification (cross-validation).

[Step S9]
As a result of verifying the model in the previous step S8, if the accuracy of the modeling is not less than a predetermined threshold value in view of the purpose of control or simulation, the model is adopted. If it is determined that the accuracy is insufficient, the process returns from step S2 to any one of steps S4, and the process of the subsequent steps is performed again. If a satisfactory model is still not obtained, the process returns to step S1 and starts again from where the input / output waveform data is collected. Here, the return destination of the processing may be determined on a case-by-case basis.

[Step S10]
After adopting the obtained model parameters and storing them in the data memory 23, the Bode diagram (gain diagram and phase diagram) of the obtained model is automatically plotted and displayed on the CRT display 43, for example. Here, we examine whether a reasonable model is obtained even when viewed in the frequency domain, that is, whether sufficient characteristics are obtained in an important frequency band for control and simulation, and the obtained model and actual A time response simulation is performed using the input waveform data of and compared with the actual output waveform. For example, if the mean square error (Root Mean Square) between the actual output waveform and the simulation waveform is larger than the allowable value, the process may return to any of steps S1 to S4, and the model may be obtained for each of a plurality of data. The one with the highest accuracy may be finally adopted. Alternatively, analyze the residual of the model, that is, obtain the autocorrelation function of the residual, or obtain the cross-correlation function of the residual and the input, and check whether the residual is white and independent of the input Validate.

  By modeling based on the above steps, it is possible to realize highly accurate and reliable modeling utilizing a priori knowledge about time constants and steady gains.

  In order to show the effectiveness of the model parameter estimation calculation method of the model parameter estimation calculation apparatus 10 according to the present embodiment, an example will be shown below.

  FIG. 4 is a waveform diagram showing input waveform data and input waveform data after disturbance addition in the system according to the first embodiment. FIG. 5 is a waveform diagram showing output waveform data in the system according to the first embodiment. Further, FIG. 6 illustrates output waveform data of a true model, output waveform data that is an estimation calculation result by the model parameter estimation calculation device 10 of the present embodiment, and model parameter estimation according to the prior art in the system according to the first embodiment. It is a wave form diagram which shows the output waveform data which is a calculation result.

  In the first embodiment, the true modeling target is assumed to be the steady gain K = 1 and the time constant T = 1 in the equation (1). In other words, the true transfer function to be modeled is expressed by the following equation.

  Here, assuming that the sampling time Δt = 0.1 [seconds], the true model discretized as shown in the equation (6) is expressed by the following equation.

の範囲にあり、定常利得Ｋが

の範囲にあることが分かっているものとする。この場合、（２５）式の制約条件は次式で表される。 Now, as a priori information, the time constant T is

And the steady-state gain K is

It is assumed that it is in the range of In this case, the constraint condition of the equation (25) is expressed by the following equation.

  For the modeling object having the characteristic of the equation (29), an input (a sine wave having a period of 1 second and an amplitude of 1, u (0) to u (9), u (0) = It is assumed that u (5) = 0) is applied as input waveform data. However, as a result of the disturbance added to the input, it is assumed that the input represented by the ♦ mark in FIG. 4 is added to the actual modeling target. Actually, it cannot be measured what disturbance is applied to the input. Note that the disturbance added here is a random sequence that follows a normal distribution with an average value of 0 and a standard deviation of 1.2, and uses the first 10 points among them. 4 is added, the system having the characteristic of equation (29) outputs response waveforms (y (1) to y (10)) as shown in FIG. 5 (however, the initial value y ( 0) = 0).

を二次計画問題の解法を利用して求め、（２６）式と（２７）式を利用して時定数と定常利得を求めると、時定数Ｔ＝１．１００４、定常利得Ｋ＝１．２のモデルを求めることができる。これらの値は、（３０）式と（３１）式の制約条件を満たしていることに注意する。このモデルに対し、図４の●印の正弦波入力を加えると、シミュレーションの結果、出力波形は図６の□印となる。また、（２９）式の特性を持つ真のモデル（時定数Ｔ＝１、定常利得Ｋ＝１）に、図４の●印の正弦波入力を加えると、その出力波形は、図６の●印となる。これらふたつの波形（図６の□印と●印）は、ほとんど一致しており、図４の◆印のような大きな入力外乱が加わったような場合でも、時定数と定常利得に関する先験的知識を活用することによって、高精度にモデリングができることが分かる。 Now, in actual modeling, since the disturbance applied to the input is unknown, using the sine wave input waveform data represented by ● in FIG. 4 and the output waveform data marked with ◆ in FIG. You will be modeling. Model parameters for minimizing the evaluation function (N = 10) of equation (20) under the constraints of equation (32)

Is obtained using the solution of the quadratic programming problem, and the time constant and the steady gain are obtained using the equations (26) and (27), the time constant T = 1.004 and the steady gain K = 1.2. Can be obtained. Note that these values satisfy the constraints of equations (30) and (31). When a sine wave input indicated by ● in FIG. 4 is added to this model, the output waveform is indicated by □ in FIG. 6 as a result of simulation. Further, when a sine wave input indicated by ● in FIG. 4 is added to a true model (time constant T = 1, steady gain K = 1) having the characteristics of the equation (29), the output waveform is as shown in FIG. It becomes a mark. These two waveforms (□ and ● in Fig. 6) are almost the same, and even when a large input disturbance such as the ◆ in Fig. 4 is applied, a priori with respect to time constant and steady gain. It can be seen that modeling can be performed with high accuracy by utilizing knowledge.

  On the other hand, when the model parameter is obtained so as to minimize only the evaluation function of the equation (13) without applying the constraints as in the equations (30) and (31), the model parameters in FIG. The model of time constant T = −0.48239 and steady gain K = −0.77388 is obtained from the input sine wave input and the output waveform data of ◆ in FIG. 5. The true model is the time constant. And the steady gain will be completely different. When a sine wave input indicated by ● in FIG. 4 is added to the model obtained by the prior art, the output waveform becomes Δ in FIG. 6 and is completely different from the true model waveform (● in FIG. 6). Become a thing. In order for the model to satisfy the stability condition of the equation (23), since the sampling time Δt = 0.1, the time constant must be T ≧ 0.05. That the time constant is T = −0.48239 means that an unstable model is obtained.

  Further, Table 1 shows the result of comparison in Example 1 with the magnitude of the disturbance changed.

  The disturbance added here is a random series that follows a normal distribution with an average value of 0 and a standard deviation σ, and uses the first 10 points, and the standard deviation σ is set to 0. From 0 (that is, no disturbance) to 2.0, it is increased in increments of 0.1. If there is no disturbance (σ = 0.0), the conventional method can also obtain true model parameters, but if the standard deviation σ of the input disturbance is greater than 0.2, the conventional method does not match a priori knowledge. In other words, a model is obtained (that is, the constraints of the expressions (30) and (31) are not satisfied). In some cases, an unstable model having a time constant T <0.05 is obtained, and in some cases, the sign of the steady-state gain is reversed. As described above, it can be confirmed that the model parameter estimation calculation method according to the present embodiment can obtain a highly accurate and highly reliable model by utilizing a priori knowledge about the time constant and the steady gain.

  As a result of the model parameter estimation according to the present embodiment shown in Table 1, the steady gain K is 0.8 or 1.2, and the time constant T is 0.8 <T <1.2. Even if only the steady gain is used as the knowledge (even if only the constraint condition of the steady gain is considered), the same result is obtained. To put it the other way around, it is an example that has the effect of improving modeling accuracy and reliability even if constraints are imposed only on the steady gain. On the other hand, when the time constant T is 0.8 or 1.2 and the steady gain K is 0.8 <K <1.2, even if only the time constant is used as a priori knowledge (time The same result was obtained (even considering only constant constraints). To put it the other way around, it is an example that has the effect of improving modeling accuracy and reliability even if only the time constant is constrained.

  FIG. 7 is a waveform diagram showing input waveform data and input waveform data after disturbance addition in the system according to the second embodiment. FIG. 8 is a waveform diagram showing output waveform data in the system according to the second embodiment. Furthermore, FIG. 9 shows the output waveform data of the true model, the output waveform data that is the estimation calculation result by the model parameter estimation calculation device 10 of the present embodiment, and the model parameter estimation according to the prior art in the system according to the second embodiment. It is a wave form diagram which shows the output waveform data which is a calculation result.

  In the second embodiment, for the modeling target having the characteristic of the equation (29), an input (a sine wave having a period of 1 second and an amplitude of 1, u (0) to u (9) represented by the mark ● in FIG. , U (0) = u (5) = 0) is applied as input waveform data. However, as a result of disturbance added to the input, it is assumed that the input indicated by the asterisk in FIG. 7 is added to the actual modeling target (in practice, it is impossible to measure what disturbance was added to the input. To do). The disturbance added here is a random sequence that follows a normal distribution with an average value of 0 and a standard deviation of 1.7, which is generated by commercially available calculation software, and the first 10 points are adopted. 7 is added, the system having the characteristic of equation (29) outputs response waveforms (y (1) to y (10)) as shown in FIG. 8 (however, the initial value y ( 0) = 0).

を二次計画問題の解法を利用して求め、（２６）式と（２７）式を利用して時定数と定常利得を求めると、時定数Ｔ＝１．２、定常利得Ｋ＝０．８のモデルが求まめることができる。このモデルに対し、図７の●印の正弦波入力を加えると、シミュレーションの結果、出力波形は図９の□印となる。また、（２９）式の特性を持つ真のモデル（時定数Ｔ＝１、定常利得Ｋ＝１）に、図７の●印の正弦波入力を加えると、その出力波形は、図９の●印となる。これらふたつの波形（図９の□印と●印）は、およそ一致しており、図７の◆印のように非常に大きな入力外乱が加わったような場合でも、時定数と定常利得に関する先験的知識を活用することによって、精度良くモデリングができることが分かる。 Now, in actual modeling, the disturbance applied to the input is unknown, so using the sine wave input waveform data represented by ● in FIG. 7 and the output waveform data marked with ◆ in FIG. You will be modeling. Model parameters for minimizing the evaluation function (N = 10) of equation (20) under the constraints of equation (32)

Is obtained using the solution of the quadratic programming problem, and the time constant and the steady gain are obtained using the equations (26) and (27), the time constant T = 1.2 and the steady gain K = 0.8. Can be obtained. When a sine wave input indicated by ● in FIG. 7 is added to this model, the output waveform is indicated by □ in FIG. 9 as a result of simulation. Further, when a sine wave input indicated by ● in FIG. 7 is added to a true model (time constant T = 1, steady gain K = 1) having the characteristics of the equation (29), the output waveform is as shown in FIG. It becomes a mark. These two waveforms (indicated by □ and ● in FIG. 9) are approximately the same, and even when a very large input disturbance is applied as in the case of ◆ in FIG. It can be seen that modeling can be performed with high accuracy by utilizing experimental knowledge.

  On the other hand, using the method according to the prior art, that is, the model parameters were obtained so as to minimize only the evaluation function of equation (13) without applying the constraints as in equations (30) and (31). In this case, a model with a time constant T = −0.69412 and a steady gain K = 0.155578 is obtained from the sine wave input indicated by ● in FIG. 7 and the output waveform data indicated by ♦ in FIG. Means that the time constant and the steady-state gain are completely different (this is called a conventional example).

  When a sine wave input indicated by ● in FIG. 7 is added to the model obtained in the conventional example, the output waveform becomes Δ in FIG. 9, which is completely different from the true model waveform (● in FIG. 9). Become a thing. Note that the time constant T = −0.69412 means that the stability condition, that is, the time constant T ≧ 0.05 is not satisfied and an unstable model is obtained.

  Furthermore, as in the method disclosed in Non-Patent Document 3, consider a case where constraints are imposed on the stability of the model and the sign of the steady gain. Here, let us consider a case where not only the sign of the steady gain but also the upper and lower limits of the steady gain are known as in equation (31) (this is a modified example). Note that not only the sign of the steady gain but also the upper and lower limit values as constraints can be expected to improve the accuracy and reliability of the model. The modification corresponds to a case in which the first and second lines are excluded from the inequalities in the constraint condition of the expression (32) (a case in which restrictions on time constants are not considered). In the modified example, a model having a time constant T = 35.4271 and a steady gain K = 1.2 is obtained from the sine wave input indicated by ● in FIG. 7 and the output waveform data indicated by ◆ in FIG. The model has a completely different time constant. When the sine wave input indicated by ● in FIG. 7 is added to the model obtained in the modified example, the output waveform becomes ◇ in FIG. 9 and greatly deviates from the true model waveform (● in FIG. 9). It will become a thing. As described above, it can be understood that the constraint condition regarding the steady state gain and the stability may lose the accuracy and reliability of the model, and it is also important to consider the constraint condition regarding the time constant.

  Furthermore, Table 2 shows the results of comparison in Example 2 with the magnitude of the disturbance changed.

  The disturbance added here is a random series that follows a normal distribution with an average value of 0 and a standard deviation σ, and uses the first 10 points, and the standard deviation σ is set to 0. From 0 (that is, no disturbance) to 2.0, it is increased in increments of 0.1. When there is no disturbance (standard deviation σ = 0.0), a true model parameter can be obtained even by the conventional method. However, when the standard deviation σ of the input disturbance becomes large, the conventional technique does not match a priori knowledge (that is, A model that does not satisfy the constraints of (30) and (31) is obtained. In some conventional examples, an unstable model having a time constant T <0.05 has been obtained, and the sign of the steady gain is reversed. Further, in the modified examples, there are many models that do not satisfy the constraint condition of the time constant of Expression (30). As described above, the present technology can confirm that a highly accurate and reliable model is obtained by utilizing a priori knowledge about the time constant and the steady gain.

  As a result of model parameter estimation according to the present embodiment in Table 2, the steady gain K is 0.8 or 1.2, and the time constant T is 0.8 <T <1.2. Even if only the steady gain is used as the knowledge (even if only the constraint condition of the steady gain is considered), the same result is obtained. To put it the other way around, it is an example that has the effect of improving modeling accuracy and reliability even if only the steady gain is constrained (in this case, the same result can be obtained in the modified example). On the other hand, when the time constant T is 0.8 or 1.2 and the steady gain K is 0.8 <K <1.2, even if only the time constant is used as a priori knowledge (time The same result was obtained (even considering only constant constraints). In other words, it is an example of improving the modeling accuracy and reliability even if only the time constant is constrained (by using a priori information on the time constant, compared to the modified example) , You can find a model with high accuracy and reliability).

  Note that the a priori information regarding the steady gain is considered as a constraint condition for the time constant, but it can be considered that the same effect can be obtained even if it is considered as an evaluation function. For example, an error from the average value of the upper and lower limits of the time constant may be entered into the evaluation function, or an error from the average value of the upper and lower limits of the steady gain may be entered into the evaluation function. When considering as an evaluation function, a weighting factor is added to adjust the balance (which evaluation item should be emphasized) with an expression that evaluates the degree of fitting to input / output data (for example, Expression (20)) Is good. Specific examples are shown below.

又は

又は

here,

Or

Or

Here, w _i1 , w _i2 and w _i3 are weighting factors.

  Furthermore, a priori information may be considered in both the evaluation function and the constraint condition. By attaching the constraint condition, even in the worst case, the model parameter can be accommodated in the range, and there is an effect of preventing the model parameter from sticking to the boundary of the constraint condition. Table 3 shows the results of calculation under the same conditions (same data) as in the example of Table 1 using the above evaluation function.

を用いて、ｗ_ｉ１＝ｗ_ｉ２＝ｗ_ｉ３＝１とした。表１と表３を比較して分かるように、表３の事例では、表１の事例に比べて、時定数Ｔや定常利得Kが制約条件の境界（上下限値）に張り付く頻度が少なくなっていることが確認できる。モデルパラメータの真値が制約条件の境界には存在せず、真値が上下限値の中ほどに存在するような場合には、このような評価関数と制約条件を考慮することによって、より真値に近いパラメータが得られる可能性を高くすることができる。 In the above formula,

And w _i1 = w _i2 = w _i3 = 1. As can be seen by comparing Table 1 and Table 3, in the case of Table 3, the frequency of sticking the time constant T and steady gain K to the boundary (upper and lower limit values) of the constraint conditions is lower than in the case of Table 1. Can be confirmed. When the true value of the model parameter does not exist at the boundary of the constraint condition and the true value exists in the middle of the upper and lower limit values, it is more true by considering such an evaluation function and the constraint condition. The possibility of obtaining a parameter close to the value can be increased.

<Modification>
In the above embodiment or example, when the model parameter estimation calculation processing program data of FIG. 3 is stored in the CD-ROM 45a and executed, it is loaded into the program memory 24 and executed. However, the recording medium may be stored in various recording media such as an optical disk such as a CD-R, a CD-RW, a DVD, or an MO, a magneto-optical disk, or a magnetic disk such as a floppy disk. These recording media are computer-readable recording media. Further, the model parameter estimation calculation process data of FIG. 3 may be stored in the program memory 24 in advance and the process may be executed.

As described above in detail, the model parameter estimation calculation apparatus and method according to the present invention have the following specific effects.
(1) A priori information on time constant (upper and lower limits of time constant) Modeling in consideration of a priori information (upper and lower limits of steady gain) on at least one of steady gains, the quality is low and the amount is low A model with high accuracy and reliability can be obtained from the output waveform data.
(2) Since a control system can be designed and a simulation can be performed using a model with high accuracy and reliability, the accuracy and reliability of the control system and simulation are also improved.
(3) When a priori information on time constants or steady gains is used as at least one of constraints and evaluation functions, the obtained model necessarily matches these a priori information. . In the conventional method, after obtaining a model from input / output waveform data, it is verified whether the model matches a priori information, and if it does not match, the modeling process is restarted (from the beginning or from the middle). There was a need. In particular, it is difficult to build models that match a priori information in processes with large disturbances and noise, processes with limited input waveforms that can be applied, and processes with a small number of data that can be collected. Sampling, data section selection, sampling time adjustment, data preprocessing, etc. must be repeated through trial and error, and the modeling process required a lot of time and effort. By taking into account a priori information in advance, the number of trial and error / redo in the modeling process can be greatly reduced, and a model with high accuracy and reliability can be constructed in a short period of time. As a result, the lead time for study by control system construction and simulation is shortened, and the effects can be enjoyed from an early stage.

It is a block diagram which shows the structure of the model parameter estimation calculating apparatus 10 which concerns on one Embodiment of this invention. It is a wave form diagram which shows the input-output waveform data which shows the step response waveform of the primary delay type | system | group based on a prior art. It is a flowchart which shows the model parameter estimation calculation process performed by the model parameter estimation calculation apparatus 10 of FIG. It is a waveform diagram which shows the input waveform data and the input waveform data after disturbance addition in the system which concerns on Example 1. FIG. 6 is a waveform diagram showing output waveform data in the system according to Embodiment 1. FIG. In the system according to the first embodiment, the output waveform data of the true model, the output waveform data that is the estimation calculation result by the model parameter estimation calculation device 10 of the present embodiment, and the output that is the result of the model parameter estimation calculation according to the prior art It is a wave form diagram which shows waveform data. It is a wave form diagram which shows the input waveform data and the input waveform data after disturbance addition in the system which concerns on Example 2. FIG. 6 is a waveform diagram showing output waveform data in a system according to Embodiment 2. FIG. In the system according to Example 2, the output waveform data of the true model, the output waveform data that is the estimation calculation result by the model parameter estimation calculation device 10 of the present embodiment, and the output that is the result of the model parameter estimation calculation according to the prior art It is a wave form diagram which shows waveform data.

Explanation of symbols

10 ... Model parameter estimation calculation device,
20 ... CPU,
21 ... ROM,
22 ... RAM,
23: Data memory,
24 ... Program memory,
25 ... Input interface,
25a, 25b ... A / D converter,
30 ... Bus
31 ... Keyboard interface,
32 ... Mouse interface,
33 ... Display interface,
34 ... Printer interface,
35 ... Drive device interface,
41 ... Keyboard,
42 ... mouse,
43 ... CRT display
44 ... Printer,
45 ... CD-ROM drive device,
45a ... CD-ROM,
50 ... modeling target system,
51 ... Input signal generator,
52. Output signal detector.

Claims (10)

In a model parameter estimation calculation device that estimates and calculates a model parameter of a first order lag system of a one-input multiple-output system based on input / output data,
Based on the input / output data, a priori information including at least one of a time constant and a steady-state gain of the first-order lag system of the one-input / multiple-output system is considered as at least one of the constraint condition and the evaluation function. And a model parameter estimation / calculation device comprising a calculation means for estimating and calculating the model parameter.   2. The model parameter estimation calculation apparatus according to claim 1, wherein the a priori information is at least one of an upper limit value and a lower limit value of a time constant and a steady gain.   The calculation means uses the a priori information as linear constraint information based on the input / output data, and outputs when the output data included in the input / output data and the model of the model parameter are estimated and calculated The model parameter estimation calculation apparatus according to claim 1, wherein the model parameter is estimated and calculated so that an evaluation function indicating an error from data is minimized. The evaluation function is a square error between the output data included in the input / output data and the output data when the model parameter is estimated and calculated,
4. The calculation means according to claim 1, wherein a quadratic programming problem method is used to estimate and calculate a model parameter that satisfies the linear constraints and minimizes the evaluation function. The model parameter estimation calculation apparatus as described in one. In a model parameter estimation calculation method for estimating and calculating a model parameter of a first order lag system based on input and output data,
Based on the input / output data, a priori information including at least one of a time constant and a steady-state gain of the first-order lag system of the one-input / multiple-output system is considered as at least one of the constraint condition and the evaluation function. A model parameter estimation calculation method comprising a calculation step of estimating and calculating the model parameter.   6. The model parameter estimation calculation method according to claim 5, wherein the a priori information is at least one of an upper limit value and a lower limit value of a time constant and a steady gain.   The calculation step uses the a priori information as linear constraint information based on the input / output data, and outputs when the output data included in the input / output data and the model parameter model are estimated and calculated 7. The model parameter estimation calculation method according to claim 5, wherein the model parameter is estimated and calculated so that an evaluation function indicating an error from data is minimized. The evaluation function is a square error between the output data included in the input / output data and the output data when the model parameter is estimated and calculated,
8. The calculation step according to claim 5, wherein a model parameter that satisfies the linear constraint condition and minimizes the evaluation function is estimated using a quadratic programming problem method. The model parameter estimation calculation method according to one.   9. A model parameter estimation calculation processing program comprising a calculation step of the model parameter estimation calculation method according to claim 5.   10. A computer-readable recording medium on which the model parameter estimation calculation processing program according to claim 9 is recorded.

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