Output device, output method, and program

Abstract

To enable a user to use a model appropriately expressing a nonlinear system.SOLUTION: Classifying a plurality of first models into a plurality of classes, the plurality of first models being represented by a product of a plurality of control inputs of a nonlinear system in a predetermined period, a disturbance input when each control input is input, and a matrix made of a plurality of basis functions randomly selected from some basis functions among a plurality of basis functions to which an output of the nonlinear system is input, the plurality of first models being capable of predicting the output of the nonlinear system; A degree of coincidence between an estimated value of an output based on a second model obtained by aggregating first models of each class and an actual output, and the number of elements having a larger influence on the estimated value than other elements among all elements of a coefficient matrix of each of the two models are output.SELECTED DRAWING: Figure 3

Description

[Technical Field]

[0001] The present invention relates to an output device, an output method, and a program for outputting information about a model capable of predicting the output of a nonlinear system. [Background technology]

[0002] There are known techniques for expressing the output of a nonlinear system using multiple functions. Patent Document 1 discloses a technique for generating a model that expresses a nonlinear system by randomly selecting a function from a list containing multiple functions. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] Special Publication No. 2005-521158 Summary of the Invention [Problem to be solved by the invention]

[0004] However, in the technology of Patent Document 1, because a function is selected randomly from multiple functions, a function physically related to the nonlinear system to be represented may not be selected. In this case, the generated model may not be able to properly represent the nonlinear system, and the output of the model may diverge from the output of the nonlinear system. This poses a problem in that users cannot use a model that properly represents the nonlinear system.

[0005] The present invention has been made in view of these points, and has as its object to enable users to use a model that appropriately represents a nonlinear system. [Means for solving the problem]

[0006] In a first aspect of the present invention, there is provided an acquisition unit that acquires a plurality of control inputs input to a nonlinear system within a predetermined period, a disturbance input that affects the nonlinear system when each control input is input to the nonlinear system, and an output of the nonlinear system when each control input and each disturbance input are input to the nonlinear system; a first generation unit that randomly selects some basis functions from a plurality of basis functions to which at least one of the control input, the disturbance input, and the output is input, thereby generating a plurality of first models that are expressed as a product of a coefficient matrix in which coefficients of the selected plurality of basis functions are arranged and a matrix made of the randomly selected plurality of basis functions, and that can predict the output of the nonlinear system when the control input is input; and a determination unit that determines the coefficient matrix corresponding to each first model by optimizing the coefficient matrix of each of the plurality of first models generated by the first generation unit. a classification unit that classifies the coefficient matrix of each first model into each class, thereby classifying each of the plurality of first models into a plurality of classes; a second generation unit that generates, for each class, a second model represented by the product of a new coefficient matrix determined by statistics of the coefficient matrix of one or more of the first models included in each class and a matrix made of the plurality of basis functions of the one or more first models included in each class; an estimation unit that estimates, for each second model, an estimate of an output of the nonlinear system within the predetermined period using the coefficient matrix, the control input, and the disturbance input of each second model generated by the second generation unit; and an output unit that outputs, in association with the second model, the number of elements that have a greater influence on the estimate than other elements among all elements included in the coefficient matrix of the second model, and the degree of agreement between the output acquired by the acquisition unit and the estimate estimated by the estimation unit.

[0007] The classifying unit may classify each of the first models into classes by performing k-means clustering on the coefficient matrix of each of the first models.

[0008] The classification unit may classify the first models among the plurality of first models whose degree of matching is greater than a predetermined value into each class, and discard the first models among the plurality of first models whose degree of matching is less than or equal to the predetermined value without classifying them into any of the plurality of classes.

[0009] The apparatus may further include a calculation unit that calculates, as the degree of agreement, a coefficient of determination obtained by subtracting the ratio of the sum of squares of the difference between the output acquired by the acquisition unit and the average value of the output to the sum of squares of the difference between the output and the estimated value estimated by the estimation unit from 1.

[0010] The determination unit may determine the coefficient matrix for each of the plurality of first models by determining, for each first model, a coefficient matrix that minimizes the sum of squares of the difference between the output and the estimated value of the first model.

[0011] The second generation unit may generate, for each class, the second model represented by the product of a new coefficient matrix determined by an average value or a median value of the coefficient matrices of the one or more first models included in each class and a matrix made of the plurality of basis functions of the one or more first models.

[0012] The output unit may output the number of elements having an influence on the estimated value equal to or greater than a threshold value among all the elements of the coefficient matrix of the second model, and the degree of agreement, in association with the second model.

[0013] The output unit may output the number of elements having a degree of influence on the estimated value greater than zero among all the elements of the coefficient matrix of the second model and the degree of agreement in association with the second model.

[0014] The acquisition unit may acquire, as the control input, the pressure of air delivered by a turbocharger to the engine and the ratio of exhaust gas supplied to an intake pipe of the engine to exhaust gas discharged by the engine, acquire, as the disturbance input, the amount of fuel injected into a combustion chamber of the engine and the rotation speed of the engine, and acquire, as the output, the opening of a nozzle vane that controls the flow rate of the exhaust gas supplied to a turbine of the turbocharger and the opening of a valve that supplies the exhaust gas to the intake pipe, and the estimation unit may estimate the estimated values ​​of the opening of the nozzle vane and the opening of the valve as estimated values ​​of the output of the nonlinear system.

[0015] In a second aspect of the present invention, a computer mounted on a vehicle performs the steps of acquiring a plurality of control inputs input to a nonlinear system within a predetermined period, a disturbance input that affects the nonlinear system when each control input is input to the nonlinear system, and an output of the nonlinear system when each control input and each disturbance input are input to the nonlinear system; randomly selecting some basis functions from a plurality of basis functions to which at least one of the control input, the disturbance input, and the output is input, thereby generating a plurality of first models that are expressed as a product of a coefficient matrix in which coefficients of the selected plurality of basis functions are arranged and a matrix made up of the randomly selected plurality of basis functions, and that are capable of predicting the output of the nonlinear system when the control input is input; and optimizing the coefficient matrix of each of the generated plurality of first models. determining the coefficient matrix corresponding to each first model; classifying each of the plurality of first models into a plurality of classes; generating, for each class, a second model represented by the product of a new coefficient matrix determined by statistics of the coefficient matrix of one or more of the first models included in each class and a matrix made of the plurality of basis functions of the one or more first models included in each class; estimating, for each second model, an estimate of the output of the nonlinear system within the predetermined period using the coefficient matrix of each generated second model, the control input, and the disturbance input; and outputting, in association with the second model, the number of elements that have a greater influence on the estimate than other elements among all elements included in the coefficient matrix of the second model, and the degree of agreement between the output acquired by the acquisition unit and the estimate estimated by the estimation unit.

[0016] In a third aspect of the present invention, a computer mounted on a vehicle includes an acquisition unit that acquires a plurality of control inputs input to a nonlinear system within a predetermined period, a disturbance input that affects the nonlinear system when each control input is input to the nonlinear system, and an output of the nonlinear system when each control input and each disturbance input are input to the nonlinear system; a first generation unit that randomly selects some basis functions from a plurality of basis functions to which at least one of the control input, the disturbance input, and the output is input, and generates a plurality of first models that are expressed as a product of a coefficient matrix in which coefficients of the selected plurality of basis functions are arranged and a matrix made of the randomly selected plurality of basis functions, and that can predict the output of the nonlinear system when the control input is input; and a first generation unit that optimizes the coefficient matrix of each of the plurality of first models generated by the first generation unit, and generates a plurality of first models that can predict the output of the nonlinear system when the control input is input. a classifier that classifies each of the plurality of first models into a plurality of classes; a second generator that generates, for each class, a second model represented by the product of a new coefficient matrix determined by statistics of the coefficient matrix of one or more of the first models included in each class and a matrix made of the plurality of basis functions of the one or more first models included in each class; an estimator that estimates, for each of the second models, an estimate of an output of the nonlinear system within the predetermined period using the coefficient matrix of each of the second models generated by the second generator, the control input, and the disturbance input; and an output unit that outputs, in association with the second model, the number of elements that have a greater influence on the estimate than other elements among all elements included in the coefficient matrix of the second model, and the degree of agreement between the output acquired by the acquisition unit and the estimate estimated by the estimator. [Effects of the Invention]

[0017] The present invention has the effect of enabling a user to use a model that appropriately represents a nonlinear system. [Brief explanation of the drawings]

[0018] [Figure 1]1 is a diagram for explaining the configuration of an intake and exhaust system S of a diesel engine. [Figure 2] FIG. 2 is a diagram illustrating a configuration of an output device. [Figure 3] FIG. 10 is a diagram illustrating a processing flow. [Figure 4] FIG. 4 is a diagram for explaining time-series data of input and output of an intake and exhaust system. [Figure 5] FIG. 10 is a diagram for explaining a classification result. [Figure 6] 1 is a table showing the coefficient of determination and the number of parameters for each class. [Figure 7] FIG. 10 is a diagram illustrating an example of time-series data. [Figure 8] This is an enlarged view of the range of 1250 seconds to 1260 seconds of the time series data. [Figure 9] This is an example of a simulation result of multi-step prediction in the basic SINDy model based on the initial state and time series data. [Figure 10] 10 is a graph showing a simulation result using an optimum model according to the present embodiment. [Figure 11] 10 is a graph showing an example of an output. [Figure 12] 10 is a table showing simulation results at different noise levels. [Figure 13] 10 is a flowchart illustrating an example of a process for outputting the number of parameters and the degree of match. DETAILED DESCRIPTION OF THE INVENTION

[0019] [Configuration of diesel engine intake and exhaust system S] The present invention provides a model that can appropriately predict the output of an industrial system having a nonlinearity. In this embodiment, a model that can appropriately predict the output of an intake and exhaust system of a diesel engine will be described as an example.

[0020] 1 is a diagram illustrating the configuration of an intake and exhaust system S of a diesel engine. The intake and exhaust system S is a nonlinear system. The intake and exhaust system S has an intake manifold 110, an intake pipe 111, an intercooler 112, an intake throttle 113, an exhaust manifold 120, an EGR pipe 121, an EGR cooler 122, an EGR valve 123, a supercharger 130, a turbine 131, a compressor 132, and an injector 140.

[0021] An intake manifold 110 supplies intake air to each of the multiple cylinders 100 of the engine. An intake pipe 111 is connected to the intake manifold 110 and supplies air (fresh air) taken in from the outside to the intake manifold 110. An intercooler 112 and an intake throttle 113 are provided in the intake pipe 111. The intercooler 112 is a heat exchanger that cools the intake air by exchanging heat between the intake air and engine cooling water or outside air. The intake throttle 113 is a valve that adjusts the amount of intake air supplied to the engine.

[0022] The exhaust manifold 120 collects exhaust gas discharged from each cylinder 100 and discharges it to the outside. An EGR pipe 121 is connected to the exhaust manifold 120. The EGR pipe 121 is connected to the intake manifold 110. A portion of the exhaust gas discharged from the engine passes through the EGR pipe 121 and reaches the intake manifold 110. The EGR pipe 121 is provided with an EGR cooler 122 and an EGR valve 123. The EGR cooler 122 is a heat exchanger that cools the exhaust gas reaching the intake manifold 110 by exchanging heat between engine cooling water or outside air and the exhaust gas reaching the intake manifold 110. The EGR valve 123 is a valve that adjusts the amount of exhaust gas supplied to the intake manifold 110.

[0023] The intake / exhaust system S can adjust the proportion of exhaust gas supplied to the intake manifold 110 by adjusting the opening degree (or closing rate) of the EGR valve 123. Hereinafter, the opening degree of the EGR valve 123 will be referred to as the "EGR valve opening degree." This allows the intake / exhaust system S to adjust the concentration of oxygen reaching the cylinder 100. That is, by mixing fresh air with exhaust gas having a lower oxygen concentration than the fresh air, the intake / exhaust system S can lower the oxygen concentration of the intake air supplied to the cylinder 100 compared to when only fresh air is supplied to the cylinder 100. By lowering the oxygen concentration of the intake air, the intake / exhaust system S can lower the combustion temperature when the intake air and fuel are combusted. As a result, the intake / exhaust system S can suppress the amount of harmful nitrogen oxides (NOx), which are more likely to be generated as the combustion temperature increases. Note that the other part of the exhaust gas passes through the turbine 131 of the turbocharger 130 and is discharged to the outside. In the following description, the proportion of the exhaust gas supplied to the intake manifold 110 to the exhaust gas discharged by the engine will be referred to as the EGR rate.

[0024] The supercharger 130 pressurizes fresh air supplied to the engine by rotating a turbine 131 with engine exhaust air to drive a compressor 132. The supercharger 130 is, for example, a variable geometry turbocharger (hereinafter referred to as "VGT"), but is not limited to this. In the following description, the pressure of the air (fresh air) delivered by the supercharger 130 to the cylinder 100 of the engine is referred to as boost pressure.

[0025] The turbocharger 130 can control the flow velocity of the exhaust gas supplied to the turbine 131. For example, the turbocharger 130 has nozzle vanes for controlling the flow velocity of the exhaust gas, and controls the flow velocity of the exhaust gas by adjusting the opening degree (or closing rate) of the nozzle vanes by changing the position or angle of the nozzle vanes. As a specific example, the turbocharger 130 increases the flow velocity by decreasing the opening degree of the nozzle vanes (hereinafter referred to as "VGT vane opening degree").

[0026] Injector 140 injects a predetermined amount of fuel into the combustion chamber of cylinder 100 of the engine. Injector 140 injects an amount of fuel into the combustion chamber based on, for example, the operation of the vehicle driver. Specifically, injector 140 injects an amount of fuel into the combustion chamber that corresponds to the amount of depression of the accelerator pedal by the driver. In the following description, the amount of fuel injected by injector 140 into the combustion chamber is referred to as the fuel injection amount.

[0027] The amount of intake air supplied to the cylinder 100 varies depending on the compression ratio of the compressor 132 of the turbocharger 130 and the amount of exhaust gas supplied from the EGR pipe 121. The amount of exhaust gas discharged from the cylinder 100 varies depending on the amount of intake air supplied to the cylinder 100, the fuel injection amount, and the engine speed. Thus, the intake / exhaust system S has strong interference between the intake and exhaust. Furthermore, the boost pressure and EGR rate of the intake / exhaust system S vary depending on the fuel injection amount and the engine speed (disturbance input) even if the EGR valve opening and VGT vane opening are constant. Therefore, the intake / exhaust system S has strong nonlinearity between the EGR valve opening and VGT vane opening (control input) and the boost pressure and EGR rate (output). When designing a controller for an industrial system with strong nonlinearity, such as the intake / exhaust system S, it is difficult to perform model-based design based on a mathematical model.

[0028] Therefore, the output device according to this embodiment outputs a model capable of predicting the output of the intake and exhaust system S using data-driven modeling that actively utilizes time-series data of the input and output of the intake and exhaust system S.

[0029] [Configuration of output device 200] The configuration of the output device 200 according to this embodiment will be described below with reference to Fig. 2 and Fig. 3. Fig. 2 is a diagram for explaining the configuration of the output device 200. Fig. 3 is a diagram for explaining the flow of processing.

[0030] The output device 200 has a storage unit 210 and a control unit 220. The storage unit 210 is a storage medium including a ROM (Read Only Memory), a RAM (Random Access Memory), a hard disk, etc. The storage unit 210 stores a program executed by the control unit 220.

[0031] The control unit 220 is a computational resource including a processor such as a CPU (Central Processing Unit). The control unit 220 executes a program stored in the storage unit 210 to realize functions as an acquisition unit 221, a first generation unit 222, a determination unit 223, a first estimation unit 224, an extraction unit 225, a classification unit 226, a second generation unit 227, a second estimation unit 228, and an output unit 229.

[0032] The acquisition unit 221 first acquires time-series data of the input and output of the intake and exhaust system S, and performs preprocessing on the time-series data for creating a model ((i) in FIG. 3). Specifically, the acquisition unit 221 acquires time-series data of the control input, disturbance input, and output of the intake and exhaust system S.

[0033] FIG. 4 is a diagram for explaining time-series data of input and output of the intake and exhaust system S. u1 is the VGT vane opening degree, and its unit is percentage [%]. Note that u1 may also be the VGT vane closing rate. u2 is the EGR valve opening degree, and its unit is percentage [%]. u2 may also be the EGR valve closing rate. The acquisition unit 221 acquires the VGT vane opening degree and the EGR valve opening degree as multiple control inputs input to the intake and exhaust system S within a predetermined period. Specifically, the acquisition unit 221 acquires a control input vector (u1∈R, u2∈R) in which multiple control inputs input to the intake and exhaust system S within a predetermined period are arranged in time series.

[0034] d1 is the fuel injection amount, and its unit is the injection amount per injection (cubic millimeters) [mm 3 / st]. d2 is the engine speed, measured in revolutions per minute [rpm]. The fuel injection amount and engine speed are determined according to the driver's operation, and are defined as disturbance inputs that affect the intake and exhaust system S when each control input is input to the intake and exhaust system S. In other words, the acquisition unit 221 acquires the fuel injection amount and engine speed as disturbance inputs. Specifically, the acquisition unit 221 acquires a disturbance input vector (d1∈R, d2∈R) in which multiple disturbance inputs input to the intake and exhaust system S within a predetermined period are arranged in chronological order.

[0035] y1 is the boost pressure in pascals [Pa]. y2 is the EGR rate in percent [%]. The acquisition unit 221 acquires the boost pressure and the EGR rate as outputs of the intake and exhaust system S when each control input (VGT vane opening and EGR valve opening) and each disturbance input (fuel injection amount and engine speed) are input to the intake and exhaust system S. Specifically, the acquisition unit 221 acquires an output vector (y1∈R, y2∈R) in which multiple outputs of the intake and exhaust system S within a predetermined period are arranged in chronological order.

[0036] The following formula (1) is obtained based on the control input vector, the disturbance input vector, and the output vector acquired by the acquisition unit 221. Each function in formula (1) is defined by formula (2).

number

[0037] f in equation (1) p is a nonlinear function that represents the intake and exhaust system S. u ∈Z is the order of control inputs for the intake and exhaust system S. d ∈Z is the order of disturbance input to the intake and exhaust system S. n y ∈Z is the order of the output of the intake and exhaust system S. t is time.

[0038] The acquisition unit 221 performs preprocessing on the acquired time-series data. For example, the acquisition unit 221 performs centering on the acquired time-series data. Specifically, the acquisition unit 221 performs centering to correct the ranges of the acquired control input, disturbance input, and output so that they are the same. More specifically, the acquisition unit 221 corrects each of the control input, disturbance input, and output so that the maximum and minimum values ​​of each of the control input, disturbance input, and output are the same.

[0039] The acquisition unit 221 may perform other preprocessing, such as filtering to remove unnecessary data or normalization to convert data of different scales into a common scale.

[0040] Here, we will describe a basic SINDy with control and exogenous inputs for modeling the intake and exhaust system S. Controller design, including Model Predictive Control (MPC), is generally performed using a discrete model, so the model formulation is introduced in discrete-time form. If the model is described in continuous time, an integration such as the Runge-Kutta method is required in the MPC implementation. In addition, the input and output signals are obtained by a zero-order Holder (ZOH) that converts analog signals to digital signals. In this embodiment, we consider a discrete nonlinear dynamic system expressed by the following equation (3):

number

number

[0041] The dynamics (equation of motion) of the intake and exhaust system S is expressed by the following equation (8) by introducing a basis function Θ(X, Γ, D) called a library or dictionary. Ξ in equation (8) is defined by equation (9).

number

number

number

[0042] The determining unit 223 determines the coefficient matrix ξ corresponding to the basis functions of the first model.i For example, the determination unit 223 determines the coefficient matrix ξ of the first model. i By optimizing the coefficient matrix ξ i Specifically, the determination unit 223 determines a coefficient matrix ξ that minimizes the sum of squares of the difference between the output and the estimated value of the first model. i More specifically, the determination unit 223 determines the coefficient matrix ξ by solving (performing convex calculation) the optimization problem of the following equation (12): i Determine.

number

[0043] X in equation (12) i + is X + represents the i-th row component of the equation. The regularization term λ0 is a sparsity promoting hyperparameter that is adjusted to obtain the best estimate of the output of the intake and exhaust system S. λ0 may be automatically adjusted by the determination unit 223, or a user value may be determined. In this embodiment, the specific value of λ0 is 30, but is not limited to this.

[0044] In the optimization problem, the coefficient matrix ξ i is obtained by LASSO (least absolute shrinkage and selection operator) regression or STLS (sequentially thresholded least-squares). The obtained coefficient matrix ξ i Using the above, the differential equation (dynamic system) of the intake and exhaust system S is expressed as equation (13). Θ in equation (13) is defined by equation (14). TIFF2026005738000009.tif25162

[0045] When a model is generated using a rich or excessive number of basis functions (library) such as that in equation (10), combinations of basis functions with high correlation coefficients may occur, which may result in multicollinearity. When multicollinearity occurs, the estimated values ​​of the coefficients change irregularly in response to small changes in the time series data, making it difficult to determine appropriate coefficients.

[0046] Therefore, in order to promote sparsity and reduce the risk of multicollinearity, the first generation unit 222 performs library bagging (bootstrap aggregating) to randomly select some basis functions from the multiple basis functions ((ii) in FIG. 3). In other words, the first generation unit 222 generates a set of multiple basis functions by randomly selecting multiple basis functions from the multiple basis functions (library) shown in equation (10). Specifically, the first generation unit 222 randomly determines the number of basis functions to select, and randomly selects the determined number of basis functions.

[0047] The first generation unit 222 generates a matrix made up of a plurality of randomly selected basis functions and a coefficient matrix ξ corresponding to the plurality of basis functions. i The first generation unit 222 generates a first model represented by the product of x and y. The first generation unit 222 executes random selection of basis functions multiple times to generate multiple first models. The first generation unit 222 generates, for example, 100 first models, but the number of first models generated is not limited to this. The number of first models generated by the first generation unit 222 may be set appropriately by the user.

[0048] The determination unit 223 determines the coefficient matrix of each first model by solving an optimization problem of SINDy. The determination unit 223 determines the coefficient matrix of each first model by using the above-mentioned STSL.

[0049] Conventional basic SINDy is applicable to complex nonlinear systems. However, while conventional basic SINDy can provide a model that can realize one-step prediction, it may not be able to provide a model that can realize multi-step prediction. Therefore, we will explain a new algorithm for SINDy (introduction of time-delay coordinates) that realizes multi-step prediction.

[0050] The state x(t) includes multiple states of the intake and exhaust system S. In the conventional basic SINDy, it is assumed that all of the multiple states are measurable. However, in the intake and exhaust system S, while some of the multiple states are measurable, many of the multiple states are not measurable. In fact, the only measurable states (outputs) in the intake and exhaust system S are the boost pressure and the EGR rate. Therefore, it is necessary to expand the coordinates (phase space) from the measurable states given by the following equation (15).

number

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[0051] Unlike linear systems, in nonlinear systems such as the intake and exhaust system S, even if accurate one-step prediction is achieved, multi-step prediction may not be achieved (diverge). To evaluate whether multi-step prediction is achieved, one-step prediction and multi-step prediction are defined. One-step prediction is defined by the following equation (17).

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number

[0052] As described above, the multi-step prediction is performed from the control input u, the disturbance input d, the estimated value (y), and the initial output x(0). That is, the first estimation unit 224 estimates an estimated value of the output of the intake / exhaust system S multi-steps ahead (within a predetermined period) from time t=0 using the coefficient matrix, control input, and disturbance input of the first model. The first estimation unit 224 estimates an estimated value of the output of the intake / exhaust system S within a predetermined period for each of a plurality of first models generated by library bagging. In this embodiment, the first estimation unit 224 estimates estimated values ​​of the EGR valve opening and the VGT vane opening within a predetermined period from the initial state (time t=0) to 2500 seconds.

[0053] The extraction unit 225 extracts, from among the multiple first models generated, a first model whose estimated value based on the first model closely matches the actual output of the intake and exhaust system S as an elite model ((iii) in FIG. 3). The extraction unit 225 first calculates, for each first model, the degree of agreement between the estimated value of the multi-step (long-term) prediction of the first model and the output. Specifically, the extraction unit 225 calculates the degree of agreement as a coefficient of determination obtained by subtracting from 1 the ratio of the sum of squares of the difference between the output and the estimated value to the sum of squares of the difference between the output and the average value of the output. The coefficient of determination R of y 2 is defined by the following equation (19).

number

[0054] The y bar (the horizontal bar symbol above y) in equation (19) is the average value of m outputs. 2The closer to 1, the better the estimated value of the output of the intake and exhaust system S estimated using the first model matches the actual output of the intake and exhaust system S.

[0055] The extraction unit 225 extracts the coefficient of determination R of the first model. 2 For example, the extraction unit 225 extracts an elite model based on the coefficient of determination R 2 Specifically, the extraction unit 225 extracts an elite model having a coefficient of determination R 2 The first model having a value greater than a predetermined value is extracted as an elite model. From an engineering viewpoint, the predetermined value is preferably 0.8, and more preferably 0.9, but is not limited to this.

[0056] The extraction unit 225 extracts at least a predetermined number of elite models. The predetermined number may be set appropriately by the user, and is, for example, 50. If the number of extracted elite models is less than 50, the extraction unit 225 causes the first generation unit 222 to perform library bagging again. In other words, the output device 200 repeats library bagging, determination of coefficient matrices, calculation of coincidence, and extraction of elite models until the number of extracted elite models reaches 50.

[0057] If a predetermined number of elite models or more are not extracted even after repeating library bagging a predetermined number of times, the first generation unit 222 increases the number of first models to be generated. For example, if a predetermined number (50) of elite models or more are not extracted even after repeating library bagging 500 times, the first generation unit 222 increases the number of first models to be generated from 100 to 200.

[0058] By doing this, the extraction unit 225 can exclude first models whose estimated values ​​of the output of the intake and exhaust system S do not match the actual output of the intake and exhaust system S, and extract a predetermined number or more of elite models whose estimated values ​​closely match the actual output of the intake and exhaust system S.

[0059] However, because the first generation unit 222 randomly selects basis functions from a plurality of basis functions (library), the selected basis functions may not be physically motivated (unrelated) to the intake and exhaust system S. As a result, each of the plurality of elite models extracted by the extraction unit 225 may tend to have different positions and numbers of zero elements in the coefficient matrix, and different sizes of elements within zeros. Aggregating (e.g., averaging) a plurality of elite models with different tendencies may result in a decrease in the number of zero elements among the plurality of elements in the coefficient matrix (decreased sparsity). Furthermore, a coefficient matrix obtained by averaging coefficients of different sizes (e.g., 1 and 100) may result in a model that cannot adequately represent the intake and exhaust system S.

[0060] Therefore, the classifier 226 classifies the elite models into a plurality of classes ((iv) in FIG. 3). For example, the classifier 226 classifies the elite models into a plurality of classes using the coefficient of determination R 2 The classifier 226 classifies each of the elite models into a plurality of classes by classifying the coefficient matrix of the elite model for which is greater than a predetermined value into each class. Specifically, the classifier 226 classifies each of the elite models into a plurality of classes by classifying the coefficient matrix of each elite model using k-means clustering.

[0061] In this embodiment, the classification unit 226 performs k-means clustering using the MATLAB (registered trademark) function "kmeans" to classify the elite models into four classes, but the number of classes to be classified is not limited to this. FIG. 5 is a diagram for explaining the classification results. The vertical axis of FIG. 5 represents each class. Since the classification unit 226 classified the elite models into four classes, numbers 1 to 4 are shown. The horizontal axis represents the silhouette value. The horizontal bars indicate the silhouette values ​​of the elite models classified into each class. As a specific example, two elite models are classified into the third class. Furthermore, three elite models are classified into the fourth class. More elite models are classified into the first and second classes than into the fourth class. The silhouette value ranges from -1 to +1, with values ​​closer to 1 indicating better classification. Since all silhouette values ​​are positive, each elite model is classified into the appropriate class.

[0062] The classification unit 226 can classify the elite models using any known classification method, not limited to the above k-means clustering. The classification unit 226 discards any first model among the multiple first models whose degree of match is less than a threshold without classifying it into any of the multiple classes. In other words, the classification unit 226 discards any first model that is not an elite model without classifying it into a class. This allows the classification unit 226 to exclude any first model that does not match the actual output of the intake and exhaust system S.

[0063] The second generation unit 227 aggregates the elite models for each class ((v) in FIG. 3). The second generation unit 227 generates a statistical model (second model) for each class by aggregating the elite models for each class. For example, the second generation unit 227 generates a statistical model for each class expressed as the product of a new coefficient matrix determined by statistics of the coefficient matrices of one or more elite models included in each class and a matrix made up of multiple basis functions of one or more first models included in each class. The statistics is, for example, the average or median of the coefficient matrices of one or more elite models included in each class. As a specific example, the second generation unit 227 generates a statistical model for each of four classes expressed as the product of the average of the coefficient matrices of multiple elite models included in each class and a matrix made up of basis functions of the elite models included in each class.

[0064] The second estimation unit 228 estimates an estimated value of the output of the intake and exhaust system S using the statistical model generated by the second generation unit 227. Specifically, the second estimation unit 228 estimates an estimated value of the output of the intake and exhaust system S within a predetermined period for each statistical model using the coefficient matrix, control input, and disturbance input of each statistical model. The method by which the second estimation unit 228 estimates the estimated value of the output of the intake and exhaust system S is the same as that of the first estimation unit 224. Note that the second estimation unit 228 is an estimation unit recited in the claims.

[0065] The output unit 229 associates the number of elements in the coefficient matrix of the statistical model and the degree of agreement between the estimated value estimated by the second estimation unit 228 and the output with the statistical model and outputs them to an external device. The external device is, for example, a display device such as a liquid crystal display, but is not limited thereto. The output unit 229 outputs the number of parameters of elements that have a greater influence on the estimated value than other elements, among all elements included in the coefficient matrix of each statistical model, to the external device. Specifically, the output unit 229 associates the number of parameters whose influence on the estimated value is greater than zero, among all elements of the coefficient matrix of the statistical model, and the degree of agreement, among all elements of the coefficient matrix of the statistical model, with the second model and outputs them. More specifically, the output unit 229 associates the number of parameters whose influence on the estimated value is equal to or greater than a threshold, among all elements of the coefficient matrix of the second model, and the degree of agreement, among all elements of the coefficient matrix of the second model, with the statistical model and output them to the external device. The threshold is a value for determining whether the influence on the estimated value is large, and is, for example, 0.001, but is not limited thereto.

[0066] The output unit 229 outputs the coefficient of determination as the degree of agreement between the estimated value and the output. For example, the output unit 229 uses equation (19) to output the coefficient of determination between the estimated value estimated by the second estimating unit 228 and the output in association with each class.

[0067] Fig. 6 is a table showing the coefficient of determination and the number of parameters for each class. As shown in the table in Fig. 6, the coefficient of determination for one-step prediction and the coefficient of determination for multi-step prediction for each class are greater than a predetermined value (0.9). In other words, the statistical model for each class generated by the second generation unit 227 is a model that can appropriately predict the output of the intake and exhaust system S.

[0068] "Unclassified" represents the simulation results when elite models are aggregated (averaged) without classification. When elite models are aggregated without classification, the coefficient of determination for one-step prediction is greater than the predetermined value, but the coefficient of determination for multi-step prediction is unstable. In other words, a model in which elite models are aggregated without classification cannot achieve multi-step prediction. In this way, the output device 200 allows the classification unit 226 to classify multiple elite models into multiple classes, thereby enabling the aggregation of elite models with similar tendencies, thereby enabling the generation of a statistical model capable of achieving multi-step prediction.

[0069] By checking the table in Fig. 6, a user of the output device 200 can select a model that appropriately represents the intake and exhaust system S from among the models corresponding to the four classes. For example, if the user places importance on the degree of agreement with the output (coefficient of determination), the user can use the statistical model of the second class, in which both the coefficient of determination for y1 and the coefficient of determination for y2 are higher than those of the other classes. In this way, the output device 200 allows the user to use a model that appropriately represents the intake and exhaust system S.

[0070] Furthermore, when implementing model predictive control in an actual vehicle, users can use statistical models corresponding to the third or fourth class, which have a smaller number of parameters than the other models. The smaller the number of parameters, the less calculations are required for predictive control, so statistical models with a small number of parameters are suitable for predictive control.

[0071] Below, we will explain the results of estimating the output of the intake and exhaust system S using a statistical model. In other words, we will explain the simulation results of the intake and exhaust system S. The simulation was performed using a PC (CPU: Intel (registered trademark) Xeon (registered trademark) w7-3465X 2.5 GHz; RAM: 128 GB). The programming language for the simulation was MATLAB / Simulink (2021b), but is not limited to this.

[0072] The simulation model of the intake and exhaust system S is a mean value engine model. Before library bagging, the basis functions are set to quadratic polynomials. The user-defined time delay order of the states is σ u = 1, i.e., the time delay embedding is m (t)=[x(t) x(t-1)] T The state x is the output (y) described above. The sparsity promotion parameter λ is 30. The sampling period is 0.1 seconds. The simulation period (predetermined period) is 2500 seconds.

[0073] Note that noisy signal data reduces modeling accuracy. Therefore, in this simulation, a noisy case is considered to perform noise robustness analysis. That is, in this simulation, noise expressed by the following equation (20) is added to the output.

number

[0074] FIG. 7 is a diagram showing an example of time series data. The time series data in FIG. 7 is input / output time series data from 0 to 2500 seconds obtained by inputting control inputs and disturbance inputs to the intake / exhaust system S under noise-free conditions. The horizontal axis of the six graphs in FIG. 7 indicates time, measured in seconds. The time series data in FIG. 7 is assumed to be obtained every 0.1 seconds.

[0075] The vertical axis in Figure 7 indicates the magnitude of the signal. Note that each signal has been centered as a pre-processing step. Of the six graphs, graph y1 in the upper left is a graph showing the boost pressure. graph y2 in the upper right is a graph showing the EGR rate. graph u1 in the middle left is a graph showing the VGT vane opening. graph u2 in the middle right is a graph showing the EGR valve opening. graph d1 in the lower left is a graph showing the fuel injection amount. graph d2 in the lower right is a graph showing the engine speed.

[0076] Figure 8 is an enlarged view of the range from 1250 seconds to 1260 seconds of the time series data. The horizontal axis of the six graphs in Figure 8 indicates time, in seconds. The relative positions of the six graphs are the same as in Figure 7.

[0077] First, we will explain the simulation results of the conventional basic SINDy model using the above time series data. The coefficient of determination R for one-step prediction of y1 and y2 in the conventional basic SINDy model is 2 were 0.988 and 0.991, respectively. The coefficient of determination R for multi-step prediction of y1 and y2 in the conventional basic SINDy model 2 could not be calculated because the estimated values ​​diverged.

[0078] Figure 9 shows an example of the simulation results of multi-step prediction in the basic SINDy model based on the initial state and time series data. As shown in Figure 9, the estimated values ​​(shown by the solid line) in the conventional basic SINDy model become unstable around 200 seconds (i.e., the estimated values ​​diverge). Thus, in the basic SINDy model, the coefficient of determination R 2 Even if is larger than the predetermined value (0.9), the estimated value of the multi-step prediction may become unstable.

[0079] Next, a simulation based on the statistical model of this embodiment will be described. As described above, the output device 200 performs multi-step prediction for each of the multiple first models generated by library bagging, and calculates the coefficient of determination R 2The output device 200 collects 50 elite models with a coefficient of determination R of greater than 0.9. The calculation time for the output device 200 to collect the 50 elite models was 164 seconds, and the number of iterations was 415. Next, the output device 200 classifies the 50 elite models into four classes. The output device 200 then aggregates (averages) the elite models of each class, and calculates the coefficient of determination R for each statistical model that aggregates the elite models. 2 (See the table in Figure 6).

[0080] The user can check the table in Figure 6 and select the second class statistical model as the optimal model that properly represents the intake and exhaust system S. Hereinafter, the second class statistical model will be referred to as the optimal model. The coefficient of determination R for one-step prediction of y1 and y2 in the optimal model is 2 were 0.989 and 0.992, respectively. The coefficients of determination R for the multi-step prediction of y1 and y2 in the optimal model were 2 were 0.959 and 0.982, respectively.

[0081] Fig. 10 is a graph showing the simulation results using the optimum model of this embodiment. The horizontal axis in Fig. 10 represents time. Fig. 10 is a graph enlarging the range of the simulation results from 2120 seconds to 2170 seconds.

[0082] Of the four graphs in Figure 10, the top left graph shows the estimated boost pressure (solid black line) and the actual boost pressure output (dashed gray line). The top right graph shows the estimated EGR rate (solid black line) and the actual EGR rate output (dashed gray line). The vertical axes of the top two graphs indicate signal magnitude. The bottom left graph shows the error between the boost pressure output and the estimated value. The bottom right graph shows the error between the EGR rate output and the estimated value. The vertical axes of the bottom two graphs indicate the error value.

[0083] As shown in FIG. 10, the optimal model of this embodiment can perform stable multi-step prediction simulations without diverging estimated values ​​within a predetermined period, which was not possible with the conventional basic SINDy. Furthermore, as shown in FIG. 10, the estimated values ​​of the optimal model of this embodiment fit well with the actual output. In other words, the second-class statistical model generated by the output device 200 of this embodiment can appropriately predict the output of the intake and exhaust system S. This allows the user to use a model that appropriately represents the intake and exhaust system S.

[0084] Furthermore, in order to evaluate the noise robustness (stability against noise or resistance to the influence of noise) of the optimal model of this embodiment, we will explain the results of a simulation in which noise was intentionally added to the output time series data. First, we will explain new output data (noisy data) in which 5%, 10%, 15%, and 20% noise was added to the noise-free output time series data.

[0085] Figure 11 is a graph showing an example of output. The left graph in Figure 11 shows a graph of noise-free boost pressure (y1) (black solid line) and a graph of noise-free boost pressure with 20% noise added (gray dashed dotted line). The right graph in Figure 11 shows a graph of noise-free EGR rate (y2) (black solid line) and a graph of noise-free EGR rate (y2) with 20% noise added (gray dashed dotted line).

[0086] Figure 12 is a table showing the simulation results for different noise levels. The simulation in Figure 12 was performed using a statistical model corresponding to the second class. The table in Figure 12 shows the coefficient of determination R for one-step and multi-step predictions when a given noise level is added. 2 Figure 12 also shows the computation time required to complete multi-step prediction and the number of iterations required to collect a predetermined number (50) of elite models. Because library bagging is performed randomly, the computation time and number of iterations vary.

[0087] As shown in FIG. 12, the coefficient of determination R 2 is greater than the predetermined value (0.9) even under noisy conditions. In this way, users can utilize a discrete model that can realize multi-step predictions even under noisy conditions in complex nonlinear systems such as the intake and exhaust system S.

[0088] [Process to output the number of parameters and the degree of match] 13 is a flowchart showing an example of a process for outputting the number of parameters and the degree of coincidence. First, the acquisition unit 221 acquires time-series data (step S1). Specifically, the acquisition unit 221 acquires a control input vector indicating the boost pressure and EGR rate, a disturbance input vector indicating the fuel injection amount and engine speed, and an output vector indicating the VGT vane opening and EGR valve opening.

[0089] The first generation unit 222 generates a first model capable of predicting the output of the intake and exhaust system S when a control input is input based on the time-series data acquired by the acquisition unit 221 (step S2). The first generation unit 222 generates the first model by performing library bagging, which randomly selects some basis functions from among a plurality of basis functions. Specifically, the first generation unit 222 generates the first model expressed as the product of a matrix made up of the plurality of basis functions randomly selected by library bagging and a coefficient matrix in which the coefficients of each of the plurality of basis functions are arranged.

[0090] The determination unit 223 determines the coefficient matrix of each first model (step S3). Specifically, the determination unit 223 determines the coefficient matrix by solving the optimization problem of equation (12) to optimize the coefficient matrix of the first model.

[0091] The first estimation unit 224 estimates an estimated value of the output of the intake and exhaust system S (step S4). The first estimation unit 224 estimates an estimated value of the output of the intake and exhaust system S within a predetermined period using the time-series data and the first model. The extraction unit 225 calculates a coefficient of determination of the first model, which indicates the degree of agreement between the estimated value by the first model and the actual output (step S5). Specifically, the extraction unit 225 determines the coefficient of determination of each first model using the above-mentioned equation (19).

[0092] The extraction unit 225 extracts an elite model based on the coefficient of determination of each first model (step S6). For example, the extraction unit 225 extracts an elite model based on the coefficient of determination R 2 Extract the elite models whose is larger than the other first models.

[0093] The classification unit 226 classifies the plurality of elite models into a plurality of classes (step S7). The classification unit 226 classifies each of the plurality of elite models into a plurality of classes by classifying the coefficient matrices of the elite models whose coefficient of determination R2 is greater than a predetermined value into each class. As a specific example, the classification unit 226 classifies the plurality of elite models into four classes.

[0094] The second generation unit 227 generates a statistical model for each class based on the elite model for each class (step S8). The second generation unit 227 generates a statistical model for each of the four classes, which is expressed as the product of the average value of the coefficient matrix of the elite model and a matrix made up of the basis functions of the elite model.

[0095] The second estimation unit 228 estimates an estimated value of the output of the intake / exhaust system S within a predetermined period using the statistical model (step S9). Specifically, the second estimation unit 228 estimates an estimated value of the output of the nonlinear system within a predetermined period for each statistical model using the coefficient matrix, control input, and disturbance input of each statistical model.

[0096] The output unit 229 outputs the number of parameters and the degree of agreement for each class (step S10). The number of parameters is the number of elements that have a greater influence on the estimated value than other elements among all elements included in the coefficient matrix of the statistical model, and more specifically, the number of elements greater than zero. The degree of agreement is the coefficient of determination between the output and the estimated value. For example, the output unit 229 outputs a table including the number of parameters and the degree of agreement corresponding to each class to a display device, thereby displaying the table shown in FIG. 5 on the display device.

[0097] (Variation) In the above embodiment, the classification unit 226 classified the elite models extracted by the extraction unit 225. However, this is not limiting, and the classification unit 226 may classify multiple first models into multiple classes. Specifically, the classification unit 226 classifies each first model into each class by classifying the coefficient matrix of each first model determined by the determination unit 223 using k-means clustering. In this case, the second generation unit 227 generates a statistical model for each class based on the first model of each class. In this way, the second generation unit 227 can aggregate (average) first models having the same tendency. The second estimation unit 228 estimates an estimated value of the intake / exhaust system S based on the statistical model aggregated for each class. In this way, the output device 200 can omit calculations related to the extraction of elite models, thereby reducing computational resources and shortening the overall computation time.

[0098] [Effects of output device 200] As described above, the output device 200 of this embodiment first acquires time-series data of the intake and exhaust system S. The time-series data includes a plurality of control inputs input to the nonlinear system within a predetermined period, disturbance inputs that affect the intake and exhaust system S when each control input is input to the intake and exhaust system S, and the output of the intake and exhaust system S when each control input and each disturbance input is input to the intake and exhaust system S. Next, the output device 200 generates a plurality of first models that are expressed as the product of a matrix made up of a plurality of basis functions randomly selected from a plurality of basis functions and a coefficient matrix in which coefficients of each of the selected basis functions are arranged, and that can predict the output of the intake and exhaust system S when a control input is input. The plurality of basis functions are functions to which at least one of the control input, the disturbance input, and the output is input.

[0099] Next, the output device 200 classifies each of the generated first models into a plurality of classes by classifying the coefficient matrices corresponding to each first model determined by optimizing the coefficient matrix of each of the generated first models into each class. Subsequently, the output device 200 generates a statistical model (second model) for each class, which is expressed as the product of a new coefficient matrix determined by statistics of the coefficient matrices of one or more first models included in each class and a matrix made up of multiple basis functions of one or more first models included in each class. The output device 200 then estimates an estimated value of the output of the nonlinear system within a predetermined period for each statistical model using the coefficient matrix, control input, and disturbance input of each statistical model. Finally, the output device 200 outputs, in association with the statistical model, the number of elements among all elements included in the coefficient matrix of the statistical model that have a greater influence on the estimated value than other elements, and the degree of agreement between the output and the estimated value.

[0100] A user of the output device 200 can check the number of elements and the degree of match output by the output device 200 and select a model that appropriately represents the intake and exhaust system S from among the statistical models corresponding to each class. For example, the user can prioritize the degree of match with the output and select and use a statistical model from among multiple statistical models that has a higher degree of match than other statistical models. In this way, the output device 200 can enable the user to use a statistical model that appropriately represents the intake and exhaust system S and has a higher degree of match between the estimated value and the output than other statistical models.

[0101] The present invention has been described above using embodiments, but the technical scope of the present invention is not limited to the scope described in the above embodiments, and various modifications and changes are possible within the scope of the gist of the present invention. For example, all or part of the device can be configured by functionally or physically distributing or integrating any unit. Furthermore, new embodiments resulting from any combination of multiple embodiments are also included in the embodiments of the present invention. The effects of the new embodiments resulting from the combination also have the effects of the original embodiments. [Explanation of symbols]

[0102] 100 cylinders 110 intake manifold 111 Intake pipe 112 Intercooler 113 Intake throttle 120 exhaust manifold 121 EGR line 122 EGR cooler 123 EGR valve 130 Supercharger 131 Turbine 132 Compressor 140 Injector 200 Output Device 210 Storage section 220 Control Unit 221 Acquisition Department 222 1st generation part 223 Decision Section 224 1st estimation part 225 Extraction Section 226 Classification Department 227 Second Generation Section 228 Presumption Part 2 229 Output Department

Claims

1. an acquisition unit that acquires a plurality of control inputs input to a nonlinear system within a predetermined period, a disturbance input that affects the nonlinear system when each control input is input to the nonlinear system, and an output of the nonlinear system when each control input and each disturbance input are input to the nonlinear system; a first generation unit that randomly selects some basis functions from a plurality of basis functions to which at least one of the control input, the disturbance input, and the output is input, to generate a plurality of first models that are expressed as products of a coefficient matrix in which coefficients of the selected plurality of basis functions are arranged and a matrix made of the randomly selected plurality of basis functions, and that can predict the output of the nonlinear system when the control input is input; a determination unit that determines the coefficient matrix corresponding to each of the plurality of first models generated by the first generation unit by optimizing the coefficient matrix of each of the plurality of first models; a classifier that classifies each of the first models into a plurality of classes by classifying the coefficient matrix of each first model into each class; a second generation unit that generates, for each class, a second model represented by the product of a new coefficient matrix determined by statistics of the coefficient matrix of one or more of the first models included in each class and a matrix made of the plurality of basis functions of the one or more first models included in each class; an estimation unit that estimates an estimate of an output of the nonlinear system within the predetermined period for each second model using the coefficient matrix, the control input, and the disturbance input of each second model generated by the second generation unit; an output unit that outputs, in association with the second model, the number of elements among all elements included in the coefficient matrix of the second model that have a greater influence on the estimated value than other elements, and a degree of agreement between the output acquired by the acquisition unit and the estimated value estimated by the estimation unit; An output device having:

2. the classifying unit classifies each first model into each class by performing k-means clustering on the coefficient matrix of each first model; The output device according to claim 1 .

3. The classification unit classifying the first models among the plurality of first models whose degree of matching is greater than a predetermined value into classes; discarding the first models among the plurality of first models whose degree of match is equal to or less than the predetermined value without classifying them into any of the plurality of classes; 3. The output device according to claim 1 or 2.

4. a calculation unit that calculates, as the degree of agreement, a coefficient of determination obtained by subtracting a ratio of a sum of squares of a difference between the output and the estimated value estimated by the estimation unit to a sum of squares of a difference between the output acquired by the acquisition unit and an average value of the output, from 1; The output device according to claim 3 .

5. the determination unit determines the coefficient matrix for each of the plurality of first models by determining, for each of the first models, a coefficient matrix that minimizes a sum of squares of a difference between the output and the estimated value of the first model. The output device according to claim 1 .

6. the second generation unit generates, for each class, the second model represented by a product of a new coefficient matrix determined by an average value or a median value of the coefficient matrices of the one or more first models included in each class and a matrix made of the plurality of basis functions of the one or more first models; The output device according to claim 1 .

7. the output unit outputs, in association with the second model, the number of elements of the coefficient matrix of the second model whose influence on the estimated value is equal to or greater than a threshold value, and the degree of match. The output device according to claim 1 .

8. the output unit outputs, in association with the second model, the number of elements having a degree of influence on the estimated value greater than zero among all the elements of the coefficient matrix of the second model and the degree of match. The output device according to claim 1 .

9. The acquisition unit The pressure of the air delivered to the engine by the turbocharger and the ratio of the exhaust gas supplied to the intake pipe of the engine to the exhaust gas discharged by the engine are acquired as the control inputs; an amount of fuel injected into a combustion chamber of the engine and a rotation speed of the engine are acquired as the disturbance input; acquiring, as the output, an opening degree of a nozzle vane that controls a flow velocity of the exhaust gas supplied to a turbine of the turbocharger and an opening degree of a valve that supplies the exhaust gas to the intake pipe; the estimator estimates an opening degree of the nozzle vane and an opening degree of the valve as estimated values ​​of the output of the nonlinear system. The output device according to claim 1 .

10. Executed by a vehicle-mounted computer, A step of acquiring a plurality of control inputs input to a nonlinear system within a predetermined period of time, a disturbance input that affects the nonlinear system when each control input is input to the nonlinear system, and an output of the nonlinear system when each control input and each disturbance input are input to the nonlinear system; generating a plurality of first models, each of which is represented by a product of a coefficient matrix in which coefficients of the selected basis functions are arranged and a matrix made up of the randomly selected basis functions, by randomly selecting some basis functions from a plurality of basis functions to which at least one of the control input, the disturbance input, and the output is input, and which can predict the output of the nonlinear system when the control input is input; determining the coefficient matrix corresponding to each of the generated first models by optimizing the coefficient matrix of each of the generated first models; classifying each of the plurality of first models into a plurality of classes; generating, for each class, a second model represented by the product of a new coefficient matrix determined by statistics of the coefficient matrix of one or more of the first models included in each class and a matrix made of the plurality of basis functions of the one or more first models included in each class; estimating an estimate of an output of the nonlinear system within the predetermined period for each of the second models using the coefficient matrix, the control input, and the disturbance input of each of the generated second models; outputting, in association with the second model, the number of elements among all elements included in the coefficient matrix of the second model that have a greater influence on the estimated value than other elements, and the degree of agreement between the output acquired by the acquisition unit and the estimated value estimated by the estimation unit; An output method having the following structure.

11. The vehicle's on-board computer an acquisition unit that acquires a plurality of control inputs input to a nonlinear system within a predetermined period, a disturbance input that affects the nonlinear system when each control input is input to the nonlinear system, and an output of the nonlinear system when each control input and each disturbance input are input to the nonlinear system; a first generation unit that randomly selects some basis functions from a plurality of basis functions to which at least one of the control input, the disturbance input, and the output is input, to generate a plurality of first models that are expressed as a product of a coefficient matrix in which coefficients of the selected plurality of basis functions are arranged and a matrix made of the randomly selected plurality of basis functions, and that can predict the output of the nonlinear system when the control input is input; a determination unit that determines the coefficient matrix corresponding to each of the plurality of first models generated by the first generation unit by optimizing the coefficient matrix of each of the plurality of first models; a classification unit that classifies each of the plurality of first models into a plurality of classes; a second generation unit that generates, for each class, a second model represented by the product of a new coefficient matrix determined by statistics of the coefficient matrix of one or more of the first models included in each class and a matrix made of the plurality of basis functions of the one or more first models included in each class; an estimation unit that estimates an estimate of an output of the nonlinear system within the predetermined period for each second model using the coefficient matrix, the control input, and the disturbance input of each second model generated by the second generation unit; and an output unit that outputs, in association with the second model, the number of elements among all elements included in the coefficient matrix of the second model that have a greater influence on the estimated value than other elements, and the degree of agreement between the output acquired by the acquisition unit and the estimated value estimated by the estimation unit; A program that functions as a

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