DE102020211931A1 - Determination of a transfer function of a technical system - Google Patents

Abstract

Machine learning system, in particular a recurrent neural network, for determining a transfer function of a technical system, comprising at least one artificial neural layer, wherein one of the at least one artificial neural layer maps at least two difference equations in parallel.

Description

The present invention relates to a system, a method, a use, a computer program and a device for determining a transfer function for a technical system.

State of the art

Traditionally, dynamic technical systems (e.g. electrical circuits, hydraulic systems, etc.) are mapped in system theory and control engineering using parameterized differential equation systems. If unknown, the exact equation parameters are determined by experiments.

In the case of complex systems, it is often not possible with this method to set up transfer functions simply by applying a load and observing the system response. For a successful system description, it is always necessary to know the structure of the system to be modeled in advance and to model it as precisely as possible.

Machine learning systems, including artificial neural networks in particular, are a common method for modeling complex data-based systems. Previously known architectures for mapping signal series to signal series, ie for determining the transfer function (system function determination), however, according to the prior art, learn models that are not understandable to humans when dealing with this problem. For this reason, in practice, machine learning systems are mostly used to reduce the error of existing simulation models. The reasons for this are that, on the one hand, due to the high dimensionality, it is not possible to understand which function the system has learned: Up until now, there has been no method that applies models to fundamental building blocks for system dimensioning that are already known from system theory (e.g. P- , I, D, PT1 elements, etc.) or to map polynomial functions. On the other hand, the interpretability of these networks for humans is not easily possible, but requires additional application-dependent methods. By using the systems to improve the simulation, the maximum influence of the systems on the end result and thus the maximum error is limited, but cannot be completely ruled out.

Advantages of the Invention

Against this background, the present invention creates a machine learning system for determining a transfer function of a technical system.

The system includes at least one artificial neural layer.

The system is characterized in that one of the at least one artificial neural layer maps at least two difference equations in parallel.

The invention is based on the finding that, to map dynamic systems, a layer for a machine learning system, in particular an artificial neural network, can be constructed in such a way that internally at least two difference equations can be mapped parallel to one another.

Such a layer is suitable for building recurrent artificial neural networks. Artificial neural networks of this type can be used to predict time series (set value I y) depending on time series (input | x).

Depending on what types of system behavior are expected, the artificial neural layer according to the present invention may include appropriate difference formulas.

für diskrete dynamische Systeme verwendet werden.So the general difference formula $$y\left(k\right)=ƒ\left(y\left(k-1\right),y\left(k-2\right),...,y\left(k-n\right),x\left(k-1\right),y\left(k-2\right),...,y\left(k-n\right)\right)$$

can be used for discrete dynamical systems.

Furthermore, the difference equations known from system theory can be used.

Due to the fact that essentially all linear systems can be described using the difference equations known from systems theory, in particular those that are the same as the basic elements, namely those for P, I, D, PT1 or PD elements, when these difference equations are used the contribution of functions that may not be required in the model is minimized as part of the training process. This in turn leads to comparatively small and less complex networks that can be executed on simple hardware at inference time. This enables the invention to be implemented simply and inexpensively.

In addition, it is conceivable that a concatenation of basic elements, for example a PID element, is used instead of the basic elements. The characteristic layer of the machine learning system according to the present invention can have a corresponding mapping thereto.

In the present case, a system of machine learning can be understood to mean an artificial neural network.

The system can be a recurrent neural network.

A technical system can e.g. a system for at least partially automated driving, a system for the automated inspection of a production process, for example.

Furthermore, in the present case, a technical system can be an industrial mechanical, hydraulic and electrical system based on analog input and output signals, in particular braking systems (hydraulic, electrical).

Using the system of machine learning according to the present invention, transfer functions of a technical system can be determined without having to set up system equations beforehand.

With the help of the present invention, system models can be constructed and automatically parameterized based on their input and output measurement series without (or in addition to) numerical simulation models.

The resulting models are easier for those skilled in the art to understand. The learned parameters can be validated using previously known techniques from systems theory. For example, in contrast to conventional machine learning systems, they can be converted into the well-known Laplace or z-representation. It is thus also possible to derive the stability of the system or similar properties using the methods already known.

The present invention and its use accordingly offer a whole range of advantages.

In this way, the system model no longer has to be specified explicitly, since the parameters are determined numerically.

Compared to usual parameter-based machine learning systems, in particular artificial neural networks, the number of parameters to be learned can be smaller. This allows the resulting machine learning system to be used for inference with simple hardware. This enables the invention to be implemented simply and inexpensively.

The machine learning system according to the present invention can be implemented using the interfaces known from Recurrent Neural Network. As a result, the machine learning system according to the present invention can easily be used as part of a comprehensive machine learning system, e.g. an artificial neural network.

$$i\left(t\right)=\frac{1}{T_I}{\displaystyle {\int }_0^{\tau }x\left(t\right)dt}$$

$$d\left(t\right)=K_D\frac{d}{dt}u\left(t\right)$$

$$pt1\left(t\right)=K_{PT1}x\left(t\right)-T_{PT1}\frac{d}{dt}y\left(t\right)$$

$$pd\left(t\right)=K_{PD}\left({\text{T}}_{PD}\frac{d}{dt}y\left(t\right)+y\left(t\right)\right)$$

$$y\left(t\right)=\left[i\left(t\right),d\left(t\right),pt1\left(t\right),pd\left(t\right),p\left(t\right)\right]$$

If the characterizing layer of the system of machine learning according to the present invention maps the basic elements known from system theory, the following applies to the output y(t) depending on the input x(t): $$p\left(t\right)=K_{P}x\left(t\right)$$

$$i\left(t\right)=\frac{1}{T_I}{\displaystyle {\int }_0^{\tau }x\left(t\right)\mathrm{i.e}t}$$

$$\mathrm{i.e}\left(t\right)=K_D\frac{\mathrm{i.e}}{\mathrm{i.e}t}\mathrm{and}\left(t\right)$$

$$pt1\left(t\right)=K_{PT1}x\left(t\right)-{\mathrm{<figure-callout\; id="1"\; label="T\; P\; T"\; filenames="DE102020211931A1\_0020.png"\; state="\{\{state\}\}">T</figure-callout>}}_{PT}\frac{\mathrm{i.e}}{\mathrm{i.e}t}y\left(t\right)$$

$$p\mathrm{i.e}\left(t\right)=K_{PD}\left({\text{T}}_{PD}\frac{\mathrm{i.e}}{\mathrm{i.e}t}y\left(t\right)+y\left(t\right)\right)$$

$$y\left(t\right)=\left[i\left(t\right),\mathrm{i.e}\left(t\right),pt1\left(t\right),p\mathrm{i.e}\left(t\right),p\left(t\right)\right]$$

The formation of the difference equation of the terms takes place via a discretization, for example via Euler 1st order. This step depends on the sampling rate, which makes it part of the artificial neural network as _Δt . For the modeling of multi-input-multi-output systems, the constant factors K and time constants T are formulated as matrices and represent the trainable weights in the shift.

Since, in contrast to previous sequential models, the parameter Δ _t (sampling rate) occurs in the difference equations, the resulting model can be trained at a first sampling rate and used for inference at a second sampling rate that is different from the first. Furthermore, the sampling rate used can be changed between training steps will.

It was observed that the number of differences in the result with a changed sampling rate indicates the stability of the model.

This leads to the further advantage that the machine learning system according to the present invention can be trained with a first, comparatively coarse, sampling rate and can be executed at a second, comparatively fine, sampling rate at inference time. This property of the machine learning system according to the present invention can significantly reduce the training time of the system.

According to one embodiment of the system according to the present invention, one of the at least two difference equations represents a P element or an I element or a D element or a PT1 element or a PD element, which are derived from system theory or known to control engineering.

According to an embodiment of the system according to the present invention, the system comprises at least one artificial neural network layer for merging the outputs of the at least one artificial neural network layer, i. H. an artificial neural network layer capable of reducing the output dimensions. Such a layer can be, for example, a fully connected layer, a convolutional neural network layer or a recurrent neural network layer.

A further aspect of the present invention is a method for determining a transfer function of a technical system as a function of a first series of input data and a second series of output data, using a system according to the present invention.

This aspect of the present invention is based on the finding that the transfer function of the technical system is learned and thus determined by the training of the machine learning system according to the present invention. Based on the inventive design of the characteristic layer of the system according to the present invention, the transfer function can be derived.

A further aspect of the present invention is the use of a machine learning system according to the present invention for status monitoring of the technical system.

This aspect is based on the finding that the transfer function of the technical system to be monitored that is learned by the machine learning system according to the invention reflects the correct state transitions.

In order to monitor the technical system, the input data can now be supplied both to the technical system and to the correspondingly trained machine learning system used to monitor it. The status of the technical system can be monitored by means of the machine learning system by comparing the initial data and, if necessary, providing a tolerance range for any deviations.

Another aspect of the present invention is the use of a machine learning system according to the present invention as a digital controller.

This aspect is based on the knowledge that a machine learning system according to the present invention learns a transfer function of a technical system. This can be used to apply input data to a trained machine learning system according to the present invention and to use the output data as control variables for a digital controller.

A further aspect of the present invention is a computer program which is set up to carry out all the steps of the method according to the present invention.

Another aspect of the present invention is a machine-readable storage medium on which the computer program according to an aspect of the present invention is stored.

A further aspect of the present invention is a device which is set up to carry out all the steps of the method according to the present invention.

Embodiments of the present invention are explained in more detail below with reference to drawings.

• 1 eine schematische Darstellung einer Schicht für ein künstliches neuronales Netz gemäß der vorliegenden Erfindung;
• 2 eine schematische Darstellung von Schichten für ein künstliches neuronales Netz gemäß der vorliegenden Erfindung;
• 3 eine schematische Darstellung einer Architektur eines künstlichen neuronalen Netzes;
• 4 eine schematische Darstellung eines trainierten künstlichen neuronalen Netzes.

Show it

• 1 Figure 12 is a schematic representation of an artificial neural network layer according to the present invention;
• 2 Figure 12 shows a schematic representation of layers for an artificial neural network according to the present invention;
• 3 a schematic representation of an architecture of an artificial neural network;
• 4 a schematic representation of a trained artificial neural network.

1 shows a schematic representation of a layer for an artificial neural network, which maps the behavior of P, I, D, PT1 and PD elements from system theory.

$$i\left(t\right)=\frac{1}{T_I}{\displaystyle {\int }_0^{\tau }x\left(t\right)dt}t$$

$$d\left(t\right)=K_D\frac{d}{dt}u\left(t\right)$$

$$pt1\left(t\right)=K_{PT1}x\left(t\right)-T_{PT1}\frac{d}{dt}y\left(t\right)$$

$$pd\left(t\right)=K_{PD}\left({\text{T}}_{PD}\frac{d}{dt}y\left(t\right)+y\left(t\right)\right)$$

$$y\left(t\right)=\left[i\left(t\right),d\left(t\right),pt1\left(t\right),pd\left(t\right),p\left(t\right)\right]\left(Konkatination\right)$$

In the time domain, the following would apply to the output y(t) of the layer depending on the input x(t): $$p\left(t\right)=K_{P}x\left(t\right)$$

$$i\left(t\right)=\frac{1}{T_I}{\displaystyle {\int }_0^{\tau }x\left(t\right)\mathrm{i.e}t}t$$

$$\mathrm{i.e}\left(t\right)=K_D\frac{\mathrm{i.e}}{\mathrm{i.e}t}\mathrm{and}\left(t\right)$$

$$pt1\left(t\right)=K_{PT1}x\left(t\right)-{\mathrm{<figure-callout\; id="1"\; label="T\; P\; T"\; filenames="DE102020211931A1\_0020.png"\; state="\{\{state\}\}">T</figure-callout>}}_{PT}\frac{\mathrm{i.e}}{\mathrm{i.e}t}y\left(t\right)$$

$$p\mathrm{i.e}\left(t\right)=K_{PD}\left({\text{T}}_{PD}\frac{\mathrm{i.e}}{\mathrm{i.e}t}y\left(t\right)+y\left(t\right)\right)$$

$$y\left(t\right)=\left[i\left(t\right),\mathrm{i.e}\left(t\right),pt1\left(t\right),p\mathrm{i.e}\left(t\right),p\left(t\right)\right]\left(KOnkatinatiOn\right)$$

The formation of the difference equation of the terms as given in 1 can be seen, takes place via a discretization, for example via Euler 1st order.

As usual in engineering, y(t) is replaced by: $$\dot{y}\left(t\right)=\frac{y\left(k\right)-y\left(k-1\right)}{\Delta t}$$

This step depends on the sampling rate Δt, which makes it part of the artificial neural network as Δt.

$$i\left(k\right)=i\left(k-1\right)+\frac{\Delta t}{T_I}\cdot x\left(k\right)$$

$$d\left(k\right)=\frac{T_D}{\Delta t}\cdot \left(x\left(k\right)-x\left(k-1\right)\right)$$

$$pd\left(k\right)=K_{PD}\cdot \left(x\left(k\right)+\frac{T_{PD}}{\Delta t}\cdot \left(x\left(k\right)-x\left(k-1\right)\right)\right)$$

$$pt\left(k\right)=pt\left(k-1\right)+\left(K_{PT1}\cdot x\left(k\right)-pt\left(k-1\right)\right)\cdot \frac{\Delta t}{\Delta t+T_{PT1}}$$

Wobei alle K, T > 0 seien.The result is the following discrete recurrent equations with a sample k as input and a time step between samples of Δt: $$p\left(k\right)=K_P\cdot x\left(k\right)$$

$$i\left(k\right)=i\left(k-1\right)+\frac{\Delta t}{T_I}\cdot x\left(k\right)$$

$$\mathrm{i.e}\left(k\right)=\frac{T_D}{\Delta t}\cdot \left(x\left(k\right)-x\left(k-1\right)\right)$$

$$p\mathrm{i.e}\left(k\right)=K_{PD}\cdot \left(x\left(k\right)+\frac{T_{PD}}{\Delta t}\cdot \left(x\left(k\right)-x\left(k-1\right)\right)\right)$$

$$pt\left(k\right)=pt\left(k-1\right)+\left({\mathrm{<figure-callout\; id="1"\; label="K\; P\; T"\; filenames="DE102020211931A1\_0020.png"\; state="\{\{state\}\}">K</figure-callout>}}_{PT}\cdot x\left(k\right)-pt\left(k-1\right)\right)\cdot \frac{\Delta t}{\Delta t+{\mathrm{<figure-callout\; id="1"\; label="T\; P\; T"\; filenames="DE102020211931A1\_0020.png"\; state="\{\{state\}\}">T</figure-callout>}}_{PT}}$$

Where all K, T > 0.

For the modeling of multi-input-multi-output systems, the factors K and T are formulated as matrices and represent the trainable weights in the shift.

As is usual with artificial neural networks, the values of the model can be learned using, for example, the gradient descent method (eg via an Adam optimizer or stochastic gradient descent). Training over several batches at the same time is also conceivable.

2 Fig. 12 shows a schematic representation of an artificial neural network or a part of an artificial neural network in which layers for an artificial neural network according to the present invention are used.

Concrete shows 2 thereby the juxtaposition of two layers for an artificial neural network according to the present invention.

The illustration makes it clear that the layers for an artificial neural network according to the present invention can be lined up in a simple manner. The respective outputs of the parallel mappings are treated as inputs for a further layer. This makes it clear that the dimensions fan out accordingly.

Depending on the application, suitable measures must therefore be taken to counteract the diversification of the dimensions. For example, the well-known Fully Connected (FC) layers can be used for this.

Furthermore, non-linear activation functions can be applied in the architecture of a corresponding artificial neural network. This allows an architecture known from the Hammerstein-Wiener models for modeling static non-linearities in data to be achieved.

If noisy input data is to be expected, e.g. due to noisy sensor data, additional convolutional layers can be used.

3 shows a schematic representation of an artificial neural network or a part of an artificial neural network comprising a cascade of layers for an artificial neural network according to the present invention.

The output of the cascade of layers for an artificial neural network according to the present invention is converted into a fully connected artificial neural network.

4 shows a schematic representation of an artificial neural network or a part of an artificial neural network comprising a cascade of layers for an artificial neural network according to the present invention.

The output of the cascade of layers for an artificial neural network according to the present invention is thereby converted into a fully connected artificial neural network.

The representation shows a resulting artificial neural network after training. At this time the weights of the activation functions lie. Based on the evaluation of the weights or their distribution, information about the modeled dynamic system can be derived, based on which the resulting artificial neural network can be interpreted.

The interpretability stems from the fact that the layers for an artificial neural network according to the present invention represent the time-dependent system behavior of the modeled system, while the fully connected part of the resulting artificial neural network represents the relevant influence on the dynamics of the model.

The higher the corresponding weight of the fully connected layer or the fully connected part of the resulting artificial neural network, the greater the significant influence of the corresponding mapping for the modeling of the dynamic system.

Due to the fact that the weights of the fully connected layer or the fully connected part of the resulting artificial neural network are to be interpreted globally and in a time-independent manner, the respective edge weights can be interpreted as the global meaning of the corresponding path of the part of the resulting artificial neural network .

In the e.g. artificial neural network shown in 4 the P, I and PT1 terms with 0.5, 1.5 and 0.8 have a significantly higher influence on the modeled system than the D and I terms with only 0.1 and 0.2. From this it can be deduced that the modeled system mainly has the properties of P, I and PT1 elements.

Claims (9)

Machine learning system, in particular a recurrent neural network, for determining a transfer function of a technical system, comprising at least one artificial neural layer, characterized in that one of the at least one artificial neural layer maps at least two difference equations in parallel. system after claim 1 , wherein one of the at least two difference equations represents a P element and/or an I element and/or a D element and/or a PT1 element and/or a PD element. system after claim 1 or 2 , wherein the system has at least one layer of an artificial neural network for merging the outputs of the at least one other artificial neural layer, for example a fully connected layer and/or a recurrent neural network layer or a convolutional neural network layer. Method for determining a transfer function of a technical system as a function of a first series of input data and a second series of output data, using a machine learning system according to one of Claims 1 until 3 , especially by training the sys machine learning systems depending on the first set of input data and/or the second set of output data. Using a machine learning system according to any of the Claims 1 until 3 for status monitoring of the technical system. Using a machine learning system according to any of the Claims 1 until 3 as a digital controller. Computer program that is set up to follow all the steps of a method claim 4 to execute. Machine-readable storage medium on which a computer program claim 6 is saved. Device which is set up to follow all the steps of a method claim 4 to execute.

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