DE102022202693A1 - Testing technical systems for complete describability using sensor data and an associated differential equation - Google Patents

Abstract

Method (100) for checking whether the combination of recording the state of a technical system (1) with several sensors (2a-2c) and a differential equation in the measured values (3a-3c) supplied by these sensors (2a-2c). Dynamics of this technical system (1) completely describes, with the steps: • At least one trajectory Y1, ..., YN is sampled at N discrete times in one by the measured values (3a-3c) supplied by the sensors (2a-2c). spanned phase space (4) provided (110); • based on this trajectory Y1, ..., YN, a Kolmogorov entropy K, which is a measure of how much additional information the measured values supplied for each point in time contain, is at least approximately determined (120 );• in response to the fact that this Kolmogorov entropy K, and/or a quantity derived from it, fulfills a predetermined criterion (5) (130), it is determined (140a) that the detection with the sensors (2a-2c) and the differential equation fully describes the dynamics of the system (1).

Description

The present invention relates to the monitoring of technical systems whose status is recorded with one or more sensors and whose dynamics are modeled with a differential equation in one or more variables.

State of the art

For the control of many technical systems, it is important to determine a forecast for the dynamics of the system in the future from the sensory observation of the system over a past period of time. This forecast is determined using a model from the measurement data recorded by sensors and serves as the basis for further control of the respective system. For example, a vehicle can use sensor data to predict its future driving dynamics at least for a few time units or time steps in advance, and if necessary, a control system can intervene to change these driving dynamics in the desired way. An example of such a control system is the electronic stability program, ESP, according to the EP 0 339 056 B1 .

In order for such control systems to function properly, it is necessary that the respective technical system actually behaves as the model used predicts based on the sensor data. For more complex systems, it is difficult to prove that the measurement data provided by a given sensor configuration in conjunction with a given model completely describes the dynamics of the system at all times and under all circumstances and that the system does not have a completely different behavior than the prediction ( (e.g. chaotic) behavior.

Disclosure of the invention

The invention provides a method for checking whether the combination of recording the state of a technical system with several sensors and a differential equation in the measured values supplied by these sensors completely describes the dynamics of this technical system. The differential equation can couple its variables in any way and can also be subject to initial conditions, district conditions and/or boundary conditions, which can also be expressed at least approximately as a differential equation. For the sake of clarity, the need to fulfill such initial conditions, district conditions and/or boundary conditions will no longer be mentioned in the following whenever the differential equation is mentioned.

The method begins by providing at least one trajectory Y ₁ , ..., Y _N sampled at N discrete times in a phase space. This phase space is spanned by the measured values provided by the sensors and optionally also by one or more temporal dimensions. Each point Y ₁ , ... , Y _N on the trajectory can therefore be viewed as a vector, matrix or tensor, the components of which contain the measured values recorded for the respective time 1, ...,N. The entire trajectory can therefore also be written as a tensor Y, in which the dimensions of the measured values around the time axis are supplemented with the times 1, ..., N as a further dimension. At any point in time, a trajectory Y ₁ , ..., Y _N can also, for example, couple measured values that relate to exactly this point in time with measured values of the same measured variable that are a predetermined offset in time.

Based on the trajectory, a Kolmogorov entropy K is at least approximately determined. The Kolmogorov entropy is a measure of how much additional information the measured values provided for each point in time contain.

In response to the fact that this Kolmogorov entropy K, and/or a quantity derived from it, meets a given criterion, it is determined that the detection with the sensors and the differential equation completely describe the dynamics of the system.

It was recognized that the Kolmogorov entropy K can be determined directly from the trajectory Y ₁ , ..., Y _N without considering the differential equation used to model the technical system. Nevertheless, the Kolmogorov entropy K contains information as to the extent to which the system is completely described by the measured values recorded by sensors in combination with the differential equation.

In particular, real-time evaluation of the trajectory is also possible on embedded systems with economical hardware, such as those used on board vehicles. The fundamental dependence of the Kolmogorov entropy K on the measured values can be solved in advance, so that concrete values for the Kolmogorov entropy K can then be obtained from concrete measured values with comparatively simple arithmetic operations and low memory requirements.

Starting from the previous course of the trajectory, the differential equation is provided in the form of: Solutions one or more forecasts for the progression of the dynamics. There are usually several possible solutions. When newly recorded measured values arrive, information is obtained as to which of the previous forecasts about the progress of the dynamic has occurred. This information gain has a finite amount, that is, the Kolmogorov entropy K is finite in this case: $$0<K<\infty .$$

From a formal point of view, a chaotic system is still deterministic, but at the same time unpredictable. In particular, the differential equation can still be fully valid for the behavior of the system. However, since the smallest changes in the initial conditions lead to very strong changes in the dynamics, conversely, it is no longer possible to identify a single solution of the differential equation that fits these newly supplied measured values based on newly supplied measured values. The information gain, and thus the Kolmogorov entropy K, is zero in this case.

However, if the behavior of the system is random, when new measured values become known, something occurs that was in no way predictable based on the previous measured values in conjunction with the differential equation. The information gain, and thus the Kolmogorov entropy K, is infinitely large in this case.

In particular, for example, 0 < K < ∞ can be chosen as the predetermined criterion for the Kolmogorov entropy K. However, approximate solutions for the Kolmogorov entropy K do not necessarily take exactly the value 0 when the system behaves chaotically and do not necessarily take exactly the value infinity when it behaves randomly. It may then become necessary to choose another predetermined criterion.

ermittelt werden. K₀ ist die topologische Entropie, K₁ ist die Kolmogorov-Entropie K, und K₂ ist die K₂-Entropie, die als Näherung für die Kolmogorov-Entropie K verwendet werden kann. Der Parameter q bestimmt über die zu berücksichtigende Potenz der Wahrscheinlichkeit P_i1 ,_i2 , ..., i_N. Diese Art der Berechnung ermöglicht gute Näherungslösungen.In a particularly advantageous embodiment, the phase space is discretized into hypercubes with an edge length r. Here, “hypercube” means that the cube can have one dimension for each measured value delivered at the relevant time and thus significantly more than three dimensions in total. For a sequence of times 1, ..., N, which can in particular, for example, be equidistant from each other by intervals Δt. The probability that the trajectory in the sampled time course from time 1 to time N passes through a certain sequence i ₁ , i ₂ , ..., i _N of hypercubes, where indices of hypercubes can also occur multiple times, is P _{i 1} , _{i 2} , ..., i _N . The Kolmogorov entropy K can be determined from such probabilities. Different entropy measures can be used, for example, according to the formula $$K_q=-\underset{r\to 0}{\text{lim}}\underset{\Delta t\to 0}{\text{lim}}\underset{N\to \infty }{\text{lim}}\frac{1}{N\Delta t}\frac{1}{q-1}\text{ln}{\displaystyle \sum _{{\mathrm{<figure-callout\; id="1"\; label="i"\; filenames="DE102022202693A1\_0017.png,DE102022202693A1\_0018.png"\; state="\{\{state\}\}">i</figure-callout>}}_1,i_2,\dots ,i_{\mathrm{<figure-callout\; id="1"\; label="N\; P\; i"\; filenames="DE102022202693A1\_0017.png,DE102022202693A1\_0018.png"\; state="\{\{state\}\}">N</figure-callout>}}}{P}_{i_{}}^{}{{}_1,i_2,\dots ,i_N}_q^{}.}$$

be determined. K ₀ is the topological entropy, K ₁ is the Kolmogorov entropy K, and K ₂ is the K ₂ entropy, which can be used as an approximation for the Kolmogorov entropy K. The parameter q determines the power of the probability P _i to be taken into account ₁ , _{i 2} , ..., i _N . This type of calculation enables good approximate solutions.

mit $$K_2^m\left(r\right)=\frac{1}{k\Delta t}\text{ln}\frac{C^m\left(r\right)}{C^{m+k}\left(r\right)}$$

verwendet werden, worin k eine hinreichend kleine ganze Zahl ist und m die Embedded-Dimension des technischen Systems ist. Die Embedded-Dimension m ist derjenige Wert von m, bei dem der Ausdruck für $K_2^m$

in die Sättigung geht, und dieses $K_2^m$

kann dann als Näherung für K₂ verwendet werden. Um diesen Wert von m zu ermitteln, kann beispielswiese $K_2^m$

für Werte zwischen m = 0 und m = 300 berechnet werden. Das Korrelationsintegral C^m(r) ist definiert durch $$C^m\left(r\right)=\frac{1}{M-1}{\displaystyle \sum _{{}_{i\ne j}^{j=1}}^M\theta \left(r-\Vert Y_i-Y_j\Vert \right)},$$

worin M = N - (m - 1)/Δt ist. Je größer also die Embedded-Dimension m des technischen Systems ist, desto mehr Summanden bleiben am Ende der Summe unberücksichtigt. θ ist die Heaviside-Stufenfunktion, die gegeben ist durch $$\theta \left(x\right)=\{\begin{array}{cc}1& x\ge 0\\ 0& x<0\end{array},$$

The Kolmogorov entropy can be approximated particularly advantageously by evaluating how much a correlation integral determined from points Y ₁ , ... , Y _M of the trajectory changes when M decreases below N in accordance with an embedded dimension m of the technical system becomes. m is also called the embedding dimension. In particular, the K ₂ form can be used here, for example $$K_2\sim \underset{m\to \infty }{\text{lim}}{K}_2^m\left(r\right)$$

with $$K_2^m\left(r\right)=\frac{1}{k\Delta t}\text{ln}\frac{C^m\left(r\right)}{C^{m+k}\left(r\right)}$$

can be used, where k is a sufficiently small integer and m is the embedded dimension of the technical system. The embedded dimension m is the value of m at which the expression for $K_2^m$

goes into saturation, and this $K_2^m$

can then be used as an approximation for K ₂ . To determine this value of m, for example $K_2^m$

can be calculated for values between m = 0 and m = 300. The correlation integral C ^m (r) is defined by $$C^m\left(r\right)=\frac{1}{M-1}{\displaystyle \sum _{{}_{i\ne j}^{j=1}}^M\theta \left(r-\Vert Y_i-Y_j\Vert \right)},$$

where M = N - (m - 1)/Δt. The larger the embedded dimension m of the technical system, the more summands remain unconsidered at the end of the sum. θ is the Heaviside step function given by $$\theta \left(x\right)=\{\begin{array}{cc}1& x\ge 0\\ 0& x<0\end{array},$$

For example, the embedded dimension m can also be determined by increasing m step by step and calculating the fractal dimension d _F (or another invariant) of the technical system for each value of m. m is increased until the fractal dimension d _F is almost preserved. This value of d _F is then the fractal Dimension of the reconstructed attractor, which can also be viewed as the fractal dimension of the original attractor. Usually m ≥ 2d _F is sufficient.

It is particularly advantageous to choose a technical system whose dynamics can be described by an Euler-Lagrange equation and optionally also by initial conditions, district conditions and/or boundary conditions. Solutions to the Euler-Lagrange equation provide a particularly good way to see whether the technical system is completely described by the equation and the sensory information processed in it: An analytical solution to the Euler-Lagrange equation corresponds to at least one trajectory under certain initial conditions and boundary conditions , for which a statement can be made to the effect that, starting from this previous trajectory, the system is currently completely described by the Euler-Lagrange equation in conjunction with the initial conditions and boundary conditions as well as the measured values. However, the analysis of the Kolmogorov entropy K is numerically much easier to carry out than an examination of the analytical solutions of the Euler-Lagrange equation.

über mehrere, für unterschiedliche Embedded-Dimensionen erhaltene Werte gemittelt werden: $$K_2^m\left(r\right)\equiv \frac{1}{L}{\displaystyle \sum _{l=1}^L\frac{1}{2l\mathrm{\Delta }t}}\text{ln}\frac{C^{m-2l}\left(r\right)}{C^m\left(r\right)}.$$

In order to reduce fluctuations and improve the statistics, the expression for can then be used $K_2^m$

can be averaged over several values obtained for different embedded dimensions: $$K_2^m\left(r\right)\equiv \frac{1}{L}{\displaystyle \sum _{l=1}^{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="DE102022202693A1\_0017.png,DE102022202693A1\_0018.png"\; state="\{\{state\}\}">L</figure-callout>}}\frac{1}{2l\mathrm{\Delta }t}}\text{ln}\frac{C^{m-2l}\left(r\right)}{C^m\left(r\right)}.$$

Here L is the number of values to be averaged over. In the inventors' experiments, numbers around L = 5 have proven to be particularly advantageous.

von m durch einen Least-Squares-Fit der Funktion $$ƒ\left(x\right)=a+\frac{b}{x^c}$$

an die Messwerte ermittelt werden. Für c > 0 und x → ∞ konvergiert diese Funktion gegen a. Damit ist $$K_2^m\left(r\right)=K_2\left(r\right)+\frac{b}{m^c},c>0.$$

b und c sind reellwertige Parameter, m ist die Embedded-Dimension des technischen Systems. Die Parameter a, b und c können aus Experimenten und/oder Simulationen des technischen Systems erhalten werden.In a further particularly advantageous embodiment, the Kolmogorov entropy K is determined by fitting a parameterized approach to the measured values supplied by the sensors. Such a fit can be calculated quickly and easily, especially on embedded systems. For example, the dependency on $K_2^m$

of m by a least squares fit of the function $$ƒ\left(x\right)=a+\frac{b}{x^c}$$

to the measured values can be determined. For c > 0 and x → ∞ this function converges to a. That's it $$K_2^m\left(r\right)=K_2\left(r\right)+\frac{b}{m^c},c>0.$$

b and c are real-valued parameters, m is the embedded dimension of the technical system. The parameters a, b and c can be obtained from experiments and/or simulations of the technical system.

As a predetermined criterion for ensuring that the detection with the sensors and the differential equation fully describe the dynamics of the system, the doubling time T ₂ can be a deviation between the actual dynamics of the system on the one hand and the prediction by the differential equation on the other hand. This doubling time T ₂ is given by $$T_2=\frac{\text{ln 2}}{K_2}.$$

If the doubling time T ₂ is below a critical limit, it can be concluded that the detection with the sensors and the differential equation fully describe the dynamics of the system. However, if the critical limit is exceeded, it can be concluded that the description is not complete. The critical limit can be determined for any technical system using experiments or simulations. For example, for traffic situations as technical systems it is 1.3.

This condition is not mathematically equivalent to the initially stated condition 0 < K < ∞ for the Kolmogov entropy K in the sense that one condition can be clearly converted into the other through mathematical transformation steps. However, the conditions are correlated with each other in the sense that if T ₂ is below the critical limit, then with a high probability 0 < K < ∞ also applies and vice versa.

An important application of the method is that a control logic for the technical system, which predicts the dynamics of the system based on the detection with the sensors and the differential equation and determines control interventions based on this prediction, can be checked for safe functioning. It can therefore be checked whether the system will behave as the differential equation predicts, or whether there are indications based on the measured values provided by the sensors that the technical system will behave differently. In the latter case, a control intervention determined by the control logic based on the prediction may not be appropriate to the situation in which the technical system actually finds itself. Such a test can be carried out both as part of the delivery and approval (release) of the control logic and online during ongoing operation.

The background is that every description of the technical system with a differential equation and measured values recorded by sensors is based on a modeling of the technical system. The level of detail of these modeling is chosen so that the phenomena relevant to the operation of the technical system are captured, while at the same time the model can be calculated with given hardware resources in a given time. For example, modeling the Earth as a point of mass is sufficient for a rough description of the movement of planets. To plan a flight to the moon, the Earth must already be modeled as a massive sphere. This is no longer sufficient to accurately describe the dynamics of satellites, because both the topography of the earth's surface and the inhomogeneous mass distribution have an impact here.

There may also be situations during ongoing operation of the technical system in which modeling is suddenly no longer sufficient and detailed modeling would be required to describe the dynamics of the system with sufficient precision. For example, when driving over long distances on a country road, modeling the traffic situation without precise depth information can be sufficient because you only have to pay attention to staying in your own lane and following traffic signs. But if a vehicle in oncoming traffic suddenly pulls out to overtake and comes head-on towards your own vehicle, the further development of the situation depends on how quickly the overtaking person approaches and whether he or she can overtake the overtaken vehicle in time to get it back into its lane. This requires reliable depth information about the traffic situation.

Therefore, in a particularly advantageous embodiment, in response to the determination that the detection with the sensors and the differential equation completely describe the system, a forecast for the state of the technical system in the future is determined based on the differential equation. A control signal is determined based on this forecast. The system is controlled with this control signal. It can then be expected with a very high degree of probability that this control of the technical system is appropriate to the actual situation in which the technical system finds itself.

However, if it is determined that the detection with the sensors and the differential equation do not fully describe the system, there are various ways to react to this.

For example, a new configuration of sensors and/or a new differential equation may be established. The test procedure can then be started again based on the new combination of sensor configuration and a differential equation formed. In particular, this can be used to achieve the goal of completely describing the dynamics of the technical system using detailed sensory recording and/or modeling. In the aforementioned example with the traffic situation in which the overtaking person pulls out, a radar sensor can, for example, be switched on in addition to one or more previously used cameras in order to precisely record the dynamics of the overtaking person and the overtaken in the direction of travel.

For example, an operating point of the technical system can be changed and the method can be restarted based on measured values supplied by the sensors at the new operating point. For example, if the exposure setting for a camera is inappropriate and a portion of the image saturates at the top or bottom of the intensity scale, this may result in the images provided by the camera containing insufficient information to fully capture a traffic situation.

Alternatively or in combination, in response to the determination that the detection with the sensors and the differential equation do not fully describe the system, the functionality of the technical system can be restricted, put into a safe state or deactivated. For example, the driving speed of an at least partially automated vehicle can be reduced, or overtaking maneuvers can be prevented. The vehicle can also be brought to a standstill, for example, on a pre-planned emergency stop trajectory. For example, an operator, such as a driver, can also be asked to take control of the system.

In a further particularly advantageous embodiment, the behavior of the technical system is observed and/or simulated in a variety of operating situations. For each operating situation, the Kolmogorov entropy and a chaos indicator are determined to determine the extent to which the technical system behaves chaotically. These Kolmogorov entropies and chaos indicators are used to determine the specified criterion for ensuring that the detection with the sensors and the differential equation fully describe the dynamics of the system.

For example, a largest Kolmogorov entropy, for which the technical system certainly does not yet show chaotic behavior, can be set as a critical limit. If based on late If a Kolmogorov entropy is determined from the measured values that lies below this critical limit, it can then be determined that the detection with the sensors and the differential equation completely describe the dynamics of the system.

verwendet werden.For the Kolmogorov entropy K, one of the previously described approximations, such as K ₂ or $K_2^m,$

be used.

As explained above, in particular, for example, a vehicle and/or a traffic situation with several road users can be selected as the technical system. Traffic situations in particular can quickly develop during ongoing operations from normal situations to critical situations in which simple modeling that was previously used may no longer be valid. The hardware resources for checking for a complete description are also scarce, especially when used in vehicles. The method proposed here can work on the basis of a ring buffer, which stores a history of the previous measured values over a limited period of time, and can make do with arithmetic operations that can be carried out quickly even on an embedded system with little computing capacity and memory. The method is therefore advantageously carried out in real time on an embedded system that is contained in the technical system or is carried by it.

The method can in particular be implemented entirely or partially by computer. Therefore, the invention also relates to a computer program with machine-readable instructions which, when executed on one or more computers, cause the computer or computers to carry out the method described. In this sense, control devices for vehicles and embedded systems for technical devices that are also capable of executing machine-readable instructions are also considered computers.

The invention also relates to a machine-readable data carrier and/or to a download product with the computer program. A download product is a digital product that can be transferred via a data network, i.e. downloadable by a user of the data network and which can be offered for sale in an online shop for immediate download, for example.

Furthermore, one or more computers can be equipped with the computer program, with the machine-readable data carrier or with the download product.

Further measures improving the invention are shown in more detail below together with the description of the preferred exemplary embodiments of the invention using figures.

Examples of embodiments

• 1 Ausführungsbeispiel des Verfahrens 100;
• 2 Beispiele für Trajektorien in Phasenräumen 4 für eine Verkehrssituation als technisches System 1.

It shows:

• 1 Embodiment of the method 100;
• 2 Examples of trajectories in phase spaces 4 for a traffic situation as a technical system 1.

1 is a schematic flowchart of an exemplary embodiment of the method 100 for checking whether the combination of recording the state of a technical system 1 with several sensors 2a-2c and a differential equation (as well as optionally initial conditions, district conditions and / or boundary conditions) in the sensors 2a -2c supplied measured values 3a-3c completely describe the dynamics of this technical system 1.

In step 110, at least one trajectory Y ₁ , ..., Y _N sampled at N discrete times is provided in a phase space 4 spanned by the measured values 3a-3c supplied by the sensors 2a-2c. Here, according to block 111, the phase space 4 can be discretized into hypercubes with an edge length r.

In step 120, based on this trajectory Y ₁ , ..., Y _N , a Kolmogorov entropy K, which is a measure of how much additional information the measured values supplied for each point in time contain, is at least approximately determined.

According to block 121, the Kolmogorov entropy K can be derived from probabilities P _{i 1} , _{i 2} , ..., i _N can be determined so that the trajectory in the sampled time course runs through certain sequences i ₁ , i ₂ , ..., i _N of hypercubes in phase space 4. The Kolmogorov entropy K can then be approximated by evaluating how much a correlation integral determined from points Y ₁ , ...,Y _M of the trajectory changes when M decreases below N according to an embedded dimension m of the technical system becomes.

According to block 123, Kolmogorov entropy K can be determined by fitting a parameterized approach to the measured values 3a-3c provided by the sensors 2a-2c.

In step 130 it is checked whether this Kolmogorov entropy K, and/or a quantity derived from it, fulfills a predetermined criterion 5. Is this the case (truth value 1), it is determined in step 140a that the detection with the sensors 2a-2c and the differential equation completely describe the dynamics of the system 1. Otherwise (truth value 0), it is determined in step 140b that the detection with the sensors 2a-2c and the differential equation do not fully describe the dynamics of the system 1.

In order to determine the specified criterion 5, the behavior of the technical system 1 can be observed and/or simulated in a variety of operating situations, for example according to block 107. According to block 108, the Kolmogorov entropy K and, on the other hand, a chaos indicator 10 can then be determined for each operating situation to determine the extent to which the technical system 1 behaves chaotically. The specified criterion 5 can finally be determined from these Kolmogorov entropies K and chaos indicators 10 according to block 109.

According to block 105, a technical system 1 can be selected as the technical system 1, for example, the dynamics of which can be described by at least one Euler-Lagrange equation.

According to block 106, in particular, for example, a vehicle and/or a traffic situation with several road users can be selected as technical system 1.

In response to the determination in step 140a that the detection with the sensors 2a-2c and the differential equation completely describe the system 1, a forecast 6 for the state of the technical system 1 in the future can be determined in step 150 using the differential equation. Based on this forecast, a control signal 7 can be determined in step 160. In step 170, the system 1 can then be controlled with this control signal 7.

In response to the determination in step 140b that the detection with the sensors 2a-2c and the differential equation do not fully describe the system 1, various reactions are possible.

In step 180, a new configuration 8a of sensors 2a-2c, and/or a new differential equation 8b, may be established. In step 190, the method 100 can then be started again based on the new combination of sensor configuration and a differential equation formed in this way.

In step 200, an operating point 9 of the technical system 1 can be changed. In step 210, the method 100 can then be restarted based on measured values 3a-3c supplied by the sensors 2a-2c in the new operating point 9.

In step 220, the technical system 1 can be restricted in its functionality, put into a safe state or deactivated. Alternatively or in combination with this, an operator can be requested to take control of the system 1 in step 230.

2a shows a traffic situation 20 at an intersection as a technical system 1. The traffic situation 20 contains three road users 21, 22 and 23, each moving in directions 21a, 22a and 23a.

The 2 B and 2c show exemplary trajectories of individual time-dependent variables x(t) of the traffic situation 20 in a two-dimensional phase space 4. This phase space 4 is spanned by the variable x(t) and its value x(t - 0.5 s) half a second before.

The comparison of the 2 B and 2c illustrates that the extent to which the dynamics of the technical system 1 is completely described by the sensory detection of certain variables and a differential equation in these variables can be easily evaluated using trajectories Y ₁ , ... , Y _N and the associated Kolmogorov entropies K.

In 2 B The trajectory spreads very widely around its attractor, so that it can no longer be clearly determined, especially in the circled area, whether a particular pair of values x(t) and x(t - 0.5 s) belongs to the left or right branch of the Trajectory belongs. Here, the evaluation of the Kolmogorov entropy will lead to the conclusion that the dynamics of System 1 are not fully described.

On the other hand, it runs in 2c the trajectory is very close to its attractor, and with the exception of a few outliers, all pairs of values x(t) and x(t - 0.5 s) can be clearly assigned to locations along the course of the trajectory in phase space 4. Thus, the evaluation of the Kolmogorov entropy here will lead to the result that the dynamics of system 1 are completely described.

QUOTES INCLUDED IN THE DESCRIPTION

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Cited patent literature

• EP 0339056 B1 [0002]

Claims (15)

Method (100) for checking whether the combination of recording the state of a technical system (1) with several sensors (2a-2c) and a differential equation in the measured values (3a-3c) supplied by these sensors (2a-2c). Dynamics of this technical system (1) completely describes, with the steps: • at least one trajectory Y ₁ , ..., Y _N is sampled at N discrete times in a measured value (3a.) supplied by the sensors (2a-2c). -3c) spanned phase space (4) provided (110); • based on this trajectory Y ₁ , ..., Y _N , a Kolmogorov entropy K, which is a measure of how much additional information the measured values supplied for each point in time contain, is at least approximately determined (120); • in response to the fact that this Kolmogorov entropy K, and/or a quantity derived from it, fulfills a predetermined criterion (5) (130), it is determined (140a) that the detection with the sensors (2a-2c) and the Differential equation fully describes the dynamics of the system (1). Procedure (100) according to Claim 1 , where the phase space (4) is discretized into hypercubes of edge length r (111) and the Kolmogorov entropy K from probabilities P _{i 1} , _{i 2} , ..., i _N is determined (121) so that the trajectory in the sampled time course runs through certain sequences i ₁ , i ₂ , ..., i _N of these hypercubes. Procedure (100) according to Claim 2 , whereby the Kolmogorov entropy K is approximated by an evaluation (122) of how much a correlation integral determined from points Y ₁ , ..., Y _M of the trajectory changes if M according to an embedded dimension m of the technical system is reduced under N. Method (100) according to one of Claims 1 until 3 , where the Kolmogorov entropy K is determined by fitting a parameterized approach to the measured values (3a-3c) provided by the sensors (2a-2c) (123). Method (100) according to one of Claims 1 until 4 , whereby in response to the statement (140a) that the detection with the sensors (2a-2c) and the differential equation completely describe the system (1), • based on the differential equation, a forecast (6) for the state of the technical system (1 ) is determined in the future (150), • based on this forecast (6) a control signal (7) is determined (160) and • the system (1) is controlled with this control signal (7) (170). Method (100) according to one of Claims 1 until 5 , where in response to the statement (140b) that the detection with the sensors (2a-2c) and the differential equation do not fully describe the system (1), • a new configuration (8a) of sensors (2a-2c), and /or a new differential equation (8b) is drawn up (180), and • the method (100) is started again (190) based on the new combination of sensor configuration and a differential equation formed in this way. Method (100) according to one of Claims 1 until 6 , whereby in response to the statement (140b) that the detection with the sensors (2a-2c) and the differential equation do not completely describe the system (1), • an operating point (9) of the technical system (1) is changed (200 ) and • the method (100) is restarted (210) based on measured values (3a-3c) supplied by the sensors (2a-2c) at the new operating point (9). Method (100) according to one of Claims 1 until 7 , whereby in response to the statement (140b) that the detection with the sensors (2a-2c) and the differential equation do not completely describe the system (1), • the technical system (1) is limited in its functionality, in a safe state is moved or deactivated (220), and/or • an operator is requested to take control of the system (1) (230). Method (100) according to one of Claims 1 until 5 , whereby a technical system (1) is selected (105), the dynamics of which can be described by an Euler-Lagrange equation. Method (100) according to one of Claims 1 until 9 , whereby • the behavior of the technical system (1) is observed and/or simulated in a variety of operating situations (107), • for each operating situation, on the one hand, the Kolmogorov entropy K and, on the other hand, a chaos indicator (10) to the extent that technical system (1) behaves chaotically, is determined (108); and • from these Kolmogorov entropies K and chaos indicators (10) the specified criterion (5) is determined for the fact that the detection with the sensors (2a-2c) and the differential equation completely describe the dynamics of the system (1) ( 109). Method (100) according to one of Claims 1 until 10 , wherein a vehicle (21-23), and / or a traffic situation (20) with several traffic parts participants, is chosen as the technical system (1) (106). Method (100) according to one of Claims 1 until 11 , wherein the method (100) is carried out in real time on an embedded system that is contained in the technical system (1) or is carried by it. Computer program containing machine-readable instructions which, when executed on one or more computers, cause the computer or computers to carry out the method (100) according to one of the Claims 1 until 12 to carry out. Machine-readable data carrier and/or download product with the computer program Claim 13 . One or more computers with the computer program Claim 13 , and/or with the machine-readable data carrier and/or download product Claim 14 .

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