Method for identifying parameter distributions for modeling technical systems

Abstract

The invention relates to a method (100) for identifying a parameter distribution for modeling a technical system (50), comprising the following steps: - determining (101) a nominal parameter estimate on the basis of provided measurement data and at least one pre-given system dynamics model of the technical system (50); - determining (102) a distribution of parameter variations on the basis of a quantification of the uncertainty due to parameters of the nominal parameter estimate; - providing (103) the nominal parameter estimate and the distribution of parameter variations as a basis for modeling the technical system (50) taking into account the uncertainty due to parameters.

Description

Technical Field

[0001] This invention relates to a method for identifying parameters used in modeling a technical system. The invention also relates to computer programs, devices, and storage media used for this purpose. Background Technology

[0002] Determining system parameters is a core element of model-based design. Crucially, this involves determining how well the model can describe reality. Many statements, such as performance guarantees, safety statements, constraint compliance, and failure probabilities, are decisively dependent on this.

[0003] Therefore, several methods for identifying system parameters based on measurement data are known from the prior art. Typically, only a nominal set of parameters is determined, which represents the solution to the optimization problem. However, the remaining uncertainties are usually not considered or are only implicitly considered, even though these uncertainties are relevant to many subsequent decisions.

[0004] Furthermore, inverse "uncertainty quantification" methods use Bayesian statistics to determine the distribution of parameters. These methods can be implemented within the framework of "probabilistic programming." Applications require a high level of expertise. Other requirements for model implementation further limit usability. Applying these methods is associated with significant time and cost expenditures and is difficult to integrate into existing processes used for model identification. Summary of the Invention

[0005] The subject of this invention is a method having the features of claim 1, a computer program having the features of claim 9, an apparatus having the features of claim 10, and a computer-readable storage medium having the features of claim 11. Other features and details of the invention derive from the corresponding dependent claims, the description, and the drawings. Herein, the features and details described in conjunction with the method according to the invention also apply in conjunction with the computer program according to the invention, the apparatus according to the invention, and the computer-readable storage medium according to the invention, and vice versa, so that mutual reference is always possible in all aspects of the disclosure of the invention.

[0006] The subject of this invention is, in particular, a method for identifying parameters and / or parameter distributions and / or for performing uncertainty quantification and / or for modeling technical systems.

[0007] The method may include determining nominal parameter estimates based on provided measurement data and at least one pre-given system dynamics model of the technical system.

[0008] The method may further include determining the distribution of parameter variation based on the quantification (or random component) of the uncertainty caused by the parameter in the estimate of the nominal parameter.

[0009] The method may further include providing the nominal parameter estimate and the distribution of the parameter variation as a basis for modeling the technical system, taking into account the uncertainties caused by the parameters.

[0010] The present invention offers the advantage of providing a model of a technical system that considers not only nominal parameter values ​​but also their uncertainties. This enables more accurate and reliable system modeling because potential fluctuations in parameters are detected. Considering uncertainties in the model leads to more robust control strategies and improved reliability of the technical system, particularly in demanding applications such as the automotive industry.

[0011] According to the present invention, a two-stage scheme is proposed for identifying more reliable model parameters. In the first step, the nominal error between the available data and the model can be minimized without considering uncertainties. Therefore, the remaining model uncertainty surrounding this estimate can then be characterized using linear Bayesian methods.

[0012] Compared to traditional nonlinear Bayesian methods, the estimates obtained according to the present invention can be more conservative when necessary due to linearization. However, significant advantages are also derived, which are further elaborated below.

[0013] Thus, this invention offers the advantage of not requiring a specific model structure as a prerequisite. The model interface only needs pre-defined input parameters, the parameters to be identified, and the corresponding system response. Therefore, this method can be applied to complex numerical models without requiring an analytical description.

[0014] The additional structural assumptions enable efficient evaluation of the required linearization and Bayesian regression. The latter allows for the use of the examined technical system at runtime.

[0015] Furthermore, Bayesian statistics can be encapsulated, enabling broad applicability even without explicit expert knowledge. Using second-order statistics, where parameters are internally described by mean and (co)variance, allows for the realization of closed-loop solutions to linear Bayesian regression problems. The resulting linear error can also be quantified and considered within the scope of model uncertainty in the draft.

[0016] Furthermore, existing nominal parameter estimates can be integrated into this method. These estimates can be confirmed, improved, or rejected within the scope of this invention.

[0017] One possible application of the invention is, for example, system identification of vehicle models used for lateral guidance, longitudinal guidance, driving dynamics control, or component development.

[0018] Furthermore, it is advantageous to use this method to identify parameters in the form of model parameters for closed-loop control loops used for vehicle control and, preferably, for lateral guidance of the vehicle. Thus, the proposed solution for parameter identification can be directly applied to real vehicle systems to, for example, more accurately model the behavior of the vehicle during cornering and thereby improve stability and safety.

[0019] Furthermore, within the scope of this invention, the method may be configured to identify parameters for a vehicle model, preferably used for vehicle lateral guidance and / or vehicle longitudinal guidance and / or driving dynamics control and / or component development and / or steering adjustment and / or braking adjustment. This, in particular, enables improved identification of parameters and / or parameter distributions in different areas of vehicle technology.

[0020] Optionally, it is conceivable to use the method described above to study the transfer characteristics (Übertragungsverhalten) from the desired curvature of the vehicle trajectory to the actual curvature of the vehicle trajectory, wherein the following steps are set up: - Provide the pre-given model in the form of a closed-loop control loop for lateral vehicle guidance; and / or - Perform the nominal parameter estimation to make a first estimate of at least one parameter of the model based on the provided measurement data; and / or - Perform quantification of the uncertainty caused by the parameters, which provides the residual uncertainty in the estimated parameters; and / or - Determine the distribution of parameter variations based on the performed quantization; and / or - To perform modeling of the technical system based on the nominal parameter estimates and the distribution of the parameter variations; and / or - Based on the modeling, a pre-given adjustment for vehicle lateral guidance is matched.

[0021] Therefore, this method can be used to determine the parameters and / or parameter distributions that influence the transfer characteristics from the desired curvature of the vehicle trajectory to the actual curvature of the vehicle trajectory. Modeling the system while considering uncertainties in the parameters allows for matching the pre-given adjustments for vehicle lateral guidance, resulting in more accurate behavior and improved safety.

[0022] Alternatively, it is conceivable that determining the nominal parameter estimate includes minimizing the nominal error between the provided measurement data and the pre-given model, particularly without considering uncertainties. In other words, calculating the nominal parameter values ​​involves minimizing the minimum deviation between the available measurement data, and in particular the measured values, and the model.

[0023] According to an advantageous improvement of the invention, determining the distribution of the parameter variations includes characterizing the uncertainty caused by the parameters in the form of the model uncertainty remaining from the nominal parameter estimates. This enables a quantitative evaluation of the impact of each parameter on the system behavior.

[0024] According to another possibility, it can be stipulated that the determination of the distribution and, in particular, the characterization of the uncertainty caused by the parameters are performed using a linear Bayesian method. In other words, the method preferably uses a linear approximation of the system to characterize the parameter distribution.

[0025] Alternatively, it is conceivable that, considering the uncertainties caused by the parameters, the modeling of the technical system is configured such that the nominal parameter estimates and the distribution of parameter variations are combined into a total estimate. In other words, the nominal parameter estimates and the distribution of parameter variations are preferably combined to obtain a comprehensive estimate of the system that considers not only the average value but also the uncertainty of the parameters. This enables the realization of a more realistic and robust model of the technical system.

[0026] It is possible to use the method according to the invention in the case of a vehicle. The vehicle may, for example, be configured as a motor vehicle and / or a passenger car and / or at least a partially automated / autonomous vehicle. The vehicle may have, for example, vehicle devices and / or driver assistance systems for providing autonomous driving functions. The vehicle devices may be implemented for at least partially automated control and / or acceleration and / or braking and / or steering of the vehicle.

[0027] The subject of this invention is also a computer program, and more particularly a computer program product, comprising instructions that, when executed by at least one computer, cause the computer to perform the method according to the invention. Therefore, the computer program according to the invention results in the same advantages as described in detail with respect to the method according to the invention.

[0028] The subject of this invention is also an apparatus for performing data processing, the apparatus being configured to implement the method according to the invention. For example, at least one computer executing a computer program according to the invention may be provided as the apparatus. The computer may have at least one processor for executing the computer program. A non-volatile data memory may also be provided, in which the computer program may be stored, and which the processor may read from the non-volatile data memory for execution.

[0029] The subject of this invention is also a computer-readable storage medium having a computer program according to the invention and / or including instructions that, when executed by at least one computer, cause the computer to perform the method according to the invention. For example, the storage medium is configured as a data storage device, such as a hard disk and / or non-volatile memory and / or a memory card. For example, the storage medium may be integrated into a computer.

[0030] Furthermore, the method according to the invention can also be implemented as a computer-based method. Alternatively or additionally, at least one of the disclosed method steps can be computer-based and / or can be performed automatically. Attached Figure Description

[0031] Other advantages, features, and details of the invention will become apparent from the following description, in which embodiments of the invention are described in detail with reference to the accompanying drawings. Here, features mentioned in the claims and the specification may be important to the invention individually or in any combination. Wherein: Figure 1 A schematic visualization illustrating a method, apparatus, storage medium, and computer program according to embodiments of the present invention.

[0032] Figure 2 The system identification of a variant of the present invention is shown embedded in the V-model.

[0033] Figure 3 An exemplary flowchart for identifying parameters and their uncertainties according to an embodiment of the present invention is shown.

[0034] Figure 4 This illustrates the technical system on which the embodiments of the present invention are based and a visualization of the problems.

[0035] Figure 5 shows another exemplary visualization of the application of a variant of the present invention. Detailed Implementation

[0036] exist Figure 1The diagram schematically illustrates a method 100, an apparatus 10, a storage medium 15, and a computer program 20 according to an embodiment of the present invention.

[0037] According to an embodiment of the invention, a method 100 is provided for identifying parameters and / or parameter distributions for modeling a technical system 50. According to a first method step 101, in this case, nominal parameter estimates are determined based on provided measurement data D and at least one pre-given system dynamics model of the technical system 50. As further described below, the technical system 50 preferably relates to a vehicle. System dynamics may, for example, involve lateral stiffness and / or vehicle parameters and / or environmental parameters and / or tire characteristic curves.

[0038] According to step 102 of the second method, the distribution of parameter variation is determined based on the quantification of the uncertainty caused by the parameters in the nominal parameter estimate. In particular, the distribution for the random component is determined in this case.

[0039] According to the third method step 103, taking into account the uncertainties caused by the parameters, nominal parameter estimates and parameter variation distributions are provided as a basis for modeling the technical system 50. In other words, modeling can be performed based on the results of the aforementioned method steps 101 and 102. Embodiments of the present invention can be a central component of system identification (see, for example, see...). Figure 2 (206 in the text) makes it possible, in other words, to consider using modeling for system identification of technical system 50.

[0040] exist Figure 2 The diagram illustrates the V-model, which describes the various stages and processes in the development cycle. Here, the steps are named as follows: Step 201 describes the formulation and definition of the requirements. Then, in step 202, system design is performed, followed by system sample definition in step 203. In the next step, 204, system samples are built ("System Sample Setup"). Subsequently, measurements occur at the system level in step 205 ("System Measurements"). Based on these results, a credible system model is created in step 206, which is then verified and validated by simulation in step 207 ("Verification & Validation by Simulation").

[0041] The system is then validated in several phases: Step 208 includes a physical system sample (“System Sample”), followed by laboratory testing in Step 209 (“Validation by Lab Tests”) and testing under real-world conditions in Step 210 (“Validation by Real World Tests”). Finally, original equipment manufacturer (OEM) testing (“OEM Tests”) is performed in Step 211.

[0042] Typically, based on system measurements 205, not only should the nominal parameter set be determined, but its uncertainty should also be characterized by a random distribution. This information can be utilized in the left path of the V-model (system synthesis, including 201 and 202) to achieve a trade-off between the performance and robustness of the regulating function. This can be done at design time or runtime. In the right path of the V-model (system analysis, including 207 to 211), the characterized uncertainty can be used for release justification based on a reliable model.

[0043] The embodiments of this invention are divided into five key steps, the technical background of which is described in more detail below. Figure 3 These steps are shown in further detail below.

[0044] exist Figure 3 In the diagram, 301 represents the processing of the system model, and 302 represents the calculation of linear... The mechanism is as follows: 303 represents nominal identification; 304 represents the process of updating the nominal identification when necessary; 305 represents the optimization problem; 306 represents the numerical solver; 307 represents constructing a prior estimate for Δp; 308 represents the process of updating the prior estimate when necessary; 309 represents linear Bayesian regression; and 310 represents the mechanism for the posterior distribution. The calculation method is defined as follows: 311 represents the result and application; 312 represents applying the algorithm to subsequent or new data packets; 313 represents the verification of the result; and 314 represents the creation of a proxy model. Additionally, 351 represents optional paths, and 352 represents externally required paths.

[0045] The basic concept for determining a credible parameter estimate within the scope of embodiments of this invention begins with the breakdown: (1) Here, p0 is the result of nominal identification 303 without considering uncertainty, and is the core of step 2. Based on this, the random component Δp is determined, which enables the determination of confidence intervals for the parameters and output parameters.

[0046] 1. System Model.

[0047] according to Figure 3 Step 301 in the process begins with the system model, namely the model of the technical system 50. This model is established on the structural relationship between the parameter p to be identified, the given system excitation u(t), and the output parameter y(t) corresponding to the measurement parameter: (2) This model typically has an additional internal state parameter x(t), and exists in the form of a system of differential equations: (3) The following section presents the time series of the input and output parameters. Assuming the data is from measurements, a constant sampling time is used here to simplify the notation method. ,in and .

[0048] By capital letters and To label measurement signals that have been aggregated into vectors or matrices.

[0049] A variation of this invention requires the partial derivatives of the model with respect to the parameters in the neighborhood of a fixed value p0. This can be done numerically, for example, using the finite difference method, as long as the model does not exist in analytical form. To avoid implicit differentiation, it is desirable to make the model... Transformation into form (4) The input and output representations are shown. Transforming them into a discrete-time representation has the advantage that internal state parameters can be replaced by past values ​​of the inputs and outputs, which are directly present in the data D. Therefore, the partial derivatives required for the model at a given point p0 can be determined as follows: (5) Due to the model The memory properties that are not clearly apparent in (1) are that the value of the Jacobian matrix also depends on past input and output values, as can be seen in (4).

[0050] General case (numerical approximation): Here, the Jacobian matrix is ​​obtained using numerical methods, such as finite difference finite differences. The approximation is that the number of evaluations required for model (2) depends on the number of parameters and output parameters.

[0051] Special Case 2 (Sensitivity ODE): Here, the model exists in the form of an analytic function (3), and the Jacobian matrix... The current value can be obtained by applying the sensitivity differential equation to the following: To determine this, we can apply the chain rule to derive the following relationship: To calculate the Jacobian matrix At the required time point The time-varying process of the system necessitates a cosimulation (kosimuliert) of the dynamic system in addition to (3). Further evaluation of the model, as is typically required, is unnecessary.

[0052] Special Case 1 (LPV System): In which and In the case of linear dynamics with nonlinear parameter intervention, a closed-loop relation for the input-output model (4) can be derived over a complex time-discrete image domain. In the exemplary case of using the zero-order hold method, the time-discrete transfer function follows... (6) It is an image variable The rational fractional polynomial in the equation. Therefore, in the time domain, (6) corresponds to the difference equation. (7) This makes it possible to pass the guidelines ,in The required Jacobian matrix can be efficiently and analytically computed from (5).

[0053] 2. Nominal identification.

[0054] Given a model (1) and data Identification of the nominal parameter set can be achieved using known existing techniques, for example, see Ljung, Lennart. "System identification." Signal analysis and prediction. Boston, MA: Birkhäuser Boston, 1998 und Bishop, Christopher, and Nasser Nasrabadi. Pattern recognition and machine learning. NY: Springer, 2006.

[0055] The goal is to compare the measurement data with the residuals. Error metric between set data Minimize. For the complete dataset under consideration, derive the error vector. and the optimal nominal parameter set: (8) Special cases: This method is not limited to a specific algorithm for determining p0. However, combining it with the linear Bayesian regression from step 4 offers advantages in choosing between maximum likelihood or maximum a posteriori estimation (Ridge Regression). This is based on the regularization parameter. This leads to the nonlinear optimization problem: (9) 3. Construct a prior estimate.

[0056] To apply Bayesian regression, it is necessary to construct a method for the random components. The prior distribution. Here, it's important to distinguish whether the algorithm is used initially or iteratively. For initializing the distribution, an example choice can be used: (10) This method is not limited to specific distributional assumptions for the prior distribution. However, Equation (10) enables the realization of closed-loop solutions to Bayesian regression problems in further processes, and thus enables runtime applicability. The covariance in (10) It can be directly derived from the known standard dispersion. Either by known (physical) boundaries To construct.

[0057] If the algorithm has already traversed a portion of the available data or new data exists in between, the posterior estimate from step 4 can be used as a new prior estimate, i.e., .

[0058] The average can be either set to It is either considered in the deterministic component. and .

[0059] The latter corresponds to the update of the linear points and is associated with the re-evaluation of the Jacobian matrix (5) when necessary.

[0060] 4. Linear Bayesian Regression.

[0061] The basis of Bayesian regression for the residual component Δp of the parameter to be identified is the residual error. Despite the nominal parameter estimate p0, the residual error still remains. To obtain a linear regression problem, a model linearized around p0 is used. (11). Here, This is the error term, which represents the variance. Measurement noise and through accuracy parameters The degree of confidence in the model. The latter can be measured by the standard error. Linearity error is also considered. The resulting regression problem is obtained through... (12) Given. Applying Bayes' theorem provides the results for a given (sub)dataset. Prior distribution under the condition The following relationship with the posterior distribution (13) For general distributions, numerical approximations are possible. Since the algorithm can also be used recursively, it is explicit across the entire dataset at this point. With sub-dataset This means that not all available data must be used, or existing estimates can be expanded with new data.

[0062] In contrast, for choice (7), a normal distribution is derived from (13). Having a closed-loop solution (14) For the system shown in (6)-(7) representing special case 2, the simplification is as follows: (15) This enables direct posterior evaluation, i.e., evaluation without numerical sampling. This is due to the Jacobian matrix. Since it can be pre-computed, the computation of the posterior distribution (13) is reduced to matrix-vector multiplication.

[0063] 5. Results and Applications.

[0064] The final step of the embodiment of the invention reassembles the two parts of the total estimate and combines them from step 2. Combined with Δp from step 4, they form the overall estimate.

[0065] This may include the following sub-aspects, depending on the circumstances: The linear assumption implies that the mean of the Bayesian posterior distribution is... It should be smaller, that is If this is not the case, then this is the metric for the estimates that need improvement in step 2 and subsequent steps, i.e., the optimization problem should be adapted, for example, by matching the regularization parameter λ in (9) or the accuracy parameter β in the regression problem (11).

[0066] Another indicator is the significant bias of the measurement data derived from an appropriate confidence interval. (Selection) Furthermore, there is consistent parameterization for both steps. Here, t can be chosen as the trace operator and can involve prior variance. or posterior variance .

[0067] As described in step 3, the implementation variations of the present invention can be applied recursively to subsets of data. If this is the case, then the next data segment can be selected here, and the previous steps can be repeated. This also includes runtime implementation, i.e., if new data has been collected since the last call, the algorithm can be made to re-traverse if necessary.

[0068] For a given model structure, it is possible to generate a deterministic surrogate model from the identified parameter distribution. Known methods can be used for this. The resulting model with the identified parameter distribution can then be applied to: the synthesis or updating of tuning algorithms during design or runtime, and sensitivity or reliability analysis during verification or validation.

[0069] The optimal input signal is determined through the steps described above.

[0070] If the nominal estimate is already known, step 2, "Nonlinear Identification," can be skipped if necessary. In this case, It must be given in advance either externally or by the user.

[0071] Implementation variations of this invention can be used in any technical context based on model-based adjustment capabilities and / or reliable model-based verification. The initial application is the identification of model parameters for a vehicle model with lateral dynamics.

[0072] Another application is identifying the longitudinal guidance behavior of a vehicle, steering adjustment (e.g., rack and pinion position adjustment), or braking adjustment loops (e.g., ESP, decoupled / integrated power brake, by-wire actuator, etc.). Other similar applications include driving dynamics control, (large-space) robotics, and motor regulation.

[0073] An important implementation variation is identifying the closed-loop control loop used for lateral guidance of the vehicle. To this end, Figure 4 Further details are shown below. A road segment model satisfying special case 1 from step 1 is used here. The objective of this study is to investigate the transfer characteristics from the desired curvature 402 of the vehicle trajectory to the actual curvature 401 of the vehicle trajectory. Based on existing measurement data (see...), Figure 5a Within the scope of step 2 of this invention, nominal parameter estimates describing key behaviors can be determined. The remaining uncertainty is addressed through steps 4. The distribution is used for quantization. The resulting distribution can be represented in the parameter space (see [reference]). Figure 5b ) or represented as a confidence interval of the system output (see Figure 5c ).

[0074] The above description of the embodiments is merely an example within the scope of the invention. Of course, the various features of the embodiments can be freely combined with each other without departing from the scope of the invention, as long as it is technically meaningful.

Claims

1. A method (100) for identifying parameter distributions for modeling a technical system (50), comprising the following steps: - Determine the nominal parameter estimate (101) based on the provided measurement data and at least one pre-given system dynamics model of the technical system (50); - The distribution of parameter variation (102) is determined based on the quantification of the uncertainty caused by the parameters in the nominal parameter estimate; - Taking into account the uncertainties caused by the parameters, the nominal parameter estimates and the distribution of the parameter variations are provided (103) as the basis for modeling the technical system (50).

2. The method (100) according to claim 1. Its features are, The method (100) is used to identify parameters in the form of model parameters for a closed-loop regulation loop used for vehicle control, and preferably for lateral guidance of the vehicle.

3. The method (100) according to any one of the preceding claims. Its features are, The method (100) is configured to identify parameters for a vehicle model, which is preferably used for vehicle lateral guidance and / or vehicle longitudinal guidance and / or driving dynamics control and / or component development and / or steering adjustment and / or braking adjustment.

4. The method (100) according to any one of the preceding claims. Its features are, The method (100) is configured to study the transfer characteristics from the desired curvature (402) of the vehicle trajectory to the actual curvature (401) of the vehicle trajectory, wherein the following steps are included: - The model is provided in the form of a model of a closed-loop control loop for lateral guidance of the vehicle. - Perform the nominal parameter estimation to make a first estimate of at least one parameter of the model based on the provided measurement data; - Perform quantification of the uncertainty caused by the parameters, which provides the residual uncertainty in the estimated parameters; - Determine the distribution of parameter variations based on the performed quantization; - Modeling of the technical system (50) is performed based on the nominal parameter estimates and the distribution of parameter variations; - Based on the modeling, a pre-given adjustment for vehicle lateral guidance is matched.

5. The method (100) according to any one of the preceding claims. Its features are, Determining the nominal parameter estimate includes: - Especially when uncertainty is not taken into account, the nominal error between the provided measurement data and the pre-given model is minimized.

6. The method (100) according to any one of the preceding claims. Its features are, Determining the distribution of the parameter variation includes: - The uncertainty caused by the parameters is characterized in the form of the model uncertainty remaining from the nominal parameter estimate.

7. The method (100) according to any one of the preceding claims. Its features are, The determination of the distribution and, in particular, the characterization of the uncertainty caused by the parameters are performed using a linear Bayesian method.

8. The method (100) according to any one of the preceding claims. Its features are, The modeling of the technical system (50) is set up in such a way that the nominal parameter estimate and the distribution of the parameter variation are combined into a total estimate, taking into account the uncertainty caused by the parameters.

9. A computer program (20) comprising instructions that, when executed by at least one computer (10), cause the computer to perform the method (100) according to any one of the preceding claims.

10. An apparatus (10) for performing data processing, the apparatus being configured to implement the method (100) according to any one of claims 1 to 8.

11. A computer-readable storage medium (15) comprising instructions that, when executed by at least one computer (10), cause the computer to perform the method (100) according to any one of claims 1 to 8.

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