DE102023206821A1 - COMPUTER-IMPLEMENTED METHOD FOR DETERMINING A SURROGATE MODEL OF A STATE-SPACE MODEL - Google Patents

Abstract

A general aspect of the present disclosure relates to a computer-implemented method for determining a surrogate model of a state space model. The method comprises receiving a state space model for describing a system to be controlled, wherein the state space model comprises a first system matrix and a first input matrix, wherein at least the first system matrix is dependent on a parameter vector comprising one or more entries each described by a stochastic distribution. The method further comprises receiving the one or more stochastic distributions of the one or more entries of the parameter vector and determining the surrogate model of the state space model.

Description

State of the art

Highly automated or autonomous systems are increasingly the focus of attention in robotics and the automotive industry, for example. Control systems in particular are becoming increasingly important in the operation of autonomous or highly automated systems. In highly automated and autonomous driving, the lateral guidance of the vehicle plays a central role. The task of lateral guidance is to keep the lateral distance of the vehicle to a given route or the lane and/or the edge of the road stable. Although numerous control methods have been proposed, these are mostly based on nominal models, i.e. uncertainties such as external disturbances, parameter uncertainties or model errors are not taken into account.

Uncertainties in the initial states of the control system and the resulting variance in the feedback of the control loop are the result of these uncertainties. The latter are unknown or difficult to quantify in the development phase and the parameterization of the controller. Therefore, later in the application phase, the controller must be intensively tested and adapted against the uncertainties. This is time-consuming and costly. In particular, the fact that development cycles are becoming increasingly shorter, also due to the increasing proportion of software, results in market-specific disadvantages due to long product delivery times. Another challenge is the high computing or implementation effort required to obtain reliable information about the spread of the target variables. This can, for example, include sampling-based simulation methods (e.g. Monte Carlo simulation, non-intrusive methods of "Uncertainty Quantification" (UQ)), which require a large number of function calls. For these reasons, there is a need for new techniques to determine uncertainties of controlled systems in order to improve the design of the state controller.

disclosure of the invention

Computer-implemented method for determining a surrogate model of a state space model. The method comprises receiving a state space model for describing a system to be controlled, wherein the state space model comprises a first system matrix and a first input matrix, wherein at least the first system matrix is dependent on a parameter vector comprising one or more entries each described by a stochastic distribution. The method further comprises receiving the one or more stochastic distributions of the one or more entries of the parameter vector and determining the surrogate model of the state space model.

A second general aspect of the present disclosure relates to a computer system configured to perform the computer-implemented method for determining a surrogate model of a state space model according to the first general aspect (or an embodiment thereof).

A third general aspect of the present disclosure relates to a computer program configured to execute the computer-implemented method for determining a surrogate model of a state space model according to the first general aspect (or an embodiment thereof).

A fourth general aspect of the present disclosure relates to a computer-readable medium or signal storing and/or containing the computer program according to the third general aspect (or an embodiment thereof).

The method proposed in this disclosure according to the first general aspect (or an embodiment thereof) can serve to provide a computer-implemented method for determining a surrogate model of a state space model. The method can serve to determine a surrogate model of a state space model for the design of a state controller. One advantage can be that the method is not limited to the lateral guidance of a vehicle, but can be applied to a very broad class of systems and used in a variety of applications. Examples of this can be vehicle longitudinal guidance, the motor control/regulation of electrical machines, distributed systems, building automation, etc. The method can be provided in a further encapsulated manner and can thus enable use by an extended user group, even if they or individual members of the user group do not know the underlying mathematical principles. layers are unknown. Since the structure of the surrogate model is similar to or identical to the structure of the original system to a certain degree, integration into an existing software structure can be enabled. The method can also enable the output of variance and/or confidence intervals of the target variables. A further advantage can be that the surrogate model can be determined before use and used in a system at runtime. This also enables use on less powerful computers, such as a control unit. This can increase safety or enable control methods to be adapted to the existing uncertainties at runtime. This enables improved control compared to sampling-based UQ methods used to date. The techniques of the present disclosure can enable system variances to be taken into account as early as in the system design (so-called “safe-by-design approach”). A further advantage can be to reduce the time to market of highly automated control functions, since relevant uncertainties can already be taken into account in the design phase. This can reduce the effort that would have to be spent in the verification/validation phase. Furthermore, a trade-off can be achieved between the robustness of the control system against parametric uncertainties and disturbances and the performance of the control system. A further advantage can be seen in the fact that a confidence interval of the closed control loop can be derived in order to be able to make stochastic statements about the performance. The techniques of the present disclosure can also be advantageous for checking existing control systems for their robustness or for quantifying their deviation from optimal control. Furthermore, the techniques of the present disclosure are not limited to the lateral control of a vehicle, but can be advantageous for a wide variety of controls. In addition to the lateral control of the vehicle, the longitudinal control can also benefit from the techniques of the present disclosure. The present techniques are also advantageous for an approach that simultaneously controls the longitudinal and lateral control.

• Ein „Surrogatmodell“ kann ein Modell sein, das eine komplexe Funktion oder ein komplexes Originalsystem approximiert oder repräsentiert. Beispielsweise kann das Surrogatmodell weniger rechenintensiv sein als das Originalsystem. In einem Beispiel kann das Surrogatmodell mithilfe eines datengesteuerten Bottom-up-Ansatz erstellt werden. In einem Beispiel kann die exakte, innere Funktionsweise unbekannt sein. Beispielsweise kann der Fokus auf dem Eingabe-Ausgabe-Verhalten des Surrogatmodells liegen. In einem Beispiel kann das Surrogatmodell auf Basis der Reaktion eines Simulators auf eine begrenzte Anzahl von Datenpunkten erstellt werden. Beispielsweise kann das Surrogatmodell mithilfe von Methoden des maschinellen Lernens und/oder statistischer Verfahren bestimmt werden.

Some terms are used in this disclosure as follows:

• A “surrogate model” can be a model that approximates or represents a complex function or a complex original system. For example, the surrogate model can be less computationally intensive than the original system. In one example, the surrogate model can be built using a bottom-up, data-driven approach. In one example, the exact inner workings can be unknown. For example, the focus can be on the input-output behavior of the surrogate model. In one example, the surrogate model can be built based on the response of a simulator to a limited number of data points. For example, the surrogate model can be determined using machine learning methods and/or statistical techniques.

A "state controller" may include an algorithm, i.e. a calculation rule, that feeds a complete or partial state variable (i.e. the internal state of the controlled system) back to an input variable. A state controller may include parameters that can weight the state variable. In examples, a state controller may be executed on a computer system. A state controller may, for example, include or be part of a hardware module with inputs and outputs.

A "vehicle" can be any device that transports passengers and/or cargo. A vehicle can be a motor vehicle (for example a car or a truck), but also a rail vehicle. A vehicle can also be a motorized two- or three-wheeler. However, floating and flying devices can also be vehicles. Vehicles can be at least partially autonomous or assisted.

Short description of the characters

• 1-A to 1-C schematically illustrates a computer-implemented method for determining a surrogate model of a state space model.
• 2 schematically illustrates an exemplary control loop according to one or more embodiments of the present disclosure.
• 3-A to 3-B illustrates exemplary confidence intervals resulting from application of the surrogate model according to techniques of the present disclosure in the linear case and nonlinear case.

Detailed description

First, with regard to the 1 -A, 1 -B, 1-C and 2 the techniques of the present disclosure are discussed. With respect to 3-A to 3-B possible results and advantages resulting from the method disclosed herein for determining a state space model are discussed.

beschreiben. In einem Beispiel und wie beispielhaft in der Gleichung gezeigt, kann das Zustandsraummodell weiter einen Zustandsvektor x(t,ξ), der eine oder mehr Zustandsgrößen x₁(t,ξ), x₂(t,ξ), x₃(t,ξ, ... enthält, einen Eingangsgrößenvektor (u) und/oder eine Störgröße (w(p)) umfassen. Dabei kann w(p) eine Störgröße sein. In einem Beispiel kann die Störgröße eine parametervariante additive Störgröße sein. In einem Beispiel kann x(t,ξ) ∈ ℝ^m gelten, wobei in einem Beispiel m ≥ 1, 2, 3, 4, 5, ....10, 11, 12, ... 100, 101, 102, 103, ... 1000,1001, ... sein kann. Eine hinreichend stetige Abhängigkeit der Systemmatrix A(p) und der Eingangsmatrix B(p) von dem Parametervektor p(ξ) kann dabei vorteilhaft für das Verfahren der vorliegenden Offenbarung sein. Die parametrischen Unsicherheiten werden in der vorliegenden Offenbarung durch das stochastische Argument ξ ausgedrückt. 1 is a flow chart showing possible method steps of the computer-implemented method 100 for determining a surrogate model of a state space model. The computer-implemented method 100 for determining a surrogate model of a state space model comprises receiving 110 a state space model for describing a system 10 to be controlled, wherein the state space model comprises a first system matrix A(p) and a first input matrix B(p), wherein at least the first system matrix A(p) is dependent on a parameter vector p(ξ) comprising one or more entries each described by a stochastic distribution. The method further comprises receiving 120 the one or more stochastic distributions of the one or more entries of the parameter vector p(ξ), and determining 130 the surrogate model of the state space model. In one example, the state space model is used to design a state controller 20. In one example, the state space model of the original system can be described by the equation $$\dot{x}=A\left(p\right)x+B\left(p\right)\text{u}+w\left(p\right),\text{wobei }t>{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102023206821A1\_0016.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,x\left(t_0\right)=x_0\left(p\right)$$

describe. In an example and as shown by way of example in the equation, the state space model can further comprise a state vector x(t,ξ) containing one or more state variables x ₁ (t,ξ), x ₂ (t,ξ), x ₃ (t,ξ, ...), an input variable vector (u) and/or a disturbance variable (w(p)). Here, w(p) can be a disturbance variable. In an example, the disturbance variable can be a parameter-variant additive disturbance variable. In an example, x(t,ξ) ∈ ℝ ^m can apply, where in an example m ≥ 1, 2, 3, 4, 5, ....10, 11, 12, ... 100, 101, 102, 103, ... 1000,1001, ... A sufficiently continuous dependence of the system matrix A(p) and the input matrix B(p) on the parameter vector p(ξ) can be advantageous for the method of the present disclosure. The parametric uncertainties are expressed in the present disclosure by the stochastic argument ξ.

beschreiben. In einem Beispiel und wie voranstehend beispielhaft/optional in der voranstehenden Gleichung gezeigt, kann die zweite Eingangsmatrix (B') des Surrogatmodells zu einer Eingangsgröße korrespondieren, die auf der einen oder den mehreren stochastischen Verteilungen basiert. Zum Beispiel kann das Surrogatmodell eine dritte Eingangsmatrix (B*) umfassen, die zu einer deterministischen Eingangsgröße korrespondiert. In einem Beispiel kann die deterministische Eingangsgröße u_det(t) = u_Steuer(t) sein und beispielsweise Vorsteueranteile umfassen. In einem Beispiel kann die Eingangsgröße, die auf der einen oder den mehreren stochastischen Verteilungen basiert, eine Ausgangsgröße des Zustandsregler 20 sein. In einem Beispiel kann die Rückführung der Zustandsgrößen mithilfe des Regelgesetzes u_stoch(t,ξ) = u_Regel = Kx(t,ξ) beschrieben werden. In einem Beispiel kann die Rückführmatrix K zeitvariant, zeitinvariant, oder konstant sein. In Beispielen kann die Rückführung mittels einem oder mehreren Sensoren durchgeführt werden. Zum Beispiel können der eine oder die mehreren Sensoren konfiguriert sein, physikalische (Umgebungs-)Größen des Systems 10 zu erfassen und in elektrische Signale umzuwandeln, die dem Regelsystem (u.a. dem Zustandsregler) direkt oder indirekt (in verarbeiteter Form) als eine Eingangsgröße dienen. In Beispielen können die einen oder die mehreren Sensoren Radarsensoren, Lidar-Sensoren, Ultraschall-Sensoren, und/oder kamerabasierte Sensoren umfassen. Beispielsweise kann die stochastische Eingangsgröße u_stoch die Rückführung des unsicheren Zustands x(t,ξ) (d.h. auf der einen oder den mehreren stochastischen Verteilungen basierend) sein und durch die Matrix u_stoch dargestellt werden. In einem Beispiel kann die Zustandsmatrix X die Zustandsgrößen des Surrogatmodells umfassen. In einem Beispiel können die unsicheren Zustände x(t,ξ) in Abhängigkeit der deterministischen erweiterten Zustände X(t) dargestellt werden. Zum Beispiel kann für den Mittelwert E[x] = MX oder die Varianz V[x] = VX² gelten. In einem Beispiel ist die Matrix M eine zu den Mittelwerten der Zustandsgrößen der Zustandsmatrix X korrespondierende Mittelwerts-Koeffizientenmatrix. In einem Beispiel ist die Matrix V eine zu den Varianzen der Zustandsgrößen der Zustandsmatrix X korrespondierende Varianz-Koeffizientenmatrix.In one example, the surrogate model may include a plurality of matrices. In one example, the plurality of matrices may include a second system matrix (A'), a state matrix (X), a second input matrix (B'), and optionally a disturbance matrix (W'). For example, the existence of the disturbance matrix W' may depend on whether the state space model of the original system includes a disturbance w(p). In one example, the surrogate model may be described using the equation $$\dot{x}=A\left(p\right)x+B\text{'}{\text{u}}_{det}+B*U_{stoch}+W\text{'},\text{wobei }t>{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102023206821A1\_0016.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,X\left(t_0\right)=X\text{'}$$

describe. In an example and as shown above by way of example/optionally in the above equation, the second input matrix (B') of the surrogate model can correspond to an input variable based on the one or more stochastic distributions. For example, the surrogate model can include a third input matrix (B*) that corresponds to a deterministic input variable. In an example, the deterministic input variable can be u _det (t) = u _Steuer (t) and can include, for example, pre-control components. In an example, the input variable based on the one or more stochastic distributions can be an output variable of the state controller 20. In an example, the feedback of the state variables can be described using the control law u _stoch (t,ξ) = u _Regel = Kx(t,ξ). In an example, the feedback matrix K can be time-variant, time-invariant, or constant. In examples, the feedback can be carried out using one or more sensors. For example, the one or more sensors may be configured to sense physical (environmental) quantities of the system 10 and convert them into electrical signals that serve as an input quantity to the control system (including the state controller) directly or indirectly (in processed form). In examples, the one or more sensors may include radar sensors, lidar sensors, ultrasonic sensors, and/or camera-based sensors. For example, the stochastic input quantity u _stoch may be the feedback of the uncertain state x(t,ξ) (i.e. based on the one or more stochastic distributions) and may be represented by the matrix u _stoch . In an example, the state matrix X may include the state quantities of the surrogate model. In an example, the uncertain states x(t,ξ) may be represented as a function of the deterministic extended states X(t). For example, the mean value E[x] = MX or the variance V[x] = VX ² may apply. In one example, the matrix M is a mean coefficient matrix corresponding to the mean values of the state variables of the state matrix X. In In an example, the matrix V is a variance coefficient matrix corresponding to the variances of the state variables of the state matrix X.

gelten, wobei X_i(t) ∈ ℝ^mN sogenannte Chaos-Koeffizienten und wie voranstehend erläutert die Zustände des Surrogatmodells darstellen. Die polynomielle Chaosentwicklung kann eine Art Approximation darstellen. In einem Beispiel kann die Bestimmung der Größe N Teil des Verfahrens 100 sein. In einem Beispiel kann die Größe N vorbestimmt sein. Zum Beispiel kann ein größeres N zu einer genaueren Approximation führen, aber eine längere Rechenzeit notwendig machen und ein kleineres N zu einer ungenaueren Approximation führen, aber eine kürzere Rechenzeit ermöglichen. In einem Beispiel kann eine polynomiellen Chaosentwicklung für die Mehrzahl von Matrizen des Surrogatmodells durchgeführt werden. Dabei umfassen die Mehrzahl von Matrizen des Surrogatmodells jeweils die Chaos-Koeffizienten der jeweiligen polynomiellen Chaosentwicklung. In einem Beispiel kann für die Bestimmung der Mehrzahl von Matrizen des Surrogatmodells im Einzelnen gelten: $$F={\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right){\mathrm{\Phi }}^T}\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)$$

$$A\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)A\left(p\left(\mathrm{\xi }\right)\right)}{\mathrm{\Phi }}^T\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)$$

$$B\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)B\left(p\left(\mathrm{\xi }\right)\right)}dw\left(\mathrm{\xi }\right)$$

$$X\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)x_0\left(p\left(\mathrm{\xi }\right)\right)}dw\left(\mathrm{\xi }\right)$$

$$W\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)A\left(p\left(\mathrm{\xi }\right)\right)}dw\left(\mathrm{\xi }\right)$$

$$B*=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)B\left(p\left(\mathrm{\xi }\right)\right)}{\mathrm{\Phi }}^T\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)$$

In one example, determining 130 the surrogate model may be performed based on the one or more stochastic distributions. In one example, determining 130 the surrogate model may further comprise determining 131 one or more polynomial bases based on the one or more stochastic distributions, wherein the surrogate model is determined using the one or more polynomial bases. In one example, determining 130 may be performed using a polynomial chaos expansion (PCE). In one example, the stochastic distributions of each entry p _i of the parameter vector p(ξ) are known. Polynomial bases may correspond to these distributions. In one example, each polynomial basis may be orthogonal to the stochastic distribution it represents. In an example, the one or more polynomial bases may be at least one of Hermitian polynomials, Legendre polynomials, Jacobi polynomials, and/or generalized Laguerre polynomials. In an example, for the state vector $x\left(t,\mathrm{\xi }\right)={\displaystyle \sum {}_{n=1}^N{X}_i}\left(t\right){\mathrm{\Phi }}_i\left(\mathrm{\xi }\right)=X^T\left(t\right)\mathrm{\Phi }\left(\mathrm{\xi }\right)$

apply, where X _i (t) ∈ ℝ ^mN represent so-called chaos coefficients and, as explained above, the states of the surrogate model. The polynomial chaos expansion can represent a type of approximation. In one example, the determination of the size N can be part of the method 100. In one example, the size N can be predetermined. For example, a larger N can lead to a more accurate approximation, but require a longer computing time, and a smaller N can lead to a less accurate approximation, but enable a shorter computing time. In one example, a polynomial chaos expansion can be carried out for the majority of matrices of the surrogate model. The majority of matrices of the surrogate model each comprise the chaos coefficients of the respective polynomial chaos expansion. In one example, the following can apply in detail for the determination of the majority of matrices of the surrogate model: $$F={\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right){\mathrm{\Phi }}^T}\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)$$

$$A\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)A\left(p\left(\mathrm{\xi }\right)\right)}{\mathrm{\Phi }}^T\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)$$

$$B\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)B\left(p\left(\mathrm{\xi }\right)\right)}dw\left(\mathrm{\xi }\right)$$

$$X\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)x_0\left(p\left(\mathrm{\xi }\right)\right)}dw\left(\mathrm{\xi }\right)$$

$$W\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)A\left(p\left(\mathrm{\xi }\right)\right)}dw\left(\mathrm{\xi }\right)$$

$$B*=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)B\left(p\left(\mathrm{\xi }\right)\right)}{\mathrm{\Phi }}^T\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)$$

In one example, determining 130 the surrogate model may include specifying 132 a plurality of quadrature points and a plurality of quadrature weights and calculating 133 the surrogate model using numerical quadrature methods. Using a numerical quadrature method (also called numerical integration), the integrals at the plurality of quadrature points may be approximated using the plurality of quadrature weights. For example, the matrix A' may be calculated using the following equation: $$A\text{'}=F^{-1}{\displaystyle \underset{\Omega }{\int }\mathrm{\Phi }\left(\mathrm{\xi }\right)A\left(p\left(\mathrm{\xi }\right)\right){\mathrm{\Phi }}^T\left(\mathrm{\xi }\right)dw\left(\mathrm{\xi }\right)\approx F^{-1}\left[{\displaystyle \sum _{j=1}^J{w}_j\alpha \left({\mathrm{\xi }}_j\right)}\right]}$$

In some examples, the determination of the number of quadrature points J may be part of the method 100. In one example, the size J may be predetermined. For example, a larger J may lead to a more accurate approximation but require a longer computation time and a smaller J may lead to a less accurate approximation but enable a shorter computation time. In one example, so-called “sparse quadrature schemes” (e.g. Smolyak quadrature) may be used to, for example, also calculate the approximation in the case of some uncertain parameters, i.e. entries of the parameter vector (p(ξ)) that are caused by a sto chastic distribution to obtain a manageable (ie with reasonable computing time and acceptable cost) number of quadrature points.

In some examples, the system 10 may be a linear system or a non-linear system. In some examples, the surrogate model for a linear system may be determined according to the method steps described above. In some examples, additional method steps may be necessary for a non-linear system. These will be explained in more detail below.

In some examples, when the system 10 is a nonlinear system, one or more state variables may include a nonlinear part. The method 100 may further include polynomializing 140 the one or more state variables that include a nonlinear part before performing the determining 130 of the surrogate model. For example, the polynomializing 140 of the one or more state variables that include a nonlinear part may include substituting 141 the nonlinear part with a respective substitution variable and introducing 142 the respective substitution variable as an additional state variable. Further, the determining 130 of the surrogate model may further be based on the additional state variable.

For example, the state space model of a nonlinear original system can have the following structure: $$\dot{x}=Ax+H\left(x\otimes x\right)+T\left(x\otimes x\otimes x\right)+Bu+C\left(x\otimes u\right)+D\left(x\otimes x\otimes u\right)$$

The corresponding surrogate model can have the following form due to the extended state variables: $$\dot{x}=A\text{'}X+H\left(X\otimes X\right)+T\text{'}\left(X\otimes X\otimes X\right)+B\text{'}u+C\text{'}\left(X\otimes u\right)+D\text{'}\left(X\otimes X\otimes u\right)$$

errechnen.The second system matrix A' and the second input matrix B' are known from the linear case. The additional state variables may require the calculation of additional matrices compared to the linear case. For example, H' can be calculated by $$H\text{'}=F^{-1}{\displaystyle \int {}_{\Omega }\mathrm{\Phi }H\left(p\right)\left[{\mathrm{\Phi }}^T\otimes {\mathrm{\Phi }}^T\right]dw}$$

calculate.

In one example, the system 10 to be controlled can be arranged for placement in a vehicle and/or robot, and/or designed to control and/or monitor a vehicle function and/or a robot function (in particular to control and/or monitor a driving function). In examples, a robot can comprise a (autonomously or assisted driving) vehicle.

For example, the vehicle function may be a function for autonomous and/or assisted driving. In some examples, the state controller 20 may be designed to be executed on a computer system of a vehicle (e.g. an autonomous, highly automated or assisted driving vehicle). For example, the computer system may be implemented locally in the vehicle or (at least partially) implemented in a backend that is communicatively connected to the vehicle. For example, the computer system may include a control unit on which the state controller 20 can be executed. In some examples, the vehicle may include a computer system with a communication interface that enables communication with a backend. For example, the state controller 20 can be executed in this backend. In one example, the system 10 to be controlled may be a system for lateral guidance and/or longitudinal guidance of the vehicle. Uncertainties described by means of the one or more stochastic distributions may include, for example, the load-dependent yaw inertia and the mass of the vehicle, the distances of the vehicle's center of gravity to the axles, or the road friction influencing the stiffnesses (or a combination thereof). In an example, the one or more state variables x ₁ (t,ξ), x ₂ (t,ξ), x ₃ (t,ξ, ...) may be variables for at least one of a steering angle, an orientation angle, a yaw rate, a slip angle, and/or a lateral error. In an example, an entry of the input variable vector u _det (t) = u _control (t) may be the steering speed or a steering angle.

The present disclosure also relates to methods for controlling a vehicle using a state controller configured using the methods of the present disclosure. In some examples, the state controller 20 and/or the system 10 to be controlled may be configured as described above.

In other examples and as indicated above, the system 10 to be controlled may be arranged in a robot and/or designed to control and/or monitor a robot function (in particular to control and/or monitor a movement function of a robot). For example, the system to be controlled may be a system for lateral guidance and/or longitudinal guidance of the robot. In some examples, the state controller 20 may be executed on a computer system of a robot. For example, the computer system may be implemented locally in the robot or (at least partially) implemented in a backend that is communicatively connected to the robot.

In the following, the 3 -A and 3 -B the advantages of the techniques of the present disclosure over the prior art are explained. In the 3 -A shows the results of the UQ analysis for a deterministic steering scenario. The area 60 is the confidence interval that results from applying the surrogate model according to the method of the present disclosure and the curve 50 that results from applying a Monte Carlo simulation. For example, using the surrogate model, variances and confidence intervals of the parameters p _i of the parameter vector p(ξ) can be determined as shown. Analogously, 3 -B exemplary confidence intervals 61 resulting in the nonlinear case by applying the surrogate model according to the method of the present disclosure in comparison to a sampling-based method 51.

The present disclosure also relates to methods for controlling a robot using a state controller designed using the methods of the present disclosure. In some examples, the state controller 20 and/or the system 10 to be controlled may be designed as described above.

Furthermore, a computer system is disclosed which is designed to carry out the computer-implemented method 100 for determining a surrogate model of a state space model. The computer system can comprise at least one processor and/or at least one main memory. The computer system can further comprise a (non-volatile) memory.

Furthermore, a computer program is disclosed which is designed to carry out the computer-implemented method 100 for determining a surrogate model of a state space model. The computer program can be present, for example, in interpretable or compiled form. It can be loaded (also in parts) into the RAM of a computer for execution, for example as a bit or byte sequence.

Also disclosed is a computer-readable medium or signal that stores and/or contains the computer program or at least a part thereof. The medium may comprise, for example, one of RAM, ROM, EPROM, HDD, SDD, ... on/in which the signal is stored.

Claims (11)

Computer-implemented method (100) for determining a surrogate model of a state space model, the method (100) comprising the following steps: - receiving (110) a state space model for describing a system to be controlled (10), the state space model comprising a first system matrix (A(p)) and a first input matrix (B(p)), at least the first system matrix (A(p)) being dependent on a parameter vector (p(ξ)) comprising one or more entries each described by a stochastic distribution, - receiving (120) the one or more stochastic distributions of the one or more entries of the parameter vector (p(ξ)), and - determining (130) a surrogate model of the state space model. Computer-implemented method (100) according to claim 1 , wherein the state space model further comprises a state vector (x(t, ξ)) containing one or more state variables (x ₁ (t,ξ), x ₂ (t,ξ), x ₃ (t,ξ), ...), an input variable vector ((u)) and/or a disturbance variable (w(p)). Computer-implemented method (100) according to claim 1 or 2 , wherein the surrogate model comprises a plurality of matrices, the plurality of matrices comprising a second system matrix (A'), a state matrix (X), a second input matrix (B'), and optionally a disturbance matrix (W'). Computer-implemented method (100) according to claim 3 , wherein the second input matrix (B') of the surrogate model corresponds to an input variable based on the one or more stochastic distributions, and wherein the surrogate model comprises a third input matrix (B*) corresponding to a deterministic input variable. Computer-implemented method (100) according to one of the preceding claims, wherein determining (130) the surrogate model further comprises - establishing (132) a plurality of quadrature points and a plurality of quadrature weights and - calculating (133) the surrogate model using numerical quadrature methods. Computer-implemented method (100) according to one of the preceding claims, wherein determining (130) the surrogate model is performed using a polynomial chaos expansion. Computer-implemented method (100) according to one of the Claims 2 until 6 , wherein, if the system (10) is a nonlinear system, one or more state variables comprise a nonlinear part, the method (100) further comprising - polynomializing (140) the one or more state variables comprising a nonlinear part before determining (130) the surrogate model. Computer-implemented method (100) according to one of the preceding claims, wherein the system to be controlled (10) is designed for arrangement in a vehicle and/or robot, and/or is designed for controlling and/or monitoring a vehicle function and/or a robot function. Computer system adapted to perform the computer-implemented method (100) for determining a surrogate model of a state space model according to one of the preceding Claims 1 until 8 to execute. Computer program comprising instructions which, when the computer program is executed by a computer system, cause the computer system to execute the computer-implemented method (100) for determining a surrogate model of a state space model according to one of the preceding Claims 1 until 8 to execute. Computer-readable medium or signal containing the computer program according to claim 10 stores and/or contains.

Images