Computer-implemented method for determining a surrogate model for a state space model

Abstract

A general aspect of the present disclosure relates to a computer-implemented method for determining a surrogate model for 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 also comprises receiving the one or more stochastic distributions of the one or more entries of the parameter vector and determining the surrogate model for the state space model.

Description

[0001]R. 405906 - 1 - Description Title COMPUTER-IMPLEMENTED METHOD FOR DETERMINING A SURROGATE MODEL OF A STATE-SPACE MODEL State of the art Highly automated or autonomous systems are increasingly in focus, for example in robotics and the automotive industry. 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 specified route or the lane and / or road edge 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 a consequence of these uncertainties. The latter are unknown or difficult to quantify during the development phase and during the parameterization of the controller. Therefore, later in the application phase, the controller must be intensively tested and adapted against these uncertainties. This is time-consuming and costly. Particularly due to the fact that development cycles are becoming increasingly shorter, also due to the increasing proportion of software, market-specific disadvantages arise due to long product delivery times. A further challenge is the high computational or implementation effort required to obtain reliable information about the dispersion of the target variables. This can be achieved, for example, by 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 for determining uncertainties in 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 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 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 be used to determine a surrogate model of a state-space model for designing a state controller. One advantage may 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 form, thus enabling use by an expanded user group, even if this or individual members of the user group are not familiar with the underlying mathematical principles. Since the structure of the surrogate model is similar to or identical to the structure of the original system to a certain extent, 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 deployed in a system at runtime.This enables use even on less powerful computers, such as a control unit. This can increase safety or enable adaptation of control methods to existing uncertainties at runtime. This enables improved control compared to previously used sampling-based UQ methods. The techniques of the present disclosure can enable the consideration of system variances already in the system design (so-called "safe-by-design approach"). A further advantage can be a reduction in the time-to-market of highly automated control functions, since relevant uncertainties can already be considered in the design phase. This can reduce the effort that would otherwise be required in the verification / validation phase.Furthermore, a trade-off can be achieved between the robustness of the control system with respect to 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 make stochastic statements about the performance. The techniques of the present disclosure can also be advantageous for testing 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 control systems. In addition to the lateral control of the vehicle, longitudinal control can also benefit from the techniques of the present R. 405906 - 4 - disclosure.The present techniques are also advantageous for an approach that simultaneously controls longitudinal and lateral guidance. Some terms are used in the present 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 created using a data-driven, bottom-up 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 created 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" can comprise 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 can comprise parameters that can weight the state variable. In examples, a state controller can be executed on a computer system. A state controller can, for example, comprise or be part of a hardware module with inputs and outputs. A "vehicle" can be any device that transports passengers and / or freight. A vehicle can be a motor vehicle (e.g., 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. Brief description of the figures R. 405906 - 5 - Fig.1-A to 1-C schematically illustrate a computer-implemented method for determining a surrogate model of a state-space model. Fig. 2 schematically illustrates an exemplary control loop according to one or more embodiments of the present disclosure. Fig. 3-A to 3-B illustrate exemplary confidence intervals resulting when applying the surrogate model according to techniques of the present disclosure in the linear case and the non-linear case. Detailed Description First, the techniques of the present disclosure are discussed with reference to Fig. 1-A, Fig. 1-B, Fig. 1-C and Fig. 2. Possible results and advantages resulting from the method disclosed herein for determining a state-space model are discussed with reference to Fig. 3-A to 3-B. Fig. 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 ^(^) and a first input matrix ^(^), wherein at least the first system matrix ^(^) is derived from a parameter vector ^. ( ξ ) , which comprises 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 ^ ( 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 ^̇ = ^ ( ^) x + ^(^)u + ^(^), where ^ > ^ ^ , ^ ( ^ ^ ) = ^ ^ (^) R. 405906 - 6 - In one example, and as exemplified in the equation, the state space model may further comprise a state vector ^(^, ξ) representing one or more state variables ^ ^ ( ^, ξ ) , contains an input variable vector ( ^ ) and / or a disturbance ( ^(^) ) include. Here, ^(^) can be a disturbance variable. In one example, the disturbance variable can be a parameter-variant additive disturbance variable. In one example, ^ ( ^, ξ ) ∈ ℝ ^ hold, where in an example ^ ≥ 1, 2, 3, 4, 5, … .10, 11, 12, … 100, 101, 102, 103, … 1000, 1001, … A sufficiently continuous dependence of the system matrix ^ ( ^ ) and the input matrix ^ ( ^ ) from the parameter vector ^ ( 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. In one example, the surrogate model may comprise a plurality of matrices. In one example, the plurality of matrices may comprise a second system matrix (^′), a state matrix (^), a second input matrix (^′), and optionally a disturbance matrix (^′). For example, the existence of the disturbance matrix ^′ may depend on whether the state space model of the original system includes a disturbance ^(^). In one example, the surrogate model can be expressed using the equation ^̇ = ^ ^ (^)x + ^ ^ ^ ^^^ + ^ ∗ ^ ^^^^^ + ^′, where ^ > ^ ^ , ^ ( ^ ^ )= ^′. In one example, and as shown above by way of example / option in the preceding equation, the second input matrix (^′) of the surrogate model may correspond to an input variable based on the one or more stochastic distributions. For example, the surrogate model may have a third input matrix (^ ∗ ) that corresponds to a deterministic input variable. In one example, the deterministic input variable may be ^ ^^^ (^) = ^ ^^^^^^ (^) and may include, for example, input control components. In one example, the input variable based on the one or more stochastic distributions may be an output variable of the state controller 20. In one example, the feedback of the state variables may be achieved using the control law ^ ^^^^^ ( ^, ξ ) = ^ ^^^^^= ^^(^, ξ). In one example, the feedback matrix ^ can be time-variant, time-invariant, or constant. In examples, the feedback can be performed using one or more sensors. For example, the one or more R. 405906 - 7 - sensors can be configured to detect 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 can include radar sensors, lidar sensors, ultrasonic sensors, and / or camera-based sensors. For example, the stochastic input quantity ^ ^^^^^ the feedback of the uncertain state ^(^, ξ) (ie based on the one or more stochastic distributions) and represented by the matrix ^ ^^^^^In one example, the state matrix ^ can comprise the state variables of the surrogate model. In one example, the uncertain states ^(^, ξ) can be represented as a function of the deterministic extended states ^(^). For example, for the mean ^[^] = ^^ or the variance ^[^] = ^^ ^apply. In one example, the matrix ^ is a mean coefficient matrix corresponding to the mean values ​​of the state variables of the state matrix ^. In one example, the matrix ^ is a variance coefficient matrix corresponding to the variances of the state variables of the state matrix ^. 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 ^ ^ of the parameter vector ^ ( ξ )known. Polynomial bases may correspond to these distributions. In one example, each polynomial base may be orthogonal to the stochastic distribution it represents. In one example, the one or more polynomial bases may be at least one of Hermitian polynomials, Legendre polynomials, Jacobian polynomials, and / or generalized Laguerre polynomials. In one example, for the state vector ^ ( ^, ξ ) = ∑ ^ ^^ where ^ ^(^) ∈ ℝ 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 quantity ^ can be part of the method 100. In one example, the quantity ^ can be predetermined R. 405906 - 8 -. For example, a larger ^ can lead to a more accurate approximation, but require a longer computation time, and a smaller ^ can lead to a less accurate approximation, but enable a shorter computation time. In one example, a polynomial chaos expansion can be performed 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 to the determination of the majority of matrices of the surrogate model: ^ ^′ ^′ ^′ ^ ∗ 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 ^′ may be calculated using the following equation: ^ ^′ R. 405906 - 9 - In some examples, the determination of the number of quadrature points ^ may be part of the method 100. In one example, the size ^ may be predetermined. For example, a larger ^ may lead to a more accurate approximation, but require a longer computation time, and a smaller ^ 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, for example, even with some uncertain parameters, ie, entries of the parameter vector (^ ( ξ )) described by a stochastic distribution, to obtain a manageable (i.e., with justifiable computing time at acceptable costs) 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, further method steps may be necessary for a non-linear system. These will be explained in more detail below. In some examples, if the system 10 is a non-linear system, one or more state variables may include a non-linear part. The method 100 may further include polynomializing 140 the one or more state variables that include a non-linear part before determining 130 the surrogate model.For example, the polynomialization 140 of the one or more state variables that comprise a nonlinear part can comprise substituting 141 each of the nonlinear part by a respective substitution variable and introducing 142 the respective substitution variable as an additional state variable. Furthermore, the determination 130 of the surrogate model can be further based on the additional state variable. For example, the state space model of a nonlinear original system can have the following structure: ^̇ = ^^ + ^. ( ^ ⊗ ^ ) + ^ ( ^ ⊗ ^ ⊗ ^ ) + ^^ + ^ ( ^ ⊗ ^ ) + ^(^ ⊗ ^ ⊗ ^) The corresponding surrogate model can have the following form due to the extended state variables: R. 405906 - 10 - ^̇ = ^ ^ ^ + ^ ^( ^ ⊗ ^ ) + ^ ^( ^ ⊗ ^ ⊗ ^ ) + ^ ^ ^ + ^ ^( ^ ⊗ ^ ) + ^′(^ ⊗ ^ ⊗ ^) Where the second system matrix ^ ^and the second input matrix ^ ^ known from the linear case. The additional state variables may require the calculation of additional matrices compared to the linear case. For example, ^ ^ through In one example, the system 10 to be controlled may be arranged for arrangement in a vehicle and / or robot, and / or be 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 may 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 execute 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 may 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 may 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 one example, the one or more state variables. Variables for at least one of a steering angle, an orientation angle, a yaw rate, a slip angle, and / or a lateral error. In one example, an entry of the input variable vector ^ ^^^ (^) = ^ ^^^^^^(^) the steering speed or a steering angle. The present disclosure also relates to methods for controlling a vehicle 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 can be designed as described above. In other examples and as indicated above, the system 10 to be controlled can be arranged in a robot and / or can be 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 can be a system for lateral guidance and / or longitudinal guidance of the robot. In some examples, the state controller 20 can be executed on a computer system of a robot.For example, the computer system can be implemented locally in the robot or (at least partially) in a backend that is communicatively connected to the robot. The advantages of the techniques of the present disclosure over the prior art will now be explained using Figs. 3-A and 3-B. Fig. 3-A shows the results of the UQ analysis for a deterministic steering scenario. The area 60 is the confidence interval resulting from application of the surrogate model according to the method of the present disclosure and the curve 50 is the one resulting from application of a Monte Carlo simulation. For example, the surrogate model can be used to determine variances and confidence intervals of the parameters ^. ^ of the parameter vector ^ ( ξ ), as shown. Analogously, Fig. 3-B shows exemplary confidence intervals 61 resulting in the non-linear case through the application of 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 R. 405906 - 12 - state controller 20 and / or the system 10 to be controlled can be designed as described above. Also disclosed is a computer system 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.Also disclosed is a computer program configured to execute the computer-implemented method 100 for determining a surrogate model of a state-space model. The computer program may, for example, be in interpretable or compiled form. It may be loaded (even in part) into the RAM of a computer for execution, e.g., 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, for example, comprise one of RAM, ROM, EPROM, HDD, SDD, etc., on / in which the signal is stored.

Claims

R. 405906 - 13 - Claims 1. 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 (10) to be controlled, the state-space model comprising a first system matrix (^ ( ^ ) ) and a first input matrix (^ ( ^ ) ), where at least the first system matrix (^ ( ^ ) ) from a parameter vector (^ ( ξ ) ), which comprises 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 (^ ( ξ )), and - determining (130) a surrogate model of the state space model.

2. The computer-implemented method (100) according to claim 1, wherein the state space model further comprises a state vector (^ ( ^, ξ ) ), which has one or more state variables (^ ^ ( ^, ξ ) , ) contains an input variable vector ( ( ^ ) ) and / or a disturbance ( ^(^) )3. The computer-implemented method (100) according to claim 1 or 2, wherein the surrogate model comprises a plurality of matrices, wherein the plurality of matrices comprises a second system matrix (^′), a state matrix (^), a second input matrix (^′), and optionally a disturbance matrix (^′).

4. The computer-implemented method (100) according to claim 3, wherein the second input matrix (^′) 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 (^ ∗ ) corresponding to a deterministic input variable. R. 405906 - 14 - 5. The computer-implemented method (100) according to any 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.

6. The computer-implemented method (100) according to any one of the preceding claims, wherein determining (130) the surrogate model is performed using a polynomial chaos expansion.

7. The computer-implemented method (100) according to any one of claims 2 to 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. 8.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.

9. Computer system designed to carry out the computer-implemented method (100) for determining a surrogate model of a state-space model according to one of the preceding claims 1 to 8.

10. Computer program comprising instructions which, when the computer program is executed by a computer system, cause the computer system to carry out the computer-implemented method (100) for determining a surrogate model of a state-space model according to one of the preceding claims 1 to 8. R. 405906 - 15 - 11. A computer-readable medium or signal storing and / or containing the computer program according to claim 10.

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