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

Abstract

A general aspect of the present disclosure relates to a method for determining a surrogate model for a state space model. The method comprises receiving a continuous-time state space model for describing a system to be controlled. The method further comprises forming a discrete-time representation of the state space model for describing the system to be controlled, wherein at least the discrete-time representation of the state space model exhibits a time dependency based on a stochastic distribution. The method further comprises determining a surrogate model of the discrete-time representation of the state space model.

Description

[0001] Description

[0002] Title: COMPUTER-IMPLEMENTED METHOD FOR DETERMINING A SURROGATE MODEL OF A STATE-SPACE MODEL

[0003] State of the art

[0004] 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. Problems arise in distributed systems where, for example, the control algorithm is executed physically separately from the system to be controlled, for example, using cloud or edge computing, resulting in delays in data transmission.

[0005] Due to different physical conditions that influence signal transmission or different channel utilization in vehicle-to-vehicle (V2V) communication, time delays or fluctuating sampling times that are non-deterministic and therefore uncertain can occur. For example, networked adaptive cruise control, group start, or cooperative lane merging can be situations in which channel utilization is unusually high compared to other situations. For example, non-deterministic delays can arise in the feedback loop, in bus communication when distributed E / E architectures are involved in vehicle control functions, and / or in signal processing and transmission in sensor systems. In control systems, such as the longitudinal guidance of a vehicle, the non-deterministic time delay can have a detrimental impact on the control success.Many state-of-the-art solutions inadequately consider the influence of uncertain time effects and their representation in the control system. There is also a lack of efficient methods for quantifying uncertain time effects.

[0006] As a result, the systems under consideration must either be tuned very conservatively or optimized using worst-case analyses, which can lead to poor performance. Alternatively, the only available methods are inefficient and computationally intensive uncertainty quantification (UQ) methods (standard Monte Carlo or Latin Hypercube sampling), which require a large number of simulations (>3000). This makes it difficult to execute UQ algorithms online, i.e., at runtime. For example, monitoring the influence of uncertain time effects can often only be performed offline at design time. Therefore, there is a need for improved methods to quantify uncertain time effects in the design of control systems.

[0007] Disclosure of the invention

[0008] A first general aspect of the present disclosure relates to a method for determining a surrogate model of a state-space model. The method comprises receiving a continuous-time state-space model for describing a system to be controlled. The method further comprises forming a discrete-time representation of the state-space model for describing the system to be controlled, wherein at least the discrete-time representation of the state-space model has a time dependence based on a stochastic distribution. The method further comprises determining a surrogate model of the discrete-time representation of the state-space model.

[0009] A second general aspect of the present disclosure relates to a computer system configured to perform the method for determining a surrogate model of a state space model according to the first general aspect (or an embodiment thereof).

[0010] A third general aspect of the present disclosure relates to a computer program configured to execute the 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).

[0011] The method proposed in this disclosure according to the first general aspect (or an embodiment thereof) can serve to provide a 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 designing a state controller.

[0012] Furthermore, the proposed method can enable a single simulation to suffice to characterize the influence of uncertain time effects on a control system, which can represent a significant efficiency gain compared to a large number of simulations using inefficient UQ methods. This can lead to savings in terms of runtime on the one hand and in terms of the computing resources required for the simulation on the other. Furthermore, the calculation of the surrogate model can be performed prior to the design of the state controller, which can reduce the computing power required for the design of the state controller. The discrete-time representation of the continuous-time state space model, taking into account the time dependence based on the stochastic distribution, can create a better representation of reality, particularly in distributed control systems, and improve the design of the state controller.Furthermore, the techniques of the present disclosure can be used to map the uncertain timing effects in a data transmission channel and / or to account for the uncertain timing effects when sampling transmitted signals. Furthermore, the resource-efficient computation allows for uncertainty quantification to be performed at runtime (C, online). This allows confidence intervals of the target variables to be monitored and / or predicted at runtime.

[0013] One advantage may be that the method can be applied to a variety of control systems. One example of this may be the longitudinal guidance of a vehicle. The longitudinal guidance of a vehicle refers to the control and stability of its movement along the direction of travel. It may include aspects such as acceleration, deceleration, and speed control, which ensure that the vehicle can move forward or backward on the road in the desired manner. Particularly with regard to autonomous and / or assisted driving, in which a plurality of vehicles communicate via the same communication points and delays in data transmission may occur, the techniques of the present disclosure can lead to improved control behavior. In this context, the disclosed method can also lead to improved control in vehicle lateral guidance.Further examples can be found in various fields of control engineering, such as robotics, motor control of electrical machines, 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 degree, integration into an existing software structure is possible. The method can also enable the output of variance and / or confidence intervals of the target variables.

[0014] Some terms are used in this disclosure as follows:

[0015] A “surrogate model” can be a model that approximates or represents a complex function or a complex original model. For example, the surrogate model may be less computationally intensive than the original model. In one example, the surrogate model may be built using a data-driven, bottom-up approach. In one example, the exact inner workings may be unknown. For example, the focus may be on the input-output behavior of the surrogate model. In one example, the surrogate model may be built based on the response of a simulator to a limited number of data points. For example, the surrogate model may be determined using machine learning methods and / or statistical techniques.

[0016] A "state controller" may comprise an algorithm, i.e., a calculation rule, that maps a complete or partial state variable (i.e., the internal state of the controlled system) back to an input variable. A state controller may comprise parameters that may weight the state variable. In examples, a state controller may be executed on a computer system. In examples, a state controller may be executed in an ECU of a vehicle, a cloud, or an edge. A state controller may, for example, comprise or be part of a hardware module with inputs and outputs.

[0017] A "vehicle" can be any device that transports passengers and / or cargo. 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.

[0018] Short description of the characters

[0019] Fig. 1-A to 1-B schematically illustrates a method for determining a surrogate model of a state space model.

[0020] Fig. 2 schematically illustrates an exemplary control loop according to one or more embodiments of the present disclosure.

[0021] Fig. 3-A to 3-B illustrate exemplary closed-loop control responses and their confidence intervals when applying the surrogate model according to techniques of the present disclosure.

[0022] Detailed description

[0023] First, the techniques of the present disclosure are discussed with reference to Figs. 1-A, 1-B, and 2. Possible results and advantages resulting from the method for determining a state-space model disclosed herein are discussed with reference to Figs. 3-A to 3-B. Fig. 1 is a flowchart illustrating possible steps of the method 100 for determining a surrogate model of a state-space model. The method 100 for determining a surrogate model of a state-space model includes receiving 110 a continuous-time state-space model for describing a system 10 to be controlled.The method comprises forming 120 a time-discrete representation of the state-space model for describing the system to be controlled, wherein at least the time-discrete representation of the state-space model has a time dependence based on a stochastic distribution. The method further comprises determining 130 a surrogate model of the time-discrete representation of the state-space model. In one example, the method comprises designing a state controller based on the surrogate model. In one example, the method comprises applying the state controller in a function for controlling and / or monitoring a vehicle and / or a robot.

[0024] In one example, the continuous-time state-space model may have a time dependence dependent on a delay parameter T(^). In one example, the delay parameter T(^) may be based on the stochastic distribution.

[0025] In one example, the continuous state-space model may include an input variable vector u(t). In examples, the input variable vector u(t) may include target specifications for acceleration and / or braking processes of a vehicle and / or robot. In examples, the input variable vector may include voltage signals and / or current signals that serve as target specifications for a motor controller and / or a braking device. In examples, the input variable vector may include target specifications for a steering unit of the vehicle and / or the robot. In one example, the input variable vector u(t) may depend on the delay parameter T(^). The delay parameter may represent a temporally uncertain delay of the input variable vector u(t - x©). In one example, the input variable vector u(t - T©) may be calculated at a different location than the system to be controlled, so that a time delay may occur due to the data transmission.The temporal uncertainties are expressed in the present disclosure by the stochastic argument. In one example, the continuous-time state-space model can be described by the equation x = Ax(t, ) + Bu(t — T©), where t > 0, x(0) = x0. In one example, the input variable vector u(t) can be determined by the control law u(t) = Kx(t). In one example, the gain matrix K can be time-variant, time-invariant, or constant. In one example, the gain matrix K can be deterministic. In one example, the state-space model can include a state variable vector x(t, ) that depends on the stochastic distribution. In one example, sensor data, such as from camera systems, can be delayed in time and result in the state variable vector x(t, ) that depends on the stochastic distribution.

[0026] In one example, the time dependence of the discrete-time representation may include a sampling time based on the stochastic distribution. In one example, the sampling time h(^) of a sample of the continuous-time states and signals of the continuous-time state space model may be based on the stochastic distribution. In one example, the sampling may be based on a plurality of journals t k A respective period of the majority of time steps can be formed by a multiple of the sampling time h©. The multiple can be expressed by a running index parameter k. For the majority of time steps t k may apply: t k= kh Q,k = 0.12, .... The time t in the continuous-time state-space model can assume a value that lies between a first sampling time and a second sampling time. In examples, the second sampling time can be formed by adding the first sampling time and the sampling time. In examples, the time t in the continuous-time state-space model can be: te [t k , t k + h©]. In examples, the delay parameter T© can be uniformly distributed. In examples, the following can apply: r In examples, the sampling time h© can be uniformly distributed. In examples, the following can apply: he Z(h _ , h +). In examples, the state-space model may include a system matrix A and an input matrix B. Fig. 2 schematically illustrates an example closed control loop 10. In examples, the state variable vector may include a plurality of state variables based on a plurality of sensor signals. In examples, the plurality of sensor signals may be generated by a plurality of sensors, such as lidar sensors, radar sensors, cameras, ultrasonic sensors, GPS, acceleration sensors, temperature sensors, and the like. In examples, the state variable vector may include state variables that include information from a virtual model, such as a digital twin. The closed control loop may further include an observer that serves to emulate a state variable vector x(t) based on one or more measurable output signals, for example, the plurality of sensor signals.

[0027] Forming 120 the time-discrete representation of the state space model can comprise converting a general solution of the time-continuous state space model into a time-discrete time domain. In examples, a piecewise constant input variable vector between an initial time t0 and a time t can be assumed in the discrete time domain. In one example, the initial time t0 can be set to any first (integer) multiple k of the sampling time h©. The time t can be set to a second (integer) multiple +1 of the sampling time h©. The second multiple can be a multiple following the first multiple. In examples, the following can apply: t0 = h©, t = (k - l - l)h©. In one example, the time-discrete representation of the state space model can be be described.

[0028] An advantage of the method disclosed here may be that the dependence on the uncertain sampling time h© is shifted by forming 120 the time-discrete representation of the state-space model into the system matrix (h©,x©) and the input matrix B(h©,r©). This may enable the determination 130 of the surrogate model.

[0029] In some examples, the surrogate model may include a plurality of surrogate matrices. The plurality of surrogate matrices may include a surrogate system matrix A', a surrogate state matrix Z k , a surrogate input matrix B' and optionally a surrogate disturbance matrix. In examples, the surrogate model may include a surrogate gain matrix K'. For example, the existence of the disturbance matrix may depend on whether the state-space model of the original system includes a disturbance. In some examples, the surrogate model can be described using the equation

[0030] Z k+1= (A' + B'K')Z k , where K' = KC) ® I N,k = 1,2,3. In one example, determining 130 the surrogate model may be performed based on the stochastic distribution. In one example, determining 130 may be performed using a polynomial chaos expansion (PCE). In one example, determining 130 the surrogate model may further comprise determining 131 one or more polynomial bases based on the stochastic distribution, wherein the surrogate model is determined using the one or more polynomial bases. In one example, each surrogate matrix of the plurality of surrogate matrices may be calculated using a polynomial chaos expansion (PCE). Polynomial bases may correspond to the stochastic distribution. In one example, each polynomial basis 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, the state variable vector may be, for example, k Q = apply, where Z k , nso-called chaos coefficients and represent the states of the surrogate model. In one example, the polynomial chaos expansion may include an approximation with a number N of journals. In one example, the method 100 may include determining the number N of journals. In one example, determining the number N of journals may be predetermined. For example, a larger number N of journals may lead to a more accurate approximation, but require a longer computation time, and a smaller number N of journals may lead to a less accurate approximation, but enable a shorter computation time. In one example, a polynomial chaos expansion may 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.

[0031] In an example, the following can apply to the determination of the majority of surrogate matrices of the surrogate model:

[0032] 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 F may be calculated using the following equation:

[0033] In some examples, the method 100 may include determining the number of quadrature points J. In one example, the size J may be predetermined. For example, a larger number of quadrature points J may result in a more accurate approximation but require a longer computation time, and a smaller number of quadrature points J may result in a less accurate approximation but allow for a shorter computation time.

[0034] In some examples, the method 100 may include designing a state controller based on the surrogate model. In some examples, the design of the state controller may include stochastic optimization. The stochastic optimization may be performed using an objective function based on the surrogate model. The stochastic optimization may include minimizing the objective function. An advantage of the proposed method can be seen in the fact that the majority of surrogate matrices can be determined independently of a specific input variable vector. Furthermore, in examples, confidence intervals of state trajectories of a closed control loop can be calculated with only one simulation based on the surrogate model. In some examples, the expected value of the state variable x kthe discrete-time representation of the state space model based on the surrogate model. In one example, the expected value of a state x k by E[x k ] = CMZ k where M is an auxiliary matrix resulting from the polynomial chaos expansion. In some examples, the variance of the state variable x k the discrete-time representation of the state space model based on the surrogate model. In one example, the variance of a state x k by V x k = CVZ k where V is an auxiliary matrix resulting from the polynomial chaos expansion.

[0035] The present disclosure also relates to methods for controlling a vehicle using a state controller 20 configured using the methods of the present disclosure. In some examples, the state controller 20 and / or the system to be controlled may be configured as described above.

[0036] In one example, the system to be controlled may be designed for arrangement in a vehicle and / or for controlling a vehicle function (in particular for controlling a driving function). 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 for execution 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 can be executed in this backend. In examples, the temporal uncertainties can arise from the data transmission between the system to be controlled and the computer system executing the state controller. In one example, the system to be controlled can be a system for lateral and / or longitudinal guidance of the vehicle. In examples, the state variable vector x(t, Q) can be based on speed information or distance information. In examples, the state variable vector can include a relative speed and / or a distance between a first vehicle, a second vehicle, a human, and / or a stationary object. In one example, a state variable vector x(t, ) of the state space model can include variables based on at least one of a steering angle, an orientation angle, a yaw rate, a slip angle, and / or a lateral error.In examples, the state variable vector x(t, ) can include information from a network, such as movement and / or direction information from other vehicles. In examples, this information can be provided by vehicle-to-vehicle communication (V2V communication) or by a backend (V2X communication). In one example, an input variable of the input variable vector u(t) can include a steering speed or target specifications for acceleration and / or braking processes. In examples, the system to be controlled can be designed for arrangement in a drive control or a drive unit and / or can serve to control a motor-related function (in particular for motor control). In examples, the system to be controlled can be arranged for arrangement in a drive control of an electric machine.For example, the state vector x(t, Q) of the state-space model may contain variables based on at least one of a control signal, an operating mode, or a power setting of the electric machine.

[0037] The present disclosure also relates to methods for controlling a robot under

[0038] Use of a state controller 20 configured using the methods of the present disclosure. In some examples, the state controller 20 and / or the system to be controlled may be configured as described above.

[0039] In other examples, the system to be controlled may be arranged in a robot and / or configured to control a robot function (in particular, to control 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 some examples, the state controller 20 may be executed in a backend. In examples, the state variable vector x(t, ) may be based on velocity information or distance information.In examples, the state variable vector may include a relative velocity and / or a distance between a first robot, a human, another mobile device, and / or a stationary object. In one example, a state variable vector x(t, ) of the state-space model may include variables based on at least one of a steering angle, an orientation angle, a yaw rate, a slip angle, and / or a lateral error. In examples, the state variable vector x(t, ) may include information from a network, such as movement and / or direction information from other robots, mobile devices, and / or humans. In examples, this information may be provided via direct communication or via a backend. In one example, an input of the input variable vector u(t) may include a steering velocity or target specifications for acceleration and / or braking.

[0040] The present disclosure also relates to methods for controlling functions in building automation using a state controller 20 configured using the methods of the present disclosure. In some examples, the state controller 20 and / or the system to be controlled may be configured as described above.

[0041] In one example, the system to be controlled may be designed for placement in a building and / or for controlling building functions (in particular for controlling building automation functions). For example, the building function may be a function for controlling room temperature, lighting, and / or security equipment. In some examples, the state controller 20 may be designed for execution on a computer system within the building. For example, the computer system may be implemented locally in the building or implemented (at least partially) in a backend communicatively connected to the building. For example, the computer system may comprise a control system or a building automation controller on which the state controller 20 may be executed. In examples, the building may have a computer system with a communication interface that enables communication with an external backend.For example, the state controller 20 can be executed in this backend. In examples, the temporal uncertainties can arise from the data transfer between the system to be controlled and the computer system executing the state controller. In examples, a state variable vector x(t, ) of the state-space model can contain variables based on information such as room temperature, brightness, or the presence of people. In some cases, the state variable vector can include a relative temperature difference, illuminance, or distance to a specific location or object in the building. An example of a state variable vector x(t, ) in the context of building automation could contain variables based on parameters such as heating control, lighting settings, ventilation speed, or security alarms.In examples, information can originate from a network, such as sensor data or settings from other buildings or building components. This information can be provided through communication between buildings or building components or via an external backend. In one example, an input variable of the input variable vector u(t) could include, for example, a temperature and / or lighting setting, for example, in the form of a voltage and / or current signal.

[0042] The following illustrates the advantages of the techniques of the present disclosure over the prior art using Fig. 3-A and Fig. 3-B. The example of a group start of several vehicles is shown. A state variable vector x = [e, w] T includes on the one hand a distance error e = d re f — d between two vehicles. The distance error is the difference between a reference distance d refand an actual distance d between two vehicles. On the other hand, the state variable vector includes the relative speed between the two vehicles. The results of a corresponding control with a controller K and a second controller K2 are compared, particularly with regard to the probability of a potentially safety-critical situation occurring in a closed-loop system. A safety-critical situation can occur if the actual distance is smaller than the reference distance (d ref> d), or when the distance error is greater than zero (e > 0). In the present example, the first controller Z^ is a conventional controller designed according to the prior art techniques and the second controller K2 is a controller designed using the method of the present disclosure. Fig. 3-A shows the distance error of both exemplary controls. In this example, line 50 represents the distance error caused by the first controller K. In this example, line 60 represents the distance error caused by the second controller K2. Areas 51 and 61 represent the 6a confidence area. It can be seen that control with the first controller leads to a higher proportion of distance errors greater than zero than control with the second controller. This can pose a safety risk, especially in situations with a larger number of vehicles. In Fig.3-B shows that the first controller drives the system faster (line 52) but introduces more oscillations than the second controller (line 62).

[0043] Furthermore, a computer system is disclosed that is configured to execute the method 100 for determining a surrogate model of a state-space model. The computer system may comprise at least one processor and / or at least one main memory. The computer system may further comprise a (non-volatile) memory. In examples, all steps of the method 100 may be executed by the computer system. In some examples, individual steps of the method 100 may be executed by the computer system. Optionally, results of individual method steps that are not executed by the computer system may be received by the computer system.

[0044] Also disclosed is a computer program designed to execute the method 100 for determining a surrogate model of a state-space model. The computer program can be in interpretable or compiled form, for example. It can be loaded (even in parts) into the RAM of a computer for execution, for example, as a bit or byte sequence.

[0045] Further disclosed is a computer-readable medium or signal that stores and / or contains the computer program or at least a portion thereof. The medium may, for example, comprise one of RAM, ROM, EPROM, HDD, SDD, etc., on / in which the signal is stored.

Claims

Claims 1 . Method (100) for determining a surrogate model of a state space model, the method (100) comprising the following steps: receiving (110) a time-continuous state space model for describing a system to be controlled, Forming (120) a time-discrete representation of the state space model for describing the system to be controlled, wherein at least the time-discrete representation of the state space model has a time dependence based on a stochastic distribution, Determining (130) a surrogate model of the discrete-time representation of the state space model.

2. The method (100) according to claim 1, wherein the continuous-time state space model has a time dependence that depends on a delay parameter based on the stochastic distribution.

3. The method (100) of claim 2, wherein the continuous-time state space model comprises an input quantity vector that is dependent on the delay parameter.

4. The method (100) according to claim 1, 2 or 3, wherein the time dependence of the discrete-time representation comprises a sampling time based on the stochastic distribution.

5. The method (100) according to any one of claims 1 to 4, wherein the surrogate model comprises a plurality of surrogate matrices, the plurality of surrogate matrices comprising a surrogate system matrix, a surrogate state matrix, a surrogate input matrix, and optionally a surrogate disturbance matrix.

6. The method (100) according to any one of claims 1 to 5, wherein determining (130) the surrogate model further comprises Determining (132) a plurality of quadrature points and a plurality of quadrature weights, and Calculate (133) the surrogate model using numerical quadrature methods.

7. The method (100) according to any one of claims 1 to 6, wherein the determination of the surrogate model is carried out using a polynomial chaos expansion.

8. Method (100) according to one of the preceding claims, wherein the system to be controlled 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. A computer system configured to execute the method (100) for determining a surrogate model of a state space model according to any one of the preceding claims 1 to 8.

10. A computer program comprising instructions which, when the computer program is executed by a computer system, cause the computer system to execute the method (100) for determining a surrogate model of a state space model according to one of the preceding claims 1 to 8.

11. A computer-readable medium or signal storing and / or containing the computer program according to claim 10.

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