Model predictive control of technical systems

Abstract

The presently disclosed subject matter provides a state-space model composed of one or more neural networks. The state-space model is configured to randomly model the technical system by modeling uncertainty in both hidden states of the technical system and weights of the one or more neural networks. Thus, the state-space model may be capable of capturing both accidental uncertainty (inherent unpredictability in observations) and cognitive uncertainty (uncertainty in parameters or weights of the model). During training and subsequent use for model predictive control, moment matching across neural network layers is used, which can ensure that the predictions of the model are consistent and close to real system behavior. The above measures can improve the control and security of many technical systems.

Description

Technical Field

[0001] The presently disclosed subject matter relates to systems and computer-implemented methods for generating a state-space model of a technical system to implement model predictive control of the technical system; to systems and computer-implemented control methods for model predictive control of a technical system; to technical systems including the above-mentioned systems as control systems; and to computer-readable media including data representing instructions, wherein the instructions are arranged to cause a processor system to perform any one of the computer-implemented methods. Background Art In recent years, the ability to control technical systems has emerged as an indispensable aspect in numerous industries and applications. An illustrative example can be found in the automotive industry, where the operation and coordination of complex systems such as adaptive cruise control, lane keeping assist, and automatic emergency braking are crucial to ensure the safety of vehicle occupants and other road users. Another illustrative example is in the field of industrial robotics, where precise control mechanisms are extremely important to prevent catastrophic failures that could result in significant damage, financial loss, or even personal injury.

[0003] Model Predictive Control (MPC) is an advanced process control method that uses optimization algorithms to obtain optimal control inputs that will drive the system's output to follow a desired trajectory in the future time frame. The core of MPC's success lies in the use of accurate models to predict future system outputs, for example, based on hypothetical future control inputs. In this field, state-space models have been widely used to represent and analyze the dynamics of technical systems. Therefore, these state-space models can be used to represent dynamic systems that can be the technical system itself and / or its interaction with its environment in the following way: define the state variables of the dynamic system and the relationship between the state variables, so that future states can be predicted based on current and past states and inputs. Recently, there has been a growing interest in augmenting state-space models using machine learning techniques, particularly neural networks. Such neural network-based state-space models can automatically learn intricate system behaviors from data, thereby potentially capturing nonlinearities and complexities that may be difficult or impractical to model using traditional methods.

[0004] However, despite the progress in state-space modeling technology, current models still face challenges in representing certain types of uncertainty inherent in real-world systems. Specifically, they often have difficulty capturing both aleatoric uncertainty and epistemic uncertainty at the same time. Aleatoric uncertainty, sometimes referred to as intrinsic uncertainty, stems from the inherent variability or randomness in the system or its environment. For example, the variability of the material properties of manufactured parts or the unpredictability of gusts that affect drone flight are examples of aleatoric uncertainty. On the other hand, epistemic uncertainty, sometimes referred to as model uncertainty, stems from the lack of knowledge about the system to be modeled. This may be due to missing data, approximate errors in modeling, or other unknown factors. The ability to capture these two types of uncertainty in state-space models is very desirable because it can ensure more robust predictions, especially in safety-critical applications. For example, in scenarios where models are used to predict potential hazards, understanding the inherent variability of the system and potential unknown factors can lead to better and safer control decisions. Summary of the invention

[0005] According to a first aspect of the present invention, as defined in claim 1, there is provided a computer-implemented method for generating a state-space model of a technical system for model predictive control of the technical system. According to a further aspect of the present invention, as defined in claim 8, there is provided a computer-implemented method for model predictive control of a technical system. According to a further aspect of the present invention, as defined in claim 11, there is provided a computer-readable medium. According to a further aspect of the present invention, as defined in claim 12, there is provided a training system for training a state-space model for model predictive control of a technical system. According to a further aspect of the present invention, as defined in claim 13, there is provided a control system for model predictive control of a technical system. According to a further aspect of the present invention, as defined in claim 15, there is provided a technical system comprising said control system.

[0006] The above measures involve generating a state-space model of the technical system and using the state-space model in a model predictive control of the technical system. Thus, a technical system, which may be, for example, a computer-controlled machine or a component thereof, may be controlled based on the generated state-space model. As is known, a state-space model may be a mathematical representation of a system, wherein the behavior of the system may be characterized by a set of state variables and the relations between these state variables, thereby capturing the evolution of these variables over time based on both internal dynamics and external inputs. Thus, a state-space model may capture both internal as well as external behavior of a technical system, wherein the term "external behavior" includes the interaction of the technical system with its environment.

[0007] The above measures involve integrating neural networks and state-space models for advanced model predictive control (MPC) of technical systems.

[0008] During the training phase, a state-space model is generated to model the technical system for use in model predictive control of the technical system. The state-space model uses one or more neural networks to represent a transition function that delineates how the technical system evolves from one state to the next, and an observation function that relates the underlying state of the technical system to a discernible output. For training, partial observations of the concealed or latent state of the system across different time intervals can be used, for example in the form of sensor data.

[0009] The state-space model can be specifically configured to represent the technical system stochastically, thereby encapsulating uncertainty in both the hidden state and the neural network parameters (and specifically its weights). For this purpose, the state-space model uses an augmented state that combines the hidden state of the system with the weights of the neural network (or neural networks) (e.g., by cascading). Therefore, the transfer function and the observation function can be modified to apply to the augmented state.

[0010] In addition, both the transfer function and the observation function, as well as the filter distributions used in the prediction and update phases of training, can be approximated by corresponding normal density functions. This approach can also be referred to as an "assumed density" approximation, since each of these functions can be assumed to have a normal density distribution. The parameters of the normal distribution (which may include the mean (or first-order moment) and variance (or second-order moment)) can be recursively adjusted at each time step of training by using moment matching across neural network layers to ensure that the predictions made by the state-space model fully match the actual observed data.

[0011] After an initial training phase, the trained state-space model can be used to control a technical system based on sensor data. The sensor data may provide past observations of hidden states at different times. Based on the sensor data, a state-space model may be employed to make predictions about the current or future hidden state of the technical system, and again, the predictions may be given as partial observations. In order to generate the predictions, a predictive distribution may be generated, which may be considered as an estimate of a possible current or future state. The predictive distribution may be generated based on a transfer function, an observation function, and a filtered distribution, all of which are applied using moment matching methods across neural network layers. The predictions may then be used to control the technical system, for example, by controlling an actuator of the technical system or an actuator acting on the technical system.

[0012] The above measures enable state-space models to capture both aleatory uncertainty (the inherent unpredictability in observations) and epistemic uncertainty (uncertainty in model parameters, and specifically their weights). By stochastically modeling both the hidden states of a technical system and the weights of a neural network, a state-space model provides a more comprehensive depiction of the real-world dynamics and uncertainty of a system. That is, a state-space model can capture not only aleatory uncertainty by capturing the probability distribution across hidden states, and therefore uncertainty, but also epistemic uncertainty by capturing the probability distribution across neural network weights, and therefore uncertainty. In addition, by using neural networks together with state-space models, a more detailed depiction of the behavior of a technical system can be given. That is, neural networks are good at showing complex relationships, which may be difficult using traditional models. Overall, the above measures can improve the control and safety of many technical systems.

[0013] It is worth noting that it is known to capture aleatoric uncertainty and epistemic uncertainty by sampling, for example, by executing the model multiple times and constructing a predicted probability distribution based on the resulting output. However, executing the model multiple times can be computationally expensive. Advantageously, the above measures, which can be considered to represent a sampling-free approach, can avoid the computational complexity of the aforementioned sampling-based approaches.

[0014] The following aspects may be described in the context of a computer-implemented method for generating a state-space model, but may be equally applicable, mutatis mutandis, to a training system. Furthermore, although described with respect to training, the following aspects may represent corresponding limitations of a computer-implemented method and control system for model predictive control of a technical system.

[0015] In one embodiment, the method further includes providing and training a separate neural network to represent each of the first and second moments of the transfer function, and each of the first and second moments of the observation function. Thus, the state-space model may include at least four neural networks, namely, a first neural network representing the mean of the transfer function, a second neural network representing the variance of the transfer function, a third neural network representing the mean of the observation function, and a fourth neural network representing the variance of the observation function. This may provide specialization, as each network may be specialized to capture the characteristics of the specific moment it represents.

[0016] In one embodiment, the method further includes resampling the weights of one or more neural networks at each time step. In a state-space model, the number of weights may exceed the number of latent dimensions. Training and using the state-space model may include calculating the cross-covariance between the latent states and the weights. The covariance may become zero at the resampling step at each time step. Therefore, when the weights are resampled at each time step, the runtime and memory complexity may be reduced compared to when the resampling is omitted.

[0017] In one embodiment, the method further includes sampling weights of one or more neural networks at an initial time step while omitting resampling the weights at subsequent time steps.

[0018] In one embodiment, the method further includes using a deterministic training objective during training, for example, based on a Type II maximum a posteriori probability criterion. This objective may also be referred to as predictive variational Bayesian inference because it can directly minimize the Kullback-Leibler divergence between the true data generating distribution and the predictive distribution to be learned. Advantageously, better predictive performance and more robustness to model misspecification can be obtained compared to other learning objectives, and a beneficial implicit regularization effect can be provided for over-parameterized state-space models.

[0019] In one embodiment, the method further comprises: - Determine the predictive distribution as an integration function of the transfer function, the observation function and the filtered distribution and by using moment matching across neural network layers; - derive prediction uncertainty from the predictive distribution; -If the prediction uncertainty exceeds a threshold, prompt or explore additional training data to reduce the prediction uncertainty.

[0020] According to the above measures, prediction uncertainty can be determined from the predictive distribution. Since high prediction uncertainty is generally undesirable, if the prediction uncertainty exceeds a threshold, additional training data can be provided by the user or obtained in an automatic manner.

[0021] In one embodiment, the training data includes one or more time series of sensor data representing partial observations of latent states of the technical system, wherein the sensor data is obtained from internal sensors of the technical system and / or from external sensors observing the technical system or its environment.

[0022] The following aspects may be described in the context of a computer-implemented method for model predictive control of a technical system, but may be equally applicable, mutatis mutandis, to control systems. Furthermore, although described with respect to control, the following aspects may represent corresponding limitations of the computer-implemented method for training a state-space model and the training system.

[0023] In an embodiment, the method further comprises deriving a prediction uncertainty from the predictive distribution, wherein the control of the technical system is further based on the prediction uncertainty. By determining the prediction uncertainty, the way in which the prediction is used in the control of the technical system can be adjusted. For example, in case of high uncertainty, the system can be controlled more conservatively, for example, in a way that the consequences of a wrong prediction have less impact.

[0024] In one embodiment, the method further comprises, if the prediction uncertainty exceeds a threshold: - refrain from performing the action associated with the prediction; - Operate technical systems in safe mode; - trigger an alarm; - Increase the sampling rate of sensor data; and / or -Switch from model predictive control to another type of control.

[0025] It will be appreciated by those skilled in the art that two or more of the above-mentioned embodiments, implementations, and / or alternative aspects of the invention may be combined in any way deemed useful.

[0026] Those skilled in the art may make modifications and changes to any system, method or computer program based on the present description, which modifications and changes correspond to the modifications and changes of another of the described entities. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Further details, aspects and embodiments are described by way of example only with reference to the accompanying drawings. For simplicity and clarity, the elements in the various figures are illustrated and are not necessarily drawn to scale. In the various figures, elements corresponding to elements already described may have the same reference numerals. In the accompanying drawings, Figure 1 A training system for training a state-space model to implement model predictive control of a technical system is shown; Figure 2 The steps of a computer-implemented method for training a state-space model to achieve model predictive control of a technical system are shown; Figure 3 A control system for model predictive control of a technical system using a trained state-space model is shown; Figure 4A control system configured to control a (semi-)autonomous vehicle based on a prediction of the state of the vehicle and / or its environment is shown; Figure 5 The steps of a computer-implemented method for model predictive control of a technical system using a trained state-space model are shown; Figure 6 A computer readable medium including data is shown; Figure 7 A comparison between a deterministic approximation scheme and a Monte Carlo simulation for the dynamic system state is shown; Figure 8 A comparison between the filtered distribution of the deterministic approximation scheme and the true hidden state is shown; Fig. 9 Similar to Figure 7 , but shows a deterministic approximation scheme that includes resampling the weights at each time step; Fig.10 shows the running time of a deterministic approximation scheme as a function of dimensionality, and the running time of a Monte Carlo simulation for a deterministic approximation scheme that resamples the weights at each time step; Fig.11 Similar to Fig.10 , but shows an approximation that does not resample the weights at each time step; and Figures 12A-12C The epistemic uncertainty as a function of noise level is shown; It should be noted that the figures are purely schematic and not drawn to scale.In the figures, elements corresponding to elements already described may have the same reference numerals.

[0028] [Reference Symbol List] The following list of references and abbreviations is provided to facilitate explanation of the drawings and should not be construed as limiting the claims.

[0029] 100 Training system for training state-space models 120 processor subsystem 140 data storage interface 150 Data storage device 150 training data 154 Data Representation of State Space Models 200 Methods for training state-space models 210 Provide state space model 220 Provide training data 230Training the state-space model on the training data 240 Moment Propagation for Transfer Functions 245 Moment Propagation for Observation Functions 250 Moment Propagation for Filtered Distributions 300 Control Systems for Model Predictive Control Using State-Space Models 320 processor subsystem 340 data storage interface 350 Data storage device 352 Data Representation of State Space Models 360 sensor data interface 362 sensor data 370 control interface 372 Control Data 400 Environment 410 (Semi)Autonomous Vehicles 420 Sensor 422 Camera 430 Actuator 432 Electric Motor 500 Methods for model predictive control using state-space models 510 provides state space model 520 Get sensor data 530 Generate predictions about the state of technical systems 540Controlling technical systems based on predictions 600 Non-transitory computer readable medium 610 Data 700 hours 710 value 720-724 Assuming density approximation 730-734 Monte Carlo Simulation 740-744 95% confidence interval 800 Dimensions 810 Time 820 particles 830-832 Local Determinism 840-842 Global Determinism 900-904 The expected value of the learned mean function 910-914 True mean function 920-924 95% confidence interval DETAILED DESCRIPTION

[0030] While the presently disclosed subject matter is susceptible of embodiment in many different forms, one or more specific embodiments are shown in the drawings and will be described in detail herein, with it being understood that this disclosure is considered an exemplification of the principles of the presently disclosed subject matter and is not intended to be limited to the specific embodiments shown and described.

[0031] Hereinafter, for ease of understanding, the elements of the embodiments are described in operation. However, it will be apparent that the respective elements are arranged to perform the functions described as being performed by them.

[0032] Furthermore, the presently disclosed subject matter is not limited only to the embodiments, but also comprises every other combination of the features described herein or in mutually different dependent claims.

[0033] The following content is for reference Figure 1 and 2 A system and a computer-implemented method for generating a state-space model of a technical system for implementing model predictive control of the technical system are described, with reference to Figure 3 and 5 Systems and computer-implemented methods for model predictive control of technical systems are described and reference is made to Figure 4 Describes the Figure 3 Systems for (semi-)autonomous vehicles. Figure 6 Computer readable media used in embodiments of the claimed invention are shown. Figure 7-12C Performance aspects of the state-space model are illustrated.

[0034] Figure 1 A system 100 for generating a state-space model of a technical system to implement model predictive control of the technical system is shown. The system 100 (which may also be referred to as a training system) may include an input interface subsystem for accessing training data 152 for training an untrained or partially trained state-space model, thereby training the state-space model for model predictive control of the technical system. As will be discussed in detail elsewhere in this specification, the training data may include partial observations of hidden states of the technical system at multiple time steps. Figure 1 As shown, the input interface subsystem may include or be comprised of a data storage interface 140 that may provide access to training data 152 on a data storage device 150. For example, the data storage interface 140 may be a memory interface or a permanent storage interface, such as a hard disk or solid state disk interface, but may also be a personal area network, a local area network, or a wide area network interface, such as a Wi-Fi interface or an Ethernet or fiber optic interface. The data storage device 150 may be an internal data storage device of the system 100, such as a memory, a hard disk drive, or an SSD, but may also be an external data storage device, such as a network accessible data storage device.

[0035] In some embodiments, the data storage device 150 may further include a data representation 154 of the state-space model, which will be discussed in detail below and can be accessed by the system 100 from the data storage device 150. The state-space model may be composed of one or more neural networks to represent the transfer function and observation function of the state-space model. For example, a separate neural network may be provided for each function. As previously explained, the data representation 154 of the state-space model may represent an untrained or partially trained state-space model, because the parameters of the model, such as the weights of (one or more) neural networks, can still be further optimized. It will be appreciated that the data representation 154 of the state-space model and the training data 152 may also each be accessed from different data storage devices, for example, via different data storage interfaces. Each data storage interface may be of the type described above for the data storage interface 140. In other embodiments, the data representation 154 of the state-space model may be generated internally by the system 100, for example, based on design parameters or design specifications, and may therefore not be explicitly stored on the data storage device 150.

[0036] The system 100 may further include a processor subsystem 120, which may be configured to train a state space model on the training data 152 during operation of the system 100. In particular, the system 100 may train a state space model on the training data so as to be able to predict the hidden state of the technical system based on partial observations of the past. The prediction of the hidden state may be in the form of partial observations of the hidden state. The state space model may be configured to stochastically model the technical system by modeling the uncertainty in both the hidden state of the technical system and the weights of one or more neural networks. For this purpose, the transfer function may be configured to map the augmented state to the next augmented state at the next time step, wherein the augmented state is composed of the hidden state of the technical system and the weights of one or more neural networks. In addition, the observation function may be configured to map the augmented state to the partial observations, and the filtering distribution that may be used during the prediction and update steps of the training may be configured to represent the distribution of the augmented state. The transfer function, the observation function, and the filtering distribution may each be approximated by a normal probability distribution. Training may include recursively computing first and second order moments of each of the transfer function, the observation function, and the filtered distribution at each time step by moment matching across neural network layers.

[0037] refer to Figure 7-12C , these and other aspects of the training of state-space models can be further elucidated.

[0038] The system 100 may further include an output interface for outputting a data representation of the trained state-space model. Figure 1As illustrated, the output interface may be comprised of a data storage interface 140, which in these embodiments is an input / output ("IO") interface, via which the trained scale estimator may be stored in a data storage device 150. For example, a data representation 154 defining an untrained state-space model may be at least partially replaced by a data representation of a trained state-space model during or after training, as parameters of the state-space model 154 may be adapted to reflect the training on the training data 152. In other embodiments, the data representation of the trained state-space model may be stored separately from the data representation 154 of the "untrained" state-space model. In some embodiments, the output interface may be separate from the data storage interface 140, but may generally be of the type described above.

[0039] Figure 2 A computer-implemented method 200 for generating a state-space model of a technical system to implement model predictive control of the technical system is shown. The method 200 may correspond to Figure 1 , but need not be so, as it may also correspond to the operation of another type of system, apparatus, device, or entity, or as it may correspond to steps of a computer program. Method 200 is shown to include: in a step entitled "Providing a state-space model", providing 210 a state-space model; and in a step entitled "Providing training data", providing 220 training data. Each of the state-space model and the training data may be of a type as described elsewhere in this specification. Method 200 may further include: in a step entitled "Training the state-space model on the training data", training 230 the state-space model on the training data to be able to predict a hidden state of the technical system based on past partial observations, wherein the prediction of the hidden state is in the form of partial observations of the hidden state. Training 230 may include: in a step entitled "Moment Propagation for Transfer Function", at each time step of training, performing moment propagation 240 for the transfer function across the neural network layers; in a step entitled "Moment Propagation for Observation Function", at each time step, performing moment propagation 245 for the observation function across the neural network layers; and in a step entitled "Moment Propagation for Filtered Distribution", at each time step, performing moment propagation 250 for the filtered distribution across the neural network layers.

[0040] Figure 3 A system 300 for model predictive control using a trained state-space model as described elsewhere in this specification is shown. In particular, the system 300 can be configured to control a technical system, and thus can also be referred to as a control system. The system 300 can include an input interface subsystem for accessing a data representation of the state-space model. For example, also as Figure 3As shown, the input interface subsystem may include a data storage interface 340 to a data storage device 350, which may include a data representation 352 of the state-space model. In some examples, the data storage interface 340 and the data storage device 350 may be connected to a reference Figure 1 The same types as described for the data storage interface 140 and the data storage device 150 .

[0041] The system 300 may further include a processor subsystem 320, which may be configured to: obtain sensor data during operation of the system 300, the sensor data representing past partial observations of hidden states of the technical system at multiple time steps; generate a prediction of the hidden state of the technical system in the form of a prediction of the partial observations of the hidden state based on the past partial observations. The processor subsystem 320 may be further configured to generate the prediction as an integrated function of the transfer function, the observation function, and the filtered distribution and by approximating the predictive distribution using moment matching across neural network layers, and deriving the prediction from the predictive distribution. The processor subsystem 320 may be further configured to control the technical system based on the prediction.

[0042] Figure 3 Further illustrated are various optional components of the system 300. For example, in some embodiments, the system 300 may include a sensor data interface 360 ​​for accessing sensor data 362 acquired by a sensor 420. The sensor 420 may, for example, be an internal sensor of the technical system ( Figure 3 ) or external sensors. Sensor 420 can observe the technical system and / or its environment 400, and thereby provide at least partial observations of the state of the technical system. Sensor 420 can have any suitable form, such as image, lidar, radar, temperature, pressure, proximity, light, humidity, motion, infrared, ultrasound, voltage or current sensor. In some embodiments, system 300 can access sensor data of multiple sensors. Multiple sensors may include different types of sensors of the aforementioned types. Sensor data interface 360 ​​can have any suitable form that corresponds in type to the type of (one or more) sensors, including but not limited to low-level communication interfaces, electronic buses, etc. In some examples, system 300 can access sensor data 362 from a data storage device. In such an example, sensor data interface 360 ​​can be, for example, as described above for Figure 1 The data storage interface 140 describes a data storage interface of the type.

[0043] In some embodiments, the system 300 may include an output interface, such as a control interface 370, for providing control data 372 to, for example, an actuator 430. The actuator 430 may be an actuator of a technical system, or may be provided in the environment 400 and may be configured to act on the technical system. The control data 372 may be generated by the processor subsystem 320 to control the actuator 430 based on a prediction of a state of the technical system. For example, the actuator 430 may be an electric, hydraulic, pneumatic, thermal, magnetic and / or mechanical actuator. Specific but non-limiting examples include electric motors, electroactive polymers, hydraulic cylinders, piezoelectric actuators, pneumatic actuators, servo mechanisms, solenoids, stepper motors, etc. Thus, the system 300 may act in response to a prediction of a state of a technical system, for example controlling a technical system in the form of a computer-controlled machine, such as a robotic system, a vehicle, a household appliance, a power tool, a manufacturing machine, a personal assistant, or an access control system.

[0044] In other embodiments ( Figure 3 ), the system 300 may include an output interface to a presentation device, such as a display, a light source, a speaker, a vibration motor, etc., which may be used to generate a sensory perceptible output signal that may be generated based on the predicted state of the technical system. The sensory perceptible output signal may directly indicate the predicted state of the technical system, but may also represent a derived sensory perceptible output signal. Using the presentation device, the system 300 may provide sensory perceptible feedback to the user.

[0045] Generally speaking, each system described in this specification (including but not limited to Figure 1 System 100 and Figure 3The system 300 of the corresponding system may be embodied as a single device or apparatus, or in a single device or apparatus. The device may be an embedded device. The device or apparatus may include one or more microprocessors that execute appropriate software. For example, the processor subsystem of the corresponding system may be embodied by a single central processing unit (CPU), but may also be embodied by a combination or system of such a CPU and / or other types of processing units (such as a graphics processing unit (GPU)). The software may have been downloaded and / or stored in a corresponding memory, such as a volatile memory such as RAM or a non-volatile memory such as flash memory. Alternatively, the processor subsystem of the corresponding system may be implemented in the device or apparatus in the form of programmable logic, for example, as a field programmable gate array (FPGA), or as an application-specific integrated circuit (ASIC). In general, each functional unit of the corresponding system may be implemented in the form of a circuit. The corresponding system may also be implemented in a distributed manner, for example, involving different devices or apparatuses, such as distributed local servers or cloud-based servers. In some embodiments, the system 300 may be an internal component of the technical system to be controlled.

[0046] Figure 4 An example of the above is shown, where system 300 is shown as a control system of a (semi-)autonomous vehicle 410 operating in environment 400. Autonomous vehicle 400 may incorporate system 300 to control aspects of the autonomous vehicle, such as steering and braking, based on sensor data obtained from camera 422 integrated into vehicle 400. For example, in the event that the autonomous vehicle is predicted to suffer a failure in the near future, then system 300 may control motor 432 to guide autonomous vehicle 300 to a safe location.

[0047] Figure 5 A computer-implemented method 500 for model predictive control of a technical system is shown. The method 500 may correspond to Figure 3The method 500 is shown to include the operation of the system 300, but may also be performed using or by any other system, machine, device or apparatus. The method 500 is shown to include: in a step entitled "Providing a state-space model", providing 510 a state-space model of a type and that has been trained in a manner described elsewhere in this specification; in a step entitled "Obtaining sensor data", obtaining 520 sensor data representing past partial observations of hidden states of the technical system at multiple time steps; in a step entitled "Generating predictions of the state of the technical system", generating 530 predictions of hidden states of the technical system in the form of predictions of partial observations of hidden states based on past partial observations, wherein generating the prediction includes approximating a predictive distribution as an integration function of a transfer function, an observation function and a filtered distribution and deriving the prediction from the predictive distribution by using moment matching across neural network layers; and in a step entitled "Controlling the technical system based on the prediction", controlling 540 the technical system based on the prediction. It will be appreciated that, generally speaking, Figure 2 and 5 The operations or steps of the computer-implemented methods 200 and 500 , respectively, shown may be performed in any suitable order, such as serially, concurrently, or a combination thereof, subject to a specific order, such as by input / output relationships, where applicable.

[0048] Each method, algorithm or pseudo code described in this specification can be implemented on a computer as a computer-implemented method, dedicated hardware, or a combination of both. Figure 6 As shown, instructions for a computer (e.g., executable code) can be stored on a computer-readable medium 600, for example, as data 610, and in the form of a machine-readable sequence of physical indicia 610 and / or as a sequence of elements having different electrical, magnetic, or optical properties or values. The executable code can be stored in a temporary or non-temporary manner. Examples of computer-readable media include memory devices, optical storage devices, integrated circuits, servers, online software, etc. Figure 6 A memory card 600 is shown. In an alternative embodiment, the computer-readable medium 600 may include as stored data 610 a data representation of a trained state-space model as described elsewhere in this specification.

[0049] With further reference to state-space models and their training and subsequent use (eg, for inference), note the following.

[0050] Modeling unknown dynamics from data. Modeling unknown dynamics from data (e.g., the internal dynamics of a technical system and / or the dynamics of the interaction of a technical system with its environment) is challenging because it may involve accounting for intrinsic uncertainty in the underlying processes and uncertainty about model parameters. Parameter uncertainty or epistemic uncertainty can be used to address uncertainty arising from incomplete data. Intrinsic uncertainty, also known as aleatory uncertainty, can be used to represent the inherent randomness of a system.

[0051] (Deep) state-space models can provide a principled solution for modeling the inherent uncertainty of unidentified dynamic processes. Such deep state-space models can assign hidden variables to each data point, which represent the underlying state and change over time, while accounting for uncertainty in both observations and state transitions. Neural networks with deterministic weights can describe the nonlinear relationship between hidden states and observations. Although providing considerable model flexibility, these deterministic weights may limit the model's ability to capture epistemic uncertainty.

[0052] On the other hand, known methods that take weight uncertainty into account either make simplifying assumptions that the transfer dynamics are noise-free or that the dynamics are fully observable. Many real-world applications do not satisfy these two assumptions, and these two assumptions may lead to miscalibrated uncertainties.

[0053] Other approaches use Gaussian processes to model state transition kernels instead of probabilistic neural networks. While these approaches may respect both sources of uncertainty, they do not scale well with the size of the latent space. Finally, there is a notable exception, namely, "Normalizing Kalman Filters for Multivariate Time Series Analysis" by de Bézenac et al., NeurIPS 2020, which aims to learn deep dynamic systems that jointly respect both sources of uncertainty. However, this approach requires marginalizing both the latent time state and the neural network weights via ordinary Monte Carlo, which is not feasible for noisy transition dynamics.

[0054] The following approaches address the problem of learning dynamic models that account for both epistemic and aleatory uncertainty. These approaches account for epistemic uncertainty by attaching uncertainty to the neural network weights, and account for aleatory uncertainty by using a deep state-space formulation. While this type of model promises a flexible predictive distribution, inference can be doubly-intractable due to uncertainty about the weights and the latent dynamics. To address this issue, a sample-free inference scheme is described that allows for efficient propagation of uncertainty along trajectories. This deterministic approximation is computationally efficient and can accurately capture the first two moments of the predictive distribution. This deterministic approximation can be used as a building block for multi-step-ahead predictions and Gaussian filtering. Furthermore, the deterministic approximation can be used as a fully deterministic training target.

[0055] The above measures excel particularly in demanding situations, such as those involving noisy transfer dynamics or high-dimensional outputs.

[0056] Figure 7 A comparison between a deterministic approximation of the state of a dynamic system and its Monte Carlo simulation is shown. This involves using a t ~p(w t ) has an impact on the uncertainty of the dynamic system p(x t+1 |x t , w t ), a dynamic system such as the internal workings of a technical system and / or the interaction of a technical system with its environment is simulated. The predictions made by the deterministic approximation scheme described in this specification are shown as a solid line 720 depicting the mean, while the shaded area 740 is the 95% confidence interval. The deterministic approximation scheme is compared with the predictions made by the Monte Carlo simulation 730 for multi-step ahead predictions. It can be seen that the deterministic approximation described in this specification accurately captures the first two moments of the samples generated by the Monte Carlo.

[0057] Figure 8 The filter distribution of the deterministic approximation scheme (e.g., the emission function p(y t |x t )) Comparison between the true hidden state. The filtered distribution is compared to the true hidden state 732, which is shown as a solid line mean 722 depicting the mean and a shaded area 742 representing a 95% confidence interval. It can be seen that the true hidden trajectory 732 lies within the 95% confidence interval 742 of the approximate filtered distribution.

[0058] Deep State Space Models. A state space model (SSM) can describe partially observable dynamic systems, such as the aforementioned internal dynamics of a technical system and / or the dynamics of the interaction of a technical system with its environment. More formally, a state space model with hidden states The true underlying process can emit observations at each time step t The latent dynamics can follow a Markov structure, for example, the state x at time t+1 may only depend on the state x at the previous time point t .

[0059] More formally, the generative model of SSM can be expressed as: x0~p(x0). (1) x t+1 ~p(x t+1 |x t ) (2) y t ~p(y t |x t ). (3) Above, p(x0) is the initial distribution, p(x t+1 |x t ) is the transfer density, and p(y t |x t ) is the emission density.

[0060] A deep state space model (DSSM) can be an SSM with neural transfer density and firing density. Typically, these densities can be modeled as Gaussians that depend on the input.

[0061] Assuming the density is approximated, the t-step transition kernel can propagate the hidden state forward in time and can be recursively calculated as: p(x t++1 |x0)=Jp(x t+1 |x t )p(x t |x0)dx t (4) Where p(x t+1 |x t ) follows from equation (2). There is no analytical solution except for the linear transfer function.

[0062] Various approximations to the transition kernel have been proposed, which can be roughly divided into two groups: (a) Monte Carlo (MC) based methods, and (b) deterministic approximations based on assumed density (AD). Although MC based methods can approximate arbitrarily complex distributions in the limit of infinite number of samples, in practice they are often slow and their convergence is difficult to assess. In contrast, deterministic methods are often based on the assumption that the t-step transition kernel can be approximated by a Gaussian distribution. In the context of machine learning, AD methods have recently been used in various applications such as deterministic variational inference or traffic forecasting.

[0063] The presently disclosed subject matter follows the AD approach and approximates the t-step transition kernel from equation (4) as: Among them, the hidden state x t can be recursively approximated as having mean and covariance Gaussian distribution of . This simplifies the calculations for solving equation (5) to approximate the first two output moments. For DSSM, there are general approximation methods as well as specialized algorithms. The currently disclosed subject matter approximates the first two output moments via moment propagation across neural network layers.

[0064] Gaussian filtering. In filtering applications, it is possible to x t|y 1：t ) is of interest, where y 1：t ={y1, ..., y t} represents the past observations. For deep state-space models, the filtering distribution is not tractable and can be approximated using a general Gaussian filter by repeating the following two steps at all time points. t |y 1：t-1 ) is called the prior, and p(x t ,y t |y 1：t-1 ) is called the joint prior.

[0065] Prediction: Use the following formula to approximate the prior p(x t |y 1：t-1 ): Where p(x t+1 |x t ) refers to the transfer model defined in equation (2). Equation (6) can be derived by multiple rounds of moment matching. First, the filter distribution can be approximated as a normal distribution, and then the one-step transfer kernel can be approximated as another normal distribution. Here, the index t|t′ explicitly represents the prior moment, for example, the moment at time step t, which is conditioned on the observations up to time step t′. If t == t′, the double index can be omitted.

[0066] Update: Approximate joint prior p(x t ,y t |y 1：t-1 ): in is x t and tThe cross-covariance between them, and the density P(y t |x t ) can be defined in Equation (3). The Gaussian approximation to the joint prior in Equation (7) can be performed by a moment matching scheme similar to that discussed elsewhere. Thereafter, the observation y can be obtained by t Conditional on calculating the posterior p(x t |y 1：t ): where equation (8) can be obtained from equation (7) by standard Gaussian conditioning. The resulting distribution has the following moments: in is the Kalman gain

[0067] Probabilistic Deep State-Space Models. The presently disclosed subject matter describes a probabilistic deep state-space model (ProDSSM). The model can account for epistemic uncertainty by attaching uncertainty to the weights of a neural network, and for aleatory uncertainty by building on top of a deep state-space formalism. By integrating these two sources of uncertainty, the family of models provides well-calibrated uncertainty. For joint marginalization on the weights and latent dynamics of a neural network, algorithms for assumed density approximation and Gaussian filtering that jointly treat latent states and weights are proposed below. Both algorithms are tailored for the ProDSSM, allowing fast and sample-free inference with low computation, and laying the foundation for deterministic training objectives.

[0068] Uncertainty weight propagation. Two variants of propagating weight uncertainty along a trajectory can be used: a local approach and a global approach. For the local approach, the weights can be propagated at each time step. In contrast, for a global approach, the weights may be sampled only once at the initial time step and kept fixed for all remaining time steps. Figure 7 The predictions made by ProDSSM using a global approach were shown previously, whereas Fig. 9 The predictions made by ProDSSM using a local approach (i.e., resampling with weights at each time step) are shown as a solid line 744 depicting the mean, while the shaded area 740 is also the 95% confidence interval. As previously discussed in Figure 7 As in , the Monte Carlo simulation 734 is shown again as a reference.

[0069] Assuming Gaussian additive noise, the transfer and emission models of ProDSSM can be defined as follows: x0~p(x0), (12) w0~p(w0|φ), (13) in Modeled the shifted mean, The transfer variance is modeled, The mean emission is modeled, and The emission variance is modeled. The weight distribution p(w t |φ) is modeled as a Gaussian distribution: It has a mean and diagonal covariance Together they define the hyperparameters of the model. Where T is the horizon.

[0070] In order to avoid cluttered representation, we can introduce an extended state z t =[x t ,w t ], the expanded state is the hidden state x t and weight w t cascade of, where the dimension is D z =D x +D w . Expanded state z t The transfer density can be followed The mean function and covariance function is defined as: In the following, the moment matching algorithm is extended towards ProDSSM and Gaussian filters. These algorithmic advances are general and can be combined with two weighted uncertainty propagation schemes.

[0071] The density approximation is assumed. The following describes the t-step transfer kernel p(z t+1 |z0). This approximation uses a hypothetical density approach and propagates moments along the time direction and across neural network layers. t As a result, a Gaussian approximation to the t-step transition kernel can be obtained. It approximates the hidden state x t and weight w t The joint density on . The mean and covariance have the following structure: in is x tThe covariance of is x t and w t The mutual covariance between .

[0072] For the standard DSSM architecture, the number of weights may exceed the number of latent dimensions. Since the mean and covariance of the weights are not updated over time, the calculation The computational burden is mainly composed of the cross-covariance For local methods, this covariance becomes zero due to the resampling step at each time point. Therefore, local methods exhibit reduced runtime and memory complexity compared to global methods.

[0073] The following describes how to efficiently compute the remaining terms by propagating moments through the neural network layers. We can start by applying the unconscious statistician's law, which states that the moments of the expanded state at time step t+1 can be obtained as a function of the previous moments at time step t: The rest is to calculate the expanded mean F(z t ) and covariance update L(z t ). In the following, we discuss the expansion of F(z t ), while omitting the update of the augmented covariance L(z t ), since its moments can be similarly approximated. Typically, a neural network is composed of L simple functions (layers), which allows the output to be written as in is the expanded state of the layer at time point t. The input can be represented as The function at the layer Receives the augmented state from the previous layer And the output Calculated as: in is the layer at time t, the state at , and is the function that updates the state. Weight are not modified in the intermediate layers, and the last layer returns the weights for the global method or their means for the local method The output distribution of each layer can be recursively approximated as: in and yes The mean and covariance of each layer can be calculated and It is called layer-wise moment propagation. In the following, the output moments of linear layers and ReLU activation functions for global methods as well as local methods are proposed.

[0074] Output distance of the linear layer. The linear layer applies an affine transformation: The transformation matrix and deviation Both are weights It should be noted that the set of all transformation matrices and biases is Defined weights Since the cross-covariance matrix For global weights to be non-zero, we assume that the transformation matrix deviation and status are jointly normally distributed.

[0075] Due to the identity function, the weight w t The mean and covariance of are equal to the input moments. The residual output moments of the affine transformation can be calculated as: This is a direct consequence of the linearity of the Cov[·,·] operator. To compute the above moments, it is necessary to compute the related normal variable and Surprisingly, as long as and Following the normal distribution, these calculations can be performed in closed form for both local and global weights. For the case of local weights, the cross-covariance matrix becomes zero, i.e., the weights and states are unrelated. In addition, the calculation of the remaining terms is significantly simplified.

[0076] The ReLU activation function applies the max operator element-wise to the hidden state, while the weights remain unaffected.

[0077] state The mean and covariance Available in the relevant literature. Status The mean and covariance Equal to the input moment and For the case of global weights, calculate the cross-covariance Still open. Using Stein’s lemma (1emma), the cross-covariance after ReLU activation can be calculated as: in is the expected Jacobian matrix of the ReLU activation. The expected Jacobian matrix is ​​equal to the expectation of the Heaviside function, which can be approximated closely.

[0078] Gaussian filtering. For the filter distribution p(z t |y 1：t ) follows from a Gaussian filter as previously described. The presently disclosed subject matter extends the filtering step to an augmented state consisting of latent dynamics and weights. In standard architectures, the number of latent states is small compared to the number of weights, which makes filtering in this new scenario more demanding. This challenge can be addressed by applying a deterministic moment matching scheme as described elsewhere in this specification, which propagates moments across neural network layers. Furthermore, this scheme can be combined with the previously derived t-step transfer kernel p(z t+1 |z0) approximations are combined.

[0079] The Gaussian filter alternates between a prediction step and an update step. The following describes in more detail how the deterministic moment matching scheme can be integrated into these two steps. For the prediction step, equation (6), the derived hypothesis density method can be reused to calculate the predictive distribution p(z t |y 1：t-1 ) is a Gaussian approximation of .

[0080] For the update step, we may first need to find the augmented state z t And y 1：t-1 The observed value y is conditional on t The Gaussian approximation of the joint distribution of (see also Eq. (7)) Hidden state z t The mean and covariance of are known from the prediction step, and the emission y t The equivalent of the mean and covariance of can be obtained as: in

[0081] These moments can be approximated using layer-wise moment propagation as described in the previous section. Finally, the cross-covariance can be facilitated by using Stein's lemma Calculation: The expected Jacobian matrix of the mean emission function is cannot be computed analytically. By approximation, this computation can be reduced to estimating the expected Jacobian matrix for each layer. The latter is often obtained in closed form, or a close approximation exists.

[0082] Once the joint distribution is calculated, the conditional distribution can be approximated as another normal distribution As shown in Equation (11). For the global method, the Kalman gain has the structure And expand the state z t The updated covariance matrix is dense. Therefore, the weight w t has a non-zero correlation after the update, and the overall variance is reduced. For the local method, since the lower block of the gain matrix is ​​zero, only the state x will be updated t Therefore, the weight distribution and the cross-covariance between states and weights are not affected by the Kalman step.

[0083] Training. This can be done by fitting the hyperparameter φ to the dataset To train ProDSSM, the hyperparameter φ describes the weight distribution. For simplicity, we introduce the shorthand notation p(w 0：T |φ)=p(w|φ) to refer to the weights at all time steps with arbitrary range T. ProDSSM can be trained on a Type II Maximum A Posteriori (MAP) objective: This objective is also known as predictive variational Bayesian inference because it directly minimizes the Kullback-Leibler divergence between the true data generating distribution and the predictive distribution to be learned. Compared to other learning objectives, Equation (32) provides better predictive performance, is more robust to model misspecification, and provides a beneficial implicit regularization effect for over-parameterized models.

[0084] Likelihood is often difficult to assess can be closely approximated using a deterministic moment matching routine.The exact form of the likelihood depends on the task at hand, and it is shown elsewhere in this specification how to closely approximate the likelihood for regression problems and dynamic system modeling.

[0085] All that remains is to define the hyper-prior p(φ). Here, φ is defined by its two first-order moments and The weight distribution is defined. To achieve the goal of analyticity, each entry in p(φ) can be modeled independently. The hyper-prior for the i-th entry of the mean can be defined as a standard normal distribution: And assuming the covariance is diagonal, a gamma distribution is chosen for the (i, i)th covariance entry: where α is the shape parameter and β is the rate parameter. The above hyperpriors for the mean and covariance can be plugged into logp(φ) and yield: This results in a total of 2D w For example, one hyperparameter for the mean of each weight and one hyperparameter for the variance of each weight.

[0086] In contrast, the classical Bayesian formalism keeps the prior p(w|φ) constant during learning and the posterior is the quantity of interest. Since an analytical solution for the posterior is intractable, one can use Markov Chain Monte Carlo (MCMC) or variational inference (VI).

[0087] Predictive distribution. During test time, i.e. for inference purposes, the observed value y -H：0 ={y -H ....., y0} is the condition (where the condition range is ) at time step t, the predictive distribution p(yt|y -H：0 ) is of interest. The predictive distribution is calculated as follows: Above, p(z0|y -t：0) is the filtered distribution, p(z t |z0) is the t-step transfer kernel, and p(z t |y -H：0) is the t-step margin.

[0088] The calculation of the predictive distribution can be performed via a series of Gaussian approximations: The density approximate filter distribution. Its computation is described in this specification. The density can be obtained by propagating the augmented hidden state forward in time as described elsewhere As the marginal kernel p(z t |y -H：0 ). Finally, we can approximate the density in equation (37) The predictive distribution p(y t |y -H：0 ), which can be accomplished by another round of moment matching, also as stated in equation (30).

[0089] The pseudocode for approximating the predictive distribution in Algorithm 1 is provided below. Algorithm 1 relies on Algorithm 2 to approximate the filtering distribution Both algorithms explicitly perform a resampling step for local weight settings. In practice, this is not necessary and the computation can be omitted.

[0090] Algorithm 1: Deterministic Inference (DetInf) enter: f(x t , w t ) Mean Update l(x t , w t ) Covariance Update g(x t ) Mean emission r Covariance emission p(z -H ) Initial distribution y -H：0 Observations Output: Algorithm 2: Deterministic Filtering (DetFilt) enter: f(x t , w t ) Mean Update l(x t , w t ) Covariance Update g(x t ) Mean emission r Covariance emission p(z0) initial distribution y 1：T Observations Output: The measured running time. Fig.10 and 11 In the above example, for the observed value y t+1 The mean and covariance The running time of the approximation is visualized, and the observed value y t+1 The mean at the previous time step and covariance The expansion state z t is the condition. On the x-axis 800, the dimension D is changing. For the observed value y t and hidden state xt Using the same dimension, D x =D y = D. Randomly initialized transfer and emission functions are used with hidden layers of width H = D. Solid lines 830, 832 represent the running time of the deterministic approximation for local weights, while dashed lines 840, 842 represent the deterministic approximation for global weights. As shown on scale 820, lines with different intensities represent the running time of the MC approximation with varying number of particles S as a function of the dimension D. Fig.10 In , the running time of the weight sampling process is considered for the MC baseline, while in Fig.11 , the running time of the weight sampling process is ignored.

[0091] Experiments. The currently available family of models, ProDSSM, is a natural choice for modeling dynamic systems, where the goal is to extract For simplicity, assume that each trajectory The length of is T. Using the chain rule, the likelihood term in equation (32) It can be written as: The predictive distribution It can be approximated in a deterministic manner, as discussed elsewhere in the specification.

[0092] The currently disclosed family of models is benchmarked on two different datasets: the first dataset is a well-established learning task with synthetic nonlinear dynamics, and the second dataset is a challenging real-world dataset.

[0093] i) Kink [https: / / arxiv.org / pdf / 1906.05828.pdf]: constructs datasets with varying degrees of difficulty by varying the emission noise level. The transfer density is given by Given, where is the kink function. The emission density is defined as where r varied between {0.008, 0.08, 0.8}. For each value of r, 10 trajectories of length T = 120 were simulated. 10 training runs were performed, where each run used data from only a single simulated trajectory. The mean function was implemented using a neural network with one hidden layer and 50 hidden units, and the variance was used as a trainable constant. For the MC-based ProDSSM variant, 64 samples were used during training. The cost of the deterministic approximation for the local method was approximately equal to 50 samples.

[0094] By evaluating the learned transfer model p(x t+1 |x t ) covers the ground truth dynamics to compare the performance of different methods with respect to epistemic uncertainty (i.e., parameter uncertainty). To compute NLL and MSE, 70 evaluation points are placed on an equidistant grid between the minimum and maximum hidden states of the ground truth time series, and 256 Monte Carlo samples are used for each point x t To approximate the mean and variance

[0095] ii) Mocap: The data is available at http: / / mocap,cs.cmu.edu / here. It consists of 23 sequences from a single person. 16 sequences are used for training, 3 for validation, and 4 for testing. Each sequence consists of measurements from 50 different sensors. A residual connection is added to the transfer density, i.e., x is used in Eq. 14. t +f(x t , w t ) to replace f(x t , w t ). For the MC-based ProDSSM variant, 32 samples were used during training and 256 samples were used during testing. The cost of the deterministic approximation for the local method is approximately 24 samples. For numerical comparisons, NLL and MSE are computed on the test sequences.

[0096] Baselines. The same ProDSSM variant as described previously with reference to deep random layers is used. In addition, the performance is compared to the following well-established baselines from the GP- and neural network-based dynamic modeling literature: VCDT, LaplaceGP, ODE2VAE, and E-PAC-Bayes-Hybrid.

[0097] For the kink dataset, Figures 12A-12C The learned transfer model of the ProDSSM model is visualized in Figures 12A-12C Epistemic uncertainty as a function of noise level is shown 900. In particular, Figures 12A-12C Each of the two shows the true mean function f(x t ) 910-914, the expected value of the learned mean function 900-904, and the 95% confidence interval 920-924. It can be seen that the confidence interval 920-924 captures the true transfer function well, and as the noise level increases, the epistemic uncertainty also increases.

[0098] In general, for low emission noise (r=0.008) and moderate emission noise (r=0.08), all ProDSSM variants achieve performance comparable to existing GP-based dynamic models and outperform ODE2VAE. For high emission noise (r=0.08), ProDSSM variants perform significantly better than previous methods. For low and moderate noise levels, the MC variants achieve the same performance as the deterministic variants. Since the noise is low, there is almost no functional uncertainty and a small number of MC samples are sufficient for an accurate approximation of the moments. If the emission noise is high, marginalization on the hidden states and weights becomes more demanding and the MC variants perform worse than their deterministic counterparts. In addition, it is observed that for high observation noise, the local weight variants of the ProDSSM model achieve lower NLL than the global variants.

[0099] On the Mocap dataset, the best performing ProDSSM variant from previous experiments, which is a local weight variant together with a deterministic inference algorithm, is able to outperform all baselines. This is despite the fact that E-PAC-Bayes-Hybrid uses an additional dataset from another motion capture task. The differences between MC and deterministic ProDSSM variants become more prominent when compared to the Kink dataset: the Mocap dataset is high-dimensional and thus requires more MC samples for accurate approximation.

[0100] Experiments have demonstrated that the currently disclosed family of models, ProDSSM, performs well in a wide range of scenarios compared to prior art alternatives. Its benefits become particularly prominent when processing complex data sets characterized by high noise levels or a high number of output dimensions. Examples, embodiments or optional features - whether or not indicated as non-limiting - should not be construed as limiting the invention as claimed.

[0101] Mathematical symbols and representations are provided to facilitate explanation of the present invention and should not be construed as limiting the claims.

[0102] It should be noted that the above embodiments illustrate but not limit the present invention, and those skilled in the art will be able to design many alternative embodiments without departing from the scope of the appended claims. In the claims, any reference symbol placed between brackets should not be interpreted as limiting the claims. The use of the verb "comprise" and its inflections does not exclude the existence of elements or stages other than those stated in the claims. The article "a or an" in front of the element does not exclude the existence of multiple such elements. Expressions such as "at least one of ..." when before an element list or element group indicate that all elements or any subset of elements are selected from the list or group. For example, the expression "at least one of A, B and C" should be understood as: only including A, only including B, only including C, including both A and B, including both A and C, including both B and C, or including all of A, B and C. The present invention can be implemented by means of hardware including several different elements, and by means of a properly programmed computer. In the device claim that lists several devices, several of these devices can be embodied by the same hardware item. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

Claims

1. A computer-implemented method (200) for generating a state-space model of a technical system to implement model predictive control of the technical system, comprising: - providing a state-space model (210), the state-space model consisting of one or more neural networks to represent the transfer function and observation function of the state-space model; - obtaining training data (220), the training data comprising partial observations of hidden states of the technical system at a plurality of time steps; - training (230) the state-space model on the training data to be able to predict a hidden state of the technical system based on past partial observations, wherein the prediction of the hidden state is in the form of partial observations of the hidden state, wherein the state-space model is configured to stochastically model the technical system by modeling uncertainty in both the hidden state of the technical system and the weights of the one or more neural networks, wherein: The transfer function is configured to map an augmented state to a next augmented state at a following time step, wherein the augmented state is composed of a hidden state of the technical system and weights of the one or more neural networks, The observation function is configured to map the augmented state to partial observations, The filter distribution used during the prediction and update steps of said training is configured to represent a distribution of augmented states; Wherein each of the transfer function, the observation function and the filter distribution is approximated by a normal probability distribution, and the training includes recursively computing the first and second moments of each of the transfer function, the observation function and the filter distribution by moment matching (240-250) across neural network layers at each time step.

2. The method (200) of claim 1, further comprising providing and training a separate neural network to represent each of the first and second moments of the transfer function and each of the first and second moments of the observation function.

3. The method (200) according to claim 1 or 2, further comprising resampling the weights of the one or more neural networks at each time step.

4. The method (200) according to claim 1 or 2 further comprises sampling the weights of the one or more neural networks at an initial time step while omitting resampling the weights at subsequent time steps.

5. The method (200) according to any one of claims 1 to 4, further comprising using a deterministic training objective during the training, for example based on a Type II maximum a posteriori probability criterion.

6. The method (200) according to any one of claims 1 to 5, further comprising: - Determine the predictive distribution as an integration function of the transfer function, the observation function and the filtered distribution and by using moment matching across neural network layers; - deriving prediction uncertainty from the predictive distribution; - If the prediction uncertainty exceeds a threshold, prompting or exploring additional training data to reduce the prediction uncertainty.

7. A method (200) according to any one of claims 1 to 6, wherein the training data comprises one or more time series of sensor data representing partial observations of hidden states of the technical system, wherein the sensor data is obtained from internal sensors of the technical system and / or from external sensors observing the technical system or its environment.

8. A computer-implemented method (500) for model predictive control of a technical system, comprising: - providing (510) a state-space model generated by the method of any one of claims 1 to 7; - obtaining (220) sensor data representing past partial observations of hidden states of the technical system at a plurality of time steps; - generating (530) a prediction of a hidden state of the technical system in the form of a prediction of a partial observation of the hidden state based on the past partial observations, including as an integration function of the transfer function, the observation function and the filtered distribution and approximating the predictive distribution by using moment matching across neural network layers, and deriving the prediction from the predictive distribution; as well as - controlling (540) a technical system based on said prediction.

9. The method (500) according to claim 8, further comprising deriving a prediction uncertainty from the predictive distribution, wherein the control of the technical system is further based on the prediction uncertainty.

10. The method (500) of claim 9, further comprising, if the prediction uncertainty exceeds a threshold: - refraining from performing an action associated with said prediction; - Operate technical systems in safe mode; - trigger an alarm; - increasing the sampling rate of the sensor data; and / or -Switch from model predictive control to another type of control.

11. A transitory or non-transitory computer-readable medium (600) comprising data (610) representing instructions arranged to cause a processor system to perform a computer-implemented method according to any one of claims 1 to 10.

12. A training system (100) for training a state-space model for model predictive control of a technical system, wherein the training system comprises a processor subsystem (120) configured to perform the method according to any one of claims 1 to 7.

13. A control system (300) for model predictive control of a technical system, wherein the control system comprises a processor subsystem (320) configured to perform the method according to any one of claims 8 to 11.

14. The control system (300) of claim 13, further comprising at least one of the following: - a sensor interface (360) for acquiring sensor data; and - A control interface (370) for controlling actuators of the technical system or acting on actuators of the technical system.

15. A technical system comprising a control system (300) according to claim 13 or 14, wherein the technical system is a computer-controlled machine or a component thereof, such as a robotic system, a vehicle (410), a household appliance, a power tool, a manufacturing machine, a personal assistant or an access control system.

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