ROBUST ADAPTIVE CONTROL FOR INDEFINITELY TIME-DELAYED CONTROL SYSTEMS WITH TIME-VARIABLE DISTRIBUTION FUNCTION

Abstract

Disclosed is a computer-implemented method for robust adaptive control of an indefinitely time-delayed system, in particular wherein the indefinitely time-delayed system comprises a controller and a technical system to be controlled, wherein the controller and the technical system to be controlled communicate at least partially via a communication network, the method comprising: receiving sub-time series of a time series of time delays determined, in particular measured, for the indefinitely time-delayed system, wherein the sub-time series form a partition of the time series or of a further sub-time series thereof and each sub-time series comprises at least two time delays; determining, for each sub-time series, one or more transition probabilities for transitions of time delays based on the time delays of the respective sub-time series, wherein in each case one or more sub-time series-related transition probabilities result;and determining one or more transition probabilities for transitions of time lags based on the sub-time series transition probabilities;

Description

Prior ArtOver the past decades, the rapid development of computer capacity and communication technology has greatly affected the present infrastructure of control systems. While conventional control systems consist of sensors, controllers and actuators (i.e. actuators) interconnected by wires, a networked control system (NCS) replaces at least part or all of these end-to-end connections with a communication network. NCS may, but need not, be based on wireless connections. Furthermore, centralized and / or zone-based I / O architectures are also included by NCS, for example.The advantages of an NCS are varied, e.g. the cabling and / or maintenance effort is reduced, while the communication topology enables more flexibility with respect to distributed systems.However, when using a communication network, for example, the following network-related problems may occur:random transmission delay in the sensor-to-controller (S2C) and controller-to-actuator (C2A) channels, e.g. due to long distances between two nodes and / or time-varying traffic in the communication network.random packet losses, e.g. due to network overloads.In particular, the communication network may entail stochastic (in particular unpredictable) end-to-end time delays, which are referred to below briefly as time delays or simply as delays, and furthermore have the result that the controller has to make it clear predominantly with old data when synthesizing its controller input. Such a control system can be referred to as an indeterminate time-delayed control system. The two problems mentioned above, if not actively treated, may degrade the performance of the control system and even result in control loop instability.In the past, various control approaches have been developed to deal with random time delays and / or packet dropouts ranging from deterministic to stochastic approaches. While deterministic (robust) approaches rely on the worst case uncertainty description, stochastic approaches use additional information in the form of statistics and / or probability distributions to obtain less conservative control. However, the stochastic approach generally presupposes complete information about the statistics and / or probability distribution of the time delay in a specific control system, which is almost never present in practice. If, on the other hand, empirical estimates of the statistics and / or the probability distribution are used, then, on the other hand, no guarantees can be given for the control system. Such guarantees are required, however, especially for the release of the control system.It is therefore desirable to provide a method which-despite an unknown probability distribution of time delays in the control system-enables a control which is not too conservative but nevertheless robust for an indefinitely delayed control system.A common assumption for the communication network is that the underlying probability distribution (also: distribution function) of the delays would be time-homogeneous, i.e. it is simply postulated that the probability distribution remains constant over time. However, this is rarely guaranteed in practical situations where the delays are affected by external factors, especially not when e.g. the NCS uses a wireless public communication network such as 5G, 6G, etc. where the number of current users that can change in the communication network has a great influence on its performance. In this case, the probability distribution of the delays usually changes with time, i.e. it is time-inhomogeneous.Most stochastic control approaches for an NCS nevertheless proceed from a time-homogeneous probability distribution for the delays. The NCS can be modeled here, for example, as a Markov Jump Linear System (MJLS) with an unknown but constant Markov transition matrix-a specific form of the probability distributions. The empirical Markov transition matrix and associated uncertainty limits (one or more so-called ambiguity sets) can then be derived directly from observed delays. Although this approach can be applied to transition matrices that are variable over time, it leads to conservative uncertainty limits that are too large for practical significance. Such controllers are then not ideally designed at all times.The document "Lotfi M. Chaouach et al.: Uncertain uncertainy in data-driven stochastic optimization: towards structured ambiguity sets. In: 61st Conference on Decision and Control, 2022, 4776-4781", a method for designing "structured waterstone ambiguity sets" for data driven stochastic optimization problems that utilizes independence between the components of the probability distributions to speed up the rate of shrinkage of the ambiguity sets with the number of samples and thus reduce conservativeness. The method enables manageable reformulations of the stochastic optimization problems for cost functions, which can be represented as sums or products of functions that depend only on the individual components of the distribution.Therefore, in particular, a problem to be solved underlying the disclosure can be seen, for example, in providing a method which, despite a time-variable and unknown probability distribution of time delays in the control system, enables a control which is not too conservative but nevertheless robust for an indefinitely time-delayed control system.Disclosure of the InventionA first general aspect of the present disclosure relates to a computer-implemented method for robust adaptive control of an indefinitely delayed system. The indefinitely delayed system may comprise a controller and a technical system to be controlled, the controller and the technical system to be controlled communicating at least partially or exclusively via a communication network. The method comprises receiving sub-time series of a time series of time delays determined, in particular measured, for the indefinitely time-delayed system, wherein the sub-time series form a partition of the time series or a further sub-time series thereof and each sub-time series comprises at least two time delays. The method further comprises determining, per sub-time series, one or more transition probabilities for transitions of time delays based on the time delays of the respective sub-time series, wherein one or more sub-time series-related transition probabilities result in each case. The method further comprises determining one or more transition probabilities for transitions of time delays based on the sub-time-series related transition probabilities.A second general aspect of the present disclosure relates to a controller configured to control a technical system, wherein the controller and the technical system to be controlled communicate at least partially or exclusively via a communication network. The controller comprises a second communication interface for transmitting information to a computing unit from which time delays for the indefinitely delayed system, in particular sub-time series of a time series of time delays or a further sub-time series thereof, can be determined. The computing unit can, but need not, be a computing unit outside an indefinitely delayed system comprising the controller, i.e. the computing unit can also be within the indefinitely delayed system. The second communication interface is furthermore designed to receive update information from the computing unit, wherein the controller is designed to be adapted by the update information. The controller may be adapted by the update information to one or more transition probabilities for transitions of time delays determined in the method according to the first general aspect (or an embodiment thereof) based on the sub-time series-related transition probabilities.A third general aspect of the present disclosure relates to an indeterminate time-delayed system comprising the controller according to the second general aspect (or an embodiment thereof) and a technical system to be controlled, wherein the controller and the technical system to be controlled communicate at least partially via a communication network.A fourth general aspect of the present disclosure relates to a computing unit (thus a computer system) configured to execute the computer-implemented method for robust adaptive control of an indefinitely delayed system according to the first general aspect (or an embodiment thereof). The computing unit comprises a first communication interface.A fifth general aspect of the present disclosure relates to an overall system including the indefinitely delayed system according to the third general aspect (or an embodiment thereof) and the computing unit according to the fourth general aspect (or an embodiment thereof).A sixth general aspect of the present disclosure relates to a computer program configured to execute the computer-implemented method for robust adaptive control of an indefinitely delayed system according to the first general aspect (or an embodiment thereof).A seventh general aspect of the present disclosure relates to a computer readable medium or signal storing and / or containing the computer program according to the sixth general aspect (or an embodiment thereof).By means of the method proposed here according to the first general aspect (or an embodiment thereof), a robust adaptive control of an indeterminate time-delayed system, wherein indeterminate time delays can arise through the communication network between the controller (also referred to below as stochastic controller) and the technical system to be controlled, can be successively adapted, and therefore the controller can be continuously redefined. The continuously newly defined controller is robust because it can bypass at any point in time with a plurality of probability distributions of the time delays, namely on the one hand with such probability distributions of the time delays which are fluctuations about a fixed distribution function (on a shorter time scale) and on the other hand with such probability distributions of the time delays which are based on a time variability of the distribution function - and in turn the fluctuations about this distribution function - (on a longer time scale). At the same time, the continuously newly defined controller is not too conservative, since it is not designed equally for all theoretically possible time delays (and their sequences), but rather for those statistically relevant in each case in the application at the current point in time.A fundamental problem in the design of stochastic controllers is that the true statistics(s) and / or probability distribution(s) of time delays at the time of the design of the controller are usually unknown in practice. Usually only approximations remain, e.g. in the form of empirical distribution, but unfortunately are subject to sample-induced errors resulting from the sample mean approximation. In order to be able to synthesize guaranteed stabilizing controllers despite these random errors, a first distribution-robustness approach can be followed on shorter time scales, in which the statistics(s) and / or probability distribution(s) do not change too much. In this case, for example, an ambiguity set ("abbigency set") can be constructed at one time in each case, i.e. an uncertainty set in the space of the probability distributions, around an empirical estimate which contains the true probability distribution with a high confidence. This makes it possible to ensure that the resulting controller--and therefore the indefinitely delayed control system--is robust at any time on the shorter time scale with respect to a whole family of probability distributions. In the closed loop, data (in particular actually occurring time delays, i.e. time delay data) are continuously collected by the control system. The empirical distribution and the amount of ambiguity can then be continuously adapted from the data, whereby stepwise improved regulators can be synthesized. This learning algorithm can guarantee that the ambiguity set shrinks to a point (also referred to as singleton) that contains only the true distribution if the sample size is sufficiently large on the shorter time scale.For example, a possible shorter time scale control strategy may utilize a general modeling framework for NCS-referred to as Markov Jump Linear System (MJLS)-and combine it with a learning-based, distribution-robust controller design. Here, time delay data, i.e. measured time delays, can be used to construct a Markov transition matrix as well as, if applicable, an ambiguity set containing the true probability distribution with a user-defined confidence level per discrete time step on the shorter time scale. By continuously collecting further time delay data on the shorter time scale, the Markov transition matrix can be recursively updated and the amount of ambiguity reduced while maintaining the same user-defined confidence level. This adaptation of the Markov transition matrix and the ambiguity set on the shorter time scale represents a learning-based controller design. This can achieve the result that too conservative a design of the regulation is reduced by successive reduction of the ambiguity quantity and nevertheless the probabilistic stability guarantees (on the basis of the confidence level) are maintained on the shorter time scale. As an alternative to recursion on the shorter time scale, transition probabilities and, if appropriate, ambiguity amounts can be determined once for the shorter time scale.Moreover, a distinction can be made between the control loop itself, which is closed by a network, and the learning algorithm. The latter may be understood as a microservice, which may be located in another network (e.g. in a cloud), with best effort requirements for availability and reliability.An exemplary control strategy on the shorter time scale may be summarized as follows:A control system can be operated with a (first) randomly stabilizing local controller to generate time delay data on the shorter time scale. Alternatively, the control system may be operated with a controller previously adjusted for a previous shorter time scale according to the method of the first general aspect (or an embodiment thereof). This is a time-critical signal path which is closed via a network. The local controller continuously transfers the time stamps into the learning algorithm, which is located e.g. in a cloud with non-time critical processing time. Subsequently, the time stamps may be converted into a set of integer time delays which may be updated each time a new time stamp arrives. The Markov transition matrix and optionally the ambiguity set may then be updated based on the new integer time delay information, for example. Finally, the controller is updated based on the new Markov transition matrix and the ambiguity amount and fed back to the control system. After the update, the control performance on the shorter time scale is equal to or better than that of the previous local controller.A problem underlying this disclosure in the design of stochastic controllers is that the temporal variability of the statistics(s) and / or probability distribution(s) of time delays at the time of the design of the controller is usually unknown in practice. This time variability can be on a longer time scale, in contrast to the shorter time scale. For example, the shorter time scale may comprise a few seconds, the longer time scale may comprise several minutes (e.g. changes between radio networks or changes in the load of a radio network).The prior art is limited here only to constant and known distributions for delay models. In many real scenarios, in particular in public reliable distributed systems (RDS), the quality of the network transmission usually depends strongly on external parameters (e.g. network coverage, number of users and / or current network traffic).Thanks to the method proposed here according to the first general aspect (or an embodiment thereof), the controller can also be adapted to such time-inhomogeneous probability distributions. In particular, the ambiguity amounts can also be adjusted in each case by the time-inhomogeneous probability distributions. It is therefore now possible to take account of time-variable network conditions. This includes learning the stochastic distribution online while the process is running, adapting the controller to the current situation and / or compensating the strength of the resulting guarantees by the adaptation speed.The controller which is continuously newly defined on the longer time scale and in particular, for example, the controller which results after a predetermined test cycle can then already be considered to be enabled. Alternatively, the continuously newly defined controller can be further examined in a release process and then released as appropriate. Once again, the controller can be released as a necessary part for the release of the technical system to be controlled.The controller can be adapted to a predetermined test cycle exclusively in the development phase. Alternatively, the controller can also be continuously adapted after the development phase, in particular during operation.Brief Description of the FiguresFIG. 1 illustrates an example indefinitely delayed system comprising three vehicles performing platooning and, in particular, group launch. FIG. 2 schematically illustrates exemplary embodiments of a computer-implemented method for robust adaptive control of an indefinitely delayed system. FIG. 3 illustrates an exemplary embodiment of an overall system comprising an indefinitely delayed system and a computing unit configured to perform the computer implemented method for robust adaptive control of an indefinitely delayed system. FIG. 4 illustrates an example partition of a time series of time delays determined for the indefinitely delayed system into sub-time series. FIG. 5 a illustrates an example predefined time-varying Markov transition matrix. FIG. 5 b illustrates, in an exemplary simulation, the adaptive adaptation by the method proposed here to a predefined time-variable transition probability of a Markov transition matrix and the difference from the conventional method. FIGS. 6 a- b illustrate exemplary simulations of the adaptive adaptation by the method proposed here to a predefined time-varying transition probability of a Markov transition matrix for different numbers of sub-time series (i.e. for different filter lengths), wherein the number of time delays per sub-time series remains constant. FIG. 7 a illustrates exemplary vectors for the state, the measurable output and the input of a controller. FIG. 7 b illustrates an example Markov transition matrix with transition probabilities for transitions between time delays. FIG. 7 c illustrates an example Markov Jump Linear System with extended state ξ for the indefinitely delayed system. FIG. 7 dillustrates an example set of integer time delays determined for the indeterminate time delayed system (successively). FIG. 8 illustrates example successive transition probabilities P i= ( p i1, p i2, p i3) and associated small, increasing ambiguity sets. FIG. 9 a illustrates exemplary control processes for a second vehicle during platooning group start at three different times in the method and associated 3σ confidence intervals. FIG. 9 b illustrates exemplary control processes for a third vehicle during platooning group start at three different times in the method and associated 3σ confidence intervals.Detailed DescriptionDisclosed first is a computer-implemented method 100 for robust adaptive control of an indeterminate (i.e., stochastic) time-delayed system 40. The controller 10 can communicate at least partially (i.e. partially or exclusively) via a communication network 30 with the technical system 20 to be controlled. The communication network may be, for example, a wireless network. For example, the communication network may be a wireless wide area network (WWAN), in particular 5G or 6G. The communication network can also be, for example, a wireless local area network (WLAN).The method 100 proposed in this disclosure can therefore be directed to a robust adaptive control of an indeterminate (i.e. stochastic) time-delayed system.As shown by way of example in FIG. 1, the time-delayed system 40 can comprise a multiplicity of vehicles which overall form the technical system 20 to be regulated and here form, for example, a convoli (i.e. execute platooning) and in particular carry out a group start. The plurality of vehicles includes at least two vehicles or three vehicles as exemplarily shown in FIG. 1. All vehicles except for the first (leading) vehicle are controlled via the communication network 30 by a controller 10 outside the technical system 20 to be controlled. In this case, it is reasonable that the communication network 30 is a wireless network. Here, v 0 denotes a speed of the first vehicle, v 1, denotes a speed of the second vehicle immediately following the first vehicle, and v 2 denotes a speed of the third vehicle immediately following the second vehicle. Furthermore, d safe denotes a respectively required safety distance which should ideally be set between successive vehicles or at least should not be undershot. Furthermore, e 1 denotes a distance error from the ideal safety distance d safe between the first vehicle and the second vehicle, and e 2 denotes a distance error from the ideal safety distance d safe between the second vehicle and the third vehicle.The time-delayed system 40, however, is not limited to this application. Another application is, for example, the control and / or coordination of guided automated vehicles by a local network (local edge, 5G network,... ) in a limited area, e.g. in logistics centers or production facilities. Yet another application is, for example, the (partial) removal of the control algorithms of robot arms from local networks, e.g. for manufacturing systems. Yet another application is lateral and / or longitudinal motion control of vehicles at traffic hubs or other control zones, which is offloaded to a local edge or cloud system (e.g., a roadside unit).In the case of a time-homogeneous probability distribution (also: distribution function) of the delays, a networked control system (NCS) can be reduced to a time-discrete Markov Jump Linear System (MJLS), wherein the jumps (jumps) between different subsystems follow a changeover signal (comprising discrete Markov modes θ k ∈ Θ={0,..., M}, see e.g. FIG. 4 ) with discrete time k ∈ N. A Markov mode may each represent a time delay in the networked control system, e.g., as multiples of a sampling time Δ ∈ R >0. The Markov modes may be transformed by a Markov process {θ k}k>o with Markov transition matrix {p ij} i,j∈θ, where p ij= P(θ k+1= j|θ k= i) for each i,j ∈θ is a time-invariant transition probability from the Markov mode i at time k to the Markov mode j at time k+1. Since i, j each represent a time delay, each of these transition probabilities is a transition probability for transitions between delays.In the case of a time-invariant probability distribution of the delays, the time-invariant transition probabilities can be extended to time-varying transition probabilities p ij( σ(k))=Pσ(k)(θ k+1= j| θ k= i), wherein Pσ (k) is a time-varying probability measure as a function of σ(k). In other words, the function σ(k) maps the discrete time k to a probability measure. For example, the range of values of σ(k) may be discrete (e.g., {1, 2}) such that between predetermined probability measures, e.g., P 1 and P 2, is selected. σ may also be referred to as a selector function. In general, the function σ(k) is unknown.Time-varying transition probabilities occur, for example, in a public communication network, wherein, for example, a low or high probability of a time delay (up to packet loss) may occur depending on the number of users of the communication network. Here, for example, the selector function σ(k)={1,2} may be selected, and the Markov transition matrices may be defined as follows by way of example. In the low load scenario, the Markov transition matrix P is 1 e.g., wherein the first element p 11= 0,95 denotes the probability of transmitting a data packet without a large delay (or successful), and p 12= = 0,05 denotes the probability of transmitting a data packet with a large delay (or not at all). In the high load scenario, the Markov transition matrix P 1 is e.g. wherein the first element p 11= 0,7 again denotes the probability of transmitting a data packet without a large delay (or successful) and p 12= 0.3 again denotes the probability of transmitting a data packet with a large delay (or not at all).Unfortunately, neither the time-varying transition probabilities p ij( σ(k)) nor the function σ(k) are known in practice.However, the transition probabilities p ij( σ(k)) can be adapted at different times by the method 100 proposed here, and therefore the time-varying transition probabilities p ij( σ(k)), in particular the time-varying Markov transition matrix, can be estimated.As schematically illustrated in FIG. 2, for example, the method 100 comprises receiving 110 sub-time series of a time series of time delays determined, in particular measured, for the indefinitely time-delayed system 40. The method 100 may in particular comprise receiving 110 at least two sub-time series of the time series.A time series may be a time ordered sequence of time delays. A time series may comprise, for example, an ordered set of relative or absolute discrete time points, wherein a time delay is associated with each discrete time point.A sub-time series may be a subsequence of the time series. For example, a sub-time series can be a sub-sequence of the time series with substantially temporally directly consecutive time delays. In particular, a sub-time series can be a subsequence of the time series with time delays that are directly consecutive. Essentially, on the other hand, it can also mean that there is no need to arrive at a few time delays of the time series, so that, apart from these few time delays, the time delays of the subsequence directly follow one another.The sub-time series may at least substantially form a partition of the time series (or a further sub-time series thereof). In particular, the sub-time series may form a partition of the time series or a further sub-time series thereof. Essentially, on the other hand, can also mean that a small number of time delays of the time series can be underturned, i.e. need not be included in the sub-time series.Each sub-time series may include at least two time delays. The receiving 110 of the sub-time series may comprise successively receiving the time delays of the respective sub-time series per sub-time series.The time delays may be discrete. They may be Markov modes θ k ∈ θ = (0,...,m}.FIG. 4 shows a time series of time delays determined, in particular measured, for the indefinitely time-delayed system 40, which time series is partitioned into three sub-time series 1, 2, 3, wherein each of the three sub-time series comprises 5 time delays, by way of example. The time can be discrete ("discrete time k") and equidistant here. The time delays are shown discretely here as Markov mode θ k ∈ θ = {0,...,m}. In this example, the further sub-time series of the time series can therefore consist of 15 time delays (one per time), which are divided directly one after the other into three partitions (h=3), each partition comprising 5 time delays (n=5).Each sub-time series may define a shorter time scale. The union of the sub-time series may define a longer time scale.As schematically illustrated in FIG. 2, for example, the method 100 further comprises determining 120, per sub-time series, one or more transition probabilities for transitions of time delays based on the time delays of the respective sub-time series, wherein one or more sub-time series-related transition probabilities result in each case. In this case, for example, a Markov transition matrix can be determined 120 per sub-time series, in which one or more transition probabilities for transitions of time delays are respectively contained. In other words, the one or more sub-time series-related transition probabilities may each be arranged in a Markov transition matrix (one per sub-time series).In FIG. 4, for example, for each sub-time series 1, 2, 3, a Markov transition matrix P̂ 1, P̂ 2, P̂ 3 is determined 120.As schematically illustrated in FIG. 2, for example, the method 100 further comprises determining 130 one or more transition probabilities for transitions of time delays based on the sub-time series related transition probabilities.The one or more transition probabilities may also be arranged in a Markov transition matrix.For example, as schematically illustrated in FIG. 2 as an option, the method 100 may further include determining 140 one or more ambiguity sets around the one or more determined 130 transition probabilities (or around subsets thereof, e.g., each around a row of a Markov transition matrix) based on the time delays of the sub-time series. In particular, the method 100 may include determining 140 each an ambiguity set around a row of the Markov transition matrix, wherein the determined 130 one or more transition probabilities are included in the Markov transition matrix.An ambiguity set around one or more transition probabilities contains these one or more transition probabilities and represents an uncertainty set in the space of these one or more transition probabilities that contains the true probability distribution with a predetermined and, for example, high confidence.For example, as schematically illustrated in FIG. 2 as an option, the method 100 may further include adjusting 150 the controller 10 based on the determined 130 one or more transition probabilities. Alternatively or additionally, the method 100 may include adjusting 150 the controller 10 based on the determined 140 one or more ambiguity amounts. In particular, the method 100 may include adjusting 150 the controller 10 based on the determined 130 one or more transition probabilities and on the determined 140 one or more ambiguity amounts.As schematically illustrated in FIG. 2 as an option, for example, adapting 150 controller 10 may include sending 151 update information, in particular via a first communication interface, to indefinitely delayed system 40 so that controller 10 is adapted.The adaptation 150 of the controller may be effected in such a way that the controller is designed to be functionally and robust in accordance with the one or more transition probabilities and / or the one or more ambiguity quantities.The method 100 may be performed during operation of the indefinitely delayed system 40. In particular, the controller 10 may be adjusted 150 during operation of the indefinitely delayed system 40 (i.e., online).The method 100 can be executed in a computing unit 50 outside the indefinitely delayed system 40, in particular in a cloud. On the other hand, the method 100 can also be executed in a computing unit within the indefinitely delayed system 40.As schematically illustrated in FIG. 2, for example, as an option, the receiving 110 of the sub-time series (or, more generally, the method 100) may comprise receiving 111, in particular via the first communication interface, the time series or the further sub-time series.For example, as schematically illustrated in FIG. 2 as an option, receiving 110 the sub-time series (or more generally the method 100) may include partitioning 112 the time series or the further sub-time series into the sub-time series.The sub-time series may include the same number of time delays. This number can be designated n>=2.In the exemplary illustration of FIG. 3, steps 111 and 112 are shown as steps upstream of step 110. Alternatively, however, they can also be part of step 110.The determination 130 of the one or more transition probabilities for transitions of time delays can take place at different points in time and can each be based on the same number of sub-time series. This number can be designated as h>=2. In other words, the determination 130 of the one or more transition probabilities for transitions of time delays may be repeated at different times. In particular, the sub-time series can be successively received 110 and processed according to the principle "first-in-first-ouf". This may be advantageous at the memory level. An advantage is also that by repeating the determination 130, a moving filter can be realized over a predetermined number of sub-time series each.This moving filter may be referred to as a moving average distribution filter (MAD filter). The moving filter may be defined over a moving horizon of h>=2 empirical distribution functions, in particular Markov transition matrices, wherein each empirical distribution function is based on n>=2 observations, i.e. time delays.Thus, for example, initially, as long as there are no hn observations present yet, a set of observations can be received. In such an embodiment, both the sub-time series and the time delays per sub-time series are thus successively received. This set may be partitioned 112 into h ordered subsets ( which are the ordered values of the respective sub-time series, each comprising n observations, i.e.:The partitioning can explicitly form the subsets from the set with hn observations. Alternatively, partitioning can be done implicitly by creating a new subset after another n observations when receiving the observations. Then, for each subset one or more transition probabilities for transitions of time delays, in particular a Markov transition matrix, can be determined. If method 100 is subsequently used in the operation of the indefinitely delayed system 40 (i.e. online), observations θ k may be stored continuously in a for example until this set contains (exactly) n elements. Then, the moving filter can be adjusted according to the principle of "first-in-first-out", wherein, for example, is set for l=1,..., h-1 and . Such an index shift can be efficiently implemented. Then, for each subset one or more transition probabilities for transitions of time delays, in particular a Markov transition matrix, can be determined. The set can now be reset to the empty set and the method repeated. An advantage of such a moving filter is that for h-1 ordered subsets the associated one or more transition probabilities are already known from the previous step, i.e. these can also be efficiently determined by index shifting.Since h sub-time series are always available in such embodiments, the transition probabilities can be weighted via these sub-time series.For example, as schematically illustrated in FIG. 2 as an option, determining 130 the one or more transition probabilities for transitions of time delays may include weights 131 of the sub-time-series related transition probabilities. In particular, the weighting 131 can be effected with weights which are constant per sub-time series. This case is particularly relevant if the number of time delays per sub-time series is always the same, i.e. if all sub-time series have exactly the same number (n) of time delays. In other words, transition probabilities may be weighted depending on which sub-time series they come from. Given h sub-time series with transition probabilities P̂ l for l=1,..., h (these can be, e.g., respectively Markov transition matrices), weighted transition probabilities can be calculated, for example, as follows, where ω l for l=1,..., h are the weights. These may be defined as, for example, ≅ l ∈ {1,..., h} where 0<α<1. Consequently, one or more weighted transition probabilities can be determined in each case for the last hn observations, in particular in each case a weighted Markov transition matrix P̂. The weighted transition probabilities can be considered a barycentre estimate. The one or more weighted transition probabilities may be the one or more transition probabilities determined in method step 130. All n observations can therefore be used to determine one or more weighted transition probabilities, in particular a Markov transition matrix.Thus, for example, each time n new observations have been received, a new weighted 131 distribution function may be determined. The selector function σ can then jump to the new weighted 131 distribution function, respectively.The method 100 may include receiving a weight value α where 0<α<1. The weight value may remain constant. Alternatively, the weighting value can be adapted in the method 100.Determining 140 the one or more ambiguity sets around the one or more determined 130 transition probabilities based on the time delays of the sub-time series may be further based on the one or more weighted transition probabilities.The one or more transition probabilities P̂ ij per h sub-time series (i.e. after n further observations in each case) in the moving filter for a transition from the time delay Δ k= i (i.e. Δt k= iΔ) determined for a first time k (i.e. t k= k Δ) to a possible time delay at a second time k+1 (i.e. t k+1= ( k+1) Δ) can be a point in a finite-dimensional vector space. In particular, a line of a Markov transition matrix can be seen as a point in a finite-dimensional vector space. The ambiguity set for these transition probabilities can be comprised by a sphere around this point with a radius in a finite-dimensional norm (e.g. p norm with p=1 or p=∞, i.e. maximum norm), wherein the radius can be based on a confidence level β and / or a total number γ i-1( or its inverse, i.e. the inverse sample size γ i) of the time delay determined for the first time in the indefinitely delayed system 40. The sphere is to be understood in the p-norm, i.e. for example in the case of the maximum norm the sphere can be a hypercubus. The radius can decrease at a constant confidence level, for example, and an increasing total number of the time delay determined for the first time in the indefinitely delayed system. For example, the radius may be determined by a concentration imbalance such as a McDi inequality. For example, the ambiguity set A(P̂ i) for all i can be determined as follows, where the subscript p is norm 1, 2,..., or ∞, and is an ith probability simplex. Here, r i( β, γ i-1) ∈(0,2) is the radius for the Markov state i, which may depend on a user-defined confidence level β ∈(0,1) and on a total number γ i-1 of the Markov mode i in the last hn observations, i.e. the radius may be specified in closed form, e.g. via concentration equations such as the McDi inequality. The radius based on a concentration imbalance presupposes that the underlying Markov transition matrix is constant piece by piece. Since this may not be the case in practice, it can be assumed that the true Markov transition matrix changes much more slowly than the time series. This can argument that the concentration imbalance-based radius is a practically useful approximation that varies depending on the sample size. Alternatively, the radius can also be r i constant in each case (the arguments β and γ i-1 in the equation for A(P̂ i) are then omitted). In this case, the radius then does not need to be determined by a concentration imbalance such as the McDi Ungleich, thereby increasing the efficiency.Thus, in method 100, after n observations in each case, one or more ambiguity sets in the one or more determined 130 transition probabilities can be determined 140 on the basis of the time delays of the sub-time series.The method 100 may include receiving the radius or radii r i or r i( β, y i-1) per Markov mode i. The radii may be, but need not be, different between the Markov modes. Alternatively, the method 100 may include receiving a confidence level (e.g., β ∈(0,1)). The method 100 may then comprise calculating the radius or radii r i( β, γ i-1).The ambiguity set may be used in the method 100 to synthesize a distributionally robust controller that stabilizes the Markov Jump Linear System (MJLS), e.g., with confidence β. An ambiguity set A(P̂ i) may refer to the transition probabilities for transitions from a Markov state i to a (generally different) Markov state j. In this respect, the ambiguity set A(P̂ i) can be referred to as the ith ambiguity set or as the state-dependent ambiguity set.In the following, results of simulations are disclosed which are based on an exemplary predefined time-inhomogeneous probability distributions, namely on a time-varying Markov transition matrix from FIG. 5 a, wherein and μ̅=0.6, wherein t stands for the (continuous) time. In this example, there are thus 3 different Markov modes (i.e. M=2). After discretisation, μ can be seen as the selector function σ.FIG. 5 b shows firstly an exact temporal profile 4 of the component p 11( also: reference curve) of the Markov transition matrix shown in FIG. 5 a. If a Markov transition matrix were successively estimated (i.e. in particular without moving filters) according to a conventional method, a temporal profile 5 and ambiguity quantities (the uncertainty band of the component p 11), which is derived therefrom and which shows only a slight agreement with the exact temporal profile 4, would result. In this example, only 24.3% of the simulated points in time is the exact temporal profile for p 11 within the uncertainty band around the temporal profile 5; if, on the other hand, the moving filter with weighting (h=5, n=250, α=0.9 and β=0.9) is used according to the method 100, a temporal profile 6 results which has a great agreement with the exact temporal profile 4. In this case, the exact temporal profile 4 is always within the uncertainty band (again derived from the ambiguity sets comprising the component p 11) around the temporal profile 6. However, the assumption here does not apply in the current situation, since μ(t)≠const. Therefore, the proposed method 100 is more suitable here.This example illustrates that the use of a conventional approach at time varying probability distributions will not converge to the actual reference curve. Asymptotically, the conventional approach converges with the mean of the reference signal, which in this example is given by μ̅=0.6. This means that the uncertainty band of the conventional approach almost never captures the actual reference curve in the limit case (in only 0.79% of all cases, μ(t)≈0.6).In Figures 6a-b (Figure 6 shall be Figure 6a), the number h of sub-time series is varied. FIG. 6 aincludes the curves shown separately in FIG. 6 b. FIGS. 6 a- beach initially contain the reference signal (dashed curve). In this simulation, the number n of delays per sub-time series is fixed, namely at n=250. Curve 7 bounds an uncertainty band around a curve with h=2, curve 8 bounds an uncertainty band around a curve with h=5, and curve 9 bounds an uncertainty band with h=10. As h increases, the moving filter contains more observations, leading to narrower uncertainty bands. However, this results in the response to distribution changes decreasing with increasing h, which may be compensated by weighting in determining the one or more transition probabilities. For example, sub-time series that are longer behind can be weighted significantly less than more recent sub-time series.Disclosed is also a controller 10 configured to control a technical system 20, wherein the controller 10 and the technical system 20 to be controlled communicate at least partially (i.e., partially or exclusively) via a communication network 30. The communication network 30 may comprise or be a wireless network, for example. The controller 10 comprises a second communication interface in order to send information to a computing unit 50, in particular to a computing unit 50 outside an indefinitely delayed system 40 comprising the controller 10, from which time delays for the indefinitely delayed system 40, in particular sub-time series of a time series of time delays or a further sub-time series thereof, can be determined. The second communication interface may further be configured to receive update information from the computing unit 50. The controller 10 may be configured to be adjusted 150 by the update information.The controller 10 may be adjusted by the update information to one or more 130 transition probabilities determined in the method 100 for transitions of time delays based on the sub-time series-related transition probabilities. Alternatively or additionally, the controller 10 can be adapted by the update information to one or more 140 ambiguity quantities determined in the method 100. In particular, the controller 10 can be adapted by the update information both to the transition probabilities and to the ambiguity quantities.Disclosed is furthermore an indeterminate time-delayed system 40 comprising the controller 10 and a technical system 20 to be controlled. The controller 10 and the technical system 20 to be controlled can communicate at least partially (i.e. partially or exclusively) via a communication network 30.Disclosed is furthermore a computing unit 50 which is designed to execute the computer-implemented method 100 for robust adaptive control of an indefinitely delayed system 40. The computing unit 50 can furthermore comprise the first communication interface. The computing unit 50 can furthermore comprise a processor and / or a working memory. The computing unit 50 may be part of the indefinitely delayed system 40. Alternatively, the computing unit 50 can be outside the indefinitely delayed system 40, in particular in a cloud.Also disclosed is an overall system 60 comprising the indefinitely delayed system 40 and the arithmetic unit 50.Furthermore, a computer program is disclosed which is designed to execute the computer-implemented method 100 for robust adaptive control of an indeterminate (i.e. stochastic) time-delayed system. The computer program can be present, for example, in interpretable or compiled form. It can be loaded (also in parts) into the RAM of a computer for execution, for example, as a bit or byte sequence.Also disclosed is a computer readable medium or signal storing and / or containing the computer program. The medium may include, for example, one of RAM, ROM, EPROM, HDD, SSD,... on / in which the signal is stored.An example method 200 for determining 120 the one or more transition probabilities of transitions between time delays per sub-time series is disclosed below.The old method can comprise receiving 210, in particular via a first communication interface (e.g. the computing unit 50), a time delay determined, in particular measured, for a first point in time for the indefinitely delayed system 40, and a time delay determined, in particular measured, for a second point in time for the indefinitely delayed system 40. The second time may be later than the first time. The time delay determined for the first time for the indefinitely delayed system 40 and the time delay determined for the second time for the indefinitely delayed system 40 may be discrete, i.e., classified into a predetermined number of classes, for example.The method 200 may further include determining 220 one or more transition probabilities for transitions from time delays at the first time to time delays at the second time based on the time delay determined for the first time for the indeterminate time-delayed system 40 and the time delay determined for the second time for the indeterminate time-delayed system 40. Determining 220 the one or more transition probabilities for the transitions from the time delays at the first time to the time delays at the second time may also be based on one or more time delays determined earlier than the first time for the indefinitely time delayed system 40.The method 200 may further comprise determining 230 one or more ambiguity sets around the one or more determined 220 transition probabilities. This step may be unnecessary if the one or more ambiguity sets are determined 140 only by the determined 130 one or more transition probabilities in the method 100. On the other hand, one or more ambiguity sets around the determined 120 transition probabilities can also be determined per sub-time series.The indeterminate time-delayed system 40 may be initially modeled as a linear continuous time system having an indeterminate time delay τ ∈R >0 and a known sampling time.DELTA.∈R >0 as exemplified by the following equations:Here, x is a state vector, y is a measurable output vector and u is an input vector. By discretized the time, for example, according to Euler integration, the first two equations can be expressed as follows:Here, the continuous time delay τ has been translated into a discrete-time Markov chain θ(k) ∈{0,..., M}. Here, therefore, each Markov state (also: Markov mode) i ∈{0,..., M} represents an integer time delay Δt k= iΔ, wherein M E N denotes the maximum time delay which is of interest per application.The equation system can be transcribed as in FIG. 7 c, wherein ξ is an extended state vector, wherein ξ k in turn, this state vector is k= kΔ (or short: k) at discrete time t. σ̂ ij is the Kronecker delta. In addition, l are unit matrices (having appropriate dimensions), respectively. This illustration is a Markov Jump Linear System (MJLS). It results from the combination of the indefinitely delayed system 40 with the Markov state θ(k). It extends the technical system 20 to be regulated with an imperfect communication network 30 that causes up to M discrete time delays.Points in time and / or possible time delays can therefore be discrete. The one or more transition probabilities for transitions from time delays at the first time to time delays at the second time may be arranged in a Markov transition matrix (see, e.g., FIG. 7 b). The Markov chain can be described by the Markov transition matrix.The receiving 210, in particular via the first communication interface, the time delay determined, in particular measured, for the first point in time for the indefinitely delayed system 40 and a time delay determined, in particular measured, for a second point in time for the indefinitely delayed system 40 can comprise receiving, in particular via the first communication interface, information from which a time delay for a first point in time for the indefinitely delayed system 40 can be determined and information from which a time delay for a second point in time for the indefinitely delayed system 40 can be determined. This information may include one or more time stamps passed by the controller 10, e.g. for calculating the integer delay. Such a calculation can be carried out, for example, in step 51 of FIG. 3.The first time may be, for example, t k= kΔ, where Δ is a sampling time and k ∈N 0 is zero or a positive integer. This time can be identified with the integer k, i.e. one can speak of the kth time, for example. The second time point can be, for example, the time point t k-1= ( k+1) Δ=t k+ Δ immediately following the first time point. This point in time can in turn be identified with the integer k+1. Here, k can be incremented further over sub-time series or reset again to k=0 after a sub-time series.The time delay determined for the first time for the indefinitely delayed system 40 and the time delay determined for the second time for the indefinitely delayed system 40 may already be received 210 discretely or may be discretized after the receiving 210 (so that they also become discrete). The discrete time delays may be expressed in multiples of the sampling time Δ, for example. For example, the following discrete time delays Δt k= iΔmay be considered for all i ∈ {0,..., M} where M is a predetermined positive integer. Such discrete time delays can be seen as (discrete) Markov states, wherein these can be identified with the integers i ∈ {0,..., M}. Discrete time delays may be determined, for example, by the formula in FIG. 7 d, where the bracket limited down may mean rounding down to the next integer. t k and t k+1 here denote (as opposed to above) the time stamps from which the difference the time delay may be calculated. N ∈ N here denotes the sample size over all points in time. In the denominator, the h is to be understood as the sampling time Δ and not as the number of sub-time series.Determining 220 one or more transition probabilities for transitions from time delays at the first time to time delays at the second time may be determining one or more transition probabilities for transitions from discrete time delays at the first time to discrete time delays at the second time. For example, transition probabilities p ij= P(θ(k+1)=j| θ(k)=i) for transitions from a Markov state i to a (generally different) Markov state i for i,j ∈{0,...,M} may be determined here. Such transition probabilities may be components in the Markov transition matrix. Through step 220 and repeating method 200, i.e., through further steps 220 for the next points in time, the one or more transition probabilities, in particular, e.g., the Markov transition matrix, are empirically estimated based on the determined and / or measured time delays of the indefinitely delayed system 40. An estimate for the transition probability p ij is referred to below as P̂ ij. Thus, for each sub-time series, a P̂ l for I=1,..., h can be calculated.An ambiguity set may be a set of transition probabilities, with the 220 transition probability determined in method 200 (by construction) included in that set. For example, transition probabilities for a respective transition from a fixed Markov state i for iE {0,...,M} to all possible Markov states j for j ∈ {0,...,M} can be determined. Generally (for M>0), these determined 220 transition probabilities form a point in a higher-dimensional space. In FIG. 8, at three times of the method 200 (i.e., at three times within a sub-time series), such example points are shown in the form of a cross. The ambiguity set here may be a non-punctiform subset of this higher-dimensional space, which comprises the point. In FIG. 8, starting from an exemplary set of ambiguity that forms a simplex in the space of the transition probabilities p i1, p i2 and p i3 further increasingly smaller sets of ambiguity are drawn, wherein all three exemplary sets of ambiguity each comprise the point identified by the cross (and a point for the true transition probabilities identified by a solid circle). The ambiguity quantity is a measure of the uncertainty of the associated transition probabilities.Determining 230 the one or more ambiguity sets around the one or more transition probabilities is not required for the method 100. However, in the method 100, the determination 140 of the one or more ambiguity sets around the one or more determined 130 transition probabilities can be carried out according to similar rules. Differences can consist, for example, only in that in the method, ambiguity quantities are determined 140 by weighted 131 transition probabilities, namely for the moving filter, i.e. not over all previous observations, but only for respective hn observations.The determination 220 of the one or more transition probabilities can comprise in particular determining one or more transition probabilities for a transition in each case from the time delay determined for the first time to a possible time delay at the second time.Determining 230 the one or more ambiguity sets about the one or more determined 220 transition probabilities may comprise determining an ambiguity set about the one or more transition probabilities for a respective transition from the time delay determined for the first time to a possible time delay at the second time.Determining 220 the one or more transition probabilities may include determining a transition probability for a transition from the time delay determined for the first time to the time delay determined for the second time.Determining 220 the one or more transition probabilities (per sub-time series) can comprise the following steps or correspond to these if the transitions originate from the time delay determined for the first time in the indefinitely delayed system 40:inverting a total number of the time delay determined for the first time for the respective sub-time series in the indefinitely delayed system 40, resulting in an inverse sample size;scaling one or more transition probabilities for transitions from time delays at a time earlier than the first time to time delays at the first time with a scaling factor, the scaling factor being one minus the inverse sample size, resulting in one or more first transition probabilities;scaling the transition probability for a transition from the time delay determined for the first time to the time delay determined for the second time with the inverse sample size, resulting in a second transition probability;optionally, adding the one or more first transition probabilities and the second transition probability.The total number of time delays Δ k= i (i.e. Δt k= iΔ) determined for the first time k (i.e. t k= kh) for the respective sub-time series in the indefinitely time-delayed system 40 can be referred to as γ i-1( k) ∈ N, and the inverse sample size can then be referred to as γ i( k). This intermediate variable per sub-time series can be, but does not have to be, indexed by the index l=1,..., h. The total number of time delays determined for the first time k (i.e. t k= kΔ) in the indefinitely delayed system 40 is incremented by one if the time delay determined for the second time k+1 (i.e. t k+1= ( k+1)Δ) in the indefinitely delayed system 40 is also Δ k+1= i(i.e. Δt k+1= iΔ), the total number γ i-1( k+1) of time delays determined for the second time in the indefinitely delayed system 40 resulting. The inverse sample size can thus be determined recursively as follows in this case,Otherwise it is maintained unchanged,If Δ k= i(i.e. Δt k= iΔ) is the time delay determined for the first time k (i.e. t k= kΔ) in the indefinitely delayed system 40, the transition probabilities P̂ i( k+1) (per sub-time series) can thus be estimated recursively as follows for a transition in each case from the time delay determined for the first time to a possible time delay at the second time k+1 (i.e. t k+1= ( k+1) h) (equation 1), where P̂ i{ / ) and %0020̂P̂ i( k+1) are each the i-th series of a Markov transition matrix at the times k and k+1, respectively, and eΔk+1 ∈ R M+1 is a standard base vector with components (eΔk+1) = δ (k+1)i, i.e. the components are all zero except for the (k+1)-th component, which is one.The determination 220 of the one or more transition probabilities may, on the other hand, comprise the following step or correspond thereto if the transitions do not originate from the time delay determined for the first point in time in the indefinitely delayed system 40:maintaining one or more transition probabilities for transitions from time delays at the earlier than the first time to time delays at the first time.If, on the other hand, Δ k ≠ i(i.e. Δt k ≠ iΔ) is the time delay determined for the first time k (i.e. t k= kΔ) in the indefinitely delayed system 40, the transition probabilities P̂ i( k+1) (per sub-time series) can therefore be estimated recursively as follows for a transition in each case from the time delay determined for the first time to a possible time delay at the second time k+1 (i.e. t k+1= ( k+1)Δ) (equation 2),Since either Δ k= i or Δ k ≠ i applies at each point in time, either equation 1 or equation 2 is used for an i at each point in time. This can be performed for all Markov states i. In this case, exactly one row of the Markov transition matrix may be adjusted in a non-trivial manner (i.e., according to Equation 1), while the other rows of the Markov transition matrix may be retained, i.e., not adjusted or adjusted in a trivial manner (i.e., according to Equation 2). As a result, a Markov transition matrix can be determined 220 for each sub-time series.The determination 220 of the one or more transition probabilities can thus be recursive within a respective sub-time series and can also be based on one or more initial transition probabilities P̂ i(0) for all Markov states i for transitions of time delays at an initial point in time, e.g. t 0= 0 Δ=0 within the sub-time series in an operation of the time-delayed system to time delays at the point in time t 1= 1 Δ=Δ immediately following the initial point in time. The one or more initial transition probabilities P̂ i(0) for all Markov states i may be partially zero. They can correspond, for example, to the Markov transition matrix P shown by way of example in FIG. 7 b, wherein their components are denoted by p ij for i,j ∈{0,1,..., M}. In this case, the upper triangular diagonal is zero except for the first secondary diagonal. This can, for example, specify that time delays can only increase incrementally by one and not abruptly.For each sub-time series, at least one transition probability for a respective transition from one time delay to another time delay at an initial point in time during operation of the time-delayed system 40 or for all points in time (within the sub-time series) can be zero. The transition probabilities P̂ il( k) for transitions from the Markov state i must be normalized to one at each time k, i.e. it applies,The transition probabilities for a transition from a time delay at the initial time t 0= 0 Δ=0 for example to a possible time delay at the time t 1= 1 Δ=Δ immediately following the initial time can be evenly distributed. For example, p O0= 1 / 2, p 01= 1 / 2, p 10, = 1 / 3, p 11= 1 / 3, p 12= 1 / 3, etc. may apply.The determination 230 of the one or more ambiguity sets per sub-time series around the one or more determined 220 transition probabilities can also be effected recursively within the respective sub-time series and can additionally be based on the one or more initial transition probabilities for transitions from time delays at an initial point in time in an operation of the time-delayed system to time delays at the point in time immediately following the initial point in time. In method 100, this step 230 is not necessary.The one or more transition probabilities P̂ ij( k) for a transition from the time delay Δ k= i (i.e. Δt k= ih) determined for the first time k (i.e. t k= kh) to a possible time delay at the second time k+1 (i.e. t k+1= ( k+1) h) within a sub-time series may be a point in a finite-dimensional vector space. The ambiguity set for these transition probabilities may be comprised by a sphere around this point with a radius in a finite-dimensional norm (e.g., p norm with p=1 or p=∞, i.e., maximum norm), wherein the radius may be based on, e.g., a confidence level and / or a total number γ i-1( k) within the sub-time series (or its inverse, i.e., the inverse sample size γi(k)) of the time delay determined for the first time in the indefinitely delayed system 40. The radius may decrease at a constant confidence level and (within the sub-time series) an increasing total number of the time delay determined for the first time in the indefinitely delayed system. For example, the radius may be determined by a concentration imbalance such as a McDi inequality. For example, the ambiguity set A(P̂ i( k)) may be determined as follows, where an ith probability simplex is. Here, r i( β, γ i-1( k)) ∈( 0,2] is the radius for the Markov state i, which may depend on a user-defined confidence level β ∈( 0,1) and on the total number y i-1( k). The radius may be given in closed form, e.g., via concentration equations such as the McDi inequality. An ambiguity set A(P̂ i( k)) may refer to the transition probabilities for transitions from a Markov state i to a (generally different) Markov state j. In this respect, the ambiguity set A(P̂ i( k)) can be referred to as the ith ambiguity set or as the state-dependent ambiguity set.FIG. 8 illustrates how learning by repeating method 200 within the respective sub-time series influences the empirically determined transition probabilities and the associated ambiguity quantity. Starting from the simplex, which comprises both the point identified by the cross for the estimated transition probabilities and the true transition probabilities identified by a filled circle, at later points in time within the sub-time series, decreasing ambiguity quantity is defined, which comprise both the cross and the filled circle and, after a sufficiently large number of points in time, converges to a single tone (to the filled circle, which then coincides with the cross). If the regulator 10 is designed with respect to such a smaller amount of ambiguity, too conservative a design is avoided. This effect is also achieved by the weighting for more recent sub-time series.As shown by way of example in FIG. 3, the methods 100 and / or 200 can be executed in a computing unit 50 outside the indefinitely delayed system 40, in particular in a cloud.FIG. 3 shows an exemplary communication between an indefinitely delayed system 40 and the computing unit 50. the indefinitely delayed system 40 comprises a controller 10 and a technical system 20 to be controlled. The indefinitely delayed system 40 here also comprises the communication network 30 via which the controller 10 communicates at least partially with the technical system 20 to be controlled. The step 210 of the method 200 here may comprise, for example, the step 51 (calculating integer delays), i.e. for example the calculation of the discrete time delays according to the formula in FIG. 7 d. In step 52, the update information to the controller 10 may be determined, which is sent 151 to the controller and is directed to adjusting 150 the controller 10.In particular, the computing unit 50 can also be designed to execute the computer-implemented method 200. The method 200 (and its repetitions per sub-time series) may be part of the method 100.For illustration, the example with the group start of three vehicles with unreliable communication via a network from FIG. 1 is explained once again. The leading vehicle is uncontrolled (with respect to the distance control) and travels, for example, at a constant speed v 0, followed by two controlled (with respect to the respective distance control) vehicles having control inputs a 1 and a 2. The aim may be to regulate the distance error e 1 and e 2 to the origin, i.e. the dashed vertical lines in FIG. 1 indicate that the necessary safety distance d safe is complied with. It can be assumed here that the vehicles are controlled by their accelerations a 1 and a 2 while the state is only partially measurable, i.e. an observer is required to estimate the state vector. The overall state, the output and input vectors are given, for example, in Figure 7a, where v 1 and v 2 are the speeds of the second and third vehicles, respectively (the leading vehicle here is the first vehicle).FIG. 9 a illustrates exemplary control processes for the second vehicle during the platooning group start at three different exemplary times in the method, namely for sample sizes N s= 1, N s= 102 and N s= 103, and associated 3σ confidence intervals. FIG. 9 b illustrates exemplary control processes for the third vehicle at the same platooning group start at the three different exemplary times in the method and associated 3σ confidence intervals. Here, 1-β=0.9 was selected as confidence for the ambiguity quantity. It can be seen that each controller-observer pair stabilizes the control system even when there is no information about structural disturbances (i.e. the empirical Markov transition matrix reflects an even distribution). As the sample size is increased, the state variance, i.e., the ambiguity set (represented as enveloping curves), decreases, reflecting an increase in performance. In this way, it is possible to converge reliably to the optimum control gain without thereby giving up stability guarantees.As shown in FIGS. 9 a- b, transition probabilities and, if appropriate, ambiguity sets can therefore be determined for a sub-time series. The ability to determine the ambiguity quantities within a sub-time series at the same time shows that ambiguity quantities can also be determined 140 for the moving filter on the basis of observations.Furthermore, for example, the following expansion is conceivable:A special case of time-variable time delay distributions is a discrete changeover. In this case, event driven re-initialization may be incorporated which resets the proposed learning method if the new data does not fit into the previously learned Markov transition matrix. A possible triggering condition can be derived from the total variation distance between two distributions.

Claims

Computer-implemented method (100) for robust adaptive control of an indeterminate time-delayed system, in particular wherein the indeterminate time-delayed system (40) comprises a controller (10) and a technical system (20) to be controlled, wherein the controller and the technical system to be controlled communicate at least partially via a communication network (30), the method (100) comprising: - receiving (110) sub-time series (1, 2, 3) a time series of time delays determined, in particular measured, for the indeterminate time-delayed system (40), wherein the sub-time series form a partition of the time series or a further sub-time series thereof and each sub-time series comprises at least two time delays; determining (120), per sub-time series, one or more transition probabilities for transitions of time delays based on the time delays of the respective sub-time series, wherein one or more sub-time series-related transition probabilities result in each case; determining (130) one or more transition probabilities for transitions of time delays based on the sub-time series-related transition probabilities.The method (100) of claim 1, comprising: - determining (140) one or more ambiguity sets around the one or more determined (130) transition probabilities based on the time delays of the sub-time series.Method (100) according to claim 1 or 2, comprising: - adjusting (150) the controller (10) based on the determined (130) one or more transition probabilities and / or, if dependent on claim 2, on the determined (140) one or more ambiguity sets; optionally wherein the adjusting (150) the controller (10) comprises: - sending (151), in particular via the first communication interface, an update information to the indefinitely delayed system (40), such that the controller (10) is adjusted.The method (100) of any preceding claim, wherein the method (100) is performed during operation of the indefinitely delayed system (40); optionally wherein, when dependent on claim 3, the controller (10) is adjusted (150) during operation of the indefinitely delayed system (40).Method (100) according to one of the preceding claims, wherein the method (100) is carried out in a computing unit (50) outside the indefinitely delayed system (40), in particular in a cloud.The method (100) according to claim 1, wherein the receiving (110) the sub-time series comprises: - receiving (111), in particular via a first communication interface, the time series or the further sub-time series; - partitioning (112) the time series or the further sub-time series into the sub-time series.The method (100) of any preceding claim, wherein the sub-time series comprises the same number of time delays.The method (100) according to any one of the preceding claims, wherein the determining (130) of the one or more transition probabilities for transitions of time delays is performed at different times and is respectively based on the same number of sub-time series.The method (100) of claim 8 when dependent on claim 7, wherein the sub-time series are successively received (110) and processed first-in-first-out.The method (100) of any preceding claim, wherein determining (130) the one or more transition probabilities for transitions of time delays comprises weights (131) of the sub-time series related transition probabilities, in particular with weights that are constant per sub-time series.The method (100) according to any one of the preceding claims, wherein the time delays are discrete, in particular wherein the one or more sub-time series related transition probabilities and / or the one or more transition probabilities are each arranged in a Markov transition matrix.Controller (10) configured to control a technical system (20), wherein the controller (10) and the technical system (20) to be controlled communicate at least partially via a communication network (30), the controller (10) comprising: - a second communication interface for sending information to a computing unit (50), in particular to a computing unit (50) outside an indeterminate time-delayed system (40) comprising the controller (10), from which time delays for the indeterminate time-delayed system (40), in particular sub-time series of a time series of time delays or a further sub-time series thereof, can be determined; wherein the second communication interface is further configured to receive update information from the computing unit (50), wherein the controller (10) is configured to be adjusted (150) by the update information.An indefinitely delayed system (40) comprising: - the controller (10) of claim 12; - a technical system (20) to be controlled; wherein the controller (10) and the technical system (20) to be controlled communicate at least partially over a communication network (30).Computing unit (50) designed to carry out the computer-implemented method (100) for robust adaptive control of an indefinitely delayed system (40) according to any one of the preceding claims 1 to 11, the computing unit (50) comprising: - a first communication interface.An overall system (60) comprising: - the indefinitely delayed system (40) of claim 13; and - the computing unit (50) of claim 14.

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