DE102024204002A1 - ROBUST ADAPTIVE CONTROL FOR INDEFINITE TIME-DELAYED CONTROL SYSTEMS - Google Patents

Abstract

A computer-implemented method for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system is disclosed, in particular wherein the indefinitely time-delayed system comprises a controller, optionally an observer, and a controlled technical system, wherein the controller and/or the observer communicate at least partially with the controlled technical system via a communication network, the method comprising: receiving a time delay determined for a first time point for the indefinitely time-delayed system and a time delay determined for a second time point for the indefinitely time-delayed system, wherein the second time point is later than the first time point; determining one or more transition probabilities for transitions from time delays at the first time point to time delays at the second time point based on the time delay determined for the first time point for the indefinitely time-delayed system and the time delay determined for the second time point for the indefinitely time-delayed system; and determining one or more ambiguity sets around the one or more determined transition probabilities.

Description

State of the art

In recent decades, the rapid development of computing power and communication technology has significantly influenced the infrastructure of today's control systems. While conventional control systems consist of sensors, controllers, and actuators connected by wires, a networked control system (NCS) replaces at least some or all of these end-to-end connections with a communication network. NCS can be based on wireless connections, but this is not mandatory. Furthermore, centralized and/or zone-based E/E architectures, for example, are also encompassed by NCS.

• - Zufällige Übertragungsverzögerung in den Kanälen Sensor-zu-Steuerung (S2C) und Steuerung-zu-Aktor (C2A), z. B. aufgrund großer Entfernungen zwischen zwei Knoten und/oder zeitlich variierendem Verkehr im Netz.
• - Zufällige Paketverluste, z. B. aufgrund von Netzüberlastungen.

The advantages of an NCS are numerous; for example, it reduces cabling and/or maintenance costs, while the communication topology allows for greater flexibility with regard to distributed systems. However, the use of a communication network can also lead to network-related problems such as:

• - Random transmission delay in the sensor-to-controller (S2C) and controller-to-actuator (C2A) channels, e.g. due to large distances between two nodes and/or time-varying traffic in the network.
• - Random packet loss, e.g. due to network congestion.

Such a control system can be described as an indefinitely time-delayed control system.

Both problems, if left unaddressed, can degrade the performance of the control system and even lead to instability of the control loop. Similarly, in this case, an observer within the control system may malfunction.

In the past, various control approaches have been developed to deal with random time delays and/or packet loss, ranging from deterministic to stochastic approaches. While deterministic (robust) approaches rely on describing the worst-case uncertainty, stochastic approaches use additional information in the form of statistics and/or probability distributions to achieve less conservative control. However, the stochastic approach typically requires complete information about the statistics and/or probability distribution of the time delay in a given control system, which unfortunately makes it rarely applicable in practice. Conversely, if empirical estimates of the statistics and/or probability distribution are used, no guarantees can be given for the control system. Such guarantees, however, are precisely what is needed to release the control system.

Therefore, a problem to be solved underlying the disclosure can be seen, for example, as providing a method that—despite initial lack of knowledge of the statistics and/or probability distribution of time delays in the control system—enables a not overly conservative, yet robust, control system for an indefinitely time-delayed control system. Alternatively or additionally, a problem to be solved can also be seen as providing a method that—despite initial lack of knowledge of the statistics and/or probability distribution of time delays in the control system—enables a not overly conservative, yet robust, design of an observer in an indefinitely time-delayed control system. A problem to be solved can also be seen as providing a control system and an observer for an indefinitely time-delayed system that are, overall, not overly conservative, yet robust.

Disclosure of the invention

A first general aspect of the present disclosure relates to a computer-implemented method for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system. The indefinitely time-delayed system can comprise a controller and optionally an observer, as well as a technical system to be controlled, wherein the controller and/or the observer communicate with the technical system to be controlled, at least partially, via a communication network. The method includes receiving, in particular via a first communication interface, a time delay determined, in particular measured, for a first time point for the indefinitely time-delayed system, and a time delay determined, in particular measured, for a second time point for the indefinitely time-delayed system, wherein the second time point is later than the first time point. The time delay determined for the first time point for the indefinitely time-delayed system and the time delay determined for the second time point for the indefinitely time-delayed system can be discrete. The procedure further includes determining one or more transition probabilities for transitions from time delays at the first time to time delays at the second time based on The procedure involves determining the time delay for the first time point for the indefinitely time-delayed system and the time delay for the second time point for the indefinitely time-delayed system. Additionally, determining one or more transition probabilities can be based on one or more time delays determined for earlier times than the first time point for the indefinitely time-delayed system. The procedure further includes determining one or more ambiguity sets around the one or more determined transition probabilities.

A second general aspect of the present disclosure relates to a controller and/or observer configured to control a technical system, wherein the controller and/or observer communicate with the technical system to be controlled, at least partially, via a communication network. The controller and/or observer comprise a second communication interface for sending information to a computing unit, in particular a computing unit outside an indefinitely time-delayed system comprising the controller and/or observer, from which a time delay for a first time point for the indefinitely time-delayed system and a time delay for a second time point for the indefinitely time-delayed system can be determined, the second time point being later than the first time point. The second communication interface can further be configured to receive an update from the computing unit, wherein the update comprises a modified controller and/or an modified observer. The controller and/or observer can be configured to implement the update.

A third general aspect of the present disclosure relates to a computing unit (i.e., a computer system) designed to execute the computer-implemented method for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system according to the first general aspect (or an embodiment thereof). The computing unit includes a first communication interface. A fourth general aspect of the present disclosure relates to a computer program designed to execute the computer-implemented method for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system according to the first general aspect (or an embodiment thereof).

A fifth general aspect of the present disclosure relates to a computer-readable medium or signal that stores and/or contains the computer program according to the fourth general aspect (or an embodiment thereof).

The method proposed here, according to the first general aspect (or an embodiment thereof), allows for the successive adaptation and eventual definition of a robust adaptive controller and/or a robust adaptive observer of an indeterminate time-delayed system. Indeterminate time delays can arise from the communication network between the controller/observer (hereinafter also referred to as stochastic controllers) and the controlled technical system. The defined controller is robust because it can handle a multitude of probability distributions of the time delays. At the same time, it is not overly conservative, as it is not designed to accommodate all theoretically possible time delays (and their sequences) equally, but rather those that are statistically relevant in the application. The defined controller and/or the defined observer can then be considered enabled or further investigated in an enablement process and subsequently enabled, if necessary. Enabling the controller and/or the observer can be a necessary step for enabling the controlled technical system.

The main problem in designing stochastic controllers in practice is the need to know the true statistics and/or probability distributions of time delays at the time of the control system design. However, these are almost never available in reality and require approximations, e.g., in the form of an empirical distribution, which unfortunately is subject to sampling-induced errors resulting from the sample mean approximation. To synthesize guaranteed stabilizing controllers despite these sampling errors, the invention employs a distributionally robust approach. Specifically, an ambiguity set—that is, a set of uncertainty in the space of probability distributions—is constructed at each point in time around an empirical estimate that contains the true probability distribution with high confidence. This ensures that the resulting controller and/or observer—that is, the indeterminate time-delayed control system—is robust against an entire family of probability distributions at any given time. In a closed-loop control system, data (especially actual time delays, i.e., time delay data) are continuously collected by the control system. The empirical distribution and the ambiguity set are then continuously adapted from this data, resulting in progressively improved controllers and/or observations. These can be synthesized. This learning algorithm guarantees that the ambiguity set shrinks to a point (also called a singleton) containing only the true distribution as the sample size approaches infinity. In practice, it suffices to have taken a sufficiently large number of samples.

The control strategy proposed in the procedure according to the first general aspect (or an embodiment thereof) utilizes a general modeling framework for NCS—referred to as the Markov Jump Linear System (MJLS)—and combines it with a learning-based, distributionally robust controller and/or observer design. In contrast to conventional methods, time-delay data, i.e., in the form of measurements, can be used to construct, for each discrete time step, a Markov transition matrix and an ambiguity set containing the true probability distribution with a user-defined confidence level. By continuously collecting further time-delay data, the Markov transition matrix can be recursively updated and the ambiguity set reduced while maintaining the same user-defined confidence level. This adaptation of the Markov transition matrix and the ambiguity set represents a learning-based controller and/or observer design. This ensures that an overly conservative control design is reduced by successively decreasing the ambiguity set, while still maintaining probabilistic stability guarantees (due to the confidence level).

Furthermore, a distinction can be made between the control loop itself, which is closed by a network, and the learning algorithm. The latter can be understood as a microservice that may reside in a different network (e.g., in a cloud), with best-effort requirements for availability and reliability.

• Ein Regelsystem wird mit einem (zunächst) willkürlich stabilisierenden lokalen Regler betrieben, um Zeitverzögerungsdaten zu erzeugen. Dabei handelt es sich um einen zeitkritischen Signalweg, der über ein Netzwerk geschlossen wird. Der lokale Regler überträgt die Zeitstempel kontinuierlich in den lernenden Algorithmus, der sich z.B. in einer Cloud mit nicht zeitkritischer Verarbeitungszeit befindet. Anschließend können die Zeitstempel in eine Menge von ganzzahligen Zeitverzögerungen umgewandelt werden, die bei jedem Eintreffen eines neuen Zeitstempels aktualisiert wird. Die Markov-Übergangsmatrix und die Ambiguitätsmenge können dann zum Beispiel auf der Grundlage der neuen ganzzahligen Zeitverzögerungsinformationen aktualisiert werden. Schließlich wird der Regler auf der Grundlage der neuen Markov-Übergangsmatrix und der Ambiguitätsmenge aktualisiert und wieder in das Regelsystem eingespeist. Nach der Aktualisierung ist die Regelungsleistung gleich oder besser als die des vorherigen lokalen Reglers.

An exemplary embodiment of the method proposed in this disclosure can be summarized as follows:

• A control system is operated with a (initially) arbitrarily stabilizing local controller to generate time delay data. This involves a time-critical signal path closed over a network. The local controller continuously transmits the timestamps to the learning algorithm, which resides, for example, in a cloud with non-time-critical processing time. The timestamps can then be converted into a set of integer time delays, which is updated with each new timestamp. The Markov transition matrix and the ambiguity set can then be updated, for example, based on the new integer time delay information. Finally, the controller is updated based on the new Markov transition matrix and the ambiguity set and fed back into the control system. After the update, the control performance is equal to or better than that of the previous local controller.

Brief description of the characters

• 1 illustrates an exemplary, indefinitely time-delayed system comprising three vehicles performing platooning and, in particular, a group start.
• 2 schematically illustrates exemplary embodiments of a computer-implemented method for robust adaptive control and/or a robust adaptive observer of an indeterminate time-delayed system.
• 3 illustrates an exemplary embodiment of a complete system comprising an indefinitely time-delayed system and a computing unit designed to execute the computer-implemented method for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system.
• 4a illustrates exemplary vectors for the state, the measurable output and the input of a controller.
• 4b illustrates an exemplary Markov transition matrix with transition probabilities for transitions between delays.
• 4c illustrates an exemplary Markov Jump Linear System with extended state ξ for the indefinitely time-delayed system.
• 4d illustrates an exemplary set of integer delays that are determined (successively) for the indefinitely time-delayed system.
• 5 illustrates exemplary successive transition probabilities P _i: = (P _i1 ,P _i2 ,P _i3 ) and corresponding small decreasing ambiguity sets.
• 6a illustrates exemplary control processes for a second vehicle during the platooning group start at three different times in the procedure, as well as the corresponding 3σ confidence intervals.
• 6b illustrates exemplary control processes for a third vehicle during a platooning group start to three different Time points in the procedure and their corresponding 3σ confidence intervals.

Detailed description

First, a computer-implemented method 100 for robust adaptive control and/or a robust adaptive observer of an indeterminately (i.e., stochastically) time-delayed system 40 is disclosed. The indeterminately time-delayed system 40 can comprise a controller 10 (also referred to as a local controller) and a controlled technical system 20. Optionally, the indeterminately time-delayed system 40 can also include an observer 11 (also referred to as a local observer). The controller 10 and/or the observer 11 can communicate, at least partially, with the controlled technical system 20 via a communication network 30.

The method 100 proposed in this disclosure can therefore be directed to a robust adaptive control and/or a robust adaptive observer of an indeterminate (i.e. stochastically) time-delayed system.

As exemplified in 1 As shown, the time-delayed system 40 can comprise a multitude of vehicles, which together form the technical system 20 to be controlled and, for example, form a convoy (i.e., perform platooning) and, in particular, carry out a group start. The multitude of vehicles comprises at least two vehicles or, as shown in 1 Three vehicles are shown as an example. All vehicles except the first (leading) vehicle are controlled via the communication network 30 by a controller 10 and/or an observer 11 outside the controlled technical system 20. In this case, it is advantageous for the communication network 30 to be a wireless network. Here, _v0 denotes the speed of the first vehicle, _v1 the speed of the second vehicle immediately following the first, and _v2 the speed of the third vehicle immediately following the second. Furthermore, _dsafe denotes the required safety distance, which should ideally be maintained between successive vehicles or at least not be undercut. _e1 denotes a distance error from the ideal safety distance _dsafe between the first and second vehicles, and _e2 denotes a distance error from the ideal safety distance _dsafe between the second and third vehicles. However, the time-delayed system 40 is not limited to this application. Another use case is, for example, the control and/or coordination of guided automated vehicles via a local network (local edge, 5G network, etc.) within a limited area, such as logistics centers or production facilities. Yet another use case is the (partial) outsourcing of robot arm control algorithms to local networks, for example, for manufacturing systems. A further use case is the lateral and/or longitudinal control of vehicles at traffic intersections or other control zones, which is outsourced to a local edge or cloud system (e.g., a roadside unit).

For example, in 2 Schematically illustrated, the procedure comprises 100 receptions 110, in particular via a first communication interface (e.g., of the computing unit 50), of a time delay determined, in particular measured, for a first time point in time for the indefinitely time-delayed system 40, and a time delay determined, in particular measured, for a second time point in time for the indefinitely time-delayed system 40. The second time point in time can be later than the first time point in time. The time delay determined for the first time point in time for the indefinitely time-delayed system 40 and the time delay determined for the second time point in time for the indefinitely time-delayed system 40 can be discrete, i.e., classified into a predefined number of classes.

As also, for example, in 2 Schematically illustrated, the procedure 100 further comprises determining 120 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 indefinitely time-delayed system 40 and the time delay determined for the second time for the indefinitely time-delayed system 40. Determining 120 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 can also be based on one or more time delays determined for the indefinitely time-delayed system 40 at times earlier than the first time.

As also, for example, in 2 Schematically illustrated, the procedure 100 further includes determining 130 one or more ambiguity sets around the one or more determined 120 transition probabilities.

The indefinitely time-delayed system 40 can initially be modeled as a linear time-continuous system with indefinite time delay τ ∈ ℝ _≥0 and known sampling time h ∈ ℝ _>0 , e.g. by the following equations: $$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t-\tau \right)\forall t>\mathrm{0,}x\left(0\right)=x_0$$ $$y\left(t\right)=Cx\left(t\right)$$ $$u\left(t\right)=u_k\forall t\in \left[t_k,t_k+h\right],t_k=kh,h\in {\mathbb{N}}_0$$

Here, x is a state vector, y a measurable output vector, and u an input vector. By discretizing time, e.g., using Euler integration, the first two equations can be expressed as follows: $$x_{k+1}=A_d{x}_k+B_d{u}_{k-\theta \left(k\right)}$$ $$y_k=C{x}_k$$

Here, the continuous time delay τ was translated into a discrete-time Markov chain θ(k) ∈ {0,...,M}. Thus, each Markov state (also: Markov mode) i ∈ {0,...,M} represents an integer time delay Δt _k = ih, where ME ℕ denotes the maximum time delay of interest in each application.

The system of equations can be solved as in 4c can be rewritten where ξ is an extended state vector, and ξ_{_k} is this state vector at discrete time t _{_k} = kh (or simply: k). δ_{_ij} is the Kronecker delta. Furthermore, I_i are identity matrices with appropriate dimensions. This representation is a Markov Jump Linear System (MJLS). It results from the combination of the indeterminate time-delayed system 40 with the Markov state θ(k). It extends the controlled technical system 20 with an imperfect communication network 30, which causes discrete time delays of up to M.

Time points and/or possible time delays can therefore be discrete. The one or more transition probabilities for transitions from time delays at the first time point to time delays at the second time point are represented in a Markov transition matrix (see, for example, [reference]). 4b) The Markov transition matrix can be used to describe the Markov chain.

Receiving 110, in particular via the first communication interface, the time delay determined, in particular measured, for the first time point for the indefinitely time-delayed system 40, and a time delay determined, in particular measured, for a second time point for the indefinitely time-delayed system 40, can include receiving, in particular via the first communication interface, information from which a time delay for a first time point for the indefinitely time-delayed system 40 can be determined, and information from which a time delay for a second time point for the indefinitely time-delayed system 40 can be determined. As, for example, in 3 In schematic terms, this information can include one or more timestamps that are passed from controller 10 and/or observer 11 to step 110 (e.g., to calculate integer delays).

The first time point can be, for example, t _{_k} = kh, where h is a sampling time and k ∈ ℕ_₀ is zero or a positive integer. This time point can be identified with the integer k; that is, one can speak, for example, of the k-th time point. The second time point can be, for example, the time point immediately following the first time point, t_{_k+1} = (k + 1)_h = t_{_k +h}. This time point can again be identified with the integer k + 1.

The time delay determined for the first time point for the indefinitely time-delayed system 40 and the time delay determined for the second time point for the indefinitely time-delayed system 40 can already be received discretely or discretized after receipt (so that they also become discrete). The discrete time delays can, for example, be expressed in multiples of the sampling time h. For example, the following discrete time delays Δt _k = ih can 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, which can be identified with the integers i ∈ {0,...,M}. Discrete time delays can be expressed, for example, by the formula in 4d The values are determined, where the lower bounding bracket can mean rounding down to the nearest integer. Here, t _{_k} and t_{_k+1} (in contrast to above) denote the timestamps, from whose difference the time delay can be calculated. Here, N _{_S} ∈ ℕ denotes the sample size over all time points.

Determining one or more transition probabilities for transitions from time delays at the first time to time delays at the second time can be determined as a process for 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 j for i,j ∈ {0,...,M} can be determined. Such transition probabilities can be components in the Markov transition matrix. By step 120 and repeating procedure 100—that is, by further steps 120 for At the next time points, one or more transition probabilities, in particular, for example, the Markov transition matrix, are empirically estimated based on the determined and/or measured time delays of the indeterminately time-delayed system 40. An estimate for the transition probability p_{_ij} is subsequently referred to as P̂_{_ij} .

An ambiguity set can be a set of transition probabilities, where the transition probability determined in Procedure 100 (by construction) is contained in this set. For example, transition probabilities can be determined for each transition from a fixed Markov state i for i ∈ {0,...,M} to all possible Markov states j for j ∈ {0,...,M}. In general (for M > 0), these determined transition probabilities form a point in a higher-dimensional space. 5 At three points in time during the procedure, 100 such exemplary points in the form of a cross are shown. The ambiguity set can be a non-point-shaped subset of this higher-dimensional space that includes the point. In 5 Starting from an exemplary ambiguity set that forms a simplex in the space of transition probabilities p _{_i1}, p _{_i2}, and p_{_i3} , further progressively smaller ambiguity sets are drawn, with each of the three exemplary ambiguity sets encompassing the point marked by the cross. The ambiguity set is a measure of the uncertainty of the corresponding transition probabilities.

Determining 120 of the one or more transition probabilities may in particular include determining one or more transition probabilities for each transition from the time delay determined for the first time point to a possible time delay at the second time point.

Determining 130 of the one or more ambiguity sets around the one or more determined 120 transition probabilities can include determining an ambiguity set around the one or more transition probabilities for each transition from the time delay determined for the first time point to a possible time delay at the second time point.

Determining 120 of one or more transition probabilities may include determining a transition probability for a transition from the time delay determined for the first time point to the time delay determined for the second time point.

• - Invertieren einer Gesamtanzahl der für den ersten Zeitpunkt im unbestimmt zeitverzögerten System 40 ermittelten Zeitverzögerung, wobei eine inverse Samplegröße resultiert;
• - Skalieren einer oder mehrerer Übergangswahrscheinlichkeiten für Übergänge von Zeitverzögerungen zu einem früheren als den ersten Zeitpunkt zu Zeitverzögerungen zum ersten Zeitpunkt mit einem Skalierungsfaktor, wobei der Skalierungsfaktor Eins minus der inversen Samplegröße ist, wobei ein oder mehrere erste Übergangswahrscheinlichkeiten resultieren;
• - Skalieren der Übergangswahrscheinlichkeit für einen Übergang von der für den ersten Zeitpunkt ermittelten Zeitverzögerung zu der für den zweiten Zeitpunkt ermittelten Zeitverzögerung mit der inversen Samplegröße, wobei eine zweite Übergangswahrscheinlichkeit resultiert;
• - optional, Addieren der ein oder mehreren ersten Übergangswahrscheinlichkeiten und der zweiten Übergangswahrscheinlichkeit.

Determining 120 of one or more transition probabilities may include or correspond to the following steps if the transitions start from the time delay determined for the first time point in the indefinitely time-delayed system 40:

• - Inverting a total number of the time delays determined for the first time point in the indefinitely time-delayed system 40, resulting in an inverse sample size;
• - Scaling one or more transition probabilities for transitions from time delays to an earlier time than the first time to time delays to the first time by a scaling factor, where the scaling factor is 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 point to the time delay determined for the second time point using the inverse sample size, resulting in a second transition probability;
• - optional, adding the one or more first transition probabilities and the second transition probability.

The total number of time delays _Δk = i (i.e., _Δtk = ih) determined for the first time k (i.e., _tk = kh) in the indefinitely time-delayed system 40 can be denoted as γi _- ¹ (k) ∈ ℕ, and the inverse sample size then as _γi (k). The total number of time delays determined for the first time k (i.e., _tk = kh) in the indefinitely time-delayed system 40 is incremented by one if the time delay determined for the second time k + 1 (i.e., _tk+1 = (k + 1)h) in the indefinitely time-delayed system 40 is also Δk ₊₁ = i (i.e., _Δtk+1 = ih), resulting in the total number γi _- ¹ (k+1) of time delays determined for the second time in the indefinitely time-delayed system 40. The inverse sample size can therefore be determined recursively in this case as follows: $${\gamma }_i^{-1}\left(k+1\right)={\gamma }_i^{-1}\left(k\right)+\mathrm{1,}wenn{\Delta }_{k+1}=i.$$ Otherwise, it will remain unchanged. $${\gamma }_i^{-1}\left(k+1\right)={\gamma }_i^{-1}\left(k\right),wenn{\Delta }_{k+1}\ne i.$$ If _Δk = i (i.e., _Δtk = ih) is the time delay determined for the first time k (i.e., _tk = kh) in the indeterminate time-delayed system 40, then the transition probabilities P̂i _: (k + 1) for each transition from the time delay determined for the first time to a possible time can be determined. The delay at the second time point k + 1 (i.e., t _k+1 = (k + 1) h) can be estimated recursively as follows (Equation 1). $${\widehat{P}}_{i:}\left(k+1\right)=\left(1-{\gamma }_i\left(k\right)\right){\widehat{P}}_i\left(k\right)+{\gamma }_i\left(k\right)e_{{\Delta }_{k+1}},wenn{\Delta }_k=i,$$ where P̂ _i :(k) and P̂ _i: (k + 1) are respectively the i-th series of a Markov transition matrix at times k and k + 1 respectively, and e _Δk+1 ∈ ℝ ^M+1 is a standard basis vector with components (e _Δk+1 ) _i = δ _(k+1) i, i.e. the components are all zero except for the (k + 1)-th component, which is one.

• - Beibehalten einer oder mehrerer Übergangswahrscheinlichkeiten für Übergänge von Zeitverzögerungen zu dem früheren als den ersten Zeitpunkt zu Zeitverzögerungen zum ersten Zeitpunkt.

Ist dagegen Δ_k # i (d.h. Δt_k # i h) die für den ersten Zeitpunkt k (d.h. t_k = k h) im unbestimmt zeitverzögerten System 40 ermittelte Zeitverzögerung, können also die Übergangswahrscheinlichkeiten P̂_i:(k + 1) für jeweils einen Übergang von der für den ersten Zeitpunkt ermittelten Zeitverzögerung zu einer möglichen Zeitverzögerung an dem zweiten Zeitpunkt k + 1 (d.h. t_k+1 = (k + 1) h) rekursiv wie folgt geschätzt werden (Gleichung 2), $${\widehat{P}}_{i:}\left(k+1\right)={\widehat{P}}_{i:}\left(k\right),wenn{\Delta }_k\ne i.$$ Determining 120 of one or more transition probabilities may, on the other hand, include or correspond to the following step if the transitions do not start from the time delay determined for the first time point in the indefinitely time-delayed system 40:

• - Maintaining one or more transition probabilities for transitions from time delays to the earlier than the first time to time delays to the first time.

If, on the other hand, Δ _k # i (i.e. Δt _k # ih) is the time delay determined for the first time k (i.e. t _k = kh) in the indeterminate time-delayed system 40, then the transition probabilities P̂ _i: (k + 1) for each transition 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) can be estimated recursively as follows (Equation 2), $${\widehat{P}}_{i:}\left(k+1\right)={\widehat{P}}_{i:}\left(k\right),wenn{\Delta }_k\ne i.$$

Since at any given time either _Δk = i or _Δk ≤ i, at any given time either Equation 1 or Equation 2 applies to some i. This can be done for all Markov states i. In this case, exactly one row of the Markov transition matrix can be fitted in a non-trivial way (i.e., according to Equation 1), while the other rows of the Markov transition matrix are either left unfitted or fitted in a trivial way (i.e., according to Equation 2).

Determining the one or more transition probabilities can thus be done recursively and can also be based on one or more initial transition probabilities P̂ _i :(0) for all Markov states i for transitions from time delays at an initial time t ₀ = 0 h = 0 in an operation of the time-delayed system to time delays at the time t ₁ = 1 h = h immediately following the initial time. The one or more initial transition probabilities P̂ _i: (0) for all Markov states i can be partially zero. They can, for example, be the one in 4b The exemplary Markov transition matrix P shown corresponds to this, where its components are denoted by p_{_ij} for i,j ∈ {0,1,...,M}. In this case, the upper triangle diagonal is zero except for the first sub-diagonal. This allows, for example, the specification that time delays can only increase incrementally by one and not abruptly.

At least one transition probability for each transition from one time delay to another at an initial time point in the operation of the time-delayed system 40, or for all times, can be zero. The transition probabilities P̂ _i: (k) for transitions from the Markov state i must be normalized to one at every time k, i.e., $${\displaystyle {\sum }_{j=0}^M{\widehat{P}}_{ij}\left(k\right)=1.}$$

The transition probabilities for each transition from a time delay at the initial time t _₀ = 0 h = 0 to a possible time delay at the time immediately following the initial time t _₁ = 1 h = h can be uniformly distributed. For example, p ₀₀ = 1/2, p_₀₁ = 1/2, p_₁₀ = 1/3, p₁₁ = 1/3, p_₁₂ = 1/3, etc. _, can hold.

Determining 130 of the one or more ambiguity sets around the one or more determined 120 transition probabilities can also be done recursively and can also be based on the one or more initial transition probabilities for transitions from time delays at an initial time point in an operation of the time-delayed system to time delays at the time point immediately following the initial time point.

The one or more transition probabilities P̂ ij (k) for each transition from the time delay Δk = 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) can be a point in a finite-dimensional vector space. The ambiguity set for these transition probabilities can be encompassed by a sphere around this point with a radius in a finite-dimensional norm (e.g., a p-norm with p = 1), where the radius can be based on a confidence level and/or a total number γ i - 1 (k) (or its inverse, i.e., the inverse sample size γ i (k)) of the time delay determined for the first time in the indefinitely time-delayed system 40. The radius can decrease with a constant confidence level and an increasing total number of time delays determined for the first time point in the indeterminately time-delayed system. For example, The radius can be determined using a concentration inequality such as a McDiarmid inequality. For example, the ambiguity set A(P̂ i :(k)) can be determined as follows: $$A\left({\widehat{P}}_{i:}\left(k\right)\right)=\left\{v\in S_i|||v-{\widehat{P}}_{i:}\left(k\right)\Vert {}_1\le r_i\left(\beta ,{\gamma }_i^{-1}\left(k\right)\right)\right\},$$ where $$S_i=\left\{p_{i:}\in {ℝ}^{M+1}|p_{ij}\ge \mathrm{0,}{\displaystyle {\sum }_{j=1}^{M+1}{p}_{ij}=1}\right\}$$ is an i-th probability simplex. 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 γ i- 1 (k). The radius can be specified in closed form, e.g., via concentration inequalities such as the McDiarmid inequality. The ambiguity set is used in Procedure 100 to synthesize a distributionally robust controller and/or observer that stabilizes the Markov Jump Linear System (MJLS) with confidence β. 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 considered as It can be called the i-th ambiguity set or the state-dependent ambiguity set.

In 5 This section describes how learning by repetition of procedure 100 influences the empirically determined transition probabilities and the associated ambiguity set. Starting with the simplex, which includes both the point marked by the cross for the estimated transition probabilities and the true transition probabilities marked by a filled circle, progressively smaller ambiguity sets are defined at later time points. These sets encompass both the cross and the filled circle and converge to a singleton (the cross coinciding with the filled circle) after a sufficiently large number of time points. Designing the controller 10 and/or observer 11 with respect to such a smaller ambiguity set avoids an overly conservative design.

For example, in 2 Illustrated schematically as an option, the procedure 100, adjusting 140 of the controller 10 and/or the observer 11 based on the determined 120, can include one or more transition probabilities. Alternatively or additionally, the procedure 100, adjusting 140 of the controller 10 and/or the observer 11 based on the determined 130, can include one or more ambiguity sets. In particular, the procedure 100, adjusting 140 of the controller 10 and/or the observer 11 based on the determined 120, can include one or more transition probabilities and one or more ambiguity sets based on the determined 130.

As also, for example, in 2 Schematically illustrated as an option, the procedure 100 can send 141, in particular via the first communication interface, an update to the indefinitely time-delayed system 40, comprising the adapted 140 controller and/or the adapted 140 observer. The update can be configured such that the (previous local) controller 10 and/or the (previous local) observer 11 are replaced by the adapted 140 controller and/or the adapted 140 observer. Such a process is exemplified in 3 This is illustrated, where in boxes 140 and 141 an update of the controller and/or observer is defined and transmitted to controller 10 and/or observer 11. Here, for example, one or more settings of the controller and/or observer can be changed, in particular the controller gain and/or observer gain.

As exemplified in 3 As shown, the procedure 100 can be executed in a computing unit 50 outside the indefinitely time-delayed system 40, in particular in a cloud.

3 Figure 1 shows an example of communication between an indefinitely time-delayed system 40 and the computing unit 50. The indefinitely time-delayed system 40 comprises a controller 10 and, optionally, an observer 11, as well as a technical system 20 to be controlled. Here, the indefinitely time-delayed system 40 also includes the communication network 30, via which the controller 10 and/or the observer 11 communicate, at least partially, with the technical system 20 to be controlled. Step 110 of the procedure 100 can, for example, include the step "calculate integer delays," i.e., for example, the calculation of the discrete time delays according to the formula in [reference missing]. 4d . Steps 120 and 130 of procedure 100 can, for example, include the step "adapting the Markov transition matrix and at least one ambiguity set".

The goal can be to design a distribution-robust controller of the form u _k = K ξ̂ _k that stabilizes the Markov Jump Linear System (MJLS) for all possible Markov states θ(k) ∈ {0,...,M} based on a state estimate ξ̂ _k derived from a distribution-robust observer 11 with the following dynamics, $${\widehat{\xi }}_{k+1}=A_{\theta \left(k+1\right)}{\widehat{\xi }}_k+B_{\theta \left(k+1\right)}{u}_k-L\left(y_k-C{\widehat{\xi }}_k\right)$$ where L is the observer gain (i.e., the gain of observer 11).

A distribution-robust LQR design approach can then be used to synthesize the controller gain K, which can be used analogously to derive the distribution-robust observer gain L due to the dual relationship between the control and observation problems. If the complete state vector ξ is measurable, the observer is superfluous and a controller of the form u _{_k} = K ξ_{_k} can be used instead.

To illustrate this, the example of a group start of three vehicles with unreliable communication over a network is shown again here. 1 further explained. The leading vehicle is uncontrolled (with respect to distance control) and travels, for example, at a constant speed _v0 , followed by two controlled vehicles (with respect to their respective distance control) with control inputs _α1 and _α2 , respectively. The goal can be to control the distance error _e1 and _e2 to the origin, i.e., the dashed vertical lines in 1 This indicates that the necessary safety distance d _safe is maintained. It can be assumed here that the vehicles are controlled by their accelerations _α1 and _α2 , respectively, while the state is only partially measurable; that is, an observer is needed to estimate the state vector. The overall state, the output and input vectors are, for example, in 4a given where _v1 and _v2 are the speeds of the second and third vehicles respectively (the leading vehicle is the first vehicle).

6a illustrates exemplary control processes for the second vehicle during the platooning group start at three different exemplary times in the procedure, namely for sample sizes N _S = 1, N _S = 10 ² and N _S = 10 ³ , as well as corresponding 3σ confidence intervals. 6b This illustrates exemplary control processes for the third vehicle at the same platooning group start at three different example times in the procedure, along with their corresponding 3σ confidence intervals. Here, 1 - β = 0.9 was chosen as the confidence level for the ambiguity set. It can be seen that each controller-observer pair stabilizes the control system, even without information about structural disturbances (i.e., the empirical Markov transition matrix reflects a uniform distribution). As the sample size increases, the state variance, i.e., the ambiguity set (represented as envelope curves), decreases, reflecting an improvement in performance. In this way, it is possible to converge safely to the optimal control gain without sacrificing stability guarantees.

• In einigen Netzregelungssystemen ändert sich die zugrundeliegende Zeitverzögerungsverteilung im Laufe der Zeit aufgrund unterschiedlicher Netzlasten. In diesem Fall kann der vorgeschlagene Regelungsansatz erweitert werden, um diese Trends zu erfassen, indem ein fester, modusabhängiger Stichprobenumfang für das Lernverfahren vorgesehen wird.
• Ein Sonderfall zeitlich veränderlicher Zeitverzögerungsverteilungen ist eine diskrete Umschaltung. In diesem Fall kann eine ereignisgesteuerte Reinitialisierung eingebaut werden, die das vorgeschlagene Lernverfahren zurücksetzt, wenn die neuen Daten nicht in die zuvor gelernte Markov-Übergangsmatrix passen. Eine mögliche Auslösebedingung lässt sich aus dem Gesamtvariationsabstand zwischen zwei Verteilungen ableiten.

Furthermore, the following extensions are conceivable, for example:

• In some grid control systems, the underlying time delay distribution changes over time due to varying grid loads. In this case, the proposed control approach can be extended to capture these trends by providing a fixed, mode-dependent sample size for the learning process.
• A special case of time-varying time-delay distributions is a discrete switching operation. In this case, an event-driven reinitialization can be implemented, which resets the proposed learning algorithm if the new data does not fit into the previously learned Markov transition matrix. A possible trigger condition can be derived from the total variation between two distributions.

Furthermore, a controller 10 and/or an observer 11 are disclosed, which are configured to control a technical system 20, wherein the controller 10 and/or the observer 11 communicate with the technical system 20 to be controlled at least partially via a communication network 30. The communication network 30 can, for example, comprise or be a wireless network. The controller 10 and/or the observer 11 comprise a second communication interface for sending information to a computing unit 50, in particular to a computing unit 50 outside of an indefinitely time-delayed system 40 comprising the controller 10 and/or the observer 11, from which a time delay for a first time point for the indefinitely time-delayed system 40 and a time delay for a second time point for the indefinitely time-delayed system 40 can be determined, the second time point being later than the first time point. The second communication interface can still be configured to receive an update from the computing unit 50, the update comprising a customized controller 140 and/or a customized observer 140. The controller 10 and/or the observer 11 can be configured to implement (i.e., install) the update (or updates).

Furthermore, an indefinitely time-delayed system 40 comprising the controller 10 and/or the observer 11, a technical system 20 to be controlled, and a communication network 30 is disclosed. The controller 10 and/or the observer 11 can be controlled at least partially via the communication network 30. network 30 communicates with the technical system 20 to be controlled.

Furthermore, a computing unit 50 (i.e., a computer system) is disclosed, which is designed to execute the computer-implemented method 100 for robust adaptive control and/or a robust adaptive observer of an indeterminately (i.e., stochastically) time-delayed system 40. The computing unit 50 includes a first communication interface. The computing unit 50 may further include a processor and/or main memory.

Furthermore, a total system 60 is revealed, comprising the indefinitely time-delayed system 40 and the computing unit 50. An exemplary total system is shown in 3 shown.

Furthermore, a computer program is disclosed that is designed to execute the computer-implemented method 100 for robust adaptive control and/or a robust adaptive observer of an indeterminately (i.e., stochastically) time-delayed system. The computer program can be in interpretable or compiled form, for example. It can be loaded (even partially) into a computer's RAM for execution, for example, as a bit or byte sequence.

Furthermore, a computer-readable medium or signal that stores and/or contains the computer program is disclosed. The medium can include, for example, RAM, ROM, EPROM, HDD, SSD, etc., on/in which the signal is stored.

Claims (15)

A computer-implemented method (100) for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system (40), in particular wherein the indefinitely time-delayed system (40) comprises a controller (10), optionally an observer (11), and a technical system (20) to be controlled, wherein the controller (10) and/or the observer (11) communicate at least partially with the technical system (20) to be controlled via a communication network (30), the method (100) comprising: - Receiving (110), in particular via a first communication interface, a time delay determined, in particular measured, for a first time point for the indefinitely time-delayed system (40) and a time delay determined, in particular measured, for a second time point for the indefinitely time-delayed system (40), wherein the second time point is later than the first time point, optionally wherein the time delay determined for the first time point for the indefinitely time-delayed system (40) The time delay and the time delay determined for the second time point for the indefinitely time-delayed system (40) are discrete; - Determine (120) one or more transition probabilities for transitions from time delays at the first time point to time delays at the second time point based on the time delay determined for the first time point for the indefinitely time-delayed system (40) and the time delay determined for the second time point for the indefinitely time-delayed system (40), optionally based on one or more time delays determined for earlier times than the first time point for the indefinitely time-delayed system (40); - Determine (130) one or more ambiguity sets around the one or more determined (120) transition probabilities. Procedure (100) according to Claim 1 , wherein determining (120) the one or more transition probabilities includes: - Determining one or more transition probabilities for each transition from the time delay determined for the first time point to a possible time delay at the second time point. Procedure (100) according to Claim 2 , wherein determining (130) the one or more ambiguity sets around the one or more determined (120) transition probabilities includes: - Determining an ambiguity set around the one or more transition probabilities for each transition from the time delay determined for the first time point to a possible time delay at the second time point. Method (100) according to any of the preceding claims, wherein determining (120) the one or more transition probabilities comprises: - Determining a transition probability for a transition from the time delay determined for the first time point to the time delay determined for the second time point. Method (100) according to one of the preceding claims, wherein the determination (120) of the one or more transition probabilities and the determination (130) of the one or more ambiguity sets around the one or more determined (120) transition probabilities are performed recursively and are based on one or more initial transition probabilities for transitions from time delays at an initial time point in an operation of the time-delayed system to time delays wrestling to the point in time immediately following the initial point in time are based on. Procedure (100) according to Claim 5 , where the transition probabilities for each transition from a time delay at the initial time to a possible time delay at the time immediately following the initial time are equally distributed. Method (100) according to one of the preceding claims, wherein possible time delays and/or time points are discrete, in particular wherein the one or more transition probabilities for transitions from time delays at the first time point to time delays at the second time point are arranged in a Markov transition matrix. Procedure (100) according to Claim 7 , if dependent on Claim 3 , wherein the one or more transition probabilities for each transition from the time delay determined for the first time to a possible time delay at the second time are a point in a finite-dimensional vector space and the ambiguity set for these transition probabilities is encompassed by a sphere around the point with a radius in a finite-dimensional norm, the radius being based on a confidence level and/or a total number of time delays determined for the first time in the indefinitely time-delayed system (40). Procedure (100) according to Claim 8 , where the radius decreases with a constant confidence level and an increasing total number of time delays determined for the first time point in the indefinitely time-delayed system. Method (100) according to any of the preceding claims, comprising: - Adjusting (140) the controller (10) and/or the observer (11) based on the determined (120) one or more transition probabilities and/or on the determined (130) one or more ambiguity sets. Procedure (100) according to Claim 10 , comprising: - Sending (141), in particular via the first communication interface, an update to the indefinitely time-delayed system, comprising the adapted (140) controller and/or the adapted (140) observer, in particular wherein the update is configured such that the controller (10) and/or the observer (11) are replaced by the adapted (140) controller and/or the adapted (140) observer. Method (100) according to one of the preceding claims, wherein the method (100) is executed in a computing unit (50) outside the indefinitely time-delayed system (40), in particular in a cloud. Controller (10) and/or observer (11), configured to control a technical system (20), wherein the controller (10) and/or the observer (11) communicate at least partially with the technical system (20) to be controlled via a communication network (30), the controller (10) and/or the observer (11) comprising: - a second communication interface for sending information to a computing unit (50), in particular to a computing unit (50) outside an indefinitely time-delayed system (40) comprising the controller (10) and/or the observer (11), from which a time delay for a first time point for the indefinitely time-delayed system (40) and a time delay for a second time point for the indefinitely time-delayed system (40) can be determined, wherein the second time point is later than the first time point; wherein the second communication interface is further configured to receive an update from the computing unit (50), wherein the update includes a customized (140) controller and/or a customized (140) observer; where the controller (10) and/or the observer (11) are configured to implement the update. indefinitely time-delayed system (40) comprising: - the controller (10) and/or the observer (11) according to Claim 13 ; - a technical system to be controlled (20); - a communication network (30); wherein the controller (10) and/or the observer (11) communicate at least partially with the technical system to be controlled (20) via the communication network (30). Computing unit (50), designed to implement the computer-implemented method (100) for robust adaptive control and/or a robust adaptive observer of an indefinitely time-delayed system (40) according to one of the preceding Claims 1 until 12 to execute, comprising the computing unit (50): - a first communication interface.