Robust adaptive control function for uncertain time delay control system

Abstract

To provide a robust adaptive control function for an uncertain time delay control system.SOLUTION: An uncertain time delay system includes a controller, optionally an observer, and a controlled technical system, in which the controller or the observer communicates with the controlled technical system via a communication network, and a computer implementation method includes: a step of receiving time delay at a first time point for the uncertain time delay system and time delay at a second time point for the uncertain time delay system, in which the second time point is later than the first time point; a step of determining, based on the time delay at the first time point for the uncertain time delay system and the time delay at the second time point for the uncertain time delay system, transition probability for transition from the time delay at the first time point to the time delay at the second time point; and a step of determining an ambiguity set around the determined transition probability.SELECTED DRAWING: Figure 1

Description

[Background technology]

[0001] In recent decades, rapid developments in computing power and communications technology have had a major impact on today's control system infrastructure. While traditional control systems consist of sensors, controllers, and actuators connected to each other via wires, networked control systems (NCSs) replace at least some or all of these end-to-end connections with communications networks. NCSs can be, but do not necessarily have to be, based on wireless connections. Furthermore, centralized and / or zone-based E / E architectures, for example, are also encompassed by NCSs.

[0002] The advantages of NCS are manifold, for example, it reduces cabling and / or maintenance costs and the communication topology can be more flexible for distributed systems. However, when using a communication network, network-related problems can arise, such as: - random transmission delays in the sensor-controller (S2C) and controller-actuator (C2A) channels, e.g., due to large separation distances between two nodes and / or time-varying traffic in the network; - Random packet loss, for example due to network overload.

[0003] Such a control system can be called an uncertain time delay control system. If not proactively addressed, both of these issues can degrade control system performance and cause control loop instability, potentially even causing observers within the control system to malfunction.

[0004] Traditionally, various control approaches have been developed to deal with random time delays and / or packet loss, ranging from deterministic to probabilistic approaches. While deterministic (robust) approaches rely on worst-case uncertainty descriptions, probabilistic approaches use additional information in the form of statistics and / or probability distributions to maintain less conservative control. However, probabilistic approaches typically assume complete information about the statistics and / or probability distributions of time delays in a particular control system, which is unfortunately rarely applicable in practice. In contrast, when using empirical estimates of statistics and / or probability distributions, no guarantees about the control system are given. However, such guarantees are precisely what is needed for the release of the control system. Summary of the Invention [Problem to be solved by the invention]

[0005] Thus, the problem to be solved underlying this disclosure can be considered to be that of providing a method that allows for less conservative, yet still robust, control of an uncertain time-delay control system, e.g., despite an initial lack of knowledge of the statistics and / or probability distribution of the time delays in the control system. Alternatively or additionally, the problem to be solved can be considered to be that of providing a method that allows for less conservative, yet still robust design of an observer in an uncertain time-delay control system, e.g., despite an initial lack of knowledge of the statistics and / or probability distribution of the time delays in the control system. The problem to be solved can also be considered to be that of providing a control function and observer for an uncertain time-delay system that is not too conservative overall, yet still robust. [Means for solving the problem]

[0006] A first general aspect of the present disclosure relates to a computer-implemented method for a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system. The uncertain time-delay system may include a controller, optionally an observer, and a controlled technical system, where the controller and / or observer communicate with the controlled technical system at least in part via a communication network. The method includes receiving, particularly via a first communication interface, a determined, particularly measured, time delay for a first point in time for the uncertain time-delay system and a determined, particularly measured, time delay for a second point in time for the uncertain time-delay system, where the second point in time is later than the first point in time. The determined time delay for the first point in time for the uncertain time-delay system and the determined time delay for the second point in time for the uncertain time-delay system may be discrete. The method further includes determining one or more transition probabilities for transitioning from a time delay at a first time point to a time delay at a second time point based on a time delay identified for a first time point for the uncertain time delay system and a time delay identified for a second time point for the uncertain time delay system. Further, determining the one or more transition probabilities may be based on one or more time delays identified for the uncertain time delay system at a time point prior to the first time point. The method further includes determining one or more ambiguity sets around the one or more determined transition probabilities.

[0007] A second general aspect of the present disclosure relates to a controller and / or observer configured to control a technical system, the controller and / or observer communicating with the controlled technical system at least in part via a communication network. The controller and / or observer includes a second communication interface for transmitting information to a computing unit, in particular a computing unit external to the uncertain time-delay system including the controller and / or observer, from which a time delay for a first point in time for the uncertain time-delay system and a time delay for a second point in time for the uncertain time-delay system can be determined, the second point in time being later than the first point in time. The second communication interface can be further configured to receive updates from the computing unit, the updates including the adapted controller and / or the adapted observer. The controller and / or observer can be configured to implement the updates.

[0008] A third general aspect of the present disclosure relates to a computing unit (and thus a computer system) designed to perform a computer-implemented method for a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system according to the first general aspect (or an embodiment thereof). The computing unit includes a first communication interface.

[0009] A fourth general aspect of the present disclosure relates to a computer program designed to perform a computer-implemented method for a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system according to the first general aspect (or an embodiment thereof).

[0010] A fifth general aspect of the present disclosure relates to a computer-readable medium or signal storing and / or including a computer program according to the fourth general aspect (or an embodiment thereof).

[0011] The method according to the first general aspect (or an embodiment thereof) proposed herein allows for the stepwise adaptation and eventual definition of a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system, where uncertain time delays may occur due to a communication network between the controller / observer (hereinafter also referred to as a stochastic controller) and the controlled technical system. The defined controllers are robust because they can accommodate multiple probability distributions of time delays. At the same time, they are not designed equally for all theoretically possible time delays (and their sequences), but rather for those that are statistically significant in the application, so are not very conservative. At this point, the defined controller and / or the defined observer can be considered already released, or can be further tested in a release process and then released if necessary. The release of the controller and / or observer can be a necessary part of the release of the controlled technical system.

[0012] A key problem in designing a stochastic controller in practice is the need to know the true statistics and / or probability distribution of the time delays already at the time of designing the control function. In practice, this is rarely the case, requiring approximations, for example in the form of empirical distributions. Unfortunately, such approximations are subject to sample-induced errors resulting from sample-average approximations. To ensure that guaranteed stabilization controllers can be synthesized despite these sample-related errors, a distribution-robust approach is pursued according to the present invention. Specifically, an ambiguity set, i.e., a set of uncertainties in the space of probability distributions, is constructed around an empirical estimate that contains the true probability distribution with a high degree of confidence at each time point. This ensures that the resulting controller and / or observer (and thus the uncertain time-delay control system) is always robust to the entire family of probability distributions. In a closed control loop, data (especially the actual time delays, i.e., time-delay data) is continuously collected by the control system. Subsequently, the empirical distributions and ambiguity sets from the data are continuously adapted, thereby enabling the synthesis of progressively improved controllers and / or observers. This learning algorithm guarantees that as the sample size approaches infinity, the ambiguity set shrinks to the point where it contains only the true distribution (also called a singleton), where in practice it is sufficient to have a sufficiently large sample.

[0013] The control strategy proposed by the method according to the first general aspect (or an embodiment thereof) uses a general modeling framework for NCSs called Markov Jump Linear Systems (MJLS) ​​and combines it with learning-based, distribution-robust controller and / or observer design. Compared to conventional methods, time-delayed data (i.e., in the form of measurements) can be used to construct a Markov transition matrix and ambiguity set containing a true probability distribution with a user-defined confidence level at each discrete time step. By continuously collecting additional time-delayed data, the Markov transition matrix can be recursively updated to reduce the ambiguity set while maintaining the same user-defined confidence level. This adaptation of the Markov transition matrix and ambiguity set represents the learning-based controller and / or observer design. This achieves that the gradual reduction of the ambiguity set reduces overly conservative design of the control function while maintaining the probabilistic stability guarantee (due to the confidence level).

[0014] Furthermore, a distinction can be made between the control loop itself, closed by the network, and the learning algorithm, which can be understood as a microservice that can reside on a separate network (e.g., the cloud) with best-effort requirements for availability and reliability.

[0015] An exemplary embodiment of the method proposed in this disclosure can be summarized as follows: The control system is operated using (initially) an arbitrarily stabilizing local controller to generate time-delay data, which is a closed, time-critical signal path through the network. The local controller continuously transmits timestamps to a learning algorithm, e.g., located in the cloud, which has non-time-critical processing time. The timestamps can then be converted into a set of integer time delays that are updated each time a new timestamp arrives. The Markov transition matrix and ambiguity set can then be updated, e.g., based on the new integer time delay information. Finally, the controller is updated based on the new Markov transition matrix and ambiguity set and re-input into the control system. After the update, the control performance is at or above that of the previous local controller. [Brief explanation of the drawings]

[0016] [Figure 1] FIG. 1 illustrates an exemplary uncertain time delay system including three vehicles performing platooning and, in particular, a group start. [Figure 2] FIG. 1 is a schematic diagram of an exemplary embodiment of a computer-implemented method for a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system. [Figure 3] FIG. 1 illustrates an exemplary embodiment of a complete system including an uncertain time-delay system and a computational unit designed to execute a computer-implemented method for a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system. [Figure 4a] FIG. 4a is a diagram illustrating exemplary vectors for controller states, measurable outputs, and inputs. [Figure 4b] FIG. 4b illustrates an exemplary Markov transition matrix containing transition probabilities for transitions between delays. [Figure 4c] FIG. 4c illustrates an exemplary Markov jump linear system with an expanded state ξ for an uncertain time-delay system. [Figure 4d]FIG. 4d illustrates an exemplary set of integer delays that are (continuously) specified for an uncertain time delay system. [Figure 5] FIG. 1 illustrates exemplary successive transition probabilities Pi:=(pi1, pi2, pi3) and the associated shrinking ambiguity sets. [Figure 6a] FIG. 6a illustrates an exemplary control process for the second vehicle at the start of a platooning group at three different times during the process and associated 3σ confidence intervals. [Figure 6b] FIG. 6b illustrates an exemplary control process for the third vehicle at the start of a platooning group at three different times during the process and associated 3σ confidence intervals. DETAILED DESCRIPTION OF THE INVENTION

[0017] First, a computer-implemented method 100 for robust adaptive control and / or robust adaptive observer of an uncertain (i.e., stochastic) time-delay system 40 is disclosed. The uncertain time-delay system 40 may include a controller 10 (also referred to as a local controller) and a controlled technical system 20. Optionally, the uncertain time-delay system 40 may also include an observer 11 (also referred to as a local observer). The controller 10 and / or the observer 11 may communicate with the controlled technical system 20, at least in part, via a communication network 30.

[0018] Thus, the method 100 proposed in this disclosure may be directed to robust adaptive control functions and / or robust adaptive observers for uncertain (ie, stochastic) time-delay systems.

[0019] As shown in FIG. 1 by way of example, the time delay system 40 may include a plurality of vehicles, which together form the technical system 20 to be controlled, for example forming a convoy (i.e., performing platooning), in particular performing a group start. The plurality of vehicles includes at least two vehicles, or three vehicles as shown by way of example in FIG. 1. Here, all vehicles except the first (leading) vehicle are controlled via a communication network 30 by a controller 10 and / or an observer 11 external to the technical system 20 to be controlled. In this case, the communication network 30 is preferably a wireless network. Here, v0 denotes the speed of the first vehicle, v1 denotes the speed of the second vehicle immediately following the first vehicle, and v2 denotes the speed of the third vehicle immediately following the second vehicle. Furthermore, d safe and d indicate the required safety distance, which should ideally be controlled between adjacent vehicles, or at least not be less than that. Furthermore, e1 is the ideal safety distance d between the first and second vehicles. safe where e denotes the distance error from the ideal safe distance d between the second and third vehicles. safe Indicates the distance error from

[0020] However, the time delay system 40 is not limited to this application. Another application is, for example, the control and / or coordination of guided automated vehicles within a limited area, such as a logistics center or production facility, via a local network (such as a Local Edge or 5G network). Yet another application is, for example, (partially) outsourcing the control algorithms of robotic arms, for example for manufacturing systems, to a local network. Yet another application is the lateral and / or longitudinal movement control of vehicles at traffic junctions or other control areas, where this control is outsourced to a local edge or cloud system (such as a roadside unit).

[0021] 2, the method 100 includes receiving 110, particularly via a first communication interface (e.g., of the computing unit 50), a determined, particularly measured, time delay for a first point in time for the uncertain time delay system 40 and a determined, particularly measured, time delay for a second point in time for the uncertain time delay system 40. The second point in time may be later than the first point in time. The determined time delay for the first point in time for the uncertain time delay system 40 and the determined time delay for the second point in time for the uncertain time delay system 40 may be discrete, i.e., may be classified into, for example, a predetermined number of classes.

[0022] 2, the method 100 further includes determining 120 one or more transition probabilities for a transition from a time delay at a first time point to a time delay at a second time point based on the time delay identified for the uncertain time delay system 40 for the first time point and the time delay identified for the second time point for the uncertain time delay system 40. The determining 120 one or more transition probabilities for a transition from a time delay at a first time point to a time delay at a second time point may be based on one or more time delays identified for the uncertain time delay system 40 at a time point prior to the first time point.

[0023] As also shown, for example, schematically in FIG. 2, the method 100 further includes determining 130 one or more ambiguity sets around the one or more determined 120 transition probabilities.

[0024] The uncertain time delay system 40 first

[0025]

number

[0026] and a known sampling time

[0027]

number

[0028] and can be modeled, for example, by the following equation:

[0029]

number

[0030] y(t)=Cx(t)

[0031]

number

[0032] where x is the state vector, y is the measurable output vector, and u is the input vector. By discretizing time, for example by Euler integrals, the first two equations can be written as: x k+1 =A d x k +B d u k-θ(k) y k =Cx k Here, the continuous time delay τ has been transformed into a time-discrete Markov chain θ(k)∈{0,…,M}, i.e., where each Markov state (also called Markov mode) i∈{0,…,M} is separated by an integer time delay Δt k =ih, where:

[0033]

number

[0034] indicates the maximum time delay that is important for each application. This system of equations can be rewritten as shown in Figure 4c, where ξ is the expanded state vector and ξ k is the discrete time t k= kh (or k for short) is this state vector. ij are Kronecker deltas. Furthermore, each I is an identity matrix with the appropriate dimensions. This representation is a Markov jump linear system (MJLS). It is obtained from the combination of an uncertain time-delay system 40 and a Markov state θ(k). It augments the technical system 20 to be controlled with an imperfect communication network 30 that introduces up to M discrete time delays.

[0035] Thus, the time points and / or possible time delays may be discrete. One or more transition probabilities for transitioning from a time delay at a first time point to a time delay at a second time point may be arranged in a Markov transition matrix (see, for example, FIG. 4b). The Markov transition matrix may describe a Markov chain.

[0036] The step of receiving 110, particularly via the first communications interface, a determined, particularly measured, time delay for a first point in time with respect to the uncertain time delay system 40 and a determined, particularly measured, time delay for a second point in time with respect to the uncertain time delay system 40 may include receiving, particularly via the first communications interface, information from which the time delay for the first point in time with respect to the uncertain time delay system 40 can be determined and information from which the time delay for the second point in time with respect to the uncertain time delay system 40 can be determined. For example, as shown schematically in FIG. 3 , this information may include one or more timestamps, which are passed from the controller 10 and / or the observer 11 to step 110 (e.g., calculating the integer delay).

[0037] The first time point is, for example, t k = kh, where h is the sampling time,

[0038]

number

[0039] is 0 or a positive integer. This time point can be identified by an integer k, i.e., can be referred to as the kth time point. The second time point can be, for example, the time point t immediately after the first time point. k+1 =(k+1)h=t k +h. This point can also be identified by the integer k+1.

[0040] The time delay specified for the first time point for the uncertain time delay system 40 and the time delay specified for the second time point for the uncertain time delay system 40 may be received 110 discretely from the beginning, or may be discretized (and thus also be discrete) after receiving 110. The discrete time delays may be expressed, for example, in multiples of the sampling time h. For example, the discrete time delays Δt for all i∈{0,...,M} may be expressed in sequence as k = ih, where M is a predetermined positive integer. Such discrete time delays can be considered as (discrete) Markov states, which can be identified by integers i∈{0,…,M}. Discrete time delays can be specified, for example, by the formula in Figure 4d, where the lower-only brackets can mean rounding to the nearest integer. Here, t k and t k+1 represents a timestamp (unlike the above), the difference of which allows the time delay to be calculated, where

[0041]

number

[0042] indicates sample size for all time points. Determining 120 one or more transition probabilities for transitioning from a time delay at a first time point to a time delay at a second time point may be determining one or more transition probabilities for transitioning from a discrete time delay at a first time point to a discrete time delay at a second time point, where, for example, transition probability p for transitioning from Markov state i to (generally another) Markov state j, for i,j∈{0,...,M} ij=P(θ(k+1)=j|θ(k)=i). Such transition probabilities can be elements of a Markov transition matrix. By repeating step 120 and method 100, i.e., by a further step 120 for a next time point, one or more transition probabilities, in particular, for example, a Markov transition matrix, are empirically estimated based on the specified and / or measured time delays of the uncertain time delay system 40. The transition probabilities p ij The estimate for

[0043]

number

[0044] It is expressed as: The ambiguity set may be a set of transition probabilities, and the transition probabilities determined 120 in the method 100 (for each structure) are included in this set. For example, a transition probability may 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 120 transition probabilities form points in a high-dimensional space. In Figure 5, such exemplary points are shown with crosses at three points in this method 100. The ambiguity set may then be a non-point-shaped subset of this high-dimensional space that contains the points. Figure 5 also shows the transition probabilities p i1 , p i2 , and p i3 Starting with an exemplary ambiguity set that forms a simplex in the space of , further smaller ambiguity sets are shown, with all three exemplary ambiguity sets each containing a point represented by an X. The ambiguity sets are a measure of the uncertainty of the associated transition probabilities.

[0045] Determining 120 the one or more transition probabilities may include, among other things, determining one or more transition probabilities for each transition from the time delay identified for the first time point to a possible time delay at the second time point.

[0046] Determining 130 one or more ambiguity sets around the one or more determined 120 transition probabilities may include determining an ambiguity set around one or more transition probabilities for each transition from the identified time delay for the first time point to a possible time delay at the second time point.

[0047] Determining 120 one or more transition probabilities may include determining a transition probability for a transition from a time delay identified for a first time point to a time delay identified for a second time point.

[0048] Determining 120 one or more transition probabilities when a transition begins from a time delay identified for a first point in time in the uncertain time delay system 40 may include or correspond to the following steps:

[0049] - taking the inverse of the total number of time delays identified for the first time point in the uncertain time delay system 40 to obtain the inverse sample size; - scaling one or more transition probabilities for transitioning from a time delay at a time point prior to the first time point to a time delay at the first time point with a scaling factor, where the scaling factor is 1 minus the inverse sample size, to obtain one or more first transition probabilities; - scaling the transition probability for a transition from the time delay specified for the first time point to the time delay specified for the second time point by the inverse sample size to obtain a second transition probability; - optionally adding one or more first transition probabilities and a second transition probability.

[0050] In the uncertain time delay system 40, at a first time point k (i.e., t k = kh) k =i (i.e., Δt k =ih) is

[0051]

number

[0052] and therefore the inverse sample size is γ i (k). At a first time point k (i.e., t k =kh) is determined in the uncertain time delay system 40 at the second time instant k+1 (i.e., t k+1 =(k+1)h), the time delay specified for k+1 =i (i.e., Δt k+1 =ih), where i is the total number of time delays specified for the second point in time in the uncertain time delay system 40.

[0053]

number

[0054] Therefore, in this case, the inverse sample size can be determined recursively as follows:

[0055]

number

[0056] Otherwise, the inverse sample size remains unchanged.

[0057]

number

[0058] Therefore, Δ k =i (i.e., Δt k =ih) occurs at a first time point k (i.e., t k = k h ), then the time delay specified for the first time point is taken to be the time delay from the time specified for the first time point to the second time point k+1 (i.e., t k+1 = (k+1)h) for each possible transition to a time delay

[0059]

number

[0060] can be estimated recursively as follows (Equation 1):

[0061]

number

[0062] where:

[0063]

number

[0064] or

[0065]

number

[0066] is the i-th row of the Markov transition matrix at time instant k or k+1, respectively,

[0067]

number

[0068] is the component

[0069]

number

[0070] , i.e., the components are all 0 except for the (k+1)th component, which is 1. On the other hand, determining 120 one or more transition probabilities when the transition does not begin at the time delay specified for the first point in time in the uncertain time delay system 40 may include or correspond to the following steps: - maintaining one or more transition probabilities for transitioning from a time delay at a time point prior to the first time point to a time delay at the first time point;

[0071] In contrast, Δ k ≠i (i.e., Δt k ≠ih) occurs at the first time point k (i.e., t k = k h ), then the time delay specified for the first time point is taken to be the time delay from the time specified for the first time point to the second time point k+1 (i.e., t k+1 = (k+1)h) for each possible transition to a time delay

[0072]

number

[0073] can be estimated recursively as follows (Equation 2):

[0074]

number

[0075] At each time point, k = i or Δ k Since ≠ i, at each point in time, either Equation 1 or Equation 2 is applied for i. This can be done for all Markov states i. In this case, only one row of the Markov transition matrix can be adapted in a nontrivial way (i.e., according to Equation 1), while the other rows of the Markov transition matrix are either retained, i.e., not adapted, or adapted in a trivial way (i.e., according to Equation 2).

[0076] Thus, determining 120 one or more transition probabilities may be performed recursively, further determining one or more initial transition probabilities for all Markov states i for a transition from a time delay at an initial time point t0=0h=0 to a time delay at a time point t1=1h=h immediately following the initial time point during operation of the time delay system.

[0077]

number

[0078] One or more initial transition probabilities for all Markov states i

[0079]

number

[0080] can be partially zero. These can correspond, for example, to the Markov transition matrix P shown as an example in Figure 4b, whose elements are p for i,j∈{0,1,…,M}. ij In this case, the components in the triangular part above the diagonal are 0 except for the first subdiagonal. This allows you to predetermine, for example, that the time delay can only be increased incrementally by 1, and not abruptly.

[0081] Initially, or for all times during operation of the time delay system 40, at least one transition probability for each transition from one time delay to another may be zero.

[0082]

number

[0083] must be normalized to 1 at each time point k, i.e.

[0084]

number

[0085] holds true. The transition probability for each transition from a time delay at an initial time t0=0h=0 to a possible time delay at a time t1=1h=h immediately following the initial time may be a uniform distribution. For example, p 00 =1 / 2, p 01 =1 / 2, p 10 =1 / 3, p 11 =1 / 3, p 12 =1 / 3, etc., can be true.

[0086] Determining 130 one or more ambiguity sets around one or more determined 120 transition probabilities may also be done recursively and may further be based on one or more initial transition probabilities for transitioning from a time delay at an initial point in time during operation of the time delay system to a time delay at a point immediately following the initial point in time.

[0087] At the first time point k (i.e., t k = kh) k =i (i.e., Δt k =ih) to the second time point k+1 (i.e., t k+1 = (k+1)h) one or more transition probabilities for each possible time delay transition

[0088]

number

[0089] may be a point in a finite-dimensional vector space. The set of ambiguities for these transition probabilities may be encompassed by a sphere with a radius of a finite-dimensional norm (e.g., p-norm with p=1) around this point, where the radius is a function of the confidence level and / or total number of time delays identified for the first time point of the uncertain time delay system 40.

[0090]

number

[0091] (or its inverse, i.e., the inverse sample size γ i (k)). The radius may decrease when the confidence level of the time delay determined for the first time point in the uncertain time delay system is constant and the total number increases. For example, the radius can be determined by a cardinality inequality such as the McDiarmid inequality. For example, the ambiguity set

[0092]

number

[0093] can be determined as follows:

[0094]

number

[0095] where:

[0096]

number

[0097] is the i-th probability simplex, where

[0098]

number

[0099] is the radius for Markov state i, with a user-defined confidence level β∈(0,1) and the total number

[0100]

number

[0101] The radius can be specified in closed form using a concentration inequality, such as the McDiarmid inequality. The ambiguity set is used in method 100 to synthesize a distributionally robust controller and / or observer that stabilizes a Markov jump linear system (MJLS) ​​with confidence β.

[0102]

number

[0103] can be related to the transition probability for a transition from Markov state i to (generally another) Markov state j. In this respect, the ambiguity set

[0104]

number

[0105] can be called the i-th ambiguity set or the state-dependent ambiguity set. 5 illustrates how learning by repeating method 100 affects the empirically determined transition probabilities and associated ambiguity sets. Starting with a simplex containing both points for estimated transition probabilities, represented by crosses, and actual transition probabilities, represented by filled circles, an ambiguity set containing both crosses and filled circles is defined, which becomes smaller at later time points, converging to a singleton (where the crosses coincide with the filled circles) after a sufficiently large number of time points. When controller 10 and / or observer 11 are designed with respect to such small ambiguity sets, overly conservative designs are avoided.

[0106] 2, the method 100 may include adapting 140 the controller 10 and / or the observer 11 based on the one or more transition probabilities determined 120. Alternatively or additionally, the method 100 may include adapting 140 the controller 10 and / or the observer 11 based on the one or more ambiguity sets determined 130. In particular, the method 100 may include adapting 140 the controller 10 and / or the observer 11 based on the one or more transition probabilities determined 120 and the one or more ambiguity sets determined 130.

[0107] 2, the method 100 may also include a step of transmitting 141, in particular via the first communication interface, an update comprising an adapted 140 controller and / or an adapted 140 observer to the indeterminate time delay system 40. The update may be configured such that the (previously local) controller 10 and / or the (previously local) observer 11 are replaced by the adapted 140 controller and / or the adapted 140 observer. Such a process is shown by way of example in FIG. 3, where controller and / or observer updates are defined in boxes 140, 141 and transmitted to the controller 10 and / or the observer 11. Here, for example, one or more settings of the controller and / or observer, in particular the settings of the controller gain and / or the observer gain, may be changed.

[0108] As an example, as shown in FIG. 3, the method 100 may be performed on a computing unit 50 external to the uncertain time delay system 40, particularly in the cloud. 3 shows exemplary communication between an uncertain time-delay system 40 and a computing unit 50. The uncertain time-delay system 40 includes a controller 10 and, optionally, an observer 11 and a controlled technical system 20. The uncertain time-delay system 40 also includes a communication network 30 via which the controller 10 and / or the observer 11 communicate, at least in part, with the controlled technical system 20. Step 110 of the method 100 may here, for example, include a step of "calculating integer delays," i.e., computing discrete time delays, for example, according to the formula of FIG. 4d. Steps 120, 130 of the method 100 may here, for example, include a step of "adapting a Markov transition matrix and at least one ambiguity set."

[0109] The goal is format

[0110]

number

[0111] The goal is to design a controller that is robust to the distribution of the state estimate

[0112]

number

[0113] Based on

[0026] , we stabilize a Markov jump linear system (MJLS) ​​for all possible Markov states θ(k)∈{0,...,M}, where the state estimates are obtained from an observer 11 that is robust to a distribution with the following dynamics:

[0114]

number

[0115] Here, L is the observer gain (i.e., the gain of the observer 11). A distribution-robust LQR design approach can then be used to synthesize the controller gain K, which in turn can be used to derive the distribution-robust observer gain L due to the dual relationship between the control problem and the observation problem. If the complete state vector ξ is measurable, the observer becomes redundant and can instead be of the form u k =Kξ k The following controllers can be used:

[0116] For illustrative purposes, we will now again consider the example of a group start of three vehicles with unreliable communication over the network of Figure 1. The lead vehicle is uncontrolled (in terms of distance control), e.g., travelling at a constant speed v0, and is followed by two controlled vehicles (in terms of distance control, respectively), each with control inputs a1 and a2. The objective may be to control the distance errors e1 and e2 to the origin, i.e., the dashed vertical lines in Figure 1 represent the required safety distance d safe It can be assumed here that the vehicles are controlled by their accelerations a1 and a2, but that the state is only partially measurable, i.e., an observer is required to estimate the state vector. The overall state, output vector, and input vector are given, for example, in Figure 4a, where v1 and v2 are the velocities of the second and third vehicles, respectively (here the lead vehicle is the first vehicle).

[0117] FIG. 6a illustrates an exemplary control process for the second vehicle at the start of a platooning group at three different exemplary points in time during the process, namely, for a sample size of N s =1, N s =10 2 , and N s =10 3, and the associated 3σ confidence intervals. Figure 6b illustrates an example control process for a third vehicle at the start of the same platooning group, at three different example points in the process and the associated 3σ confidence intervals. Here, a confidence level of 1-β=0.9 was chosen for the ambiguity set. It can be seen that each controller-observer pair stabilizes the control system (i.e., the empirical Markov transition matrix reflects a uniform distribution) even in the absence of information about structural disturbances. As the sample size increases, the state variance, i.e., the ambiguity set (represented as an envelope curve), shrinks, reflecting improved performance. In this way, we can now reliably converge to the optimal control gains without compromising stability guarantees.

[0118] Further extensions are possible, for example: In some net-controlled systems, the underlying time delay distribution changes over time due to differences in net loads. In this case, the proposed control approach can be extended to capture these trends by providing a fixed, mode-dependent sampling size for the learning process.

[0119] A special case of time-varying time delay distributions is discrete switching. In this case, event-driven reinitialization can be incorporated, where the proposed learning process is reset if new data does not fit the previously learned Markov transition matrix. Possible trigger conditions can be derived from the total variation distance between the two distributions.

[0120] Further disclosed is a controller 10 and / or an observer 11 configured to control a technical system 20, the controller 10 and / or the observer 11 communicating with the controlled technical system 20 at least in part via a communication network 30. The communication network 30 may include or be, for example, a wireless network. The controller 10 and / or the observer 11 include a second communication interface for transmitting information to a computing unit 50, in particular to a computing unit 50 external to the uncertain time-delay system 40 including the controller 10 and / or the observer 11, from which a time delay for a first point in time for the uncertain time-delay system 40 and a time delay for a second point in time for the uncertain time-delay system 40 can be determined, the second point in time being later than the first point in time. The second communication interface can further be designed to receive updates from the computing unit 50, the updates including an adapted 140 controller and / or an adapted 140 observer. The controller 10 and / or observer 11 may be designed to implement (ie, colloquially install) the update(s).

[0121] Also disclosed is an uncertain time delay system 40 including a controller 10 and / or an observer 11, a controlled technical system 20, and a communication network 30. The controller 10 and / or the observer 11 may communicate with the controlled technical system 20, at least in part, via the communication network 30.

[0122] Also disclosed is a computing unit 50 (and thus a computer system) designed to execute the computer-implemented method 100 for a robust adaptive control function and / or a robust adaptive observer for an uncertain (i.e., stochastic) time-delay system 40. The computing unit 50 includes a first communication interface. The computing unit 50 may further include a processor and / or a main memory.

[0123] Also disclosed is an overall system 60 that includes the uncertain time delay system 40 and the computation unit 50. An exemplary overall system is shown in FIG. Also disclosed is a computer program designed to perform the computer-implemented method 100 for robust adaptive control and / or robust adaptive observer functions for uncertain (i.e., stochastic) time-delay systems. The computer program may exist, for example, in an interpretable or compiled form. The computer program may be loaded (even partially) into a computer's RAM for execution, for example, as a bit or byte sequence.

[0124] Also disclosed is a computer-readable medium or signal that stores and / or includes a computer program. The medium may include, for example, one of a RAM, a ROM, an EPROM, a HDD, an SSD, etc., on which the signal is stored. [Explanation of symbols]

[0125] 10 Controller 11 Observer 20 Controlled Technical Systems 30 Communication Network 40 Uncertain Time Delay Systems 50 compute units 60 All Systems 100 Computer Implementation Methods 110 Time-delayed reception / discrete reception 120 Determining one or more transition probabilities 130 Determining one or more ambiguity sets around transition probabilities 140 Controller and / or Observer Adaptation 141 Sending updates including adapted controllers and / or adapted observers to uncertain time delay systems

Claims

1. A computer-implemented method (100) for a robust adaptive control function and / or a robust adaptive observer of an uncertain time-delay system (40), in particular said uncertain time-delay system (40) comprising a controller (10), optionally an observer (11), and a technical system (20) to be controlled, said controller (10) and / or said observer (11) communicating with said technical system (20) to be controlled at least in part via a communication network (30), said method (100) comprising: receiving (110), in particular via a first communication interface, a time delay determined, in particular measured, for a first point in time for the uncertain time delay system (40) and a time delay determined, in particular measured, for a second point in time for the uncertain time delay system (40), the second point in time being later than the first point in time, and optionally the time delay determined, in respect of the uncertain time delay system (40), for the first point in time and the time delay determined, in respect of the uncertain time delay system (40), for the second point in time being discrete; determining (120) one or more transition probabilities for a transition from a time delay at the first time point to a time delay at the second time point based on the time delay identified for the uncertain time delay system (40) for the first time point and the time delay identified for the uncertain time delay system (40) for the second time point, optionally based on one or more time delays identified for the uncertain time delay system (40) at a time point prior to the first time point; determining (130) one or more ambiguity sets around the one or more determined (120) transition probabilities; A method (100) comprising:

2. said determining (120) said one or more transition probabilities determining one or more transition probabilities for each transition from the time delay identified for the first time point to a possible time delay at the second time point; The method (100) of claim 1, comprising:

3. said determining (130) one or more ambiguity sets around said one or more determined (120) transition probabilities; determining an ambiguity set around the one or more transition probabilities for each transition from the time delay identified for the first time point to a possible time delay at the second time point; The method (100) of claim 2, comprising:

4. said determining (120) said one or more transition probabilities determining a transition probability for a transition from the time delay specified for the first time point to the time delay specified for the second time point; The method (100) of any one of claims 1 to 3, comprising:

5. 5. The method of claim 1, wherein the determination of the one or more transition probabilities and the determination of the one or more ambiguity sets around the one or more determined transition probabilities are performed recursively and are based on one or more initial transition probabilities for a transition from a time delay at an initial point in time during operation of the time delay system to a time delay at a point immediately following the initial point in time.

6. 6. The method (100) of claim 5, wherein the transition probability for each transition from a time delay at the initial time point to a possible time delay at the time point immediately following the initial time point is a uniform distribution.

7. 7. The method (100) of claim 1, wherein the possible time delays and / or points in time are discrete, and in particular the one or more transition probabilities for transitioning from a time delay at the first point in time to a time delay at the second point in time are arranged in a Markov transition matrix.

8. 8. The method (100) of claim 7 when dependent on claim 3, wherein the one or more transition probabilities for each transition from the time delay identified for the first time point to a possible time delay at the second time point are points in a finite-dimensional vector space, and the ambiguity set for the transition probabilities is encompassed by a sphere having a radius of a finite-dimensional norm around the point, the radius being based on a confidence level and / or a total number of the time delays identified for the first time point in the uncertain time delay system (40).

9. 9. The method (100) of claim 8, wherein the radius decreases when the confidence level of the time delay identified for the first time point in the uncertain time delay system remains constant and the total number increases.

10. adapting (140) the controller (10) and / or the observer (11) based on the determined (120) one or more transition probabilities and / or the determined (130) one or more ambiguity sets. The method (100) of any one of claims 1 to 9.

11. transmitting (141), in particular via the first communication interface, an update comprising the adapted (140) controller and / or the adapted (140) observer to the uncertain time delay system, in particular configured such that the controller (10) and / or the observer (11) is replaced by the adapted (140) controller and / or the adapted (140) observer; The method (100) of claim 10.

12. The method (100) according to any one of claims 1 to 11, executed on a computing unit (50) external to the uncertain time delay system (40), in particular on a cloud.

13. A controller (10) and / or an observer (11) configured to control a technical system (20), said controller (10) and / or said observer (11) communicating with said controlled technical system (20) at least in part via a communication network (30), said controller (10) and / or said observer (11) comprising: a second communication interface for transmitting information to a calculation unit (50), in particular to a calculation unit (50) external to the uncertain time delay system (40) including the controller (10) and / or the observer (11), from which a time delay for a first point in time for the uncertain time delay system (40) and a time delay for a second point in time for the uncertain time delay system (40) can be determined, the second point in time being later than the first point in time; the second communication interface is further designed to receive updates from the computing unit (50), the updates including an adapted (140) controller and / or an adapted (140) observer; The controller (10) and / or the observer (11) are designed to implement the updates. A controller (10) and / or an observer (11).

14. A controller (10) and / or observer (11) according to claim 13, a technical system (20) to be controlled; a communication network (30); Including, the controller (10) and / or the observer (11) communicate with the controlled technical system (20) at least in part via the communication network (30); An uncertain time delay system (40).

15. A computing unit (50) designed to perform a computer-implemented method (100) for a robust adaptive control function and / or a robust adaptive observer for an uncertain time-delay system (40) according to any one of claims 1 to 12, comprising: a first communication interface; A calculation unit (50).

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