Contact Person:
Jörg Raisch
Abstraction-based, hierarchical, and distributed control can be seen as different, but related, attempts to handle complex control synthesis problems. Abstraction-based control aims at providing approximations, or abstractions, of given plant models such that a controller that "works" for an abstraction will, provably, also "work" for the underlying, more complex model. Hierarchical control can be interpreted as a multi-level decomposition of a given control problem, where solutions of low-level problems are "tied together" by higherlevel control, with the latter typically based on abstractions of the plant under the respective low-level control. Finally, distributed control refers to a scenario where a process, or a set of interacting processes, is to be controlled via a set of local controllers which share a common goal but lack a central decision unit.
l-complete approximation [Moor and Raisch, 1999; Moor et al., 2002], and its extension, asynchronous l-complete approximation [Schmuck and Raisch, 2014b], were developed to facilitate control synthesis for hybrid dynamical systems, i.e., systems where continuous and discrete-event components interact. They provide purely discrete conservative approximations of the considered hybrid system, where conservativeness is in the sense of overapproximating the latter's external behaviour. In [Schmuck et al., 2015], the relation between different realisations of this class of abstractions and widely used quotient based abstractions (QBA) was investigated. It was shown that, in general, they are incomparable both in terms of behavioural inclusion and similarity relations.
Based on l-complete (asynchronous) approximations, standard methods from discrete event systems theory can be used to synthesise least restrictive control. This scenario was extended in [Park and Raisch, 2015] to cover the case where some of the external signals included in the abstraction are not visible for the controller.
If controller synthesis on the basis of the employed abstraction fails, l-complete (asynchronous) approximation allows for global abstraction refinement. This, however, may be computationally infeasible. For this reason, we have investigated a method for local refinement that focuses on aspects of the current abstraction preventing the controller synthesis step from being successful [Yang et al., 2018, 2020].
A formal hierarchical control synthesis framework which is general enough to encompass both continuous and discrete levels was outlined in [Moor et al., 2003; Raisch and Moor, 2005]. It guarantees that the control layers interact properly and do indeed enforce the overall specifications for the considered plant model. Its elegance stems from the fact that the specifications for lower control levels can be considered suitable abstractions for the plant under low-level control which may be used as a basis for the synthesis of high-level controllers.
An essential task within a hierarchical control synthesis procedure is then to come up with a suitable choice of specifications for the individual control layers. Because of the dual role of these specifications, this typically involves a non-trivial trade-off. E.g., imposing a less strict specification for a control layer will facilitate the control synthesis task for this layer, but will make the control synthesis task for higher level control more dificult. In [David-Henriet et al., 2012], this trade-off was formally investigated for a specific scenario, where the top control layer is only responsible for the timing of certain discrete events, and where the abstraction it is based on can be represented by a timed event graph.
A specific two-layer scenario involving a purely discrete top layer control system was developed in [Baldissera et al., 2016] and applied to the control of gene regulatory networks. This involves abstractions of gene networks in the form of finite state machines, where each state corresponds to a set of gene expression levels and the events are associated with the activation/repression of genes.
In a distributed control scenario, a process, or a set of interacting processes, is to be controlled via a set of local control agents which share a common goal but lack a central decision unit.
Consensus plays an important role in distributed control. We have investigated a number of different consensus scenarios. For example, in [Goldin and Raisch, 2014], agents with double integrator dynamics exchange position and velocity information via different undirected communication networks. It turns out that consensus can then be achieved even if neither of the two networks is connected. In [Arun Kumar et al., 2017], we have investigated consensus problems where information is only exchanged when triggered by discrete events and the triggering mechanism may either be static or dynamic. We have proposed an approach that uses results from the field of max-plus algebra to analyse max-consensus, which is especially important in applications such as minimum time rendezvous, leader election, and distributed synchronisation, for both timeinvariant and time-variant communication topologies [Monajemi Nejad et al., 2009, 2010]. Moreover, it was analysed when the application of a consensusbased control protocol to a network of Timed Event Graphs (TEGs) leads to a synchronised and stable overall TEG [Monajemi Nejad and Raisch, 2014].
In the context of the DFG priority programme 1914 (Cyber-Physical Networking), in two joint projects with S. Stanczak from the Network Information Theory Group at TU Berlin, we have investigated the interplay between control and wireless communication in consensus problems for multi-agent systems. In particular, we have been interested in how to exploit the wireless superposition property to dramatically reduce information exchange for different consensus algorithms. Results for linear consensus were reported in [Molinari et al., 2018a]. Results for max-consensus problems can be found in [Molinari et al., 2018b, 2021, 2022]. The superposition property of the wireless channel was also exploited to obtain efficient consensus-based formation control for multi-agent systems [Molinari and Raisch, 2019] and for distributively solving systems of linear equations [Molinari and Raisch, 2020].
In a case study [Molinari and Raisch, 2018], we investigated the use of consensus-based auction algorithms for the automation of road intersections. In [Molinari et al., 2019b; Molinari et al., 2020], these results were extended and combined with local model-predictive control to obtain a distributed control scheme for road intersections. Average consensus, consensus-based auctioning, and distributed model-predictive control were also combined to automate lane changes and collision avoidance on highways [Molinari et al., 2019a].
The application of consensus-based methods to control problems in electrical power grids has been explored in [Schiffer et al., 2013, 2016; Parada Contzen and Raisch, 2015, 2016; Krishna et al., 2018, 2020].