Control Hierarchies and Tropical Algebras

Cyber-physical systems are often characterised by large scale, distributed and heterogeneous dynamics. Hierarchical control has been a popular and widely used approach to deal with such systems. It can be interpreted as an attempt to handle complex problems by decomposing them into smaller subproblems and reassembling their solutions into a ”functioning” hierarchical structure. It typically involves a number of control layers operating on different time scales that may be clock driven or event driven. Signals on different levels of the control hierarchy may be of different granularity representing phenomena like measurement aggregation when passing from lower to higher level control. In prior work, which is set within Willems’ behavioural systems theory (e.g., [7, 8]), we have proposed a formal hierarchical framework that captures the notion of causal signal aggregation (both in time and signal space) for measurements propagating from the plant through the various control layers, and which provides certain consistency conditions that guarantee conflict-free interaction of the different control layers for a given inclusion-type specification [5, 6]. The proposed design procedure is bottom-up, and complexity reduction is achieved by interpreting specifications for lower control levels as abstractions of the plant under low level control. An essential task within the proposed hierarchical control synthesis procedure is then to come up with a suitable choice of specifications for the individual control layers, which, in total, have to be at least as restrictive as the provided overall specification. Because of the dual role of these low-level 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 difficult. In this talk, this intuitive trade-off will be formally investigated for a specific scenario: we assume that the top-level controller’s sole task is to decide about the timing of certain input events and that all other (discrete and continuous) control inputs are provided by lower control layers. The synthesis of the top-layer controller can then be based on an abstraction of the plant under low-level control in the form of a Timed Event Graph (TEG). It is a well known fact that the temporal evolution of timed event graphs becomes linear in certain tropical algebras. Tropical algebras, or dioid alge-