Colloquium Mathematics - Brayan Shali, University of Groningen

Title: Contract theory for linear control system

Abstract: Modern engineering systems, such as intelligent transportation systems, smart grids, smart manufacturing systems, etc., often comprise a large number of interconnected components. The design requirements for the components of such systems are becoming increasingly complex. Consequently, the development of components usually requires specialized expertise and is, thus, handled by different (independent) teams. However, independent teams seldom have the facilities to collaborate effectively, which can cause long and costly delays. This can be avoided by adopting a method for specifying design requirements that is inherently modular, i.e., that allows components to be considered independently. One such method is based on using so-called contracts, first introduced in the field of computer science. While contract theories have been developed for various system classes, these are generally restricted to systems with discrete variables evolving in discrete time.

Motivated by this, we develop a contract theory for a class of dynamical control systems with continuous variables in continuous time. In particular, we introduce assume-guarantee contracts for linear time-invariant dynamical systems with inputs and outputs. These contracts are defined as a pair of linear systems called assumptions and guarantees, and they serve as specifications for components through two aspects. First, the assumptions capture the expected dynamics of the environment in which the component operates. Second, the guarantees capture the required dynamics of the component when interconnected with a compatible environment. We formally develop notions of contract implementation, refinement and conjunction, which allow us to express, compare and combine specifications using contracts. In addition, we develop notions of contract composition, which, together with the notion of refinement, enables the independent design of components within interconnected systems.