A geometric approach to differential-algebraic systems: from bisimulation to control by interconnection

Network modeling of complex systems often leads to models involving both differential equations and algebraic equations in the state variables, called differential-algebraic equation (DAE) systems. In this thesis we use concepts and results from geometric control theory to study several problems in DAE systems. The first problem that we study is the equivalence of DAE systems by bisimulation. As a preliminary result we characterize the set of consistent states and the set of solution trajectories. In the second problem we study a different notion of bisimulation relation for DAE systems having a regular matrix pencil. Under the assumption of regularity, we develop the notion of bisimulation relation as the direct product of two partial bisimulation relations corresponding to the fast and the slow subsystems. The third problem concerns disturbance decoupling for linear systems with complementarity switching. We present necessary and sufficient conditions for the linear system with complementarity switching to be disturbance decoupled. In general these two conditions do not coincide.

In the last problem, we study the control by interconnection problem of a standard input-state-output system, based on an abstraction system. An abstraction system is a lower-dimensional system whose external behavior contains the external behavior of the original system. First, we study the problem of constructing a controller for the abstraction system such that the interconnection of the abstraction system and the controller is bisimilar to a given specification system. Next we consider the problem of applying the controller system derived for the abstraction system to the original plant system.