University of Groningen
A geometric approach to differential-algebraic systems
Megawati, Noorma
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Publication date: 2017
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Megawati, N. (2017). A geometric approach to differential-algebraic systems: from bisimulation to control by interconnection. University of Groningen.
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Summary
In this thesis we use concepts and results from geometric control theory to study several problems in differential-algebraic equation (DAE) systems. The first problem is the definition and characterization of the notion of bisimulation relation for DAE systems. As a preliminary result we fully characterize the solution set of DAE systems, where we restrict to (ordinary) continuous and piecewise-differentiable solutions. We use geometric control theory in order to explicitly describe the set of consistent states and the set of solution trajectories. Next, we define the notion of bisimulation relation for general DAE systems including internal disturbances. These internal disturbances are used for modelling ‘non-determinism’ in the system dynamics. ‘Non-determinism’ means that from a given initial state and for a given input function, the state may evolve into different solution trajectories. Such non-determinism generally occurs for abstraction systems, where the state space of the original deterministic systems is reduced to a lower-dimensional state space. A linear-algebraic algorithm for computing the maximal bisimulation relation between two DAE systems with internal disturbances is provided. We also specialize the general definition of bisimulation relation for DAE systems to the case where the matrix pencil of the DAE system is regular. Furthermore, we develop the notion of simulation relation for DAE systems as a one-sided version of bisimulation relation. In particular, abstraction systems are defined as systems by which the original DAE system is simulated.
In the second problem we study a different notion of bisimulation relation for DAE systems, where we also consider states outside the consistent subset. For this case, we restrict attention to DAE systems with a regular matrix pencil. It is well-known that under this regularity assumption the state vector of the DAE system can be decoupled into two parts: one part corresponding to an ordinary input-state-output behavior with a standard proper transfer matrix, called the slow subsystem, and the other one related to a polynomial transfer matrix, called the fast subsystem. The solution trajectories of the system are the direct sum
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106 Summary
of the solution trajectories of slow and the fast subsystem. Using notions from geometric control theory we develop the notion of bisimulation relation as the sum of two partial bisimulation relations; one corresponding to the fast subsystem and one corresponding to the slow subsystem. We also provide an algorithm for computing the maximal bisimulation relation in this sense. This is finally extended to simulation relations and abstraction systems.
The third problem concerns disturbance decoupling for linear systems with complementarity switching. These systems are obtained by taking a standard linear input-state-output system and imposing complementarity zero constraints on the inputs and outputs. Often of the modes of the resulting hybrid system are DAE systems. Considering additional outputs and external disturbances we obtain a necessary condition and a sufficient condition for the linear system with complementarity switching to be disturbance decoupled.
In the last problem, we consider the control by interconnection problem for standard input-state-output systems, basing the controller design on a lower-dimensional abstraction system of the original system. First, we study the problem of constructing a controller for the abstraction system (including internal distur-bances) such that the DAE system resulting from interconnecting the controller system with the abstraction system is bisimilar to a given specification system. Next we consider the problem of applying the controller system derived for the abstrac-tion system to the original plant system. Here we make a distincabstrac-tion between the situation where the set of control variables of the abstraction system is equal to the set of control variables of the plant system, and the more general situation where this is not anymore the case. In this last case we need to modify the interconnection of the controller to the original plant system. The main theorem consists of showing that the resulting interconnection of the original plant system and the controller system derived for the abstraction system is simulated by the specification system.
