Modeling and Analysis of Higher-Order Switched Linear Systems

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Switched linear dynamics arise e.g. in power electronics, distributed power systems, multi-controller schemes, etc.; classically they are modeled by state space equations of the same order. However, such representations may be unnecessarily restrictive or complex: first-principles models are often of higher order, and often the modes often do not really share a common global state space. Consider e.g. distributed power systems, where the loads connected to the source have dynamics of different complexity: the state space changes depending on which loads are actually connected. Modelling such systems using a global state variable results in a more complex model than necessary, reduces modularity, and thus is done mainly to satisfy a priori-defined structural properties. I am developing a new framework where the dynamical modes are described by systems of higher-order linear constant coefficient differential equations. The system trajectories satisfy these equations on time-intervals determined by a switching signal. "Gluing conditions", i.e. algebraic equations involving the system trajectories and their derivatives before and after the switching instant, specify whether the piecewise restrictions can be concatenated to form an admissible trajectory. This framework is based on polynomial algebra and is thus conducive to the use of computer algebra techniques for modelling, analysis, and control. It is efficient in the number of variables and equations used to model a system, completely modular, and integrates perfectly with hierarchical modelling. The final aim of my research is contributing to the creation of a modelling, simulation and design environment based on a sound mathematical methodology aligned with efficient simulation techniques.In this framework I obtained encouraging results on Lyapunov stability, but much work remains to be done. I propose here to investigate 3 areas: - detection of impulsive phenomena:Gluing conditions may implicitly specify that certain trajectories cannot be concatenated smoothly at switching. This may imply instantaneous surges in the values of the system variables, which may lead to component breakdown. I aim at developing algebraic tests to ascertain when such situations may occur. These tests could be used to detect automatically the presence of impulsive behavior from the equations describing the systems, and thus would be useful for implementing my framework in a computer-aided design environment. - dissipative switched systems:I want to extend my framework to open systems and to modelling the interaction between dynamics and environment associated with energy exchange. This is a first step towards the investigation in this new framework of control techniques based on dissipation ideas for switched systems. - polynomial methods for differential variational inequalities. In many situations (e.g. in circuits, chemical processes, genetics, hydraulics, etc.) switching depends on the satisfaction of sets of algebraic inequalities, rather than on an external switching signal. Such a point of view can also efficiently overcome the combinatorial complexity associated with modelling transitions via switches. I want to investigate how to represent inequality-based transition rules in a polynomial setting; the well-posedness of solutions in a polynomial setting; the algebraic characterization of stability and the computation of Lyapunov functionals. This is a completely new area of application of polynomial algebraic techniques to the description of dynamical systems. The three areas described above constitute challenging test fields for the soundness of my approach, and offer the opportunity for developing it further in directions important for applications.