Modeling and Analysis of Higher-Order Switched Linear Systems

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.

ACADEMIC AND SCIENTIFIC IMPACT

The relevance of the proposed research for the academic community, both national and international, lies in the absolute originality of the issues considered here. With the exception of a single conference publication recently produced by Prof. S. Trenn of Kaiserslautern, there has been no attempt at developing a higher-order approach to switched systems. Scientifically speaking, the problems we aim to investigate are by and large unexplored, given the narrow scope (state-space systems sharing the same state space) of previous research in this area. Successful results in addressing such issues would thus effect a considerable impact on the switched systems community, and would perhaps contribute to a re-evaluation and a re-alignment of research in this area to problems concerned with real-life situations (such as those posed by power distribution networks, control, genetics, and so forth), rather than academic abstractions thereof.

We plan to increase the impact on the academic community by aiming to publish in top control and systems journals (e.g. IEEE Transactions on Automatic Control, Automatica) and spreading our message in leading international conferences (e.g. IEEE CDC, American Control Conference). National-level seminars will disseminate the project outcomes on a more local basis.

IMPACT ON THE ENGINEERING COMMUNITY

The impact of this research will also be felt outside of academia. Given the importance within the project of the development of algorithms and given our determination on modelling naturally and effectively real switched systems, as opposed to dealing only with academic examples, the relevance of our research for the engineering community is very high. The possibility of modelling in a natural, modular way complex switched systems is important for applications, and our final aim to contribute to an integrated computer-aided design environment for switched systems informs every aspect of our work. Specifically, one of the proposed work packages aims at providing automatic identification of impulsive behaviours in switched systems, an important feature of any design tool. The other work packages aim at developing the theory so as to encompass larger classes of systems, and making use of efficient representations, thus enlarging the scope of application of our approach to more complex systems.

We have especially in our aim to contribute to the modelling and analysis of distributed power networks, that are of increasing importance for environmental and energetic reasons. We believe that our modular, parsimonious and efficient (less variables and less equations) approach has much to recommend itself for in such area, where accurate models for effective simulation are necessary to address the upcoming challenges set by an ageing infrastructure and increased demand. Preliminary contacts with the Electrical Power Engineering group at Southampton, a leader in research concerning distribution and transmission networks, are being further developed so as to provide our framework with a challenging test field for present, current and future research problems and their solution.

We plan to develop software implementing some of the algorithms devised during this research, so as to put them to the test of actual engineering situations, and to provide the scientific and engineering community with computer-algebra-based software tools usable in practice. Such software will be publicly available through the Web pages of the University of Southampton.