Switched Linear Systems: Control and Design

Traditionally, most of the research work in systems and control has been concerned predominantly with dynamical systems that are described purely as either time-driven continuous variable dynamics or event-driven discrete logic dynamics. Yet, there are numerous dynamical systems that are of heterogeneous nature. By heterogeneity, we mean that both continuous variable and discrete-event dynamics are present and interacting with each other to generate complex dynamical behaviors. These are termed hybrid systems and have been identified in a wide variety of natural and man-made systems, for example gene regulatory networks, embedded systems, process control, communication networks, aircraft and traffic control, and among many other fields. To study such kind of heterogeneous dynamical systems, one needs to develop new methods, since the traditional approaches may become unsuitable. For example, if one intends to capture, or approximate, the hybrid phenomena, like jumping and switching, via a continuous variable model, then it may result in a very complicated differential equation that is difficult to handle. In contrast, the discrete models might be too abstract for some applications, especially when the plant to be controlled has tight interactions of continuous variable and discrete event dynamics, or when the requirements are represented by time and event-based behaviors. The history of hybrid system research can be traced back at least to the 1950s with the study of engineering systems that contain relays and/or hysteresis. However, hybrid systems began to attract people’s attentions in the early 1990s, mainly because of the vast development and implementation of digital micro controllers and embedded devices. The last decade has seen considerable research activities in modeling, analysis and synthesis of hybrid systems involving researchers from a number of traditionally distinct fields, such as computer science and control system engineering [3], [2]. Therefore, it is not surprising that there are a variety of approaches for modeling, analysis, and synthesis of hybrid systems. On the one hand, computer scientists look at hybrid systems primarily as discrete (computer) programs interacting with the analog environment. They extend their computational models, such as finite state machine, automaton and Petri net, from discrete systems to hybrid systems by embedding the continuous variable dynamics into these discrete models. Typically, these approaches are able to deal with complex discrete dynamics and emphasize analysis results (verification) and simulation methodologies. On the other hand, researchers from the areas of dynamical systems and control theory extend the models and methodologies for traditional continuous variable systems, such as ordinary differential/difference equations, by including discrete variables so as to describe the jumping or switching phenomena. Typically, these approaches are able to deal with complex continuous variable dynamics and mainly concern stability, robustness and synthesis issues. This book falls into the second category and focuses on control and design for switched linear systems.