Lyapunov methods in the analysis of complex dynamical systems

Control systems today are becoming increasingly logic-based, and therefore incorporate discrete state dynamics as well as continuous state dynamics. Moreover, due to advances in communication systems and data networks, more and more systems are built over asynchronous (packet-switched) networks where signals can be lost or delayed by varying amounts. Examples include timing circuits, computer disk drives, parallelized numerical algorithms, and queuing networks. These systems do not fall within the framework of continuous synchronous control theory, and therefore, a new theory and control methodology is required for analyzing them.; These so-called complex systems can exhibit very complex behavior, so it is not surprising that many problems in this area are known or conjectured to be theoretically difficult (NP-hard) or even impossible (undecidable) to solve. Therefore, it is not expected to solve these problems exactly in practice. It is expected, however, to develop methods that are effective for most instances of these problems. One such method is to search over a fixed, finite dimensional class of Lyapunov functions that guarantee some specification for a given system—it may not be possible to find such a function, but if one is found, the result is unambiguous. This research develops Lyapunov function based analysis methods for the more complex dynamical systems described above.; Specifically, in this thesis, we provide a new Lyapunov-based theory and practice for three very important classes of complex systems: (1)  piecewise-linear systems, which can model many nonlinear systems, (2) hybrid systems, which are continuous systems with discrete logic and memory, and (3) asynchronous systems , which are continuous systems driven by discrete events. In each case, we present a class of Lyapunov functions for analyzing such systems. In addition, we show that searching over the proposed class of Lyapunov functions to prove some specification (e.g., stability) can be cast as optimization problems involving linear matrix inequalities, which can then be solved efficiently using widely available software. Examples are also included that demonstrate the effectiveness of the approach and significant improvement over the very few other existing methods.