Constructive global analysis of hybrid systems

Abstract

Many systems of interest are dynamic systems whose behavior is determined by the interaction of continuous and discrete dynamics. These systems typically contain variables or signals that take values from a continuous set and also variables that take values from a discrete, typically finite set. These continuous or discrete-valued variables or signals depend on independent variables such as time, which may also be continuous or discrete. Such systems are known as Hybrid Systems. Although widely used, not much is known about analysis of hybrid systems. This thesis attempts to take a step forward in understanding and developing tools to systematically analyze certain classes of hybrid systems. In particular, it focuses on a class of hybrid systems known as Piecewise Linear Systems (PLS). These are characterized by a finite number of affine linear dynamical models together with a set of rules for switching among these models. Even for simple classes of PLS, very little theoretical results are known. More precisely, one typically cannot assess a priori the guaranteed stability, robustness, and performance properties of PLS designs. Rather, any such properties are inferred from extensive computer simulations. In other words, complete and systematic analysis and design methodologies have yet - emerge. In this thesis, we develop an entirely new constructive global analysis methodology for PLS. This methodology consists in inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. These maps are proven globally stable by constructing quadratic Lyapunov functions on switching surfaces. Impact maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an LTI flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. This representation of impact maps allows the search for quadratic Lyapunov functions on switching surfaces to be done by simply solving a set of LMIs. Global asymptotic stability of limit cycles and equilibrium points of PLS can this way be efficiently checked. The classes of PLS analyzed in this thesis are LTI systems in feedback with an hysteresis, an on/off controller, or a saturation. Although this analysis methodology yields only sufficient criteria of stability, it has shown to be very successful in globally analyzing a large number of examples with a locally stable limit cycle or equilibrium point. In fact, existence of an example with a globally stable limit cycle or equilibrium point that could not be successfully analyzed with this new methodology is still an open problem. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. We have shown that this methodology can be efficiently applied to not only globally analyze stability of limit cycles and equilibrium points, but also robustness, and performance of PLS. Using the same ideas, performance of on/off systems in the sense that bounded inputs generate bounded outputs, can also be checked. Among those on/off and saturation systems analyzed are systems with unstable nonlinearity sectors for which classical methods like Popov criterion, Zames- Falb criterion, IQCs, fail to analyze. This success in globally analyzing stability, robustness, and performance of certain classes of PLS has shown the power of this new methodology, and suggests its potential towards the analysis of larger and more complex PLS.