Research By Area

Zeno behavior occurs in a hybrid system when there are an infinite
number of transitions (discrete events) in a finite amount of time. This behavior can prevent a simulation from running or a control law from being successfully implemented, so it is of great practical importance.  Since this behavior is unique to hybrid systems, it also provides an important proving ground for hybrid systems theory. There are two areas in which to explore Zeno phenomena: detection and elimination.

In our work, we provide the first sufficient conditions for the existence of Zeno behavior in a class of hybrid systems (first-quadrant hybrid systems) with non-trivial dynamics, and demonstrate that the existence of Zeno behavior is related to the stability of a certain type of equilibria, termed Zeno equilibria. This observation motivates a Lyapunov-type theorem for the global stability of Zeno equilibria in first-quadrant hybrid systems; hence, this theorem gives conditions on the global existence of Zeno behavior.  Using hybrid category theory, this result can be extended to general hybrid systems.  If there is a morphism from a hybrid system to a first-quadrant hybrid system with a (globally) stable Zeno equilibria set, then the hybrid system also has a stable Zeno equilibria set and so is also Zeno.

We also investigated two distinct methods for eliminating Zeno behavior: we provide a constructive method for universally eliminating Zeno behavior in a class of hybrid systems while preserving stability, and we give a method for carrying Zeno executions (trajectories) past a Zeno point by completing the hybrid system model. We argue that at the Zeno point, a hybrid system should switch to a holonomically constrained dynamical system.  That is, we extend the behavior of a hybrid system past the Zeno point by ``composing'' the hybrid system with a specific dynamical system obtained from the original hybrid system.

Underlying every discrete or continuous dynamical system is a topological space---the space on which the evolution of the system occurs. The topology of this space can be used to infer important properties about the system. For example, Morse theory makes this connection explicit via homology and can be used to give conditions on when a system is not globally stabilizable.

Underlying every hybrid system is a hybrid topological space. Using preexisting mathematical constructions, we can associate to a hybrid system a single topological space: its underlying topological space.   By considering the homology of this space, we establish a homology theory for hybrid systems termed hybrid homology.  Moreover, we provide a ``Morse type theorem'' which says that when the hybrid homology of a hybrid topological space is trivial, any hybrid system with this hybrid topology cannot be Zeno.  We can show, therefore, that the underlying topological space of a hybrid system gives useful informationabout the behavior of the hybrid system, especially with respect to Zeno behavior.

Reduction of mechanical systems---systems whose dynamics are described by Lagrangians or Hamiltonians---with symmetries plays a fundamental role in understanding the many important and interesting properties of these systems.  Most significantly, it allows us to reduce the dimensionality of a mechanical system, thus circumventing the curse of dimensionality that often plagues the modeling, analysis and simulation of these systems.  Since the curse of dimensionality is even more prevalent in hybrid systems, reduction is arguably more important for these systems than for their continuous counterparts.

We have derived a unifying framework in which to carry out hybrid reduction, generalizing classical reduction to a hybrid setting. Utilizing hybrid category theory, we can hybridize all of the major ingredients necessary for classical reduction: Hamiltonians, symplectic manifolds, differential forms, Lie groups, Lie algebras, group actions and momentum maps.  That is, we can show (building upon the work of Marsden and Weinstein) that when there is a hybrid symplectic manifold on which a hybrid Lie group acts symplectically, we can reduce the hybrid phase space to another hybrid symplectic manifold by "dividing out" the hybrid symmetries; moreover, hybrid trajectories on the hybrid phase space correspond to hybrid trajectories on the reduced hybrid phase space.  These general results have been applied to robotic and mechanical systems.

Robotic and mechanical systems provide the quintessential examples of dynamical systems. Over time, they have been the catalyst for many important theoretical developments because of the clear connection between mathematical models of these systems and physical reality. Robotic and mechanical systems undergoing impacts are naturally modeled as hybrid systems. Systems of this form provide a rich and interesting class of hybrid systems.

Our research has focused on understanding unilaterally constrained
mechanical systems, both with respect to reduction and Zeno behavior.  Systems of this form typically consists of a Lagrangian
(or Hamiltonian) together with a set of admissible configurations
(often dictated by physical constraints on the system); this data
defines a hybrid Lagrangian (or a simple hybrid mechanical system). The term ``hybrid'' is used because there is an instantaneous non-smooth change in the velocity of the system when the boundary of the admissible configuration space is reached, which models impacts in a physical system. Moreover, from hybrid Lagrangians, we obtain simple (Lagrangian or Hamiltonian) hybrid systems; these systems have the ability to model a wide range of robotic systems, e.g., robotic bipedal walkers.

We are able to give conditions on when Routhian and cotangent bundle reduction can be extended to a hybrid setting for simple hybrid systems; this is an important special case of general hybrid reduction.  Moreover, we give a method for carrying Zeno executions (flows) past a Zeno point for hybrid systems obtained from hybrid Lagrangians.

Embedded systems commonly refer to systems in which hardware and software interact with physical environments.  In an embedded
system, therefore, different components of the system evolve according to processes local to the specific components.  Across the entire system, these typically heterogeneous processes may not be compatible, i.e., answering questions regarding the concurrency, timing and causality of the entire system---all of which are vital in the actual physical implementation of the system---can be challenging even if these questions can be answered for specific components.

A heterogeneous network of embedded systems can be modeled mathematically by a network of tagged systems (which provide a specific denotational semantics). Taking the heterogeneous composition of this network results in a single, homogeneous, tagged system. We address the following question is: when is semantics (behavior) preserved by composition? To answer this question, we use the framework of hybrid category theory to reason about heterogeneous system composition and derive results that are as general as possible. In particular, we show that composition is endowed with a universal property by demonstrating that it  corresponds to the limit of a diagram. Using this universality, we can derive verifiable necessary and sufficient conditions on when composition preserves semantics.

Modeling and simulating complex systems plays an important role in their design and implementation.  This is especially true for hybrid systems. Unfortunately, numerical issues that are prevalent in the simulation of continuous systems are amplified in hybrid systems due to the necessity of accurately detecting state-dependent events.  In addition, the existence of pathological behavior unique to hybrid systems, e.g., Zeno behavior and grazing phenomenon, often prevents the accurate simulation of these systems.

Our research has centered on detecting and eliminating such phenomenon.  For example, our results on Zeno detection have practical implications to modeling and simulation; they have been used to a priori detect Zeno behavior in communication networks modeled as hybrid systems.   We also investigate stochastic approximations of deterministic hybrid systems based on the error-bounds produced by numerical integration methods; these approximations are provably non-Zeno and can handle both grazing phenomenon and event detection. Finally, addresses the interchangeability of hybrid system tools based on the semantics of different models; this has applications to the construction of a hybrid system toolbox.