Joao P. Hespanha

As computers, digital networks, and embedded systems become ubiquitous and increasingly complex, one needs to understand the coupling between logic-based components and continuous physical systems. This prompted a shift in the standard control paradigm-in which dynamical systems were typically described by differential or difference equations-to allow the modeling, analysis, and design of systems that combine continuous dynamics with discrete logic. This new paradigm is often called hybrid or switched control.

This talk deals precisely with systems that result from the interconnection of differential equations with logic-based decision rules. Such systems are hybrid in the sense that some of the variables that describe their behavior take continuous values (e.g., the state of a differential equation) whereas others take discrete values (e.g., a Boolean value, or the state of a finite automaton). We are particularly interested in switched system. These are systems for which the continuous dynamics are effectively determined by the values of one or more discrete variables.

In the talk, we present several mathematical tools that have been developed to understand the behavior of switched systems. These tools are introduced in the context of specific applications where both logic and differential equations arise naturally. We draw these examples from areas as diverse as computer networks, vision-based robotics, and adaptive control. The goal of this talk is twofold: (i) demonstrate that switched systems are ubiquitous and of significant practical application, and (ii) show that a unified theory of switched systems is becoming available.

The time evolution of chemically reacting molecules is sometimes modeled using a stochastic formulation, which takes into account the inherent randomness of molecular motion. This formulation is especially useful for complex reactions inside living cells, where small populations of key reactants can set the stage for significant stochastic effects. In this talk, we show how Stochastic Hybrid Systems can be used to construct stochastic models for chemical reactions.

Hybrid systems combine continuous-time dynamics with discrete modes of operation. The states of such system usually have two distinct components: one that evolves continuously, typically according to a differential equation; and another one that only changes through instantaneous jumps. To model chemical reactions, we actually need Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events, much like transitions between states of a continuous-time Markov chains. However, the rate at which transitions occur is allowed to depend on both the continuous and the discrete states of the SHS.

Several tools are available to analyze SHSs. Among these, we discuss the use of the extended generator, infinite-dimensional moment dynamics, and finite-dimensional truncations by moment closure. The application of these tools is illustrated in the context of modeling the evolution of populations of molecules undergoing a system of chemical reactions.

Networked Control Systems (NCSs) are spatially distributed systems for which the communication between processes, sensors, actuators, and/or controllers is supported by a digital communication network. This type of systems exhibits several characteristics that make them unique from a control perspective.

In this talk we address the effect of limited communication bandwidth and network latency in the overall performance of a closed-loop NCS. Not surprisingly, there is a trade-off between the amount of communication resources utilized and the control performance achievable. For prototypical examples (linear processes and quadratic costs) we construct optimal communication logics that achieve optimal performance with minimal communication. The effect of network latency is also investigated in this context.