Lecture Notes on Choice & Control

In the Lecture 2 on Differential Equations & Domains, we have seen the beginning of cyber-physical systems, yet emphasized their continuous part in the form of differential equations x′ = f(x). The sole interface between continuous physical capabilities and cyber capabilities was by way of their evolution domain. The evolution domain Q in a continuous program x′ = f(x)&Q imposes restrictions on how far or how long the system can evolve along that differential equation. Suppose a continuous evolution has succeeded and the system stops following its differential equation, e.g., because the state would otherwise leave the evolution domain Q if it had kept going. Then what happens now? How does the cyber take control? How do we describe what the cyber elements compute and how they interact with physics? This lecture extends the model of continuous programs for continuous dynamics to the model of hybrid programs for hybrid dynamics, which is exactly the model we need to describe the hybrid system dynamics of cyber-physical systems. Hybrid programs combine discrete and continuous dynamics in a seamless way. This lecture is based on material on cyber-physical systems and hybrid programs [Pla12b, Pla10, Pla08, Pla07]. Continuous programs x′ = f(x)&Q are very powerful for modeling continuous processes. They cannot—on their own—model discrete changes of variables, however1, which is useful to understand the impact of computer decisions on cyber-physical systems. During the evolution along a differential equation, all variables change continuously in time, because the solution of a differential equation is (sufficiently) smooth.