While I was hoping to upload this post by the weekend, one thing (laziness) led to another (watching copious amounts of Battlestar) and my to-do list persists! So it goes. With a new semester upon us, I figure I should give just a hint of the fascinating mathematics that I’ve been delving into the past couple of weeks. Some of the most fascinating bits of hybrid control theory that I’ve been exposed to thus far have been illustrated with seemingly trivial problems. (Hybrid control, in a nutshell, is the control of continuous system which encounter discrete switched states. Hopefully this is clarified with the bouncing ball example). However, the results of the analysis prove to be profound. One simple example, a bouncing ball, illustrates how naïve modeling results in physically impossible implications. In this case, Zeno behavior. 
Zeno behavior arises in low fidelity hybrid system models in which the state trajectory gets “stuck” in a switched area in finite time. (This behavior is aptly named after the paradox suggested by the Greek philosopher, Zeno. He conjectured that one was to step halfway to a wall, then half way again, and so on ad infinitum… you would never actually reach the wall… well, his true example involved Achilles and a tortoise). In the bouncing ball example, with a simple model of the ball that attenuates velocity each bounce we encounter a point where the ball bounces an infinitely many times in a limited time. Physically, we know that this is impossible. However, if we’re not careful with our models, Zeno behavior can crop up and destroy both control design efforts and simulation ability.
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