Hierarchies, spikes, and hybrid systems : physiologically inspired control problems

Abstract

In animal motor control and locomotion, neurons process information, muscles are the actuators, and the body is the plant. Control theory is the standard mathematical language for describing motor control and locomotion, but many phenomena in physiological control remain outside of the scope of control theoretic reasoning. Unlike traditional engineering control systems, nearly all the components of physiological control systems have complex dynamics. Instead of a fast centralized computer, an animal implements controllers using a distributed network of slow, nonlinear, and noisy neurons. Rather than having linear plants and actuators, the animal must control limbs with nonlinear and hybrid dynamics. This dissertation develops basic control theory motivated by physiological systems. Dynamical phenomena that arise in physiology but remain outside the scope of mathematical methods are isolated and studied in general control theoretic frameworks. In particular, three problems are discussed: distributed linear quadratic Gaussian (LQG) control with communication delays, control over communication channels modeled after spiking neurons, and Zeno stability of hybrid systems. Motivated by the presence of delays in the human motor system, Chapter 2 explores the architecture of distributed LQG controllers when communication between subsystems is limited by delays. Sensory and motor command information is processed in several different regions throughout the nervous system. Since processing speed in neurons is limited, information from different sensory and motor regions can only be integrated after a time delay. In spite of this difficulty, humans make efficient and reliable motions that are well-described by optimal feedback control. Optimal delay compensation is studied in a distributed LQG framework. The structure that emerges as the result of optimization resembles a management hierarchy, bearing similarities with the organization of the motor system. Networked control systems, in which communication between the controller and the plant occurs over a special neuron-inspired channel, are analyzed in Chapter 3. In addition to being the basic computing elements, neurons are the long-range communication channels of the body. Neurons transmit information in the form of short-lived voltage spikes, called action potentials. Sufficient conditions for stable control over the spiking channel are presented, along with bounds on tracking error and data rates. The final technical chapter studies the connections between Zeno behavior and Lyapunov stability. Zeno behavior occurs in a hybrid system when an infinite number of discrete transitions occurs in a finite amount of time. While Zeno behavior results from modeling abstractions, it is commonly observed in models of mechanical systems undergoing impacts, including models important for locomotion. Often, Zeno behavior is associated with dynamical mode transitions, such as knee locking and the transition between bouncing and sliding. To reason about such transitions without modifying the models, the chapter on hybrid systems gives Lyapunov-like sufficient conditions for Zeno behavior. A technique for constructing the Lyapunov-like certificates is presented for a general class of mechanical systems undergoing impacts.