What I Study, Part 1: Control Theory

My research as a PhD student is in the field of control theory. When I say this, most people don’t know what it means. I am often asked if that means I am learning “the psychology of how to manipulate people” (I’m not). This page provides a brief introduction to the topic that doesn’t require any prior knowledge about math or engineering.

In order to understand control theory, it’s helpful to first introduce the concept of a dynamical system. A dynamical system is a system that changes over time. Some examples of dynamical systems are a pendulum (mechanical), a power transformer (electronic), a stock market (economic), and populations of predators and prey (ecological). A dynamical system is described using a list of numbers that change over time. We call the list of numbers the state of the system. The main questions we ask about a dynamical system is how it behaves over time.

• Is the state attracted to a particular point?
• Does state periodically return to the same point?
• Does the state remain in a particular region?

Consider, for example, an ecosystem with a population of a predator species and the population of its prey. The state of the system has two values at each moment in time: the population of the predator and the population of the prey. If either of these values goes to zero, then that species goes extinct.

In some dynamical systems, there are inputs that affect the behavior of the system. An input is a value that can be directly chosen at each moment in time. For a car, the inputs are the throttle (gas pedal), the brake, and the steering wheel. The position and velocity of the car cannot be controlled directly—to move the car to a new location, one must use the throttle and steering wheel to maneuver there. A dynamical system with inputs is a control system and the study of how to pick the inputs achieve various goals is called control theory. In general, our goal is to design the inputs so that the system

• goes where we want it,
• avoids obstacles, and
• minimizes energy use.