# 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.

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