Overview

Control Theory is the hidden technology that keeps planes in the air, cars in their lanes, and robots standing up. It’s about forcing a system to behave the way you want it to, despite disturbances.

Core Idea

The core idea is correcting error. You measure the difference between where you are and where you want to be (the error), and you apply a force to reduce that error.

Formal Definition

The study of dynamical systems with inputs (controls) and outputs (measurements). The goal is to design a Controller that manipulates the inputs to achieve a desired output.

Intuition

  • Cruise Control:
    1. Set speed: 60 mph.
    2. Car goes up hill, slows to 55.
    3. Error: -5 mph.
    4. Controller: Give more gas.
    5. Car speeds up.
  • Balancing a Broom: You constantly move your hand to keep the broom upright. You are the controller.

Examples

  • PID Controller: The most common control algorithm (Proportional-Integral-Derivative). It looks at:
    • P: Current error (Where am I?).
    • I: Past error (Have I been wrong for a long time?).
    • D: Future error (How fast am I approaching the target?).
  • Segway: Uses gyroscopes and control theory to stay upright.

Common Misconceptions

  • Misconception: It’s just “on/off.”
    • Correction: Bang-bang control (on/off) is crude (like a cheap heater). Good control is smooth and proportional.
  • Misconception: You can control anything.
    • Correction: Some systems are Uncontrollable (you don’t have the right levers) or Unobservable (you can’t see what’s happening).

Applications

  • Aerospace: Autopilots, rocket guidance.
  • Robotics: Walking, grasping.
  • Economics: Central banks trying to control inflation (interest rates are the control input).

Criticism and Limitations

  • Model Dependency: If your mathematical model of the system is wrong, your controller will fail (sometimes catastrophically).

Further Reading

  • Feedback Control of Dynamic Systems by Franklin et al.