Overview

How does a cruise control keep your car at 60 mph even up a hill? How does a drone stay stable in the wind? Control theory is the math of keeping things steady.

Core Idea

Feedback Loop: Measuring the output (speed), comparing it to the desired setpoint (60 mph), and adjusting the input (gas pedal) to minimize the error.

Formal Definition (if applicable)

PID Controller (Proportional-Integral-Derivative): The most common control algorithm.

  • P: React to current error.
  • I: React to accumulation of past errors.
  • D: React to the rate of change of error (prediction).

Intuition

Driving a car.

  • You see you are drifting left (Error).
  • You turn the wheel right (Control Action).
  • You check again (Feedback). If you overcorrect, you swerve (Instability).

Examples

  • Thermostat: Turns heat on/off to maintain temperature.
  • Autopilot: Keeps a plane on course.
  • Homeostasis: The body regulating blood sugar and temperature.

Common Misconceptions

  • “Automation is easy.” (Tuning a controller to be fast but stable is an art.)
  • “Open loop is fine.” (Open loop—like a toaster timer—doesn’t check the result. If the bread is frozen, it won’t toast enough. Closed loop is better.)
  • Stability: Will the system settle down or oscillate forever?
  • Transfer Function: A mathematical representation of the system’s input-output relationship (Laplace Transform).
  • Robustness: Can the system handle disturbances?

Applications

  • Robotics: Balancing a walking robot.
  • Manufacturing: Precision assembly.
  • Space: Rocket guidance.

Criticism / Limitations

Linear control theory works well for simple systems, but real-world systems are often non-linear and chaotic.

Further Reading

  • Ogata, Modern Control Engineering
  • Franklin et al., Feedback Control of Dynamic Systems