Feedback Loops and PID Control

Lesson 1: Bridging Code and Physical Motion

The "Cruise Control" Challenge

Imagine you are a car's computer. Your goal: Maintain exactly 60 mph.

You hit a steep hill. Speed drops to 52 mph. What do you do?
You crest the hill and go down. Speed hits 68 mph. Now what?
How fast and smoothly do you react?

Feedback Loop Simulation

SETPOINT

60

MEASURED

52

Anatomy of a Feedback Loop

Setpoint

Desired State
\(r(t)\)

Σ

Controller (PID)

The Brain
\(u(t)\)

Plant / Robot

The Physics
\(y(t)\)

Feedback / Sensor

Proportional Term (P)

Correcting the Present Error

01

\[ u_p(t) = K_p \cdot e(t) \]

where \(e(t) = \text{setpoint} - \text{measured}\)

The harder you are away from the target, the harder you push.

The Downside:

High \(K_p\) leads to oscillation. Low \(K_p\) results in steady-state error.

Response with High \(K_p\)

Fast reaction, but overshoots wildly.

Response with Low \(K_p\)

Stable, but never actually reaches the target.

Integral Term (I)

Correcting Persistent Error

02

\[ u_i(t) = K_i \int_{0}^{t} e(\tau) d\tau \]

Summing the error over time.

It "notices" if the target isn't being met and adds more power until the gap closes.

Eliminates Steady-State Error

Perfect for systems with friction or constant disturbances (gravity).

The "I-Windup" Problem

If the motor is physically stuck, the Integral keeps growing... and growing. When the obstacle is removed, the robot catapults uncontrollably.

// Solution: Anti-Windup Clamping
if (integral > max) integral = max;

Derivative Term (D)

Predicting the Trajectory

03

\[ u_d(t) = K_d \frac{de(t)}{dt} \]

Rate of change of the error.

The "brakes." If the error is closing quickly, the D-term reduces output to prevent overshoot.

The Noise Trap:

Derivative is sensitive to sensor noise. It can make a motor "chatter" or jitter.

The Damper

Makes the system smooth. Like a shock absorber on a car or a door closer.

The PID Control Law

\(u(t) = K_p e(t) + K_i \int e dt + K_d \frac{de}{dt}\)

P-Term

Fast action. Reacts to today.

I-Term

Accuracy. Remembers yesterday.

D-Term

Stability. Predicts tomorrow.

Discrete Implementation

// Run this in your main control loop (e.g. every 10ms)

error = setpoint - sensor_value;

integral += error * dt;

derivative = (error - last_error) / dt;

output = (Kp * error) + (Ki * integral) + (Kd * derivative);

last_error = error;

In today's lab, you will implement this code on your microcontrollers to spin a DC motor at a constant RPM.

Feedback Control Systems for Robotics

Feedback Loops and PID Control

Tuning and Stability Analysis

State Space Representation

LQR Control Design

Trajectory Tracking and Path Following

Feedback Basics Slides

PID Tuning Lab Worksheet

Stability Analysis Slides

Destructive Tuning Lab Worksheet

State Space Slides

Matrix Modeling Worksheet

LQR Design Slides

LQR Optimization Lab Worksheet

Path Following Slides

Robot Racing Tournament Guide

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