Balancing an Inverted Pendulum on a cart using a simple neural network

Описание: The control problem of balancing a pole on a cart, often referred to as the inverted pendulum problem, is a classical problem in control theory and robotics. Here's the background:

🔧 The Problem Setup
Imagine a pole hinged to a cart that can move left and right on a track.

The pole starts in an upright (unstable) position.

The goal is to apply horizontal forces to the cart to keep the pole balanced vertically (i.e., prevent it from falling over).

🧠 Why It's Important
Classic Benchmark in Control Theory:
The inverted pendulum is a standard benchmark for testing and demonstrating:

Feedback control algorithms

Stability analysis

Real-time control

Real-World Analogs:

Rockets balancing during launch

Segway and self-balancing robots

Human posture and walking (biomechanics)

Nonlinear, Unstable Dynamics:
The system is inherently nonlinear and open-loop unstable — small disturbances grow unless actively corrected. This makes it challenging and interesting.

📘 Historical and Academic Context
First studied in the early 20th century in physics and engineering.

Formalized in the mid-20th century with the rise of modern control theory.

Has been a core teaching example since the development of state-space control, PID control, optimal control (LQR), and modern AI techniques like reinforcement learning.

🧪 Control Techniques Used
Linear Control:

PID controllers

Linear Quadratic Regulator (LQR)

Nonlinear Control:

Feedback linearization

Sliding mode control

Modern Approaches:

Reinforcement learning

Neural network controllers

Model Predictive Control (MPC)

🔍 Why It's a Good Learning Tool
Simple to model (with two main variables: angle and position)

Easy to simulate and build physically

Deep insights into the challenges of control system design:

Sensing and state estimation

Actuator limits and time delays

Tradeoffs between stability and performance