🧠 Optimal Control Tutorial

🎥 YouTube Lecture Playlist

[(https://youtu.be/bp_qtED03vY?si=esJ1FsKYW3IAbkP6)]

Python implementations of fundamental Optimal Control algorithms:

• LQR – Linear Quadratic Regulator
• iLQR / DDP – Iterative LQR / Differential Dynamic Programming
• MPC – Model Predictive Control

This repository is organized for learning / teaching optimal control, with clean and minimal Python implementations.

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📂 Repository Structure

Optimal_Control/
│
├── LQR/
│ ├── lqr.py # Continuous-time LQR (solve CARE)
│ ├── double_integrator_lqr.py # LQR demo on double integrator
│ └── manipulator_lqr.py # LQR for n-DOF manipulator (via linearization)
│
├── iLQR/
│ ├── iLQR.py # iLQR / iLQG solver
│ ├── boxQP.py # Box-constrained QP solver for control limits
│ └── demo_inverted_pendulum.py# iLQR demo: inverted pendulum
│
├── MPC/
│ ├── qpmpc/
│ │ ├── mpc_problem.py # Define linear MPC problem
│ │ ├── mpc_qp.py # Convert MPC → QP
│ │ ├── plan.py # Container for MPC results
│ │ └── solve_mpc.py # Solve MPC using qpsolvers
│ └── examples/ (TODO)
│
└── viz/
├── LQR_manipulator.gif
├── iLQR_inverted_pendulum.gif
├── bipedal_mpc_onestep.gif
└── bipedal_mpc_multistep.gif

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🎯 Algorithms Overview

1️⃣ LQR — Linear Quadratic Regulator

LQR is an optimal control method for linear systems. It computes a state-feedback controller that minimizes a quadratic cost on states and control inputs.

Key ideas:

• Assumes linear dynamics

• Penalizes deviation from desired state

• Produces an optimal control law of the form u = -Kx

• Very fast and widely used in robotics & control

✔️ LQR Manipulator Example

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2️⃣ iLQR / DDP — Iterative LQR

iLQR generalizes LQR to nonlinear systems.

Main procedure:

• Linearize nonlinear dynamics locally

• Quadratically approximate the cost

• Perform an LQR backward pass to compute gains

• Apply line-search updates to refine the solution

• Repeat until convergence

This makes iLQR suitable for pendulums, manipulators, and complex nonlinear robots.

✔️ iLQR Inverted Pendulum Demo

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3️⃣ MPC — Model Predictive Control

MPC solves a finite-horizon optimal control problem at every timestep.

Characteristics:

• Predicts future states over a horizon

• Optimizes control inputs while respecting constraints

• Applies only the first control input

• Repeats the process at the next timestep

• Great for robots that must follow trajectories or stay within limits

This repository converts MPC into a Quadratic Program (QP) and solves it using QP solvers.

✔️ MPC Bipedal Example (one-step)

✔️ MPC Bipedal Example (multi-step)

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⚙️ Requirements

Install dependencies:

pip install numpy scipy matplotlib qpsolvers