Akreon Tutorial Series
From Dynamics to Feedback with Uncertainty
A Cartpole-based tutorial arc for understanding decision systems under uncertainty.
Overview
One system, the full decision loop
This tutorial arc uses the cartpole system as a compact example of Akreon's broader decision engine thesis. Across the series, we move from nonlinear dynamics to state-space models, feedback synthesis, observer design, optimal control, and uncertainty modeling.
The goal is not just to stabilize a toy system. The goal is to expose the recurring structure behind real decision problems: state, dynamics, objectives, constraints, uncertainty, and feedback.
Rendered Notebook
Technical reference
The consolidated notebook contains the nonlinear model, state-space derivation, controller and observer synthesis, numerical results, diagnostics, and closed-loop simulations for the complete arc.
Read the rendered notebookThe notebook is the detailed technical reference. The chapter summaries below provide a guided path through its equations, code, plots, and results.
Video Series
Watch the tutorial arc
The six-part companion series walks through the derivations, design choices, results, and limitations developed in the notebook.
Watch the YouTube playlistChapters
Six parts
- Part 01
Equations of Motion
Build the coupled nonlinear cartpole dynamics from cart translation, pole rotation, actuator force, and gravity. This establishes the physical model and identifies the nonlinear terms that shape the control problem.
- Part 02
Linearization to State Space
Linearize the dynamics around the unstable upright equilibrium, assemble the continuous-time state-space model, and examine its poles, controllability matrix, and controllability Gramian.
- Part 03
State Feedback via LMIs
Convert the bilinear Lyapunov stability condition into a convex LMI using a change of variables. Recover the feedback gain, close the loop, and verify that the resulting poles are strictly stable.
- Part 04
Observer Feedback
Use position and angle measurements to reconstruct the full state with a Luenberger observer. Observability diagnostics and dual LMI synthesis lead to an observer-based dynamic compensator.
- Part 05
Optimal Control Design
Compare LQR and LQG designs with H₂ state-feedback and observer synthesis. The chapter connects cost weighting, disturbance channels, estimator design, and packed dynamic compensators.
- Part 06
Uncertainty Modeling
Represent pendulum-mass variation with a polytopic model and synthesize gains that satisfy every vertex. Solver diagnostics and nonlinear rollouts compare nominal and robust controllers under disturbance.