01 Notes
Every note in the knowledge base. Use the search box or the field chips to narrow things down — each note carries several tags, so you can search by field, subtopic, method, or tool. New notes slot straight into this list.
Symplectic Matrices: From Phase Space to Optimal Control
What survives when a phase-space circle is stretched into an ellipse? Follow canonical pairing through Hamiltonian mechanics, symplectic integrators, ray optics, and quantum quadratures, then derive continuous- and discrete-time LQR as stable Riccati graphs inside a symplectic state–costate flow. Includes worked examples, three figures, and reproducible Octave checks.
Read the note (PDF) MathematicsThe Structure-Preserving Doubling Algorithm
Solve a discrete-time Riccati equation by joining optimal-control blocks: eliminate their shared boundary and double the horizon at every iteration. Derive the three SDA updates, see why symmetry and positivity survive, and follow a scalar convergence example through to a practical Octave/MATLAB implementation.
Read the note (PDF) Control TheoryLinear Matrix Inequalities for Robust Control
Six questions about certifying a plant whose matrices depend on an unknown bounded operator — what a certificate must quantify over, why no finite sample decides it, the multiplier that makes the "for all" constraint affine, and what changes when the uncertainty is structured. Answers are deliberately withheld — the note is the question map.
Read the note (PDF) Control TheoryTube Model Predictive Control
Six questions about making a nominal MPC plan robust to a bounded disturbance — why the nominal plan guarantees nothing, what set keeps the deviation from drifting, how that set tightens the constraints, why feasibility survives, what sets the tube size, and what changes when the tube shape is chosen online. Answers are deliberately withheld — the note is the question map.
Read the note (PDF) Control TheoryPID Control: Design, Tuning, and Modern Practice
How a PID controller works, how the gains interact, and how to move beyond textbook tuning to modern practice.
Read the note Control TheoryConstructive Nonlinear Control
A guided tour of Sepulchre, Janković & Kokotović for engineers who know linear control and want the mechanisms, not the folklore.
Read the note Control TheoryModel Identification in Control Theory
Building a trustworthy dynamical model from measured input–output data — least squares, prediction-error, recursive, and subspace methods.
Read the note Control TheoryThe Kalman Decomposition
A system has an external language (the transfer function) and an internal one (a state-space model) — the Kalman decomposition shows what the transfer function cannot see.
Read the note Control TheoryThe Pontryagin Maximum Principle
How to steer a system optimally when actuators have limits — a braking example, the brachistochrone, everything verified exactly.
Read the note Control TheoryControl as Inference
Six questions about planning with noisy trajectories instead of a nominal one — what a stochastic cost actually compares, the tilt that makes path weights computable, what the normalizer means, the change of variable that linearizes HJB, and which part of the weight makes Monte Carlo error grow with the horizon. Answers are deliberately withheld — the note is the question map.
Read the note (PDF) Control TheoryReinforcement Learning and Feedback Control
How an agent can learn the optimal controller online, along the plant’s own trajectories, without knowing its dynamics.
Read the note Control TheoryInside acados: a real-time NMPC solver
How an open-source NMPC solver squeezes a nonlinear optimal-control problem into a 1 millisecond budget: discretization, Gauss–Newton SQP, structure-exploiting QP solving, the real-time iteration, and the code-generation stack underneath.
Read the note Control TheoryDesigning a Filter for a Signal in a Control Loop
A controller is a noise amplifier, and a filter inside the loop is never free: its gain at crossover is nearly one but its phase comes straight out of your margin. Corner frequency, steepness, group delay, notch versus low-pass, and how to place all of it against a real PID speed loop.
Read the note Control TheoryContingency Planning: Keeping the Important Thing Working
Plan from functions and promises, not from component lists: what must keep running, how degraded it may safely become, and in what order recovery happens under scarce time and staff — with contingency MPC as the control-theoretic version of keeping an escape trajectory alive.
Read the note Control TheoryInterior-Point Methods: HPIPM & IPOPT
The algorithm that quietly powers HPIPM (QP for MPC) and IPOPT (large-scale NLPs): why it works, the mathematics, and how each solver adapts the core idea.
Read the note OptimizationSNOPT: Sparse Sequential Quadratic Programming
A tour of the engine inside one of the most battle-tested solvers for large-scale constrained nonlinear optimization — and when to reach for it.
Read the note OptimizationZeroth-Order Optimization
Descending without gradients — how to minimize a function you can only query, never differentiate.
Read the note OptimizationLinear Matrix Inequalities, From Zero
What an LMI is, why control engineers fell in love with them, and the semidefinite-programming machinery that solves them.
Read the note OptimizationInformation Geometry Optimization
Gradient-free black-box optimization by pushing a distribution around a manifold — and how CMA-ES, NES and PBIL are one algorithm in disguise.
Read the note OptimizationA Mathematical Introduction to Robotic Manipulation
After Murray, Li & Sastry: a robot is a curve in a group of matrices — kinematics, Jacobians, dynamics and control all follow from the exponential map.
Read the note RoboticsROS 2: Distributed Real-Time Robotics
Distributed real-time robotics taught to an engineer — nodes, topics, the execution model, and quality of service.
Read the note RoboticsALTRO: The Augmented Lagrangian Trajectory Optimizer
iLQR wrapped in an augmented-Lagrangian loop, with an active-set solution-polishing phase — built from first principles for constrained optimal control.
Read the note RoboticsMotion Planning via Optimal Transport
Around every waypoint of a batch of trajectories, probe the cost along a direction — the Sinkhorn step that plans robot motions without cost-function gradients.
Read the note RoboticsPOMDP: Acting on a Belief, Not a Guess
An MDP hands the agent the state; a POMDP hides it and returns an observation. This note builds the belief with Bayes’ rule, shows why acting on the most likely state can be far worse than paying to observe, and writes the belief-space Bellman equation whose tree explodes. Then it keeps the part a real planner keeps: the factored ego-times-driver transition that turns prediction into the belief-update step inside planning, and the guided branching that makes it run in real time.
Read the note (PDF) RoboticsEPSILON: Planning When Everyone Else Is Deciding Too
An engineer's tour of a multi-agent driving planner: closed-loop maneuver policies instead of open-loop control sequences, conditional focused branching that keeps only the scenarios that change the answer, and spatio-temporal corridors that turn every road semantic into a box — then into one QP via Bézier hulls.
Read the note RoboticsLie Groups for State Estimation in Robotics
A robot's pose lives on a curved set, so averaging, subtracting and nudging it with vector arithmetic leave that set. Nine questions rebuild estimation on the manifold — the Lie algebra, the exp/log maps, the plus and minus operators, the adjoint, manifold Jacobians, covariance, and the error-state filter. Answers are deliberately withheld — the note is the question map.
Read the note (PDF) RoboticsOptimal Transport, from the Ground Up
Move four spoonfuls of sand to build a transport plan, then turn the cheapest rearrangement into a distance between distributions. Pictures explain conservation, Wasserstein distance, cumulative totals on a line, and Sinkhorn scaling.
Read the note (PDF) MathematicsOrdinary Differential Equations
From a law of change to a predicted future — the mathematics every engineer uses daily, whether or not they name it.
Read the note MathematicsVisual Differential Geometry and Forms
The core ideas of Tristan Needham’s book, rebuilt as one visual lesson for engineers.
Read the note MathematicsA Micro Lie Theory for State Estimation
A walk-through of Lie groups, tangent spaces, and the exp/log maps that power state estimation in robotics.
Read the note MathematicsBayesian Inference: Ten Coins and a Stamped Box
A box of coins carries a label: each coin's bias is drawn from Beta(3, 5). Six questions about what that label does to the arithmetic — the single number every posterior shares and what it does to the spread of the answers, what one coin can and cannot say about the box it came from, why the answer reduces to adding the label's own counts to each coin's tally, where the label's answer beats a coin's own frequency and where it must lose, what the label prices for a coin nobody has flipped yet, and what this arithmetic cannot see when a second box holds two kinds of coin. Every number is a fraction of small integers, reproduced by one script beside the note; the answers are withheld — the note is the question map.
Read the note (PDF) MathematicsStable Diffusion, from First Principles
How a neural network learns to turn pure noise into images — and why doing it in a compressed latent space runs on your GPU.
Read the note AI & Machine LearningVariational Inference: Ten Flips and an Unknown Bias
Six questions about the one thing a posterior density always lacks: the number that turns a value per unit of coordinate into a probability. One coin model whose exact answer you can look up, then a prior that breaks it — what the bound guarantees, what a product of twelve factors cannot carry, and what a one-bump fit does to a two-bump truth. Answers are deliberately withheld — the note is the question map.
Read the note (PDF) AI & Machine LearningMathematica for Engineers
Calculate, solve, plot — how Mathematica turns a scientific computation into a single, readable line.
Read the note Programming & Tools02 Fields
The territory this knowledge base covers. Each field's count is computed from the notes on this page, so it stays accurate as new notes land.
Algebra
Matrix pencils, Schur and QZ forms, and the algebra underneath Riccati equations and structure-preserving solvers.
Control Theory
Feedback, identification, model-predictive, robust and stochastic optimal control.
Optimization
Interior-point, SQP, LMI and derivative-free methods for real problems.
Robotics
Manipulation, motion planning, trajectory optimization, state estimation and middleware.
Mathematics
ODEs, differential geometry, Lie theory and the analysis underneath.
AI & ML
Diffusion models, variational inference, reinforcement learning and decision-making under uncertainty.
Programming & Tools
Computer algebra, languages and software craft that turn ideas into running code.
03 About
What is this?
A personal, evolving knowledge base. The idea is to write down things the way I'd want them explained — with the motivation first, the reasoning laid out, and the details where they're actually needed.
- Questions first. Each note is built around the questions it sets out to answer, and a new note is published as a question map before it is written up.
- Intuition before formalism. Build the mental model, then nail the equations.
- Two formats. Recent notes are written in Typst and read as typeset PDFs, older ones are hand-written pages, and the grid above links both.
- Breadth is real. Control, optimization, numerical linear algebra, robotics, mathematics, AI, and tools.
- Always growing. This is a living document, not a finished product.
If something reads as unclear, it's probably a note that needs another pass — the gaps are as much part of the map as the notes themselves.