Author(s): Maxwell’s Demon

Originally published on Towards AI.

State estimation for a nonlinear system

Every system that evolves with time has a state which is the set of quantities that describe it at a given moment. Examples are the position and velocity of a vehicle, the temperature and pressure in a reactor, or the charge left in a battery. State estimation is the problem of recovering that state when you cannot measure it directly. Such a state changes over time, and you never observe it in any way. What you do observe are noisy measurements related to it, arriving one at a time. The job is to combine those measurements with a mathematical model of how the state evolves with time, and produce the best estimate of the state at each step.

After introducing the nonlinear state estimation problem and comparing it to the standard (linear) Kalman filter, the article recaps the general filter structure and shows that nonlinear dynamics require computing moments of nonlinear functions of a Gaussian, which is where linear and nonlinear cases diverge. It then explains the unscented transform as a Gaussian-quadrature method: by choosing “sigma points” around the prior mean (with user-selected spread) and using weighted sums, the method approximates the needed expectations, means, variances, and covariances. Using these approximations, the piece derives the UKF recursion: generate sigma points from the current Gaussian prior, propagate them through the nonlinear process and measurement models, compute the six required statistical moments, and plug them into the estimator equations to obtain the filtered estimate and the one-step-ahead prediction. The article concludes with a concrete scalar example implemented in Python, demonstrating that the UKF tracks the true state well in the presence of nonlinear dynamics and measurement noise.

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Published via Towards AI