stateestimate

A state estimate is an estimate of the internal state variables of a dynamical system based on a mathematical model and observed data. The true state is often not directly measurable, so the estimate combines prior knowledge from a model with information contained in measurements that are typically corrupted by noise. In a common discrete-time state-space formulation, the system is described by x_k+1 = f(x_k, u_k) + w_k and y_k = h(x_k) + v_k, where x is the state, u is the input, y is the measurement, and w and v represent process and measurement noise.

The estimation task is to compute x̂_k, an estimate of x_k, using measurements up to time k. When the estimate is updated after each new measurement, it is called filtering. When all measurements (past and future) are used to refine past states, it is called smoothing. When the model is used to forecast future states, it is called prediction. A key performance metric is the estimation error and its uncertainty, often summarized by a covariance matrix P_k.

Common methods include the Kalman filter for linear Gaussian models, the extended Kalman filter and unscented Kalman filter for nonlinear systems, and particle filters for highly non-Gaussian cases. Deterministic observers, such as Luenberger observers, achieve state estimation using known inputs and a system model. The feasibility and quality of state estimation depend on observability, model accuracy, and noise characteristics.

Applications span robotics, navigation and tracking, aerospace, economics, and any domain requiring inference of hidden system states from noisy measurements.