State Estimation

Although stochastic differential equations provide accessible mathematical models and have become standard models in many areas of science and engineering, features extraction from high‐order processes is usually much more complex than from first‐order processes such as the Langevin equation. Estimation performed using the state‐space model solves simultaneously two problems: filtering measurement noise and, in some cases, process noise and thus filter, and solving state‐space equations with respect to the process state and thus state estimator. Methods of linear state estimation can be extended to nonlinear problems to obtain acceptable estimates in the case of smooth nonlinearities. Both optimal filtering and optimal smoothing minimize the mean square error using data taken from the past up to the current time index. Since many systems are nonlinear in nature and therefore have nonlinear dynamics, their mathematical representations require nonlinear ODEs and algebraic equations.