A Kalman-filtering derivation of simultaneous input and state estimation

Abstract

Simultaneous input and state estimation algorithms are studied as particular limits of Kalman filtering problems. This admits interpretation of the algorithm properties and critical analysis of their claims to being partly model-free and to providing unbiased estimates. A disturbance model, white noise of unbounded variance, is provided and the bias feature is shown to be a geometric projection property rather than probabilistic in nature. As a consequence of this analysis, the algorithm is connected, in the stationary case, to Algebraic Riccati equation computations for the gains, estimate covariances and filter frequency response.

Optimal Sensor Placement for Partially Known Power System Dynamic Estimation

“…We first describe the filtering method. A summary of KF-SISE, presented in [9], is given in the following subsection.…”

“…where x t , w t ∈ R n are the state vector and process noise at time step t, d t ∈ R m is the disturbance signal from the unknown part of the system, and y t , v t ∈ R m are the measurements and the measurement noise. The following assumptions are required to use the Kalman filtering formulation of SISE [9]. Assumption 1: The following is assumed to hold: i w t ∼ N (0, Q), v t ∼ N (0, R) and initial condition x O ∼ N (x 0|0 , P O ) are mutually independent Gaussian white noises, ii R is diagonal positive definite matrix, iii the pair (A, C) is observable, and rank CG = rank G = m. By defining X t+1 , K t+1 , M t+1 , D t and P t+1 respectively as the prior state covariance matrix, the Kalman gain for the state vector, the Kalman gain for the unknown input vector, the posterior disturbance covariance matrix and the posterior state covariance matrix, and xt+1|t+1 as the posterior estimates for the state and dt|t+1 as the estimate for the unknown input of the system, and the measurement sequence, Y t+1 {y t+1 , y t , .…”

“…In this subsection, we derive the information matrix for KF-SISE [9], which can simplify the analysis of estimation error and its covariance further in this study. The following lemma provides us with the information version of the KF-SISE.…”

“…Hence, the partially known power system concept has been recently introduced in [8]. In this context, the Kalman Filtering -Simultaneous Input and State Estimation (KF-SISE), proposed and investigated in [9], [10], is utilized to obtain the estimates of the states and unknown inputs of the partially known power grid. It should be borne in mind that various sensor placement methods based on steady-state/quasi-steady-state conditions, including those proposed in [11], [12], [13] are not applicable for modern electrical grids facing fast and dynamic changes.…”

Equivalence of recursive three‐step filter and infinity augmented Kalman filter for linear discrete‐time stochastic systems with direct feedthrough

“…Intuitively, when tends to be infinity, the result of AKF will be equal to that of the RTSF method. More recently, the equivalence between AKF in the infinity condition and MVU filter has been proved, respectively, in [19, 20] for the system that unknown input only affects the state. However, in the direct feedthrough systems, whether or not the result of corresponding AKF in the infinity condition will be equal to that of RTSF is still unknown.…”