Array Algorithms for H 2 and H ∞ Estimation

Abstract

Currently, the preferred method for implementing H 2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms which a r e t ypically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically o er an order of magnitude reduction in the computational e ort. Using our recent o b s e r v ation that H 1 ltering coincides with Kalman ltering in Krein space, in this paper we d e v elop array algorithms for H 1 ltering. These can be regarded as natural generalizations of their H 2 counterparts, and involve propagating the inde nite square-roots of the quantities of interest. The H 1 square-root and fast array algorithms bothhave the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H 1 lters. These conditions are built into the algorithms themselves so that an H 1 estimator of the desired level exists if, and only if, the algorithms can be executed. However, since H 1 square-root algorithms predominantly use Junitary transformations, rather than the unitary transformations required in the H 2 case, further investigation is needed to determine the numerical behaviour of such algorithms.

Information filtering and array algorithms for discrete-time Markovian jump linear systems

“…To avoid this problem, in usual state space systems, square-root array algorithms have been proposed to solve this numerical problem [14], [3], [15], [16], [19], [20], [25], [23]. Besides providing the guarantee that the final covariance matrix will be positive at each step of the recursion, array algorithms reduce the dynamic range in fixed-point implementations and assure better condition numbers than the conventional Kalman filter algorithm.…”

A fast‐array Kalman filter solution to active noise control

“…with y 0 ¼ y 1 0 þ y 2 0 and yðkÞ ¼ y 1 ðkÞ þ y 2 ðkÞ: Substituting (7)-(9) into the state-space equations (16)- (17) gives the final state-space description of the active control system considered in this paper…”

“…Such a robustness analysis is well known for the (modified) FxLMS type algorithms, for a recent contribution see Reference [16]. We will also investigate the potential of exploiting H 1 state-estimation [17] and robust Kalman filtering [18].…”

Algoritmo Array Rápido para Filtragem Robusta de Sistemas Singulares com Variação Estruturada dos Parâmetros no Tempo

“…O filtro de Kalman calculado via algoritmo array rápido apresenta algumas vantagens se comparado ao cálculo tradicional, tendo um aumento da eficiência e da estabilidade numéricas, devido ao uso de transformações ortogonais nos cálculos, além de uma redução da faixa dinâmica dos valores calculados em implementações por aritmética de ponto fixo, veja [1] e as referências contidas nela.…”