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

Abstract

Abstract-This paper develops information filter and array algorithms for the linear minimum mean square error estimator (LMMSE) for discrete-time Markovian jump linear systems (MJLSs). A numerical example to show the advantage of the array algorithm in fix-point implementations is presented in order to demonstrate the advantage of this approach.

Fast array algorithm for filtering of Markovian jump linear systems

“…where B.Z/ is given in Equation (15). Moreover, Q Z i C1ji can be computed through the following equation:…”

“…Markovian jump linear systems (MJLSs) have received great attention in the literature, mainly in the last 20 years. Several fundamental concepts developed for standard state-space systems have been extended to this class of systems, as for example, controllability, observability, optimal control, and optimal filtering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. As it was pointed out by [1,7], MJLSs have been widely used in several fields of signal processing such as impulsive deconvolution, digital communications, and target tracking.…”

“…However, when Kalman filters are computed via Riccati equations, some fundamental properties to guarantee their stability, as for example, Hermitian solutions, are no longer guaranteed (see for instance, [17] for more details). Although the computational and numerical issues have been studied for standard state-space systems for a long time, for MJLSs, only [15,18] have dealt with this kind of problem. These two works proposed array algorithms to compute the linear minimum mean square error estimator (LMMSE) for MJLSs, which provide more accuracy and numerical stability if compared with solutions on the basis of Riccati equations.…”

“…The first part solves a Lyapunov equation whose result is used in the second part, which solves a simplified Riccati equation. The better performance of this algorithm, if compared with the array algorithms of [15,18], is because the algorithm exploits the properties that all parameters of the estimator developed in [19] can be frozen. Its solution converges with the assumptions that the system is mean square stable and the Markov chain is ergodic.…”

“…Its solution converges with the assumptions that the system is mean square stable and the Markov chain is ergodic. As an example of the effectiveness of the fast array algorithm proposed, we can compare the dimensions of the pre-array algorithm proposed in Theorem 3.1 and the pre-array developed in [15]. They are .m C N n/ .m C˛/ and .m C N n/ .m C 3N n/, respectively, where m is the dimension of the output variable, n is the dimension of the state variable, N is the number of state spaces of the Markov chain, and˛6 N n.…”