The$H_infty$Fixed-Interval Smoothing Problem for Continuous Systems

Abstract

The smoothing problem for continuous systems is treated in a state space representation by means of variational calculus techniques. The smoothing problem is introduced in an criterion by means of an artificial discontinuity that splits the problem in term of forward and backward filtering problems. Hence, the smoother design is realized in three steps. First, a forward filter is developed. Secondly, a backward filter is developed taking into account the backward Markovian model. The third step consists of combining the two previous steps in order to compute the smoothed estimate. An example shows the efficiency of this proposed smoother.

An Approach to Hybrid Optimal Smoothing for Linear Discrete-Time Systems with Non-Gaussian Noises

“…Much work has been done for smoothing theory for both of continuous-and discrete-time systems ( [1,2,3,6,8,9,10,12,13,14,15,18,19,20,22,23] and so on). Various researchers have studied the smoothing problems by various approach, for example, maximum likeli-hood approach ( [9,13,19]), projection approach ( [14]) and so on.…”

“…We adopt most probable trajectory (MPT) approach to guarantee the optimality of estimation methods. On this approach, given information of observation, we consider optimal control problems where we seek optimal control by which averaged noises Proceedings of the 44th ISCIE International Symposium on Stochastic Systems Theory and Its Applications Tokyo, Nov. [1][2]2012 energies are minimized for averaged systems throughout the mode distributions. In [24,25] Zhang has presented hybrid filtering algorithm by MPT approach for the continuous-and discrete-time hybrid systems.…”

“…In the smoothing case we can expect better estimation performance by taking into consideration noncausal information of observations. The hybrid smoother is realized by two filters approach ( [2,8,12,20,22,23]). Finally we give numerical examples and verify that we can obtain better estimation performance by smoothing than filtering.…”

$H_{\infty}$ Fixed-Interval Smoothing Estimation for Time-Delay Systems

“…One of the most common is the smoother that was presented by [16], called Rauch-Tung-Streibel smoother which is computationally less costly than others. In the sense, it has been known that the fixed-interval smoothing estimation for linear delay-free systems is identical to the fixed-interval smoothing problem, significant contributions being [3]- [5], [7], [14], [15]. By using variational calculus approach, the fixed-interval smoother has been designed in [3] where the techniques in [6] are used to construct a smoothing estimate.…”

“…In the sense, it has been known that the fixed-interval smoothing estimation for linear delay-free systems is identical to the fixed-interval smoothing problem, significant contributions being [3]- [5], [7], [14], [15]. By using variational calculus approach, the fixed-interval smoother has been designed in [3] where the techniques in [6] are used to construct a smoothing estimate. Alternatively, the minimum-variance fixed-interval smoother is introduced by calculating a Riccati equation in [4], [5].…”