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**Chapter 7**

**An Approach to Hybrid Smoothing**

**for Linear Discrete-Time Systems**

Additional information is available at the end of the chapter

It is very important to consider simultaneous estimation of both system states and inaccessi‐ ble modes for hybrid systems with unknown modes [4,7,24,25]. This estimation is called hy‐ brid estimation. By the hybrid estimation we often want to know both a current mode and system state at each time through information of observation. However there exist cases that we want to know distributions of modes on long run time interval rather than each estimate of the modes themselves at each time to grasp global performance over long time intervals, for example, distributions of active modes in solar systems [5,21], distributions of active

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 stud‐ ied the smoothing problems by various approach, for example, maximum likelihood ap‐ proach [9,13,19], projection approach [14] and so on. It is well known that smoothers (noncausal estimators) more effectively estimates the states than filters (causal estimators)

It is well known that utilization of accumulated information of observation improves esti‐ mation performance. Nevertheless, on research of estimation for hybrid systems, little work has been done from the point of view of the noncausal information of observation, i.e., smoothing. In [9] Helmick et al. have presented a fixed-interval smoothing algorithm for discrete-time Markovian jump systems by maximum likelihood (ML) approach. However they have considered only the case with fully accessible modes and their approach is based on approximate approach to probability density functions (PDFs). Therefore they have pre‐

> © 2012 Nakura; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 Nakura; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

agents on formation or consensus via hybrid systems representation and so on.

**with Non-Gaussian Noises**

because of more information of observation.

Gou Nakura

**1. Introduction**

http://dx.doi.org/10.5772/51385

**Chapter 7**
