Brief paperA multiple model multiple hypothesis filter for Markovian switching systems☆
Introduction
In many applications, the state of a dynamic system with time-varying dynamics must be estimated. Applications of these types of systems can be found in target tracking, change detection, digital communications and navigation, see e.g. (Gustafsson, 2000).
A specific class of these systems is the class of Markovian switching systems, where the modes are not directly observed. This type of system is also referred to as a hybrid system. The state of such a system is hybrid, i.e it consists of a continuous part, the kinematics and a discrete part, the mode. We will consider systems in which the mode is not directly observed and is thus said to evolute according to a hidden Markov model.
For the class of hybrid Markovian switching systems, a matching sequence of extended Kalman filters can be used for every hypothesis of the mode history. It is well known that the number of hypotheses, and therefore the computational load, associated with the optimal solution, grows exponentially in time. The computational load is proportional to , where M is the number of modes and k is the time index. Thus, the computational load of the optimal filter grows exponentially in time. Therefore, approximate methods are needed. Obviously, all these methods are suboptimal.
The generalized pseudo-Bayesian (GPB) algorithm has been developed, see (Tugnait, 1982, Blackman and Popoli, 1999). A more efficient and now widely used filter, that has been developed, is the interacting multiple model (IMM) filter. The original idea for this filter can be attributed to Henk Blom, see (Blom, 1984, Blom, 1990; Blom & Bar-Shalom, 1988; Mazor, Averbuch, Bar-Shalom, & Dayan, 1998; Blackman & Popoli, 1999). The IMM filter is the workhorse of many real-life applications that involve hybrid systems and is extensively applied in air traffic control systems and target tracking. In Mazor, Averbuch, Bar-Shalom, and Dayan (1998) the reader will find a complete and fairly recent overview of the IMM literature.
In Chapter 10 of Gustafsson (2000), pruning and merging strategies for hypothesis management of Markovian switching systems are discussed. A good and early treatment of pruning and merging strategies for hypothesis management is given in Pattipati and Sandell (1983). Alternative merging and hypothesis management in filters for target tracking have also been suggested in Koch, 1999a, Koch, 1999b.
In this paper a new filtering method for Markovian switching systems is proposed. The filter that is introduced in this paper, is the multiple model multiple hypothesis filter ( Filter). The basic idea is that a variable number of hypotheses on different mode histories is propagated over time. The number of hypotheses that are propagated is not fixed, but is situation dependent or data driven. The rationale behind this is that in ‘difficult situations’, i.e. situations in which a considerable amount of hypotheses have a relative high likelihood, the number of hypotheses that are maintained is increased and in ‘easy situations’, where only one or very few hypotheses have a high likelihood, the number of hypotheses is decreased substantially. This is the crucial difference with the IMM filter, where a fixed number of hypotheses is propagated over time, regardless of the situation at hand. Furthermore, the filter has the ability to deal with a variable hypotheses merging depth, where the IMM has a fixed hypotheses merging depth, i.e. a hypotheses merging depth of one in case of the standard IMM. The timing of the hypotheses merging and pruning is the same as in the IMM, i.e. before the prediction and update step. The novelty of the filter is the combination of the pruning and merging strategies and their timing.
The contribution of this paper is that it provides the user with a new filter, the filter, for a class of hybrid Markovian switching systems. This filter can be used as an excellent alternative for the popular IMM filter.
Moreover, for a realistic application, i.e. a target tracking application, it is shown that the filter outperforms the IMM filter in terms of performance and computational load.
The outline of the remainder of the paper is as follows: in Section 2 we provide and explain the system setup and problem formulation, in Section 3 we will present the filter, in Section 4 we give an example from target tracking to illustrate the newly proposed algorithm and finally in Section 5 we will draw some conclusions and point out some possible additional applications of the filter.
Section snippets
The system setup and basic problem formulation
Consider a discrete time hybrid Markovian switching system perturbed by additive Gaussian noise:where
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is the base state of the system.
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, where M is a positive integer, is the modal state of the system.
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is the measurement.
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is time.
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is the process noise.
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is the measurement noise.
The
The multiple model multiple hypothesis filter
In this section the filter is presented by providing an algorithmic filter description. The various steps are given in some detail, many of the steps involve similar operations as in the IMM filter, very detailed and explicit descriptions of these operations are given in (Mazor, Averbuch, Bar-Shalom, & Dayan, 1998).
In the algorithm, certain parameters have to be chosen or set by the designer in advance. These are a pruning threshold, , this represents a threshold on the hypothesis
A radar target tracking example
This section compares the performance and computational load of the filter to the performance and computational load of the IMM filter for a typical target tracking application of a long-range surveillance radar.
Although in a true target tracking application also other aspects, like data association and extraction play important roles and have a huge influence on the computational load and performance of the total system, see (Blackman & Popoli, 1999), we will only consider the filtering
Conclusions
In this paper a new filter for Markovian switching systems has been presented. The filter uses both pruning and merging techniques. An adaptive computational load is a powerful feature of the new filter. In difficult situations the load is increased and in easy situations the load is decreased. A realistic target tracking example shows that in different representative scenarios the new filter outperforms the popular and widely used IMM filter at a lower computational load. The filter can be
Acknowledgements
The authors wish to thank Ms. Mary Ann Bjorklund for the careful proof reading of the manuscript and her suggestions for grammatical improvements. Any errors left, grammatical or otherwise, are the sole responsibility of the authors. Furthermore, the authors like to thank Prof. Arun Bagchi and Dr. Wolfgang Koch for the discussions and suggestions.
Yvo Boers received the M.Sc. degree in applied mathematics from Twente University, The Netherlands, in 1994 and the Ph.D. degree in electrical engineering from the Technical University Eindhoven, The Netherlands, in 1999. Since 1999 he has been employed at THALES Nederland B.V. He works as a scientist and consultant in the area of system design and signal and data processing. His interests are in the areas of detection, filtering, target tracking, signal and data processing and control.
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Yvo Boers received the M.Sc. degree in applied mathematics from Twente University, The Netherlands, in 1994 and the Ph.D. degree in electrical engineering from the Technical University Eindhoven, The Netherlands, in 1999. Since 1999 he has been employed at THALES Nederland B.V. He works as a scientist and consultant in the area of system design and signal and data processing. His interests are in the areas of detection, filtering, target tracking, signal and data processing and control.
Hans Driessen received the M.Sc. and Ph.D. degrees in 1987 and 1992, respectively, both in electrical engineering from the Delft University of Technology. In 1993 he joined THALES Nederland B.V. (formerly known as Hollandse Signaalapparaten B.V.) as system design engineer. He is currently working as a scientist and consultant in the area of radar system design and the associated signal and data processing. His interests are in the area of application of stochastic detection, estimation and classification theory.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor J. Schoukens under the direction of Editor T. Soderstrom.