ARTICLE IN PRESS Signal Processing 89 (2009) 2171–2177 Contents lists available at ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro A novel interacting multiple model algorithm$ Qu HongQuana,, Pang LiPingb, Li ShaoHongb a b College of Information Engineering, North China University of Technology, Beijing 100144, China School of Electronics and Information Engineering, Beihang University, Beijing 100191, China a r t i c l e in fo abstract Article history: Received 1 August 2008 Received in revised form 28 April 2009 Accepted 28 April 2009 Available online 6 May 2009 For maneuvering target tracking, the interacting multiple model (IMM) algorithm employs a ﬁxed model set. The performance of this algorithm depends on the model set adopted. The result of using too many models is as bad as the case of too few models. Therefore, a variable structure IMM (VSIMM) was presented and applied to ground target tracking. This algorithm improves performance and reduces computational load with using auxiliary information. But it is difﬁcult to extend the VSIMM to other scenario (for example, aerial target), where there is not auxiliary information such as a map. A novel interacting multiple model (Novel-IMM) algorithm was presented to solve the problem of model set adaptation without auxiliary information. The Novel-IMM algorithm consists of N independent IMM ﬁlters operating in parallel, and each independent IMM ﬁlter also consists of multiple sub-ﬁlters, which operate interactively. In every time index, only one IMM output of a certain model set is used; but for a long time, the algorithm will alternatively choose an output of the model set to be the optimum ﬁnal output. The Novel-IMM approach was illustrated in detail with an aerial complex maneuvering target tracking example. & 2009 Elsevier B.V. All rights reserved. Keywords: Interacting multiple model Variable structure IMM Model set adaptation Novel-IMM 1. Introduction Computing the optimal state estimate for a jump Markov system requires exponential complexity, and hence, practical ﬁltering algorithms are necessarily suboptimal. In target tracking literatures, suboptimal multiple model ﬁltering algorithms, such as the interacting multiple model (IMM) method, are widely used for state estimation of such systems [1]. The IMM estimation has received a great deal of attention in recent years due to its unique power and great success in handling problems with both structural and parametric uncertainties and/or changes, and in decomposing a complex problem into simpler sub-problems, ranging from target tracking to fault detection and isolation, and from biomedical signal $ Supported by National Science Foundation of China (no. 50808007) and the NCUT Young Major Research Foundation (no. 200802). Corresponding author. Tel.: +86 10 88801519. E-mail address: qhqphd@sina.com (H.Q. Qu). 0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2009.04.033 processing to process control [2,3]. The application of IMM ﬁlters for detection and diagnosis of anticipated reaction wheel failures in the attitude control system has been described and developed in Refs. [4,5]. The IMM method for fault detection of railway vehicle from the measurement of the lateral acceleration of bogie and body and from rate of bogie has been described in Refs. [6,7]. The IMM algorithm, which uses a ﬁxed set of models, usually performs reasonably well for problems with a small model set. Many practical problems, however, involve more than just a small number of models [8]. When these algorithms are applied to solve target tracking problems, it happens that only using a small number of models cannot get satisfying result. At any time, the target trajectory evolves according to one of a ﬁnite number of predetermined model set. This requires that the model set include as many models as necessary to handle the varying target motion characteristics. But this cannot improve the performance successfully. The result of using too many models is as bad as the case of too few ARTICLE IN PRESS 2172 H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177 models. As demonstrated in Ref. [9], using more models makes it no better. In fact, the performance will deteriorate if too many models are used due to the excessive ‘‘competition’’ from the ‘‘unnecessary’’ (excess) models. The dilemma is that more models have to be used to improve the accuracy, but the use of too many models would degrade the performance [10]. Several modiﬁed IMM algorithms were proposed for improvement of performance or computation efﬁciency in recent years. A selected ﬁlter IMM (SFIMM) algorithm which uses a subset of ﬁlters with the speciﬁc subset chosen using decision rules was presented to improve computation efﬁciency of maneuvering target tracking [11]. A variable structure IMM (VSIMM) algorithm was presented to solve the dilemma where the model set not only differs across targets, but also varies with time for a given target [2,3]. Not only does VSIMM inherit the effective cooperation strategies of the IMM and the superior output processing of the AMM (autonomous multiple model), but it also adapts to the outside world by producing new elemental ﬁlters if the existing ones are not good enough and by eliminating those elemental ﬁlters that are harmful. This algorithm works successfully in ground target tracking where the model set adapts sequentially according to the target position and the road network conﬁguration [12–15]. The major objective of the MM is to achieve best modeling accuracy with a minimum number of models. Here VSIMM estimation has certain advantages. In a general setting, the problem of efﬁcient model set design for MM estimation is still open, Researches in the ﬁeld of VSIMM focus on the efﬁcient model set design [16–18] and model set adaptation algorithms [1,3,19,20], which includes the decision for activating a candidate model as well as terminating the model in effect. In general, those decisions should consist of a set of complex rules based on both a priori and a posteriori information about the current system mode in effect, but there are not uniﬁed theories and practical methods to guide us how to use these rules. These theories about model adaptation is much more challenging [21,22]. So the advantages in all presented model set adaptation methods based on the statistical hypothesis testing have been surprisingly limited. In addition, it is very difﬁcult to realize the VSIMM without the auxiliary information. A novel interacting multiple model (Novel-IMM) algorithm has been presented in this paper to solve the problem of model set adaptation without auxiliary information. This method adopts independent parallel model set method but not a serial model set adaptation which is adopted in VSIMM. It consists of N independent IMM ﬁlters operating in parallel, and each independent IMM ﬁlter also consists of multiple sub-ﬁlters, which operate interactively. In every time index, only one IMM output of a certain model set is used; but for a long time, the algorithm alternatively chooses an output of the model set to be the optimum ﬁnal output. The method does not use the decision for model activation and the decision for termination of the model in effect. The computer simulations illustrate the Novel-IMM could improve the performance of target tracking. 2. The Novel-IMM algorithm 2.1. The IMM algorithm Multiple model (MM) estimation is a powerful approach to adaptive estimation. It is particularly good for systems subject to structural as well as parametric changes. In this approach a model set is selected to represent (or ‘‘cover’’) the possible system behavior patterns and the overall estimate is obtained by a certain combination of the estimates based on these models. The multiple model approach is best described in terms of stochastic hybrid systems. The IMM algorithm consists of r interacting ﬁlters operating in parallel. In the IMM approach, at time k the state estimate is computed under each possible current model using r ﬁlters, with each ﬁlter using a different combination of the previous model-conditioned estimates (mixed initial condition). Fig. 1 describes the IMM algorithm, which consists of r interacting ﬁlters operating in parallel, one cycle of the algorithm consists of the following [23–25]: (1) Calculation of mixing probabilities: lk ðijjÞ ¼ cj ¼ r X 1 p mi ; cj ij k i; j ¼ 1; 2; . . . ; r pij mik (1) i¼1 where lk(i|j) is mixing probability, [pij] is the model transition probability matrix, mik is the model probability, and cj is normalizing constant. Fig. 1. IMM algorithm (r ¼ 2). ARTICLE IN PRESS H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177 (2) Interaction: 0j X^ kjk ¼ r X 2.2. The VSIMM algorithm i X^ kjk lk ðijjÞ; j ¼ 1; . . . ; r (2) i¼1 0j X^ kjk is mixed initial condition for the ﬁlter where matched to model j. The covariance corresponding to the above is ¼ P 0j kjk r X i j i j lk ðijjÞfP ikjk þ ðX^ kjk X^ kjk ÞðX^ kjk X^ kjk Þ0 g i¼1 j ¼ 1; :::; r (3) 1 (3) Model-matched ﬁltering: Compute X^ kþ1jkþ1 P1kþ1jkþ1 2 2 ^ X kþ1jkþ1 P kþ1jkþ1 using Kalman ﬁlter. The likelihood function of each model is expððd j Þ2 =2Þ kþ1 Ljkþ1 ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð2pÞM Sjkþ1 j ¼ 1; . . . ; r (4) j where ðdkþ1 Þ2 ¼ vjkþ1 ðSjkþ1 Þ1 ðvjkþ1 Þ0 , vjkþ1 , and Sjkþ1 are the innovation vector and its variance matrix, and M is the measurement dimension. (4) Model probability update: Ljkþ1 cj mjkþ1 ¼ Pr j j¼1 Lkþ1 cj ; j ¼ 1; . . . ; r (5) (5) Estimate and covariance combination: X^ kþ1jkþ1 ¼ r X j X^ kþ1jkþ1 mjkþ1 ; j ¼ 1; . . . ; r (6) j¼1 P kþ1jkþ1 ¼ r X 2173 j mjkþ1 fPjkþ1jkþ1 þ ½X^ kþ1jkþ1 X^ kþ1jkþ1 j¼1 j ½X^ kþ1jkþ1 X^ kþ1jkþ1 0 g (7) Note that this combination is only for output purposes and it is not part of the algorithm recursions. It can be got from above equations that the performance will deteriorate if too many models are used due to the excessive competition from the unnecessary models. The IMM approach computes the state estimation that accounts for each possible model using a suitable mixing of the model-conditioned estimations depending on the model probability. These algorithms are decision free (no maneuver detection decision is needed) and undergo a soft switching according to the latest updated mode probabilities. When target motion is complex, small number of models is insufﬁcient, worse still, using more models will result in excessive competition from the unnecessary models. In Ref. [9] the conclusion indicates that two models is the best solution in normal cases when model transition matrix is effective. Considering model match, three models are often used in IMM algorithms. One model set cannot solve the diploma, so the multiple model set algorithms are proposed and applied to complex target motion tracking. Most of the work on IMM estimator has considered only ﬁxed model sets, i.e. it is assumed that, at any time, the target trajectory evolves according to one of a ﬁnite number of predetermined model set. This requires that the model set include as many models as necessary to handle the varying target motion characteristics. All the models in the estimator are kept throughout the entire tracking period and, when it is necessary for a large number of models, it brings extra computational load, still more, it may degrade estimation accuracy. One remedy for the above shortcomings is to vary the model set in the IMM estimator based on some criteria to yield better estimation. This results in a variable structure IMM (VSIMM) estimator where the model set not only differs across targets, but also varies with time for a given target. In [10], a general framework for variable structure hybrid estimation based on a graph theoretic formulation was proposed and applied to ground target tracking [12–15]. It assumes that the total models can be covered by a number of model set, and a particular model set is running at any given time determined by a hard decision. One cycle of the simple and practical algorithm is described in Fig. 2. Model set adaptation is done by using additive auxiliary information. The targets under track are moving along a constrained path, for example, a highway, with varying obscuration due to changing terrain conditions. In addition, the roads on which the targets travel can branch, merge, or cross. Some of the targets can also move in an open ﬁeld. Because of the varying terrain conditions, the possible ways (models) in which a target trajectory can evolve also vary. For example, at junctions, target motion uncertainty increases—the target can move along any of the roads meeting at the junction. Another example of terrain-conditioned motion is that of a target along a highway or a bridge, where the motion of the target is orthogonal to the road or the bridge is restricted. In this case, the motion of a target is highly directional. Similarly, another constraint is imposed when the targets are moving in a ﬁle, where the target motion is unidirectional and the target velocity is such that ‘‘passing’’ is not allowed. Kirubarajan presents a VS-IMM estimator, where ﬁlter model set is adaptively modiﬁed depend on the terrain topography (auxiliary information) [10]. For example, the added uncertainty at junctions is handled with model set Model Set 1 Model Set 2 ... Model Set r Model Set Selection (Auxiliary Information) Model Set n General IMM Fig. 2. VSIMM algorithm. ARTICLE IN PRESS 2174 H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177 which represents motion along the possible roads (hard decision using additive auxiliary information). These additional model set is replaced by the primitive model set after the target passes through the junction. At each scan, the structure of the estimator for every target is individually modiﬁed based on the known topography. This enables the estimator to handle the variation in the possible motion modes across targets as well as with time for each target. The VSIMM algorithm eliminates the need for carrying all the possible models throughout the entire tracking period, signiﬁcantly improving performance. But the model set adaptation is very difﬁcult to extend to aerial target without auxiliary information. 2.3. The Novel-IMM algorithm Although much effort has been made and several ways have been tried, the IMM and VSIMM are not very successful for complex motion track without prior information. The objective of this paper is to develop a better method to solve the dilemma about how many models to be used and how to interact with them. The IMM algorithm consists of r interacting ﬁlters operating in parallel. In the IMM approach, at time k the state estimation is computed under each possible current model using r ﬁlters, with each ﬁlter using a different combination of the previous model-conditioned estimates (mixed initial condition). The Novel-IMM algorithm consists of N independent IMM ﬁlters operating in parallel, and these multiple IMM ﬁlters are independent. But each independent IMM ﬁlter also consists of multiple sub-ﬁlters, which operate interactively. At time k, the state estimation is computed with all possible models in the N model sets, and in these possible compute, every IMM ﬁlter uses their own independent initial condition without being combined. One of outputs of the IMM ﬁlter of the N model sets is used to match target motion, but the chosen model set would vary with time. Therefore, it is very important to set up a method to choose the optimum output in these independent IMM ﬁlters. To determine the optimum ﬁnal output, the Novel-IMM algorithm performs a continuous MSPT (model set probability test) for N hypotheses (H1: model set 1 is chosen to match target motion, H2: model set 2 is chosen, y, HN: model set N is chosen), and only one IMM output with its corresponding model set is chosen as the optimum output or ﬁnal estimation. So model set adaptation is done by using MSPT without any auxiliary information. Independence between different IMM ﬁlters can solve the excessive competition from the unnecessary models. The MSPT algorithm helps to choose the model set which matches the target motion best. When target motion changed, the MSPT algorithm will choose another model set accordingly. For each IMM, the model set does not change with time. The ﬂow chart for one cycle of the Novel-IMM logic is shown in Fig. 3. We consider two model sets. A recursion of the Novel-IMM algorithm consists of four fundamental steps: (1) Parallel independent IMM: X^ kþ1jkþ1 ðiÞ, Pkþ1jkþ1 ðiÞ, mkþ1 ðiÞ, and Lkþ1 ðiÞ are computed for each model set using 1 1 general IMM (see Fig. 1). X^ kjk ð1Þ and P^ kjk ð1Þ are target state and its covariance estimation of model 1 in 2 2 model set 1 at k time index, while X^ kjk ð1Þ and P^ kjk ð1Þ are target state and its covariance estimation of model 2 in model set 1. PT(1) and mk(1) are transition matrix and model probability vector of model set 1. 1 1 2 2 X^ kjk ð2Þ,P^ kjk ð2Þ,X^ kjk ð2Þ,P^ kjk ð2Þ,P T ð2Þ, and mk(2) are corresponding terms of model set 2. PT(1) and PT(2) are as same as Pij in Eq. (1). mk(1) and mk(2) are model probability. Fig. 3. Novel-IMM algorithm (two model sets and two models in each model set). ARTICLE IN PRESS H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177 (2) The likelihood function value of every model set: C kþ1 ðnÞ ¼ r X Ljkþ1 ðnÞmjk ðnÞ; n ¼ 1; . . . ; N multiple model set algorithm and its model set is adaptive not using any additional auxiliary information. (8) j¼1 3. Simulations (3) Normalized likelihood function value of every model set: The weight probability of model set kk+1(n) is updated from Ck+1(n) and kk(n): C 2175 kkþ1 ðnÞ ¼ PN kþ1 ðnÞkk ðnÞ n¼1 C kþ1 ðnÞkk ðnÞ ; n ¼ 1; . . . ; N The new approach is illustrated in detail with two examples of complex aerial maneuvering target tracking. The sensor sampling period T is 1 s. The trajectory 1 is a target ﬂying in the (x,y) plane, starting with an initial position [10 km, 40 km]0 and an initial velocity [300 m/s, 0 m/s]0 , Fig. 4(a) shows trajectory 1 that executes a 5-motion sequences (CV–CA–CV– CT–CV): (9) (4) Estimate and covariance: According to MSPT, the output of IMM, whose sequential likelihood function value is maximum, is chosen as target state output. So the ﬁnal output, X^ kþ1jkþ1 and P kþ1jkjþ1 , are computed as follows: kkþ1 ðnm Þ ¼ max fkkþ1 ðnÞg (10) X^ kþ1jkþ1 ¼ X^ kþ1jkþ1 ðnm Þ (11) n¼1;...;N P kþ1jkjþ1 ¼ P kþ1jkþ1 ðnm Þ (1) CV motion in 30 s; (2) CA motion in 30 s, its acceleration is (10 m/s2, 10 m/s2); (3) CV motion in 30 s; (4) coordinated turn motion in 8 s, its radius is 1.5 km, its acceleration is 60 m/s2; and (5) CV motion in 30 s. The trajectory 2 is a target ﬂying in the (x,y) plane, starting with an initial position [10 km, 40 km]0 and an initial velocity [0 m/s, 300 m/s]0 , Fig. 4(b) shows trajectory 2 that executes a 9-motion sequences (CV–CA–CV– CA–CV–CT–CV–CT–CV): (12) From the above interpretation, we insure the independence of all IMM ﬁlters, so the deterioration of performance due to the excessive competition from the unnecessary models is solved. While in every IMM ﬁlter, ﬁlter results of every model in every model set are interacted, so the good performance of general IMM is preserved. Therefore, the Novel-IMM algorithm not only inherits the merit of IMM method, but also overcomes the weakness of IMM. The VSIMM is a series of multiple model set adaptation algorithm and its model set selection depends on the auxiliary information. While the Novel-IMM is a parallel 45 (1) CV motion in 50 s; (2) CA motion in 15 s, its acceleration is (20 m/s2, 20 m/s2); (3) CV motion in 50 s; (4) CA motion in 10 s, its acceleration is (30 m/s2, 30 m/s2); (5) CV motion in 50 s; (6) coordinated turn motion in 31 s, its radius is 1.5 km, its acceleration is 60 m/s2 (trajectory 2); 45 40 40 2 CA 30 Y (km) Y (km) 35 3 CV 30 5 CV 25 1 CV 35 1 CV 2 CA 25 15 7 CV 9 CV 5 CV 5 0 20 4 CA 8 CT 20 10 4 CT 3 CV 5 10 15 X (km) 20 25 -5 6 CT 0 10 20 X (km) Fig. 4. Target trajectory: (a) trajectory 1 and (b) trajectory 2. 30 40 ARTICLE IN PRESS 2176 H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177 (7) CV motion in 50 s; (8) coordinated turn motion in 31 s, its radius is 1.5 km, its acceleration is 60 m/s2 (trajectory 2); and (9) CV motion in 50 s. probabilities of model set CVCA2 and CVCT2, respectively. The transitions between the model sets are quite clear. During the CA portion, the probability of model set CVCA2 is much higher than CVCT2, but during the coordinated turn motion portion, the probability of CVCT2 is much higher than CVCA2. The RMSE of IMM and Novel-IMM are given in Fig. 6. During the CA maneuvering, two algorithms get the same performance, but during the coordinated turn motion, Novel-IMM algorithm has better performance than IMM. In addition, for a ground target track with the map auxiliary information, the Novel-IMM algorithm can also use the map auxiliary information for the hard decision, now only one model set is used for Novel-IMM algorithm at a given time. In this case, Novel-IMM algorithm equals to VSIMM algorithm. Therefore, Novel-IMM algorithm can be used in more situations with or without auxiliary information. But VSIMM algorithm cannot be used in above simulation case without any prior auxiliary information. To validate the performance of the method, the RMSE of VSIMM and Novel-IMM are given in Fig. 7. The trajectory of the target is as shown in Fig. 4(b). In VSIMM method the maneuvering onsets are known as the prior auxiliary condition while there is not prior information in Novel-IMM method. Performance of the Novel-IMM is nearly same as VSIMM, but VSIMM could only be used The Novel-IMM algorithm includes two model sets (CVCA2 and CVCT2). The ﬁrst model set CVCA2 consists of three models: CV model, CA model with large process noise covariance [26] and CA model with small process noise covariance [26]. The second model set CVCT2 consists of three models: CV model, clockwise CT (coordinated turn) model with known turn rate [27], counter clockwise CT model with known turn rate [27]. The standard covariance of models’ process noise is sv ¼ 2 m/s, while the standard deviation of measurement noise is sx ¼ sy ¼ 50 m. The matrix of the probabilities should have a large diagonal element, and it is as follows: 2 3 0:95 0:025 0:025 6 0:025 0:95 0:025 7 P¼4 5 0:025 0:025 0:95 1 CVCA2 CVCT2 0.5 0 30 40 50 60 70 80 90 Time (s) 100 110 120 130 Weight Probability Weight Probability The IMM algorithm includes 1 model set CVCA2 as the ﬁrst model set in the Novel-IMM algorithms. The average model set probabilities for the Novel-IMM are given in Fig. 5, where CVCA2 and CVCT2 denote the 1 CVCA2 CVCT2 0.5 0 0 50 100 150 200 Time (s) 250 300 350 Fig. 5. Model set weight probability: (a) trajectory 1 and (b) trajectory 2. 50 30 40 50 60 70 80 90 Time (s) 150 50 0 30 40 50 60 70 80 90 Time (s) 100 110 120 130 IMM Novel-IMM 50 0 100 110 120 130 IMM Novel-IMM 100 100 RMSEy (m) 100 0 RMSEVx (m/s) IMM Novel-IMM RMSEVy (m/s) RMSEx (m) 150 0 50 100 150 200 Time (s) 250 80 300 350 IMM Novel-IMM 60 40 20 0 0 50 100 150 200 Time (s) Fig. 6. Position RMSE for IMM and Novel-IMM: (a) trajectory 1 and (b) trajectory 2. 250 300 350 ARTICLE IN PRESS RMSEy (m) 80 Novel-IMM VSIMM 60 40 20 0 0 50 100 150 200 250 Time (s) 300 350 400 RMSEVy (m/s) H.Q. Qu et al. / Signal Processing 89 (2009) 2171–2177 2177 60 Novel-IMM VSIMM 40 20 0 0 50 100 150 200 250 Time (s) 300 350 400 Fig. 7. Position RMSE for VSIMM and Novel-IMM (trajectory 2). under the assumption that the true model sequence is known while Novel-IMM could be used without the true model sequence. Indeed, the Novel-IMM algorithm must cost more computational resources than IMM for real time tracking system, but this problem can be solved easily with the development of high speed DSP and computer. 4. Conclusion A Novel-IMM algorithm was presented which leads to a systematic treatment of model set adaptation without additional auxiliary information. The Novel-IMM algorithm consists of N independent IMM ﬁlters operating in parallel, and these multiple IMM ﬁlters are independent. 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