The behavior of model probability in multiple model algorithms

This research aims at improving Detect and Avoid (DAA) functions in Unmanned Aircraft Systems (UAS) using a Multiple Model Estimation algorithm to track maneuvering intruders. This research builds on previous work that used predefined aircraft encounter trajectories. An established encounter model generates the intruder trajectories while a multiple model algorithm is introduced to improve intruder dynamics estimation. A new method based on the Kalman prediction phase inside the Interactive Multiple Model (IMM) algorithm is presented to estimate time to closest point of approach, horizontal miss distance, and vertical separation. An analysis of the sensor error on the algorithm estimation and the sensor field of regard requirement from the Air-to-Air Radar Minimum Operational Performance Standards (MOPS) is performed. The efficiency of the trajectory estimation has direct implication on the estimation of the intruder trajectory in relation to the own aircraft. The methods described in this research can aid a certification authority in determining if a DAA system is sufficient for safely integration of UAS into the National Airspace System.

The generalized Shiryayev sequential probability ratio test (SSPRT) is applied to linear dynamic systems for singlefault isolation and estimation. The algorithm turns out to be the multiple model (MM) algorithm considering all thepossible model trajectories. In real application, this algorithm must be approximated due to its increasing computationcomplexity and the unknown parameters of the fault severeness. The Gaussian mixture reduction is employed toaddress the problem of computation complexity. The unknown parameters are estimated in real time by modelaugmentation based on maximum likelihood estimation (MLE) or expectation. Hence, the system state estimation,fault identification and estimation can be fulfilled simultaneously by a multiple model algorithm incorporating these twotechniques. The performance of the proposed algorithm is demonstrated by Monte Carlo simulation. Although ouralgorithm is developed under the assumption of single fault, it can be generalized to deal with the case of (infrequent)sequential multiple faults. The case of simultaneous faults is more complicated and will be considered in future work.

Multiple-model approach provides the state-of-the-art solutions to many problems involving estimation, filtering, control, and/or modeling. One of the most important problems in the application of the multiple-model approach is the design of the model set used in a multiple-model algorithm. To our knowledge, however, it has never been addressed systematically in the literature. This paper deals with this challenging topic in a general setting. General problems of model-set design are considered. A concept of a random model is introduced. In other words, modeling of models used in a multiple model (MM) algorithm as well as the true model as random variables is proposed. Three classes of general methods for optimal design of model sets-by minimizing distribution mismatch, minimizing modal distance, and moment matching, respectively-are proposed. Theoretical results that address many of the associated issues are presented. Examples that demonstrate how some of these theoretical results can be used as well as their effectiveness are given. Many of the general results presented in this paper are also useful for performance evaluation of MM algorithms.

In this paper, the multiple model algorithm is used in deriving recursive algorithms for the identification of nonlinear systems. The radial basis function (RBF) networks with only linear weights requiring estimation combined with the Kalman filter algorithm forms the essence of the identification algorithm. Multiple networks are used to identify the multi-modes of the system under a Markovian assumption, the model estimation and selection being carried out on-line. Both, 'hard' and 'soft' competition based estimation schemes are developed where in the former, the most probable network is adapted by the Kalman filter and in the latter all networks are adapted by appropriate weighting of the observation.

In order to solve the problem of an uncertain initial state and big errors for hypersonic glide vehicle (HGV) tracking, a hybrid model algorithm is proposed by combining a single model algorithm with a multiple model algorithm. To develop the tracking algorithm with the Cubature Kalman filter, in every model filter the process equation is established based on the HGV aerodynamic model and the measurement equation is established based on the radar measurement principle. The proposed hybrid model algorithm is developed by using the multiple model algorithm in the initial tracking stage and using the single model algorithm in the stable tracking stage, and they are divided by a proposed parameter. The former can avoid divergence and reduce the errors caused by the uncertain initial state. The latter can track the HGV at higher accuracy. The simulation indicates that the proposed hybrid model has high speed accuracy in the whole tracking stage and high position accuracy in the stable tracking stage. The average position root mean square error (RMSE) using the hybrid model algorithm is almost the same as that using the single model algorithm but the average speed RMSE using the single model algorithm is about 30% greater than that using the hybrid model algorithm. In a system for defending the HGV, the speed accuracy has more effect on the trajectory prediction as time goes on. Thus, the hybrid model algorithm is an engineering algorithm for HGVs with high accuracy. In future research, the hybrid model algorithm will be studied for general maneuvering target tracking.

This paper considers the decline problem of tracking accuracy caused by acceleration estimation inaccuracy in hybrid grid multiple model (HGMM) algorithm. A new improved tracking algorithm of maneuvering targets based on CSHGMM is proposed. Grid partition is performed by estimation of the acceleration mean in current statistic (CS) model, where a fading factor is adopted to improve tracking accuracy by adjusting the gain matrix under the condition of the abrupt change of acceleration. Then possible model sets of the target are generated online, and interactive multiple model algorithm is adopted to estimate the target state. The proposed algorithm performance is tested and compared in general and strong maneuvering scenarios, and simulation results demonstrate that the proposed algorithm not only improves the accuracy for maneuvering target tracking, but also can be applied to most of the tracking scenarios.

The quality of multiple model estimators can be improved with multisensor fusion. This paper contrasts the performance of three multiple model algorithms. It is shown that the simplest is adequate in high signal-to-noise environments. The more sophisticated warrant attention when the observations are ambiguous.

The Interacting Multiple Model (IMM) algorithm has been used in various applications with good performance. However, to the best of our knowledge, stability conditions of the IMM algorithm have not been shown in the control literature. This technical note presents a sufficient condition for the exponential stability of the IMM algorithm for a class of Markov jump linear systems. We derive a lower bound and an upper bound for the error covariances of the IMM algorithm, and then derive the stability condition by the Lyapunov approach.

The grid points of the Adaptive Grid Interactive Multiple Model (AGIMM) algorithm change slowly at the target high maneuvering and fluctuate when the target is moving at a constant speed. The paper proposes an improved AGIMM algorithm based on maneuverability of target. Firstly, divide the Maneuverability zones by Filter residuals. Secondly, according to different zones, the grid points are updated in different ways to speed up the convergence of the model when the target is high maneuvering, and reduce the fluctuation of the model when the target is low maneuvering. Realize the model update method according to the target maneuverability adaptive adjustment. The simulation shows that the improved algorithm significantly reduces the tracking error regardless of whether the target turns or not.

The paper compares the performance of two different configurations of the interacting multiple model (IMM) algorithm in tracking a civil aircraft. The first configuration of the IMM algorithm uses one straight line motion model and different constant turn rate models whereas the other configuration also employs a straight line motion model and a single turn rate model which uses an estimate of the turn rate. In the first algorithm, results have been obtained using different process noise values and different numbers of models. From these results, recommendations are made regarding the choice of both these variable parameters.

For maneuvering target tracking, general interacting multiple model (IMM) algorithms have a fixed structure. It has been overlooked that how the performance of these algorithms depends on the set of models used. The use of too many models is as bad as that of too few models. Therefore, a variable structure IMM (VSIMM) was presented and applied to ground target tracking. This algorithm eliminates the need for carrying all the possible models throughout the entire tracking period as in the standard IMM estimator, significantly improving performance and reducing computational load. But it is difficult to apply the VSIMM to other scenario (for example, aerial target) without auxiliary information such as map for ground target. This paper then presents a model set multiple hypotheses IMM algorithm (MS-MHIMM) which leads to a systematic treatment of model-set adaptation without using any additional auxiliary information. The new approach is illustrated in detail with an aerial complex maneuvering target tracking example.

In this paper, a fault detection, identification and estimation approach has been developed for the condition monitoring of the electro-hydraulic actuator (EHA) system using the multiple-model (MM) estimation algorithm. The MM estimation algorithm makes use of the extended Kalman filter (EKF) technique to generate estimates of states and key physical parameters, which are related to faults in the EHA system. The proposed fault detection and identification (FDI) is formulated as a hybrid interacting multiple-model estimation scheme. The interaction scheme between multiple models is introduced into the MM estimation algorithm to yield more robust detection and estimation. Estimates of the key physical parameters in the EHA system are assessed against baseline values and fused with the FDI results for higher level monitoring purposes. Two parameters of interests, namely torque motor equivalent resistance and the effective bulk modulus are investigated for the EHA system condition monitoring purpose. The simulation results highlight the considerable potential of the proposed technique for achieving improved condition monitoring of the EHA system.

We provide background information to allow a heuristic understanding of two types of criteria used in selecting a model for making inferences from ringing data. The first type of criteria (e.g. AIC, AlCc QAICc and TIC) are estimates of (relative) Kullback-Leibler information or distance and attempt to select a good approximating model for inference, based on the principle of parsimony. The second type of criteria (e.g. BIC, MDL, HQ) are 'dimension consistent' in that they attempt to consistently estimate the dimension of the true model. These latter criteria assume that a true model exists, that it is in the set of candidate models and that the goal of model selection is to find the true model, which in turn requires that the sample size is very large. The Kullback-Leibler based criteria do not assume a true model exists, let alone that it is in the set of models being considered. Based on a review of these criteria, we recommend use of criteria that are based on Kullback-Leibler information in the biological sciences.

Markov-Switching Model Selection Using Kullback-Leibler Divergence

The wind as a natural phenomenon would cause the derivation of the pesticide drops during the operation of agricultural unmanned aerial vehicles (UAV). In particular, the changeable wind makes it difficult for the precision agriculture. For accurate spraying of pesticide, it is necessary to estimate the real-time wind parameters to provide the correction reference for the UAV path. Most estimation algorithms are model based, and as such, serious errors can arise when the models fail to properly fit the physical wind motions. To address this problem, a robust estimation model is proposed in this paper. Considering the diversity of the wind, three elemental time-related Markov models with carefully designed parameter α are adopted in the interacting multiple model (IMM) algorithm, to accomplish the estimation of the wind parameters. Furthermore, the estimation accuracy is dependent as well on the filtering technique. In that regard, the sparse grid quadrature Kalman filter (SGQKF) is employed to comprise the computation load and high filtering accuracy. Finally, the proposed algorithm is ran using simulation tests which results demonstrate its effectiveness and superiority in tracking the wind change.