A novel estimator with moving ants

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Abstract

In the traditional ant colony optimization (ACO) algorithm, ants are utilized to solve various combinatorial optimization problems regardless of their individual velocities. In this work, however, a moving ant estimator (MAE) is developed on the premise that the issue of moving velocity of each ant is considered. The determination of the velocity of each ant depends directly on the normalized weights between the one-step prediction of the ant starting position and the selected one-step position. Besides this, the velocity of each ant is further adjusted locally based on its individual moving ability, which is defined in the corresponding “pheromone” update process just as the traditional ACO algorithm. To improve the accuracy of the proposed estimator, two improved versions are investigated. Numerical simulations show that the moving ant estimator, as well as its two improved versions, could estimate adaptively the state of maneuvering or non-maneuvering target. To verify the effectiveness of the MAE, the obtained results are compared with those from PF, IMMPF, etc.

Introduction

To the best of our knowledge, recursive state estimation of dynamic stochastic systems is of fundamental importance in many other engineering applications besides target tracking. For the linear/Gaussian estimation, the required density is assumed to be Gaussian, and the best solution is usually Kalman filter. However, when such an assumption is extended to the nonlinear/non-Gaussian case, such as bearings-only target tracking problem [1], [2], [3], the problem has no analytic closed-form solution. Fortunately, the extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF) [4], [5], [6], [7] are recognized popular and effective means to solve the problem. In EKF, the state distribution is approximated by a Gaussian random variable (GRV), which is then propagated analytically through the first-order linearization of the nonlinear system, and this can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergences of the filter. The UKF address this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, capture the posterior mean and covariance accurately to the third order for any nonlinearity. Particle filter, also known as Sequential Monte Carlo (SMC) method, has attracted more attentions in recent years over a wide range of applications. In PF, the target state variable distribution is approximated by a weighted set of samples. These samples are propagated and updated once new measurements are available. Using these samples, the target state can be estimated by standard Monte-Carlo integration techniques. To avoid the phenomenon of degeneration, the re-sampling strategy is usually adopted [7].

ACO utilizes artificial ants that cooperate to find good solutions for discrete or continuous optimization problems [8], [9], [10]. These artificial ants mimic the foraging behavior of their biological counterparts in finding the shortest-path to the food source. In generic ACO, such as Ant System in the discrete domain, ants iteratively build solutions and deposit pheromone to the paths they have traveled. Path selection is characterized by a stochastic procedure based usually on two parameters, the pheromone and heuristic values. The pheromone value gives an indication of the number of ants that chose the trail recently, while the heuristic value is a problem-dependent quality measure. When an ant reaches a decision point, it is more likely to choose the trail with the higher pheromone and heuristic values. Once the ant arrives at its destination, the solution corresponding to the ant’s followed path is evaluated and the pheromone value of the path is updated accordingly. Additionally, evaporation causes the pheromone level of all trails to diminish gradually. Hence, trails that are not reinforced gradually lose pheromone and will in turn have a lower probability of being chosen by subsequent ants.

So far, the ACO algorithm is widely utilized in many occasions, such as traveling salesman problem (TSP) [11], job shopping [12], telecommunications networks [13], track initiation problem [14] and so on [15], [16], [17], [18]. However, there are few reports on the ACO algorithm employed into the parameter estimate filed except [19]. In this work, the movement behavior of each ant is considered and utilized to determine the parameter state. Similar to the traditional ACO algorithm, the “pheromone” update process is also defined in the proposed moving ant estimator, which is helpful to guiding all ants moving toward the solution to the problem.

The remainder of this paper is organized as follows: Section 2 presents in detail the moving ant estimator, as well as its improved versions. In Section 3, numerical Monte-Carlo runs are conducted and corresponding results are also analysized. Finally, conclusions and future research directions are given in Section 4.

Section snippets

Dynamic state-space model

Suppose that the states follow the first Markov process and the observations are assumed to be independent given the states. Therefore, for a general nonlinear non-Gaussian system, the model (neglecting control input terms) can be formulated as belowx(t)=f(x(t-1),w(t))z(t)=h(x(t),v(t))where x(t) denotes the state of the system at time t; z(t) is the observation of the interested state; w(t) and v(t) are defined as the process noise and measurement noise with constant covariances Q(t) and R(t),

Algorithm performance and comparison

Several different scenarios are now presented to demonstrate the performance of the proposed moving ant estimator, and the improved versions as well. In the first we provide an example of tracking non-maneuvering target, and the performance comparisons are conducted with PF, PF–EKF, and PF–UKF. In the second scenario we give an example of maneuvering target tracking, and present the comparative simulations with a standard Interacting Multiple Model Particle Filter (IMMPF), IMMPF–EKF, and

Conclusions

A novel recursive parameter estimator, moving ant estimator, is proposed in this paper, which is implemented by regulating movements of a group of ants. In addition, two improved versions of the MAE are also proposed. Numerical simulation results show that the proposed MAE including its two improved versions is capable of tracking not only the non-maneuvering target but also the maneuvering one without any maneuver detection methods and model assumptions. Meanwhile, their performances are

Acknowledgments

This work is supported by National Natural Science Foundation of China (No. 60804068), by Natural Science Fundamental Research Program of Higher Education Colleges in Jiangsu Province (No. 07KJB510001) and by Suzhou Municipal Science Foundation (No. SG0704).

References (21)

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