Don A. Grundel

Theory and algorithms for cooperative systems

Over the past several years, cooperative control and optimization have increasingly played a larger and more important role in many aspects of military sciences, biology, communications, robotics, and decision making. At the same time, cooperative systems are notoriously difficult to model, analyze, and solve - while intuitively understood, they are not axiomatically defined in any commonly accepted manner. The works in this volume provide outstanding insights into this very complex area of research. They are the result of invited papers and selected presentations at the Fourth Annual Conference on Cooperative Control and Optimization held in Destin, Florida, November 2003.

Test Problem Generator for the Multidimensional Assignment Problem

The multidimensional assignment problem (MAPs) is a higher dimensional version of the standard linear assignment problem. Test problems of known solution are useful in exercising solution methods. A method of generating an axial MAP of controllable size with a known unique solution is presented. Certain characteristics of the generated MAPs that determine realism and difficulty are investigated.

Asymptotic behavior of the expected optimal value of the multidimensional assignment problem

The Multidimensional Assignment Problem (MAP) is a higher-dimensional version of the Linear Assignment Problem that arises in the areas of data association, target tracking, resource allocation, etc. This paper elucidates the question of asymptotical behavior of the expected optimal value of the large-scale MAP whose assignment costs are independent identically distributed random variables with a prescribed probability distribution. We demonstrate that for a broad class of continuous distributions the limiting value of the expected optimal cost of the MAP is determined by the location of the left endpoint of the support set of the distribution, and construct asymptotical bounds for the expected optimal cost.

On the number of local minima for the multidimensional assignment problem

The Multidimensional Assignment Problem (MAP) is an NP-hard combinatorial optimization problem occurring in many applications, such as data association, target tracking, and resource planning. As many solution approaches to this problem rely, at least partly, on local neighborhood search algorithms, the number of local minima affects solution difficulty for these algorithms. This paper investigates the expected number of local minima in randomly generated instances of the MAP. Lower and upper bounds are developed for the expected number of local minima, E[M], in an MAP with iid standard normal coefficients. In a special case of the MAP, a closed-form expression for E[M] is obtained when costs are iid continuous random variables. These results imply that the expected number of local minima is exponential in the number of dimensions of the MAP. Our numerical experiments indicate that larger numbers of local minima have a statistically significant negative effect on the quality of solutions produced by several heuristic algorithms that involve local neighborhood search.

Formulation and Solution of the Target Visitation Problem

This paper presents the Target Visitation Problem (TVP) for a single unmanned aerial vehicle (UAV). The ability to effectively plan a path for a UAV to visit multiple targets is an increasingly important capability in a variety of applications, including surveillance, attack, assessment, search and rescue, disaster relief, and environmental cleanup. The TVP is related to both the Traveling Salesman Problem and the Linear Ordering Problem, with an objective function that combines elements of both problems. The TVP considers both total travel distance and the order of targets visited. In this paper, we formulate the target visitation problem for a single vehicle, describe a heuristic solution procedure, and report results for problems of various size.

Optimization of Spatiotemporal Clustering for Target Tracking From Multisensor Data

This study focuses on the information extraction from reported sensor data in the communication system of wide-area-search munitions (WASMs). Such sensor data could be erroneous and inconsistent. For example, two WASMs might detect the same target, but associate it with two different targets and tracks. Similarly, two WASMs might detect two distinct targets, but recognize them as the same target. The research challenge is how to fuse both accurate and inaccurate information broadcasted from WASMs, and reconstruct the battle space for accurate target tracking. For each of the detected target points, WASMs provide its location information, detection time, and directional velocity. We, herein, propose a target clustering approach to group target points detected by WASMs and identify the track of individual targets. Our approach differs from traditional clustering techniques as it performs clustering using the time and orientation information, in addition to the distance in the Euclidean space. Our approach employs a network modeling technique to reconstruct all target points and their feasible movement, and a new optimization technique to find the most probable target tracks. Our approach can also determine the optimal number of clusters (targets) automatically from the input data. In this study, distributed interactive simulation, a real-time simulation of a networks information exchange, is used to generate battle space test instances that are used in evaluating the proposed framework. Based on seven realistically simulated instances, the computational results show that our approach provides extremely accurate target-tracking results in a timely fashion. We also compare our results with those obtained using the k-means clustering technique. On average, our approach reconstructs the real target tracks with about 95% accuracy in less than 10 s, while the k-means clustering results yields about 80% accuracy in a similar computational time.

APPLYING SIMULATED ANNEALING TO THE MULTIDIMENSIONAL ASSIGNMENT PROBLEM

A divider plate having a strainer basket depending therefrom is positioned in a skimmer apparatus intermediate the skimming inlet and the lower end. A conduit passes through the basket and communicates with the drain inlet in the bottom of the skimmer which also includes an outlet from which water is drawn by the swimming pool pump. The conduit is aligned with a water passage in the divider plate. A coupling member affixed to the hose associated with vacuum cleaning equipment is passed through the water passage and engaged with the conduit. The coupling blocks water flow from the drain inlet and has an exit therethrough for movement of water from the hose to the outlet of the skimmer. Valve means are provided for selectively closing the water passage and for selectively retarding water flow from the drain inlet.

Asymptotic Results for Random Multidimensional Assignment Problems

The multidimensional assignment problem (MAP) is an NP-hard combinatorial optimization problem occurring in applications such as data association and target tracking. In this paper, we investigate characteristics of the mean optimal solution values for random MAPs with axial constraints. Throughout the study, we consider cost coefficients taken from three different random distributions: uniform, exponential and standard normal. In the cases of uniform and exponential costs, experimental data indicates that the mean optimal value converges to zero when the problem size increases. We give a short proof of this result for the case of exponentially distributed costs when the number of elements in each dimension is restricted to two. In the case of standard normal costs, experimental data indicates the mean optimal value goes to negative infinity with increasing problem size. Using curve fitting techniques, we develop numerical estimates of the mean optimal value for various sized problems. The experiments indicate that numerical estimates are quite accurate in predicting the optimal solution value of a random instance of the MAP.

Constrained search for a moving target

We consider the problem of constrained path planning for one or two agents in search of a single randomly moving target such that we maximize the probability of intercepting the target at some time in its trajectory. We assume the agents operate in a receding-horizon optimization framework with some finite planning horizon. We present and compare two search path planning methods. This problem is particularly applicable in the case of wide area search munitions searching and engaging moving ground targets

Nonlinear Dynamics of Sea Clutters and Detection of Small Targets

This chapter presents the problem of determining sea clutter dynamics with application to detecting and classifying small targets. A systematic method of reconstruction of a sea clutter attractor is considered. We explore the use of dynamical system techniques, optimization methods and statistical methods to estimate the dynamical characteristics of sea clutters. We assume that the radar information is in the form of a nonlinear time series. Then we sequentially apply a dynamical approach for characterizing radar signals, based on nonlinear estimation of dynamical characteristics, forming the vector of these characteristics, and modelling the evolution of dynamical processes over time. We consider an optimization method for reconstructing parameter spaces of dynamical systems. These techniques can be applied to systems with one or more hidden variables, and can be used to reconstruct maps or differential equations of the sea clutter dynamics. The possible use of chaotic models for development of classical and quantum detector signals is discussed. The systems analysis based methods are illustrated using numerically generated data and radar data previously recorded from experimental radar systems. The dynamical characteristics can be used to better visualize the ‘state vector’ of the radar signal and for the purpose of pattern recognition