[+] Team 1: Modeling Planarization in Chemical-Mechanical Polishing

**Mentor** Leonard Borucki, Motorola- Dilek Alagoz, University of Kentucky
- Stephanie Hoogendoorn, University of Pittsburgh
- Satyanarayana Kakollu, Mississippi State University
- Richard Schugart, North Carolina State University
- Michael Sostarecz, The Pennsylvania State University
- Maria Westdickenberg, New York University

Modeling of Planarization in Chemical-Mechanical Polishing

[+] Team 2: Modeling Networked Control Systems

**Mentor** Sonja Glavaski, Honeywell- Madalena Chaves, Rutgers, The State University of New Jersey
- Robert Day, University of Maryland
- Lucia Gomez-Ramos, University of Wisconsin, Milwaukee
- Parathasarathi Nag, Washington State University
- Anca Williams, Portland State University
- Wei Zhang, University of Kentucky

Recently modeling and control of networked control systems with limited communication capability has emerged as a topic of significant interest to controls community. Nature and level of information flow throughout the system is central to a discussion of cooperative control. Applications span wide range, including traffic control, satellite clusters, mobile robotics.

A natural way to model such interconnection topology is as a graph. Each system can be modeled as a node, and arc joins node i and node j if vehicle j is receiving information from vehicle i. To consider all possible topologies it is advisable to use directed graphs, meaning that bi-directional communication is not assumed. In reference [1] it has been demonstrated that a minimal exchange of information between systems can be designed to realize a dynamical system which supplies each individual system with shared reference trajectory. The sensing paths were modeled as a graph, and eigenvalues of the Laplacian matrix of a graph determine the stability of a whole system. In this study quality of communication has not been incorporated.

For coordination of individual systems within networked control system one is especially concerned with acceptable limits of communication network performance. Knowledge of bounds on acceptable network performance is crucial to making networked control system robust in realistic environments. In reference [2] the effects of communication packet losses in the feedback loop of a control system is studied. Motivation for this study has been derived from vehicle control problems where information is communicated via a wireless local area network. A Linear Matrix Inequality condition is developed for the existence of a stabilizing feedback controller. This result can also be used to give a worst-case performance specification (in terms of packet loss rate) for an acceptable communications system.

The focus of our project would be to investigate a possibility of combining these two approaches. Starting form typical directed graph representation of a networked control system we will investigate how to introduce into it a dynamical representation of individual systems and communication network performance. This would allow to systematically address various system performance issues (e.g. stability, observability and controllability).

**References:**

[1] A. Fax, R.M. Murray. Information Flow and Cooperative Control of Vehicle Formations. Accepted for 2002 IFAC World Congress

[2] P. Seiler and R. Sengupta. Analysis of Communication Losses in Vehicle Control Problems. In Proceedings of the 2001 American Control Conference.

[3] D. Cvetkovic, P. Rowlins, and S. Simic. Eigenspaces Of Graphs, volume 66 of Encyclopedia of Mathematics and Applications, Cambridge University Press, 1997.

[4] Fan R.K. Chung. Spectral Graph Theory

[+] Team 3: Designing Airplane Engine Struts Using Minimal Surfaces

**Mentor** Thomas Grandine, The Boeing Company- Sara Del, The University of Iowa
- Todd Moeller, Georgia Institute of Technology
- Siva Natarajan, Utah State University
- Gergina Pencheva, University of Pittsburgh
- Jason Sherman, Kent State University
- Steven Wise, University of Virginia

Airplane drag is a complex function of many variables. One way to reduce drag is to reduce the surface area of the airplane components over which outside air must pass. A simple means, at least conceptually, of determining an initial design of an airplane strut is to construct a surface of minimal area which attaches to both wing and engine in a prescribed way.

In this workshop, we will investigate some of the extensive literature on minimal surfaces and attempt to find relevant references for this problem. Our goal will be to find or develop a formulation of the problem which leads to a practical means of constructing such a surface. Techniques involving the calculus of variations and the numerical solution of Euler equations are possible candidates for practical methods, but other considerations will surely arise as we become more familiar with the literature.

[+] Team 4: Mobility Management in Cellular Telephony

**Mentor** David Shallcross, Telcordia- Benjamin Cooke, Duke University
- Dmitry Glotov, Purdue University
- Darongsae Kwon, Seoul National University
- Simon Schurr, University of Maryland
- Daniel Taylor, Washington State University
- Todd Wittman, University of Minnesota, Twin Cities

We will look at a problem arising in the operation of cellular telephone systems. A mobile station (i.e. cell phone) communicates directly with the Base Transceiver Station (BTS) for the cell it occupies. Several BTSs are associated with a single Base Station Controller (BSC). In turn, several BSCs are associated with a single Mobile Switching Center (MSC). If a mobile station moves from one cell to another, work must be done by the BTSs, BSCs, and possibly the MSCs, for the old and new cells. This amount of work will vary depending on whether the two cells use the same BSC, or even the same MSC. Given an estimate of this handover traffic, we would like to avoid overloading the BSCs and MSCs. To this end we may reassign BTSs among the BSCs at the same MSC.

This problem may be formulated as a variant of the Quadratic Assignment Problem (QAP), a well-known NP-hard problem. We will investigate applying modifications of methods for the QAP to this cell phone problem. I intend for the students to implement and test one such method of their choosing. I hope to provide something like realistic data for this.

Students should have some experience with optimization, and implementing optimization algorithms.

**References:**

For the technology: Understanding Cellular Radio, William Webb, Artech House, 1998

A general introduction: Combinatorial Optimization, Cook, Cunningham, Schrijver, + Pulleyblank, John Wiley and Sons, 1998

One possible class of methods:

Tabu Search, F. Glover + M. Laguna, Kluwer Academic Publishers, 1997

One model:

Quadratic Assignment and Related Problems, Pardalos + Wolkowicz, eds., DIMACS vol 16, American Methematical Society, 1994

Also see the QAPLIB home page: http://www.opt.math.tu-graz.ac.at/qaplib/.

[+] Team 5: Optimal Design for a Varying Environment

**Mentor** David Misemer, 3M- Serguei Lapin, University of Houston
- Xuan Hien Nguyen, University of Wisconsin, Madison
- Jiyeon Oh, University of Cincinnati
- Daniel Vasiliu, Michigan State University
- Pei Yin, University of Missouri
- Ningyi Zhang, University of Delaware

**Project Description**

pdf Word

[+] Team 6: Optimal Pricing Strategy in Differentiated Durable-Goods Market

**Mentor** Suzhou Huang, Ford Research Laboratory- Miyuki Breen, Case Western Reserve University
- George Chikhladze, University of Missouri
- Jose Figueroa-Lopez, Georgia Institute of Technology
- Yaniv Gershon, Wayne State University
- Yanto Muliadi, Stanford University
- Ivy Prendergast, The University of Iowa

We consider the problem of how to device an economic mechanism that implements the optimal pricing strategy for differentiated durable-goods in certain markets with imperfect competition. The pricing strategy has to be incentive compatible for all consumers, in the sense that each consumer, who has private information on his/her own preference, will end up choosing voluntarily the product that is intended for him/her by its producer. The optimality concept in a monopoly setting will be extended by the concept of Nash equilibrium in a competitive environment. Differentiated products imply that each product has a varied appeal to different consumers. So, the pricing strategy needs to accomodate substitutions and potential cannibalizations among products. Additional challenges arise when products are durable goods. Durability implies that any decision made today has a future ramification. Therefore, all decision making processes have to be modeled to avoid regrets for every market participant. Furthermore, a durable good can be sold, leased or bought as used, and often entails a non-negligible cost for each transaction.

Finding the solution to the problem posed above will rely heavily on game theory and optimization with inequality constraints. The complex setting for the problem provides an excellent opportunity to practice modeling skills: abstracting the essence of a realistic problem that would otherwise be too complicated to solve and then articulating it into mathematical equations.

**References:**

1) For background in economics: Hal Varian, "Microeconomic Analysis", 3rd Edition, Chapter 14 (Monopoly), Chapter 16 (Oligopoly)

2) For background in game theory: Hal Varian, "Microeconomic Analysis", 3rd Edition, Chapter 15 (Game Theory), Chapter 25 (Information)

3) For background in optimization: Hal Varian, "Microeconomic Analysis", 3rd Edition, Chapter 27 (Optimization)

4) For a more specific and detailed reference: S. Huang and D. Kuzyutin, "Incentive Compatible Pricing Strategies for Product Differentiation in Durable-Goods Markets", Ford Technical Report. PDF file attached. Students will be asked to consider situations that were not explicitly addressed by this paper.