Assessment of accuracies of some approximate solutions

The classic work of Richardson and Gaunt [1 ], has provided an effective means of extrapolating the limiting result in an approximate analysis. From the authors' work on "Bounds for eigenvalues" [2-4] an interesting alternate method has emerged for assessing monotonically convergent approximate solutions by generating close bounds. Whereas further investigation is needed to put this work on sound theoretical foundation, we intend this letter to announce a possibility, which was confirmed by an exhaustive set of examples.

Estimation of Execution Times of Process Control Algorithms on Microcomputers

An important question which has to be answered in evaluting the suitability of a microcomputer for a control application is the time it would take to execute the specified control algorithm. In this paper, we present a method of obtaining closed-form formulas to estimate this time. These formulas are applicable to control algorithms in which arithmetic operations and matrix manipulations dominate. The method does not require writing detailed programs for implementing the control algorithm. Using this method, the execution times of a variety of control algorithms on a range of 16-bit mini- and recently announced microcomputers are calculated. The formulas have been verified independently by an analysis program, which computes the execution time bounds of control algorithms coded in Pascal when they are run on a specified micro- or minicomputer.

Automatic Path Planning and Control Design for Autonomous Landing of UAVs using Dynamic Inversion

In this paper a nonlinear control has been designed using the dynamic inversion approach for automatic landing of unmanned aerial vehicles (UAVs), along with associated path planning. This is a difficult problem because of light weight of UAVs and strong coupling between longitudinal and lateral modes. The landing maneuver of the UAV is divided into approach, glideslope and flare. In the approach UAV aligns with the centerline of the runway by heading angle correction. In glideslope and flare the UAV follows straight line and exponential curves respectively in the pitch plane with no lateral deviations. The glideslope and flare path are scheduled as a function of approach distance from runway. The trajectory parameters are calculated such that the sink rate at touchdown remains within specified bounds. It is also ensured that the transition from the glideslope to flare path is smooth by ensuring C-1 continuity at the transition. In the outer loop, the roll rate command is generated by assuring a coordinated turn in the alignment segment and by assuring zero bank angle in the glideslope and flare segments. The pitch rate command is generated from the error in altitude to control the deviations from the landing trajectory. The yaw rate command is generated from the required heading correction. In the inner loop, the aileron, elevator and rudder deflections are computed together to track the required body rate commands. Moreover, it is also ensured that the forward velocity of the UAV at the touch down remains close to a desired value by manipulating the thrust of the vehicle. A nonlinear six-DOF model, which has been developed from extensive wind-tunnel testing, is used both for control design as well as to validate it.

Cochannel Interference in Partially Coherent Communication

Effects of cochannel interference and synchronization error of the carrier phase on the probability of error in binary communications are considered. Several bounds on the probability of error are proposed. The bounds are easy to compute and do not require complete statistical characterization of the errors. They turn out to be simple linear combinations of error probabilities with no cochannel interferences and no phase errors. Several illustrative examples are given which show that the bounds can be tight.

Popular Mixed Matchings

We study the problem of matching applicants to jobs under one-sided preferences: that is, each applicant ranks a non-empty subset of jobs under an order of preference, possibly involving ties. A matching M is said to be rnore popular than T if the applicants that prefer M to T outnumber those that prefer T to M. A matching is said to be popular if there is no matching more popular than it. Equivalently, a matching M is popular if phi(M,T) >= phi(T, M) for all matchings T, where phi(X, Y) is the number of applicants that prefer X to Y. Previously studied solution concepts based oil the popularity criterion are either not guaranteed to exist for every instance (e.g., popular matchings) or are NP-hard to compute (e.g., least unpopular matchings). This paper addresses this issue by considering mixed matchings. A mixed matching is simply a probability distributions over matchings in the input graph. The function phi that compares two matchings generalizes in a natural manner to mixed matchings by taking expectation. A mixed matching P is popular if phi(P,Q) >= phi(Q,P) for all mixed matchings Q. We show that popular mixed matchings always exist. and we design polynomial time algorithms for finding them. Then we study their efficiency and give tight bounds on the price of anarchy and price of stability of the popular matching problem.

Greedy Scheduling in Multihop Wireless Networks

We study the performance of greedy scheduling in multihop wireless networks where the objective is aggregate utility maximization. Following standard approaches, we consider the dual of the original optimization problem. Optimal scheduling requires selecting independent sets of maximum aggregate price, but this problem is known to be NP-hard. We propose and evaluate a simple greedy heuristic. Analytical bounds on performance are provided and simulations indicate that the greedy heuristic performs well in practice.

The alpha method for solving differential algebraic inequality (DAI) systems

This paper describes an algorithm for ``direct numerical integration'' of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation's (DAE) Jacobian and this difficulty is addressed by a regularization property of the alpha method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in C-2 near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. Thisn condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range-roughly speaking-from being mildly infinite-type'' to very flat at the infinite-type points.

On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

A unit cube in k dimensions (k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), a(i) + 1] on the real line. A graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of C can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes. An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We give an O(bw . n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(Delta) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and cocomparability graphs which have O(Delta) bandwidth. Thus we have: 1. cub(G) <= 3 Delta - 1, if G is an AT-free graph. 2. cub(G) <= 2 Delta + 1, if G is a circular-arc graph. 3. cub(G) <= 2 Delta, if G is a cocomparability graph. Also for these graph classes, there axe constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(Delta) width. We can thus generate the cube representation of such graphs in O(Delta) dimensions in polynomial time.

DMT of Multi-hop Cooperative Networks-Part II: Layered and Multi-Antenna Networks

We consider single-source, single-sink (ss-ss) multi-hop relay networks, with slow-fading Rayleigh links. This two part paper aims at giving explicit protocols and codes to achieve the optimal diversity-multiplexing tradeoff (DMT) of two classes of multi-hop networks: K-parallel-path (KPP) networks and Layered networks. While single-antenna KPP networks were the focus of the first part, we consider layered and multi-antenna networks in this second part. We prove that a linear DMT between the maximum diversity d(max). and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks under the half-duplex constraint. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4. For the multiple-antenna case, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks. Along the way, we compute the DMT of parallel MIMO channels in terms of the DMT of the component channel. For arbitrary ss-ss single-antenna directed acyclic networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable using an amplify-and-forward (AF) protocol. Explicit short-block-length codes are provided for all the proposed protocols. Two key implications of the results in the two-part paper are that the half-duplex constraint does not necessarily entail rate loss by a factor of two as previously believed and that simple AN protocols are often sufficient to attain the best possible DMT.

DMT of Multi-hop Cooperative Networks-Part I: K-Parallel-Path Networks

We consider single-source, single-sink multi-hop relay networks, with slow-fading Rayleigh fading links and single-antenna relay nodes operating under the half-duplex constraint. While two hop relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this two-part paper, we identify two families of networks that are multi-hop generalizations of the two hop network: K-Parallel-Path (KPP) networks and Layered networks. In the first part, we initially consider KPP networks, which can be viewed as the union of K node-disjoint parallel paths, each of length > 1. The results are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the optimal DMT of KPP(D) networks with K >= 4, and KPP(I) networks with K >= 3. Along the way, we derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. As a special case, the DMT of two-hop relay network without direct link is obtained. Two key implications of the results in the two-part paper are that the half-duplex constraint does not necessarily entail rate loss by a factor of two, as previously believed and that, simple AF protocols are often sufficient to attain the best possible DMT.

On the Average Case Communication Complexity for Detection in Sensor Networks

The problem of sensor-network-based distributed intrusion detection in the presence of clutter is considered. It is argued that sensing is best regarded as a local phenomenon in that only sensors in the immediate vicinity of an intruder are triggered. In such a setting, lack of knowledge of intruder location gives rise to correlated sensor readings. A signal-space viewpoint is introduced in which the noise-free sensor readings associated to intruder and clutter appear as surfaces $\mathcal{S_I}$ and $\mathcal{S_C}$ and the problem reduces to one of determining in distributed fashion, whether the current noisy sensor reading is best classified as intruder or clutter. Two approaches to distributed detection are pursued. In the first, a decision surface separating $\mathcal{S_I}$ and $\mathcal{S_C}$ is identified using Neyman-Pearson criteria. Thereafter, the individual sensor nodes interactively exchange bits to determine whether the sensor readings are on one side or the other of the decision surface. Bounds on the number of bits needed to be exchanged are derived, based on communication complexity (CC) theory. A lower bound derived for the two-party average case CC of general functions is compared against the performance of a greedy algorithm. The average case CC of the relevant greater-than (GT) function is characterized within two bits. In the second approach, each sensor node broadcasts a single bit arising from appropriate two-level quantization of its own sensor reading, keeping in mind the fusion rule to be subsequently applied at a local fusion center. The optimality of a threshold test as a quantization rule is proved under simplifying assumptions. Finally, results from a QualNet simulation of the algorithms are presented that include intruder tracking using a naive polynomial-regression algorithm.

Self-interrupted regenerative metal cutting in turning

A new approach is used to study the global dynamics of regenerative metal cutting in turning. The cut surface is modeled using a partial differential equation (PDE) coupled, via boundary conditions, to an ordinary differential equation (ODE) modeling the dynamics of the cutting tool. This approach automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Taylor's 3/4 power law model for the cutting force is adopted. Lower dimensional ODE approximations are obtained for the combined tool–workpiece model using Galerkin projections, and a bifurcation diagram computed. The unstable solution branch off the subcritical Hopf bifurcation meets the stable branch involving self-interrupted dynamics in a turning point bifurcation. The tool displacement at that turning point is estimated, which helps identify cutting parameter ranges where loss of stability leads to much larger self-interrupted motions than in some other ranges. Numerical bounds are also obtained on the parameter values which guarantee global stability of steady-state cutting, i.e., parameter values for which there exist neither unstable periodic motions nor self-interrupted motions about the stable equilibrium.

Single-Symbol ML Decodable Precoded DSTBCs for Cooperative Networks

Single-symbol maximum likelihood (ML) decodable distributed orthogonal space-time block codes (DOST- BCs) have been introduced recently for cooperative networks and an upper-bound on the maximal rate of such codes along with code constructions has been presented. In this paper, we introduce a new class of distributed space-time block codes (DSTBCs) called semi-orthogonal precoded distributed single-symbol decodable space-time block codes (Semi-SSD-PDSTBCs) wherein, the source performs preceding on the information symbols before transmitting it to all the relays. A set of necessary and sufficient conditions on the relay matrices for the existence of semi-SSD- PDSTBCs is proved. It is shown that the DOSTBCs are a special case of semi-SSD-PDSTBCs. A subset of semi-SSD-PDSTBCs having diagonal covariance matrix at the destination is studied and an upper bound on the maximal rate of such codes is derived. The bounds obtained are approximately twice larger than that of the DOSTBCs. A systematic construction of Semi- SSD-PDSTBCs is presented when the number of relays K ges 4 and the constructed codes are shown to have higher rates than that of DOSTBCs.

Time and Energy Complexity of Distributed Computation of a Class of Functions in Wireless Sensor Networks

We consider a scenario in which a wireless sensor network is formed by randomly deploying n sensors to measure some spatial function over a field, with the objective of computing a function of the measurements and communicating it to an operator station. We restrict ourselves to the class of type-threshold functions (as defined in the work of Giridhar and Kumar, 2005), of which max, min, and indicator functions are important examples: our discussions are couched in terms of the max function. We view the problem as one of message-passing distributed computation over a geometric random graph. The network is assumed to be synchronous, and the sensors synchronously measure values and then collaborate to compute and deliver the function computed with these values to the operator station. Computation algorithms differ in (1) the communication topology assumed and (2) the messages that the nodes need to exchange in order to carry out the computation. The focus of our paper is to establish (in probability) scaling laws for the time and energy complexity of the distributed function computation over random wireless networks, under the assumption of centralized contention-free scheduling of packet transmissions. First, without any constraint on the computation algorithm, we establish scaling laws for the computation time and energy expenditure for one-time maximum computation. We show that for an optimal algorithm, the computation time and energy expenditure scale, respectively, as Theta(radicn/log n) and Theta(n) asymptotically as the number of sensors n rarr infin. Second, we analyze the performance of three specific computation algorithms that may be used in specific practical situations, namely, the tree algorithm, multihop transmission, and the Ripple algorithm (a type of gossip algorithm), and obtain scaling laws for the computation time and energy expenditure as n rarr infin. In particular, we show that the computation time for these algorithms scales as Theta(radicn/lo- g n), Theta(n), and Theta(radicn log n), respectively, whereas the energy expended scales as , Theta(n), Theta(radicn/log n), and Theta(radicn log n), respectively. Finally, simulation results are provided to show that our analysis indeed captures the correct scaling. The simulations also yield estimates of the constant multipliers in the scaling laws. Our analyses throughout assume a centralized optimal scheduler, and hence, our results can be viewed as providing bounds for the performance with practical distributed schedulers.