Single-Sink Fractionally Subadditive Network Design

We study a generalization of the Steiner tree problem, where we are given a weighted network G together with a collection of k subsets of its vertices and a root r. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for k=2, and give a 3/2-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known O(log n)-approximation. Despite these obstacles, we conjecture that this problem should have an O(1)-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path

Sticky Brownian rounding and its applications to constraint satisfaction problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semidefinite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations

Turbulence effects on modified state observer-based adaptive control: Black kite micro aerial vehicle

This paper presents the implementation of a modified state observer-based adaptive dynamic inverse controller for the Black Kite micro aerial vehicle. The pitch and velocity adaptations are computed by the modified state observer in the presence of turbulence to simulate atmospheric conditions. This state observer uses the estimation error to generate the adaptations and, hence, is more robust than model reference adaptive controllers which use modeling or tracking error. In prior work, a traditional proportional-integral-derivative control law was tested in simulation for its adaptive capability in the longitudinal dynamics of the Black Kite micro aerial vehicle. This controller tracks the altitude and velocity commands during normal conditions, but fails in the presence of both parameter uncertainties and system failures. The modified state observer-based adaptations, along with the proportional-integral-derivative controller enables tracking despite these conditions. To simulate flight of the micro aerial vehicle with turbulence, a Dryden turbulence model is included. The turbulence levels used are based on the absolute load factor experienced by the aircraft. The length scale was set to 2.0 meters with a turbulence intensity of 5.0 m/s that generates a moderate turbulence. Simulation results for various flight conditions show that the modified state observer-based adaptations were able to adapt to the uncertainties and the controller tracks the commanded altitude and velocity. The summary of results for all of the simulated test cases and the response plots of various states for typical flight cases are presented

Robust Stabilization of Micro Aerial Vehicles with Uncertain Actuator Nonlinearities

Low cost actuators and linkages used in Micro Aerial Vehicles will lead to unmodeled nonlinearities like dead zone and backlash in the actuator. These actuator nonlinearities are often neglected in the design of feedback control system and can detoriate the system performance remarkably. This paper presents the design of robust control laws for Micro Aerial vehicles (MAV) using sliding mode control approach to deal with nonlinear uncertain plant models. The uncertainties can be in the form of parametric variations, external disturbances or uncertain actuator nonlinearities. The uncertainties are assumed to be bounded with known values. Simulation has been carried out using MAV 6 DoF nonlinear model and the results are provided to check the performance of the control laws

Design of autopilot control laws for MAV using super twisting control

Autopilot control law has been designed for Micro Aerial Vehicle (MAV) using the second order sliding mode (SOSM) approach. The control implementation has an inner loop and an outer loop structure, outer loop converts the guidance commands to desired attitudes. Inner loop attitude tracking control is done using output feedback based higher order sliding mode technique. Finite time convergence properties of second order sliding mode technique has been used for designing tracking controller. Lyapunov based stability proof has been given to analyze the zero dynamics stability of the MAV longitudinal dynamics during tracking. Nonlinear 6DoF model of the Blackkite 300 mm wingspan fixed wing MAV, is used both for control design and as well as to verify controller performance against the classical PID control methods

Design and Hardware Implementation of Autopilot Control Laws for MAV Using Super Twisting Control

In this paper we present the design and implementation of autopilot tracking control law for Micro Aerial Vehicle using the second order sliding mode approach. The inner loop attitude tracking control design is carried out using output feedback based second order sliding mode technique, to ensure finite time convergence of the tracking error dynamics. While addressing tracking control of a time varying reference signal, it is important to investigate the stability characteristics of the internal dynamics to ensure perfect tracking. This paper mainly addresses the output tracking control problem for a MAV and investigate the stability characteristics of the longitudinal zero dynamics during tracking. We have proposed a stability proof based on Lyapunov theory to analyze the stability of the MAV longitudinal zero dynamics during tracking. A nonlinear aircraft model obtained using aerodynamic derivatives of The Blackkite 300 mm wingspan fixed MAV is used for both control design and as well as to verify its performance against the classical control methods. Extensive hardware in-loop simulation results of the proposed control algorithm implemented on the commercially available PX4 based Pixhawk autopilot board are also presented here

On the Lovász theta function for independent sets in sparse graphs

We consider the maximum independent set problem on graphs with maximum degree d. We show that the integrality gap of the Lovasz Theta function-based SDP has an integrality gap of O~(d/log3/2 d). This improves on the previous best result of O~(d/log d), and narrows the gap of this basic SDP to the integrality gap of O~(d/log2 d) recently shown for stronger SDPs, namely those obtained using poly log(d) levels of the SA+ semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r. We also show how to obtain an algorithmic version of the above-mentioned SAplus-based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of O~(d/log2 d) matches the best unique-games-based hardness result up to lower-order poly (log log d) factors