AF: Small: THEORETICAL AND ALGORITHMIC FOUNDATIONS OF CONSTRAINED PARTICLE FILTERING

许多现代技术（从相机的图像稳定，到化工厂，从电网到机器人导航）都需要计算机算法来跟踪动态系统的状态，而动态系统是用不确定性建模和测量的。粒子滤波是一种跟踪许多粒子（候选状态）以达到最佳估计的技术，即跟踪状态的平均值或平均值。该项目考虑了对最佳估计（而不仅仅是单个粒子）的约束，给出了一种确保建模正确性和底层系统安全性的新方法。当系统或约束或两者都是非线性时，实时处理动态系统中的约束是一项挑战。这个项目的新方法将约束合并到评估过程本身，避免了浪费时间，并保证以以前不可能的方式收敛。该项目还包括综合研究和学习活动，并将作为罗文大学新博士项目的重要催化剂，提供其开幕班。本研究(1)发展了一种序贯蒙特卡罗方法，迭代构造一组近似状态后验密度且满足非线性约束的粒子；Ii)建立了该方法的误差界和收敛性；Iii)推导出传统方法承认有界估计误差的充分必要条件；Iv)应用和评估理论结果来解决现实世界的应用，具有线性和非线性约束，包括手部假肢的控制，通信和生物学中时变稀疏网络的估计，以及电网中的新兴应用。

Many modern technologies (from image stabilizations in a camera, to chemical plants, from power grids to robot navigation) require computer algorithms to track the state of a dynamical system that is both modeled and measured with uncertainty. Particle filters are a technique that track many particles (candidate states) to arrive at a best estimate, which is the mean or average of tracked state. This project considers constraints on the best estimate (and not just individual particles) giving a new way to ensure correctness of the modeling, and safety of the underlying system. Handling constraints in dynamical systems in real time is challenging when either the systems or the constraints, or both, are nonlinear. The new methods of this project incorporate the constraints into the estimation process itself, avoiding wasted time and guaranteeing convergence in ways that were not possible before. The project also includes integrated research and learning activities, and will serve as a crucial catalyst to the new Ph.D. program at Rowan University by providing its inaugurating class. This research (i) develops a sequential Monte Carlo method that iteratively constructs a set of particles that approximate the posterior density of the state and also satisfy the non-linear constraints; ii) establishes error bounds and convergence properties of this method; iii) derives necessary and sufficient conditions under which traditional approaches admit a bounded estimation error; iv) applies and assesses the theoretical results to solve real-world applications, with linear and non-linear constraints, including control of hand prostheses, estimation of time-varying sparse networks in communications and biology, and emerging applications in the electric power grid.