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

许多现代技术(从相机中的图像稳定，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，而动态系统的建模和测量都带有不确定性。粒子过滤器是一种跟踪许多粒子(候选状态)以获得最佳估计的技术，该估计是跟踪状态的平均值或平均值。这个项目考虑了对最佳估计(而不仅仅是单个粒子)的约束，提供了一种新的方法来确保建模的正确性和底层系统的安全性。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具有挑战性的。该项目的新方法将约束纳入评估过程本身，避免了浪费时间，并以以前不可能实现的方式保证了收敛。该项目还包括综合研究和学习活动，并将通过提供开学班，成为罗文大学新博士项目的重要催化剂。本研究(I)发展了一种序贯蒙特卡罗方法，该方法迭代地构造一组近似于状态的后验密度且满足非线性约束的粒子集；ii)建立该方法的误差界和收敛性质；iii)推导出传统方法允许估计误差有界的充要条件；iv)将理论结果应用于解决具有线性和非线性约束的实际应用，包括手部假肢的控制、通信和生物中的时变稀疏网络的估计以及在电网中的新兴应用。

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.