This paper is concerned with finding an optimal path for an observer, or sensor, moving at a constant speed, which is to estimate the position of a stationary target, using only bearing angle measurements. The generated path is optimal in the sense that, along the path, information, and thus the efficiency of a potential estimator employed, is maximized. In other words, an observer path is deemed optimal if it maximizes information so that the location of the target is estimated with smallest uncertainty, in some sense. We formulate this problem as an optimal control problem maximizing the determinant of the Fisher information matrix, which is one of the possible measures of information. We derive analytical results for optimality using the Maximum Principle. We carry out numerical experiments and discuss the multiple (locally) optimal solutions obtained. We check graphically that the necessary conditions of optimality are verified by the numerical solutions. Finally we provide a comprehensive list of possible extensions for future work.
An Improved Distributed Nonlinear Observer for Leader-Following Consensus Via Differential Geometry Approach
研究了分布式非线性观测器框架下的领导者跟随输出一致性问题。通过引入一组几何条件，取代了对领导系统的某些假设，得到了一种新的具有较小保守性的分布式观测器策略，从而提高了已有结果的适用性.更具体地说，改进的分布式观测器可以精确地处理一些非线性领导系统的一致性问题，而传统的策略在一定的假设条件下是无效的，如弹性轴单连杆机械手（ESSLM）系统和大多数一阶非线性系统。我们证明了我们的分布式观测器的误差动态的指数稳定性的充分条件，提出了两个开创性的引理，以显示在李雅普诺夫型矩阵中出现的两个矩阵的最大特征值之间的关系。然后，利用微分几何中提出的零动态部分反馈线性化方法，设计了仿射非线性多智能体系统的纯分散控制律。利用这一改进，可以将已有的结果看作是一种特殊情况，因为跟踪器可以选择为任意的最小相位仿射光滑非线性系统。最后，将新的分布式观测器和改进的纯分散控制律应用到分布式控制框架中，构建了闭环系统。我们还证明了闭环系统的稳定性，以实现领导者跟随共识，即，证明了分布式控制框架满足确定性等价原理。以ESSLM系统和Van der Pol系统为例说明了该方法的有效性。
