This paper considers nonlinear state estimation subject to inequality constraints in the form of linear and linear-matrix inequalities. Rewriting the standard maximum likelihood objective function used to derive the Kalman filter allows the Kalman gain to be found by solving a constrained optimization problem with a linear objective function subject to a linear-matrix-inequality constraint. Additional constraints, such as weighted-norm- or linear-inequality constraints, that the state estimate must satisfy are easily augmented to the constrained optimization problem. The proposed constrained estimation methodology is applied in the extended Kalman filter (EKF) and sigma point Kalman filter (SPKF) frameworks. Motivated by estimation problems involving a vehicle that can rotate and translate in space, multiplicative versions of the constrained EKF and SPKF formulations are discussed. Simulation results for a ground-based mobile robot operating in a constrained three-dimensional terrain are presented and are compared to results that use the traditional multiplicative EKF and SPKF, as well as filters that enforce inequality constraints by simply projecting the state estimate into the constrained domain along the shortest Euclidean distance.

中文翻译：

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非线性系统的线性和线性矩阵不等式约束状态估计

本文考虑了线性和线性矩阵不等式形式的不等式约束下的非线性状态估计。重写用于推导卡尔曼滤波器的标准最大似然目标函数允许通过使用受线性矩阵不等式约束的线性目标函数求解约束优化问题来找到卡尔曼增益。状态估计必须满足的附加约束，例如加权范数或线性不等式约束，很容易扩展到约束优化问题。所提出的约束估计方法应用于扩展卡尔曼滤波器 (EKF) 和西格玛点卡尔曼滤波器 (SPKF) 框架。受到涉及可以在空间中旋转和平移的车辆的估计问题的启发，讨论了约束 EKF 和 SPKF 公式的乘法版本。展示了在受约束的三维地形中运行的基于地面的移动机器人的仿真结果，并将其与使用传统乘法 EKF 和 SPKF 的结果进行比较，以及通过简单地将状态估计投影到受约束条件来强制执行不等式约束的滤波器沿最短欧几里德距离的域。