Global Observer Design for a Class of Linear Observed Systems on Groups

2025/07/24 by Liu, Changwu, Shen, Yuan
#FOS: Electrical engineering #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2507.18493

Abstract

Linear observed systems on groups encode the geometry of a variety of practical state estimation problems. In this paper, we propose a unified observer framework for a class of linear observed systems by restricting a bi-invariant system on a Lie group to its normal subgroup. This structural property powerfully enables a system immersion of the original system into a linear time-varying system. Leveraging the immersion, an observer is constructed by first designing a Kalman-like observer for the immersed system and then reconstructing the group-valued state via optimization. Under a rank condition, global exponential stability (GES) is achieved provided one global optimum of the reconstruction optimization is found, reflecting the topological difficulties inherent to the non-Euclidean state space. Semi-global stability is guaranteed when input biases are jointly estimated. The theory is applied to the GES observer design for two-frame systems, capable of modeling a family of navigation problems. Two non-trivial examples are provided to illustrate implementation details.

Citations

• Attitude Estimation Using Scalar Measurements
• A Geometric Approach For Pose and Velocity Estimation Using IMU and Inertial/Body-Frame Measurements
• Classification of Linear Observed Systems on Multi-Frame Groups via Automorphisms
• A General Nonlinear Observer Design for Inertial Navigation Systems with Almost Global Stability Guarantees
• Pose, Velocity and Landmark Position Estimation Using IMU and Bearing Measurements
• Equivariant Symmetries for Aided Inertial Navigation
• State Estimation Using Single Body-Frame Bearing Measurements
• MSCEqF: A Multi State Constraint Equivariant Filter for Vision-aided Inertial Navigation
• Constructive Equivariant Observer Design for Inertial Navigation
• EqVIO: An Equivariant Filter for Visual Inertial Odometry
• Equivariant filter (EqF): A general filter design for systems on homogeneous spaces
• Observers Design for Inertial Navigation Systems: A Brief Tutorial
• Associating Uncertainty to Extended Poses for on Lie Group IMU\n Preintegration with Rotating Earth
• A Mathematical Framework for IMU Error Propagation with Applications to Preintegration
• Exploiting Symmetries to Design EKFs with Consistency Properties for Navigation and SLAM
• The invariant extended Kalman filter as a stable observer
• Intrinsic filtering on Lie groups with applications to attitude\n estimation

Related

• Observer design for invariant systems with homogeneous observations
• Higher-Order Nonlinear Complementary Filtering on Lie Groups
• The invariant extended Kalman filter as a stable observer
• Gradient-like observers for invariant dynamics on a Lie group
• Design of Globally Exponentially Convergent Continuous Observers for\n Velocity Bias and State for Systems on Real Matrix Groups