LMI-based ℌ2 adaptive filtering for 3D positioning and tracking systems

In this work, a cascade of two estimators is proposed as the solution for a joint parameter and state estimation problem associated with a target maneuvering in the three-dimensional space. A model for the target that depends on its angular speed is considered and only the target position is measured. A parameter identifier is used to obtain estimates of the target angular speed, which are then fed into an adaptive filter that estimates the position, linear velocity, and linear acceleration of the target. The synthesis of the parameter identifier resorts to Lyapunov techniques and the synthesis of the adaptive filter is tackled using Linear Matrix Inequalities (LMIs) and H2 optimization strategies. Under persistence of excitation conditions, the error in the angular speed identification and the error in the target state estimates provided by the H2 adaptive filter are: i) proved to converge exponentially fast to zero in the deterministic setup, i.e., in the absence of noise, and ii) proved to be bounded when bounded stochastic disturbances are considered and there is an upper bound on the target linear velocity and angular speed. To assess the proposed methods, simulations showing that the aforementioned stability and convergence properties hold, even when the estimates provided by an Extended Kalman Filter diverge, are presented.