Minimax Robust Kalman Estimation Under Uncertain Noise Variances, Multi‐Step Random Measurement Delays and Missing Measurements

ABSTRACT

The paper solves the minimax robust Kalman estimation problem for a system under linearly correlated noise, uncertain noise variances, multiplicative noise, multi‐step random measurement delays, and missing measurements. The system noise variance is uncertain but bounded above, and a set of Bernoulli distributed random variables with known probability is used to describe the multi‐step random measurement delays and missing measurements from sensor to estimator. A novel model transformation method is proposed by using Hadamard product, and then the original system is converted into one only with uncertain fictitious noise variance. The robust time‐varying Kalman predictor, filter and smoother are designed in a unified form based on the minimax robust estimation principle. A new robustness proof method, including Gerŝgorin circle theorem, matrix elementary transformation, Hadamard product theorem and generalized Lyapunov equation method, is presented to demonstrate the robustness such that the actual estimation error variance is guaranteed to have minimal upper bound for all admissible uncertainties. The computational complexity is analyzed for the robust time‐varying estimator in each step, and the robust steady‐state estimator is designed. The convergence in a realization between the time‐varying and steady state estimator is proved. A simulation example verifies the correctness and effectiveness of the proposed results.