The convergence analysis of the self-tuning Riccati equation

For the linear discrete time-invariant stochastic system with unknown transition matrix and unknown noise variances, a self-tuning Riccati equation is presented based on the on-line consistent estimations of the transition matrix and noise variances. In order to prove its convergence to the steady-state Riccati equation, a dynamic variance error system analysis (DVESA) method is presented, which transforms the convergence problem of the self-tuning Riccati equation to the stability problem of a time-varying Lyapunov equation. A stability decision criterion for the time-varying Lyapunov equation is presented. Using the DVESA method and Kalman filtering stability theory, it is proved that the solution of the self-tuning Riccati equation converges to the solution of the steady-state optimal Riccati equation. The proposed results will yield a new self-tuning Kalman filtering algorithm, and will provide the theoretical base for solving the convergence problem of the self-tuning Kalman filters. A simulation example shows the correctness of the proposed results.