Convergence of self-tuning Riccati equation with correlated noises

For the linear discrete time-invariant stochastic system with correlated noises, and with unknown model parameters and noise statistics, substituting the online consistent estimators of the model parameters and noise statistics into the optimal time-varying Riccati equation, a self-tuning Riccati equation is presented. By the dynamic variance error system analysis (DVESA) method, it is rigorously proved that the self-tuning Riccati equation converges to the optimal time-varying Riccati equation. Based on this, by the dynamic error system analysis (DESA) method, it is proved that the corresponding self-tuning Kalman filter converges to the optimal time-varying Kalman filter in a realization, so that it has asymptotic optimality. As an application to adaptive signal processing, a self-tuning Kalman signal filter with the self-tuning Riccati equation is presented. A simulation example shows the effectiveness.