CONVERGENCE OF THE DOUBLING-ALGORITHM FOR THE DISCRETE-TIME ALGEBRAIC RICCATI EQUATION

If the doubling algorithm (DA) for the discrete-time algebraic Riccati equation converges, the speed of convergence is high. However, its convergence has not yet been examined exactly. Firstly a certain matrix appearing in the DA is shown to be non-singular and therefore the algorithm is well defined. Secondly, it is found that the loss of significant digits hardly occurs in the DA. Finally, it is proved that if the time-invariant discrete-time linear system, whose state is estimated by the steady-state Kalman filter, is reachable and detectable, or stabilizable and observable, then all three matrix sequences in the DA converge.