White noise estimation for linear discrete fractional order system

The process white noise (PWN) and observation white noise (OWN) estimation problem for linear discrete fractional order systems (LDFOS) is addressed in this study. By using the Grünwald-Letnikov (G-L) operator as a definition of the discrete fractional calculus (DFC), LDFOS is transformed into a class of linear discrete time-delay systems. However, it is different from the general time-delay system, in which the time-delay part is the cumulative sum from time 0 to the previous time. Based on the orthogonal projection theorem, a suboptimal one-step predictor of LDFOS is designed. Due to the existence of cumulative sum time-delay in system, the Riccati equation has one more cumulative sum state error variance term, which is different from the classical Kalman filter (KF). Moreover, using innovation analysis technology, the filtering and fixed-lag smoothing estimators of PWN and OWN in the form of noise orthogonal projection gain matrices are derived. Finally, two simulation examples are given to verify the effectiveness of PWN and OWN estimators.