LINEAR TIME-VARIANT SYSTEM INTRODUCTION In this chapter we extend the results of Chapters 4 and 8 to systems having time-variant dynamic models and observation schemes [5, pp. 99–104]. For a time-varying observation system, the observation matrix M of () and () could be different at different times, that is, for different n. Thus the observation equation becomes Y n ¼ M nX n þ N n ð15:1-1Þ For a time-varying dynamics model the transition matrix È would be different at different times. In this case È of () is replaced by Èðt n ; t nÀ1 Þ to indicate a dependence. | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 15 LINEAR TIME-VARIANT SYSTEM INTRODUCTION In this chapter we extend the results of Chapters 4 and 8 to systems having time-variant dynamic models and observation schemes 5 pp. 99-104 . For a time-varying observation system the observation matrix M of and could be different at different times that is for different n. Thus the observation equation becomes Yn MnXn Nn For a time-varying dynamics model the transition matrix would be different at different times. In this case of is replaced by tn tn_1 to indicate a dependence of on time. Thus the transition from time n to n 1 is now given by Xn 1 tn 1 tn Xn The results of Section now apply with M T and T replaced by Mn tn tn_i and Tn respectively see through . Accordingly the least-squares and minimum-variance weight estimates given by and apply for the time-variant model when the same appropriate changes are made 5 . It should be noted that with replaced by tn t _f the results apply to the case of nonequal spacing between observations. We will now present the dynamic model differential equation and show how it can be numerically integrated to obtain tn tn_ . 354 TRANSITION MATRIX DIFFERENTIAL EQUATION 355 DYNAMIC MODEL For the linear time-variant dynamic model the differential equation becomes the following linear time-variant vector equation 5 p. 99 ddtX t A t X t where the constant A matrix is replaced by the time-varying matrix A t a matrix of parameters that change with time. For a process described by there exists a transition matrix tn tn that transforms the state vector at time tn to tn that is X tn 0 tn C tn X tn This replaces for the time-invariant case. It should be apparent that it is necessary that t tn I TRANSITION MATRIX DIFFERENTIAL EQUATION We .
