GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME Introduction In Section we defined the target dynamics model for target having a constant velocity; see (). A constant-velocity target is one whose trajectory can be expressed by a polynomial of degree 1 in time, that is, d ¼ 1, in (). (In turn, the tracking filter need only be of degree 1, ., m ¼ 1.) Alternately, it is a target for which the first derivative of its position versus time is a constant. In Section we rewrote the target. | 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 8 GENERAL FORM FOR LINEAR TIME-INVARIANT SYSTEM TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME Introduction In Section we defined the target dynamics model for target having a constant velocity see . A constant-velocity target is one whose trajectory can be expressed by a polynomial of degree 1 in time that is d 1 in . In turn the tracking filter need only be of degree 1 . m 1. Alternately it is a target for which the first derivative of its position versus time is a constant. In Section we rewrote the target dynamics model in matrix form using the transition matrix see and . In Section we gave the target dynamics model for a constant accelerating target that is a target whose trajectory follows a polynomial of degree 2 so that d 2 see . We saw that this target also can be alternatively expressed in terms of the transition equation as given by with the state vector by for m 2 and the transition matrix by see also . In general a target whose dynamics are described exactly by a dth-degree polynomial given by can also have its target dynamics expressed by which we repeat here for convenience Xn 1 Xn where the state vector Xn is now defined by with m replaced by d and the transition matrix is a generalized form of . Note that in this text d represents the true degree of the target dynamics while m is the degree used by 252 TARGET DYNAMICS DESCRIBED BY POLYNOMIAL AS A FUNCTION OF TIME 253 the tracking filter to approximate the target dynamics. For the nonlinear dynamics model case discussed briefly in Section when considering the tracking of a satellite d is the degree of the polynomial that approximates the elliptical motion of the satellite to negligible error. We shall now give three ways to .
