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**Extra resources for Control and Dynamic Systems: Advances in Theory and Applications : Advances in Aerospace Systems Dynamics and Control Systems/Part 1 of 3**

**Example text**

Random variables uniformly distributed over the validation region V , whose volume V is given by Eq. (23). The factor P [see Eq. (13)] accounts for the restriction of the normal density to the validation region. The second term on the right side of Eq. (37) is the probability of event 0 (fc) conditioned on Z ~ and the number of current validated measurements. T o simplify the notation in deriving this term, the time index k and the conditioning on Z ~ are dropped. Let m be the random variable of the number of validated measurements (at time k) and m its realization.

The second term in Eq. (62), p [ x ( / c ) I Mj(k\Z ~ ~\, is the prior for the filter matched to model Mj(k) and is related to the mixed initial conditions x (k — 11 k — 1) and P (k — 11 k — 1) mentioned previously. T o see how the mixed initial conditions are obtained, apply the total probability theorem to this prior to obtain j9 k 1 0j 0j plx(k)\Mj(k) Z -^ k 9 = = £ i= ι plxik^MjikXM^k-llZ'-^^PiM^k-l^MjiklZ^ } 1 £ p[x(k)I Mj(k),x (k l - 1 \k - 1),P'(k - 1 \k - 1)] · ^ (k u - 1 \k - 1), (63) where μ φ - 1 \k - 1) = P{M,(k - \)\Mj(k),Z - }.

S. MAYBECK, "Practical Adaptive Estimation (and Control) Algorithms," unpublished notes, A. F. Institute of Technology, Wright-Patterson AFB, Ohio (1979, updated 1983). 36. Κ. P. S. Thesis, A. F. Institute of Technology, Wright-Patterson AFB, Ohio (1984). 37. P. S. MAYBECK and K. P. HENTZ, "Investigation of Moving-Bank Multiple Model Adaptive Algorithms," Proc. IEEE Conf. Decision Control, Fort Landerdale, Florida, pp. 1874-1881 (1985). 37a. P. S. M A Y B E C K and K. P. HENTZ, "Investigation of Moving-Bank Multiple Model Adaptive Algorithms," AIAA J.