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By Cornelius T. Leondes

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Additional info for Advances in Aerospace Systems Dynamics and Control Systems: Advances in Theory and Applications : Advances in Aerospace Systems Dynamics and Control Systesm, Part 2 of 3

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Thus, (17) becomes z(t) = Aez(t) + (Aus - BeCs)xs(t) + Dluw(t) - BeD2w(t). (19) Furthermore, the explicit dependence of the estimation error (14) on the xu(i) subsystem can be eliminated by constraining (20) Ce = Lu-DeCu. Henceforth, we assume that Ae and Ce are given by (18) and (20). (0 14 o„ sXnu ^ J* To guarantee that J is finite, consider the set of asymptotically stable reducedorder estimators, ^ = {(AC9 Be, Ce,De): Ae = Au- BeCu is asymptotically stable}, which is nonempty if (Au, Cu) is detectable.

Denote *4 := 7P> *5 ·= 7E> and x6 := φΕ - φΡ. It is assumed here that the motion of E and P relative to the coordinate system (xP,yP,Zp) is perturbed by Gaussian white noise processes that model the uncertainties in the kinematics, the effect of external random perturbations arising from the environment, and noises in the measurements received by E and P. Thus, with this reasoning, Eqs. ,6, are given positive numbers such that 0 < oi « min(F E , VP\ i = 1,2,3, and 0 < σ, « π, j = 4,5,6. 43 PROPORTIONAL NAVIGATION IN 3-D Equations (22)-(27) constitute the fundamental equations of motion of players E and P.

Let Ξ be the event {before player E enters the safe zone, E enters KP at some i, 0 < t < T, or player E stays at the domain Dp — Kp for all t e [0,i 0 ], for some t0 > T}. Player E steers by choosing, at each instant, control functions in such a manner as to minimize Prob. (Ξ), whereas player P steers by choosing, at each instant, control functions in such a manner as to maximize Prob. (Ξ). In the past few years pursuit-evasion problems in three dimensions have been dealt with in, for example, [ 1 - 7 ] .

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