By Alain-Sol Sznitman
Offers an account of the non-specialist of the circle of rules, effects & innovations, which grew out within the examine of Brownian movement & random hindrances. DLC: Brownian movement techniques.
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Extra info for Brownian motion, obstacles, and random media
Example text
T N 1 = L[X(tN). U ( M r t N 1 + j-mx, j:GLf(6)k( 6 One potential choice of penalty function L, in the state of the system at some selected final time tf, such as for the air-to-air missile intercept problem mentioned previously. In particular, we will weigh the final state by a scalar penalty or loss function, L , [ x ( t , ) ] , to express our desires about possible values of x(rf). If we wanted a scalar terminal state to we might choose a penalty be inside some region bounded by xminand ,,,x, function as depicted in Fig. 6. , a continuous function or a limit of continuous functions (although the function in Fig. That requires explicit knowledge of the entire measurement history Zi and of the current time t i : a very burdensome growing memory requirement for implementation. However, under the LQG assumptions, a vast simplification occurs in the form of suficient statistics [11,12,40,41]. For the LQG controller problem, this density is Gaussian and completely specified by its mean %(ti+)and covariance P(ti+):if these are available, we need not remember all of the measurement values Zi explicitly. Thus, we can equivalently solve and P(ti+),the latter of which the optimization problem as a function of %(ti+) does not depend on Zi, as provided by a Kalman filter.