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Theory and applications by Leondes, Cornelius T.

By Leondes, Cornelius T.

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EXAMPLE 2. It can be shown that the choice of U(t) = [1, t] made in Example 1 makes the control u(t) = U(t)oc optimal in the sense of Section III. We will now apply the preceding dicussion to Example 1. Using Eqs. (18) and (21) from Example 1, it is seen that x(t) = i/j(t, t )x + M(t )x 0 0 (39) f where >P(t, t ) = (40) 0 M(0 = (41) Thus *i(t )=x -x (T-t) n xJt\=x f2 + C (T-tf 1 + C (T-t 2CJT~t)-3CJT-tY. 2 ) 3 (42) (43) where C and C are the appropriate constants; and x 2 (44) 31 FINAL VALUE CONTROL SYSTEMS As t -> T there is a first-order singularity in the velocity loop and a second-order singularity in the position loop, as predicted.

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