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By Cornelius T Leondes
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Example text
Moyer and Pinkham [11] compare the convergence properties of quasilinearization to those of gradient and second-order methods. For the orbit transfer problem considered, the convergence properties of quasilinearization are significantly better than those of the other two methods. Paine [12] extends the algorithms to permit bounded control variables, but he does not treat general forms of control variable inequality constraints. McGill [13] uses a penalty function approach to handle state variable inequality constraints.
Are sufficiently close, one can write the follow- K—l ing linear approximation. K K~*l - Y k-l ( Vl) ] = KcA(\-J +K cp [ V\-l + 2£ ) where = [δψ / Ο Υ ( τ ) ] | , . K cm er [k-1 K *c = [K cmF(a)(T) cp + = KcpF(a+l)(T The terms \ ( \ _ 2 . + 2 ε ) approximated as follows. + 2e)] ^JhY(T a n d Y 40 + (121) ( 1 2 2 ) |k-l 2£) + V ô t ] | k - l · k-i^\-i + 2ε ) c a nb e ( l 2 3 ) SUBOPTIMAL STRATEGIES USING QUASI LINEARIZATION dg Y k(T k + 2e) g (T + 2 k k- 1 g g Subarc Extrapolation e) ) (T k-1 k -1 k ( t U k-1 ) k-1 Time 'k Fig.
Evaluated using k M a) means that M(a) is Uk-1 and uk-l. Assuming that nonsingular, equation (11)+) can be solved for uk 1 uk - uk-1 (nkal / 1_ (a) Mka1 `Uk 1kal -u is k-l' - Yk- l)] (117) The 5G(a) term will normally be zero, but it is retained here in the interests of generality. Equation (117) can be substituted into equation (111) and the result solved for . ~(a) k ~(a) _F + a [Jkai Rkai kai )-1 (n kat_ Y k 1-1 M - (Jkal - P a l k (Nkal + P(a) ltal N(a) / (118) k-1 1 5G(a). (a) = F(a)• k-1 k-1 satisfies the differential constraints, this is Unless Ykal not strictly true.