By Cornelius T Leondes
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Extra resources for Control and dynamic systems : advances in theory and applications ; 10
Moyer and Pinkham  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  extends the algorithms to permit bounded control variables, but he does not treat general forms of control variable inequality constraints. McGill  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.