A method for comparing trajectories in optimum linear perturbation guidance schemes.

Abstract

where the dxj functions are perturbed values of the state at refajtf], the fty(^r) functions are the feedback gains associated with the time r, and the functions Ui(t) = uf*(t) + 5ui(l) define the optimal controls for te[r,tf] if no further disturbances occur. In Ref. 4, the Lag-range multipliers (which result from the Euler-Lagrange equations associated with the variational problem) for the perturbed trajectory are obtained as power series in the state perturbations, dxi, and the maximum principle is then used to determine the corresponding dui. In the usual case, the 5w/s are determined so that Eqs. (3) are satisfied and the perturbed trajectory is optimal in some sense. Assume that the values of x*(r), u*(r), and Gij*(t,T) are stored onboard for each £,re [/ ,£/]. Then the time r is actually a parameter which associates the feedback gain GH (t,r) with the state-control point fe*(r), . . . ,#«*(T), W]*(T), . . . jUm*(r)] of the optimal trajectory. An inherent ambiguity in these schemes is the way that the parameter re[fo,£/] is determined for a state fe, . . . ,xn) which is close.to the optimal trajectory, but not on it. At first glance it appears that the time, say TI, at which the vehicle arrives at fa, . . . ,xn) is also the value of the lookup parameter Te[t0)tf]. However, TI may be greater than tf) and/or Z(TI) may not be close to X*(TI), whereas x(n} may be close to some other point on the optimum trajectory, say £*(r2) (see Fig. 1). In Refs. 5 and 6 an unpublished suggestion by J. C. Dunn is used to alleviate this ambiguity. In Ref. 5 it is shown