M.Sc Student M.Sc Thesis Heller Carmit Pursuer Maneuver Minimization by Softly Constrained Optimal Guidance Department of Aerospace Engineering PROF. Yoseph Ben-Asher PROF. Isaac Yaesh

Abstract

When designing a guidance law, one should take into consideration both the missile limitations requiring small maneuver accelerations, and the final miss distance requiring the maneuvering to be as large as possible. The suggested new guidance law is derived from a cost function that minimizes the weighted sum of miss distance and normal accelerations both to the power of q, where q is an even number. This formulation of the cost function leads to minimization of the maximal acceleration, as the value of q increases, rather than the total energy. This cost function is referred to as "Hyper-quadratic", thus naming the new guidance law "HQN".

The problem formulation involves the relative displacement, x1, the closing velocity, x2, pursuer acceleration, u, and evader acceleration, w. When studying the case of one-sided optimization, the evader acceleration is constant: w0. The optimal controller is found by variational calculus and is found to be of similar form as APN, where the expression (2q-1)/(q-1) can be viewed as the navigation constant N':

u = -  ((2q-1)/(q-1))*(x1(t) +  x2(t) *tgo + 0.5*w0*tgo2 )/ tgo2

The optimal guidance law, that tends to minimize the maximum absolute maneuver is achieved when q tends to infinity, leading to N'=2.

Next the problem of differential game optimization, where both u and w take part in the minimization and maximization of the cost function respectively, is solved. The optimal pursuit maneuver and optimal evasion maneuver are derived using a Riccati equation approach. In this case the optimal u and w are found to be function of N' and the maneuverability ratio (MR). For q = 2 we obtain the navigation constant: N' = 3/(1- MR-1), and for q that tends to infinity the term tends to: N' = 2/(1- MR-1).

In order to gain a deeper insight into the resulting family of guidance laws, a comparison is offered to the Pareto optimal set of mixed L2/Lq guidance laws. The comparison reveals that the family of guidance laws which were obtained by even q = [2,∞ ) closely resembles the Pareto optimal L2/Lq set obtained by  numerical optimization.

Finally, HQN performance in various cases is analyzed and compared to the corresponding performance of some well known guidance laws: Proportional Navigation (PN) for one sided optimization and Differential game Law (DGL) for differential games. This study shows that HQN achieves very small miss distance values while maintaining low lateral acceleration.