Optimal guidance for high-order and acceleration constrained missile

Abstract

Explicit formulas of optimal guidance for a linear, time-invariant, arbitrary-order, and acceleration constrained missile are derived. These formulas are given in terms of the missile's transfer function and acceleration constraint. Optimal, full-state feedback guidance laws are synthesized and compared to first-order approximation and the proportional navigation for minimum and nonminimum missile dynamics. Simulations on a third-order missile model show the relative gain from using the full-order guidance law vs the acceleration constraint as well as some robustness tests. HE optimal control theory has been used to derive modern/optimal guidance laws which have improved performance. The improved performance of these homing laws is achieved by a consideration of the detailed dynamics of the threat (target) and the interceptor (missile). However, it comes at the expense of increased complexity in realization (cost), sensitivity to knowledge of various parameters, etc. An extensive study of literature on guidance laws in general, and optimal guidance laws in particular, is performed by Pastrick et al.1 In various references,2'6 optimal guidance laws are derived for first- and second-order missiles, respectively. In our previous paper,7 the structure of optimal guidance laws for a linear, arbitrary high-order missile was considered. Mainly, we derived the closed-loop, general structure formulas of the guidance law. Further, we studied the behavior of the gains for minimum and nonminimum phase missiles and compared the performance of some sub-optimal approximations of the guidance laws. The effect of the acceleration constraint, which is imposed by the structural or aerodynamic limitations, on guidance laws and performance for a first-order missile is systematicall y treated by Anderson.8 In this paper we derive an optimal/moder n guidance law, on collision course, for a linear, time-invariant, arbitrary order and acceleration constrained missile. It is shown that, for a minimum phase missile, the optimal guidance law is the guidance law for an unconstrained missile with saturation on the commanded acceleration. However, for a nonminimum phase missile, this is only a suboptimal guidance law, and the optimal controller is more complicated. In the paper comparison of the proportional navigation, first-order approximation, and full-order guidance laws is performed on a third-order minimum and nonminimum phase model of a missile. The comparison is performed on a common basis. Moreover, the robustness of these guidance laws is subject to an analysis, namely, the sensitivity to uncertainty/ variation in parameters, radome refraction slope, and acceleration constraint is checked for minimum and nonminimum phase air frames. The main conclusions are that, for a minimum phase missile, the full-order guidance law does not give improved performance with respect to the first-order approximation,