Missile guidance based on weight adjusted singular perturbation

Abstract

AbriraciThere are two major limitations in applications of the singular perturbation technique to missile guidance and other optimization problems. The first limitation is the requirement that the reduced order subproblems be solvable and may be overcome in some cases by coordinate transformation. The second limitation is the failure of the required time scale separation when applying feedback control as the final time is reduced. The weight adjusted singular perturbation method developed in this paper overcomes both limitations and provides greatly improved performance which is validated by a realistic simulation in both deterministic and stochastic environments. The proportional navigation (PN) guidance law has been widely used for air-to-air missile guidance during the fast several decades because of its effectiveness and simplicity of implementation. Moreover, it was shown that the PN guidance law was optimal under certain conditions [I]. In order to obtain better performance, many authors have attempted to derive a guidance law by using modern control theory. Since the missile motion equation is nonlinear, the resulting nonlinear two point boundary value problem (TPBVP) is difficult to solve by on line computation. In order to circumvent this computational burden, Fiske developed an optimal linear (OL) guidance law by using a simplified missile motion equation modelled as double integrator in the inertial axis [2]. The performance of the optimal linear guidance law is satisfactory. But, when the initial range is short (within 4000 ft), the performance of the optimal linear guidance law is not satisfactory. This may be due to the neglected missile body X-axis acceleration. The singular perturbation method has been developed to obtain a nearly optimal solution for a class of nonlinear systems [3][4]. The utility of the singular perturbation method when applied to a nonlinear system is that an approximate solution can be obtained by solying several lower dimensional subproblems instead of solving the full problem. Hence, in order to successfully apply the singular perturbation, the problem must be formulated such that the lower order dimensional subproblem is well posed and solvable. 'This work warr sponsored by Eglin Air Force Armament Laboratory (FXG), Eglin, Florida under contract F08635-86-K-0265. In [5], Sridar derived a nearly optimal guidance law which is approximated up to the zeroth order. In order to obtain a better approximation, Visser [6] suggested a first order correction method by using the matched asymptotic expansion method. But, it was observed that the nearly optimal guidance law derived based on the singular perturbation method failed to give a good approximation during the terminal phase in which the time scale decomposition is not valid. In the application of optimal guidance laws to missile control, feedback guidance laws must be used to compensate for target motion, measurement errors and model errors. However, this means that the optimization problem should be resolved many times as the time-to-go (final time minus initial time) changes. This causes a difficulty with the singular perturbation scheme in that the choice of a performance index which allows time scale separation depends on the time-to-go. Therefore, a performance index which will allow good time scale separation and corresponding good approximations for large time-to-go will fail to work for small time-to-go.