Multiple Stepping Solution to Linear Two Point Boundary Value Problems in Missile Integrated Control

Abstract

A multi -stepping state transition matrix approach for solving a linear two -point boundary value problem is developed. The algorithm employs partitioned state transition matrix of the Hamiltonian system, and is computationally less expensive than back ward integration of differential Riccatti equation. This fact makes it ideally suited for online implementation. The application of this technique is illustrated for a finite interval moving mass actuated missile guidance -autopilot for target intercepti on. A combination of feedback linearization and the multi -stepping linear boundary value solution algorithm is employed in the example. Closed loop simulation results are given. 1-3 . As an example, these pro blems arise in the derivation of guidance -control laws that minimize the terminal miss distance, while penalizing the control effort. Different techniques for solving the two -point boundary value problem are discussed in R eference 1. Techniques such as the shooting method require the solution to the initial value problem using either numerical forward integration of the differential equations or the use of state -transition matrix solution. The numerical integration of differential equations is a time consum ing method, while the state -transition matrix approach suffers from numerical difficulties 1 for large time intervals and ill -conditioned Hamiltonian matrices. Control computation for a finite -interval LQR problem can also be posed as a solution to the Ricc atti differential equation. However, integrating the differential Riccatti equation at each instant of time backwards may not feasible for real -time control computation. Off -line gain computation and implementation requires large memory from the on -board p rocessor. Moreover, the solution may not be useful in a dynamic setting where the boundary conditions keep changing with time. An analytical state transition matrix based solution has been discussed in Reference 1. However, the state - transition matrix can be difficult to compute for large time periods. This paper addresses the numerical ill conditioning problem by dividing the time -interval into multiple intervals and employing the transition matrix solution in each subinterval. This approach dramatically i mproves the numerical condition of the problem, while avoiding the need for numerical integration. One of the byproducts of this algorithm is the solution to the differential Riccati equation. The technique can be implemented using linear -algebraic operat ions available in software packages such as LAPACK 4 . The numerical algorithm is described in the Section II. Section III describes the application of this technique to the development of a finite -interval guidance -control system for a kinetic warhead. Enga gement simulation results for a moving mass actuated kinetic warhead are presented in section IV. Section V summarizes the conclusions from the present research.