The singular perturbation technique model of air-to-air interception in the horizontal plane

Abstract

In this work, an optimal trajectory of a pursuit missile, intercepting an evading target is investigated. The technique used to solve this problem is the singular perturbation technique (SPT). The interception problem is solved to the first order in the open loop form and in the feedback form. The open loop solution is applicable to systems with initial and/or terminal boundary layer, whereas the feedback form solution can be applied to problems with initial boundary layer only. The general aircraft performance optimization problems, described by stiff, autonomous non-linear systems do not have a "genuine" singularly perturbed structure. However, the dynamic behavior of such systems can sometimes be very similar to the behavior of singularly perturbed systems in that they contain variables that change over different time scales. An important advantage of the SPT is related to the on-board implementation of optimal control strategies as direct inputs to the automatic flight control system. Such an implementation generally requires the optimal control solution to be found in a feedback form, so that use can be made of measured information. It is well known that a linear feedback controller can be obtained using linear-quadratic analysis. Unfortunately, performance optimization problems cannot be described by a linear model. Often the performance index, such as time-of-flight or fuel-consumed has no quadratic form. It was observed that if the terminal constraints do not involve fast variables, the terminal boundary layer may disappear and then the SPT solution can be expressed in a uniformly valid feedback form. In the interception problem stated below a pursuit missile, flying at a constant speed vm, has to intercept an evading target, flying at a constant speed vt in a given direction. The relative motion of the two missiles can be described in Cartesian coordinates by : where x and y are the relative cpordinates and y is the heading angle of the pursuing missile, a = vmhm where rm is the minimum turning radius of the pursuer and equals vm21g The controller u is realized by the bank angle, @ and is defined by tg+/tg@max. The natural singular perturbation parameter for this problem is rm/R0, where Rg is the initial separation distance. If Ro is large compared with the radius of turn, the rate of change of the relative location (or the line of sight) is much slower than the change of the heading direction of the pursuer. Hence, the forced SPT model can be described by: dy --v, . sin y~ dt