Design Equations for Proportional Navigation with Parasitic Feedback

Abstract

Much of guidance and control system design involves iterative analysis with a series ofdynamic models ofincreasing complexity. To initialize this process for classical proportional navigation systems with parasitic feedback, we have developed algebraic equations that embody certain elements of synthesis and analytical design (1]. The canonic model is a seventh order system with radome aberration and imperfect seeker body motion isolation. The design equations yield fifteen parameters that describe the model, the miss budget as a function ofradome aberration, and the stochastic commanded acceleration and control surface rate due to range independent noise. Other indicators provided to guide the design include the rate loop gin and the low frequency phase margin ofthe autopilot. The effects of radome have been discussed by Peterson [21 Garnell [31 and Nesline and Zarchan [41, with emphasis on stability and performance analysis of canonic models such as biquadratic 12A and five equal time constants [41 Imperfect body motion isolation has been analyzed by Neshne and Zarchan [5]. In this paper, only the controller architecture is regarded as canonic; the distribution of the dynamics can vary over a considerable range. The dynamic representation of a homing controller can be considered as a point in multidimensional space that is converted to a zone by the effect of parasitic feedback. This zone in dynamic space has its counterpart in performance space, and it is this zone which is constrained and normalized. Then an equivalent constrained normalized zone in dynamics space is defined. Finally, the process ends with the means of getting from the point at the center of the normalized dynamics zone to a desired, but constrained, point in dynamics space. In the approach taken here, a set ofnormalized miss distance adjoints is defined for a normalized parasitic zone. All systems to be considered are scaled to pass through the points corresponding to these adjoints. The techniques for adjusting the system dynamics are order reduction by truncation, biasing, and scaling. The order of the system is expanded to provide modeling detail within the constraints imposed by the reduced order synthesis. The elements of these equations were assembled over more than a decade. The first steps involved the precomputation of a large number of normalized cubic adjoints, which were stored for use in the rapid analysis of guidance systems with parasitic feedback. Later the advent of the programmable calculator motivated the reduction ofthe number ofstored adjoints and resulted in a very simplified analysis program. The final phase of algorithm development, that of synthesis and order expansion, coincided with the appearance of the personal computer as an engineering tool.