Technical report | On Alternative Formulations for Linearised Miss Distance Analysis
In this report, techniques generally employed in the analysis of intercept guidance problems are reviewed. From the governing non-linear equations describing such problems, two basic linear models are derived. Traditionally, these linear models are utilised as a basis for preliminary intercept engagement studies. Under certain input conditions, the two models are mathematically equivalent and, hence, have been used interchangeably by weapons analysts to yield appropriate design and performance data in support of their programs. However, for a specific set of initial conditions, which includes a very important class of practical problems that may be assessed with the use of these models, it is noted herein that one of these linear models produces incorrect performance data when compared to a non-linear simulation of the engagement. In contrast, the other model produces consistent results with those generated by the non-linear simulation regardless of the initial conditions considered. To remedy this discrepancy, the necessary mathematics are derived to bring the two formulations into alignment for any form of the initial conditions and inputs to the system. Consequently, this leads to a consistency in the corresponding adjoint models which are constructed from these linear models, thus ensuring the generation of correct output data regardless of which model is employed by the analyst.
In this report, the missile-target engagement problem is analysed. As part of the analysis, the non-linear governing equations are reviewed for motion in a single plane. These non-linear equations are employed for two purposes. Firstly, the equations form the basis of a non-linear simulation program developed in Simulink. This Simulink model is controlled and executed via a graphical user interface specifically developed as an aid for the weapons analyst to study the engagement problem. Secondly, the non-linear equations are utilised as a basis for linearisation and, hence, the derivation of an approximate linear model of the engagement. Two different formulations of the linear model are derived. These are designated as Model A and Model B in the report.
Although both models are generally used interchangeably in the literature for guided missile homing loop analysis, it is demonstrated herein that, under certain input conditions, care needs to be exercised when using one of these models, Model A, for performance analysis. In order to ensure that both models yield the same performance data for all input conditions considered, a correction factor is derived. This correction factor needs to be included in the form of an added initial condition on one of the states in the state space representation of Model A. Simulation results show that the two models are in agreement when this correction factor is applied. Knowledge of this fact is important to ensure that analysts generate correct performance data when using linear techniques such as the adjoint method. The adjoint model is constructed from a knowledge of the forward linear model, that is, Model A or Model B, and is traditionally employed by analysts as part of the solution process.
To gain further insight into the nature of the missile-target engagement problem and the parameters that influence performance, also included in this report is an analytical treatment of the problem. It is well known that, when the missile guidance dynamics is represented by a first order lag (single time constant system), then the linear differential equations describing the intercept problem are readily amenable to analytical treatment. Consequently, an analytical investigation of each model (A and B) is carried out for the range of input conditions considered herein. The resulting closed form solutions from each model are compared and are shown to be mathematically equivalent for all input conditions considered provided that Model A has the proposed correction factor implemented. For simulation purposes, the corresponding adjoint model of Model A and Model B are constructed and then implemented in Simulink. As with the analytical results, outputs from the two models are shown to be in agreement. Finally, the models are extended to represent more realistic guidance system dynamics (fifth order system) and simulation results are generated and compared.
Finally, a special relationship linking two of the derived miss distance formulas is noted and explored further. This relationship highlights a connection between the miss distance due to an initial target displacement and that due to an initial heading error in the context of the linear analysis. Following verification using linear simulation, a formula based on this relationship is derived and proposed as a means for predicting performance data of more complex linear systems. This formula may also be used in connection with the non-linear simulation model for generating approximate miss distance profiles due to the effects of a step in target position prior to intercept. A step in target position typically arises in problems associated with the seeker resolution of the target in a multi-target scenario. It may also arise in the case of a single target scenario. For this case, the missile may be on a collision course with the predicted intercept point (PIP) of the target. However, at the time of seeker turn-on, the PIP may not necessarily coincide with the actual target position.