Adjoint Method for Missile Performance Analysis on State-Space Models

Abstract

The adjoint method is a method widely used in the preliminary design of guidance loops to obtain quick estimates of the performance of a guided weapon while avoiding time consuming Monte Carlo simulation experiments. Traditionally, the adjoint method is presented as a set of rules for transforming the original (linearized) model of the guidance loop to an adjoint model that is used to obtain the performance estimate for the original model in a single simulation run. An attempt is made to derive the adjoint method in the general setting of state-space models. This does not only lead to an elegant and clear exposition of the adjoint method, but it also extends the area of application of the adjoint method to more general situations that were not previously covered in the literature. One possible extension is illustrated with the analysis of the performance of a rolling missile against a target that performs a maneuver with a random start time. I. Introduction T HE adjoint method has a well-established place in the arsenal of tools available for guidance-loop performance analysis, in particular in preliminary phases of guided missile design. The success of the method is mainly because it is relatively simple to use and because it enables a quick performance assessment for a wide range of engagement conditions. The method as a general analysis method for control systems under stochastic disturbances was introduced by Laning and Battin. 1 However, some authors, such as Zarchan, 2 place the origin of the method back to the work of the 19th century mathematician Vitto Voltera. Zarchan mentions the application of the method in computing ballistic dispersions as early as the 1920s. Although the method is also applicable for the analysis of systems under deterministic disturbances, it is the stochastic case that demonstrates the real power of the method. In this case, the method can be efficiently used to avoid time consuming Monte Carlo simulations when linear approximation results can deliver sufficient accuracy. The traditional presentation of the adjoint method in the literature 2,3 is based on the input‐output (transfer function-type) system representation with a strong emphasis on the procedural aspects of the method: inverting the sense of the signal flow, substituting time with time-to-go, etc. In this work, we pursue in detail the derivation of the construction rules for the adjoint system based on state-space models. We consider both the deterministic and the stochastic case for continuoustime models. In the stochastic case, our approach is shown to extend the potential of the method beyond applications currently reported in the literature that assume uncorrelated inputs and look only at the variance of the chosen output signal. We show here that the adjoint method can perfectly accommodate correlations between the different inputs and that a possible outcome is the influence of the random inputs on the covariance of two different outputs at the a priori fixed moment of time. In this sense, it is shown that the analysis power of the adjoint method is almost identical to that of another method for preliminary design analysis called the covariance ma