Adjoint analysis of guidance systems for time-series inputs using Fourier analysis
July 1, 2020
The adjoint technique is a proven technique for analysis of linear time-varying systems and is widely used in the missile design community. It is a very efficient technique that can solve for both deterministic and stochastic disturbances and can develop a miss distance budget in a single computer solution of the differential equations without use of time-consuming Monte Carlo simulations. The adjoint technique is very valuable in both preliminary and more advanced missile design stages and is based upon the mathematical adjoint of the system dynamics matrix of the homing loop. Zarchan [1,2] describes extensive use of the technique for a variety of disturbances for homing missiles, and this author has developed its use for command guided missiles . For adjoint analysis, the usual method of modeling maneuver disturbances to a missile guidance system starts by modeling the maneuver in the forward-time system as a delta function input into a transfer function with the same second-order statistics as the maneuver, and its output is input into the guidance system; then the system is converted into its adjoint system . Bucco and Weiss  show that a set of nonstandard time-varying inputs cannot be treated in the normal fashion [2,5,6], and they present a new technique that enables these nonstandard inputs to be analyzed using adjoint analysis. This paper was inspired by and extends the results of the paper by Bucco and Weiss . This paper shows that the use of the complex digital Fourier amplitude spectrums of both the maneuver and the adjoint impulse response at the maneuver point allows adjoint analysis to address another type of nonstandard input, namely, an arbitrary time-series inputs such as specific target maneuvers that are not representable by an impulse input into a transfer function; heretofore, these time-series inputs have not been treatable with adjoint analysis. Additionally, if there are several sets of arbitrary time series of target maneuvers, each with an associated probability of occurrence, the root-mean-square (rms) value of the set of probabilistic maneuvers can be calculated, another significant new capability introduced in this paper.