Publications

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 [3]. 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 [1]. Bucco and Weiss [4] 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 [4]. 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.

It is well known that interceptor performance against a weaving or spiraling target can be improved by use of a special purpose weave guidance law. However the weave guidance law requires knowledge of the target weave frequency. When the target weave frequency is unknown an extended Kalman filter is usually considered for the problem because it can be used to estimate the target weave frequency. However, the performance of the extended Kalman filter is sensitive to initialization errors. This paper offers an unusual linear Kalman filter bank approach, where each filter is tuned to a different target weave frequency, as a potential solution for estimating the target weave frequency. Rather than combining individual filter outputs in some probabilistic sense, a straightforward algorithm is presented for choosing the filter that is most closely tuned to the actual target weave frequency. This paper demonstrates that this filter bank approach is superior to that of the extended Kalman filter for the weaving target problem.

Performance prediction of miss distance due to sensor measurement errors and random target manuevers for missiles using proportional navigation guidance has been analyzed using the adjoint technique; a normalization technique has been usedto reduce the solution of the set of differential equations describing the proportional navigationguidance problemto a set of algebraic equations using normalized steady-state adjointmiss distance coefficients. The four-state optimalguidance system is generally accepted to yield superior miss distance performance to that of proportional navigation guidance. The previously mentioned normalization technique is described and extended to the four-state optimal guidance system to calculate a new set of values for the normalized steady-state adjointmiss distance coef. cients for this con. guration. Plots of these normalized coefficients as a function of a normalized tuning parameter provide designers with insight into system performance sensitivities to design parameter and intercept parameter variations. The advantage of this technique is that the results are closed-form equations, and the analyst neither needs to perform simulations nor even to solve the adjoint differential equations. In addition, optimalguidance system results formiss distance due to target spiral maneuver are presented asmiss distance normalized to the target maneuver spiral radius, thus providing valuable insights into interceptor performance.

It is demonstrated that at high altitude the performance of a tail-controlled aerodynamic missile can degrade because of the existence of low frequency right-half plane zeroes in the airframe transfer function when either proportional navigation or optimal guidance is used. A new guidance law that accounts for the airframe zeroes is developed numerically and shown to have superior performance to existing guidance laws at the higher altitudes. Although no closed-form solution for the guidance law is presented, the resultant numerical values for the control gains of the guidance law can easily be stored as a multidimensional table in existing on-board flight control computers. Two methodologies for computing the guidance law control gains are presented.

A concise theoretical technique is presented for estimating the minimum miss distance capability of command guided missile systems using synthetic proportional navigation. The effect of the parameter values on the system capability is shown to be a function of range-to-intercept; the technique enables the system designer and analyst to quantify system performance and to develop a systematic understanding of the performance limitations of command guidance systems at each intercept range. New analytical equations based upon adjoint theory are developed for statistical miss distance caused by target maneuver, range-dependent, servo, glint and atmosphere noises for command guided systems. An optimal total system time constant is derived which yields the minimum statistical miss distance. Realistic constraints on the minimum achievable system time constant are considered. The equations derived for the optimal total system time constant are valuable to the system designer for minimizing miss distance over the ranges of system parameters and limitations, and intercept conditions.