Lead-Lag Algorithms

1. Introduction

This paper presents the derivations of recurrence formulas used in the digital simulation of continuous dynamic response, as modeled by ordinary linear differential equations. Emphasis is placed on first and second order equations, since these are the two most common "transfer functions" used in real-world applications. A general formula for the Nth order response is presented in Appendix B.

Along with the optimum recurrence formulas, a variety of specialized and simplified methods are discussed and evaluated to clarify their underlying assumptions and ranges of applicability. A brief discussion of how discrete-time recurrence formulas can be combined in a feedback loop (that may contain an arbitrary non-linear function) is provided in Appendix A.

Throughout this document we assume that the time required to perform a recurrence computation is negligible relative to the characteristic times of the dynamic functions being simulated. (This implies that the "present value" of the input signal is always available for use in computing the "present" output value.) Also, the "round off" error of the computations is assumed to be negligible. In view of the speed and word size of modern digital processors, both of these assumptions are justified when modeling most large-scale physical phenomena.