Retaining Physical Understanding Through Discretization

While digital implementation of control design is standard, the form of discrete model used is far less settled. At one end is the zero-order hold (ZOH) equivalent, which can be viewed as an "exact" model when the continuous-time (CT) system model is linear and time invariant (LTI) and driven only by outputs from one or more digital-to-analog converters (DACs) at a single sample rate. At the other end are ad-hoc methods that often discretize individual subsystems or blocks, before combining them into a single overall discrete model. The issue with the ZOH equivalent is that for all but the simplest models closed-form solutions become largely intractable. ZOH equivalents are largely computed numerically for larger problems, but this makes it hard to comprehend such basic features as the meanings of the internal states, or the effects on the model as physical parameters or sample periods change. By contrast, discretizing individual subblocks of the model - as is often done in practice - retains much of the continuous model's intuition, allowing for easier debugging of the discrete model. We propose a "best-of-both-worlds" methodology in which we use the availability of excellent numerical software such as MATLAB and the knowledge that model differences imparted by different discretization methods tend to shrink with the diminishing sample period. In the proposed methodology, the "one-block-at-a-time" (OBLAAT) discretized model is evaluated at different sample rates and each is compared to a numerically computed ZOH equivalent of the full system CT model. An error metric of the intuition preserving discrete model is then compared against the "exact" ZOH equivalent. This is used to gauge when the inaccuracy of the intuition-preserving discrete model is small enough that it can be chosen for implementation.