Simple Digital Controls from Approximate Plant Models

Two ways of designing low-order discrete-time (i.e. digital) controls for low-order plant (i.e. process) models are considered in this tutorial. The first polynomial method finds the controller coefficients that place the poles of the closed-loop feedback system at specified positions for adroit controls, i.e. for a rapid and compressed transient response, when the plant model is known precisely. The poles and zeros of the resulting controller are unconstrainted, although an integrator may be included in the controller structure as a special case to drive steady-state errors towards zero. The second frequency method ensures that the feedback system has the desired phase-margin at a specified gain cross-over frequency (for the desired bandwidth) yielding robust stability with respect to plant model uncertainty. The poles of the controller are at specified positions, e.g. for a standard Proportional-Integral (PI), Proportional-Derivative (PD), Proportional-Integral-Derivative (PID), structure or other more general configurations if necessary, and the problem is solved for the controller zeros. The poles and zeros of the resulting closed-loop feedback system are unconstrained. These complementary design procedures allow simple and effective controls to be derived analytically from a plant model, using a matrix inverse operation to solve a small set of linear simultaneous equations, as an alternative to more heuristic (e.g. trial-and-error) or empirical PID-tuning approaches. An azimuth controller for a pan-tilt-zoom camera mount is used as an illustrative example. The ways in which both procedures may be used to design controls with the desired balance between adroitness and robustness are discussed.