  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.
