DIGITAL CONTROL: Digital Controller Implementation

A digital controller is usually cascaded with the plant in a feedback system. The rest of the system can either be digital or analog.
Typically, a digital controller requires:

A/D conversion to convert analog inputs to machine readable (digital) format
D/A conversion to convert digital outputs to a form that can be input to a plant (analog)
A program that relates the outputs to the inputs

Output Program

Outputs from the digital controller are functions of current and past input samples, as well as past output samples - this can be implemented by storing relevant values of input and output in registers. The output can then be formed by a weighted sum of these stored values.

The programs can take numerous forms and perform many functions

A digital filter for low-pass filtering
A state space model of a system to act as a state observer
A telemetry system

Stability

Although a controller may be stable when implemented as an analog controller, it could be unstable when implemented as a digital controller due to a large sampling interval. During sampling the aliasing modifies the cutoff parameters. Thus the sample rate characterizes the transient response and stability of the compensated system, and must update the values at the controller input often enough so as to not cause instability.
When substituting the frequency into the z operator, regular stability criteria still apply to discrete control systems. Nyquist criteria apply to z-domain transfer functions as well as being general for complex valued functions. Bode stability criteria apply similarly. Jury criterion determines the discrete system stability about its characteristic polynomial.

Design of digital controller in s-domain

The digital controller can also be designed in the s-domain (continuous). The Tustin transformation can transform the continuous compensator to the respective digital compensator. The digital compensator will achieve an output which approaches the output of its respective analog controller as the sampling interval is decreased.
$s = \frac{2(z-1)}{T(z+1)}$

Tustin transformation deduction

Tustin is the Padé_(1,1) approximation of the exponential function $\begin{align} z &= e^{sT} \end{align}$ :

$\begin{align} z &= e^{sT} \\ &= \frac{e^{sT/2}}{e^{-sT/2}} \\ &\approx \frac{1 + s T / 2}{1 - s T / 2} \end{align}$

And its inverse

$\begin{align} s &= \frac{1}{T} \ln(z) \\ &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\ &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\ &= \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} \end{align}$

We must never forget that the digital control theory is the technique to design strategies in discrete time, (and/or) quantized amplitude (and/or) in (binary) coded form to be implemented in computer systems (microcontrollers, microprocessors) that will control the analog (continuous in time and amplitude) dynamics of analog systems. From this consideration many errors from classical digital control were identified and solved and new methods were proposed: