Definitions

# Digital control

Digital control is a branch of control theory that uses digital computers to act as a system. Depending on the requirements, a digital control system can take the form of a microcontroller to an ASIC to a standard desktop computer. Since a digital computer is a discrete system the Laplace transform is replaced with the Z-transform. Also since a digital computer has finite precision (See quantization) extra care is needed to ensure the error in coefficients, A/D conversion, D/A conversion, etc. are not producing undesired or unplanned effects.

The application of digital control can readily be understood in the use of feedback. Since the creation of the first digital computer in the early 1940s the price of digital computers has dropped considerably, which has made them key pieces to control systems for several reasons:

• Cheap: under \$5 for many microcontrollers
• Flexibility: easy to configure and reconfigure through software
• Static operation: digital computers are much less prone to environmental conditions than capacitors, inductors, etc.
• Scaling: programs can scale to the limits of the memory or storage space without extra cost
• Adaptive: parameters of the program can change with time ()

## 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. Some examples of analog systems with a digital feedback controller are:

Typically, a digital controller requires:

• 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

### Stability

Note that 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. Thus the sample rate characterises 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.

Stability of digital control systems can be checked using a specific bilinear transform to the Laplace domain, allowing the use of the Routh-Hurwitz stability criterion. This bilinear transform is application specific, and can not be used to compare system attributes such as transient responses in the s and z domains.

### 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.

$s = frac\left\{2\left(z-1\right)\right\}\left\{T\left(z+1\right)\right\}$

#### Tustin transformation deduction

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


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} &approx frac{2}{T} frac{1 - z^{-1}}{1 + z^{-1}} end{align}

We never must 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:

a) Yutaka Yamamoto and his "lifting function space model":

b) Marcelo Tredinnick and Marcelo Souza and their new type of analog-digital mapping:

and

and

c) Alexander Sesekin and his studies about impulsive systems: