Digital Control Loop

• The material covered so far has been based on continuous-time model for the process and a continuous-time control law.

• However, modern control technology is computer-based which operates in discrete time.

In this lecture, we develop a systematic approach for implementing controllers in a digital control loop.

The digital control loop is shown below:

There are two additional hardware elements:

1. Digital to Analog Converter (D/A)

2. Analog to Digital Converter (A/D)

D/A Converter

• The D/A converter reconstructs an analog signal from a digital

• This is done via a zero-order hold: the analog signal is held constant between sampling instants as shown below.

• Mathematically, the zero-order hold operation can be expressed as:

a(t) = a_{d}(k − 1)  for (k − 1)h ≤ t ≤ kh (1)

A/D Converter

• The A/D converter is a sampler: given an analog signal, it samples its values at different time instants and generates a discrete signal as shown in the figure below:

• Mathematically, the sampler operation can be expressed as:

y_{d}(k) = y(kh)  for k = 0, 1, 2, 3, ... (2)

Two Points of View

• VIEW 1: Analysis/Design is first performed in continuous time. Then the designed continuous time controller (e.g. PID) is approximately discretized so that the computer can implement this control law.

• VIEW 2: The process is exactly discretized first. Then, a discrete controller is designed that can be implemented by the computer.

No matter which point of view is followed, one has to develop a method for discretizing linear systems of the form:

\begin{aligned}\frac{dX}{dt} & = & AX + BU \\ Y & = & CX + DU \end{aligned}  (3)

Discretization Formula

Consider a system of the form

\begin{aligned}\frac{dX}{dt} & = & AX + BU \\ Y & = & CX + DU \end{aligned}  (4)

If the input U(t) is discretized as U(t) = U(k − 1), then the dynamic system is discretized as:

\begin{aligned}X(k) & = & e^{Ah}X(k − 1) + (∫_{0}^{h}e^{At}Bdt) U(k − 1) \\ Y (k) & = & CX(k) + DU(k) \end{aligned}

Example

Consider the following dynamic system

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] & = & [ \begin{matrix}−1 & 0 \\ 2 & −2 \end{matrix} ] [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] + [ \begin{matrix}1 \\ 0 \end{matrix} ] U(t) \\ Y (t) & = & [ 0 1 ] [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] \end{aligned}

(6)

If the input U(t) is discretized as U(t) = U(k − 1), then what is discretized version of the dynamic system?