AME-558

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Topic 2: Discrete Time Systems

• 2.1 Introduction and sampled data systems
• 2.2 Z-transforms
• 2.3 Block diagram manipulation

Introduction

Many control systems are implemented using digital components including a microprocessor. Provision must be made for interaction with sensors and actuators. Some sensors like angle-measuring optical encoders are inherently digital. These sensors are read via an appropriate digital interface. Other sensors like angle-measuring precision potentiometers are inherently analog. These sensors require an analog-to-digital converter (ADC). Servocontrol actuators usually require analog inputs. To this end a digital-to-analog converter (DAC) is usually used.

Control computations are best performed at regular, well-defined intervals. Some sort of timer is needed to generate regular interrupts. The timing signal may be derived from the computer system clock or a dedicated timer may be used.

A block diagram of a generic single-input, single-output digital control system is shown below. The digital computer includes the microprocessor and timer.

Digital systems are inherently discrete in two ways. First, the sensor output, control input, and other signals are discretized. This discretization is rarely accounted for. It can be ignored if the resolution of the signals is small relative to the signal amplitudes. Second, time is discretized. Sensors are read and control outputs are written only at discrete intervals. We shall study such systems in this lesson. If the sampling interval is small relative to the fastest time constant of the system, then temporal discretization can be ignored as well. In this case standard continuous control methods can be used.

Objectives: At the end of this lesson you should be able to compute transfer functions in the Z-domain for some simple feedback systems when given a block diagram.

Sampled-Data Systems

Data Sampling

Assume that the sensors are read and that the actuator commands are written at a fixed sampling period $T$ . As a practical matter it is important to first read the sensors and then write the previously computed outputs. This is to avoid exciting high-order dynamics and influencing the measurements. Let $y$ be a variable of interest. Its value at instant $kT$ may be denoted $y\left(kT\right)$ , $y\left(k\right)$ , $y_k$ , $y\left(t_k\right)$ , etc.

An ideal sampler is a switch that closes every $T$ seconds for one instant. Considering the figure below, let the input be $y\left(t\right)$ and the output be $y^{*}\left(t\right)$ .

The sampling process can be thought of ideally as impulse modulation. Formally, the sampled, output signal is

$y^{*}\left(t_k\right)=\sum _{k=0}^{\infty }y\left(kT\right)\delta (t-kT)$ ,

where $\delta \left(\right)$ is the Dirac delta function.

Is that really what happens?

Desampling

The desampling process can be modelled as an ideal sampler in series with a so-called zero-order hold (ZOH). A zero-order hold integrates the difference between two consecutive, periodic impulses. The transfer function corresponding to this is

$G_0\left(s\right)=\frac{1-e^{-\mathrm{sT}}}{s}$

Note that $\frac{1}{s}$ is the transfer function of an integrator. Subtracted from this is the transfer function $\frac{1}{s}{e}^{-sT}$ , which is the transfer function of an integrator with a delay of $T$ . The input and output of a sampler in series with a zero-order hold is shown below.

Mathematically,

$y\left(t\right)={\int }_0^t{y}^{*}\left(\tau \right)d\tau -{\int }_0^t{y}^{*}(\tau -T)d\tau$

The result is the magnitude of the most recent impulse.

What's next?

Go on to page 2, where we'll look at Z-transforms.