Файл: Microcontroller based applied digital control (D. Ibrahim, 2006).pdf
ВУЗ: Не указан
Категория: Не указан
Дисциплина: Не указана
Добавлен: 14.06.2025
Просмотров: 3306
Скачиваний: 1
6
Sampled Data Systems and the z-Transform
A sampled data system operates on discrete-time rather than continuous-time signals. A digital computer is used as the controller in such a system. A D/A converter is usually connected to the output of the computer to drive the plant. We will assume that all the signals enter and leave the computer at the same fixed times, known as the sampling times.
A typical sampled data control system is shown in Figure 6.1. The digital computer performs the controller or the compensation function within the system. The A/D converter converts the error signal, which is a continuous signal, into digital form so that it can be processed by the computer. At the computer output the D/A converter converts the digital output of the computer into a form which can be used to drive the plant.
6.1 THE SAMPLING PROCESS
A sampler is basically a switch that closes every T seconds, as shown in Figure 6.2. When a continuous signal r(t) is sampled at regular intervals T , the resulting discrete-time signal is shown in Figure 6.3, where q represents the amount of time the switch is closed.
In practice the closure time q is much smaller than the sampling time T , and the pulses can be approximated by flat-topped rectangles as shown in Figure 6.4.
In control applications the switch closure time q is much smaller than the sampling time T and can be neglected. This leads to the ideal sampler with output as shown in Figure 6.5.
The ideal sampling process can be considered as the multiplication of a pulse train with a continuous signal, i.e.
r*(t) = P(t)r(t), |
(6.1) |
|
where P(t) is the delta pulse train as shown in Figure 6.6, expressed as |
||
P(t) = n |
∞ |
|
δ(t − nT ); |
(6.2) |
|
=−∞ |
||
thus, |
||
∞ |
||
r*(t) = r(t) n δ(t − nT ) |
(6.3) |
|
=−∞ |
||
Microcontroller Based Applied Digital Control D. Ibrahim
C 2006 John Wiley & Sons, Ltd. ISBN: 0-470-86335-8
132 SAMPLED DATA SYSTEMS AND THE Z-TRANSFORM
Input |
Output |
|||||||||||||
A / D |
Digital |
D/A |
Plant |
|||||||||||
computer |
||||||||||||||
sensor
Figure 6.1 Sampled data control system
r(t) |
Sampler |
|||
r* (t) |
||||
Continuous |
Sampled |
|||
signal |
signal |
|||
Figure 6.2 |
A sampler |
|||
r(t) |
|||||
0 |
T |
2T |
3T |
4T |
5T |
r*(t) |
|||||
q |
0 |
T |
2T |
3T |
4T |
5T |
Figure 6.3 The signal r(t) after the sampling operation
THE SAMPLING PROCESS |
133 |
r(t) |
|||||
0 |
T |
2T |
3T |
4T |
5T |
r*(t) |
|||||
q |
0 |
T |
2T |
3T |
4T |
5T |
Figure 6.4 Sampled signal with flat-topped pulses
r(t) |
|||||
0 |
T |
2T |
3T |
4T |
5T |
r*(t) |
0 |
T |
2T |
3T |
4T |
5T |
Figure 6.5 Signal r(t) after ideal sampling
134 SAMPLED DATA SYSTEMS AND THE Z-TRANSFORM
P(t)
t |
|||||||||||
0 |
T |
2T |
3T |
4T |
|||||||
5T |
|||||||||||
Figure 6.6 Delta pulse train |
|||||||||||
or
r*(t) = n |
∞ |
|||||
r(nT )δ(t − nT ). |
(6.4) |
|||||
=−∞ |
||||||
Now |
||||||
r(t) = 0, |
for t < 0, |
(6.5) |
||||
and |
||||||
r*(t) = |
∞ |
|||||
r(nT )δ(t − nT ). |
(6.6) |
|||||
n |
= |
0 |
||||
Taking the Laplace transform of (6.6) gives |
||||||
R*(s) |
∞ |
|||||
(6.7) |
||||||
= |
r(nT )e−snT . |
|||||
n |
= |
0 |
||||
Equation (6.7) represents the Laplace transform of a sampled continuous signal r(t).
A D/A converter converts the sampled signal r (t) into a continuous signal y(t). The D/A can be approximated by a zero-order hold (ZOH) circuit as shown in Figure 6.7. This circuit remembers the last information until a new sample is obtained, i.e. the zero-order hold takes the value r(nT) and holds it constant for nT ≤ t < (n + 1)T , and the value r(nT) is used during the sampling period.
The impulse response of a zero-order hold is shown in Figure 6.8. The transfer function of a zero-order hold is given by
G(t) = H (t) − H (t − T ), |
(6.8) |
|||||
Sampler |
||||||
r(t) |
r*(t) |
|||||
Zero-order |
y(t) |
|||||
Continuous |
Sampled |
hold |
||||
signal |
signal |
|||||
Figure 6.7 A sampler and zero-order hold
THE SAMPLING PROCESS |
135 |
g(t)
1
t
0 |
T |
Figure 6.8 Impulse response of a zero-order hold
where H (t) is the step function, and taking the Laplace transform yields
G(s) |
1 |
e−T s |
1 − e−T s |
. |
(6.9) |
||||
= s |
− |
s |
= |
||||||
s |
|||||||||
A sampler and zero-order hold can accurately follow the input signal if the sampling time T is small compared to the transient changes in the signal. The response of a sampler and a zero-order hold to a ramp input is shown in Figure 6.9 for two different values of sampling period.
r(t) and y(t)
r(t) |
||||||
y(t) |
||||||
T=1 second |
||||||
seconds |
||||||
0 |
1 |
2 |
3 |
4 |
5 |
6 |
r(t) and y(t)
r(t) |
||||||
y(t) |
||||||
T = 0.6 second |
||||||
seconds |
||||||
0 |
1 |
2 |
3 |
4 |
5 |
6 |
Figure 6.9 Response of a sampler and a zero-order hold for a ramp input