Файл: Microcontroller based applied digital control (D. Ibrahim, 2006).pdf
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164 SAMPLED DATA SYSTEMS AND THE Z-TRANSFORM
r(s) |
e(s) |
G(s) |
y(s) |
|
+ −
y*(s)
H(s)
Figure 6.29 Closed-loop sampled data system
and |
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e(s) = r(s) − H (s)y*(s). |
(6.47) |
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Substituting (6.47) into (6.46), we obtain |
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y(s) = G(s)r(s) − G(s)H (s)y*(s) |
(6.48) |
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or |
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y*(s) = Gr*(s) − GH*(s)y*(s). |
(6.49) |
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Solving for y*(s), we obtain |
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y*(s) = |
Gr*(s) |
(6.50) |
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1 + GH*(s) |
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and |
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y(z) = |
Gr(z) |
(6.51) |
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. |
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1 |
+ |
GH(z) |
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Example 6.23
The block diagram of a closed-loop sampled data control system is shown in Figure 6.30. Derive an expression for the transfer function of the system.
Solution
The A/D converter can be approximated with an ideal sampler. Similarly, the D/A converter at the output of the digital controller can be approximated with a zero-order hold. Denoting
r(s) |
A/D |
Digital |
D/A |
Gp(s) |
y(s) |
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controller |
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Plant |
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H(s)
Sensor
Figure 6.30 Closed-loop sampled data system
PULSE TRANSFER FUNCTION AND MANIPULATION OF BLOCK DIAGRAMS |
165 |
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G(s) |
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r(s) |
e(s) |
y(s) |
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D*(s) |
1 − e−Ts |
Gp(s) |
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s |
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H(s)
Figure 6.31 Equivalent diagram for Example 6.23
the digital controller by D(s) and combining the zero-order hold and the plant into G(s), the block diagram of the system can be drawn as in Figure 6.31. For this system can write
e(s) = r(s) − H (s)y(s) |
(6.52) |
and |
|
y(s) = e*(s)D*(s)G(s). |
(6.53) |
Note that the digital computer is represented as D*(s). Using the above two equations, we can write
e(s) = r(s) − D*(s)G(s)H (s)e*(s) |
(6.54) |
or
e*(s) = r*(s) − D*(s)GH*(s)e*(s)
and, solving for e*(s), we obtain
e*(s) =
r*(s)
1 + D*(s)GH*(s)
and, from (6.53),
r*(s)
y(s) = D*(s)G(s) 1 + D*(s)GH*(s) .
The sampled output is then
y*(s) = r*(s)D*(s)G*(s) , 1 + D*(s)GH*(s)
Writing (6.57) in z-transform format, |
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y(z) = |
r(z)D(z)G(z) |
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1 |
+ |
D(z)GH(z) |
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and the transfer function is given by |
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y(z) |
= |
D(z)G(z) |
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. |
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r(z) |
1 |
+ |
D(z)GH(z) |
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(6.55)
(6.56)
(6.57)
(6.58)
(6.59)
166 SAMPLED DATA SYSTEMS AND THE Z-TRANSFORM
r (s) |
e (s) |
e*(s) |
1 |
y(s) |
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+ |
− |
s (s + 1) |
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Figure 6.32 Closed-loop system
6.3.4 Closed-Loop Time Response
The closed-loop time response of a sampled data system can be obtained by finding the inverse z-transform of the output function. Some examples are given below.
Example 6.24
A unit step signal is applied to the sampled data digital system shown in Figure 6.32. Calculate and plot the output response of the system. Assume that T = 1 s.
Solution |
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The output response of this system is given in (6.44) as |
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y(z) = |
r(z)G(z) |
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. |
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1 |
+ |
GH(z) |
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where |
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r(z) |
z |
, |
G(z) |
z(1 − e−T ) |
, |
H (z) |
1; |
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= z |
− |
= (z |
= |
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1 |
− |
1)(z |
− |
− |
T ) |
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e |
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thus, |
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y(z) |
= |
+ |
− |
z/z − 1 |
− |
− |
z(1 − e−T ) |
. |
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1 |
(z(1 |
− |
T )/(z |
− |
1)(z |
T )) (z |
− |
1)(z |
− |
− |
T ) |
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e |
e |
e |
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Simplifying, |
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y(z) |
z2(1 − e−T ) |
. |
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= (z2 − 2ze−T + e−T )(z − 1) |
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Since T = 1,
0.632z2
y(z) = . z3 − 1.736z2 + 1.104z − 0.368
After long division we obtain the first few terms
y(z) = 0.632z−1 + 1.096z−2 + 1.25z−3 + . . . .
The first 10 samples of the output response are shown in Figure 6.33.
6.4 EXERCISES
1.A function y(t) = 2 sin 4t is sampled every T = 0.1 s. Find the z-transform of the resultant number sequence.
EXERCISES 167
Figure 6.33 First 10 output samples
2.Find the z-transform of the function y(t) = 3t.
3.Find the inverse z-transform of the function
y(z) = |
z |
. |
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(z |
+ |
1)(z |
− |
1) |
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4. The output response of a system is described with the z-transform
y(z) = |
z |
. |
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(z |
+ |
0.5)(z |
− |
0.2) |
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(i)Apply the final value theorem to calculate the final value of the output when a unit step input is applied to the system.
(ii)Check your results by finding the inverse z-transform of y(z).
5.Find the inverse z-transform of the following functions using both long division and the method of partial fractions. Compare the two methods.
(i) y(z) |
= |
0.2z |
(ii) y(z) |
= |
0.1(z + 1) |
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− |
− |
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(z |
1)(z |
0.5) |
(z |
− |
0.2)(z |
− |
1) |
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(iii) y(z) |
= |
0.2 |
(iv) y(z) |
= |
z(z − 1) |
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− |
− |
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(z |
3)(z |
1) |
(z |
− |
2)2 |
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