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168 SAMPLED DATA SYSTEMS AND THE Z-TRANSFORM
r(s) |
r*(s) |
y(s) |
|||
G(s) |
|||||
Figure 6.34 Open-loop system for Exercise 6
G(s) |
||||||||||
r(s) |
r*(s) |
y(s) |
||||||||
Z.O.H |
2 |
|||||||||
(s + 1)(s |
+ 2) |
|||||||||
Figure 6.35 Open-loop system with zero-order hold for Exercise 10
6.Consider the open-loop system given in Figure 6.34. Find the output response when a unit step is applied, if
0.2 |
||||
G(s) = |
. |
|||
s(s |
+ |
1) |
||
7.Draw the output waveform of Exercise 6.
8.Find the z-transform of the following function, assuming that T = 0.5 s:
y(s) |
= (s |
s + 1 |
. |
|||
− |
1)(s |
+ |
3) |
|||
9. Find the z-transforms of the following functions, using z-transform tables:
(i) y(s) |
s + 1 |
(ii) y(s) |
s |
||||||||
= (s |
1)2 |
||||||||||
= s(s |
2) |
+ |
|||||||||
(iii) y(s) = |
+ s2 |
(iv) y(s) = |
0.4 |
||||||||
s(s + 1)(s + 2) |
|||||||||||
(s + 1)2(s + 2) |
|||||||||||
10.Figure 6.35 shows an open-loop system with a zero-order hold. Find the output response when a unit step input is applied. Assume that T = 0.1 s and
G(s) = |
2 |
. |
||||
(s |
+ |
1)(s |
+ |
2) |
||
11. Repeat Exercise 10 for the case where the plant transfer function is given by
0.1 |
2s |
|||||||||
(i) G(s) = |
(ii) G(s) = |
|||||||||
s(s |
+ |
2) |
(s |
+ |
1)(s |
+ |
4) |
|||
12.Derive an expression for the transfer function of the closed-loop system whose block diagram is shown in Figure 6.36.
r(s) |
e(s) |
e*(s) |
G(s) |
y(s) |
+ −
H(s)
Figure 6.36 Closed-loop system for Exercise 12
FURTHER READING |
169 |
||||
r(s) |
e(s) |
e*(s) |
|||
G(s) |
y*(s) |
||||
+ − |
|||||
H(s)
Figure 6.37 Closed-loop system for Exercise 13
13.Derive an expression for the output function of the closed-loop system whose block diagram is shown in Figure 6.37.
FURTHER READING
[Astrom and Wittenmark, 1984] Astrom, K.J. and Wittenmark, B. Computer Controlled Systems. Prentice Hall,
Englewood Cliffs, NJ, 1984. |
|
[D’Azzo and Houpis, 1966] |
D’Azzo, J.J. and Houpis, C.H. Feedback Control System Analysis and Synthesis, |
2nd edn., McGraw-Hill, New York, 1966. |
|
[Dorf, 1992] |
Dorf, R.C. Modern Control Systems, 6th edn., Addison-Wesley, Reading, MA, |
1992. |
|
[Evans, 1954] |
Evans, W.R. Control System Dynamics. McGraw-Hill, New York, 1954. |
[Houpis and Lamont, 1962] |
Houpis, C.H. and Lamont, G.B. Digital Control Systems: Theory, Hardware, |
Software, 2nd edn., McGraw-Hill, New York, 1962. |
|
[Hsu and Meyer, 1968] |
Hsu, J.C. and Meyer, A.U. Modern Control Principles and Applications. |
McGraw-Hill, New York, 1968. |
|
[Jury, 1958] |
Jury, E.I. Sampled-Data Control Systems. John Wiley & Sons, Inc., New York, |
1958. |
|
[Katz, 1981] |
Katz, P. Digital Control Using Microprocessors. Prentice Hall, Englewood Cliffs, |
NJ, 1981. |
|
[Kuo, 1963] |
Kuo, B.C. Analysis and Synthesis of Sampled-Data Control Systems. Englewood |
Cliffs, NJ, Prentice Hall, 1963. |
|
[Lindorff, 1965] |
Lindorff, D.P. Theory of Sampled-Data Control Systems. John Wiley & Sons, |
Inc., New York, 1965. |
|
[Ogata, 1990] |
Ogata, K. Modern Control Engineering, 2nd edn., Prentice Hall, Englewood |
Cliffs, NJ, 1990. |
|
[Phillips and Harbor, 1988] |
Phillips, C.L. and Harbor, R.D. Feedback Control Systems. Englewood Cliffs, |
NJ, Prentice Hall, 1988. |
|
[Raven, 1995] |
Raven, F.H. Automatic Control Engineering, 5th edn., McGraw-Hill, New York, |
1995. |
|
[Smith, 1972] |
Smith, C.L. Digital Computer Process Control. Intext Educational Publishers, |
Scranton, PA, 1972. |
|
[Strum and Kirk, 1988] |
Strum, R.D. and Kirk, D.E. First Principles of Discrete Systems and Digital Signal |
Processing. Addison-Wesley, Reading, MA, 1988. |
|
[Tou, 1959] |
Tou, J. Digital and Sampled-Data Control Systems. McGraw-Hill, New York, |
1959. |
7
System Time Response Characteristics
In this chapter we investigate the time response of a sampled data system and compare it with the response of a similar continuous system. In addition, the mapping between the s-domain and the z-domain is examined, the important time response characteristics of continuous systems are revised and their equivalents in the discrete domain are discussed.
7.1 TIME RESPONSE COMPARISON
An example closed-loop discrete-time system with a zero-order hold is shown in Figure 7.1(a). The continuous-time equivalent of this system is also shown in Figure 7.1(b), where the sampler (A/D converter) and the zero-order hold (D/A converter) have been removed. We shall now derive equations for the step responses of both systems and then plot and compare them.
As described in Chapter 6, the transfer function of the above discrete-time system is given
by
y(z) |
= |
G(z) |
, |
(7.1) |
|||||||||||
r (z) |
1 |
+ |
G(z) |
||||||||||||
where |
|||||||||||||||
r (z) = |
z |
(7.2) |
|||||||||||||
z |
− |
1 |
|||||||||||||
and the z-transform of the plant is given by |
|||||||||||||||
G(s) |
1 − e−s T |
. |
|||||||||||||
= s2(s |
+ |
1) |
|||||||||||||
Expanding by means of partial fractions, we obtain |
+ s 1 |
1 |
|||||||||||||
G(s) = (1 − e−s T ) s2 |
− s |
||||||||||||||
1 |
1 |
||||||||||||||
+
Microcontroller Based Applied Digital Control D. Ibrahim
C 2006 John Wiley & Sons, Ltd. ISBN: 0-470-86335-8
172 SYSTEM TIME RESPONSE CHARACTERISTICS
r(s) |
ZOH |
1 |
y(s) |
|||||||
− |
s (s + 1) |
|||||||||
+ |
||||||||||
(a)
r(s) |
1 |
y(s) |
|||||||||
+ |
− |
s (s + 1) |
|||||||||
(b)
Figure 7.1 (a) Discrete system and (b) its continuous-time equivalent
and the z-transform is |
||||||||||||||||||||||||||||
1 |
1 |
|||||||||||||||||||||||||||
G(z) = (1 − z−1)Z |
1 |
− |
+ |
. |
||||||||||||||||||||||||
s2 |
s |
s |
1 |
|||||||||||||||||||||||||
From z-transform tables we obtain |
+ |
|||||||||||||||||||||||||||
G(z) = (1 − z−1) |
T z |
− |
z |
+ |
z |
. |
||||||||||||||||||||||
(z 1)2 |
z |
1 |
z |
e |
− |
T |
||||||||||||||||||||||
Setting T = 1s and simplifying gives |
− |
− |
− |
|||||||||||||||||||||||||
0.368z + 0.264 |
||||||||||||||||||||||||||||
G(z) |
. |
|||||||||||||||||||||||||||
= z2 − 1.368z + 0.368 |
||||||||||||||||||||||||||||
Substituting into (7.1), we obtain the transfer function |
||||||||||||||||||||||||||||
y(z) |
G(z) |
0.368z + 0.264 |
, |
|||||||||||||||||||||||||
r (z) = |
1 + G(z) |
|||||||||||||||||||||||||||
= z2 − z + 0.632 |
||||||||||||||||||||||||||||
and then using (7.2) gives the output |
||||||||||||||||||||||||||||
y(z) |
= |
z(0.368z + 0.264) |
. |
|||||||||||||||||||||||||
(z − 1)(z2 − z + 0.632) |
||||||||||||||||||||||||||||
The inverse z-transform can be found by long division: the first several terms are
y(z) = 0.368z−1 + z−2 + 1.4z−3 + 1.4z−4 + 1.15z−5 + 0.9z−6 + 0.8z−7 + 0.87z−8
+0.99z−9 + . . .
and the time response is given by
y(nT ) = 0.368δ(t − 1) + δ(t − 2) + 1.4δ(t − 3) + 1.4δ(t − 4) + 1.15δ(t − 5)
+0.9δ(t − 6) + 0.8δ(t − 7) + 0.87δ(t − 8) + . . . .
TIME RESPONSE COMPARISON |
173 |
From Figure 7.1(b), the equivalent continuous-time system transfer function is
y(s) |
= |
G(s) |
= |
1/(s(s + 1)) |
1 |
. |
||||
r (s) |
1 + G(s) |
1 + (1/s(s + 1)) |
||||||||
= s2 + s + 1 |
||||||||||
Since r (s) = 1/s, the output becomes
1
y(s) = . s(s2 + s + 1)
To find the inverse Laplace transform we can write
y(s) |
1 |
s + 1 |
1 |
s + 0.5 |
0.5 |
. |
|||||
= s |
− s2 + s + 1 |
= s |
− |
(s + 0.5)2 − 0.52 |
− |
(s + 0.5)2 − 0.52 |
|||||
From inverse Laplace transform tables we find that the time response is
y(t ) = 1 − e−0.5t (cos 0.5t + 0.577 sin 0.5t ) .
Figure 7.2 shows the time responses of both the discrete-time system and its continuous-time equivalent. The response of the discrete-time system is accurate only at the sampling instants. As shown in the figure, the sampling process has a destabilizing effect on the system.
Figure 7.2 Step response of the system shown in Figure 7.1