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Unformatted text preview: ta (Hybrid) Systems r elate t he samples of t he o utput t o those of the input. But, this procedure yields
information a bout t he o utput only a t sampling instants. We c an overcome this
difficulty by taking t he samples a t i nstants in between samples using t he m odified
z -transform as explained later.
In sampled-data systems, t he discrete-time signals are often o btained as a result
o f sampling continuous-time signals. These samples are narrow pulses, which may be
considered as impulses, provided t he pulse width is small compared t o t he s ystem
time constant. Thus, in t he following discussion, a discrete-time signal, when i t
a ppears in conjunction with a continuous-time system, is a sequence o f impulses
r ather t han a sequence o f numbers. Hence, a discrete-time signal I [k] c an also be
considered as continuous-time signal I (t), where (b) I (t) = L /[k]6(t - kT)
k y (t) (c) F ig. 1 1.9 Typical sampled-data systems. i tself a t a c ertain position a nd fires. Another example is attitude-control problem
in a spacecraft, where t he i nformation a bout t he a ctual s pacecraft a ttitude is fed
b ack t o a d igital processor, which generates corrective i nput t o b e applied t o t he
s pacecraft, which is a c ontinuous-time system. I n a utomatic periodic quality check
in production line, t he d iscrete-data obtained from t he p eriodic check, after some
digital processing, generates t he corrective i nput t o b e a pplied t o a continuous-time
p lant. I n complex control systems, use o f d igital processor as a controller or a
c ompensator for continuous-time plants is growing rapidly.
I n t ime-sharing systems, where, for economic reasons, certain facilities are
s hared by several systems, t he signals are, by nature, discrete-time o r sampled.
I n r egulator t ype control systems, where a n o utput v ariable m ust b e maintained
a t a c onstant value, t he e xternal disturbance a nd p lant p arameters v ariations are
usually so slow t hat c ontinuous monitoring (or feedback) is unnecessary. I t is adequate t o s ample t he o utput periodically a nd t hen feed back this discrete-data. In
s uch cases, feedback transducers, data-processing facilities a nd possibly long a nd
e xpensive feedback communication facilities c an b e s hared a mong several control
Figure 11.9 shows some typical sampled-data systems. Figure 11.9a contains
a digital processor, whereas in Fig. 11.9b, t he s ampled signal is directly applied t o
D j A converter (the hold circuit) without further digital processing. Figure 11.9c
shows a practical system, where t he i nput signal itself is a d iscrete-time signal
J[kJ, a nd t he s ampler is in t he feedback path. T his s ystem is equivalent t o t hat
i n Fig. 11.9b. How do we analyze such hybrid systems, where continuous-time and
discrete-time signals intermingle? An effective s trategy in such a situation is t o Observe a n i nteresting fact: in this representation a discrete-time u nit impulse
6[k] is t he c...
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