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Unformatted text preview: s. Therefore, we need to combine t he c onjugate poles a nd realize
t hem as a second-order transfer function. t In t he p resent case, we can express H (s)
8+ 3 ' -v-" H 3(') = (8 + 6 ' -v-" H(8) 421 Realization of Complex Conjugate Poles 4s + 28
H (s)= s 2+6s+5 H (s) System Realization H .(8) (6.76b) = - 8-+-1 - -8-+-5 E quation (6.76) gives us t he o ption of realizing H (s) as a cascade o f HI.(8) a~d
H2(S), as shown in Fig. 6.27a, or a parallel of H3(S) a nd H4(S), as depIcted m
Fig. 6.27b. E ach of the first-order transfer functions in Figs. 6.27a o r 6.27b can be
realized by u sing a single integrator, as discussed in Example 6.18. H (8)= C~3)C2:~:13)
2 F~S) 43'+28 : s+T =s .~(S)
1 J :! (6.77a) 28 - 8 + 3 - s2 + 4s + 13 (6.77b) Now we c an realize H (s) in cascade form using Eq. (6.77a) o r in parallel form using
Realization o f Repeated Poles ( a) ( b) : F ig. 6 .27 Realization of (S:~n~!5) (a) cascade form (b) parallel form.
We have presented here t hree forms of realization (canonical, cascade, and
parallel). T he second canonical realization is developed in Appendix 6.1. However,
this discussion by no means exhausts all t he possibilities. Moreover, in t he cascade
form t here a re different ways of grouping t he factors in t he n umerator a nd the
deno:Uinator o f H (s). Accordingly, several cascade forms are possible.
From a practical viewpoint, parallel a nd cascade forms are preferable because
parallel a nd c ertain cascade forms are numerically le~s se.nsitive t~an.canonical forms
t o small p arameter variations in t he system. Qualitatively, thiS. dlffe:ence can .be
e xplained by t he fact t hat in a canonical realization all th~ coeffiCients I~teract With
each other a nd a change in any coefficient will be maglllfied t hrough ItS r epeated
influence f~om feedback a nd feed forward connections. In a parallel realization, in
contrast, t he change in a coefficient will affect only a localized segment; t he case
w ith a c ascade realization is similar.
In t he a bove examples of cascade a nd parallel realization, we have separated
H (s) into f irst-order factors. For H (s) of higher orders, we could group ~(s) int.o
factors, not a ll of which are necessarily of the first order. For example, If H (s) IS
a t hird-order transfer function, we could realize this function as a cascade (or a
parallel) combination of a first-order a nd a second-order factor. W hen repeated poles occur, t he p rocedure for canonical a nd cascade realization
is exactly t he s ame as above. In parallel realization, however, t he p rocedure requires
a special precaution, as explained in Example 6.19 below.
• E xample 6 .19 Determine the parallel realization of
H (s) = 7s 2 + 37s + 51 (s + 2)(s + 3)2
5 2 3 = - - ++3- -(- - s+2
This third-order transfer function should require no more than three integrators.
But if we try to realize each of the three partial fractions separately, we require four
integrators because one of the terms is second-...
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