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W hen t he o utput is fed back t o t he i nput, as shown I II Fig. 6.18d, t he overall
transfer function Y (s) / F (s) c an be computed as follows. T he i nputs t o ~he s ummer
are F (s) a nd - H(s)Y(s). Therefore, E (s), t he o utput of t he s ummer, IS The feedback transfer function, when K = 7, 16, and SO, can be obtained by the
following MATLAB commands:
num/den 7 K=16,c62
16 E (s) = F (s) - H(s)Y(s)
B ut y es) = G (s)E(s)
= G (s)[F(s) - H(s)Y(s)] T herefore y es) [1 + G(s)H(s)] = G (s)F(s) K =80;c62
s "2 + 8 s + 8 0 o 414 6 6 .6 Continuous-Time System Analysis Using t he Laplace Transform 6.6 System Realization 415 S ystem realization x (t)
(a) We now develop a systematic method for realization (or simulation) of a n arbitrary n th-order t ransfer function. Since realization is basically a synthesis problem,
t here is no u nique way of realizing a system. A given transfer function can b e realized in many different ways. We p resent here three different ways o f realization:
canonical, cascade a nd parallel realization. T he second form of canonical realization is d iscussed in Appendix 6.1 a t t he e nd of this chapter. A transfer function
H (s) c an b e realized by using integrators or differentiators along with summers a nd
multipliers. For practical reasons we avoid t he use of differentiators. A differentiator a ccentuates high-frequency signals, which, by their nature, have large rates
of change (large derivative). Signals, in practice, are always corrupted b y noise,
which happens t o b e a broad-band signal; t hat is, t he noise contains components of
frequencies r anging from low t o very high. In processing desired signals by a differentiator, t he high-frequency components of noise are amplified disproportionately.
Such amplified noise may swamp t he desired signal. T he i ntegrator, in contrast,
tends t o s uppress a high frequency signal by smoothing i t out. In addition, practical
differentiators b uilt w ith op amp circuits tend to be unstable. For these reasons we
avoid differentia t ors in practical realizations.
Consider a n LTIC s ystem with a transfer function ( b) b msm + b m_ls m - l + ... + b ls + bo
s n + a n_ls n - l + ... + a lS + ao (e) Hs=-"'----''---''---,:------'--~
() For large s (s ~ (0)
H (s) ' " b ms m - n F ig. 6 .20 Realization of H l(S) = Therefore, for m > n , t he s ystem acts as a n (m - n)th-order differentiator [see
Eq. (6.55)]. F or t his reason, we r estrict m ::; n for practical systems. W ith t his
restriction, the m ost general case is m = n w ith t he t ransfer function 1 83 + a2 8 2 + UI s +ao For convenience of realization, we shall express this transfer function as a cascade of two transfer functions H I(S) a nd H 2(S), as depicted in Fig. 6.19.
H (s) = ( s3+ a
28 (6.70) , 2~ als + ao ) '(b3S3+b2S2+bIS+bo), H.-(s) , (6.72) H ;(s) T he o utput of HI(8) is d enoted by X (s), as illustrated in Fig. 6.19b. Therefore
i1+b l s+bO a nd...
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