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Unformatted text preview: Y (s) = (b3S3 S J+.a?+als + ao X (s) =
(a) or ( D3 Canonical or Direct-Form Realization R ather t han realizing t he general nth-order system described by Eq. (6.70), we
b egin with a specific case of t he following third-order system a nd t hen e xtend the
results t o t he n th-order case
H (s) = + b2S2 + blS + bo
- -";,---'--=';;----"---=s3 + a2s2 + a lS + ao
b3S3 1 s3 + a2s 2 + a lS + ao F (s) (6.73)
. We shall first realize H I(S), E quation (6.74) enables us to write t he differential
equation relating x (t) t o f (t) as (b) F ig. 6 .19 Realization of a transfer function in two steps. 6 .6-1 + b2S 2 + blS + bo) X (s) (6.71) x (t) + a 2D2 + a iD + a o)x(t) + a 2x(t) + a lx(t) + a ox(t) = f (t)
= f (t) (6.75a)
(6.75b) a d eduction which yields
(6.75c) 416 6 Continuous-Time S ystem A nalysis Using t he L aplace Transform 6.6 S ystem R ealization 417 O ur t ask is t o realize a system whose i nput f (t) a nd o utput x (t) s atisfy Eq. (6.75c).
L et u s a ssume t hat x is available. Successive integration of x yields x, x, a nd x (Fig.
6.20a). We c an now generate x from f (t), x , x, a nd x a ccording t o E q. (6.75c),
using t he f eedback connections, as shown in Fig. 6.20b.t T he r eader c an verify t hat
t he s ystem i n Fig. 6.20b indeed satisfies Eq. (6.75c). Clearly, t he t ransfer function
of t his s ystem (Fig. 6.20b) is H l(S) i n Eq. (6.72). T he s ignals x (t), x (t), a nd
x (t) a re available a t p oints N l , N2, a nd N3, respectively. I f t he i nitial conditions
x(O), x (t), a nd x(O) a re nonzero, t hey s hould b e a dded a t p oints N l , N2, a nd N3,
respectively.:j: F igure 6.20c shows t he s ystem of Fig. 6.20b in frequency domain.
E ach i ntegrator is r epresented by t he t ransfer function l /s [see Eq. (6.56)]. Signals
f (t), x (t), x (t), x(t), a nd x (t) a re represented by their Laplace t ransforms F (s),
X (8), s X (8), s2 X (s), a nd 83X (s), respectively. F ig. 6 .22 Realization of the nth-order transfer function H(s) in Eq. (6.70).
G eneralization of t his r ealization for t he n th-order t ransfer f unction in Eq.
(6.70) is shown in Fig. 6.22. T his is one of t he two c anonical r ealizations (also
known as t he c ontroller c anonical o r d irect-form r ealization). T he s econd
canonical realization ( observer c anonical realization) is discussed in A ppendix 6.1
a t t he e nd o f this chapter. Observe t hat n i ntegrators a re required for a realization
o f a n n th-order t ransfer function. T he c anonical (direct-form) realization procedure
o f a n n th-order t ransfer f unction is s ystematic a nd s traightforward, as i llustrated in
Fig. 6.22. After examining t his figure, we can s ummarize t he p rocedure as follows:
1. D raw a n i nput s ummer followed by n i ntegrators in cascade. F igure 6 .20c is a realization of t he t ransfer function H l(S) i n Eq. (6.72). To
realize t he t ransfer f unction H (s), we need t o a ugment H l(S) so t hat t he final
o utput Y (s) i s g enerated from X (s) a ccording t o E q. (6.73):...
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