Signal Processing and Linear Systems-B.P.Lathi copy

F igure 61sa shows a block diagram of a system with a

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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) ( 674) . 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):...
View Full Document

Ask a homework question - tutors are online