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

2 a s ystematic p rocedure for d etermining s tate e

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Unformatted text preview: , u se [ num, d enj=ss2tf(A,B,C,D) p rintsys(num,den) 0 A G eneral Case I t is c lear t hat a s ystem h as s everal s tate-space d escriptions. N otable a mong t hese a re t he c anonical-form v ariables a nd t he d iagonalized v ariables ( in t he p arallel r ealization). S tate e quations i n t hese f orms c an b e w ritten i mmediately b y i nspection o f t he t ransfer f unction. C onsider t he g eneral n th-order t ransfer f unction H (s) = (13.22a) a nd + b m_lS m - l + ... + b ls + bo s n + a n_lS n - l + ... + a lS + ao b ms m + b m_lS m - l + ... + b ls + bo b msm ~~-'---'.::~--:---'---'--=--'-~ ( 13.25a) (s - Al)(S - A2)··· (s - An) (13.22b) 3 . Parallel Realization (Diagonal Representation) T he three integrator outputs s tate equations are Z l, Z 2, a nd Z3 in Fig. 13.5d are t he s tate variables. The ( 13.25b) F igures 1 3.6a a nd 1 3.6b s how t he r ealizations o f H (s), u sing t he c ontroller c anonical f orm [Eq. (13.25a)] a nd t he p arallel f orm [Eq. (13.25b)], respectively. T he n i ntegrator o utputs X l, X 2, . .. , X n i n F ig. 13.6a a re t he s tate v ariables. I t is clear t hat ( 13.26a) (13.23a) X n-l = Xn xn = - an-lXn - a n -2 X n-l - ..• - alx2 - aOxl +f • 13 S tate-Space A nalysis 796 • 13.2 A S ystematic P rocedure for D etermining S tate E quations y = [bo bl . .. 0 . .. bm 797 0] (13.27b) O bserve t hat t hese e quations ( state e quations a nd o utput e quation) c an b e w ritten i mmediately b y i nspection o f H (s ). T he n i ntegrator o utputs Z l, Z 2, . .. , Z n i n F ig. 13.6b a re t he s tate v ariables. I t is c lear t hat ( 13.28a) a nd ( 13.28b) or il Al 0 0 0 Zl i2 0 A2 0 0 Z2 + . ...................... f Z n-l 0 0 A n-l 0 Z n-l 1 in 0 0 0 An Zn ( 13.29a) 1 F ig. 1 3.6 Controller canonical and parallel realizations for a n n th o rder L TlC system. a nd Zl a nd o utput y is (13.26b) or ( 13.29b) Xl 0 1 0 0 0 Xl 0 X2 0 0 1 0 0 X2 0 Zn + . ..................................... f X n-l 0 0 0 0 1 X n-l 0 xn a nd Z n-l - ao - al - a2 - a n -2 - an-l xn 1 (13.27a) T he s tate e quation ( 13.29a) a nd t he o utput e quation ( 13.29b) c an b e w ritten i mmediately b y i nspection o f t he t ransfer f unction H (s) i n E q. (13.25b). O bserve t hat t he d iagonalized form o f t he s tate m atrix [Eq. (13.29a)] has t he t ransfer function p oles a s i ts d iagonal e lements. T he p resence o f r epeated p oles in H (s) will modify t he p rocedure slightly. T he h andling o f t hese c ases is discussed i n Sec. 6.6. a " 13 S tate-Space Analysis 7 98 13.3 Solution of S tate E quations 799 T hese e quations c an b e solved in b oth t he t ime d omain a nd f requency domain (Laplace t ransform). T he l atter requires fewer new concepts a nd is therefore easier t o d eal w ith t han t he t ime-domain solution. For t his reason, we shall first consider t he L aplace t ransform s olution. 13.3-1 f Laplace Transform Solution o f S tate Equations T he k th s tate e quation [Eq. (13.6a)] is of t he form Xk = a klXI y + ak2X2 + ... +...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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