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

1357a as c 1 s c 1 si a i thus e at a

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Unformatted text preview: uation for t he s ystem in Example 13.9. In t his case, G 8 16 13 S tate-Space A nalysis 1 3.4 817 L inear T ransformation O f S tate V ector W e f ound Al = - 1 a nd A2 = - 2. Hence A =[-1 o 0] -2 a nd E q. (13.74b) becomes (a) E quating t he four elements o n two sides, we o btain - PU = -2P12- (13.75a) -P12 = P u - 3P12 (13.75b) -2P21 = - 2P22 (13.75C) - 2P22 = P21 - 3P22 (13.75d) T he r eader will i mmediately recognize t hat Eqs. (13.75a) a nd (13.75b) a re identical. Similarly, Eqs. (13.75c) a nd (13.75d) a re i dentical. Hence two e quations m ay b e discarded, l eaving us with only two e quations [Eqs. (13.75a) a nd (13.75c)] a nd f our unknowns. T his o bservation m eans t here is no unique solution. T here is, in fact, a n i nfinite n umber o f solutions. We c an a ssign a ny value t o p u a nd P21 t o yield o ne p ossible solution. t I f P u = k l a nd P21 = k2, t hen from Eqs. (13.75a) a nd (13.75c) we have P12 = k l/2 a nd P22 = k2: ( b) (13.75e) F ig. 1 3.8 W e m ay assign any values t o k l a nd k2. For convenience, l et k l = 2 a nd k2 = 1. T his s ubstitution yields T wo r ealizations o f t he s econd-order s ystem i n E xample 13.10. where 13 = (13.75f) PB = [ : : ] [ :] [ :] Hence T he t ransformed v ariables [Eq. (13.73a)] a re Zl] [ Z2 [2 1 1 ] [ Xl] = [ 2XI + X2] 1 X2 Xl + X2 (13.76) or [::J [-: -:] [::J + [:] / i l = - Zl T hus, t he new s tate v ariables Zl a nd Z2 a re r elated t o Xl a nd X2 b y Eq. (13.76). T he s ystem e quation w ith z a s t he s tate v ector is given by [see Eq. (13.73c)] z= A z+Bf t If, however, we want the state equations in diagonalized form, as in E q. (13.29a), where all t he elements of B matrix are unity, there is a unique solution. The reason is t hat t he equation B = P B, where all t he elements of B are unity, im{'oses additional constraints. In the present example, this condition will yield PU = ~, Pl2 = 4 ' P21 = ~, and P22 = ~. The relationship between z and x is t hen (13.77a) + 4/ (13.77b) N ote t he d istinctive n ature o f t hese s tate e quations. E ach s tate e quation involves only o ne v ariable a nd t herefore c an b e solved by itself. A general s tate e quation h as t he d erivative o f o ne s tate v ariable e qual t o a l inear c ombination o f all s tate v ariables. S uch is n ot t he c ase w ith t he d iagonalized m atrix A. E ach s tate v ariable Z i is chosen so t hat it is uncoupled from t he r est o f t he v ariables; hence a s ystem w ith n eigenvalues is split i nto n d ecoupled systems, each w ith a n e quation o f t he form ii = A izi + ( input t erms) q 8 18 13 S tate-Space Analysis T his f act also c an b e r eadily seen from Fig. 13.8a, which is a realization o f t he s ystem r epresented b y E q. (13.77). I n c ontrast, c onsider t he o riginal s tate e quations [see Eq. 1 3.70a)] 13.5 Controllability And Observability 819 s tate [Eqs. (13.78)] is c ompletely controllable i f a nd o nly i fthe m atrix B h as n o row o f zero elements. T he o utputs [Eq.(13.78b)] are of t he form j X, = X2 + f (t) Yi = C ijZj + Ci2 Z 2 + ... + C inZn + L d imfm m =l A r ealization...
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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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