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

# T hus i f p is a nonsingular m atrix t he v ector w

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Unformatted text preview: +1 ( b) ~ s -I F ig. 1 3.9 S ystems for E xample 13.11. tWe can show t hat a system is completely controllable if and only if the n x n j composite matrix [B, A B, A 2B, . .. , A n-IB] has a rank n . Similarly, a system is completely observable if and only if the n x n k composite matrix [C', A 'C', A ,2C', . .. , A m-IC'] has a rank n . 8 20 X 2. 13 S tate-Space Analysis I n b oth cases, t he s tate v ariables are identified a s t he t wo i ntegrator o utputs, x 1 a nd T he s tate e quations for t he s ystem i n Fig. 13.9a a re 13.5 Controllability And Observability 821 T he first column o f C is zero. Hence t he m ode Z l ( corresponding t o A l = 1) is unobservable. T he s ystem is therefore controllable b ut n ot o bservable. We come t o t he s ame conclusion by realizing t he s ystem w ith t he s tate v ariables Z l a nd Z 2, w hose s tate e quations a re z= a nd Az y= (13.79) Cz + Bf A ccording t o Eqs. (13.80) a nd (13.81), we have H ence s- 1 l sI-AI T herefore = o I = (s - I -1 a nd 1)(s + 1) y= s +1 Z2 a nd a nd f (13.80) (a) y W e s hall now use t he p rocedure in Sec. 13.4-1 t o d iagonalize t his s ystem. According t o E q. (13.74b), we have [: -:] P12 ] [ PH P21 [ PH P21 Pn P12] [ 1 P22 1 -:] T he s olution o f t his e quation yields P12 C hoosing PH =0 a nd - 2P21 = P22 ( b) = 1 a nd P21 = 1, we have p= [: f ~2] a nd B =PB= A ll t he rows o f B a re nonzero. C~2] [ :] F ig. 1 3.10 C] y E quivalent o f t he s ystems i n Fig. 13.9. (13.81a) F igure 13.10a shows a realization o f t hese e quations. I t is clear t hat e ach o f t he t wo modes is controllable, b ut t he first m ode ( corresponding t o A = 1) is n ot o bservable a t t he o utput. T he s tate e quations for t he s ystem i n Fig. 13.9b a re Hence t he s ystem is controllable. Also, Y =Cx = C p- 1 z (13.82) = Cz (13.81b) a nd y a nd o -2 -1 = X2 Hence = [1 -21 [ ; _ :] = [0 11 (13.81c) C =[O I I, D =O 13 S tate-Space A nalysis 8 22 s l sI-AI = +1 I = (s + 1)(8 - o I -1 8- 1 3.6 S tate-Space A nalysis o f D iscrete- Time S ystems 823 A hat=P* A * inv(P); B hat=P*B C hat=C*inv(P) 0 1) 1 so t hat Al = - 1, A2 = 1, and 1 3.5-1 Inadequacy o f t he Transfer Function Description o f a S ystem (13.83) Diagonalizing the matrix, we have [1 0] o -1 [ PH P21 P12] = P22 T he solution of this equation yields PH P21 = 1, we o btain [PH P21 = P12] [-1 0] -2 P22 - P12 a nd P22 E xample 13.11 d emonstrates t he i nadequacy o f t he t ransfer f unction t o d escribe a n L TI s ystem i n g eneral. T he s ystems i n F igs. 13.9a a nd 1 3.9b b oth h ave t he s ame t ransfer f unction 1 H(8) = 8+1 = 1 O. Choosing P H = - 1 and Y et t he t wo s ystems a re v ery different. T heir t rue n ature is revealed i n F igs. 13.lOa a nd 1 3.lOb, respectively. B oth t he s ystems a re u nstable, b ut t heir t ransfer f unction H(8) = d oes n ot give a ny h int o f i t. T he s ystem i n F ig. 13.9a a ppears s table f rom t he e xternal t erminals, b ut i t is i nternally u nstable. T he s ystem i n F ig. 13.9b, o n t he o ther h and, will show i nstability a t t h...
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