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

# 139a a re 135 controllability and observability 821 t

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Unformatted text preview: o btain C A kU[k] + ( zl - A)-lBF[z] z-lA)-lX[O] + ( zl - A)-lBF[z] = (13.101b) ' z ero-state c omponent A c omparison o f E q. ( 13.101b) w ith E q. ( 13.93b) shows t hat (13.99b) (13.102) I t follows t hat y[k] = [ -8(3)-k + 21(2)-ku[k] + [12 + 6(3)-k - 18(2)-k]u[k - 1] (13.100a) T he o utput e quation is given b y Y[z] y[k] = [12 - 2(3)-k o + 3(2)-k]u[k] (13.100b) A =[O 1 ;-1/6 5 /6]; B =[O; 1]; C =[-1 5]; D =O; x O=[2;3]; k =0:25; u =ones(I,26); [ y,x]=dlsim(A,B,C,D,u,xO); s tem(k,y) 0 CX[z] C [(I - z-l A)-lx[O] + ( zl - A)-lBF[zJJ + DF[z] = C (I - z-l A)-lx[O] + [ C(zl - A)-lB + D]F[z] • C omputer E xample C 13.7 Solve Example 13.12 using MATLAB. + DF[z] = = T his is t he desired answer. We can simplify this answer by observing t hat 12 + 6 (3)-k IB(2)-k = 0 for k = O. Hence, u[k - 1] may b e replaced by u[k] i n Eq. (13.99b), and = ? (I - z-l A )-lx[O! + (13.103a) H[z]F[z] ' ---v--' z ero-input r esponse z ero-state r esponse w here H[z] = C (zl - A - l)B + D (13.103b) N ote t hat H[z] is t he t ransfer f unction m atrix o f t he s ystem, a nd H;j[z], t he i jth e lement ofH[z], is t he t ransfer f unction r elating t he o utput Yi(k) t o t he i nput I j(k). I f we define h[k] a s h[k] = Z -l[H[zJJ 13 S tate-Space Analysis 8 30 t hen h[kJ represents t he u nit i mpulse function response m atrix o f t he s ystem. Thus, hiAkJ, t he i jth element of h (k), r epresents t he z ero-state response Yi(k) when t he i nput f j(k) = h[kJ a nd all o ther i nputs a re zero. • E xample 1 3.13 Using the z-transform, find the response y[k] for t he system in Example 13.12. According to Eq. (13.103a) Y [z]=[-1 1 - 2. 6z 6z z (6z-5) = ~ [ -1 5] [ -z 6 z 2 -5z+1 13z 2 - 3z = 5 z2 - SZ -1 [2] _~ 5][~ 1 +S+ -1 ] + [-1 z-~ 6 3 -1 [ 0] --=z -1 6Z2~Z:Z+1l [:] +[-1 831 response a nd t he t ransfer f unction. State-variable description c an also b e e xtended t o t ime-varying p arameter s ystems a nd n onlinear systems. A n e xternal s ystem d escription m ay n ot d escribe a system completely. T he s tate e quations o f a s ystem c an b e w ritten d irectly from t he knowledge of t he s ystem s tructure, from t he s ystem equations, o r from t he block d iagram r epresentation o f t he s ystem. S tate e quations consist of a s et o f n f irst-order differential equations a nd c an b e solved by time-domain o r f requency-domain (transform) methods. Because a s et o f s tate variables is n ot unique, we c an h ave a variety of s tate-space d escriptions of t he s ame s ystem. I t is possible t o t ransform one given s et o f s tate v ariables into a nother by a linear transformation. Using such a t ransformation, we c an see clearly which of t he s ystem s tates a re controllable a nd which are observable. References 6 z 2 -5z+1 (5z - l)z 5 ( z - 1)(z 2 - SZ + 1. 21z 12z 12z 6z 18z - - - + z -! +z -1 + - - +-~- -z -! - - - - z -1 z -- z-~ Therefore + 2 1(2)-k +,12 + 6 (3)-k - '-...,--' z ero-input r esponse 18(2)-k]u[k] z ero-state r esponse , • Linear Transformation, Controllability, and Observability T he p rocedure for linear transformation is p arallel t o t hat i n t he c ontinuoustime c ase (Sec....
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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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