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

1 26 2 design a digital lowpass b utterworth filter

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Unformatted text preview: a re t he j s ystem i nputs. F or a linear system, these equations reduce t o a s impler linear form V3 = Xl - V4 = Xl - (13.2) 2X2 T his s et o f e quations is known a s t he o utput e quation o f t he s ystem. I t is clear from t his s et t hat e very possible o utput a t some i nstant t c an b e d etermined from a knowledge o f X I(t), X2(t), a nd f (t), t he s ystem s tate a nd t he i nput a t t he i nstant t. O nce we solve t he s tate e quations (13.1) t o o btain X I(t) a nd X2(t), we c an d etermine e very possible o utput f or a ny given i nput f (t) . • I f we a lready have a system equation in t he form o f a n n th-order differential e quation, we c an c onvert i t i nto a s tate e quation as follows. Consider t he s ystem e quation d ny d n - 1y +a - 1 - d tn n d tn-l dy + ... + al- +aoy = dt f (t) (13.3) k =I,2, . .. , n (13.6a) a nd t he o utput e quations a re o f t he form Ym = CmlXI + Cm2X2+" . +cmnx n + dmlfl + d m 2h+" · +dmjfj m = 1 ,2, . .. , k (13.6b) T he s et of E quations (13.6a) a nd (13.6b) is called a d ynamical e quation. W hen i t is used t o d escribe a system, i t is called t he d ynamical-equation d escription o r s tate-variable d escription of t he s ystem. T he n s imultaneous first-order s tate e quations a re also known as t he n ormal-form e quations. These equations c an b e w ritten m ore conveniently in m atrix form: .... < 13 S tate-Space A nalysis 788 13.2 A S ystematic P rocedure for D etermining S tate E quations HI a ll X2 a l2 al n a 21 Xl a 22 a 2n b ll X2 + . .................. a nI Xn an2 a nn b lj b 2j 12 ..... ..... b nl Xn A 10 h b 22 bn 2 , '-v-' x '-v-' x b l2 b 21 Xl IH 789 b nj f (13.7a) Ij ,'-v-' v F ig. 1 3.2 R LC network for Example 13.2. f B a nd 3. W rite t he l oop e quations a nd e liminate all variables o ther t han s tate variables ( and t heir first derivatives) from t he e quations d erived in S teps 2 a nd 3. C ll Y2 C i2 C in C21 YI C22 C 2n d ll X2 + . ................. C kl Yk C kn Ck2 '-v-' y c d l2 d lj h d 21 Xl d 22 d 2j 12 . ................. d kl Xn • , '-v-' x d kj d k2 (13.7b) Ij ,'-v-' D E xample 1 3.2 Write the state equations for the network shown in Fig. 13.2. S tep 1. There is one inductor and one capacitor in the network. Therefore, we shall choose the inductor current X l a nd the capacitor voltage X 2 a s t he state variables. S tep 2. T he relationship between t he loop currents and the s tate variables can be written by inspection: f (13.9a) or (13.9b) x= A x+Bf (13.8a) y = C x+Df (13.8b) S tep 3. The loop equations are (13.lOa) E quation (13.8a) is t he s tate e quation a nd E q. (13.8b) is t he o utput e quation. T he v ectors x , y , a nd f a re t he s tate vector, t he o utput v ector, a nd t he i nput v ector, respectively. For discrete-time systems, t he s tate e quations a re n s imultaneous f irst-order difference equations. Discrete-time systems a re d iscussed in Sec. 13.6. 1 3.2 A Systematic Procedure for Determining S tate Equations (13.10b) Now we eliminate...
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