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

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

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Unformatted text preview: can b e specified in m any different ways. Consequently, t he s ystem s tate c an also b e specified in many different ways. T his ~eans t hat s tate v ariables are n ot unique. T he c oncept of a system s tate is v ery Important. We know t hat a n o utput y (t) a t a ny i nstant t > to c an b e d etermined from t he i nitial s tate { x(to)} a nd a knowledge o f t he i nput f (t) d uring t he i nterval (to, t). Therefore, t he o utput y(to) ( at t = to) is d etermined from t he i nitial s tate { x(to)} a nd t he i nput f (t) d uring t he i nterval (to, to). T he l atter is f (to). Hence, t he o utput a t a ny i nstant is d etermined completely from a knowledge of t he s ystem s tate a nd t he i nput a t t hat i nstant. T his r esult is also valid for multiple-input, multipleoutput (MIMO) systems, where every possible system o utput a t a ny i nstant t is d etermined completely from a knowledge of t he s ystem s tate a nd t he i nput(s) a t t he i nstant t. T hese ideas should become clear from t he following example of a n R LC circuit. • E xample 1 3.1 Find a state-space description of the R LC circuit shown in Fig. 13.1. Verify t hat all possible system outputs a t some instant t can be determined from a knowledge of the system state and the input at that instant t . I t is known t hat inductor currents and capacitor voltages in an R LC circuit can be used as one possible choice of state variables. For this reason, we shall choose XI (the capacitor voltage) and X 2 (the inductor current) as our state variables. The node equation a t the intermediate node is but i3 = 0 .2'h, i l = 2 (J - xt), i2 = 3 Xl. 0 .2:h or 784 :h Hence = 2 (J - xt) - = - 25xl - 5X2 3Xl - X2 + 1 0f 13 S tate-Space Analysis 786 787 I ntroduction O ne possible s et o f i nitial conditions is y(O), y(O), . .. , y(n-l)(o). L et u s define y, y, ii, ... , y (n-l) as t he s tate variables a nd, for convenience, l et u s r ename t he n s tate variables as X l, X2, . .. , Xn: IH + 13.1 + Xl f =Y Y X2 = X3 ii = F ig. 1 3.1 R LC n etwork for E xample 1 3.l. Xn = y (n-l) T his is t he first s tate e quation. T o o btain t he s econd s tate e quation, we s um t he v oltages i n t he e xtreme right loop formed by C , L , a nd t he 2 !1 r esistor so t hat t hey a re e qual t o z ero: - Xl + X2 + 2X2 = 0 (13.4) According t o E q. (13.4), we have or X2 = Xl - 2X2 T hus, t he two s tate e quations a re Xl = - 25xI - 5X2 X2 = Xl - + 1 0f (13.1a) (13.1b) 2X2 a nd, a ccording t o E q. (13.3), E very possible o utput c an now b e e xpressed a s a l inear c ombination o f F rom F ig. 13.1, we have VI =f - X l, X2, a nd f · (13.5a) T hese n s imultaneous first-order differential equations are t he s tate e quations of t he s ystem. T he o utput e quation is Xl i t = 2 (J - xI) V2 i3 = il - y = Xl = Xl (13.5b) For continuous-time systems, t he s tate e quations are n s imultaneous first-order differential equations in n s tate v ariables X l, X2, . .. , Xn o f t he form i2 - X2 = 2 (J - xI) - 3XI - X2 = - 5XI - X2 + 2f i = 1 ,2, . .. , n w here f l, h , . .. , f n...
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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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