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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 multipleinput, 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 statespace 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 tateSpace Analysis 786 787 I ntroduction O ne possible s et o f i nitial conditions is y(O), y(O), . .. , y(nl)(o). L et u s define y,
y, ii, ... , y (nl) 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 (nl)
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 firstorder 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 continuoustime systems, t he s tate e quations are n s imultaneous firstorder
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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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