Unformatted text preview: i jth e lement Hij(S) o f H (s) is t he t ransfer f unction t hat r elates t he
o utput Yi(t) t o t he i nput f j(t).
• ( 8+1):<8+2) t li = (13.38b)
a nd t he z erostate r esponse is 803 (13.41) I t is i nteresting t o o bserve t hat t he d enominator o f e very t ransfer f unction in
Eq. (13.42) is (s + l )(s + 2) w ith t he e xception o f H21(S) a nd H22(S), w here t he
c ancellation o fthe f actor (s + 1) occurs. T his f act is n o c oincidence. We see t hat t he
d enominator o f e very e lement o f . (s) is I sI AI b ecause . (s) = ( sl A)I, a nd t he
i nverse o f a m atrix h as i ts d eterminant i n t he d enominator. S ince C , B , a nd D are
m atrices w ith c onstant e lements, we see from E q. (13.38b) t hat t he d enominator
o f . (8) will also b e t he d enominator o f H (s). H ence, t he d enominator o f e very
e lement o f H (s) is l sI  AI, e xcept for t he p ossible cancellation o f t he c ommon
f actors m entioned e arlier. I n o ther w ords, t he p oles o f all t ransfer f unctions of
t he s ystem a re also t he z eros o f t he p olynomial l sI  AI. Therefore, t he z eros
o f t he p olynomial l sI  AI a re t he c haracteristic roots o f t he system. H ence, t he
c haracteristic r oots o f t he s ystem a re t he r oots o f t he e quation l sI  AI = 0 ( 13.43a) a
13 S tateSpace Analysis 804 Since I sI AI is a n n thorder p olynomial in s w ith n zeros Al, A2, . .. , An, we c an
w rite Eq. (13.43a) as
l sI  AI =
= sn + n
a n_lS +! + . .. + a lS l sI  AI = A = [:
(13.43b) = s2 2 0
[2
(13.44a)
(13.44b) = ( s+l)(s+2) a nd 1] t = [0
1
2t t]
t A 2=2 1] [0
1
2 1 ]::, = [ 2
12
2 1 ]::, = [ t2
32
t2 C le Alt + C2eA2t + . .. + c ne Ant (13.45) (13.51a) (13.51b)
Note t hat t he i nfinite series o n t he r ighthand side of Eq. (13.51a) also m ay b e
e xpressed as T his fact is also obvious from E q. (13.38). T he d enominator of every element of
t he z eroinput response m atrix Cf1(s)x(O) is l sI  AI = (s  Al)(S  A2)··· (s An). Therefore, t he p artialfraction expansion a nd t he s ubsequent inverse Laplace
transform will yield a zeroinput component of t he form i n E q. (13.45). d e
dt At [ A2t2
A 3t 3
I + A t +   +   + . .. + . . . A
2!
3! Hence 13.32 TimeDomain Solution o f S tate Equations ~eAt = A e At = e At A T he s tate e quation is (13.52) dt (13.46) X = Ax+Bf (13.50) a nd so on.
We c an s how t hat t he infinite series i n E q. (13.48) is absolutely a nd u niformly
convergent for all values o f t . Consequently, i t c an b e differentiated or i ntegrated
t erm b y t erm. T hus, t o find ( djdt)e At , we differentiate t he series o n t he r ighthand
s ide o f E q. (13.48a) t erm b y term: Equation (13.43) is known as t he c haracteristic e quation o f t he m atrix A ,
a nd Al, A2, . .. , An a re t he c haracteristic roots o f A . T he t erm e igenvalue,
meaning " characteristic value" i n German, is also commonly used in t he l iterature.
Thus, we have shown t hat t he c haracteristic roots of a system are t he eigenvalues
(chara...
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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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