PageNo.

5
3 THOMAS cOLLEGE OF ENGINEERING & TECHNoLOGY CHENGANNUN &T.THOMAB COLLEGE OF ENGINEERING 4 TECHNOLoGY CHENOANNUR ST.THOMA GOLLEGE OF ENONEERING &

4 A discvel tim 6ysam given by okjenence equounM

ck+2)+ 5y (k+) +6y Ck) - ulk
YCo) = ) =0, T Sec.
a) dalunime Hhe S t a mada P.
in comt i Cal Fum.

5) Fmd Hhe Stalk fram ihm matrex

9 for wput uk)=l , k7, fmd ofp YC).

Ck+2)+ 5y (k+ t) + 6 yCk) = u(k)

Appy z-troofnm on balh Sides

Ycz) +5z Ya) + 6 Ycz) = l(z)
Ycz)[+5114] -Ucz
Yc) =
Ucz)
TRACE KTU z+5Zt6
A
(Z+2)(Zt3)

t
(Z+2)(Zt3) Zt
A=
Z43 Z--2
13 L
Z+-3
ce)
Z3
Yrz) U(z) Mz)
Zt2
Z43
Yz) I()Ya(z) hme
= (Z+2) X a) = Urz)
Zt
Z A(z)t I ( Z ) = U(z)
1 (k+ 1) t 2 C4 [k) U(k) =

l4t) -Il[k) +u (k) ()

21Z) Uz) + 3) 2 (z) = U(2)
Z4.3
Z h2rz) +3 2rz)= U(z)
 Page No 6
SHOMAS COLLE Gt OF
ENGINE ERING & TECHNOLOGY CHt NEEANNUN ST THOMAS COEO OY
ENGINININ tGh y

2(k+ t3 X3k) = U(k )

a (k) -3 3 (K) + uk)
XICk)
T2k)
Yczl () - X2()
YCK K (E) - d2Ckk)
Yck)L )
7Ck)

To Tindstniz trasithom matrX, A

A
z1-Alz -
A
Z1-A-z TRACE
o1
-3
KTU Zt 2

Zt3
Z1-A (z+2) ( z+3)
L[zs
(z1-A)E
(+2) |2
o
7 Zt 2

Zt Z13

1-A
743
rtA z
213
A (-a
(-3)
 PageNo. 7
O M A S COLLEGE OF ENGINEERING & TECHNOLoOY CHENGANNUR ST.THOMAS coLLEGE OF ENGINEERING & TECHNOLOGY CHENGANNUR ST.THOMAS cOLLEGE OF ENGINEERING &

Tohind yclk) whm wwt skp

) (z1-)zXto)+ Z (21-4) 8 )
hk Xco) = O, &Ck) = I u(z) =
nck) - (z1-A) B Ua)
(Z1-AB Uz)=
Z+

Z+3
Zt
z- (z-1) (Zt+2)
Z43

TRACE KTU Z
(z-1) (2t3)

(z-1)(z+2a

T-3
(2-1)(Z42) z- Z+2
-

(Z (Z+ A) Z-I Z+2

(z-) (Z1)

z-1) (Z+ 3) Zt3

Z431 Z-
Z

(2-1) 2 3 Z-I 4 z +3
O A S coLLEGE OF ENGINEERING &
TECHNOL OGY CHENGANNUR ST
PageNoE
THOMAS coLLEGE OF ENGINEERING & TECHNOLoGY CHENGANNUR ST THOMAS
COLLtGE 0

z ) (z+2)

(z)(Z43)
X1[e) () (-2) k

2(b)( (-3)
y( -]7]
7(k) - XiCk)

()*- (2) (3
=TTRACE KTU
- (-21k.
5) Fnd Hu reshonse of he sm
xlkt1) = Mck) + u6),
-I

Co)=_ if us)=I fr e 0, 1 2,

U(z)
LO -

z-I
nok)= z (z1-A) z, Xroj t z (11-A) Bua)

ZI-A =z 7Z-
Zt
-

2
21-A (z-1(z+1) z
=(z4)z+1)
 Page No. 1
ng&r
Engiaterung
ENGINEERING&
ST.THOMAS CcOLLEGE F
TECHNOLOGY CHENGANNUR
cOLLEGE OF ENGINEERING &
TECHNOLoGY CHENGANNUR ST.THOMAS
OMAS COLLEGE OF ENGINEERING&

(ZI-4)
(z-1) (Z+1) z-

(z-) (z+)
O

(Z1-A)Z
(z-) (Z+D
z
O
Z+f-

3-Alz? (Z-!(z) Z4 Zf

A B
z-I1-1

TRACE KTU
z
E

k k
--

A Xo)a f
(1)
(-,
-( (-)
(z1-)Buz)= Z-I (z) (Z+1
Z+
 NAAC

1 5tale wmd enplam h propertis of Stk t romwetton mtry

a) Fwd a stae trmsitram motvix fo A= o -1

Conyley-Hrmilto heovem 3

A-[
-

a 3The cha ctemstrt equhn iu1-A-0

21-42 0
-

0
3 - 2 43|
Thcuts of I chmautervwht eguahm eagen v a s
n(743) t =0 n+ 37t 0 :
(+1) (+a) -0
a =-a
Let fA)= e TRACE KTU
when 1--1Slau)=e»t
= e»t -E
a)ent

f-)=e
fCna)-ezt=_

we kno hot £C2)= _do + 2 t d2 ai..

hn M-2, 2t-do4 AL

oD -()
o-3 o
We kns _hat fe)=_e_gt-2) - eE
do-d e - (1)
(2 7 o-2d = e*.
( 2 )(2) _4 = ë -e
odoE+ d ét+ef-E= aet-e"
 NAAC PaeNo
&
tNGINEER'NG
COLIEGt O
81 HOMAB
UR

JA) do 1t d A = e

A do 1 t dy A

"- o + e-at[o -
LO L -3

a Lt t s3e*]

whwch 4 STM.
- e+ae

12
3)Fmd ftA)-_A f A=0 usng Caylay
Hrowwilfo hcave -2-3
A TRACE
-3
KTUTheohamaAtevshz e4ubhm
ni-A=0

13
a (h3 td =0 nA3 =D.
(EL)_(2t2) = d

wn A- S)=
-l=_(2)
- 2 ) - (-2 2 4036

n-
We knowo hat ica:)-dota i t d2E n-1
Cai)-_do t d Ai
whn = - 1 - ) = do -d 1
2- 7 t(-2) = do-a d_= 4036
 thunt LPiNG &iELurihuLUr CH NiANNIH S irwMAB Ft A tRING& TFCIUNC GY CMENGANAK I HOMAS coLEGE oF ENGINEERING

odo- dy (
d -a = 4036b
(3)
()-(2) o =-4035
d-4=4,. + o = 4-4035 =4034
SCA)=_Al do1 t oA
4034 0 t4035 o
-23
-4034 0 o 4035 1
-4034 -8070-12105
12
-4034 -4035
8010

4) Foy wtaim sysem_ when Xto)= e nXCt)=|

wwle Xo)=| nXtH)-e TRACE KTU -3est|

Dekimwme \hs ystm makrix A

KLt)=_eAtXto)
Let_ A A2 nd h cquohim u
Aa A4 XI)_AXtt)._
3t
No Xtt)= e hene
3 3t
CundXco)= 4 XLo]_[-3]
-3
Now XLol =_A XLo)
31 AL A
As At l-3]
 NAAC age No.
naer

N EKIMG 4 lttNoLCIGY CIIENGANNUR T THOMAS COLL&GE OtNINEERING & TECINOLUC C E NGANNUR ST THOMS COLLEGt OEGINEERRING

A-3 A& = -3 (1)

A3-3 A4 = -()
StmdlomyXct)e nn&XLE)==_|eet

Cmd Xlo) =
4
Xta)=_A_Xto)
fA AL Az1
ALtAa _4 -(3)
A3t A4=
3- 4Az= 4
Ael-Aa=l-4= 0
(A)-(2) TRACE
4A4= -8 KTU
A4 =-2
(A=Ast A4 =1L A3=l-A4 =I+2 3
S AA1 A2
LA3 A4J L
SCekain_ls rtuhm_o lhamogenisskata eqnahm
-AXhas A[1-21 nd Xtoj=p.s
to homageni.os Stnle Equasm X
Ax
Salwtiom Tct) =_pct). Xco)
t)-e= 51-)
S1-A)=s 4 -2 - 2
4 -4 S44
2
S1-A st4 t 3 = S35-4 +3 =5+35-2
 iNASGY E NGANNU OMAS LALE

51-A= S3 5-= S+35

2
a
=(S-0.561) (5 +3. 561)
(51-A Adj -6
(51-A) -D-66)(55 S+ 4 -2
S1-4 (s-0-561)(S+3.561) 4
St4 S-1J
= (S-0-56t}{S+3-3EY- S-0-56t} {S+3+561
4
-1
(S-0-561) (5t 3.S61) (S-0S61) (Si3-561
St4
5-0.561) (St3.561) S-0.56] St3 561
St4 =A St3.561 + B (s-0 561)
A=: 1D
A
S+4 11D6 e°551E 3-561
S-0-561) (S+3.56) - D-106 e

S-D561) (St3-561) TRACE

A KTU S-D 5b1 St3 561

a = A(Sf 3-561)+3(S-0.561)
A= - 6:485 Cmd B= O485
- -0.485 e
0561t -3.561t
t0:4-85 e
CS-0 561) (5+ 3 561)
A
S-D.561) (S+3.561) (S-0.561) (St3 561)
4 A (3 St3.56 + B(S-0 561
A-0:242 n 3 =-0:
242
-5 61E -3 561t
|
(s-0.561) (S+ 3.561) = 0:242e D-24 2 e
S
A
- - (S-0.56i) (S+ 3.561) S-0S61 St3 56
 NAAC
sOnROnATION CO
Page No

S- St 3.561) + 13 ( S-0-56
A = - O1b6 md B 106 3.561t
- D10b .561t I-10b e
5-1
S-0561)(5t3 56) 0.561t -3-56Tt D.56t -3.5E
- 0-1D6 e - D46Se +D-465e
eAtbt) =11b6e
0 56It .3-56t ,561E -356lE
Lo-242e-0242e -0-1Obe+p.10be

Lctl= d Xo) d¢t)[05

TE) = | 0-068 e 56|E

0.432e37561t
3.561t
0.DIS e 56|E
D985 e

TRACE KTU 12spawe v

6) FrSystemeprssenled by_X= AX A
XEl=[aetj when Ktal= Cmd

L X4)=[4e when Kco)=|4

Delanm lhe systm matrix A_md Ihe Stalk_

romsihan matzix
he solwim of homaqeni eqnah on =AX
t

Let A-AAz
Aa A4
We howe Kctl= aet t)=|-eet|
AL t=o Xtol= and
 NAAC
AIONAL ABSESSAMET AND
ACCCHEDYATKON o U N C H
PageNo.
Or
ENGINEERING
&
COLLEGt
CHENGANNYR SLOMAS
{NGNEERING4
TECHNO OGY
O
Xc)= - 8e2t
OLLEGE
HONAS
N G t R G 3 TECHNOLOG H N

+e+
(-22t

At t-0
4

We hawe X=AKtt)=fA1 A XtU

A3 A+
At teO
A AXco)
As A4
SA A2 1 2.

Fo irst case -e
4 As
2 At A2 8
2 A3 +A4 = -12)
Far Sewn came TRACE KTU
[A Az1 4
As 2
A4 L
4At A2 =-8 (3
4 A3tA4 =2 -(4
- (2) -2AL0 A= 0
8
2At A2= -8, Aa
(2)-(4 2 2A3 =-2 As
(2) 2 A3+ Af =-4 a A4-4 A4=
Sn A=A Az 0-8
Aa A4
Toad stalk rgmsihan matrix
(s1-A)= 0- -6
5 6
St6
-

51-A_s(5+4)++8 s65te =(S12) (314)

 NAAC
aara ACCREAT CJCN

LLtt
ENGINEtRING & !E CINOLOGY CHENCGANNUH ST THOMAS CLLEGf ENGiNE ERING &
itiJL CA HENGANUA OF
THOMAS COLLE GE EIPiEER

$1-A Adj(51- St6 -8

S1-4 St2) (St4) S
St6 -8
S+2)(S+4)(SF2}{sT4)
S
S+2) (5t4) (St2) (St4)
St6 A
(S+2) (St4) S+2 St4
(+4) + B (s+2)
Stb A
utS=-a t EA[2) A-
S--4 B-2) B--4
Stb e 4t
S+2) (S+4) S+2

4 TRACE KTU S+4

(S+2) (St4) St22 Stf

Ywt S=-2 1 =A2 A=2 .5

S4 _1= B2).. -

B-/20:5
05 0.5 0.5eD.5 ett
(S12) (514 Sta St
-8
St2) (St 4)
5D.5 4 4E
S

S2) (St4) Sta St4

S= A(St4) +B(Sta)
P S--3 -a= A(2 A
S=4 2 - 4 = B(-2) _8= 2
 NAAC
ADCREDTATI GoUNCA
Page No

S -2+2et
(542) S+4) 5t
4 2 4 4t

ae4t
0-5 e0.5

A mem ttmme imvoiowt sys lem y chavact evized

by Ih omaqemies state equatio

Compwle e sonhin of homogemous

equiovM, wnme he tnihal stoi vector XDs
Gaiven A=|4 Xo
TRACE KTU
trnonoqeniaws equahn X- AX, Ie scduhn
ct)=ft) Xco)_
Otl= 1S1-A)=eAt
=5-1
S-1
51-A(s-
(53-AAdi(s1-a)
61- (s-1) S-1
(s1-A-[E
te
o
Ket)- ct): Xco) - |e e
te
ee
NAAC
ATHONL AS8 NAA
CENAON COUNCH
AND Pa
ENGINEERINGa
ENGINEERING &EINOL OGY CHENGANNUR ST THOMAS COuEGF D
Lt tNGINEERIMG ECtNOOGY CHENGANNUR ST (HOMAS CCLeGE O:

3) taim I time sponse of Ihe tollcwg smM

- ) whn uci lke wmnit
skp ocving at t=0 amd

Xo)=
O

Total Respanse, Lct)= Gt) Xco) tL|$1-4BUs

We kn dromLast enample, ZIR=et7

zSB =_(61-)8 US tet

We knowe (51-4)=
TRACE KTU
ZSR L

- s(S-)
ZSk=
SS-1)
ss-)
Puut s-01 A-), A=H
= A (s-1) +BS
6-1l= BO),B-I

1+(S-)
(s-1)2
 NAAC Piage Na
AIK2BA AS hh
ACCREDATKOANCOJANh

1U1iEat ENGINEERgG
twGNt RNG 'tNCN.UG R aANNi ST T«AMAS tE D ENINT ERIN 4 1F ,HNU uY
t NGANNR THOMA

z5R et
te
(6-1)
Total Raspose.. Kt) = ZIR +ZSR
zt) = et
te te L atet

g Fwdie Vspavse of he syskm

-2 -3 LoJ
nd Ytt)=4 oXblhe ollowrin tnpuk
41
Uct)= TRACE
uct) KTU
unau Ltluwwt 3t
U2tt s p tmsh o.
A=
--3
-qct)=_et
S1-A)
61-A)[s
-2 La St3]
SI-A s(S+3) ta =Si35t2 =(5t) (St a)
(s1-A Ad(s1-4)= S#3 1
S1-A (s+)(S2-2 S
S+3
S+S+ a) (s+1) (S+2)
o

St3
t5+19LS+2) (S++}{S+2}}-
A
541)(S+3) St2
 NAAC
ATIONA AsseSE a age N
D ER
LCA ,ENGANNUR
S THMAS CAI EGE
aNEEKINGIECHNOLOG
CEGANNUR
ST TtOMAS COLLEut
9 NsiNE t Ri

5+8A(S+2) + B (5+1)
A2
PutS-1 > 3 - A )
S 1= B(H)1 3= e
-2t
St3
S+I St2
(S41)(S+2)
A + B
S+)(6+2) St S+2

Put 6-1 1_Al_E

6-2 I=B . 13 2t

6+1) (42)
TRACE KTU StI 5t2
L64)(542)
A s2
6F1) (S+2) 4|

6= A(S+2) +B(s+1)
A=
S-2 -2= B() B 2
-2E
- Etae

(s+1) f512) StI S+2

eF2
KcoJ =[oo1
ZI ct) Xco)= Lo
 NAAC
NATIONAL ASSESSMENT AND
a r CAEDTATIOM COUNCH
PageNo.
COLLEGË Of ENGINEERIK.
STHOAS LOLLEGE OF ENGINEERING
L TECHiNO CGY CHENGANN JR T THOMAb
L ut
N L RING &
TECHNOLOGY HE NGANNUR

SR = 51-4)'B Uts) Uul Ys

St3 LS+3
(S41) (S+2) (St1) (St2)
o TJLEE
(S+1) (53) 1)(+2)J
|St3
L S H )(St2) G)(3+2)5 2.t
St3
t(sH1)(S+2 (S+1) (S+2)
S+3 3 (5+3)
(S1) (Sta) (S+1)5+2) S(S+3
5

s(5+1)
TRACE 3
(s+15+2)(6t3)
KTU
6 5
(5-)5 (S+1) (542)[St3J
sing wtial rochr
t0:5 tD.5
ZSR=L S+1 S+I St2 S+3

3 0.5 5
TSt-S+3- S S t S+3

S 3-3.5Et-EZ0.5e
2ta5e+3E 2.5
5 3t
3 E

=3E
+Zs=|3-2.5 2
-e +0.5 e
=ZIR
2 t 2 5e't2e2.5e5 ate)
 ENLNLE iNG N OG7 CHENiNNUH hAAS LEU NIN E,MING &tCiil G CitNEjANHLH S AS t LEGi NGEt*

Ytt)= X6) = DX, tt)

1 LI 1 2 1)
YCE XLE)=3-a.5e 2+D 5 e
-2t
zttXTEI+Xzttl=3-2:5e-+0D53
t a : 5 e t a E 2 g . 5 es3t

Y2lt)=S=4 t e - e s

IoComioley hesta madel

2 4
Fmd. Ihe Set ofinitial conditirw Sudh_Hhot mode
2t
iaSupprSsednm Ytt
e At
Fmd_eAt A=o
TRACE KTU
61-A= 0
[s -11
Lo s S-1
SI-A s(s--a=S-5- =
(S+1) (S-2)
51-A]'= Adi (ST-A) S-1
5
S1-A (S+1) (s-2) S
S-I

(s+1) (-2)SH1) (5-2)

S
Ts+fs-2}- (S+1(s-2)
S-L A
(S+1) (5-2) StI S-
S-1=_A (S-3) + 3 (S+1)
PukSl -3 =_A-3 A=23
S= _ = B(3
u i tNGINEE.RING& " L H . G CHENGANNUR ST THOGAS OLLEGE OF =MNEERNG & YECiNO G Y CHENGANNUR ST THOMAS UOLEGE OF ENGNEERH

5- 3 -* Y3 e+2
s+1) (5-a) StI S-2 3

A S-
S+1) (5-2) S+I

4Al6-a) +B_(S+1
Pot S=2 4 B(3 B-3
= A-3 A=
S+I S-2 3
2t
(S+1) LS-a) 3

S A B
S+1)(6-2) Stl S-
S A(s-a) + 3 (St
Puk S=a TRACE
2 -B(3, KTU
B 23
5- -1=A-3 _A =Y3
S+1) (5-2) StI S-2

h Sduhic homog enirws eau rn

Xt)=_e X(b)_ Let Xto)=|
3
a
 tt)-
a3 ) 5(gEz)|
eRa-b) + 4(a+b)
2
(-2atb)(aaab)
Yc) [ 4 ] X taXa
(2a-b) + (R+b}+ae (-2a+h
(2a+2b)|
Yt)4 (20-b -4a +2b) +L e2 (Dtbt4a1 4b)
-20+bes 5aisb)
YcA)= TRACE(5a15b)
(-2atb]t KTU
at
Gwwen hat SWhhye e term acn YIE
5t5b= o atbo a
Initnl stask vestov Xo)- =+ whame vn fo
LD - m

1otrwm mw sponwe of I lwmg Veutymatvix

oifierential eqyolion K=o 4lktolu) ond
F6-5
LX2
L2whme uE) V wmn skp wp
omd ih Condlrv me X1CoJ =X2( 0) =0.
Here intial condawrys ov
Total Rspe, 2t4) =L$1-A 13 Uu Z5R
 NAAC
ATHONAL ASBESSENT KND
ACCREDITATION COUNCH
PageNo.
COLIEGE ENGiNEEkIN
ENGINEERiNG & 1E CHNO CGY CHENGANNUR ST THOMAS
CHENGANNUP S 1hOMAS COLLEGE OF
L G t GINEERING & TECHNOL OG

(s1-) -[s o1.[° -5]-[s

L6 St5

631- =S55t 6=_(5+2 (5t3)

s(S +5) +6T St5 `t5
(61- S+2)(St3)-b tsr2tst3-(st2)(sf3

- S
tGF2)(SR CS+2)[SF3
St5 A
SfA
3St3
B
(s+2) (S+3)
St 5 A (5+3)+ (S+a
Pwt s--2 3 = AU A-3
S--3 B-))
ZSR= St5
(St2)(5t3) (S+2) (5+3)
6
TRACE KTU
S+2)(St31(S2 (St3)J
= sCS+2) (St3)

612 (S13 (S+2(Sr

C
s(s+2) (5+3) St3

4-AG+2 (St3) +B S (St3)+ C s(5ta

Futs-0 = A213=6A A6
S- = B(-2)0 =-28 , 8-2
S=-3 l c(-3 (1) =3C C3

tStSEr3
(S2)(St3J
I=A [S+3) +18(s+2)
 NAAC P2

Put62 I= Alt) A-
S-3 I =B(-) 8=
3t
SF3

xt esuu s1-[14
Et_t 2
Ycb)-[ 0] Xtt) - X1tt)=- a 3

12) Givenlhe syskm =[-3 D +3 o . Fmdk

T2-J 2J
vwpw Vectw uct) m he Tolowtn meY0spanse
XH)=_6 (1-e)
Y2lt) =_3e St
he TRACE
goven ime
U.
KTU
mwt
a e2

satuty 1k stali ea"

Ysþase

We knguk X) =_A Xt) +B uct)._

6e

+
[3 0
32
8 +18
44
6-6-3 e+ae 2
-6e+18-18e
!-63 33 aebebe 3a
 GYCHENGANNF StiHOMAS
TOLIEGE
ENGINEERING
18 1&e &
TECHNOL OGY CHENGANNUR ST
HOMAG COLLEGE OF
ENGINEERNG &

|-6t12e-6ebe 32
3
o 8 18- 12et
2 -6t12 6E3 E4t =3013
L3 2 3 2
We knaus a o
3_D T
3 2 3 2
3 D
32 6 3

OT18-122e
-6t12 e 6 3 6 4
6-4 e t
TRACE KTU 3t

13 tov Syslem vepresen ed by srolk

eqy eurm Xtt)=AX)
Yspase Xtt)=[E2t when co=
2E2
Gnd Xt) = wne Kco]=
-2
-e
DelwM h syslem matvis A
The souhian of hame qemious
e4uaktmX=AX _
Xt)=et Xco)
Lek_A=AL A2
W hawe
A3 4 e 2

Xtt=-aE
 NAAC ntE
COMLEt O NGINEERS
:ENGANNUR S HMAS
E3t. OENINE a ihaJL
THOMA
o EINg d
t Ot de AE NANN PS

Kue)--2 at t Kto)=
-
and Xo)= -2

Secand cane Xct)=e

Perc Xc)=[17
At_b0 Xco)=
XLE)= AKct)=|A2|Kt)
'We haNe
As At
A X)=A A2 XLo
3 Aa

TRACE KTU
-)
4 A3-3A4 -( )
toy Seuond _caoe [-11. A A2
A3A
- = Ai=ta (3)
A3-A4 -(4)

- =A-2A2=Ai-2
AlED
_()-(a 3=-2A4t A4=-A A4
L4) hz-A4=l
Aa=1+Az=1-3=4