Assignment-5

Criterion
Jury Stability
In applying the Jury stability criterion to a characteristic equation
0127=0 ,
we construct a table whose elements are based
on the
coefficients of old

General form of characteristic equation " ""

aozhtqzh azzn
"
☐ (2) =
+ - - -

AKZ . - - -
t an -12 t an

where ao > 0
,

Jury stability table is

formed using following rules :

4) The first two rows of the table consist

of the co -

efficient of
012)
,
arranged in
ascending order
of power of 2 in

row 1 and in reverse order in row 2 .

Iii ) All even numbered rows are

simply reverse
of the

immediately preceding odd-numbered rows .

iii ) The elements through can -37

for row 3 are
given by
 bk =
An Ant K 11--011,2 n -
l
;
,
- - -

Ao AKH

CK =
bn-1 bn -2 -
K
;
K
-
- 0
, 1,2 ,
- - - - n - 2

bo bktl
:
:
:

qk= P3 th K K -
011,2
;
-

Po PKH

"

""""""
1- An Ann an -2 - -
-
-

An -

k - - -
-

Az Al do

2 90 An An -1
A, As An
-

Ak
-
-

-2
- -
-

3 bn-ibn.zbn.rs -
-
-

bn.nl - -
-

bi bo
4 bo bi bz -
-
-

bk -
- -
'

bn -2 bn-1
5 G- 2 Cn -3 Cn -4 -
-
-

Cn -
K -

L -
-
-
-

co
6 G Cz
-

Co Cn -2
-

Ck
- -
-
-

:
2n -5 p} Pz Pl
In -4 to Pl P2
2h -3 92 91 Go
?⃝
The
necessary and sufficient conditions for polynomial 0Gt
to have no roots on or outside the unit circle
in the 2- plane are :

4) out >0

Hit
{z; for
on n -

even

for n -
odd

Liii ) tant <

Idol

Ibm /

}
livl > lbol
ten -21 > 161 (m2 ) constraints
:
:

loyal >
1901

met
These conditions must for test of stability
.

Singular cases :
if either the first and the last
elements
of row are
zero ,
or a
complete
row is zero .

It resolved
is

unit circle
by expanding and
contracting
the
infinitesimally
z^= ( It E) 2 where E. is
very
small red no .
 ( ite )"zh I 11th E) 2h

when E is
positive radius
of circle Ite
radius circle 14
E is
negative of 1-

eg
.

☐ (2) =
2241-723+102't 421-1

Here ,
n -4
-

90--2
A, = 7

Az = 10

Az
-
-

4
at
-
-
l

ou )= ✗ 1- 71-101-41-1 =
24 which is so

0ft )= 2- 7+10-41-1 = 2 which is > 0

111<121 ;
which is true

AS all conditions satisfied

to construct
are

table
,
moving on

jury
Row 20 2
'
22 23 24
1
Ay Az Az Ai Ao

Z Ao Ai Az Az At
3 bz bz b ,
bo
4 bo bi bz bs
5 Ce G lo

bz
f
=
At ao = I

do At L

bz =-3

bz T
at Al
L
=
'

ao az 4

bz= -10

b, -_ at an =
" lo
ao Az g 10

b, = -
lo

bo =
Aa Az =
1 4

ao Ai 2 7

bo = -
I
 G. bz
Is
↳
?
=
=

↳ bz

↳ =
8

Gz -3 -10 =
20
-
I -10

Co =
-3 -10 = 20
-

I -10

Row 20 2
'
22 23 24
2
1 I 4 to 7

2 2 7 10 4 I
3 -3 -
to -
to -

I
4 -
l -

to
-
to -

3
g 8 20 20

constraints
Now
checking the :

Lil Ibn -11 > Ibo / =

I -31 > 1- 1)
which is true .

Iii ) 1cm / .
> Kol =
181>1201

which is not true .

Hence ,
given system is unstable