Circuits and Systems, 2016, 7, 709-717
Published Online May 2016 in SciRes. http://www.scirp.org/journal/cs
http://dx.doi.org/10.4236/cs.2016.76060
How to cite this paper: Ramesh, P. (2016) Stability Analysis of Multi-Dimensional Linear Time Invariant Discrete Systems
within the Unity Shifted Unit Circle. Circuits and Systems, 7, 709-717. http://dx.doi.org/10.4236/cs.2016.76060
Stability Analysis of Multi-Dimensional
Linear Time Invariant Discrete Systems
within the Unity Shifted Unit Circle
P. Ramesh
Department of Electrical and Electronics Engineering, Anna University, University College of Engineering,
Ramanathapuram, India
Received 21 March 2016; accepted 9 May 2016; published 12 May 2016
Copyright © 2016 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativ ecommon s.org/l icenses/by /4.0/
Abstract
This technical brief proposes a new approach to multi-dimensional linear time invariant discrete
systems within the unity shifted unit circle which is denoted in the form of characteristic equation.
The characteristic equation of multi–dimensional linear system is modified into an equivalent one-
dimensional characteristic equation. Further formation of stability in the left of the z-plane, the
roots of the characteristic equation f(z) =0 should lie within the shifted unit circle. Using the coef-
ficients of the unity shifted one dimensional equivalent characteristic equation by applying mi-
nimal shifting of coefficients either left or right and elimination of coefficient method to two tri-
angular matrixes are formed. A single square matrix is formed by adding the two triangular ma-
trices. This matrix is used for testing the sufficient condition by proposed Jury’s inner determinant
concept. Further one more indispensable condition is suggested to show the applicability of the
proposed scheme. The proposed method of construction of square matrix consumes less arith-
metic operation like shifting and eliminating of coefficients when compare to the construction of
square matrix by Jury’s and Hurwitz matrix method.
Keywords
Stability, Multi-Dimension al, Unity -Sh ifting, Characteristics Equations, Inner Determinants
1. Introduction
The stabilit y problem of multi dimensional discr ete polynomials is r eceiving more attention due to t he emergin g
widespread applications. In recent years, multivariable functions have been increasing the applications i n analyses

P. Ramesh
710
and synthesizes problem of discrete continues system. Various algebraic stability test algorithms have been
proposed for multi dimension system. However, they required huge amount of computation time for all. The
stability investigation associated w ith multidimensional digital filters used in the areas like seismology needs to be
considered. Other applications arise in obtaining reliability properties of impedances of netw orks and transmission
lines which represent multidimensional continuous systems in the form of multidimensional continuous filters.
Foremost among them is the stability of two and multi dimension system which find the application in the process
of bio medical, sonar and radar data. The study of multi dimension discrete shift invariant system has received
considerable attention am ong the researches. The stability problem is an important issue in the design and analyses
of multi dimension linear discrete system. A huge amount of research was dedicated to developing technique for
multi dimension system. Jury proposed the stability test for multi dimension system. It should be motivated by
practical a pplications. It is do ne mainly b y engi neers in the mathe matical litera te. The stab ility analysis o f mul-
tidimensional digital system is much more complex than single dimensional system. Result for the gen eral case of
any multidi mensional realizati on is still warranted which will the topic o f future investiga tion. The Jur y’s inne r
wise determinant method of stability test is useful for numerical testing of stability and for the design of linear time
invariant discrete systems. At the result of this paper multi-dimensional discrete system is converted into single
dimensional discrete system. The necessary condition is that the roots of the characteristic equation should lie
withi n the l eft ha lf o f the s -plane is verified. Then the characteristic equation forms the (n + 1) * (n + 1) matrix
and the inner determinant is calculated with the help of inner Jury’s concept and the sufficient condition is the
inner deter minant value is positive s ho uld be verified. This shows that the system is stable .
2. Literature Survey
To have a knowledge about the various stability problem in designing and analysing the multi-dimensional li-
near time invariant discrete system. Many practical tests have been developed to solve the stability problem. The
new criterion stability of the multi-dimensional system is obtained by using the functional schur co-efficient
which been discussed bySerban et al. in [1] [2] and it is illustrated by means of various examples. Benidir had
demonstrated in [3] about the stability conditions and its properties. The number of final constraints should be
represented as
( )
11
2nn+


in these constraints n will be reduced to obtain the stability criterion which was
proposed by Jury et al. in [4]. Bos e et al. presented in [5] a simplified for m to test the stability of multi-d imen-
sional syste m by as curtai ning the r ob ust stability o f linear d iscrete syste m. Sivana nda m et al. in [6] revealed the
stability tes t for multi-dimensional syste m. In t his mu lti-dimensiona l digital filter is a total of 2n different cases
are considered and by extending the Routh test. The necessary and sufficient conditions are verified as in one-
dimensional. A new tra nsfo rm, whic h i s app lied to the denominator p olynomial of unstabl e multi -dimen sional to
yield a stable polynomial and the stability of one-dimensional and two-dimensional are obtained using discrete
Hilbert transform was addressed by Dameravenkata et al. in [7]. Anderson et al. in [8] gave a simplified form
for c heckin g the st abilit y b y convert ing t he mult ivari able p olynomi al int o the s ingle d egre e pol ynomial. The re-
ducti on method whic h re duce s a non-linear n-di me nsio nal e quat ion to one -dime nsio na l no n-linea r eq uatio n the n
comparing the results using the proposed matrix norm approach was proposed by Jury et al. in [9] [10]. I n mo s t
of the cases this method provides less restrictive and it can be used for any order-dimensional if it is a direct rea-
lization. Bose had proposedin [11] a repeated application of an extended schurcohn formulation and also tested
the positive definiteness of an arbitrary binary quadratic form. serban et al. in [12] revealed a new multidi men-
sional BIBO stability algorithm. Hu Foc used in [13] on the condition for multi-dime nsional variables to te st the
stability. J ustice et al. in [14] Proved stabilit y theorem for multidi mensional s ystem. Strintzis proposed a simple
approach in [15] [16] to task the numerical testing of stabilit y condition and establishing t he stabilization. Mas-
torakis et al. in [17] made a approach to the stability problem where it reduced it into an ap propriate constrained
optimization using the Neural Network because it provides computational speed and by performing many exp e-
riments. Bose et al. in [18] explained the robust stability of discrete system which depends upon the bilinear
transformation. Kurosawa et al. had addressed in [19] the various algebraic stability testing algorithm multi-di-
mensional s ystem. To determine whether o r not a polynomial in several real variab les is positive procedure was
proposed by Bose et al. in [20] [21] gave a simplified form a reduction of multivariable polynomial into several
variable polynomials by the finite number of rational operations. Huang and Jury introduced the concept in [22]

P. Ramesh
711
[23] of inner wise matrix. The stab ility test of 2-D and mult i-dimensional system reduces to several applications
of the stabilit y test of one-di mensional s ystem. Man y results o n stabilit y tests of such syste m had rece ntly been
presented by Hertz et al. in [24] and it reveals that simplifications and computational complexity which grows
steeply with dimensionality of the system and which cause the test to become almost impractical for large n.
Ezra zeheb et al. in [25] had proposed a new theorem which can be used to derive new simplified procedure for
a multi-dimensional stability test, presented for the continuous analog case as well as for the discrete digital
case.
3. Proposed Method
A Multidimensional (M-D) linear discrete system [26] described by the trans fer function
() ( )
( )
1
11
,,
,, ,,
m
mm
Az z
Gz zBz z = 

(1)
The Multidimensional system is stable if and onl y if
()
( )
( )
( )
11
2 123
121 2
0,,0,0 for1
0,,0,,0 for1,
0,,,,0 for1,1
,,,0 for1,1
mm
mmm m
mm m
mm
Bz z
Bzzz z
Bz zzzzz
Bzz zzzz
−−
−
≠≤
≠≤
≠ ≤===
≠ ≤===




(2)
Let
( )
2
1 2331122
,, 40BZ ZZZZZZZ=+ +−+=
1. with
23
0ZZ= =
,
( )
11 111
1
40,1,the reciprocal ofand withBZ ZZZX
Z

= +=≤=


( )
1
41 0,1.Tx xx = +=<
Thus
11
4
x= <
(Satiesfied)
2. with
3
0Z=
( )
2
1 2112222
1
,40,the reciprocal ofwithBZ ZZZZZZX
z

= +−+==


( )
3
241 0,1Tx xx= +=<
Thus
3
1
4
x= −
, ind icating
1.x<
Using the Equations (1) and (2) stability checking involves more complexity and computation cost also in-
creased. Assuming that
()
1,, m
Gz z
has no no nesse ntial s ingula rity o f the se cond ki nd. T he ab ove the orem is
known as the theorem of Anderson and Jury [8]. t is well known that, for the purposes of stability testing, we
need more practical tests than the above theorem. For two dimensional systems, a great variety of practical tests
have been developed in the last three decades and some of these are Jury’s two dimensional test [9], Schur-Chon
test [1], Inner’s test [22], Zeheb-Walach test [5] [6]. There are also a variety of special results and other consid-
erations [25].
However for multi-dimensional systems (M-D) systems
( )
3m>
, we have a complete lack of such tests,
though we must refer to the contributions of [2] [3] [6] [7]. Hence, it is diffic ult to check as to whether a give n
multi-dimensional polynomial
( )
1
,,
m
Bz z
corresponds to the characteristic polynomial [27] of the stable
multi-dimensional system, when
2m>
. In this paper a simple method is proposed to find the stability of the
given hi ghe r or d er s yst e m, we mean t hat it corresponds to the characteristic polynomial of a stable (Or) unstable
multi-dimensi onal system.

P. Ramesh
712
3.1. Proposed Test Procedure
The previous nece ssary co nditio n as ment ioned i n [8] and one sufficient condition referred in [21] [23] and (2)
may be rewritten using reciprocal, since all the roots
1
n
z



are assumed to be lying within a unit circle surface
1
01
12 1231231
111111 1111 11
,,, ,,,,,,0
nn
n
nn n
BT TT
zzzzzzzzzz zZ
−
  
=+++ =
  
   
 
(3)
Afte r simplification the abo ve equation may be written as Equation (4)
( )
( )
( )
123
123
,, 0
ZZZZ
MZZ ZFZ
== ===
(4)
( )
0FZ=
Is one dimensional equation and for stability
1Z<
.
Let
()
1
10
0
nn
nn
F ZaZaZa
−
−
= +++= 
(5)
whe re
i
a
—are the coefficients and n is the degree of
( )
0.fz=
The Equa tion (5) is written as the following Eq uation (6)
( )
1
10
0
nn
n
nn
aa
FZ ZZ
aa
−
−
 
= +++=
 
 

(6)
For unity s hifted [8] unit circle state that in
( )
FZ
, z is replaced as
1Zx
= +
.
The Equa tion (6 ) can be written as
() ()()( )
1
10
11 1
nn
n
nn
aa
Fx xxfx
aa
−
−
 
+= +++++=
 
 

(7)
Then the unity shifted
( )
fx
can be analyzed by algebraic method using following necessary and sufficient
condition for stab ility. The Proof for the above equations is discussed by Jury in [26] [27].
3.2. Proposed Necessary Conditions
The Unity shifted equivalent one dimensional characteristic equation tested using following necessary condi-
tions [26] [28].
(i)
( )
00f>
(8)
(ii)
()( )
20forEvenf fx−> 

(iii)
()( )
20 forOddf fx−< 

(9)
3.3. Proposed Sufficient Conditions
Using the coe fficie nts o f unit y shifted
( )
fx
along with two triangular matrice s [26] [28] referred in Equations
(10) and (11) as given below.
[ ]
12 0
1
0
00
00 0
nn n
nn
n
n
n
aa aa
aa
a
Xa
a
−−
−
⋅ ⋅⋅


⋅ ⋅⋅⋅


⋅ ⋅⋅⋅

=⋅⋅⋅


⋅⋅⋅⋅ ⋅⋅ ⋅

⋅⋅⋅⋅ ⋅⋅ ⋅


⋅⋅⋅⋅ ⋅⋅

(10)

P. Ramesh
713
[ ]
12 0
123
23
0
0
00
0
0
0
00 0
nn n
nnn
nn
aa aa
aaa
aa
Y
a
−−
−−−
−−
⋅⋅⋅


⋅⋅⋅


⋅⋅⋅

=⋅⋅ ⋅⋅⋅⋅


⋅ ⋅ ⋅⋅⋅⋅

⋅ ⋅ ⋅⋅⋅⋅


⋅⋅⋅

(11)
Adding
[ ]
X
and
[]
Y
to the above equation, a square matrix is formed as in Equation (12).
[][][ ]
H XY= +
(12)
This square matrix
[ ]
S
is said to be positive inner wise when all the determinants with the center element
[ ]
S
and to be scram bled outwards and must be pos itiv e [23]. This procedure is used for testing the sufficient condition
that
1x<
. In Jury method [22] two matrices
1
H XY= +
and
2
H XY= −
were formed to determine inner
dete rmina nts start ing with t he centr e element s and pro ceed ing outwar ds up to the entire matrix are po sitive. The
proposed scheme accounts all the coefficient of the characteristic equation in order to form the matrix followed
by applying left shifting and right shifting principle to form X and Y matrix respectively. The single square ma-
trix
H XY= +
has been constructed from X and Y matrix with respect to Jury’s proposal in [26], hence this
proposed method reduces the number of computations compared to jury method [22].
4. Illustrations
Example 1: [3]
( )
1 1213123
15f ZZZZZZZZZ=+++ +
Convert the two dimensional equation into one dimensional equation
( )
1 12 13 123
11 15
1fZ Z ZZZZZZZ =+++ +
Put
123
ZZZZ== =
( )
223
11 1 5
1fZZZZZ =++++
( )
32
25fZZ ZZ=+++
For unity s hifti ng the unit circle to left half o f Z-plane
( )
1XZ= −
put
1ZX=+
.
()()( )( )
32
111215fX XXX+=+ +++++
()( )
32
14710 0fXfXXXX+==+++ =
Necessary Conditions:
1)
( )()
0100satisfiedf= >
2)
( )()
240 satisfiedf−=>
Sufficient conditions:
To check the requirements of sufficiency, we construct X and Y
147 1014710
0 14747100
,
0 0 1471000
0001 10000
XY
 
 
 
= =
 
 
 

P. Ramesh
714
Add matrix X and Y
2814 20
48 147
7 1014
10 001
H XY



= +=



The determinants are
( )
2
8 140.1320Not satisfied
10 1
∇==− <
( )
4
2814 20
48 147174240 satified
7 1014
10 001
∇== >
The given system satisfied the all necessar y condition but not satisfied the one sufficient condition therefore
the system is u nstable .
Example 2: [25]
()
34144 2
5fZZZZZZz=++ ++
Convert the multi-dimensional equation into one-dimensional equation
( )
341442
1111
5fZ ZZzZZZ =+ +++
Put
1234
ZZZZZ = == =
( )
22
11 11
5fZZZZZ =+++ +
( )
2
5 22 fZ Z Z= ++
For unity s hifti ng the unit circle put
1ZX=+
.
( )()()
2
151212fX XX+=+ +++
()( )
15 1290fX fXX+ ==++=
Necessary conditions:
1)
( )()
090 satisfiedf= >
2)
( )()
250 satisfiedf−=>
Sufficient conditions:
To check the requirements of sufficiency, we construct X and Y and Add matrix X and Y
5 1295129102418
0512 ,1290 ,121412
00 59 00905
XYH XY
 
 
===+=
 
 
 
The determinants are
( )()
13
102418
140satisfied,1214124160Not satified
905
∇= >∇==−<
The given syste m satisfied the all necessar y conditio ns but not satis fied the one su fficient condition therefore
the system is u nstable .

P. Ramesh
715
Example 3: [13]
Let
()
222
12 32 1211223
222 22
123123132 3
1 20.3340.0445
0.7322 23.024
f ZZZZZZZZZZZZ
ZZZZZZZ ZZ Z
=−+ ++−+−
+−+ −
Convert the multi-dimensional equation into one-dimensional equation
( )
222222 22
1232 12123
11223123132 3
2 0.3340.04450.732223.0241
1fZ Z ZZZZZZZZ
ZZZ ZZZZZ ZZ ZZ
=−+ ++−+−+−+−
Put
123
ZZZZ== =
( )
22 2 333444
20.3 340.0445 0.7322 2 3.024 1
1fZ ZZZZ Z ZZZZZZ =−+++− +−+−+−
( )
43 2
1.71.040.268 0.024fZ ZZZZ=−+−+
For unity s hifting the unit circle put
1ZX=+
.
() ()()( )( )
43 2
111.711.0410.2681 0.024fX XXXX+=+− +++−++
()( )
43 2
12.31.940.7120.096 0fXfXXXXX+==++++=
Necessary conditions:
1)
( )()
00.0960 satisfiedf= >
2)
( )()
2 4.032 satisfiedf−==
Sufficient conditions:
To check the requirements of sufficiency, we construct X and Y
1 2.31.94 0.712 0.09612.31.940.712 0.096
012.31.940.7122.31.940.7120.0960
, 0012.31.941.940.712 0.09600
00012.30.7120.0960 0 0
0000 10.0960 0 0 0
XY
 
 
 
 
= =
 
 
 
 
Add matrix X and Y
24.63.881.424 0.192
2.32.943.012 2.036 0.712
1.940.712 1.0962.31.94
0.712 0.096012.3
0.096 0001
H XY




= +=




The determinants are
()
1
1.0960 satisfied ∇= >
( )
3
2.943.012 2.036
0.7121.0962.31.52840 satified
0.096 01
∇== >
( )
5
24.63.881.424 0.192
2.32.943.012 2.036 0.712
0.026210satisfied 1.940.712 1.0962.31.94
0.712 0.096012.3
0.096 0001
∇== >

P. Ramesh
716
The given system satisfies the all necessary and sufficient conditions therefore the system is stable.
5. Conclusion
In this paper, an overview of the most important results for the stability of m-D discrete systems was made.
Various approaches to this important problem for the analysis and design of m-D discrete systems have been
studied. A simple and direct method for stability testing of m-dimensional linear discrete system has been pro-
posed. The implementation of the method has been discussed and compared to jury [22] method which r equir es
minimum arithmetic opera tion and the results illustr a te d by three examples.
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