This document describes a new approach to the modified Schur-Cohn criterion for stability analysis of discrete time invariant systems. The approach formulates the problem in the form of an array, similar to Chen-Chang array and Jury's criterion. The array provides values for the coefficients of the inverse polynomials at each step of iteration, as well as the ratios αi, which must satisfy certain conditions for stability. Examples are provided to demonstrate the application of the new approach. The approach reduces computational efforts compared to traditional methods and provides the necessary information for stability analysis in a concise manner.

1. National Conference on Soft Computing and Machine learning for Signal processing,
Control, power and Telecommunications, NCSC-2006
New Approach to Modified Schur-Cohn
Criterion for Stability Analysis of a
Discrete time System
Ritesh Kumar Keshri
Abstract-- In this paper a new approach to obtain αi in modified II. PROBLEM FORMULATION
Schur-Cohn criterion for stability analysis of discrete time The approach proposed here become clear only when the
invariant system is presented. Problem is formulated in the form traditional approach for modified Schur–Cohn criterion is
of an array. The array suggested at any step of iteration provides explained first.
−1 −1 Modified Schur–Cohn Criterion for stability analysis of
values of α i , Fi ( z ) , Fi ( z ) , ratio Fi +1 ( z ) Fi ( z ) and
discrete time in-variant system:
information for necessary and sufficient condition for roots of
characteristic equation of discrete time invariant system to lie For a discrete time invariant system of nth order characteristic
inside the unit circle. equation is given as 1 + G ( z ) = F ( z ) = 0
F ( z ) = a n z n + an−1 z n−1 + ........ + a1 z + a0 ---- (1)
Index Terms-- Discrete time invariant system, modified Schur –
Cohn criterion, system stability. Inverse of F ( z ) is given as
I. INTRODUCTION F −1 ( z ) = z n F ( z −1 )
F −1 ( z ) = a 0 z n + a1 z n −1 + ........ + a n −1 z + a n ---- (2)
T here are literatures available in the area of stability
analysis of a discrete time invariant system [1, 2, 3]. The
references [1, 2, 3], basically discuss the stability analysis of
Now dividing equation (2) by equation (1)
F −1 ( z ) F −1 ( z )
discrete time system using Routh-Hurwitz criterion and Jury’s = α0 + 1 ---- (3)
criterion [1, 2]. However if the characteristic equation is F ( z) F ( z)
−1
available for a continuous time system, a suitable bi–linear Order of F1 ( z ) is n – 1 i.e. one less than F ( z ) or F −1 ( z ) .
transformation can be applied to obtain transfer function of the
Relation given by equation (3) gives us iterative relation
system in z – domain. Hence characteristic equation in z –
−1 −1
domain is obtained. For the system to be stable all the roots of Fi ( z ) Fi +1 ( z )
= αi + ---- (4)
the characteristic equation must lie inside the unit circle [1, 2, Fi ( z ) Fi ( z )
3]. One of the traditional methods for this is modified Schur–
where i = 1, 2, ---------n-2
Cohn criterion which provides information about the relative −1
position of the roots [3]. It is well known that the modified Order of Fi +1 ( z ) is one less than order of Fi ( z )
Schur–Cohn criterion basically deals with the algebraic Modified Schur-Cohn criterion
−1 Necessary and sufficient condition for the system to be stable
manipulations involving ratio of F ( z ) to F ( z ) , where
th is
F ( z ) represents a polynomial of n order in z, when equated
1. F (1) > 0
to zero gives characteristic equation of the system.
The present paper proposes a new and quite simpler approach 2. F (−1) > 0 for n even; F (−1) < 0 for n odd
to modified Schur–Cohn. Avoiding the traditional approach 3. | α i |< 1 , i = 0, 1, 2 …., n-2.
based on iterative methods the present paper basically
Hence total number of conditions to be satisfied is (n+1). If
formulates the problem in the form of an array like that for
Chen – Chang array and Jury’s criterion. The detailed process any of the conditions fails to satisfy, the system is said to be
of formulation is presented in the next section. unstable. The recursive relation (4) requires reduction of
Fi −1 ( z ) to lower order Fi +1 ( z ) so that α i can be determined
−1
in each step.
Ritesh Kumar Keshri is 2nd semester M. Tech (Power System) student at NIT
Jamshedpur, India and is Ex – lecturer EE dept. M.P.E.C. Kanpur, India
(E-mail: riteshkeshri@gmail.com).
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Control, power and Telecommunications, NCSC-2006
A. The new approach Solution: F (1) = 0.380 > 0 Satisfied
F (-1) = 4.580 > 0 Satisfied as n is Even.
In present paper all the steps to determine αi are brought to
Determination of α i ,
an array form. Each step gives the values of the coefficient
−1 The following array has been formed by the suggested
of Fi +1 ( z ) . αi is given by the ratio of first and the last term of
approach
each step hence | α i | can be determined. Number of terms in
each of the step is one less the than previous step. Elements
ai , j of each step can be determined by the relation
α i = ai ,0 ai ,n−i ---- (5)
a(i +1), j = ai ,( j +1) − α i × ai ,( n −1−i − j ) ---- (6)
i = 0,1, 2,.....(n − 2); j = 0,1, 2,.....(n − i ) Calculated value of | α 0 | = 0.02 < 1 Satisfied
Where i is represents row number or iteration number, ai , j is | α1 | = 0.14 < 1 Satisfied
(i, j)th element of the array, in other words ai , j is the | α 2 | = 1.275 > 1 not satisfied
−1
coefficient of zj in Fi ( z ) . Number of rows in the formed Hence, the total number of conditions satisfied = 4
Number of conditions to be satisfied for stability
array will be n-1. The array formed is shown in Table-1. = n + 1 = 5, so system is unstable.
Derivation of the iterative relation for the formation of Calculations involved in determining elements of the array as
suggested array and determination of αi is detailed in per the suggested approach
Appendix – I. a0,0 = -0.02, a0,1 = - 0.1, a0,2 = 1.5, a0,3 = -2, a0,4 = 1,
TABLE I a0,0 −0.02
α0 = = = −0.02
ARRAY FORMED TO DETERMINE α i IN MODIFIED SCHUR-COHN METHOD FOR a0,4 1
ANALYSIS
a1,0 = a0,1 − α 0 a0,3 = −0.01 − ( −0.02) × ( −2)
= -0.14
a1,1 = a0,2 − α 0 a0,2 = 1.5 − ( −0.02) × 1.5 = 1.53
a1,2 = a0,3 − α 0 a0,1 = −2 − ( −0.02) × ( −1) = -2.002
a1,3 = a0,4 − α 0 a0,0 = 1 − ( −0.02) × ( −0.02) = 1.00
a1,0 −0.14
α1 = = = −0.14
a1,3 1
Similarly elements of 3rd row can be obtained.
B. Code for the formation of suggested array Example 2: F (z) = 8 z4 + 4 z3 + 2 z2 + 4 z Test stability.
for (i = 0; i < n − 1; i + + ) F (1) = 18 > 0 Satisfied
{ F (-1) = 2 > 0 and n=4 is even; Satisfied
ai ,0 Array is
αi = ;
ai ,n −i
for ( j = 0; j ≤ n − i; j + +)
ai +1, j = ai , j +1 − α i × ai ,( n −i − j −1) ;
}
Statement for (i = 0; i < n − 1; i + + ) means loop start at i=0
and continues till i<n-1 with the increment of one.
| α 0 | = 0 < 1, | α1 | = 0.5 <1, | α 2 | = 0 <1
C. Examples
Example 1: Test the stability of system by modified Schur- As all the 5 conditions are satisfied so system is stable.
Cohn criterion. F (z) = z4 – 2 z3 + 1.5 z2 – 0.1 z - 0.02
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Control, power and Telecommunications, NCSC-2006
D. Results of the computer program for the modified Schur– F (-1) = -3.128 & n = 3 ....satisfied
Cohn criterion by suggested approach α[1]=11.434414 !< 1 so |α[1]| < 1 ...is not satisfied
1. Enter order of the system :4 Values of alpha[i]'s calculated are:
Enter the coefficients a[j]’s: α[0] = -0.368000
a[0] = -.02 α[1] = 11.434414
a[1] = -.1 No. of conditions satisfied: 3
a[2] = 1.5 No. of conditions to be satisfied for stable system: 4
a[3] = -2
Hence system is unstable...
a[4] = 1
Formed Array is:
F (1) = 0.380 > 0.....Satisfied Inv F0(z) -0.368 7.700 5.940 1.000 α[0] = -0.368
F (-1) = 4.580 & n = 4 ....satisfied Inv F1(z) 9.886 8.774 0.865 α[1] = 11.434
a[2]=1.275120 !< 1 so |a[2]| < 1 ..Is not satisfied To Exit: Enter 1-9, To Continue: Enter 0: 1
Values of alpha[i]'s calculated are:
α[0] = -0.020000 III. COMPARISION
α[1] = -0.140056
Comparison of suggested array with that for Chen - Chang’s
α[2] = 1.275120
criterion
No. of conditions satisfied: 4 TABLE II
No. of conditions to be satisfied for stable system: 5 ARRAY FORMED OF CHEN CHAN’S STABILITY ANALYSIS OF DISCRETE TIME
Hence system is unstable... INVARIANT SYSTEM
Formed Array is:
Inv F0(z) -0.020 -0.100 1.500 -2.000 1.000 α[0] = -0.020
Inv F1(z) -0.140 1.530 -2.002 1.000 α[1] = -0.140
Inv F2(z) 1.250 -1.788 0.980 α[2] = 1.275
To Exit: Enter 1-9, To Continue: Enter 0: 0
2. Enter order of the system : 4
Enter the coefficients a[j]’s:
a[0] = 0
a[1] = 4
a[2] = 2
a[3] = 4
a[4] = 8
F (1) = 18.000 > 0.....Satisfied From table – 1 and table – 2 it is clear that s-rows elements in
F (-1) = 2.000 & n = 4 ....satisfied Chen - Chang’s array is same as that of rows of suggested
Values of alpha[i]'s calculated are: array for modified Schur–Cohn criterion, i.e. elements of s-
α[0] = 0.000000 −1
α[1] = 0.500000 rows are coefficients of Fi ( z ) . Elements of t-rows are
α[2] = 0.000000 coefficients of Fi ( z ) . For stability in Chen-Chang criterion 1st
No. of conditions satisfied: 5
element of all t-rows ( ai , n −i , I = 0, 1, 2 …) must be positive
No. of conditions to be satisfied for stable system: 5
Hence system is stable... whereas in modified Schur–Cohn criterion magnitude of the
Formed Array is: −1 ai ,0
Inv F0(z) 0.000 4.000 2.000 4.000 8.000 α[0] =0.000 ratio of 1st and last element (n-i)th , of Fi ( z ) ( α i = ,
ai ,n −i
Inv F1(z) 4.000 2.000 4.000 8.000 α[1] = 0.500
Inv F2(z) 0.000 3.000 6.000 α[2] = 0.000 i = 0, 1, 2, … n-2) must be less than unity. The two criterion
To Exit: Enter 1-9, To Continue: Enter 0:0 are deferring only at this step otherwise both criterion are
same.
3. Enter order of the system :3
Enter the coefficients a[j]’s:
IV. CONCLUSION
a[0] = -0.368
a[1] = 7.7 An alternative approach to determine α i for stability analysis
a[2] = 5.94 of discrete time invariant system has been proposed. The
a[3] = 1 proposed approach reduces efforts and time for computation.
F (1) = 14.272 > 0.....Satisfied Algorithm suggested here is also valid in determining s – rows
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Control, power and Telecommunications, NCSC-2006
and t – rows elements of Chen Chang’s array [Table – II], for a1,0 z n −1 + a1,1 z n − 2 + .... + a1,n − 2 z + a1,n −1
F1−1 ( z )
stability analysis of discrete time invariant system and that of = ---- (10)
Routh-Hurwitz array to stability analysis of continuous time F1 ( z ) a1, n −1 z n −1 + a1, n − 2 z n − 2 + .... + a1,1 z + a1,0
system. Further table – II also provides information for Like (4) we can reduce (10) to the form
necessary and sufficient condition for the roots of n−2 n −3
F1−1 ( z ) a2,0 z + a2,1 z + .... + a2, n −3 z + a2, n − 2
F0 ( z ) = 0 to lie inside the unit circle i.e. | z |< 1 [1]. = α1 +
F1 ( z ) a1, n −1 z n −1 + a1, n − 2 z n − 2 + .... + a1,1 z + a1,0
V. APPENDIX ---- (11)
where,
Derivation of iterative relation for the formation of suggested a1,0 a1,0
array and determination of αi α1 = ; a2,0 = a1,1 − a1,n − 2 ;
a1, n −1 a1, n −1
Let the characteristic equation for discrete time invariant a1,0 a1,0
system is given by a2,1 = a1,2 − a1, n −3 ; a2,2 = a1,3 − a1, n − 4
F ( z) = 1 + G( z) = 0 a1, n −1 a1,n −1
F ( z ) = an z n + an −1 z n −1 + ........ + a2 z 2 + a1 z + a0 ---- (1) a1,0 a1,0
…; a2, n − 3 = a1, n − 2 − a1,1 ; a2,n − 2 = a1,n −1 − a1,0
Writing F ( z ) = F0 ( z ) a1, n −1 a1,n −1
F0 ( z ) = a0, n z n + a0, n −1 z n −1 + ........ + a0,1 z + a0,0 ---- (2) So we can write a2, j = a0,( j +1) − α1 × a1,( n −1− j −1) ---- (12)
−1
as F ( z ) = z F ( z )
n −1 F1−1 ( z ) F2−1 ( z )
Thus (10) can be written as = α1 +
Therefore F1 ( z ) F1 ( z )
−1 n n −1
F ( z ) = a0,0 z + a0,1 z + ........ + a0,n −1 z + a0,n ---- (3) Here
0
For 0th row Dividing (3) and (2) we get F2−1 ( z ) = a2,0 z n − 2 + a2,1 z n − 3 + .... + a2, n −1 z + a2, n − 2 ---- (13)
F0−1 ( z ) a0,0 z n + a0,1 z n −1 + ........ + a0,n −1 z + a0,n Therefore
= ---- (4) F2 ( z ) = a2,n − 2 z n − 2 + a2,n − 3 z n −3 + .... + a2,1 z + a2,0 ---- (14)
F0 ( z ) a0,n z n + a0, n −1 z n −1 + ........ + a0,1 z + a0,0
We can proceed further in similar way for determination of
a0,0 a0,0 z n + a0,1 z n −1 + ........ + a0,n −1 z + a0, n a0,0 −1
= + n n −1
− α i and the coefficients of Fi +1 ( z ) for ith row and can get a
a0, n a0,n z + a0,n −1 z + ........ + a0,1 z + a0,0 a0, n
general iterative relation
---- (5)
Fi −1 ( z ) −1
Fi +1 ( z )
Where = αi +
a0,0 a0,0 a0,0 Fi ( z ) Fi ( z )
α0 = ; a1,0 = a0,1 − a0,n −1 ; a1,1 = a0,2 − a0, n − 2 ; Where
a0,n a0,n a0, n
ai ,0
a0,0 αi =
a1,2 = a0,3 − a0,n − 3 ai ,n −i
a0, n
ai +1, j = ai , j +1 − α i × an −i − j −1
a0,0 a0,0
…; a1, n − 2 = a0, n −1 − a0,1 ; a1,n −1 = a0, n − a0,0 i = 0,1, 2,.....(n − 2); j = 0,1, 2,.....(n − i ) ---- (15)
a0, n a0,n (15) is a required iterative relation using which we can bring
in general a1, j = a0,( j +1) − α 0 × a0,( n − 0 − j −1) ---- (6) our problem into an array form as in Table -1, suggesting
alternative approach to classical one.
Therefore (4) can be written as
F0−1 ( z ) F1−1 ( z ) VI. ACKNOWLEDGMENT
= α0 + ---- (7)
F0 ( z ) F0 ( z ) The author gratefully acknowledge Dr. A. B. Chattopadhyay
Where and Dr. R. N. Mahanty of N.I.T. Jamshedpur, India for their
valuable suggestions and support, to Dr. A. K. Singh of N.I.T.
F1−1 ( z ) = a1,0 z n −1 + a1,1 z n − 2 + ..... + a1, n − 2 z + a1,n −1 ---- (8)
Jamshedpur, India whose class lecture motivated him for
Therefore simplification of the discussed criterion.
F1 ( z ) = a1, n −1 z n −1 + a1, n − 2 z n − 2 + ..... + a1,1 z + a1,0 ---- (9)
st
For 1 row dividing (8) by (9) we get
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5. National Conference on Soft Computing and Machine learning for Signal processing,
Control, power and Telecommunications, NCSC-2006
VII. REFERENCES
[1] Jonckheere Edmund and Ma Chingwo, “A further simplification of
Jury’s stability test”, IEEE Trans. Circuits and systems, vol. 36, No 3
pp 463 – 464, 1989
[2] [Nagrath, I. J. and M. Gopal, Control System Engineering, New Delhi:
New Age International Publishers, 2001.
[3] Kuo Benjamin C., Digital Control Systems, New York: Oxford
University Press, 1992.
VIII. BIOGRAPHIES
Ritesh Kr. Keshri (B’03- ) had been a full time
lecturer in M.P.E.C., Kanpur from 2003 to 2005, is
currently M. Tech Power system student at N.I.T.
Jamshedpur. He received his B Sc (Engg) Electrical
from N.I.T. Jamshedpur, India.
His field of interest includes Control theory,
Electric drives and Power electronics.
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