New Approach to Modified Schur-Cohn Criterion for Stability Analysis of a Discrete Time System

National Conference on Soft Computing and Machine learning for Signal processing,

Control, power and Telecommunications, NCSC-2006

255

Abstract-- In this paper a new approach to obtain i

α in modified

Schur-Cohn criterion for stability analysis of discrete time

invariant system is presented. Problem is formulated in the form

of an array. The array suggested at any step of iteration provides

values ofi

α ,1( )

iF z

−

, ( )i

F z , ratio 1

1 ( ) ( )i i

F z F z−

+and

information for necessary and sufficient condition for roots of

characteristic equation of discrete time invariant system to lie

inside the unit circle.

Index Terms-- Discrete time invariant system, modified Schur –

Cohn criterion, system stability.

I. INTRODUCTION

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

discrete time system using Routh-Hurwitz criterion and Jury’s

criterion [1, 2]. However if the characteristic equation is

available for a continuous time system, a suitable bi–linear

transformation can be applied to obtain transfer function of the

system in z – domain. Hence characteristic equation in z –

domain is obtained. For the system to be stable all the roots of

the characteristic equation must lie inside the unit circle [1, 2,

3]. One of the traditional methods for this is modified Schur–

Cohn criterion which provides information about the relative

position of the roots [3]. It is well known that the modified

Schur–Cohn criterion basically deals with the algebraic

manipulations involving ratio of 1( )F z−

to ( )F z , where

( )F z represents a polynomial of nth

order in z, when equated

to zero gives characteristic equation of the system.

The present paper proposes a new and quite simpler approach

to modified Schur–Cohn. Avoiding the traditional approach

based on iterative methods the present paper basically

formulates the problem in the form of an array like that for

Chen – Chang array and Jury’s criterion. The detailed process

of formulation is presented in the next section.

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: [email protected]).

II. PROBLEM FORMULATION

The approach proposed here become clear only when the

traditional approach for modified Schur–Cohn criterion is

explained first.

Modified Schur–Cohn Criterion for stability analysis of

discrete time in-variant system:

For a discrete time invariant system of nth

order characteristic

equation is given as 1 ( ) ( ) 0G z F z+ = =

1

n 1 1 0( ) a ........n n

nF z z a z a z a

−

−= + + + + ---- (1)

Inverse of ( )F z is given as

1 1( ) ( )nF z z F z− −

=

1 1

0 1 1( ) a ........n n

n nF z z a z a z a

− −

−= + + + + ---- (2)

Now dividing equation (2) by equation (1) 11

10

( )( )

( ) ( )

F zF z

F z F zα

−−

= + ---- (3)

Order of 1

1 ( )F z−

is n – 1 i.e. one less than ( )F z or1( )F z

−

.

Relation given by equation (3) gives us iterative relation 1 1

1( ) ( )

( ) ( )

i ii

i i

F z F z

F z F zα

− −

+= + ---- (4)

where i = 1, 2, ---------n-2

Order of 1

1 ( )i

F z−

+is one less than order of ( )

iF z

Modified Schur-Cohn criterion

Necessary and sufficient condition for the system to be stable

is

1. (1) 0F >

2. ( 1) 0F − > for n even; ( 1) 0F − < for n odd

3. | | 1i

α < , i = 0, 1, 2 …., n-2.

Hence total number of conditions to be satisfied is (n+1). If

any of the conditions fails to satisfy, the system is said to be

unstable. The recursive relation (4) requires reduction of 1( )

iF z

−

to lower order 1

1 ( )i

F z−

+so that

iα can be determined

in each step.

Ritesh Kumar Keshri

New Approach to Modified Schur-Cohn

Criterion for Stability Analysis of a

Discrete time System

T



256

A. The new approach

In present paper all the steps to determine i

α are brought to

an array form. Each step gives the values of the coefficient

of1

1 ( )i

F z−

+.

iα is given by the ratio of first and the last term of

each step hence | |i

α can be determined. Number of terms in

each of the step is one less the than previous step. Elements

,i ja of each step can be determined by the relation

,0 ,i i i n ia aα

−= ---- (5)

( 1), ,( 1) ,( 1 )i j i j i i n i ja a aα

+ + − − −= − × ---- (6)

0,1, 2,.....( 2); 0,1,2,.....( )i n j n i= − = −

Where i is represents row number or iteration number, ,i ja is

(i, j)th element of the array, in other words ,i ja is the

coefficient of zj in

1( )i

F z−

. Number of rows in the formed

array will be n-1. The array formed is shown in Table-1.

Derivation of the iterative relation for the formation of

suggested array and determination of i

α is detailed in

Appendix – I.

TABLE I

ARRAY FORMED TO DETERMINEi

α IN MODIFIED SCHUR-COHN METHOD FOR

ANALYSIS

B. Code for the formation of suggested array

( 0; 1; )for i i n i= < − + +

{

,0

,

i

i

i n i

a

aα

−

= ;

( 0; ; )for j j n i j= ≤ − + +

1, , 1 ,( 1)i j i j i i n i ja a aα

+ + − − −= − × ;

}

Statement ( 0; 1; )for i i n i= < − + + means loop start at i=0

and continues till i<n-1 with the increment of one.

C. Examples

Example 1: Test the stability of system by modified Schur-

Cohn criterion. F (z) = z4 – 2 z

3 + 1.5 z

2 – 0.1 z - 0.02

Solution: F (1) = 0.380 > 0 Satisfied

F (-1) = 4.580 > 0 Satisfied as n is Even.

Determination ofi

α ,

The following array has been formed by the suggested

approach

Calculated value of |0

α | = 0.02 < 1 Satisfied

|1

α | = 0.14 < 1 Satisfied

|2

α | = 1.275 > 1 not satisfied

Hence, the total number of conditions satisfied = 4

Number of conditions to be satisfied for stability

= n + 1 = 5, so system is unstable.

Calculations involved in determining elements of the array as

per the suggested approach

0,0a = -0.02,

0,1a = - 0.1,

0,2a = 1.5,

0,3a = -2,

0,4a = 1,

0,0

0

0,4

0.020.02

1

a

aα

−

= = −=

1,0 0,1 0 0,30.01 ( 0.02) ( 2)a a aα= − = − − − × −

= -0.14

1,1 0,2 0 0,21.5 ( 0.02) 1.5a a aα = − − ×= − = 1.53

1,2 0,3 0 0,12 ( 0.02) ( 1)a a aα = − − − × −= − = -2.002

1,3 0,4 0 0,0( 0.02) ( 0.02)1a a aα − × −= −= − = 1.00

1,0

1

1,3

0.140.14

1

a

aα

−

= = −=

Similarly elements of 3rd

row can be obtained.

Example 2: F (z) = 8 z4 + 4 z

3 + 2 z

2 + 4 z Test stability.

F (1) = 18 > 0 Satisfied

F (-1) = 2 > 0 and n=4 is even; Satisfied

Array is

|0

α | = 0 < 1, |1

α | = 0.5 <1, |2

α | = 0 <1

As all the 5 conditions are satisfied so system is stable.



257

D. Results of the computer program for the modified Schur–

Cohn criterion by suggested approach

1. Enter order of the system : 4

Enter the coefficients a[j]’s:

a[0] = -.02

a[1] = -.1

a[2] = 1.5

a[3] = -2

a[4] = 1

F (1) = 0.380 > 0.....Satisfied

F (-1) = 4.580 & n = 4 ....satisfied

a[2]=1.275120 !< 1 so |a[2]| < 1 ..Is not satisfied

Values of alpha[i]'s calculated are:

α[0] = -0.020000

α[1] = -0.140056

α[2] = 1.275120

No. of conditions satisfied: 4

No. of conditions to be satisfied for stable system: 5

Hence system is unstable...

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



a[0] = 0

a[1] = 4

a[2] = 2

a[3] = 4

a[4] = 8

F (1) = 18.000 > 0.....Satisfied

F (-1) = 2.000 & n = 4 ....satisfied


α[0] = 0.000000

α[1] = 0.500000

α[2] = 0.000000



Hence system is stable...

Formed Array is:

Inv F0(z) 0.000 4.000 2.000 4.000 8.000 α[0] =0.000

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

To Exit: Enter 1-9, To Continue: Enter 0:0



a[0] = -0.368

a[1] = 7.7

a[2] = 5.94

a[3] = 1

F (1) = 14.272 > 0.....Satisfied

F (-1) = -3.128 & n = 3 ....satisfied

α[1]=11.434414 !< 1 so |α[1]| < 1 ...is not satisfied


α[0] = -0.368000

α[1] = 11.434414



Hence system is unstable...

Formed Array is:

Inv F0(z) -0.368 7.700 5.940 1.000 α[0] = -0.368

Inv F1(z) 9.886 8.774 0.865 α[1] = 11.434

To Exit: Enter 1-9, To Continue: Enter 0: 1

III. COMPARISION

Comparison of suggested array with that for Chen - Chang’s

criterion TABLE II

ARRAY FORMED OF CHEN CHAN’S STABILITY ANALYSIS OF DISCRETE TIME

INVARIANT SYSTEM

From table – 1 and table – 2 it is clear that s-rows elements in

Chen - Chang’s array is same as that of rows of suggested

array for modified Schur–Cohn criterion, i.e. elements of s-

rows are coefficients of1( )

iF z

−

. Elements of t-rows are

coefficients of ( )i

F z . For stability in Chen-Chang criterion 1st

element of all t-rows (,i n ia

−, I = 0, 1, 2 …) must be positive

whereas in modified Schur–Cohn criterion magnitude of the

ratio of 1st and last element (n-i)

th , of

1( )i

F z−

(,0

,

i

i

i n i

a

aα

−

= ,

i = 0, 1, 2, … n-2) must be less than unity. The two criterion

are deferring only at this step otherwise both criterion are

same.

IV. CONCLUSION

An alternative approach to determinei

α for stability analysis

of discrete time invariant system has been proposed. The

proposed approach reduces efforts and time for computation.

Algorithm suggested here is also valid in determining s – rows



258

and t – rows elements of Chen Chang’s array [Table – II], for

stability analysis of discrete time invariant system and that of

Routh-Hurwitz array to stability analysis of continuous time

system. Further table – II also provides information for

necessary and sufficient condition for the roots of

0 ( ) 0F z = to lie inside the unit circle i.e. | | 1z < [1].

V. APPENDIX

Derivation of iterative relation for the formation of suggested

array and determination of i

α

Let the characteristic equation for discrete time invariant

system is given by

( ) 1 ( ) 0F z G z= + =

1 2

1 2 1 0( ) ........

n n

n nF z a z a z a z a z a

−

−= + + + + + ---- (1)

Writing 0( ) ( )F z F z=

1

0 0, 0, 1 0,1 0,0( ) ........

n n

n nF z a z a z a z a

−

−= + + + + ---- (2)

as 1 1( ) ( )

nF z z F z

− −

=

Therefore 1 1

0 0,0 0,1 0, 1 0,( ) ........

n n


− −

−= + + + + ---- (3)

For 0th

row Dividing (3) and (2) we get 11

0,0 0,1 0, 1 0,0

1

0 0, 0, 1 0,1 0,0

........( )

( ) ........

n n

n n

n n

n n

a z a z a z aF z

F z a z a z a z a

−−

−

−

−

+ + + +

=

+ + + +

---- (4)

1

0,0 0,0 0,1 0, 1 0, 0,0

1

0, 0, 0, 1 0,1 0,0 0,

........

........

n n

n n

n n

n n n n

a a z a z a z a a

a a z a z a z a a

−

−

−

−

+ + + +

+ + + +

= + −

---- (5)

Where

0,0

0

0,n

a

aα = ;

0,0

1,0 0,1 0, 1

0,

n

n

aa a a

a−

= − ;0,0

1,1 0,2 0, 2

0,

n

n

aa a a

a−

= − ;

0,0

1,2 0,3 0, 3

0,

n

n

aa a a

a−

= −

…;0,0

1, 2 0, 1 0,1

0,

n n

n

aa a a

a− −

= − ; 0,0

1, 1 0, 0,0

0,

n n

n

aa a a

a−

= −

in general 1, 0,( 1) 0 0,( 0 1)j j n j

a a aα+ − − −

= − × ---- (6)

Therefore (4) can be written as 1 1

0 1

0

0 0

( ) ( )

( ) ( )

F z F z

F z F zα

− −

= + ---- (7)

Where 1 1 2

1 1,0 1,1 1, 2 1, 1( ) .....

n n


− − −

− −= + + + + ---- (8)

Therefore 1 2

1 1, 1 1, 2 1,1 1,0( ) .....

n n


− −

− −= + + + + ---- (9)

For 1st row dividing (8) by (9) we get

1 21

1,0 1,1 1, 2 1, 11

1 2

1 1, 1 1, 2 1,1 1,0

....( )

( ) ....

n n

n n

n n

n n

a z a z a z aF z

F z a z a z a z a

− −−

− −

− −

− −

+ + + +

=

+ + + +

---- (10)

Like (4) we can reduce (10) to the form 2 3

2,0 2,1 2, 3 2, 2

1 1 2

1, 1 1, 2 1,1 1,0

1

1

1

....

....

( )

( )

n n

n n

n n

n n

a z a z a z a

a z a z a z a

F z

F zα

− −

− −

− −

− −

−+ + + +

= +

+ + + +

---- (11)

where,

1,0

1

1, 1n

a

aα

−

= ;1,0

2,0 1,1 1, 2

1, 1

n

n

aa a a

a−

−

= − ;

1,0

2,1 1,2 1, 3

1, 1

n

n

aa a a

a−

−

= − ; 1,0

2,2 1,3 1, 4

1, 1

n

n

aa a a

a−

−

= −

…;1,0

2, 3 1, 2 1,1

1, 1

n n

n

aa a a

a− −

−

= − ; 1,0

2, 2 1, 1 1,0

1, 1

n n

n

aa a a

a− −

−

= −

So we can write 2, 0,( 1) 1 1,( 1 1)j j n j

a a aα+ − − −

= − × ---- (12)

Thus (10) can be written as

1 1

1 2

1

1 1

( ) ( )

( ) ( )

F z F z

F z F zα

− −

= +

Here 1 2 3

2 2,0 2,1 2, 1 2, 2( ) ....

n n


− − −

− −= + + + + ---- (13)

Therefore 2 3

2 2, 2 2, 3 2,1 2,0( ) ....

n n


− −

− −= + + + + ---- (14)

We can proceed further in similar way for determination of

iα and the coefficients of

1

1 ( )i

F z−

+for i

th row and can get a

general iterative relation 1 1

1( ) ( )

( ) ( )

i i

i

i i

F z F z

F z F zα

− −

+

= +

Where

,0

,

i

i

i n i

a

aα

−

=

1, , 1 1i j i j i n i ja a aα

+ + − − −= − ×

0,1, 2,.....( 2); 0,1,2,.....( )i n j n i= − = − ---- (15)

(15) is a required iterative relation using which we can bring

our problem into an array form as in Table -1, suggesting

alternative approach to classical one.

VI. ACKNOWLEDGMENT

The author gratefully acknowledge Dr. A. B. Chattopadhyay

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.

Jamshedpur, India whose class lecture motivated him for

simplification of the discussed criterion.



259

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.

Documents

New Approach to Modified Schur-Cohn Criterion for Stability Analysis of a Discrete Time System