Research ArticleModified Schur-Cohn Criterion for Stability of Delayed Systems

Juan Ignacio Mulero-Martiacutenez

Departamento de Ingenierıa de Sistemas y Automatica Universidad Politecnica de CartagenaCampus Muralla del Mar 30203 Cartagena Spain

Correspondence should be addressed to Juan Ignacio Mulero-Martınez juanmuleroupctes

Received 5 June 2014 Revised 11 September 2014 Accepted 14 September 2014

Academic Editor Hak-Keung Lam

Copyright copy 2015 Juan Ignacio Mulero-Martınez This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A modified Schur-Cohn criterion for time-delay linear time-invariant systems is derived The classical Schur-Cohn criterion hastwo main drawbacks namely (i) the dimension of the Schur-Cohn matrix generates some round-off errors eventually resulting ina polynomial of s with erroneous coefficients and (ii) imaginary roots are very hard to detect when numerical errors creep in Incontrast to the classical Schur-Cohn criterion an alternative approach is proposed in this paper which is based on the application oftriangular matrices over a polynomial ring in a similar way as in the Jury test of stability for discrete systemsThe advantages of theproposed approach are that it halves the dimension of the polynomial and it only requires seeking real roots making this modifiedcriterion comparable to the Rekasius substitution criterion

1 Introduction

Stability theory for linear time invariant systems with delayshave long been in existence but it still remains an activearea of research with applications in power system controlbiology human-in-the-loop control neural networks con-trol in the context of automotive engines haptic control andsynthetic biology [1]

In this work we treat with linear time invariant time-delayed systems (LTI-TDS) with commensurate delaysdescribed by the state-space equation [2]

(119905) = 119860

0119909 (119905) +

119898

sum

119896=1

119860

119896119909 (119905 minus 119896120591) 119898 isin Z

+ (1)

where 119909 isin R119899 stands for the state vector 1198600 119860

119896isin R119899times119899 are

given system matrices and 120591 gt 0 is a delay time For thesesystems the characteristic quasi-polynomial 119901

119888(119904) possesses

the form of

119901

119888(119904 120591) =

119902

sum

119896=0

119886

119896(119904) 119890

minus119896120591119904 (2)

where 119902 is the degree of commensuracy in the dynamics and119886

119896isin R[119904] are polynomials of degree at most 119899 minus 119896

It is well known that the analysis of the stability of LTI-TDS lies on the root continuity argument that is the locationof the poles of 119901

119888varies continuously with respect to delay

This means that any root crossing from the left half plane tothe right half-plane will need to pass through the imaginaryaxis as a result the computation of crossing frequencies iscrucial when analyzing the stability and also for the plot ofthe root locus of 119901

119888as 120591 varies [3ndash9]

There are five criteria in the literature to compute delaymargins of general time-delayed systems (i) Schur-Cohn(Hermite matrix formulation) [2 10ndash12] (ii) eliminationof transcendental terms in the characteristic equation [13](iii) matrix pencil Kronecker sum method [2 11 12 14](iv) Kroneckermultiplication and elementary transformation[15] (v) Rekasius substitution [4 16 17] These criteria arebased on the determination of all the imaginary roots ofthe characteristic equation and require numerical procedureswhich result in vast range of precisions in finding pureimaginary roots of polynomials Some criteria show subtleinapplicabilities mainly from a numerical implementationviewpoint [18] In particular methods (i) (iii) and (iv) leadto polynomials of degree 2119902

2 while (ii) requires findingthe roots of a polynomial of degree 1199022

119902 The Kroneckermultiplication method generates the results with the smallest

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 846124 7 pageshttpdxdoiorg1011552015846124

2 Mathematical Problems in Engineering

number of significant digits among the above five methodsIn the literature the Rekasius substitution approach is themost attractive since the generated polynomial has degree 1199022much smaller than all the other criteria Another advantageof this method is that only real roots are searched for insteadof complex ones which is very crucial from a practical viewpoint since complex roots are difficult to compute whennumerical errors are dragged and propagated For a totalcomparison of these methods demonstrating their strengthsand weakness the reader is referred to [19]

This paper is focused on improving the Schur-Cohnprocedure so as to obtain the advantageous properties ofthe Rekasius substitution criterion Specifically we modifythe Schur-Cohn criterion to generate a real polynomial ofdegree 119902

2 whose real roots determine the pure imaginaryroots of the characteristic equationThe original Schur-Cohnprocedure computes the determinant of a partitioned matrix119860 = (

Λ1(119904) Λ

2(119904)

Λlowast

2(119904) Λlowast

1(119904)) where Λ

1 Λ

2isin C[119904] are appropriate

119902 times 119902 matrices over the ring of polynomials in the unknown119904 and Λ

lowast denotes the Hermitian (or conjugated complex) ofΛ The classical Schur-Cohn procedure for high dimensions(21199022 gt 10) becomes numerically unreliable due to repeatedround-off errors in the determinant expansion procedureunless the operation is performed using very large numberof significant digits This particular point alone brings aweakness from the numerical deployment viewpoint sincethe operation ultimately yields poor precision in determiningthe desired imaginary roots because of the accumulatednumerical errors the imaginary roots shape up in the formof 119904 = 119895120596 + 120598 120598 ≪ 1 and 119895 =

radicminus1 which leads the numerical

error in real parts to be at least in the order of 10 magnitudeslarger than the computational precision Therefore it isproblematic to decide whether these roots which are veryclose to the imaginary axis are really imaginary roots ornot As an attempt to overcome the problems of the Schur-Cohn method some authors have proposed a modificationbased on the determination of the real eigenvalues of aconstant matrix [11] By contrast our modification stemsfrom the applications of some transformations on119860 by usingtriangular matrices over the polynomial ring in a similar wayas in the Jury test of stability for discrete systems [20]

The remainder of the paper is structured as follows inSection 2 the basic idea of the modified stability criterionis explained and a numerical example is provided Finally asummary and an outlook are given in Section 3

2 Delayed Systems Schur-CohnModified Method

The stability of the system (1) is determined by its character-istic quasi-polynomial in 119904 [21]

119901

119888(119904 119890

minus119904120591 119890

minus119898119904120591) = det(119904119868 minus 119860

0minus

119898

sum

119896=1

119860

119896119890

minus119904119896120591)

=

119902

sum

119896=0

119886

119896 (119904) 119890

minus119896120591119904

(3)

119886

0 (119904) = 119904

119899+

119899minus1

sum

119894=0

119886

0119894119904

119894

119886

119896(119904) =

119899minus1

sum

119894=0

119886

119896119894119904

119894 119896 = 1 119902

(4)

The characteristic quasi-polynomial contains time delays ofcommensurate nature with degree 119902 that is 119901

119888depends on

119890

minus119896120591119904 for 119896 = 0 119902 with 119886

119902(119904) = 0 and rank(119860

119896) le 119902 le

119899 = rank(1198600) The starting-point of all the stability criteria is

the determination of the roots of 119901119888on the imaginary axis

In fact it is well known from the stability theory of LTI-TDS that the pure imaginary characteristic roots are the onlypossible transition points from stable to unstable behaviorand vice versa We want to determine all the imaginary rootsof 119901119888(119904 119890

minus119904120591 119890

minus119898119904120591) and to this end it is convenient to

introduce the variable 119911 = 119890

minus119904120591 andwrite119901119888as a polynomial in

two unknowns over the field of reals119901119888(119904 119911) = sum

119902

119894=0119886

119896(119904)119911

119896isin

R[119904 119911]We define the bivariate polynomial 119903 isin C[120596 119911] as

119903(120596 119911) = sum

119902

119894=0119887

119894(120596)119911

119894 such that 119887119894(120596) = 119886

119894(119895120596) isin C[120596] With

every bivariate polynomial over the complex field 119903(120596 119911) weassociate two triangular matrices over the polynomial ringC[120596]

119880

119902(119903 (120596))

=

(

(

119887

119902(120596) 119887

119902minus1(120596) 119887

119902minus2(120596) sdot sdot sdot 119887

1(120596)

0 119887

119902(120596) 119887

119902minus1(120596) sdot sdot sdot 119887

2(120596)

0 sdot sdot sdot 0 119887

119902(120596) 119887

119902minus1(120596)

0 sdot sdot sdot 0 0 119887

119902 (120596)

)

)

119871

119902(119903 (120596))

= (

0 sdot sdot sdot 0 0 119887

0 (120596)

0 sdot sdot sdot 0 119887

0 (120596) 119887

1 (120596)

0 119887

0 (120596) 119887

1 (120596) sdot sdot sdot 119887

119902minus2 (120596)

119887

0(120596) 119887

1(120596) 119887

2(120596) sdot sdot sdot 119887

119902minus1(120596)

)

(5)

For the sake of simplicity we write 119880

119902 119871

119902instead

of 119880

119902(119903(120596)) 119871

119902(119903(120596)) The Schur-Cohn criterion solves

the problem of computing the values of 120596 such that119901

119888(119895120596 119911 119911

119902) = 0 by multiplying 119901

119888(119895120596 119911 119911

119902) by 119911

minus119896

for 119896 = 0 119902minus1 and119901119888(119895120596 119911 119911

119902) by 119911119896 for 119896 = 1 119902

This leads to a system of 2119902 homogeneous equations in theunknowns 119911119896 119896 = minus119902 0 119902minus1which has the followingmatrix representation

(

119871

119902119880

119902119869

119869119880

119902119869119871

119902119869

)(

e1

e2

) = (

0

0

) (6)

where 119869 is the counteridentity matrix (ie an antidiagonalmatrix with ones) e

1= 119869119897

119879

119902minus1(119911) e2

= 119869119897

119879

119902minus1(119911

minus1)119911

minus1 and

Mathematical Problems in Engineering 3

119897

119879

119902minus1(119911) = (1 119911 119911

2 119911

119902minus1) the homogeneous system in (7)

is the starting point of the Schur-Cohn method If we makethe change of unknowns e

119894997891rarr 119869e119894and we take into account

the idempotence of 119869 (6) can be transformed into

(

119871

119902119880

119902

119869119880

119902119869119871

119902

)(

119869e1

119869e2

) = (

0

0

) (7)

It is easily verified that119880lowast119902119871

119902is of the same structure as119871

119902 and

in particular it is symmetric Therefore 119880lowast119902119871

119902= (119880

lowast

119902119871

119902)

119879

=

119871

lowast

119902119880

119902 As a consequence we obtain

(

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119868 119874

119874 119869

)(

119871

119902119880

119902

119869119880

119902119869119871

119902

)

= (

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119871

119902119880

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119874

119880

119902119871

119902

)

(8)

The homogeneous system (7) has a nontrivial solution solong as det( 119871119902 119880119902

119869119880119902119869119871119902

) = 0 We use the fact that

det((119871lowast

119902minus119880

lowast

119902

119874 119868

)(

119868 119874

119874 119869

))

= det(119871lowast

119902minus119880

lowast

119902119869

119874 119869

) = det (119871lowast119902119869) = det (119871

119902) (minus1)

119902

(9)

and taking determinants in (8) yields

det (119871119902) (minus1)

119902 det( 119871

119902119880

119902

119869119880

119902119869119871

119902

)

= det (119871119902) det (119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902)

(10)

As a result

det( 119871

119902119880

119902

119869119880

119902119869119871

119902

) = (minus1)

119902 det (119871lowast119902119871

119902minus 119880

lowast

119902119880

119902) (11)

Therefore the problem has been reduced to find the pureimaginary roots of det(119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902) which is just the

determinant of the Jury matrix 119869119902(119903(120596)) over C[120596] defined as

119869

119902(119903 (120596)) = (

119871

119902119880

119902

119880

119902119871

119902

) isin C2119902times2119902

[120596] (12)

This can be seen by a similar argument as above in (8)

(

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119871

119902119880

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119871

lowast

119902119880

119902minus 119880

lowast

119902119871

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119874

119880

119902119871

119902

)

(13)

and det(119869119902(119903(120596))) = det(119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902)

In the sequel we define the polynomial 120577(120596) =

det(119869119902(119903(120596))) and the set of zeros of a polynomial119901 is denoted

asZ(119901)

Proposition 1 The real polynomial 120577(120596) is even

Proof We prove that 120577(120582) = 0 if and only if 120577(minus120582) = 0 Firstof all note that 119887

119894(minus120596) = 119887

119894(120596) and then 119903(minus120596 119911) = 119903(120596 119911)

As a consequence

119880

119902(119903 (120596)) = 119880

119902(119903 (120596)) = 119880

119902(119903 (minus120596))

119871

119902(119903 (120596)) = 119871

119902(119903 (120596)) = 119871

119902(119903 (minus120596))

(14)

Thus

119871

lowast

119902(119903 (minus120596)) 119871119902 (

119903 (minus120596)) minus 119880

lowast

119902(119903 (minus120596))119880119902 (

119903 (minus120596))

= 119871

119879

119902(119903 (120596)) 119871

119902(119903 (120596)) minus 119880

119879

119902(119903 (120596)) 119880

119902(119903 (120596))

(15)

and 120577(120582) = 120577(minus120582)

From the above development we obtain the followingcriterion

Criterion 1 Let 1205960isin R+ be a real root of 120577(radic119904) then plusmn119895

radic120596

0

is a pair of pure imaginary roots of the LTI-TDS system in (1)

Proof It is immediate that plusmn119895120596 is a pair of pure imaginaryroots of 119901

119888(119904 119911 119911

119902) if and only if 120596 is a positive real root

of119901119888(119895120596 119911 119911

119902) According to the Schur-Cohnmethod the

roots of 119901119888(119895120596 119911 119911

119902) can be determined by searching for

the real roots in the system (7) Additionally it was shownabove that the solution of (7) implies the computing of theroots of the even polynomial 120577(120596) henceforth if 120596

0isin R+ is

a root of 120577(radic119904) plusmnradic120596

0is a pair of real roots of 120577(119904) and plusmn119895120596

0

is a pair of pure imaginary roots of 119901119888(119904 119911 119911

119902)

Note that this criterion is advantageous since firstly weonly need to compute real roots and secondly deg(120577(radicsdot)) = 119902In the following the set of crossing frequencies is defined as

Ω = 120596 isin R+ 120596

2isin Z (120577 (radicsdot)) capR

+ (16)

Example 2 Let us consider the LTI-TDS system (119905) =

119860

0119909(119905) + 119860

1119909(119905 minus 120591) where the system matrices are defined

as

119860

0= (

minus1 135 minus1

minus3 minus1 minus2

minus2 minus1 minus4

) 119860

1= (

minus59 71 minus703

2 minus1 5

2 0 6

)

(17)

The quasi-polynomial characteristic is

119901

119888(119904 119890

minus119904120591 119890

minus119898119904120591)

= 119886

3(119904) 119890

minus3120591119904+ 119886

2(119904) 119890

minus2120591119904+ 119886

1(119904) 119890

minus120591119904+ 119886

0(119904)

(18)

4 Mathematical Problems in Engineering

with

119886

0(119904) = 119904

3+ 6119904

2+ 455119904 + 111

119886

1 (119904) = 09119904

2minus 1168119904 minus 221

119886

2(119904) = 909119904 minus 1851

119886

3(119904) = 1194

(19)

In the frequency domain (119904 = 119895120596) we define the bivariatepolynomial 119903(120596 119911) = sum

3

119894=0119886

119894(119895120596)119911

119894 and from (19) we buildthe triangular matrices 119880

3(119903) and 119871

3(119903) as follows

119880

3= (

1194 minus1851 minus09120596

2minus 221

0 1194 minus1851

0 0 1194

)

+ 119895120596(

0 minus909 1168

0 0 minus909

0 0 0

)

119871

3= (

0 0 1110 minus 60120596

2

0 1110 minus 60120596

2minus09120596

2minus 221

1110 minus 60120596

2minus09120596

2minus 221 minus1851

)

+ 119895120596(

0 0 455 minus 120596

2

0 455 minus 120596

2minus1168

455 minus 120596

2minus1168 909

)

(20)

Then we compute the polynomial 120577(radic119904) according to Crite-rion 1

120577 (radic119904) = 119904

9minus 16581119904

8minus 18682119904

7+ 24869 times 10

5119904

6

+ 35185119904

5minus 41197 times 10

9119904

4+ 94110 times 10

10119904

3

minus 71137 times 10

11119904

2+ 18933 times 10

12119904

minus 10144 times 10

12

(21)

whose positive real roots are Z(120577(radicsdot)) cap R+ =

84802 44564 92141 070634 24035 and then

plusmn119895Ω = plusmn11989529121 plusmn1198952111 plusmn11989530355 plusmn119895084044 plusmn11989515503

(22)

is the set of pure imaginary roots of the delayed system Notethat from the fundamental theoremof algebra there are only afinite number of crossing frequencies and thatZ(120577(radicsdot))capR+

represents the set of squares of crossing frequencies

The method presented above is the first step to test thestability of the characteristic quasi-polynomial 119901

119888(119904 119890

minus119904120591) and

to compute the delay margin for LTI delay systems It iscommon practice to make the following assumption on 119901

119888

Assumption 3 The characteristic quasi-polynomial 119901119888is sta-

ble at 120591 = 0 that is Z(119901

119888(sdot 0)) sub Cminus where Cminus = 119904 isin C

Re119904 lt 0

Recall that the delay margin 120591 is the smallest deviation of120591 from 120591 = 0 such that the system becomes unstable

120591 = min 120591 ge 0 119901

119888(119895120596 119890

minus119895120596120591) = 0 for some 120596 isin R

+

(23)

For any 120591 isin [0 120591) the system is stable and whenever 120591 = infinthe system is said to be stable independent of delay For eachfinite delay 120591 we can associate a crossing frequency whichrepresents the first contact or crossing of the roots of 119901

119888from

the stable region to the unstable oneThedetermination of thedelay margin requires solving the bivariate polynomial 119901

119888isin

C[120596 119911] in two phases (i) computing the positive real roots of120577(radic119904) so as to obtain the crossing frequencies 120596 isin Ω of 119901

119888

(ii) finding the roots of 119901119888(119895120596) isin C[119911] on the boundary of the

unit disk D (denoted as 120597D) at each crossing frequency 120596 Adetailed analysis of the delay margin 120591 can be found in [11]and in Theorem 211 pp 59-60 in [2] Under the assumptionthat 119901

119888is stable at 120591 = 0 the two-step stability method is

summarized in Table 1

Example 4 Taking into account the positive crossing fre-quencies 120596

119896isin Ω computed for the LTI-TDS system in

Example 2 we can derive the set of roots of 119901119888(119895120596

119896) isin C[119911]

onto the unit circle 120597D Let 119911119896

isin 120597D cap Z(119901

119888(119895120596

119896)) from

the identity 119911

119896= 119890

minus119895120596119896120591

= 119890

119895 arg(119911119896) it follows that 120591

119896=

(2120587minus arg(119911119896))120596

119896(with arg(119911

119896) isin [0 2120587]) and the system can

cross the stability boundary 120597C = 119895R from the stable half-plane Cminus to the unstable half-plane C+ = C (Cminus cup 120597C) orvice versa at 120591

119896 If the root crosses 120597C from C+ to Cminus at 120591

119896

there exists a 120591 lt 120591

119896such that makes the root to cross from

Cminus to C+ As a consequence the stability margin is defined as

120591= min1le119896le119902

2120587 minus arg (119911119896)

120596

119896

120596

119896isin Ω 119911

119896isin 120597D capZ (119901

119888(119895120596

119896))

(24)

Under Assumption 3 the system is unstable at 120591 = 120591 andat least a root crosses 120597C from Cminus to C+ at 120591 If we choose120591 isin [0 120591) then 120591120596

119896= 120579

119896and 119901

119888(119895120596

119896 119890

minus119895120596120591) = 0 for all

120596 isin R+ Therefore 119901119888(119904 119890

minus119904120591) is stable for all 120591 isin [0 120591) For

each crossing frequency120596119896a delay 120591

119896is computed as shown in

Table 2 The delay margin is then computed as the minimumvalue of these delays that is 120591 = min120591

119896 119896 = 1 5 =

016230The crossing direction of those roots onto the imaginary

axis (either from stable to unstable complex plane or viceversa) as 120591 increases is a quantity called the root tendency(RT) [7 8] For a cross frequency 120596

0and a delay 120591

0 the root

tendency is defined as

(RT)120596120591

= sgn(Re 120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

) (25)

where sgn(sdot) denotes the sign function In the followinglemma we propose a simple method to compute the roottendency based on the implicit function theorem

Lemma 5 (root tendency) Let 119901119888be a characteristic quasi-

polynomial as defined in (3) and let 120591

0be the delay for

Mathematical Problems in Engineering 5

Table 1 Stability analysis of the quasipolynomial 119901119888in terms of the delay margin 120591

Set of squares of crossing frequencies Solutions of 119901119888(120596) isin C[119911] on 120597D 120596 isin Ω Stability of the quasi-polynomial 119901

119888isin C[120596 119911]

Z (120577 (radicsdot)) capR+ =

Stable independent of delay (120591 = infin)Z (120577 (radicsdot)) capR+ = (0)

Z (120577 (radicsdot)) capR+ = Z (119902) capR+ = (0)

Z (119901

119888(119895120596)) cap 120597D =

Z (119901

119888(119895120596)) cap 120597D = Stable dependent of delay (120591 lt infin)

Table 2 Delays corresponding to the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Roots of 119901

119888(119895120596

119896) on 120597D (119911

119896= 119890

minus119895120596119896120591) Phase of 119911119896(120579119896) Delay (120591

119896)

29121 085692 minus 119895051548 57416 0185972111 minus026768 minus 119895096352 44414 08724730355 088110 minus 119895047297 57905 016230084044 097521 + 119895022129 022314 7210615503 minus095537 + 119895029541 28417 022199

Table 3 Root tendency for the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Delay (120591

119896) sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591

0119886

119896(119895120596

0))119890

minus11989512059601198961205910sum

119902

119896=1119896119886

119896(119895120596

0)119890

minus11989511989612059601205910(119877119879)

120596120591

29121 018597 minus48520 + 11989530258 74136 minus 11989558701 minus1

2111 087247 minus14062 + 11989534286 14878 minus 11989518237 +1

30355 016230 minus47511 + 11989527405 94101 minus 11989541172 +1

084044 72106 87231 minus 11989569290 minus11943 + 11989599556 minus1

15503 022199 minus90235 minus 11989591993 18255 + 11989544754 +1

which the system crosses the stability boundary at 1198951205960 for

some 120596

0isin Ω that is 119901

119888(119895120596

0 120591

0) = 0 Let us assume

that (120597119901119888(119904 120591)120597119904)|

(11989512059601205910)= sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591119886

119896(120596

0))

119890

minus11989512059601198961205910

= 0 Then the root tendency of 1198951205960is given by

( RT )

12059601205910

= minus sgn( Im((

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910)

times (

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0))

times119890

minus11989511989612059601205910)

minus1

))

(26)

Proof In virtue of the implicit function theorem there exista 120575 gt 0 a neighborhood 119880

0sub C of 119895120596

0 and a continuous

function 119904 (1205910minus120575 120591

0+120575) rarr 119880

0such that 119901

119888(119904(120591) 120591) = 0 for

all 120591 isin (120591

0minus120575 120591

0+120575) this is also obvious from the continuity

of the roots 119904(sdot)with respect to the delay 120591 Furthermore since119901

119888is differentiable and (120597119901

119888(119904 120591)120597119904)|

(11989512059601205910)

= 0 it is clear that

120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus

120597119901

119888(119904 120591) 120597120591

1003816

1003816

1003816

1003816(11989512059601205910)

120597119901

119888(119904 120591) 120597119904

1003816

1003816

1003816

1003816(11989512059601205910)

(27)

where

120597119901

119888(119904 120591)

120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

=

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0)) 119890

minus11989511989612059601205910

+ 119901

119888(119895120596

0 120591

0)

120597119901

119888(119904 120591)

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus119895120596

0

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910

(28)

Owing to 119901

119888(119895120596

0 120591

0) = 0 and from the chain rule of

derivation the result in (26) follows immediately

Example 6 Following with Example 2 and using the roottendency lemma we can compute the crossing direction foreach crossing frequency as shown in the fifth column inTable 3 RT = minus1 indicates that the crossing is from C+ toCminus and RT = +1 implies a crossing from Cminus to C+

We finish this analysis by noting that it is not possible ingeneral to separate 120577(radic119904) with a factor containing exactly allthe real roots of 120577(radic119904) as stated in the following lemma

Lemma 7 Given a polynomial 119901 isin R119899[119904] there does not exist

a general procedure of factorization of 119901 in polynomials 119901119877isin

R[119904] and 119901119862isin R[119904] 119901(119904) = 119901

119877(119904)119901

119862(119904) such that 119901

119877includes

exactly all the real roots of 119901 (and 119901

119862all the pairs of complex

conjugated roots of 119901 not in R)

6 Mathematical Problems in Engineering

Proof We prove this assertion by a counterexample withoutloss of generality we assume that 119901 is monic Let us assumethat 119901 has only real roots which are ordered as 120572

1lt 120572

2lt

sdot sdot sdot lt 120572

119898 119898 le 119899 Always we can find a value 120575 isin R such

that 1205721lt 120575 lt 120572

2 We make the change of variable 119904 997891rarr 119904 + 120575

resulting into a polynomial 119901120575(119904) = 119901(119904 + 120575) The roots of

119901

120575are located at 1205721015840

119896= 120572

119896minus 120575 and it is easy to see that

120572

1015840

1lt 0 and 120572

1015840

119896gt 0 for 119896 = 2 119898 Then we apply the map

119904 997891rarr 119904

2 which produces a polynomial 119902(119904) = 119901

120575(119904

2) with a

pair of pure conjugated complex roots at plusmniradic120575 minus 119886

1and pairs

of symmetric real roots at plusmnradic119886

119896minus 120575 for 119896 = 2 119898 Under

the hypothesis of separability 119902(119904) = 119902

119877(119904)119902

119862(119904) and undoing

in reverse order the changes of variables 119904 997891rarr radic119904 997891rarr 119904+120575 weobtain that 119902

119877(

radic

119904 minus 120575) = (119904 minus 120572

1) and 119902

119862(

radic

119904 minus 120575) = 119901(119904)(119904 minus

120572

1) Therefore we might isolate roots of 119901 by successively

repeating the procedure onto 119902119862This is a contradiction since

from the Galois theory it is well known that it is not alwayspossible to solve a polynomial equation by radicals

3 Conclusions

We overcome the drawbacks of the original Schur-Cohncriterion for delay systems namely the high degree ofthe polynomial that emerges from the Schur-Cohn matrixand the difficulty of detection of complex roots especiallypure imaginary roots due to the propagation of numericalerrors In particular we decrease the degree of the generatedpolynomial to a half and we reduce the problem numericallyto that of seeking the real roots of this polynomial notthe complex ones These properties are exactly those thatmake the Rekasius substitution criterion so attractive in thestability theory of delay systems Therefore from a practicalviewpoint the modified Schur-Cohn criterion is comparableto other widely used criteria in the literature Our methodis based on making transformations on the Schur-Cohnmatrix via triangular matrices over a polynomial ring inone unknown This leads to the determinant of the Jurymatrix over a polynomial ring which is proved to be an evenpolynomial The pure imaginary roots of the Schur-Cohnpolynomial are those of the even polynomial and in virtueof the change of variable 120596 997891rarr radic120596 we only need to seekreal roots which is more precise from a numerical viewpointwhile the degree of the polynomial is halved

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vyhlidal J-F Lafay and R Sipahi Delay Systems FromTheory to Numerics and Applications Advances in Delays andDynamics Springer New York NY USA 2014

[2] K Gu J Chen and V L Kharitonov Stability of Time-DelaySystems Birkhaauser Boston Mass USA 2003

[3] C Bonnet A R Fioravanti and J R Partington ldquoStability ofneutral systems with commensurate delays and poles asymp-totic to the imaginary axisrdquo SIAM Journal on Control andOptimization vol 49 no 2 pp 498ndash516 2011

[4] H Fazelinia R Sipahi and N Olgac ldquoStability robustness anal-ysis of multiple time-delayed systems using ldquobuilding blockrdquoconceptrdquo IEEE Transactions on Automatic Control vol 52 no5 pp 799ndash810 2007

[5] A R Fioravanti C Bonnet and H Ozbay ldquoStability offractional neutral systems with multiple delays and polesasymptotic to the imaginary axisrdquo in Proceedings of the 49thIEEEConference onDecision andControl pp 31ndash35 Atlanta GaUSA December 2010

[6] A R Fioravanti C Bonnet H Ozbay and S-I Niculescu ldquoAnumerical method to find stability windows and unstable polesfor linear neutral time-delay systemsrdquo in Proceedings of the 9thIFAC Workshop on Time Delay Systems (TDS rsquo10) pp 183ndash188Prague Czech Republic June 2010

[7] A R Fioravanti C Bonnet H Ozbay and S-I NiculesculdquoA numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systemsrdquoAutomatica vol 48 no 11 pp 2824ndash2830 2012

[8] N Olgac and R Sipahi ldquoA practical method for analyzing thestability of neutral type LTI-time delayed systemsrdquo Automaticavol 40 no 5 pp 847ndash853 2004

[9] R Sipahi ldquoDelay-margin design for the general class of single-delay retarded-type LTI systemsrdquo International Journal ofDynamics and Control vol 2 no 2 pp 198ndash209 2014

[10] S Barnett Polynomials and Linear Control Systems MarcelDekker New York NY USA 1983

[11] J Chen G Gu and C N Nett ldquoA new method for computingdelay margins for stability of linear delay systemsrdquo Systems andControl Letters vol 26 no 2 pp 107ndash117 1995

[12] P Fu S-I Niculescu and J Chen ldquoStability of linear neutraltime-delay systems exact conditions via matrix pencil solu-tionsrdquo IEEE Transactions on Automatic Control vol 51 no 6pp 1063ndash1069 2006

[13] K Walton and J E Marshall ldquoDirect Method for TDS StabilityAnalysisrdquo IEE Proceedings D Control Theory and Applicationsvol 134 no 2 pp 101ndash107 1987

[14] J H Su ldquoThe asymptotic stability of linear autonomous systemswith commensurate time delaysrdquo IEEE Transactions on Auto-matic Control vol 40 no 6 pp 1114ndash1117 1995

[15] J Louisell ldquoAmatrixmethod for determining the imaginary axiseigenvalues of a delay systemrdquo IEEE Transactions on AutomaticControl vol 46 no 12 pp 2008ndash2012 2001

[16] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5 pp 793ndash797 2002

[17] Z V Rekasius ldquoA stability test for systems with delaysrdquo inProceedings of the Joint Automatic Control Conference 1980paper no TP9-A

[18] A Thowsen ldquoThe Routh-Hurwitz method for stability deter-mination of linear differential-difference systemsrdquo InternationalJournal of Control vol 33 no 5 pp 991ndash995 1981

[19] R Sipahi and N Olgac ldquoA comparative survey in determiningthe imaginary characteristic roots of LTI time delayed systemsrdquoin Proceedings of the 16th Triennial World Congress of Interna-tional Federation of Automatic Control (IFAC rsquo05) vol 16 pp390ndash399 Prague Czech Republic July 2005

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

number of significant digits among the above five methodsIn the literature the Rekasius substitution approach is themost attractive since the generated polynomial has degree 1199022much smaller than all the other criteria Another advantageof this method is that only real roots are searched for insteadof complex ones which is very crucial from a practical viewpoint since complex roots are difficult to compute whennumerical errors are dragged and propagated For a totalcomparison of these methods demonstrating their strengthsand weakness the reader is referred to [19]

This paper is focused on improving the Schur-Cohnprocedure so as to obtain the advantageous properties ofthe Rekasius substitution criterion Specifically we modifythe Schur-Cohn criterion to generate a real polynomial ofdegree 119902

2 whose real roots determine the pure imaginaryroots of the characteristic equationThe original Schur-Cohnprocedure computes the determinant of a partitioned matrix119860 = (

Λ1(119904) Λ

2(119904)

Λlowast

2(119904) Λlowast

1(119904)) where Λ

1 Λ

2isin C[119904] are appropriate

119902 times 119902 matrices over the ring of polynomials in the unknown119904 and Λ

lowast denotes the Hermitian (or conjugated complex) ofΛ The classical Schur-Cohn procedure for high dimensions(21199022 gt 10) becomes numerically unreliable due to repeatedround-off errors in the determinant expansion procedureunless the operation is performed using very large numberof significant digits This particular point alone brings aweakness from the numerical deployment viewpoint sincethe operation ultimately yields poor precision in determiningthe desired imaginary roots because of the accumulatednumerical errors the imaginary roots shape up in the formof 119904 = 119895120596 + 120598 120598 ≪ 1 and 119895 =

radicminus1 which leads the numerical

error in real parts to be at least in the order of 10 magnitudeslarger than the computational precision Therefore it isproblematic to decide whether these roots which are veryclose to the imaginary axis are really imaginary roots ornot As an attempt to overcome the problems of the Schur-Cohn method some authors have proposed a modificationbased on the determination of the real eigenvalues of aconstant matrix [11] By contrast our modification stemsfrom the applications of some transformations on119860 by usingtriangular matrices over the polynomial ring in a similar wayas in the Jury test of stability for discrete systems [20]

The remainder of the paper is structured as follows inSection 2 the basic idea of the modified stability criterionis explained and a numerical example is provided Finally asummary and an outlook are given in Section 3

2 Delayed Systems Schur-CohnModified Method

The stability of the system (1) is determined by its character-istic quasi-polynomial in 119904 [21]

119901

119888(119904 119890

minus119904120591 119890

minus119898119904120591) = det(119904119868 minus 119860

0minus

119898

sum

119896=1

119860

119896119890

minus119904119896120591)

=

119902

sum

119896=0

119886

119896 (119904) 119890

minus119896120591119904

(3)

119886

0 (119904) = 119904

119899+

119899minus1

sum

119894=0

119886

0119894119904

119894

119886

119896(119904) =

119899minus1

sum

119894=0

119886

119896119894119904

119894 119896 = 1 119902

(4)

The characteristic quasi-polynomial contains time delays ofcommensurate nature with degree 119902 that is 119901

119888depends on

119890

minus119896120591119904 for 119896 = 0 119902 with 119886

119902(119904) = 0 and rank(119860

119896) le 119902 le

119899 = rank(1198600) The starting-point of all the stability criteria is

the determination of the roots of 119901119888on the imaginary axis

In fact it is well known from the stability theory of LTI-TDS that the pure imaginary characteristic roots are the onlypossible transition points from stable to unstable behaviorand vice versa We want to determine all the imaginary rootsof 119901119888(119904 119890

minus119904120591 119890

minus119898119904120591) and to this end it is convenient to

introduce the variable 119911 = 119890

minus119904120591 andwrite119901119888as a polynomial in

two unknowns over the field of reals119901119888(119904 119911) = sum

119902

119894=0119886

119896(119904)119911

119896isin

R[119904 119911]We define the bivariate polynomial 119903 isin C[120596 119911] as

119903(120596 119911) = sum

119902

119894=0119887

119894(120596)119911

119894 such that 119887119894(120596) = 119886

119894(119895120596) isin C[120596] With

every bivariate polynomial over the complex field 119903(120596 119911) weassociate two triangular matrices over the polynomial ringC[120596]

119880

119902(119903 (120596))

=

(

(

119887

119902(120596) 119887

119902minus1(120596) 119887

119902minus2(120596) sdot sdot sdot 119887

1(120596)

0 119887

119902(120596) 119887

119902minus1(120596) sdot sdot sdot 119887

2(120596)

0 sdot sdot sdot 0 119887

119902(120596) 119887

119902minus1(120596)

0 sdot sdot sdot 0 0 119887

119902 (120596)

)

)

119871

119902(119903 (120596))

= (

0 sdot sdot sdot 0 0 119887

0 (120596)

0 sdot sdot sdot 0 119887

0 (120596) 119887

1 (120596)

0 119887

0 (120596) 119887

1 (120596) sdot sdot sdot 119887

119902minus2 (120596)

119887

0(120596) 119887

1(120596) 119887

2(120596) sdot sdot sdot 119887

119902minus1(120596)

)

(5)

For the sake of simplicity we write 119880

119902 119871

119902instead

of 119880

119902(119903(120596)) 119871

119902(119903(120596)) The Schur-Cohn criterion solves

the problem of computing the values of 120596 such that119901

119888(119895120596 119911 119911

119902) = 0 by multiplying 119901

119888(119895120596 119911 119911

119902) by 119911

minus119896

for 119896 = 0 119902minus1 and119901119888(119895120596 119911 119911

119902) by 119911119896 for 119896 = 1 119902

This leads to a system of 2119902 homogeneous equations in theunknowns 119911119896 119896 = minus119902 0 119902minus1which has the followingmatrix representation

(

119871

119902119880

119902119869

119869119880

119902119869119871

119902119869

)(

e1

e2

) = (

0

0

) (6)

where 119869 is the counteridentity matrix (ie an antidiagonalmatrix with ones) e

1= 119869119897

119879

119902minus1(119911) e2

= 119869119897

119879

119902minus1(119911

minus1)119911

minus1 and

Mathematical Problems in Engineering 3

119897

119879

119902minus1(119911) = (1 119911 119911

2 119911

119902minus1) the homogeneous system in (7)

is the starting point of the Schur-Cohn method If we makethe change of unknowns e

119894997891rarr 119869e119894and we take into account

the idempotence of 119869 (6) can be transformed into

(

119871

119902119880

119902

119869119880

119902119869119871

119902

)(

119869e1

119869e2

) = (

0

0

) (7)

It is easily verified that119880lowast119902119871

119902is of the same structure as119871

119902 and

in particular it is symmetric Therefore 119880lowast119902119871

119902= (119880

lowast

119902119871

119902)

119879

=

119871

lowast

119902119880

119902 As a consequence we obtain

(

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119868 119874

119874 119869

)(

119871

119902119880

119902

119869119880

119902119869119871

119902

)

= (

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119871

119902119880

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119874

119880

119902119871

119902

)

(8)

The homogeneous system (7) has a nontrivial solution solong as det( 119871119902 119880119902

119869119880119902119869119871119902

) = 0 We use the fact that

det((119871lowast

119902minus119880

lowast

119902

119874 119868

)(

119868 119874

119874 119869

))

= det(119871lowast

119902minus119880

lowast

119902119869

119874 119869

) = det (119871lowast119902119869) = det (119871

119902) (minus1)

119902

(9)

and taking determinants in (8) yields

det (119871119902) (minus1)

119902 det( 119871

119902119880

119902

119869119880

119902119869119871

119902

)

= det (119871119902) det (119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902)

(10)

As a result

det( 119871

119902119880

119902

119869119880

119902119869119871

119902

) = (minus1)

119902 det (119871lowast119902119871

119902minus 119880

lowast

119902119880

119902) (11)

Therefore the problem has been reduced to find the pureimaginary roots of det(119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902) which is just the

determinant of the Jury matrix 119869119902(119903(120596)) over C[120596] defined as

119869

119902(119903 (120596)) = (

119871

119902119880

119902

119880

119902119871

119902

) isin C2119902times2119902

[120596] (12)

This can be seen by a similar argument as above in (8)

(

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119871

119902119880

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119871

lowast

119902119880

119902minus 119880

lowast

119902119871

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119874

119880

119902119871

119902

)

(13)

and det(119869119902(119903(120596))) = det(119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902)

In the sequel we define the polynomial 120577(120596) =

det(119869119902(119903(120596))) and the set of zeros of a polynomial119901 is denoted

asZ(119901)

Proposition 1 The real polynomial 120577(120596) is even

Proof We prove that 120577(120582) = 0 if and only if 120577(minus120582) = 0 Firstof all note that 119887

119894(minus120596) = 119887

119894(120596) and then 119903(minus120596 119911) = 119903(120596 119911)

As a consequence

119880

119902(119903 (120596)) = 119880

119902(119903 (120596)) = 119880

119902(119903 (minus120596))

119871

119902(119903 (120596)) = 119871

119902(119903 (120596)) = 119871

119902(119903 (minus120596))

(14)

Thus

119871

lowast

119902(119903 (minus120596)) 119871119902 (

119903 (minus120596)) minus 119880

lowast

119902(119903 (minus120596))119880119902 (

119903 (minus120596))

= 119871

119879

119902(119903 (120596)) 119871

119902(119903 (120596)) minus 119880

119879

119902(119903 (120596)) 119880

119902(119903 (120596))

(15)

and 120577(120582) = 120577(minus120582)

From the above development we obtain the followingcriterion

Criterion 1 Let 1205960isin R+ be a real root of 120577(radic119904) then plusmn119895

radic120596

0

is a pair of pure imaginary roots of the LTI-TDS system in (1)

Proof It is immediate that plusmn119895120596 is a pair of pure imaginaryroots of 119901

119888(119904 119911 119911

119902) if and only if 120596 is a positive real root

of119901119888(119895120596 119911 119911

119902) According to the Schur-Cohnmethod the

roots of 119901119888(119895120596 119911 119911

119902) can be determined by searching for

the real roots in the system (7) Additionally it was shownabove that the solution of (7) implies the computing of theroots of the even polynomial 120577(120596) henceforth if 120596

0isin R+ is

a root of 120577(radic119904) plusmnradic120596

0is a pair of real roots of 120577(119904) and plusmn119895120596

0

is a pair of pure imaginary roots of 119901119888(119904 119911 119911

119902)

Note that this criterion is advantageous since firstly weonly need to compute real roots and secondly deg(120577(radicsdot)) = 119902In the following the set of crossing frequencies is defined as

Ω = 120596 isin R+ 120596

2isin Z (120577 (radicsdot)) capR

+ (16)

Example 2 Let us consider the LTI-TDS system (119905) =

119860

0119909(119905) + 119860

1119909(119905 minus 120591) where the system matrices are defined

as

119860

0= (

minus1 135 minus1

minus3 minus1 minus2

minus2 minus1 minus4

) 119860

1= (

minus59 71 minus703

2 minus1 5

2 0 6

)

(17)

The quasi-polynomial characteristic is

119901

119888(119904 119890

minus119904120591 119890

minus119898119904120591)

= 119886

3(119904) 119890

minus3120591119904+ 119886

2(119904) 119890

minus2120591119904+ 119886

1(119904) 119890

minus120591119904+ 119886

0(119904)

(18)

4 Mathematical Problems in Engineering

with

119886

0(119904) = 119904

3+ 6119904

2+ 455119904 + 111

119886

1 (119904) = 09119904

2minus 1168119904 minus 221

119886

2(119904) = 909119904 minus 1851

119886

3(119904) = 1194

(19)

In the frequency domain (119904 = 119895120596) we define the bivariatepolynomial 119903(120596 119911) = sum

3

119894=0119886

119894(119895120596)119911

119894 and from (19) we buildthe triangular matrices 119880

3(119903) and 119871

3(119903) as follows

119880

3= (

1194 minus1851 minus09120596

2minus 221

0 1194 minus1851

0 0 1194

)

+ 119895120596(

0 minus909 1168

0 0 minus909

0 0 0

)

119871

3= (

0 0 1110 minus 60120596

2

0 1110 minus 60120596

2minus09120596

2minus 221

1110 minus 60120596

2minus09120596

2minus 221 minus1851

)

+ 119895120596(

0 0 455 minus 120596

2

0 455 minus 120596

2minus1168

455 minus 120596

2minus1168 909

)

(20)

Then we compute the polynomial 120577(radic119904) according to Crite-rion 1

120577 (radic119904) = 119904

9minus 16581119904

8minus 18682119904

7+ 24869 times 10

5119904

6

+ 35185119904

5minus 41197 times 10

9119904

4+ 94110 times 10

10119904

3

minus 71137 times 10

11119904

2+ 18933 times 10

12119904

minus 10144 times 10

12

(21)

whose positive real roots are Z(120577(radicsdot)) cap R+ =

84802 44564 92141 070634 24035 and then

plusmn119895Ω = plusmn11989529121 plusmn1198952111 plusmn11989530355 plusmn119895084044 plusmn11989515503

(22)

is the set of pure imaginary roots of the delayed system Notethat from the fundamental theoremof algebra there are only afinite number of crossing frequencies and thatZ(120577(radicsdot))capR+

represents the set of squares of crossing frequencies

The method presented above is the first step to test thestability of the characteristic quasi-polynomial 119901

119888(119904 119890

minus119904120591) and

to compute the delay margin for LTI delay systems It iscommon practice to make the following assumption on 119901

119888

Assumption 3 The characteristic quasi-polynomial 119901119888is sta-

ble at 120591 = 0 that is Z(119901

119888(sdot 0)) sub Cminus where Cminus = 119904 isin C

Re119904 lt 0

Recall that the delay margin 120591 is the smallest deviation of120591 from 120591 = 0 such that the system becomes unstable

120591 = min 120591 ge 0 119901

119888(119895120596 119890

minus119895120596120591) = 0 for some 120596 isin R

+

(23)

For any 120591 isin [0 120591) the system is stable and whenever 120591 = infinthe system is said to be stable independent of delay For eachfinite delay 120591 we can associate a crossing frequency whichrepresents the first contact or crossing of the roots of 119901

119888from

the stable region to the unstable oneThedetermination of thedelay margin requires solving the bivariate polynomial 119901

119888isin

C[120596 119911] in two phases (i) computing the positive real roots of120577(radic119904) so as to obtain the crossing frequencies 120596 isin Ω of 119901

119888

(ii) finding the roots of 119901119888(119895120596) isin C[119911] on the boundary of the

unit disk D (denoted as 120597D) at each crossing frequency 120596 Adetailed analysis of the delay margin 120591 can be found in [11]and in Theorem 211 pp 59-60 in [2] Under the assumptionthat 119901

119888is stable at 120591 = 0 the two-step stability method is

summarized in Table 1

Example 4 Taking into account the positive crossing fre-quencies 120596

119896isin Ω computed for the LTI-TDS system in

Example 2 we can derive the set of roots of 119901119888(119895120596

119896) isin C[119911]

onto the unit circle 120597D Let 119911119896

isin 120597D cap Z(119901

119888(119895120596

119896)) from

the identity 119911

119896= 119890

minus119895120596119896120591

= 119890

119895 arg(119911119896) it follows that 120591

119896=

(2120587minus arg(119911119896))120596

119896(with arg(119911

119896) isin [0 2120587]) and the system can

cross the stability boundary 120597C = 119895R from the stable half-plane Cminus to the unstable half-plane C+ = C (Cminus cup 120597C) orvice versa at 120591

119896 If the root crosses 120597C from C+ to Cminus at 120591

119896

there exists a 120591 lt 120591

119896such that makes the root to cross from

Cminus to C+ As a consequence the stability margin is defined as

120591= min1le119896le119902

2120587 minus arg (119911119896)

120596

119896

120596

119896isin Ω 119911

119896isin 120597D capZ (119901

119888(119895120596

119896))

(24)

Under Assumption 3 the system is unstable at 120591 = 120591 andat least a root crosses 120597C from Cminus to C+ at 120591 If we choose120591 isin [0 120591) then 120591120596

119896= 120579

119896and 119901

119888(119895120596

119896 119890

minus119895120596120591) = 0 for all

120596 isin R+ Therefore 119901119888(119904 119890

minus119904120591) is stable for all 120591 isin [0 120591) For

each crossing frequency120596119896a delay 120591

119896is computed as shown in

Table 2 The delay margin is then computed as the minimumvalue of these delays that is 120591 = min120591

119896 119896 = 1 5 =

016230The crossing direction of those roots onto the imaginary

axis (either from stable to unstable complex plane or viceversa) as 120591 increases is a quantity called the root tendency(RT) [7 8] For a cross frequency 120596

0and a delay 120591

0 the root

tendency is defined as

(RT)120596120591

= sgn(Re 120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

) (25)

where sgn(sdot) denotes the sign function In the followinglemma we propose a simple method to compute the roottendency based on the implicit function theorem

Lemma 5 (root tendency) Let 119901119888be a characteristic quasi-

polynomial as defined in (3) and let 120591

0be the delay for

Mathematical Problems in Engineering 5

Table 1 Stability analysis of the quasipolynomial 119901119888in terms of the delay margin 120591

Set of squares of crossing frequencies Solutions of 119901119888(120596) isin C[119911] on 120597D 120596 isin Ω Stability of the quasi-polynomial 119901

119888isin C[120596 119911]

Z (120577 (radicsdot)) capR+ =

Stable independent of delay (120591 = infin)Z (120577 (radicsdot)) capR+ = (0)

Z (120577 (radicsdot)) capR+ = Z (119902) capR+ = (0)

Z (119901

119888(119895120596)) cap 120597D =

Z (119901

119888(119895120596)) cap 120597D = Stable dependent of delay (120591 lt infin)

Table 2 Delays corresponding to the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Roots of 119901

119888(119895120596

119896) on 120597D (119911

119896= 119890

minus119895120596119896120591) Phase of 119911119896(120579119896) Delay (120591

119896)

29121 085692 minus 119895051548 57416 0185972111 minus026768 minus 119895096352 44414 08724730355 088110 minus 119895047297 57905 016230084044 097521 + 119895022129 022314 7210615503 minus095537 + 119895029541 28417 022199

Table 3 Root tendency for the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Delay (120591

119896) sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591

0119886

119896(119895120596

0))119890

minus11989512059601198961205910sum

119902

119896=1119896119886

119896(119895120596

0)119890

minus11989511989612059601205910(119877119879)

120596120591

29121 018597 minus48520 + 11989530258 74136 minus 11989558701 minus1

2111 087247 minus14062 + 11989534286 14878 minus 11989518237 +1

30355 016230 minus47511 + 11989527405 94101 minus 11989541172 +1

084044 72106 87231 minus 11989569290 minus11943 + 11989599556 minus1

15503 022199 minus90235 minus 11989591993 18255 + 11989544754 +1

which the system crosses the stability boundary at 1198951205960 for

some 120596

0isin Ω that is 119901

119888(119895120596

0 120591

0) = 0 Let us assume

that (120597119901119888(119904 120591)120597119904)|

(11989512059601205910)= sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591119886

119896(120596

0))

119890

minus11989512059601198961205910

= 0 Then the root tendency of 1198951205960is given by

( RT )

12059601205910

= minus sgn( Im((

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910)

times (

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0))

times119890

minus11989511989612059601205910)

minus1

))

(26)

Proof In virtue of the implicit function theorem there exista 120575 gt 0 a neighborhood 119880

0sub C of 119895120596

0 and a continuous

function 119904 (1205910minus120575 120591

0+120575) rarr 119880

0such that 119901

119888(119904(120591) 120591) = 0 for

all 120591 isin (120591

0minus120575 120591

0+120575) this is also obvious from the continuity

of the roots 119904(sdot)with respect to the delay 120591 Furthermore since119901

119888is differentiable and (120597119901

119888(119904 120591)120597119904)|

(11989512059601205910)

= 0 it is clear that

120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus

120597119901

119888(119904 120591) 120597120591

1003816

1003816

1003816

1003816(11989512059601205910)

120597119901

119888(119904 120591) 120597119904

1003816

1003816

1003816

1003816(11989512059601205910)

(27)

where

120597119901

119888(119904 120591)

120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

=

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0)) 119890

minus11989511989612059601205910

+ 119901

119888(119895120596

0 120591

0)

120597119901

119888(119904 120591)

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus119895120596

0

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910

(28)

Owing to 119901

119888(119895120596

0 120591

0) = 0 and from the chain rule of

derivation the result in (26) follows immediately

Example 6 Following with Example 2 and using the roottendency lemma we can compute the crossing direction foreach crossing frequency as shown in the fifth column inTable 3 RT = minus1 indicates that the crossing is from C+ toCminus and RT = +1 implies a crossing from Cminus to C+

We finish this analysis by noting that it is not possible ingeneral to separate 120577(radic119904) with a factor containing exactly allthe real roots of 120577(radic119904) as stated in the following lemma

Lemma 7 Given a polynomial 119901 isin R119899[119904] there does not exist

a general procedure of factorization of 119901 in polynomials 119901119877isin

R[119904] and 119901119862isin R[119904] 119901(119904) = 119901

119877(119904)119901

119862(119904) such that 119901

119877includes

exactly all the real roots of 119901 (and 119901

119862all the pairs of complex

conjugated roots of 119901 not in R)

6 Mathematical Problems in Engineering

Proof We prove this assertion by a counterexample withoutloss of generality we assume that 119901 is monic Let us assumethat 119901 has only real roots which are ordered as 120572

1lt 120572

2lt

sdot sdot sdot lt 120572

119898 119898 le 119899 Always we can find a value 120575 isin R such

that 1205721lt 120575 lt 120572

2 We make the change of variable 119904 997891rarr 119904 + 120575

resulting into a polynomial 119901120575(119904) = 119901(119904 + 120575) The roots of

119901

120575are located at 1205721015840

119896= 120572

119896minus 120575 and it is easy to see that

120572

1015840

1lt 0 and 120572

1015840

119896gt 0 for 119896 = 2 119898 Then we apply the map

119904 997891rarr 119904

2 which produces a polynomial 119902(119904) = 119901

120575(119904

2) with a

pair of pure conjugated complex roots at plusmniradic120575 minus 119886

1and pairs

of symmetric real roots at plusmnradic119886

119896minus 120575 for 119896 = 2 119898 Under

the hypothesis of separability 119902(119904) = 119902

119877(119904)119902

119862(119904) and undoing

in reverse order the changes of variables 119904 997891rarr radic119904 997891rarr 119904+120575 weobtain that 119902

119877(

radic

119904 minus 120575) = (119904 minus 120572

1) and 119902

119862(

radic

119904 minus 120575) = 119901(119904)(119904 minus

120572

1) Therefore we might isolate roots of 119901 by successively

repeating the procedure onto 119902119862This is a contradiction since

from the Galois theory it is well known that it is not alwayspossible to solve a polynomial equation by radicals

3 Conclusions

We overcome the drawbacks of the original Schur-Cohncriterion for delay systems namely the high degree ofthe polynomial that emerges from the Schur-Cohn matrixand the difficulty of detection of complex roots especiallypure imaginary roots due to the propagation of numericalerrors In particular we decrease the degree of the generatedpolynomial to a half and we reduce the problem numericallyto that of seeking the real roots of this polynomial notthe complex ones These properties are exactly those thatmake the Rekasius substitution criterion so attractive in thestability theory of delay systems Therefore from a practicalviewpoint the modified Schur-Cohn criterion is comparableto other widely used criteria in the literature Our methodis based on making transformations on the Schur-Cohnmatrix via triangular matrices over a polynomial ring inone unknown This leads to the determinant of the Jurymatrix over a polynomial ring which is proved to be an evenpolynomial The pure imaginary roots of the Schur-Cohnpolynomial are those of the even polynomial and in virtueof the change of variable 120596 997891rarr radic120596 we only need to seekreal roots which is more precise from a numerical viewpointwhile the degree of the polynomial is halved

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vyhlidal J-F Lafay and R Sipahi Delay Systems FromTheory to Numerics and Applications Advances in Delays andDynamics Springer New York NY USA 2014

[2] K Gu J Chen and V L Kharitonov Stability of Time-DelaySystems Birkhaauser Boston Mass USA 2003

[3] C Bonnet A R Fioravanti and J R Partington ldquoStability ofneutral systems with commensurate delays and poles asymp-totic to the imaginary axisrdquo SIAM Journal on Control andOptimization vol 49 no 2 pp 498ndash516 2011

[4] H Fazelinia R Sipahi and N Olgac ldquoStability robustness anal-ysis of multiple time-delayed systems using ldquobuilding blockrdquoconceptrdquo IEEE Transactions on Automatic Control vol 52 no5 pp 799ndash810 2007

[5] A R Fioravanti C Bonnet and H Ozbay ldquoStability offractional neutral systems with multiple delays and polesasymptotic to the imaginary axisrdquo in Proceedings of the 49thIEEEConference onDecision andControl pp 31ndash35 Atlanta GaUSA December 2010

[6] A R Fioravanti C Bonnet H Ozbay and S-I Niculescu ldquoAnumerical method to find stability windows and unstable polesfor linear neutral time-delay systemsrdquo in Proceedings of the 9thIFAC Workshop on Time Delay Systems (TDS rsquo10) pp 183ndash188Prague Czech Republic June 2010

[7] A R Fioravanti C Bonnet H Ozbay and S-I NiculesculdquoA numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systemsrdquoAutomatica vol 48 no 11 pp 2824ndash2830 2012

[8] N Olgac and R Sipahi ldquoA practical method for analyzing thestability of neutral type LTI-time delayed systemsrdquo Automaticavol 40 no 5 pp 847ndash853 2004

[9] R Sipahi ldquoDelay-margin design for the general class of single-delay retarded-type LTI systemsrdquo International Journal ofDynamics and Control vol 2 no 2 pp 198ndash209 2014

[10] S Barnett Polynomials and Linear Control Systems MarcelDekker New York NY USA 1983

[11] J Chen G Gu and C N Nett ldquoA new method for computingdelay margins for stability of linear delay systemsrdquo Systems andControl Letters vol 26 no 2 pp 107ndash117 1995

[12] P Fu S-I Niculescu and J Chen ldquoStability of linear neutraltime-delay systems exact conditions via matrix pencil solu-tionsrdquo IEEE Transactions on Automatic Control vol 51 no 6pp 1063ndash1069 2006

[13] K Walton and J E Marshall ldquoDirect Method for TDS StabilityAnalysisrdquo IEE Proceedings D Control Theory and Applicationsvol 134 no 2 pp 101ndash107 1987

[14] J H Su ldquoThe asymptotic stability of linear autonomous systemswith commensurate time delaysrdquo IEEE Transactions on Auto-matic Control vol 40 no 6 pp 1114ndash1117 1995

[15] J Louisell ldquoAmatrixmethod for determining the imaginary axiseigenvalues of a delay systemrdquo IEEE Transactions on AutomaticControl vol 46 no 12 pp 2008ndash2012 2001

[16] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5 pp 793ndash797 2002

[17] Z V Rekasius ldquoA stability test for systems with delaysrdquo inProceedings of the Joint Automatic Control Conference 1980paper no TP9-A

[18] A Thowsen ldquoThe Routh-Hurwitz method for stability deter-mination of linear differential-difference systemsrdquo InternationalJournal of Control vol 33 no 5 pp 991ndash995 1981

[19] R Sipahi and N Olgac ldquoA comparative survey in determiningthe imaginary characteristic roots of LTI time delayed systemsrdquoin Proceedings of the 16th Triennial World Congress of Interna-tional Federation of Automatic Control (IFAC rsquo05) vol 16 pp390ndash399 Prague Czech Republic July 2005

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119897

119879

119902minus1(119911) = (1 119911 119911

2 119911

119902minus1) the homogeneous system in (7)

is the starting point of the Schur-Cohn method If we makethe change of unknowns e

119894997891rarr 119869e119894and we take into account

the idempotence of 119869 (6) can be transformed into

(

119871

119902119880

119902

119869119880

119902119869119871

119902

)(

119869e1

119869e2

) = (

0

0

) (7)

It is easily verified that119880lowast119902119871

119902is of the same structure as119871

119902 and

in particular it is symmetric Therefore 119880lowast119902119871

119902= (119880

lowast

119902119871

119902)

119879

=

119871

lowast

119902119880

119902 As a consequence we obtain

(

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119868 119874

119874 119869

)(

119871

119902119880

119902

119869119880

119902119869119871

119902

)

= (

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119871

119902119880

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119874

119880

119902119871

119902

)

(8)

The homogeneous system (7) has a nontrivial solution solong as det( 119871119902 119880119902

119869119880119902119869119871119902

) = 0 We use the fact that

det((119871lowast

119902minus119880

lowast

119902

119874 119868

)(

119868 119874

119874 119869

))

= det(119871lowast

119902minus119880

lowast

119902119869

119874 119869

) = det (119871lowast119902119869) = det (119871

119902) (minus1)

119902

(9)

and taking determinants in (8) yields

det (119871119902) (minus1)

119902 det( 119871

119902119880

119902

119869119880

119902119869119871

119902

)

= det (119871119902) det (119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902)

(10)

As a result

det( 119871

119902119880

119902

119869119880

119902119869119871

119902

) = (minus1)

119902 det (119871lowast119902119871

119902minus 119880

lowast

119902119880

119902) (11)

Therefore the problem has been reduced to find the pureimaginary roots of det(119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902) which is just the

determinant of the Jury matrix 119869119902(119903(120596)) over C[120596] defined as

119869

119902(119903 (120596)) = (

119871

119902119880

119902

119880

119902119871

119902

) isin C2119902times2119902

[120596] (12)

This can be seen by a similar argument as above in (8)

(

119871

lowast

119902minus119880

lowast

119902

119874 119868

)(

119871

119902119880

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119871

lowast

119902119880

119902minus 119880

lowast

119902119871

119902

119880

119902119871

119902

)

= (

119871

lowast

119902119871

119902minus 119880

lowast

119902119880

119902119874

119880

119902119871

119902

)

(13)

and det(119869119902(119903(120596))) = det(119871lowast

119902119871

119902minus 119880

lowast

119902119880

119902)

In the sequel we define the polynomial 120577(120596) =

det(119869119902(119903(120596))) and the set of zeros of a polynomial119901 is denoted

asZ(119901)

Proposition 1 The real polynomial 120577(120596) is even

Proof We prove that 120577(120582) = 0 if and only if 120577(minus120582) = 0 Firstof all note that 119887

119894(minus120596) = 119887

119894(120596) and then 119903(minus120596 119911) = 119903(120596 119911)

As a consequence

119880

119902(119903 (120596)) = 119880

119902(119903 (120596)) = 119880

119902(119903 (minus120596))

119871

119902(119903 (120596)) = 119871

119902(119903 (120596)) = 119871

119902(119903 (minus120596))

(14)

Thus

119871

lowast

119902(119903 (minus120596)) 119871119902 (

119903 (minus120596)) minus 119880

lowast

119902(119903 (minus120596))119880119902 (

119903 (minus120596))

= 119871

119879

119902(119903 (120596)) 119871

119902(119903 (120596)) minus 119880

119879

119902(119903 (120596)) 119880

119902(119903 (120596))

(15)

and 120577(120582) = 120577(minus120582)

From the above development we obtain the followingcriterion

Criterion 1 Let 1205960isin R+ be a real root of 120577(radic119904) then plusmn119895

radic120596

0

is a pair of pure imaginary roots of the LTI-TDS system in (1)

Proof It is immediate that plusmn119895120596 is a pair of pure imaginaryroots of 119901

119888(119904 119911 119911

119902) if and only if 120596 is a positive real root

of119901119888(119895120596 119911 119911

119902) According to the Schur-Cohnmethod the

roots of 119901119888(119895120596 119911 119911

119902) can be determined by searching for

the real roots in the system (7) Additionally it was shownabove that the solution of (7) implies the computing of theroots of the even polynomial 120577(120596) henceforth if 120596

0isin R+ is

a root of 120577(radic119904) plusmnradic120596

0is a pair of real roots of 120577(119904) and plusmn119895120596

0

is a pair of pure imaginary roots of 119901119888(119904 119911 119911

119902)

Note that this criterion is advantageous since firstly weonly need to compute real roots and secondly deg(120577(radicsdot)) = 119902In the following the set of crossing frequencies is defined as

Ω = 120596 isin R+ 120596

2isin Z (120577 (radicsdot)) capR

+ (16)

Example 2 Let us consider the LTI-TDS system (119905) =

119860

0119909(119905) + 119860

1119909(119905 minus 120591) where the system matrices are defined

as

119860

0= (

minus1 135 minus1

minus3 minus1 minus2

minus2 minus1 minus4

) 119860

1= (

minus59 71 minus703

2 minus1 5

2 0 6

)

(17)

The quasi-polynomial characteristic is

119901

119888(119904 119890

minus119904120591 119890

minus119898119904120591)

= 119886

3(119904) 119890

minus3120591119904+ 119886

2(119904) 119890

minus2120591119904+ 119886

1(119904) 119890

minus120591119904+ 119886

0(119904)

(18)

4 Mathematical Problems in Engineering

with

119886

0(119904) = 119904

3+ 6119904

2+ 455119904 + 111

119886

1 (119904) = 09119904

2minus 1168119904 minus 221

119886

2(119904) = 909119904 minus 1851

119886

3(119904) = 1194

(19)

In the frequency domain (119904 = 119895120596) we define the bivariatepolynomial 119903(120596 119911) = sum

3

119894=0119886

119894(119895120596)119911

119894 and from (19) we buildthe triangular matrices 119880

3(119903) and 119871

3(119903) as follows

119880

3= (

1194 minus1851 minus09120596

2minus 221

0 1194 minus1851

0 0 1194

)

+ 119895120596(

0 minus909 1168

0 0 minus909

0 0 0

)

119871

3= (

0 0 1110 minus 60120596

2

0 1110 minus 60120596

2minus09120596

2minus 221

1110 minus 60120596

2minus09120596

2minus 221 minus1851

)

+ 119895120596(

0 0 455 minus 120596

2

0 455 minus 120596

2minus1168

455 minus 120596

2minus1168 909

)

(20)

Then we compute the polynomial 120577(radic119904) according to Crite-rion 1

120577 (radic119904) = 119904

9minus 16581119904

8minus 18682119904

7+ 24869 times 10

5119904

6

+ 35185119904

5minus 41197 times 10

9119904

4+ 94110 times 10

10119904

3

minus 71137 times 10

11119904

2+ 18933 times 10

12119904

minus 10144 times 10

12

(21)

whose positive real roots are Z(120577(radicsdot)) cap R+ =

84802 44564 92141 070634 24035 and then

plusmn119895Ω = plusmn11989529121 plusmn1198952111 plusmn11989530355 plusmn119895084044 plusmn11989515503

(22)

is the set of pure imaginary roots of the delayed system Notethat from the fundamental theoremof algebra there are only afinite number of crossing frequencies and thatZ(120577(radicsdot))capR+

represents the set of squares of crossing frequencies

The method presented above is the first step to test thestability of the characteristic quasi-polynomial 119901

119888(119904 119890

minus119904120591) and

to compute the delay margin for LTI delay systems It iscommon practice to make the following assumption on 119901

119888

Assumption 3 The characteristic quasi-polynomial 119901119888is sta-

ble at 120591 = 0 that is Z(119901

119888(sdot 0)) sub Cminus where Cminus = 119904 isin C

Re119904 lt 0

Recall that the delay margin 120591 is the smallest deviation of120591 from 120591 = 0 such that the system becomes unstable

120591 = min 120591 ge 0 119901

119888(119895120596 119890

minus119895120596120591) = 0 for some 120596 isin R

+

(23)

For any 120591 isin [0 120591) the system is stable and whenever 120591 = infinthe system is said to be stable independent of delay For eachfinite delay 120591 we can associate a crossing frequency whichrepresents the first contact or crossing of the roots of 119901

119888from

the stable region to the unstable oneThedetermination of thedelay margin requires solving the bivariate polynomial 119901

119888isin

C[120596 119911] in two phases (i) computing the positive real roots of120577(radic119904) so as to obtain the crossing frequencies 120596 isin Ω of 119901

119888

(ii) finding the roots of 119901119888(119895120596) isin C[119911] on the boundary of the

unit disk D (denoted as 120597D) at each crossing frequency 120596 Adetailed analysis of the delay margin 120591 can be found in [11]and in Theorem 211 pp 59-60 in [2] Under the assumptionthat 119901

119888is stable at 120591 = 0 the two-step stability method is

summarized in Table 1

Example 4 Taking into account the positive crossing fre-quencies 120596

119896isin Ω computed for the LTI-TDS system in

Example 2 we can derive the set of roots of 119901119888(119895120596

119896) isin C[119911]

onto the unit circle 120597D Let 119911119896

isin 120597D cap Z(119901

119888(119895120596

119896)) from

the identity 119911

119896= 119890

minus119895120596119896120591

= 119890

119895 arg(119911119896) it follows that 120591

119896=

(2120587minus arg(119911119896))120596

119896(with arg(119911

119896) isin [0 2120587]) and the system can

cross the stability boundary 120597C = 119895R from the stable half-plane Cminus to the unstable half-plane C+ = C (Cminus cup 120597C) orvice versa at 120591

119896 If the root crosses 120597C from C+ to Cminus at 120591

119896

there exists a 120591 lt 120591

119896such that makes the root to cross from

Cminus to C+ As a consequence the stability margin is defined as

120591= min1le119896le119902

2120587 minus arg (119911119896)

120596

119896

120596

119896isin Ω 119911

119896isin 120597D capZ (119901

119888(119895120596

119896))

(24)

Under Assumption 3 the system is unstable at 120591 = 120591 andat least a root crosses 120597C from Cminus to C+ at 120591 If we choose120591 isin [0 120591) then 120591120596

119896= 120579

119896and 119901

119888(119895120596

119896 119890

minus119895120596120591) = 0 for all

120596 isin R+ Therefore 119901119888(119904 119890

minus119904120591) is stable for all 120591 isin [0 120591) For

each crossing frequency120596119896a delay 120591

119896is computed as shown in

Table 2 The delay margin is then computed as the minimumvalue of these delays that is 120591 = min120591

119896 119896 = 1 5 =

016230The crossing direction of those roots onto the imaginary

axis (either from stable to unstable complex plane or viceversa) as 120591 increases is a quantity called the root tendency(RT) [7 8] For a cross frequency 120596

0and a delay 120591

0 the root

tendency is defined as

(RT)120596120591

= sgn(Re 120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

) (25)

where sgn(sdot) denotes the sign function In the followinglemma we propose a simple method to compute the roottendency based on the implicit function theorem

Lemma 5 (root tendency) Let 119901119888be a characteristic quasi-

polynomial as defined in (3) and let 120591

0be the delay for

Mathematical Problems in Engineering 5

Table 1 Stability analysis of the quasipolynomial 119901119888in terms of the delay margin 120591

Set of squares of crossing frequencies Solutions of 119901119888(120596) isin C[119911] on 120597D 120596 isin Ω Stability of the quasi-polynomial 119901

119888isin C[120596 119911]

Z (120577 (radicsdot)) capR+ =

Stable independent of delay (120591 = infin)Z (120577 (radicsdot)) capR+ = (0)

Z (120577 (radicsdot)) capR+ = Z (119902) capR+ = (0)

Z (119901

119888(119895120596)) cap 120597D =

Z (119901

119888(119895120596)) cap 120597D = Stable dependent of delay (120591 lt infin)

Table 2 Delays corresponding to the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Roots of 119901

119888(119895120596

119896) on 120597D (119911

119896= 119890

minus119895120596119896120591) Phase of 119911119896(120579119896) Delay (120591

119896)

29121 085692 minus 119895051548 57416 0185972111 minus026768 minus 119895096352 44414 08724730355 088110 minus 119895047297 57905 016230084044 097521 + 119895022129 022314 7210615503 minus095537 + 119895029541 28417 022199

Table 3 Root tendency for the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Delay (120591

119896) sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591

0119886

119896(119895120596

0))119890

minus11989512059601198961205910sum

119902

119896=1119896119886

119896(119895120596

0)119890

minus11989511989612059601205910(119877119879)

120596120591

29121 018597 minus48520 + 11989530258 74136 minus 11989558701 minus1

2111 087247 minus14062 + 11989534286 14878 minus 11989518237 +1

30355 016230 minus47511 + 11989527405 94101 minus 11989541172 +1

084044 72106 87231 minus 11989569290 minus11943 + 11989599556 minus1

15503 022199 minus90235 minus 11989591993 18255 + 11989544754 +1

which the system crosses the stability boundary at 1198951205960 for

some 120596

0isin Ω that is 119901

119888(119895120596

0 120591

0) = 0 Let us assume

that (120597119901119888(119904 120591)120597119904)|

(11989512059601205910)= sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591119886

119896(120596

0))

119890

minus11989512059601198961205910

= 0 Then the root tendency of 1198951205960is given by

( RT )

12059601205910

= minus sgn( Im((

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910)

times (

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0))

times119890

minus11989511989612059601205910)

minus1

))

(26)

Proof In virtue of the implicit function theorem there exista 120575 gt 0 a neighborhood 119880

0sub C of 119895120596

0 and a continuous

function 119904 (1205910minus120575 120591

0+120575) rarr 119880

0such that 119901

119888(119904(120591) 120591) = 0 for

all 120591 isin (120591

0minus120575 120591

0+120575) this is also obvious from the continuity

of the roots 119904(sdot)with respect to the delay 120591 Furthermore since119901

119888is differentiable and (120597119901

119888(119904 120591)120597119904)|

(11989512059601205910)

= 0 it is clear that

120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus

120597119901

119888(119904 120591) 120597120591

1003816

1003816

1003816

1003816(11989512059601205910)

120597119901

119888(119904 120591) 120597119904

1003816

1003816

1003816

1003816(11989512059601205910)

(27)

where

120597119901

119888(119904 120591)

120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

=

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0)) 119890

minus11989511989612059601205910

+ 119901

119888(119895120596

0 120591

0)

120597119901

119888(119904 120591)

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus119895120596

0

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910

(28)

Owing to 119901

119888(119895120596

0 120591

0) = 0 and from the chain rule of

derivation the result in (26) follows immediately

Example 6 Following with Example 2 and using the roottendency lemma we can compute the crossing direction foreach crossing frequency as shown in the fifth column inTable 3 RT = minus1 indicates that the crossing is from C+ toCminus and RT = +1 implies a crossing from Cminus to C+

We finish this analysis by noting that it is not possible ingeneral to separate 120577(radic119904) with a factor containing exactly allthe real roots of 120577(radic119904) as stated in the following lemma

Lemma 7 Given a polynomial 119901 isin R119899[119904] there does not exist

a general procedure of factorization of 119901 in polynomials 119901119877isin

R[119904] and 119901119862isin R[119904] 119901(119904) = 119901

119877(119904)119901

119862(119904) such that 119901

119877includes

exactly all the real roots of 119901 (and 119901

119862all the pairs of complex

conjugated roots of 119901 not in R)

6 Mathematical Problems in Engineering

Proof We prove this assertion by a counterexample withoutloss of generality we assume that 119901 is monic Let us assumethat 119901 has only real roots which are ordered as 120572

1lt 120572

2lt

sdot sdot sdot lt 120572

119898 119898 le 119899 Always we can find a value 120575 isin R such

that 1205721lt 120575 lt 120572

2 We make the change of variable 119904 997891rarr 119904 + 120575

resulting into a polynomial 119901120575(119904) = 119901(119904 + 120575) The roots of

119901

120575are located at 1205721015840

119896= 120572

119896minus 120575 and it is easy to see that

120572

1015840

1lt 0 and 120572

1015840

119896gt 0 for 119896 = 2 119898 Then we apply the map

119904 997891rarr 119904

2 which produces a polynomial 119902(119904) = 119901

120575(119904

2) with a

pair of pure conjugated complex roots at plusmniradic120575 minus 119886

1and pairs

of symmetric real roots at plusmnradic119886

119896minus 120575 for 119896 = 2 119898 Under

the hypothesis of separability 119902(119904) = 119902

119877(119904)119902

119862(119904) and undoing

in reverse order the changes of variables 119904 997891rarr radic119904 997891rarr 119904+120575 weobtain that 119902

119877(

radic

119904 minus 120575) = (119904 minus 120572

1) and 119902

119862(

radic

119904 minus 120575) = 119901(119904)(119904 minus

120572

1) Therefore we might isolate roots of 119901 by successively

repeating the procedure onto 119902119862This is a contradiction since

from the Galois theory it is well known that it is not alwayspossible to solve a polynomial equation by radicals

3 Conclusions

We overcome the drawbacks of the original Schur-Cohncriterion for delay systems namely the high degree ofthe polynomial that emerges from the Schur-Cohn matrixand the difficulty of detection of complex roots especiallypure imaginary roots due to the propagation of numericalerrors In particular we decrease the degree of the generatedpolynomial to a half and we reduce the problem numericallyto that of seeking the real roots of this polynomial notthe complex ones These properties are exactly those thatmake the Rekasius substitution criterion so attractive in thestability theory of delay systems Therefore from a practicalviewpoint the modified Schur-Cohn criterion is comparableto other widely used criteria in the literature Our methodis based on making transformations on the Schur-Cohnmatrix via triangular matrices over a polynomial ring inone unknown This leads to the determinant of the Jurymatrix over a polynomial ring which is proved to be an evenpolynomial The pure imaginary roots of the Schur-Cohnpolynomial are those of the even polynomial and in virtueof the change of variable 120596 997891rarr radic120596 we only need to seekreal roots which is more precise from a numerical viewpointwhile the degree of the polynomial is halved

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vyhlidal J-F Lafay and R Sipahi Delay Systems FromTheory to Numerics and Applications Advances in Delays andDynamics Springer New York NY USA 2014

[2] K Gu J Chen and V L Kharitonov Stability of Time-DelaySystems Birkhaauser Boston Mass USA 2003

[3] C Bonnet A R Fioravanti and J R Partington ldquoStability ofneutral systems with commensurate delays and poles asymp-totic to the imaginary axisrdquo SIAM Journal on Control andOptimization vol 49 no 2 pp 498ndash516 2011

[4] H Fazelinia R Sipahi and N Olgac ldquoStability robustness anal-ysis of multiple time-delayed systems using ldquobuilding blockrdquoconceptrdquo IEEE Transactions on Automatic Control vol 52 no5 pp 799ndash810 2007

[5] A R Fioravanti C Bonnet and H Ozbay ldquoStability offractional neutral systems with multiple delays and polesasymptotic to the imaginary axisrdquo in Proceedings of the 49thIEEEConference onDecision andControl pp 31ndash35 Atlanta GaUSA December 2010

[6] A R Fioravanti C Bonnet H Ozbay and S-I Niculescu ldquoAnumerical method to find stability windows and unstable polesfor linear neutral time-delay systemsrdquo in Proceedings of the 9thIFAC Workshop on Time Delay Systems (TDS rsquo10) pp 183ndash188Prague Czech Republic June 2010

[7] A R Fioravanti C Bonnet H Ozbay and S-I NiculesculdquoA numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systemsrdquoAutomatica vol 48 no 11 pp 2824ndash2830 2012

[8] N Olgac and R Sipahi ldquoA practical method for analyzing thestability of neutral type LTI-time delayed systemsrdquo Automaticavol 40 no 5 pp 847ndash853 2004

[9] R Sipahi ldquoDelay-margin design for the general class of single-delay retarded-type LTI systemsrdquo International Journal ofDynamics and Control vol 2 no 2 pp 198ndash209 2014

[10] S Barnett Polynomials and Linear Control Systems MarcelDekker New York NY USA 1983

[11] J Chen G Gu and C N Nett ldquoA new method for computingdelay margins for stability of linear delay systemsrdquo Systems andControl Letters vol 26 no 2 pp 107ndash117 1995

[12] P Fu S-I Niculescu and J Chen ldquoStability of linear neutraltime-delay systems exact conditions via matrix pencil solu-tionsrdquo IEEE Transactions on Automatic Control vol 51 no 6pp 1063ndash1069 2006

[13] K Walton and J E Marshall ldquoDirect Method for TDS StabilityAnalysisrdquo IEE Proceedings D Control Theory and Applicationsvol 134 no 2 pp 101ndash107 1987

[14] J H Su ldquoThe asymptotic stability of linear autonomous systemswith commensurate time delaysrdquo IEEE Transactions on Auto-matic Control vol 40 no 6 pp 1114ndash1117 1995

[15] J Louisell ldquoAmatrixmethod for determining the imaginary axiseigenvalues of a delay systemrdquo IEEE Transactions on AutomaticControl vol 46 no 12 pp 2008ndash2012 2001

[16] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5 pp 793ndash797 2002

[17] Z V Rekasius ldquoA stability test for systems with delaysrdquo inProceedings of the Joint Automatic Control Conference 1980paper no TP9-A

[18] A Thowsen ldquoThe Routh-Hurwitz method for stability deter-mination of linear differential-difference systemsrdquo InternationalJournal of Control vol 33 no 5 pp 991ndash995 1981

[19] R Sipahi and N Olgac ldquoA comparative survey in determiningthe imaginary characteristic roots of LTI time delayed systemsrdquoin Proceedings of the 16th Triennial World Congress of Interna-tional Federation of Automatic Control (IFAC rsquo05) vol 16 pp390ndash399 Prague Czech Republic July 2005

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

with

119886

0(119904) = 119904

3+ 6119904

2+ 455119904 + 111

119886

1 (119904) = 09119904

2minus 1168119904 minus 221

119886

2(119904) = 909119904 minus 1851

119886

3(119904) = 1194

(19)

In the frequency domain (119904 = 119895120596) we define the bivariatepolynomial 119903(120596 119911) = sum

3

119894=0119886

119894(119895120596)119911

119894 and from (19) we buildthe triangular matrices 119880

3(119903) and 119871

3(119903) as follows

119880

3= (

1194 minus1851 minus09120596

2minus 221

0 1194 minus1851

0 0 1194

)

+ 119895120596(

0 minus909 1168

0 0 minus909

0 0 0

)

119871

3= (

0 0 1110 minus 60120596

2

0 1110 minus 60120596

2minus09120596

2minus 221

1110 minus 60120596

2minus09120596

2minus 221 minus1851

)

+ 119895120596(

0 0 455 minus 120596

2

0 455 minus 120596

2minus1168

455 minus 120596

2minus1168 909

)

(20)

Then we compute the polynomial 120577(radic119904) according to Crite-rion 1

120577 (radic119904) = 119904

9minus 16581119904

8minus 18682119904

7+ 24869 times 10

5119904

6

+ 35185119904

5minus 41197 times 10

9119904

4+ 94110 times 10

10119904

3

minus 71137 times 10

11119904

2+ 18933 times 10

12119904

minus 10144 times 10

12

(21)

whose positive real roots are Z(120577(radicsdot)) cap R+ =

84802 44564 92141 070634 24035 and then

plusmn119895Ω = plusmn11989529121 plusmn1198952111 plusmn11989530355 plusmn119895084044 plusmn11989515503

(22)

is the set of pure imaginary roots of the delayed system Notethat from the fundamental theoremof algebra there are only afinite number of crossing frequencies and thatZ(120577(radicsdot))capR+

represents the set of squares of crossing frequencies

The method presented above is the first step to test thestability of the characteristic quasi-polynomial 119901

119888(119904 119890

minus119904120591) and

to compute the delay margin for LTI delay systems It iscommon practice to make the following assumption on 119901

119888

Assumption 3 The characteristic quasi-polynomial 119901119888is sta-

ble at 120591 = 0 that is Z(119901

119888(sdot 0)) sub Cminus where Cminus = 119904 isin C

Re119904 lt 0

Recall that the delay margin 120591 is the smallest deviation of120591 from 120591 = 0 such that the system becomes unstable

120591 = min 120591 ge 0 119901

119888(119895120596 119890

minus119895120596120591) = 0 for some 120596 isin R

+

(23)

For any 120591 isin [0 120591) the system is stable and whenever 120591 = infinthe system is said to be stable independent of delay For eachfinite delay 120591 we can associate a crossing frequency whichrepresents the first contact or crossing of the roots of 119901

119888from

the stable region to the unstable oneThedetermination of thedelay margin requires solving the bivariate polynomial 119901

119888isin

C[120596 119911] in two phases (i) computing the positive real roots of120577(radic119904) so as to obtain the crossing frequencies 120596 isin Ω of 119901

119888

(ii) finding the roots of 119901119888(119895120596) isin C[119911] on the boundary of the

unit disk D (denoted as 120597D) at each crossing frequency 120596 Adetailed analysis of the delay margin 120591 can be found in [11]and in Theorem 211 pp 59-60 in [2] Under the assumptionthat 119901

119888is stable at 120591 = 0 the two-step stability method is

summarized in Table 1

Example 4 Taking into account the positive crossing fre-quencies 120596

119896isin Ω computed for the LTI-TDS system in

Example 2 we can derive the set of roots of 119901119888(119895120596

119896) isin C[119911]

onto the unit circle 120597D Let 119911119896

isin 120597D cap Z(119901

119888(119895120596

119896)) from

the identity 119911

119896= 119890

minus119895120596119896120591

= 119890

119895 arg(119911119896) it follows that 120591

119896=

(2120587minus arg(119911119896))120596

119896(with arg(119911

119896) isin [0 2120587]) and the system can

cross the stability boundary 120597C = 119895R from the stable half-plane Cminus to the unstable half-plane C+ = C (Cminus cup 120597C) orvice versa at 120591

119896 If the root crosses 120597C from C+ to Cminus at 120591

119896

there exists a 120591 lt 120591

119896such that makes the root to cross from

Cminus to C+ As a consequence the stability margin is defined as

120591= min1le119896le119902

2120587 minus arg (119911119896)

120596

119896

120596

119896isin Ω 119911

119896isin 120597D capZ (119901

119888(119895120596

119896))

(24)

Under Assumption 3 the system is unstable at 120591 = 120591 andat least a root crosses 120597C from Cminus to C+ at 120591 If we choose120591 isin [0 120591) then 120591120596

119896= 120579

119896and 119901

119888(119895120596

119896 119890

minus119895120596120591) = 0 for all

120596 isin R+ Therefore 119901119888(119904 119890

minus119904120591) is stable for all 120591 isin [0 120591) For

each crossing frequency120596119896a delay 120591

119896is computed as shown in

Table 2 The delay margin is then computed as the minimumvalue of these delays that is 120591 = min120591

119896 119896 = 1 5 =

016230The crossing direction of those roots onto the imaginary

axis (either from stable to unstable complex plane or viceversa) as 120591 increases is a quantity called the root tendency(RT) [7 8] For a cross frequency 120596

0and a delay 120591

0 the root

tendency is defined as

(RT)120596120591

= sgn(Re 120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

) (25)

where sgn(sdot) denotes the sign function In the followinglemma we propose a simple method to compute the roottendency based on the implicit function theorem

Lemma 5 (root tendency) Let 119901119888be a characteristic quasi-

polynomial as defined in (3) and let 120591

0be the delay for

Mathematical Problems in Engineering 5

Table 1 Stability analysis of the quasipolynomial 119901119888in terms of the delay margin 120591

Set of squares of crossing frequencies Solutions of 119901119888(120596) isin C[119911] on 120597D 120596 isin Ω Stability of the quasi-polynomial 119901

119888isin C[120596 119911]

Z (120577 (radicsdot)) capR+ =

Stable independent of delay (120591 = infin)Z (120577 (radicsdot)) capR+ = (0)

Z (120577 (radicsdot)) capR+ = Z (119902) capR+ = (0)

Z (119901

119888(119895120596)) cap 120597D =

Z (119901

119888(119895120596)) cap 120597D = Stable dependent of delay (120591 lt infin)

Table 2 Delays corresponding to the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Roots of 119901

119888(119895120596

119896) on 120597D (119911

119896= 119890

minus119895120596119896120591) Phase of 119911119896(120579119896) Delay (120591

119896)

29121 085692 minus 119895051548 57416 0185972111 minus026768 minus 119895096352 44414 08724730355 088110 minus 119895047297 57905 016230084044 097521 + 119895022129 022314 7210615503 minus095537 + 119895029541 28417 022199

Table 3 Root tendency for the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Delay (120591

119896) sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591

0119886

119896(119895120596

0))119890

minus11989512059601198961205910sum

119902

119896=1119896119886

119896(119895120596

0)119890

minus11989511989612059601205910(119877119879)

120596120591

29121 018597 minus48520 + 11989530258 74136 minus 11989558701 minus1

2111 087247 minus14062 + 11989534286 14878 minus 11989518237 +1

30355 016230 minus47511 + 11989527405 94101 minus 11989541172 +1

084044 72106 87231 minus 11989569290 minus11943 + 11989599556 minus1

15503 022199 minus90235 minus 11989591993 18255 + 11989544754 +1

which the system crosses the stability boundary at 1198951205960 for

some 120596

0isin Ω that is 119901

119888(119895120596

0 120591

0) = 0 Let us assume

that (120597119901119888(119904 120591)120597119904)|

(11989512059601205910)= sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591119886

119896(120596

0))

119890

minus11989512059601198961205910

= 0 Then the root tendency of 1198951205960is given by

( RT )

12059601205910

= minus sgn( Im((

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910)

times (

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0))

times119890

minus11989511989612059601205910)

minus1

))

(26)

Proof In virtue of the implicit function theorem there exista 120575 gt 0 a neighborhood 119880

0sub C of 119895120596

0 and a continuous

function 119904 (1205910minus120575 120591

0+120575) rarr 119880

0such that 119901

119888(119904(120591) 120591) = 0 for

all 120591 isin (120591

0minus120575 120591

0+120575) this is also obvious from the continuity

of the roots 119904(sdot)with respect to the delay 120591 Furthermore since119901

119888is differentiable and (120597119901

119888(119904 120591)120597119904)|

(11989512059601205910)

= 0 it is clear that

120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus

120597119901

119888(119904 120591) 120597120591

1003816

1003816

1003816

1003816(11989512059601205910)

120597119901

119888(119904 120591) 120597119904

1003816

1003816

1003816

1003816(11989512059601205910)

(27)

where

120597119901

119888(119904 120591)

120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

=

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0)) 119890

minus11989511989612059601205910

+ 119901

119888(119895120596

0 120591

0)

120597119901

119888(119904 120591)

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus119895120596

0

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910

(28)

Owing to 119901

119888(119895120596

0 120591

0) = 0 and from the chain rule of

derivation the result in (26) follows immediately

Example 6 Following with Example 2 and using the roottendency lemma we can compute the crossing direction foreach crossing frequency as shown in the fifth column inTable 3 RT = minus1 indicates that the crossing is from C+ toCminus and RT = +1 implies a crossing from Cminus to C+

We finish this analysis by noting that it is not possible ingeneral to separate 120577(radic119904) with a factor containing exactly allthe real roots of 120577(radic119904) as stated in the following lemma

Lemma 7 Given a polynomial 119901 isin R119899[119904] there does not exist

a general procedure of factorization of 119901 in polynomials 119901119877isin

R[119904] and 119901119862isin R[119904] 119901(119904) = 119901

119877(119904)119901

119862(119904) such that 119901

119877includes

exactly all the real roots of 119901 (and 119901

119862all the pairs of complex

conjugated roots of 119901 not in R)

6 Mathematical Problems in Engineering

Proof We prove this assertion by a counterexample withoutloss of generality we assume that 119901 is monic Let us assumethat 119901 has only real roots which are ordered as 120572

1lt 120572

2lt

sdot sdot sdot lt 120572

119898 119898 le 119899 Always we can find a value 120575 isin R such

that 1205721lt 120575 lt 120572

2 We make the change of variable 119904 997891rarr 119904 + 120575

resulting into a polynomial 119901120575(119904) = 119901(119904 + 120575) The roots of

119901

120575are located at 1205721015840

119896= 120572

119896minus 120575 and it is easy to see that

120572

1015840

1lt 0 and 120572

1015840

119896gt 0 for 119896 = 2 119898 Then we apply the map

119904 997891rarr 119904

2 which produces a polynomial 119902(119904) = 119901

120575(119904

2) with a

pair of pure conjugated complex roots at plusmniradic120575 minus 119886

1and pairs

of symmetric real roots at plusmnradic119886

119896minus 120575 for 119896 = 2 119898 Under

the hypothesis of separability 119902(119904) = 119902

119877(119904)119902

119862(119904) and undoing

in reverse order the changes of variables 119904 997891rarr radic119904 997891rarr 119904+120575 weobtain that 119902

119877(

radic

119904 minus 120575) = (119904 minus 120572

1) and 119902

119862(

radic

119904 minus 120575) = 119901(119904)(119904 minus

120572

1) Therefore we might isolate roots of 119901 by successively

repeating the procedure onto 119902119862This is a contradiction since

from the Galois theory it is well known that it is not alwayspossible to solve a polynomial equation by radicals

3 Conclusions

We overcome the drawbacks of the original Schur-Cohncriterion for delay systems namely the high degree ofthe polynomial that emerges from the Schur-Cohn matrixand the difficulty of detection of complex roots especiallypure imaginary roots due to the propagation of numericalerrors In particular we decrease the degree of the generatedpolynomial to a half and we reduce the problem numericallyto that of seeking the real roots of this polynomial notthe complex ones These properties are exactly those thatmake the Rekasius substitution criterion so attractive in thestability theory of delay systems Therefore from a practicalviewpoint the modified Schur-Cohn criterion is comparableto other widely used criteria in the literature Our methodis based on making transformations on the Schur-Cohnmatrix via triangular matrices over a polynomial ring inone unknown This leads to the determinant of the Jurymatrix over a polynomial ring which is proved to be an evenpolynomial The pure imaginary roots of the Schur-Cohnpolynomial are those of the even polynomial and in virtueof the change of variable 120596 997891rarr radic120596 we only need to seekreal roots which is more precise from a numerical viewpointwhile the degree of the polynomial is halved

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vyhlidal J-F Lafay and R Sipahi Delay Systems FromTheory to Numerics and Applications Advances in Delays andDynamics Springer New York NY USA 2014

[2] K Gu J Chen and V L Kharitonov Stability of Time-DelaySystems Birkhaauser Boston Mass USA 2003

[3] C Bonnet A R Fioravanti and J R Partington ldquoStability ofneutral systems with commensurate delays and poles asymp-totic to the imaginary axisrdquo SIAM Journal on Control andOptimization vol 49 no 2 pp 498ndash516 2011

[4] H Fazelinia R Sipahi and N Olgac ldquoStability robustness anal-ysis of multiple time-delayed systems using ldquobuilding blockrdquoconceptrdquo IEEE Transactions on Automatic Control vol 52 no5 pp 799ndash810 2007

[5] A R Fioravanti C Bonnet and H Ozbay ldquoStability offractional neutral systems with multiple delays and polesasymptotic to the imaginary axisrdquo in Proceedings of the 49thIEEEConference onDecision andControl pp 31ndash35 Atlanta GaUSA December 2010

[6] A R Fioravanti C Bonnet H Ozbay and S-I Niculescu ldquoAnumerical method to find stability windows and unstable polesfor linear neutral time-delay systemsrdquo in Proceedings of the 9thIFAC Workshop on Time Delay Systems (TDS rsquo10) pp 183ndash188Prague Czech Republic June 2010

[7] A R Fioravanti C Bonnet H Ozbay and S-I NiculesculdquoA numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systemsrdquoAutomatica vol 48 no 11 pp 2824ndash2830 2012

[8] N Olgac and R Sipahi ldquoA practical method for analyzing thestability of neutral type LTI-time delayed systemsrdquo Automaticavol 40 no 5 pp 847ndash853 2004

[9] R Sipahi ldquoDelay-margin design for the general class of single-delay retarded-type LTI systemsrdquo International Journal ofDynamics and Control vol 2 no 2 pp 198ndash209 2014

[10] S Barnett Polynomials and Linear Control Systems MarcelDekker New York NY USA 1983

[11] J Chen G Gu and C N Nett ldquoA new method for computingdelay margins for stability of linear delay systemsrdquo Systems andControl Letters vol 26 no 2 pp 107ndash117 1995

[12] P Fu S-I Niculescu and J Chen ldquoStability of linear neutraltime-delay systems exact conditions via matrix pencil solu-tionsrdquo IEEE Transactions on Automatic Control vol 51 no 6pp 1063ndash1069 2006

[13] K Walton and J E Marshall ldquoDirect Method for TDS StabilityAnalysisrdquo IEE Proceedings D Control Theory and Applicationsvol 134 no 2 pp 101ndash107 1987

[14] J H Su ldquoThe asymptotic stability of linear autonomous systemswith commensurate time delaysrdquo IEEE Transactions on Auto-matic Control vol 40 no 6 pp 1114ndash1117 1995

[15] J Louisell ldquoAmatrixmethod for determining the imaginary axiseigenvalues of a delay systemrdquo IEEE Transactions on AutomaticControl vol 46 no 12 pp 2008ndash2012 2001

[16] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5 pp 793ndash797 2002

[17] Z V Rekasius ldquoA stability test for systems with delaysrdquo inProceedings of the Joint Automatic Control Conference 1980paper no TP9-A

[18] A Thowsen ldquoThe Routh-Hurwitz method for stability deter-mination of linear differential-difference systemsrdquo InternationalJournal of Control vol 33 no 5 pp 991ndash995 1981

[19] R Sipahi and N Olgac ldquoA comparative survey in determiningthe imaginary characteristic roots of LTI time delayed systemsrdquoin Proceedings of the 16th Triennial World Congress of Interna-tional Federation of Automatic Control (IFAC rsquo05) vol 16 pp390ndash399 Prague Czech Republic July 2005

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 Stability analysis of the quasipolynomial 119901119888in terms of the delay margin 120591

Set of squares of crossing frequencies Solutions of 119901119888(120596) isin C[119911] on 120597D 120596 isin Ω Stability of the quasi-polynomial 119901

119888isin C[120596 119911]

Z (120577 (radicsdot)) capR+ =

Stable independent of delay (120591 = infin)Z (120577 (radicsdot)) capR+ = (0)

Z (120577 (radicsdot)) capR+ = Z (119902) capR+ = (0)

Z (119901

119888(119895120596)) cap 120597D =

Z (119901

119888(119895120596)) cap 120597D = Stable dependent of delay (120591 lt infin)

Table 2 Delays corresponding to the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Roots of 119901

119888(119895120596

119896) on 120597D (119911

119896= 119890

minus119895120596119896120591) Phase of 119911119896(120579119896) Delay (120591

119896)

29121 085692 minus 119895051548 57416 0185972111 minus026768 minus 119895096352 44414 08724730355 088110 minus 119895047297 57905 016230084044 097521 + 119895022129 022314 7210615503 minus095537 + 119895029541 28417 022199

Table 3 Root tendency for the crossing frequencies of the LTI-TDS system of the Example 2

Crossing frequency (120596119896) Delay (120591

119896) sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591

0119886

119896(119895120596

0))119890

minus11989512059601198961205910sum

119902

119896=1119896119886

119896(119895120596

0)119890

minus11989511989612059601205910(119877119879)

120596120591

29121 018597 minus48520 + 11989530258 74136 minus 11989558701 minus1

2111 087247 minus14062 + 11989534286 14878 minus 11989518237 +1

30355 016230 minus47511 + 11989527405 94101 minus 11989541172 +1

084044 72106 87231 minus 11989569290 minus11943 + 11989599556 minus1

15503 022199 minus90235 minus 11989591993 18255 + 11989544754 +1

which the system crosses the stability boundary at 1198951205960 for

some 120596

0isin Ω that is 119901

119888(119895120596

0 120591

0) = 0 Let us assume

that (120597119901119888(119904 120591)120597119904)|

(11989512059601205910)= sum

119902

119896=0(119886

1015840

119896(119895120596

0) minus (119896 minus 1)120591119886

119896(120596

0))

119890

minus11989512059601198961205910

= 0 Then the root tendency of 1198951205960is given by

( RT )

12059601205910

= minus sgn( Im((

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910)

times (

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0))

times119890

minus11989511989612059601205910)

minus1

))

(26)

Proof In virtue of the implicit function theorem there exista 120575 gt 0 a neighborhood 119880

0sub C of 119895120596

0 and a continuous

function 119904 (1205910minus120575 120591

0+120575) rarr 119880

0such that 119901

119888(119904(120591) 120591) = 0 for

all 120591 isin (120591

0minus120575 120591

0+120575) this is also obvious from the continuity

of the roots 119904(sdot)with respect to the delay 120591 Furthermore since119901

119888is differentiable and (120597119901

119888(119904 120591)120597119904)|

(11989512059601205910)

= 0 it is clear that

120597119904

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus

120597119901

119888(119904 120591) 120597120591

1003816

1003816

1003816

1003816(11989512059601205910)

120597119901

119888(119904 120591) 120597119904

1003816

1003816

1003816

1003816(11989512059601205910)

(27)

where

120597119901

119888(119904 120591)

120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

=

119902

sum

119896=0

(119886

1015840

119896(119895120596

0) minus (119896 minus 1) 120591

0119886

119896(119895120596

0)) 119890

minus11989511989612059601205910

+ 119901

119888(119895120596

0 120591

0)

120597119901

119888(119904 120591)

120597120591

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816(11989512059601205910)

= minus119895120596

0

119902

sum

119896=1

119896119886

119896(119895120596

0) 119890

minus11989511989612059601205910

(28)

Owing to 119901

119888(119895120596

0 120591

0) = 0 and from the chain rule of

derivation the result in (26) follows immediately

Example 6 Following with Example 2 and using the roottendency lemma we can compute the crossing direction foreach crossing frequency as shown in the fifth column inTable 3 RT = minus1 indicates that the crossing is from C+ toCminus and RT = +1 implies a crossing from Cminus to C+

We finish this analysis by noting that it is not possible ingeneral to separate 120577(radic119904) with a factor containing exactly allthe real roots of 120577(radic119904) as stated in the following lemma

Lemma 7 Given a polynomial 119901 isin R119899[119904] there does not exist

a general procedure of factorization of 119901 in polynomials 119901119877isin

R[119904] and 119901119862isin R[119904] 119901(119904) = 119901

119877(119904)119901

119862(119904) such that 119901

119877includes

exactly all the real roots of 119901 (and 119901

119862all the pairs of complex

conjugated roots of 119901 not in R)

6 Mathematical Problems in Engineering

Proof We prove this assertion by a counterexample withoutloss of generality we assume that 119901 is monic Let us assumethat 119901 has only real roots which are ordered as 120572

1lt 120572

2lt

sdot sdot sdot lt 120572

119898 119898 le 119899 Always we can find a value 120575 isin R such

that 1205721lt 120575 lt 120572

2 We make the change of variable 119904 997891rarr 119904 + 120575

resulting into a polynomial 119901120575(119904) = 119901(119904 + 120575) The roots of

119901

120575are located at 1205721015840

119896= 120572

119896minus 120575 and it is easy to see that

120572

1015840

1lt 0 and 120572

1015840

119896gt 0 for 119896 = 2 119898 Then we apply the map

119904 997891rarr 119904

2 which produces a polynomial 119902(119904) = 119901

120575(119904

2) with a

pair of pure conjugated complex roots at plusmniradic120575 minus 119886

1and pairs

of symmetric real roots at plusmnradic119886

119896minus 120575 for 119896 = 2 119898 Under

the hypothesis of separability 119902(119904) = 119902

119877(119904)119902

119862(119904) and undoing

in reverse order the changes of variables 119904 997891rarr radic119904 997891rarr 119904+120575 weobtain that 119902

119877(

radic

119904 minus 120575) = (119904 minus 120572

1) and 119902

119862(

radic

119904 minus 120575) = 119901(119904)(119904 minus

120572

1) Therefore we might isolate roots of 119901 by successively

repeating the procedure onto 119902119862This is a contradiction since

from the Galois theory it is well known that it is not alwayspossible to solve a polynomial equation by radicals

3 Conclusions

We overcome the drawbacks of the original Schur-Cohncriterion for delay systems namely the high degree ofthe polynomial that emerges from the Schur-Cohn matrixand the difficulty of detection of complex roots especiallypure imaginary roots due to the propagation of numericalerrors In particular we decrease the degree of the generatedpolynomial to a half and we reduce the problem numericallyto that of seeking the real roots of this polynomial notthe complex ones These properties are exactly those thatmake the Rekasius substitution criterion so attractive in thestability theory of delay systems Therefore from a practicalviewpoint the modified Schur-Cohn criterion is comparableto other widely used criteria in the literature Our methodis based on making transformations on the Schur-Cohnmatrix via triangular matrices over a polynomial ring inone unknown This leads to the determinant of the Jurymatrix over a polynomial ring which is proved to be an evenpolynomial The pure imaginary roots of the Schur-Cohnpolynomial are those of the even polynomial and in virtueof the change of variable 120596 997891rarr radic120596 we only need to seekreal roots which is more precise from a numerical viewpointwhile the degree of the polynomial is halved

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vyhlidal J-F Lafay and R Sipahi Delay Systems FromTheory to Numerics and Applications Advances in Delays andDynamics Springer New York NY USA 2014

[2] K Gu J Chen and V L Kharitonov Stability of Time-DelaySystems Birkhaauser Boston Mass USA 2003

[3] C Bonnet A R Fioravanti and J R Partington ldquoStability ofneutral systems with commensurate delays and poles asymp-totic to the imaginary axisrdquo SIAM Journal on Control andOptimization vol 49 no 2 pp 498ndash516 2011

[4] H Fazelinia R Sipahi and N Olgac ldquoStability robustness anal-ysis of multiple time-delayed systems using ldquobuilding blockrdquoconceptrdquo IEEE Transactions on Automatic Control vol 52 no5 pp 799ndash810 2007

[5] A R Fioravanti C Bonnet and H Ozbay ldquoStability offractional neutral systems with multiple delays and polesasymptotic to the imaginary axisrdquo in Proceedings of the 49thIEEEConference onDecision andControl pp 31ndash35 Atlanta GaUSA December 2010

[6] A R Fioravanti C Bonnet H Ozbay and S-I Niculescu ldquoAnumerical method to find stability windows and unstable polesfor linear neutral time-delay systemsrdquo in Proceedings of the 9thIFAC Workshop on Time Delay Systems (TDS rsquo10) pp 183ndash188Prague Czech Republic June 2010

[7] A R Fioravanti C Bonnet H Ozbay and S-I NiculesculdquoA numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systemsrdquoAutomatica vol 48 no 11 pp 2824ndash2830 2012

[8] N Olgac and R Sipahi ldquoA practical method for analyzing thestability of neutral type LTI-time delayed systemsrdquo Automaticavol 40 no 5 pp 847ndash853 2004

[9] R Sipahi ldquoDelay-margin design for the general class of single-delay retarded-type LTI systemsrdquo International Journal ofDynamics and Control vol 2 no 2 pp 198ndash209 2014

[10] S Barnett Polynomials and Linear Control Systems MarcelDekker New York NY USA 1983

[11] J Chen G Gu and C N Nett ldquoA new method for computingdelay margins for stability of linear delay systemsrdquo Systems andControl Letters vol 26 no 2 pp 107ndash117 1995

[12] P Fu S-I Niculescu and J Chen ldquoStability of linear neutraltime-delay systems exact conditions via matrix pencil solu-tionsrdquo IEEE Transactions on Automatic Control vol 51 no 6pp 1063ndash1069 2006

[13] K Walton and J E Marshall ldquoDirect Method for TDS StabilityAnalysisrdquo IEE Proceedings D Control Theory and Applicationsvol 134 no 2 pp 101ndash107 1987

[14] J H Su ldquoThe asymptotic stability of linear autonomous systemswith commensurate time delaysrdquo IEEE Transactions on Auto-matic Control vol 40 no 6 pp 1114ndash1117 1995

[15] J Louisell ldquoAmatrixmethod for determining the imaginary axiseigenvalues of a delay systemrdquo IEEE Transactions on AutomaticControl vol 46 no 12 pp 2008ndash2012 2001

[16] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5 pp 793ndash797 2002

[17] Z V Rekasius ldquoA stability test for systems with delaysrdquo inProceedings of the Joint Automatic Control Conference 1980paper no TP9-A

[18] A Thowsen ldquoThe Routh-Hurwitz method for stability deter-mination of linear differential-difference systemsrdquo InternationalJournal of Control vol 33 no 5 pp 991ndash995 1981

[19] R Sipahi and N Olgac ldquoA comparative survey in determiningthe imaginary characteristic roots of LTI time delayed systemsrdquoin Proceedings of the 16th Triennial World Congress of Interna-tional Federation of Automatic Control (IFAC rsquo05) vol 16 pp390ndash399 Prague Czech Republic July 2005

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Proof We prove this assertion by a counterexample withoutloss of generality we assume that 119901 is monic Let us assumethat 119901 has only real roots which are ordered as 120572

1lt 120572

2lt

sdot sdot sdot lt 120572

119898 119898 le 119899 Always we can find a value 120575 isin R such

that 1205721lt 120575 lt 120572

2 We make the change of variable 119904 997891rarr 119904 + 120575

resulting into a polynomial 119901120575(119904) = 119901(119904 + 120575) The roots of

119901

120575are located at 1205721015840

119896= 120572

119896minus 120575 and it is easy to see that

120572

1015840

1lt 0 and 120572

1015840

119896gt 0 for 119896 = 2 119898 Then we apply the map

119904 997891rarr 119904

2 which produces a polynomial 119902(119904) = 119901

120575(119904

2) with a

pair of pure conjugated complex roots at plusmniradic120575 minus 119886

1and pairs

of symmetric real roots at plusmnradic119886

119896minus 120575 for 119896 = 2 119898 Under

the hypothesis of separability 119902(119904) = 119902

119877(119904)119902

119862(119904) and undoing

in reverse order the changes of variables 119904 997891rarr radic119904 997891rarr 119904+120575 weobtain that 119902

119877(

radic

119904 minus 120575) = (119904 minus 120572

1) and 119902

119862(

radic

119904 minus 120575) = 119901(119904)(119904 minus

120572

1) Therefore we might isolate roots of 119901 by successively

repeating the procedure onto 119902119862This is a contradiction since

from the Galois theory it is well known that it is not alwayspossible to solve a polynomial equation by radicals

3 Conclusions

We overcome the drawbacks of the original Schur-Cohncriterion for delay systems namely the high degree ofthe polynomial that emerges from the Schur-Cohn matrixand the difficulty of detection of complex roots especiallypure imaginary roots due to the propagation of numericalerrors In particular we decrease the degree of the generatedpolynomial to a half and we reduce the problem numericallyto that of seeking the real roots of this polynomial notthe complex ones These properties are exactly those thatmake the Rekasius substitution criterion so attractive in thestability theory of delay systems Therefore from a practicalviewpoint the modified Schur-Cohn criterion is comparableto other widely used criteria in the literature Our methodis based on making transformations on the Schur-Cohnmatrix via triangular matrices over a polynomial ring inone unknown This leads to the determinant of the Jurymatrix over a polynomial ring which is proved to be an evenpolynomial The pure imaginary roots of the Schur-Cohnpolynomial are those of the even polynomial and in virtueof the change of variable 120596 997891rarr radic120596 we only need to seekreal roots which is more precise from a numerical viewpointwhile the degree of the polynomial is halved

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Vyhlidal J-F Lafay and R Sipahi Delay Systems FromTheory to Numerics and Applications Advances in Delays andDynamics Springer New York NY USA 2014

[2] K Gu J Chen and V L Kharitonov Stability of Time-DelaySystems Birkhaauser Boston Mass USA 2003

[3] C Bonnet A R Fioravanti and J R Partington ldquoStability ofneutral systems with commensurate delays and poles asymp-totic to the imaginary axisrdquo SIAM Journal on Control andOptimization vol 49 no 2 pp 498ndash516 2011

[4] H Fazelinia R Sipahi and N Olgac ldquoStability robustness anal-ysis of multiple time-delayed systems using ldquobuilding blockrdquoconceptrdquo IEEE Transactions on Automatic Control vol 52 no5 pp 799ndash810 2007

[5] A R Fioravanti C Bonnet and H Ozbay ldquoStability offractional neutral systems with multiple delays and polesasymptotic to the imaginary axisrdquo in Proceedings of the 49thIEEEConference onDecision andControl pp 31ndash35 Atlanta GaUSA December 2010

[6] A R Fioravanti C Bonnet H Ozbay and S-I Niculescu ldquoAnumerical method to find stability windows and unstable polesfor linear neutral time-delay systemsrdquo in Proceedings of the 9thIFAC Workshop on Time Delay Systems (TDS rsquo10) pp 183ndash188Prague Czech Republic June 2010

[7] A R Fioravanti C Bonnet H Ozbay and S-I NiculesculdquoA numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systemsrdquoAutomatica vol 48 no 11 pp 2824ndash2830 2012

[8] N Olgac and R Sipahi ldquoA practical method for analyzing thestability of neutral type LTI-time delayed systemsrdquo Automaticavol 40 no 5 pp 847ndash853 2004

[9] R Sipahi ldquoDelay-margin design for the general class of single-delay retarded-type LTI systemsrdquo International Journal ofDynamics and Control vol 2 no 2 pp 198ndash209 2014

[10] S Barnett Polynomials and Linear Control Systems MarcelDekker New York NY USA 1983

[11] J Chen G Gu and C N Nett ldquoA new method for computingdelay margins for stability of linear delay systemsrdquo Systems andControl Letters vol 26 no 2 pp 107ndash117 1995

[12] P Fu S-I Niculescu and J Chen ldquoStability of linear neutraltime-delay systems exact conditions via matrix pencil solu-tionsrdquo IEEE Transactions on Automatic Control vol 51 no 6pp 1063ndash1069 2006

[13] K Walton and J E Marshall ldquoDirect Method for TDS StabilityAnalysisrdquo IEE Proceedings D Control Theory and Applicationsvol 134 no 2 pp 101ndash107 1987

[14] J H Su ldquoThe asymptotic stability of linear autonomous systemswith commensurate time delaysrdquo IEEE Transactions on Auto-matic Control vol 40 no 6 pp 1114ndash1117 1995

[15] J Louisell ldquoAmatrixmethod for determining the imaginary axiseigenvalues of a delay systemrdquo IEEE Transactions on AutomaticControl vol 46 no 12 pp 2008ndash2012 2001

[16] N Olgac and R Sipahi ldquoAn exact method for the stabilityanalysis of time-delayed linear time-invariant (LTI) systemsrdquoIEEE Transactions on Automatic Control vol 47 no 5 pp 793ndash797 2002

[17] Z V Rekasius ldquoA stability test for systems with delaysrdquo inProceedings of the Joint Automatic Control Conference 1980paper no TP9-A

[18] A Thowsen ldquoThe Routh-Hurwitz method for stability deter-mination of linear differential-difference systemsrdquo InternationalJournal of Control vol 33 no 5 pp 991ndash995 1981

[19] R Sipahi and N Olgac ldquoA comparative survey in determiningthe imaginary characteristic roots of LTI time delayed systemsrdquoin Proceedings of the 16th Triennial World Congress of Interna-tional Federation of Automatic Control (IFAC rsquo05) vol 16 pp390ndash399 Prague Czech Republic July 2005

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

[20] D Hinrichsen and A J PritchardMathematical SystemsTheoryI vol 48 of Texts in Applied Mathematics Springer BerlinGermany 2005

[21] H Gluesing-Luerssen Linear Delay-Differential Systems withCommensurate Delays An Algebraic Approach vol 1770 ofLecture Notes in Mathematics Springer 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of