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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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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
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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