Linear Algebra and its Applications

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1 Linear Algebra and its Applications 435 (011) Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa Hurwitz rational functions < Yury Barkovsky a, Mikhail Tyaglov b, a Department of Calculus Mathematics and Mathematical Physics, Faculty of Mathematics, Mechanics & Computer Science, Southern Federal University, Milchakova str 8a, Rostov-on-Don, Russia b Technische Universität Berlin, Institut für Mathematik, MA 4-5, Strasse des 17 Juni 136, 1063 Berlin, Germany ARTICLE INFO Article history: Received 5 October 010 Accepted 4 March 011 Available online 17 May 011 Submitted by V Mehrmann AMS classification: 6C15 6C10 30C15 6C05 15B05 15A15 ABSTRACT A generalization of Hurwitz stable polynomials to real rational functions is considered We establish an analog of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants 011 Elsevier Inc All rights reserved Keywords: Hurwitz stable polynomials Hurwitz rational function Root localization Pole localization Hurwitz matrix 1 Introduction It is well known that the problem of stability of a linear difference or differential system with constant coefficients reduces to the question of locating the zeroes of its characteristic polynomial in the left half-plane of the complex plane One of the most famous results from stability theory is the Hurwitz theorem, which expresses stability of a real polynomial in terms of its coefficients [5,6,3] (see also [1,7]) Namely, the Hurwitz theorem states that a real polynomial has all its zeroes in the open left half-plane if and only if some determinants constructed with the coefficients of the polynomial < This work was supported by the Sofja Kovalevskaja Research Prize of Alexander von Humboldt Foundation Corresponding author address: tyaglov@mathtu-berlinde (M Tyaglov) /$ - see front matter 011 Elsevier Inc All rights reserved doi:101016/jlaa

2 1846 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) are positive(see Theorem 5) Those determinants are now called the Hurwitz determinants due to Adolf Hurwitz who introduced them in [5] In the present work we investigate a class of real rational functions satisfying the Hurwitz conditions: the Hurwitz determinants constructed with the coefficients of the Laurent series at of a real rational function are positive up to the order n, where n is the sum of degrees of the numerator and denominator of the rational function This is a generalization of the class of real Hurwitz stable polynomials It turns out that such a class of rational functions is characterized by location of poles and zeroes: all zeroes lie in the open left half-plane of the complex plane while all poles lie in the open right half-plane (Theorem 3) Finally, we express the Hurwitz determinants of real rational functions in terms of the coefficients of the numerator and denominator (Lemma 34) Thereby we introduce a new type of determinants and describe the class of real rational functions satisfying Hurwitz conditions in terms of coefficients of their numerator and denominator (Theorem 35) As well as in case of polynomials this class of rational functions is characterized by positivity of those determinants Some simple properties of the new type of determinants that can be treated as a generalization of the Hurwitz determinants are considered Hurwitz polynomials Consider a real polynomial p(z) def = a 0 z n + a 1 z n 1 + +a n, a 1,,a n R, a 0 > 0 (1) Throughout the paper we use the following notation: [ ] l def n =, () where n = deg p, and [ρ] denotes the largest integer not exceeding ρ The polynomial p can always be represented as follows: p(z) = p 0 (z ) + zp 1 (z ), where p 0 and p 1 are the even and odd parts of the polynomial, respectively Introduce the following function: Φ(u) def = p 1(u) p 0 (u) (3) Definition 1 We call Φ the function associated with the polynomial p Associate with the polynomial p the following determinants: a 1 a 3 a 5 a 7 a j 1 a 0 a a 4 a 6 a j j (p) def 0 a 1 a 3 a 5 a j 3 =, j = 1,,n, (4) 0 a 0 a a 4 a j a j where we set a i 0fori > n Definition The determinants j (p), j = 1,,n, are called the Hurwitz determinants or the Hurwitz minors of the polynomial p

3 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) Suppose that deg p 0 deg p 1 and expand the function Φ into its Laurent series at : Φ(u) = p 1(u) p 0 (u) = s 1 + s 0 + s 1 + s + s 3 +, u u u 3 (5) u4 where s 1 = 0ifdegp 0 = deg p 1 and s 1 = 0ifdegp 0 > deg p 1 For a given infinite sequence (s j ) j=0, consider the determinants s 0 s 1 s s j 1 D j (Φ) def = s 1 s s 3 s j, j = 1,, 3,, (6) s j 1 s j s j+1 s j These determinants are referred to as the Hankel minors or Hankel determinants Together with the determinants (6) we consider one more sequence of Hankel determinants s 1 s s 3 s j Dj (Φ) def s s 3 s 4 s j+1 =, j = 1,, 3, (7) s j s j+1 s j+ s j 1 It is very well known [5,3] (see also [7]) that there are relations between the determinants D j (Φ), Dj (Φ) and the Hurwitz minors j (p): (1) If n = l, then j 1 (p) = a j 1 0 D j (Φ), j (p) = ( 1) j a j D 0 j (Φ), () If n = l + 1, then j = 1,,,l; (8) ( ) j a0 j (p) = D j (Φ), s 1 ( ) a0 j+1 (p) = ( 1) j+1 Dj j (Φ), s 1 j = 0, 1,,l, (9) where D 0 (Φ) 1 Here l is defined in () It is also well known [5,3] (see also [4]) that the number of poles of the function Φ equals the order of the last non-zero minor D j (Φ)SincethefunctionΦ has at most l poles, we have D j (Φ) = D j (Φ) = 0, j > l (10) Thus, in the sequel, we deal only with the determinants D j (Φ), D j (Φ) of order at most l Definition 3 The polynomial p defined in (1) is called Hurwitz or Hurwitz stable if all its zeroes lie in the open left half-plane of the complex plane The following criterion of Hurwitz stability of a real polynomial was (implicitly) established in [5] (see also [3,1,7])

4 1848 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) Theorem 4 Let a real polynomial p be defined by (1) The following conditions are equivalent: (1) the polynomial p is Hurwitz stable; () the following hold s 1 > 0 for n = l + 1, D j (Φ) >0, j = 1,,l, ( 1) j Dj (Φ) >0, where l = [ ] n j = 1,,l, (11) This theorem together with formulae (8) and (9) imply the following theorem which is very well known as the Hurwitz criterion of polynomial stability Theorem 5 (Hurwitz [5]) A real polynomial p of degree n as in (1) is Hurwitz stable if and only if all Hurwitz determinants j (p) are positive: 1 (p) >0, (p) >0,, n (p) >0 (1) 3 Hurwitz rational functions Consider a rational function R(z) def = h(z) g(z) = t 0z r m + t 1 z r m 1 + t z r m +, (31) where h and g arerealpolynomials h(z) def = b 0 z r + b 1 z r 1 + +b r 1 z + b r, b 0, b 1,,b r R, b 0 > 0; (3) g(z) def = c 0 z m + c 1 z m 1 + +c m 1 z + c m, c 0, c 1,,c m R, c 0 > 0 Here r, m N {0}, r + m > 0 Assume that the polynomials p and q are coprime The number n = r + m is called the order of the function R Definition 31 The real rational function R is called Hurwitz if both polynomials h(z) and g( z) are Hurwitz stable, or one of them is Hurwitz stable while the second one is a constant It turns out that one can establish a criterion for real rational functions to be Hurwitz similar to the Hurwitz criterion of polynomial stability Theorem 3 A real rational function R(z) of the form (31) and (3) is Hurwitz if and only if the leading principal minors k (R) of the infinite Hurwitz matrix t 1 t 3 t 5 t 7 t 9 t 0 t t 4 t 6 t 8 0 t 1 t 3 t 5 t 7 H(R) def = 0 t 0 t t 4 t 6 (33) 0 0 t 1 t 3 t t 0 t t 4

5 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) are positive up to the order n (inclusive): 1 (R) >0, (R) >0,, n (R) >0 Proof Consider the following auxiliary polynomial P(z) def = ( 1) m h(z)g( z) = a 0 z n + a 1 z n 1 + +a n 1 z + a n, a 0 = b 0 c 0 > 0 (34) By definition, the function R is Hurwitz if and only if the polynomial P is Hurwitz stable Consider the rational function Φ associated with the polynomial P: Φ(u) = P 1(u) P 0 (u) = s 1 + s 0 + s 1 + s + cdots, u u (35) u3 where the polynomials P 0 (u) and P 1 (u) are the even and odd parts of the polynomial P, respectively Since P 0 (z ) = P(z) + P( z) and zp 0 (z ) = P(z) P( z), wehave zφ(z ) = Henceweobtain Φ(u) = R 1(u) R 0 (u) = P 1(u) P 0 (u), where R(z) + R( z) R 0 (z ) =, P(z) P( z) h(z)g( z) h( z)g(z) R(z) R( z) = = P(z) + P( z) h(z)g( z) + h( z)g(z) R(z) + R( z) R(z) R( z) R 1 (z ) = z are the even and odd parts of the function R,respectivelyThus,forn = l + 1, we have Φ(u) = t 0 + t u 1 + t 4 u + t 1 + t 3 u 1 + t 5 u + = s 1 + s 0 u + s 1 u + s u 3 +, and for n = l, wehave Φ(u) = t 1u 1 + t 3 u + t 5 u 3 + t 0 + t u 1 + t 4 u + = s 1 + s 0 u + s 1 u + s u 3 + We recall that n = r + m is the order of the rational function RNownotethattheformulae(8) and (9) are formal, so following verbatim the proof of the formulae (8) and (9) (see, for instance, [7,4]) one can establish the following relationships (1) for n = l, j 1 (R) = t j 1 0 D j (Φ), j (R) = t j D 0 j (Φ) j = 1,,,l, (37) () for n = l + 1, (36) j (R) = ( t0 s 1 j+1 (R) = ( 1) j ( t0 ) j D j (Φ), j = 1,,,l, s 1 ) j+1 Dj (Φ), j = 0, 1,,l, (38)

6 1850 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) where the Hurwitz minors j (R) are defined as follows: t 1 t 3 t 5 t 7 t j 1 t 0 t t 4 t 6 t j j (R) def 0 t 1 t 3 t 5 t j 3 = (39) 0 t 0 t t 4 t j t j By Theorem 4, the polynomial P is Hurwitz stable if and only if the minors D j (Φ) and D j (Φ) satisfy the inequalities (11) Now since the polynomial P is Hurwitz stable if and only if R is a Hurwitz rational function, the formulae (37) and (38) and Theorem 4 imply the assertion of the theorem Note that the leading principal minors j (R) of order more than n equal zero regardless whether the function R being Hurwitz or not: n+1 (R) = n+ (R) = n+3 (R) = = 0 (310) It follows from (10) and from the formulae (37) and (38) which are obviously valid for j > l Remark 33 It is easy to see that a 0 = b 0 c 0 and t 0 = b 0 c 0 So the formulae (8) and (9) and the formulae (37) and (38) imply the equalities j (P) = c j 0 j(r), j = 1,, (311) This equalities verify (310), since j (P) = 0forj > n Recall now that one of necessary conditions for the polynomial p defined in (1) to be Hurwitz stable is the positivity of its coefficients a j > 0, j = 0, 1,,n This is the so-called Stodola theorem [5,3] It turns out that positivity of the coefficients t j in (31) is not necessary condition for the Hurwitzness of the function R Indeed,ifR = h is Hurwitz, then the polynomial g has g all zeroes in the open right half-plane, so it has coefficients of different signs Therefore, it is easy to find polynomials h and g such that R has both negative and positive coefficients For example, the following function is Hurwitz, but its Laurent series at infinity has positive, negative and zero coefficients: F(z) = z + z + 1 z z + = 1 t 0 + t 1 z 1 + t z + t 3 z 3 +, where t 0 = 1 and t 3j = t 3j 1 = ( 1) (j 1), t 3j = 0 for j = 1,, Finally, we find a connection between the coefficients of the polynomials h and g and the Hurwitz determinants j (R) This gives us a criterion of Hurwitzness of a real rational function in terms of the coefficients of its numerator and denominator

7 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) Let again the function R be defined by (31) and (3) Introduce the following determinants of order j, j = 1,, c 0 c 1 c c 3 c 4 c 5 c j 1 c j c j 3 c j c j 1 b 0 b 1 b b 3 b 4 b 5 b j 1 b j b j 3 b j b j c 0 c 1 c c 3 c j 3 c j c j 5 c j 4 c j 3 0 b 0 b 1 b b 3 b 4 b j b j 1 b j 4 b j 3 b j Ω j (h, g) def = c 0 c 1 c j 5 c j 4 c j 7 c j 6 c j 5 (31) 0 0 b 0 b 1 b b 3 b j 3 b j b j 5 b j 4 b j c 0 c b 0 b 1 b j b j 1 b j Here we set b k := 0fork > r, and c j := 0forj > m Lemma 34 For any j = 1,,, Ω j (h, g) = c j 0 j(r), where j (R) are the Hurwitz determinants of the function R defined in (39) (313) Proof First interchange the rows of the determinant (31) toobtain c 0 c 1 c c 3 c j 1 c j c j c j c 0 c 1 c j 3 c j c j 4 c j c0 c 1 Ω j (h, g) = (314) b 0 b1 b j 1 b j 0 b 0 b 1 b b j b j 1 b j 3 b j b 0 b 1 b b 3 b j 1 b j b j b j 1 This does not change the sign of Ω j (h, g) In fact, lower the jth and (j + 1)st rows (they are marked by in (314)) to their initial positions in (31) This will require an even number of transpositions The next pair of rows will then meet, the lowering operation will be applied to them, and so on (j 1 times) until we obtain the initial determinant (31) Now we rearrange the columns of the determinant (314) in the following way: 1st, 3rd,,(j 1)st, nd, 4th,,jth To move the 3rd column to the second place, we need one transposition To move the 5th column to the third place, we need two transpositions, and so on At last, to move the (j 1)st column to the jth place, we need (j 1) transpositions During all those transpositions the resulting determinant will change its sign j 1 i=1 i = j(j 1) times Next we interchange the (j + 1)st row with the jth row, the (j + )nd row with the (j 1)st row, and so on This will also require

8 185 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) j(j 1) transpositions Thus, the resulting determinant has the same sign as the initial determinant, so we have c 0 c c 4 c j c 1 c 3 c 5 c j 1 0 c 0 c c j 4 0 c 1 c 3 c j c 0 c j c 1 c j c c 1 Ω j (h, g) = b 0 b b 4 b j b 1 b 3 b 5 b j 1 (315) 0 b 1 b 3 b j 3 b 0 b b 4 b j 0 b 0 b b j 4 0 b 1 b 3 b j b 1 b j 5 0 b 0 b b j b j b j Now from each (j + i 1)st row, i = 1,,, [ ] j+1, we subtract rows ith, (i + 1)st,, jth ], multiplied by t 0, t,,t (j i), respectively Then, from each (j + i)th row, i = 1,,, [ j subtract rows (i + 1)st, (i + )nd,, jth multiplied by t 1, t 3,,t (j i) 1, respectively As a result, we have a determinant, which can be represented in a block form: C Ω j (h, g) = 0 C 1 à Â, where the upper triangular matrices C 0 and C 1 have the forms c 0 c c 4 c j c 1 c 3 c 5 c j 1 0 c 0 c c j 4 0 c 1 c 3 c j 3 C 0 = 0 0 c 0 c j 6, C 1 = 0 0 c 1 c j c c 1 Note that C 0 =c j 0 > 0 (316) In order to describe the matrices à and Â, we establish a connection between the coefficients b i s, c i s and t i s Multiplying (31) by the denominator and equating coefficients, we get b k = k c k i t i, k = 0, 1,, (317) i=0

9 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) Taking into account those formulae, we obtain the entries of the matrix à to have the form: 0 if k < i + 1, ã i 1,k = b (k i) c (k i q) t q = c (k i q)+1 t q 1 if k i + 1, q=0 q=1 where i = 1,,, [ ] j + 1 ; 0 ifk < i + 1, ã i,k = k i 1 b (k i) 1 c (k i q) t q 1 = c (k i q) 1 t q if k i + 1, q=1 q=0 where i = 1,,, [ ] j (318) From these formulae it follows that the matrix à can be represented as a product of two matrices: 0 t 1 t 3 t j 3 c 1 c 3 c 5 c j 1 0 t 0 t t j 4 0 c 1 c 3 c j 3 à = 0 0 t 1 t j c 1 c j 5 (319) t j c 1 Analogously, by (317), for the entries of the matrix Â, wehave 0 if k < i, â i 1,k = b (k i)+1 c (k i q)+1 t q = c (k i q) t q+1 if k i, q=0 q=0 where i = 1,,, [ ] j + 1 ; 0 if k < i, c 0 t 0 if k = i, â i,k = b (k i) c (k i q)+1 t q 1 = c (k i q) t q if k > i, q=1 q=0 where i = 1,,, [ ] j Therefore, the matrix  can be represented as follows: t 1 t 3 t 5 t j 1 c 0 c c 4 c j t 0 t t 4 t j 0 c 0 c c j 4  = 0 t 1 t 3 t j c 0 c j 6 =: Λ j C 0 (30) t j c 0 It is clear that Λ j is the j j leading principal submatrix of the matrix H(R) defined in (33) Taking (30) into account, we multiply both parts of the equality (316) by the following determinant:

10 1854 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) E j 0 0 C 1 = c j 0 > 0, 0 where E j is the j j identity matrix, to obtain c j 0 Ω C j(h, g) = 0 C 1 C 1 0 à Λ, j or in the entry-wise form: c 1 c 0 c c 4 c j c 0 0 c 0 c c j 4 0 c 1 c 0 c c 0 c j c 0 c j 0 Ω j(h, g) = c c (31) c 0 0 c 1 t 1 c 3 t 1 + c 1 t 3 t 1 t 3 t 5 t j 1 0 c 1 t 0 c 3 t 0 + c 1 t t 0 t t 4 t j 0 0 c 1 t 1 0 t 1 t 3 t j t j The formulae (318) (31) show that all the columns of the matrix à are linear combinations of the columns of the matrix Λ j Consequently, we are able to eliminate the entries in the lower left corner of the determinant (31) To do this, from the nd column we subtract the (j + 1)st column multiplied by c 1 Then from the 3rd column we subtract the (j + 1)st column multiplied by c 1 and the (j + )nd column multiplied by c 3, and so on As a result, we obtain the determinant c j 0 Ω j(h, g) to have the following form c 1 c 0 c 0 c 1 0 c 0 0 c 0 c c c 0 c j 0 Ω c 1 j(h, g) = c = c j c 0 0 Λ j t 1 t 3 t 5 t j t 0 t t 4 t j t 1 t 3 t j t j Since Λ j = j (R),weget Ω j (h, g) = c j 0 j(r), as required

11 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) Now Lemma 34 and Theorem 3 imply the following criterion of Hurwitzness of real rational functions in terms of the coefficients of their numerator and denominator Theorem 35 A real rational function R defined in (31) and (3) is Hurwitz if and only if the inequalities hold Ω (h, g) >0, Ω 4 (h, g) >0,, Ω n (h, g) >0 From (310) and (313) it also follows that Ω n+ (h, g) = Ω n+4 (h, g) = Ω n+6 (h, g) = = 0 Lemma 34 andtheformulae(311) imply the following corollary Corollary 36 Let the polynomials h and g be defined in (3) Then the Hurwitz minors of the polynomial P(z) = ( 1) m h(z)g( z) satisfy the equalities: j (P) = Ω j (h, g), where n = deg P = deg h + deg g j = 1,,,n, Consider now the rational function F(z) = ( 1)n R( z), (3) where R is defined in (31) and (3), and n = r + m is order of R It is clear that the function F is Hurwitz if and only if the function R is Hurwitz This fact can be verified by the following relationship between the Hurwitz minors of the functions F and R Theorem 37 Let the function R be defined in (31) and (3), and let the function F be defined in (3) Then j (R) = t j 0 j(f), j = 1,, (33) Proof Let F(z) = f (z) q(z), where f (z) = ( 1)m g( z), and q(z) = ( 1) r h( z) Consider the polynomial Q(z) = ( 1) r q( z)f (z), which is analogous to (34) We have Q(z) = ( 1) r q( z)f (z) = ( 1) r ( 1) r h(z)( 1) m g( z) = ( 1) m h(z)g( z) = P(z) Now the formulae (311)imply c j 0 j(r) = j (P) = j (Q) = b j 0 j(f), j = 1,,, that is exactly (33), since t 0 = b 0 c 0 Corollary 38 Let the polynomials h and g are defined in (3)Then Ω j (h, g) = Ω j (f, q), j = 1,,, where f (z) = ( 1) m g( z) and q(z) = ( 1) r h( z) Finally, note that if one of the polynomials h and g is a constant, the determinants Ω j (h, g) become Hurwitz determinants (up to a constant factor) of the non-constant polynomial This follows from the formulae (311)

12 1856 Y Barkovsky, M Tyaglov / Linear Algebra and its Applications 435 (011) Acknowledgment The authors are grateful to Professor Olga Holtz for helpful discussions References [1] YuS Barkovskiy, Lectures on the Routh Hurwitz Problem, 008, in press Available from: arxiv: [3] FR Gantmacher, The Theory of Matrices, Vol II, AMS Chelsea Publ, 000 [4] O Holtz, M Tyaglov, Structured matrices, continued fractions and root localization of polynomials, SIAM Review 009, accepted for publication Available from: arxiv: [5] A Hurwitz, Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math Ann 46 (1895) (in German) English transl On the conditions under which an equation has only roots with negative real parts, in stability theory (Ascona, 1995), Internat Ser Numer Math, Birkhäuser, Basel 11 (1996) [6], 1936 (in Russian) English transl MG Krein, MA Naimark, The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra 10 (4) (1981) Translated from the Russian by O Boshko and JL Howland [7] M Tyaglov, Generalized Hurwitz polynomials, 010, submitted for publication Available from: arxiv: Further reading [] G Frobenius, Über das Trägheitsgesetz der quadratischen Formen, Sitzungsber Ber Acad Wiss Phys Math Klasse Berlin (1894) (in German) ; also in: Gesammelte Abhandlungen, Band II, pp