20 ZIAD ZAHREDDINE For some interesting relationships between these two types of stability: Routh- Hurwitz for continuous-time and Schur-Cohn for disc

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1 SOOCHOW JOURNAL OF MATHEMATICS Volume 25, No. 1, pp , January 1999 ON SOME PROPERTIES OF HURWITZ POLYNOMIALS WITH APPLICATION TO STABILITY THEORY BY ZIAD ZAHREDDINE Abstract. Necessary as well as necessary and sucient conditions for the Hurwitness of a polynomial are established. The arguments involve the use of complex plane geometry techniques without invoking the theory of positive para-odd functions or continued fraction expansions methods. Some of the established properties are then applied to test for the stability of systems of dierential equations. 1. Introduction Several papers of recent vintage have originated on the eigenvalue clustering and distribution of systems of dierential equations. The results of [6] are concerned with the extension of the famous Routh array to the complex case. The extended Routh array (ERA) was then established and shown to be the natural complex counterpart of the Routh array. In [6], the central part of the argument dwells on the use of positive para-odd functions and the partial fraction and continued fraction expansions of these functions in some special forms. The notion of positive para-oddness is playing an increasingly eective role in the stability analysis of continuous-time systems of dierential equations with complex coecients. For discrete-time systems (eigenvalues inside the unit circle), the concept of positive para-oddness is replaced by complex discrete reactance functions which are the discrete-time counterpart of positive para-odd functions. Received June 17, 1996 revised February 18, AMS Subject Classication. 34E05. Key words. Hurwitz polynomials, stability theory, positive para-odd functions. 19

2 20 ZIAD ZAHREDDINE For some interesting relationships between these two types of stability: Routh- Hurwitz for continuous-time and Schur-Cohn for discrete-time systems, we refer to [5]. Intimately related to the theory of positive para-odd functions are Hurwitz polynomials which appear as the characteristic polynomials of stable continuoustime systems of dierential equations. These relations are for example highlighted by the major results of [6], they also show in[3]. It is therefore the objective of this paper to display some of the intriguing properties of Hurwitz polynomials, using complex plane geometry arguments. Some of these properties are necessary conditions for Hurwitness, others are necessary as well as sucient and can be looked at as stability criteria of systems of dierential equations. In Section 2, we give a necessary condition for Hurwitness. Other types of conditions including a necessary and sucient one are given in Section 3. Finally some applications are given in Section A Necessary Condition Let f(z) =a n z n + a n;1z n;1 + + a 1 z + a 0 be a Hurwitz polynomial with real coecients. It follows from [4] that all the coecients of f(z) should be of the same sign. We assume that a k > 0, for k =0 ::: n. The following lemma is needed: Lemma 2.1. For any complex number z such that Rez > 0, we have jf(z)j > jf(;z)j: Proof. If z k is a root of f, then for Rez > 0 as shown in Figure 1. jz ; z k j > jz +z k j

3 ON SOME PROPERTIES OF HURWITZ POLYNOMIALS 21 jz;zkj z jz+zkj zk ;zk Figure 1. An analytic proof is conceivable upon squaring both sides. It follows from the above inequality that jf(z)j > jf(;z)j for Re z > 0. Since the coecients of f are real, it is clear that jf(;z)j = jf(;z)j and the proof is complete. As a special case of Lemma 2.1, we have jf(z)j z on C > jf(;z)j z on C where C is a closed contour consisting of a segment of the vertical line z = " such that ">0 and "! 0, and a circular arc of arbitrarily radius R centered at the origin. Therefore by means of Routhe's theorem [1], the polynomial g(z) = 1 2 (f(z)+f(;z)) = a 0 + a 2 z 2 + a 4 z 4 + has the same number of zeros within C as does the polynomial f(z). C 0 " R Figure 2.

4 22 ZIAD ZAHREDDINE Since f is a Hurwitz polynomial, then it has no zeros within C. By taking R arbitrarily large and " arbitrarily small, we conclude that g has no zeros in the open right half-pale. Since g(z) consists only of even powers in z, its zeros are symmetrically distributed with respect to the imaginary axis. Therefore the zeros of g(z) must all be imaginary from which we conclude that the polynomial h(x) =a 0 + a 2 x + a 4 x 2 + (1) obtained by letting x = z 2 in g, should have only negative real zeros. In a similar way, we show that the polynomial k(x) =a 1 + a 3 x + a 5 x 2 + (2) obtained by letting x = z 2 in g(z) = 1 (f(z) ; f(;z)), should only have negative 2z real zeros. We summarize in the next theorem. Theorem 2.1. If f is a Hurwitz polynomials, then the two polynomials h and k in (1) and (2) respectively formed by alternating the coecients of f should have negative real zeros only. 3. Necessary and Sucient Conditions Suppose now that f is a monic polynomial, i.e. f(z) =z n + a n;1z n;1 + + a 1 z + a 0 : We begin by establishing the following necessary condition. Theorem 3.1. If f(z) is a Hurwitz polynomial with n 1, then arg(f(js)) is a continuous and strictly increasing function of s where j 2 = ;1: Proof. Since f is Hurwitz, then f(z) = Q n k=1(z ; z k ) where z k = Rez k + jimz k and Rez k < 0 for k =1 ::: n: So arg(f(js)) = = k=1 k=1 arg(js + jrez k j;jimz k ) arctan s ; Imzk jrez k j

5 ON SOME PROPERTIES OF HURWITZ POLYNOMIALS 23 where the summands are all continuous and strictly increasing. Im(z) s z-plane Re(z) Figure 3. Angle of f(js) for Hurwitz f. Figure 3 illustrates the proof of Theorem 3.1. If all zeros of f are in the open left half-plane, then the angle contribution of each zero increases as z moves up along the imaginary axis. Im(f (js)) f (js) s =0 Re(f (js)) s!1 Figure 4. Image of f(js) for Hurwitz f (n =7). Now, Figure 4 shows the result. As s increases from 0 to 1, f(js) starting on the positive real axis, smoothly circles strictly counterclockwise around n=2 radians as it increases to innity. Before we give our necessary and sucient condition we make the following denition. If f(js) 6= 0 for all s, we dene arg net (f) = s!1 lim farg(f(js)) ; arg(f(0))g

6 24 ZIAD ZAHREDDINE where the right-hand side denotes the net total angle, counting the encirclements, subtended by f(js) ass increases from 0 to 1. Theorem 3.2. The nth-degree polynomial f(z) is Hurwitz if and only if arg net (f) is well dened (f(js) 6= 0 for all s) and equal n(=2): Proof. Figure 3 above shows that each zero in the left half-plane contributes =2 to arg net (f), while each zero in the right half-plane contributes ;=2, and the proof is complete. 4. Applications Since necessary conditions of stability are at the same time sucient conditions of instability, we will illustrate here how the results of Section 2 can be used to easily check the instability of some systems. Example 1. Consider a system of dierential equations of order 4 with characteristic polynomial f(z) =3z 4 +2z 3 +5z 2 + z +7: Then h and k of Section 2 are given by h(z) =3z 2 +5z +7 and k(z) =2z +1: Since = 5 2 ; < 0, it follows that h has no real zeros. Therefore f is not Hurwitz and the system is unstable. Example 2. Let the characteristic polynomial of a 6th order system of dierential equations be f(z) =2z 6 +7z 5 +50z 4 +34z 3 +25z 2 +40z +5: Then h(z) =2z 3 +50z 2 +25z + 5 and k(z) =7z 2 +34z +40: The well-known Cardan's formula [2] asserts that for a cubic polynomial h(z) =a 3 z 3 + a 2 z 2 + a 1 z + a 0

7 ON SOME PROPERTIES OF HURWITZ POLYNOMIALS 25 to have only real zeros the condition a 2 1 ; 3a 2a 0 0must be satised. Since 25 2 ; < 0, h has non-real zeros and therefore f is not Hurwitz. Example 3. Consider the 8th degree polynomial f(z) =5z 8 +10z 7 +25z 6 +30z 5 +21z 4 +45z 3 +32z 2 +15z +60: Then and h(z) =5z 4 +25z 3 +21z 2 +32z +60 k(z) =10z 3 +30z 2 +45z +15: Since k satises Cardan's formula, we should look at h and split it up again into h 1 and k 1 where h 1 (z) =5z 2 +21z +60 andk 1 (z) =25z +32: In h 1,wehave =21 2 ; < 0: Therefore h 1 has non-real zeros and hence f is not Hurwitz. Obviously the power of the inspection method expressed by Theorem 2.1 will be more appreciated as the order of the system increases where long and tedious Routh array will be required to test the stability of the system. Now we illustrate the usefulness of Theorems 3.1 and 3.2, but rst we make the following denition: A set of polynomials is said to be Hurwitz if and only if every memberoftheset is Hurwitz. Fix n 1 and the real numbers a k and a k such that a k a k for k = 0 ::: n; 1: We dene the set N to be the set of monic nth-degree polynomials of the form f(z) =z n + a n;1z n;1 + :::+ a 0 for all a 0 ::: a n;1 such the a k a k a k k = 0 ::: n; 1. Then for all real numbers s we dene H(s) = ff(js) : f 2 Ng i.e. H(s) is the image of N under the evaluation map at z = js where the evaluation map at z e z () maps polynomials into the complex plane and is given by e z (f) =f(z). Next we dene the polynomials (where a n = a n =1)

8 26 ZIAD ZAHREDDINE g 1 (z)=a 0 + a 2 z 2 + a 4 z 4 + = g 2 (z)=a 0 + a 2 z 2 + a 4 z 4 + = h 1 (z)=a 1 z + a 3 z 3 + a 5 z 5 + = h 2 (z)=a 1 z + a 3 z 3 + a 5 z 5 + = Finally, we dene the polynomials k=0 even k=0 even k=1 odd k=1 odd j k minfj k a k j k a k gz k j k maxfj k a k j k a k gz k j k;1 minfj k;1 a k j k;1 a k gz k j k;1 maxfj k;1 a k j k;1 a k gz k : k kl (z) =g k (z)+h l (z) k l =1 2: It is to be noted that k 11 k 12 k 21 and k 22 are all members of N, and for all real numbers s, g 1 (js) and g 2 (js) are purely real, while h 1 (js) and h 2 (js) are purely imaginary. Furthermore, 8s 0, we have Refg 1 (js)g Reff(js)g Refg 2 (js)g Imfh 1 (js)g Imff(js)g Imfh 2 (js)g 8f 2 N 8f 2 N and for s 0we switch h 1 and h 2 in the last relation. Therefore, we see that for all real numbers s, H(s) is a level rectangle i.e., the sides are parallel to the real and imaginary axes with corners k 11 (js) k 12 (js) k 21 (js) andk 22 (js) asshown in Figure 5. h 2(js) k 12(js) k 22(js) H(s) h 1(js) k 11(js) k 21(js) g 1(js) g 2(js) Figure 5. Rectangular image of N at z = js(s >0). Before we state our main result here, we need the following lemma:

9 ON SOME PROPERTIES OF HURWITZ POLYNOMIALS 27 Lemma 4.1. If the polynomials k 11 k 12 k 21 and k 22 are Hurwitz, then 0 62 H(s), for all real s. Proof. For the set fk 11 k 12 k 21 k 22 g to be Hurwitz, wemust have a k > 0 8k because it is well-known that a monic Hurwitz polynomial has all its coecients positive. Clearly 0 62 H(0) = [a 0 a 0 ]: Since H is continuous (i.e., the four corners vary continuously with s), if 0 2 H(s) for some s > 0, then 0 must be on the boundary of H(^s) for some ^s 2 [0 s]. Since no corner may pass through zero (the corners are Hurwitz), we must have an edge containing zero in its interior. Without loss of generality, we assume it is the `bottom' edge. Then, as illustrated in Figure 6 k 11 (j^s) is on the negative real axis and k 21 (j^s) is on the positive real axis. Theorem 3.1 implies that for s > 0 suciently small, we have k 11 (j(^s+s)) in the open third quadrant andk 21 (j(^s + s)) in the open rst quadrant. Since Imfk 11 (js)g =Imfk 21 (js)g =Imfh 1 (js)g, this is clearly not possible. H(^s) k 11(j^s) 0 k 21(j^s) Figure 6. Why 0 cannot enter H(s). We note that the whole rectangle H(s) must travel counterclockwise through a total angle of n=2, always completely entering one quadrant before crossing into the next. Theorem 4.1. The class of polynomials N is Hurwitz if and only if fk 11 k 12 k 21 k 22 g is Hurwitz. Proof. The `only if' is immediate since fk 11 k 12 k 21 k 22 gn: Nowsuppose fk 11 k 12 k 21 k 22 g is Hurwitz, and f 2 N. Lemma 4.1 implies that f(js) 6= 08s, so arg net (f) iswell dened. Furthermore, f(js) 2 H(s) 8s, soarg net (f) =n=2: Now by Theorem 3.2, f has n zeros in the left half-plane and is therefore Hurwitz.

10 28 ZIAD ZAHREDDINE References [1] E. A. Guillemin, The Mathematics of Circuit Analysis, 7th edition, Oxford & IBH Publishing, [2] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw, New York, [3] N. Levinson and R. M. Redheer, Complex Variable, McGraw-Hill Publishing Company Limited, [4] M. Marden, Geometry of Polynomials, Amer. Math. Soc., 2nd edition, [5] Z. Zahreddine, Explicit relationships between Routh-Hurwitz and Schour-Cohn types of stability, Irish Math. Soc. Bull., 29(1992), [6] Z. Zahreddine, An extension of the Routh array for the asymptotic stability of a system of dierential equations with complex coecients, Applicable Analysis, 49(1993), Department of Basic Sciences, (Mathematics Division), University of Sharjah, Sharjah, P.O. Box 27272, United Arab Emirates.

Parallel Properties of Poles of. Positive Functions and those of. Discrete Reactance Functions

International Journal of Mathematical Analysis Vol. 11, 2017, no. 24, 1141-1150 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ima.2017.77106 Parallel Properties of Poles of Positive Functions and