On Schur Stable Multivariate Polynomials - IEEE Xplore

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 5, MAY 2006

On Schur Stable Multivariate Polynomials J. A. Torres-Muñoz, E. Rodríguez-Angeles, and V. L. Kharitonov

Abstract—The class of stable multivariate polynomials, recently introduced by Kaczorek (1985), is the largest class of polynomials preserving the stability property under small coefficient variations. The principal goal of the contribution is to show that the class of Schur stable multivariate polynomials is the Moebius transformation of the latter one. This fundamental relation provides a vehicle to translate results known for one class to the other one. Index Terms—Multivariate polynomials, stability.

I. INTRODUCTION

T

HE CONTRIBUTION deals with the stability of multivariate polynomials in the continuous, as well as in the discrete domains. In both domains there are different stability concepts which generate different classes of stable polynomials. A detailed account of the classes may be found in [3]–[5], [8]–[10]. In the continuous domain our main attention is oriented to the class of stable multivariate polynomials introduced in [15]. Polynomials of the class possess several properties which allow an extension of results, known for univariate Hurwitz stable polynomials, see [18], [20]. Among various stability concepts used in the discrete domain for the stability analysis of multidimensional digital filters, see [1], [11], [13], one can distinguishes the class of strict sense Schur stable polynomials, see [3], [4], [10]. But, for some technical reasons this class has been defined in such a way that, in the univariate case, it does not coincide with the standard class of Schur stable (SS) polynomials, but rather with the class of anti-Schur polynomials. This fact does not permit an extension of habitual properties of SS univariate polynomials to the multivariate ones. On the other hand, the class of SS multivariate polynomials, see [13], [14], may be considered as a natural multivariate extension of the class of SS univariate polynomials. A fundamental relation of univariate Schur and Hurwitz polynomials is that they can be transformed one into the other by means of the Moebius transformation. The transformation allows to translate any stability criterion known for Hurwitz polynomials to a similar one for Schur polynomials. Such translation becomes much more complicated when one addresses the multivariate polynomials because of the diversity of the stability

notions in this case, see [5], [12]. As a consequence, several results obtained for discrete stable multivariate polynomials can not be easily transformed to the continuous counterparts. The main goal of the contribution is to show that the class of stable multivariate polynomials introduced in [15] is the Moebius transformation of the class of SS multivariate polynomials. Once the relation is established, it provides a vehicle to connect continuous and discrete stable polynomials. In the next section, some notations and preliminary statements concerning multivariate polynomials are provided. Our main result is presented in Section III. Section IV is dedicated to reformulation of some basic properties of stable multivariate polynomials in terms of SS polynomials. Some new sets of convex directions for SS multivariate polynomials are given there as well. An illustrative example, and concluding remarks end the paper. II. MULTIVARIATE POLYNOMIALS In this section, some basic notions and concepts for multivariate polynomials are recalled, see [5]. A multivariate polynomial is a finite sum of the form (1) where , and are real (or complex) coefficients. One may define, following the lexicographic order of the indices, the coefficient vector

The maximal degree of with a non zero coefficient in (1) . The vector of partial is known as its partial degree, degrees is denoted by . Polynomial (1) may be also written, with respect to the variable , as (2)

Manuscript received March 10, 2005; revised August 30, 2005. This paper was supported by CONACyT-Mexico.This paper was recommended by Associate Editor C. Hadjicostis. J. A. Torres-Muñoz and V. L. Kharitonov are with the Automatic Control Department, Center of Research and Advanced Studies (CINVESTAV-IPN) Mexico, Mexico city, Mexico (e-mail: [email protected]; [email protected]). E. Rodríguez-Angeles is with the Engineering Faculty, Mexico State Autonomous University, CP 50130 Toluca, Mexico (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2006.870219

, polynomials. Here and are named the main and free polynomial coefficients with respect to the variable , respectively. where

1057-7122/$20.00 © 2006 IEEE

coefficients

are

-variate

TORRES-MUÑOZ et al.: ON SS MULTIVARIATE POLYNOMIALS

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Given any vector with nonnegative integer components, , let us define the set

of constant degree -variate polynomials. With polynomial one can associate the reciprocal polynomial

Here means that all coefficients of complex conjugate numbers.

B. Discrete Stable Polynomials Hereafter, a discrete multivariate polynomial of degree will be denoted as (5) where

and the coefficient vector is . Let us introduce the polydomain

are changed by the

along with its essential boundary

A. Continuous Stable Polynomials Let us define the polydomain (3) and its essential boundary (4) Definition 1: A polynomial strict sense stable (SSS) if

is called

The class of SSS polynomials plays an important role in the stability and robust stability analysis of multidimensional systems, see [2], [6], [7]. Nevertheless, polynomials of this class may lose the stability property under arbitrary small coefficient variations. To remedy this situation, the following stability concept has been recently introduced. : Definition 2: [15] Given a polynomial 1) for : polynomial of degree is called stable if it is Hurwitz stable; : polynomial is called stable if it is SSS 2) for and additionally satisfies the following conditions: • main coefficients

•

are and

Definition 3: A polynomial

is called SS

if

In the next section, we will need the following two auxiliary statements. be a SS polynomial, and Lemma 3: Let . Then

is a SS polynomial from . Proof: To simplify our notations, let us introduce vectors and . Without loss of generality we may assume that all partial degrees of are positive. We show first that is SS. If it is not the case, has a root . This implies that has a then root . Last conclusion contradicts the Schur . stability of with respect to the Assume now that partial degree of is less than . The main coefficient in the decomvariable position

-variate stable polynomials;

The class of stable polynomials is the largest class of polynomials preserving the stability property under small coefficient variations. It turns out that several useful properties of Hurwitz stable univariate polynomials have been extended to stable multivariate polynomials, see [15]. Among them the following two will be used later on. be a stable polyTheorem 1: [15] Let such that every polynomial with nomial. There exists the coefficient vector from the -neighborhood of the coefficient vector of is also stable. be a stable polyTheorem 2: [15] Let such that polynomial has no nomial. There exists . roots in the -neighborhood of the essential boundary

is not equal to zero, , otherwise the correis less than . On the other sponding partial degree of hand, according to our degree assumption the main coefficient in the decomposition

is identically zero, . Certainly, at least one of the other coefficients in the last decomposition, say , is not equal to zero, otherwise will have a . Let us select such root in the polydomain that . As is not trivial, for any in a small neighborhood of there exists a such that point and

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As a result, for sufficiently small values of polynomial

the univariate

has a root of magnitude greater than 1. And we arrive at has a root . This the conclusion that contradicts the Schur stability of . So, the partial degree with respect to the variable is equal to . Same of arguments can be applied to any other of the partial degrees, too. Corollary 4: If we fix in a SS polynomial some of the variables in such a way that the magnitude of the fixed variables is greater or equal to one, then the resulting polynomial will be SS and its partial degrees will coincide with the corresponding . partial degrees of the polynomial be a SS polynomial. Lemma 5: Let in the decompoThen, the main coefficient sition

is a SS polynomial from . Proof: Ones again let us set vectors and . Observe first that polynomial is not identically zero, otherwise the partial degree of with respect to the variable is less than . Assume by contradiction that admits a root . Define the univariate polynomial

If for all , then fixing such that we obtain a root of . This contradiction in the case when for proves the Schur stability of is not trivial, then at least one of the coefficients all . If , , say is not equal to zero, . For any in a small neighborhood say there exists a point , such that of the point and It means that for sufficiently small at least one of the zeros of lies outside the unit disc of the complex has a root . plane. As a result in this case This contradicts the Schur stability of , and we arrive at the . Schur stability of the main coefficient Let us address now the degree condition. Assume by contradiction that at least one of the partial degrees of the main , is less than the corresponding partial decoefficient, , say . On the gree of the polynomial , other hand, we know that at least one of the coefficients , say , should have the partial degree such that the folequal to . Let us select a point lowing conditions hold.

•

•

the main coefficient

in the de-

is not equal to zero. the main coefficient At the point composition

in the de-

At the point composition

is not equal to zero. Now one can factorize polynomial

as follows:

For sufficiently large all coefficients of the polynomial in the square brackets are bounded, and additionally

and

As

, then for any

there exists

such that

and It means that for a sufficiently small the univariate polyhas a root which lies outside the unit nomial disc of the complex plane. And we arrive at the contradiction because SS polynomial has a root . The contradiction proves the fact that . Same , arguments can be applied to any other partial degree of is a SS polynomial of degree so the main coefficient . Remark 1: Lemma 5 also holds when one considers decomwith respect to any other of position of the SS polynomial . the variables , for III. MAIN RESULT The main goal of the contribution is to show that, in the multivariate case, the class of stable polynomials is the Moebius transformation of the class of SS polynomials. To motivate the further analysis let us recall, firstly, that it is well known that such intimate relationship holds when dealing with univariate case. be a Hurwitz stable polynomial of deTheorem 6: Let gree , then

is a SS polynomial of degree , and vice versa.


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On the other hand, it is also true that discrete SS and continuous SSS polynomials are not associated, via the Moebius transformation, as it can be seen from the following example. Example 1: Polynomial is SSS of degree (1,1), but it is not stable in the sense of Definition 2. The transformed polynomial

is not SS because it allows a root . Even more, the class of continuous SSS polynomials is not associated in this sense with any other class of discrete stable polynomials, see [5], [12]. Before introducing our main result we need the following two auxiliary statements. be a stable polynomial. Lemma 7: Let Then

(6) . is a SS polynomial from Proof: For conciseness, define . Let in the decomposition

and

be the coefficients of

, be the coefficients of

in the decomposition

Let us consider the case when some of the components of the are equal to one, say the first of them. Polynomial root can be decomposed as follows:

where all terms constituting have no factor . It -variate polynomial is the main cois clear that with respect to the variefficient in the decomposition of ables , and by Definition it is a stable polynomial of degree . Now, we can easily check that

Last equality implies that stable polynomial

has a root

This contradiction proves the Schur stability of . be a SS polynomial. Lemma 8: Let Then

(7) . is a stable polynomial from , Proof: Let , and . We prove the statement by induction the with respect to the number of the variables. For statement follows from Theorem 6. Assume now that the variables, statement is true for polynomials with less than and consider the case of -variate polynomials, i.e., , . It follows from (7) that

It follows from (6) that

The fact that the inequality

This implies that

Polynomial is stable, so . As a result, the is equal to . degree of . Assume by contraNow we prove the Schur stability of diction that has a root . Observe first that not all of the components of the root are equal to one. This directly follows from the formula

If all the components of the root are different from one then has a root

This contradicts the stability of

.

directly follows from

Let us check now the strict sense stability of . Assume by contradiction that has a root . If all components has a root of the root are different of one, then

and we arrive at a contradiction with the Schur stability of . Consider now the case when some components of the root are equal to one, say the first of them. As components have positive real parts, one may assume, without any loss of generality, that , . Define the polynomial

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By definition . If for all , , then selecting in the right half complex plane values , we obtain a new root of which all components are different of one. Then, applying the previous arguments we arrive at the contradiction with the Schur . Let now polynomial be not trivial. Then stability of there exists a vector all components of which are different of zero, such that for sufficiently small

Now, polynomial

has a root

in a small neighborhood of the point , so is a root of . When is small enough we may conclude that this root belongs to the , and all components of the root are now difpolydomain ferent of one. Then, this again contradicts the Schur stability of , so polynomial is SSS. The main coefficient, , in the decomposition

can be written as

Polynomial

is -variate SS of degree , see Lemma 3. Therefore, by the inducis a -variate stable polynomial tion hypothesis of degree . The same is true for the main with respect to any coefficient in the decomposition of other of the variables. This ends the proof of the Lemma. Now we are ready to formulate the main result of the contribution. Theorem 9: Polynomial is stable if and only if the polynomial

is SS polynomial from . Proof: The statement directly follows from Lemmas 7 8.

be a SS polynomial. Theorem 10: Let such that every polynomial with a coefficient There exists vector from the -neighborhood of the coefficient vector of is SS. Proof: The property holds for continuous stable polynomials, see Theorem 1. This and Theorem 9 imply the result. It is worth to be mentioned that this result can be also derived from the Rudin’s Theorem, see [22]. The following result is known for the univariate Schur polynomials. be a SS polynomial Theorem 11: Let in the representation (5), then . Proof: The statement follows from Lemma 7 and Theorem 9. This theorem is also a consequence of the Rudin’s Theorem [22]. be a SS polynomial. Theorem 12: Let such that has no roots in the -neighThere exists . borhood of the essential boundary Proof: Continuous stable polynomials have no zeros in a small neighborhood of the essential boundary, see Theorem 2. Joining this with the basic Theorem 9 allow to arrive at the statement. Once again, the result can be also deduced from The Rudin’s Theorem [22]. Following statement shows the invariance of the Schur stability under differentiation as in the univariate case. be a SS polynomial. Theorem 13: Let , then If

is a SS polynomial from . Proof: Assume by contradiction that polynomial has a root . On the other hand, it is is a Schur univariate polynoknown that mial, with , see Corollary 4. Then, polynomial

is a SS univariate polynomial, so has no roots in . This contradiction proves the statement. Corollary 14: Let be a SS polynomial. , , then If

IV. SOME BASIC RESULTS FOR SCHUR STABLE POLYNOMIALS The statement of Theorem 9 provides a bridge to translate properties and stability results between the above mentioned polynomials classes. In this Section it is shown how the translation works. A. General Statements In this Subsection we apply Theorem 9 in order to derive several basic properties of SS -variate polynomials.

is a SS polynomial from . Observe that Theorem 13 does not imply that other coeffishould also cients in decomposition (2) of a SS polynomial be SS. is SS of Example 2: Polynomial degree (1,1). Its free coefficient with respect to the variable , , is not a SS univariate polynomial.


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B. Discrete Convex Directions In this subsection, we use Theorem 9 in order to derive a concept of convex directions for SS -variate polynomials. Definition 4: An -variate real (or complex) polynomial with , , is said to be a real (or complex) convex direction for SS polynomials from if the following condition is satisfied: Given any SS real (or complex) polynomial such that is also SS, then is SS for all . Following statement is an extension of a known result for univariate polynomials, see [21]. , for . A polynomial Lemma 15: Let is a convex direction for SS polynomials if and only if the polynomial from

is a convex direction for stable polynomials from . Proof: The statement follows from Theorem 9 and Theorem 27 in [17]. Theorem 16: Let , for . A real (or is a real (or complex) complex) polynomial if the convex direction for SS polynomials from inequality

is a convex direction for . SS polynomials from , for . Then any Theorem 20: Let -variate polynomial of the form , are such that for at least one , is a where , . real convex direction for SS polynomials from Proof: The statement follows from Corollary 17 in [19], and Theorem 9. is called A polynomial anti-Schur if its reciprocal polynomial is SS. Theorem 21: Let , . Then any antiis a convex direction for Schur polynomial . SS polynomials from Proof: From [7, Th. 2] and Theorem 9 one may derive a necessary phase condition for SS polynomials. The statement directly follows from this condition and the definition of antiSchur polynomials. . Theorem 22: Given a SS polynomial Polynomial is a convex . direction for -variate SS polynomials from . Proof: To simplify let us denote such Assume by contradiction there is a SS polynomial that polynomial is SS, but polynomial is not SS for some , i.e., it has a root . Point , so both polynomials and are SS univariate polynomials, see Corollary 4. By construction polynomial is antiSchur univariate polynomial. So, is a convex direction for SS univariate polynomials. This implies that polynomial may have no zeros in . The contradiction proves the statement.

(8) V. EXAMPLE , where . holds for all Proof: The statement is a consequence of the corresponding result for continuous polynomials, see Theorems 36 37 in [16], and Theorem 9. A real polynomial is said to be auto-reciprocal if , where is a complex number such than . From Theorem one may derive the following class of convex directions for SS polynomials. , for . Any autoLemma 17: Let is a complex convex reciprocal polynomial direction for SS polynomials from . More classes of convex directions are described in the following statements. . Then every -variate Theorem 18: Let polynomial with for , is a convex direction for SS -variate polynomials from . Proof: The statement follows from [17, Th. 27], and Theorem 9. Corollary 19: Let , for . If , then any -variate polynomial

The aim is to illustrate the use of convex directions. For this, let us consider the following plant:

The plant is not SS because for and the denominator, ,, has a root . The goal is to find a proportional controller such that the closed-loop system is stable. In this case, the closed-loop characteristic polynomial is given by

This polynomial has a polytopic structure, so the edge theorem may be applied. Theorem 23: [23] Let a family of invariant degree, , of bivariate polynomials with set of uncertainty, , convex polytope. Then the family is SS if and only if all its edge polynomials are SS.

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Recall that an edge is the convex combination of two vertex and we get first the vertex polynomials polynomials

introduced in [15]. This statement allows to transform stability and robust stability results obtained for one class to the polynomials of the other one.

ACKNOWLEDGMENT The authors would like to thank an anonymous reviewer for pointing out that several properties of SS polynomials can be easily deduced from the Rudin’s Theorem. so, the edge polynomials are REFERENCES

In all of these edges, the term multiplied by is a constant or a univariate polynomial, so it is a convex direction for SS bivariate polynomials, see Theorem 18. So one only has to prove Schur stability of the vertex polynomials. , For the vertex polynomials at a value , such that we obtain the roots

then the following inequalities are obtained:

Inequalities , , 2, 3, 4, are necessary and suffi, therecient in order to guarantee that there are no roots in fore the proportional controller capable to stabilize the proposal . plant must have a gain

VI. CONCLUDING REMARKS It is shown that the class of SS multivariate polynomials is the Moebius transformation of the class of stable polynomials

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J. A. Torres-Muñoz received the M.S. degree in electrical engineering from the Center of Research and Advanced Studies (CINVESTAV-IPN), Mexico city, Mexico, and the Ph. D. degree in automatic control from the Polytechnic National Institute Grenoble (INPG), Grenoble, France, in 1985 and 1990, respectively. He has been a Professor in the Department of Automatic Control, CINVESTAV-IPN since 1990. He served as the Head of the Department of Automatic Control from 1999 to 2003. He was Visiting Researcher at IRCCYN, Ecole Centrale de Nantes, Nantes, France (1997–1998). His main research interests are in the structural approach of linear systems and in the stability and robust stability of multidimensional polynomials.

E. Rodríguez-Angeles was born in Hidalgo, Mexico, in 1976. He received the B.S. degree in control and automation engineering from the Escuela Superior de Ingeniería Mecánica y Eléctrica (ESIME-IPN), Mexico City, Mexico, in 1999, and the M.S. and Ph.D. degrees in in automatic control from de Center of Research and Advanced Studies (CINVESTAV-IPN), Mexico City, Mexico, in 2001 and 2004, respectively. Since early 2005, he is working in the Engineering Faculty, Autonomous University of Mexico State (UAEM), Toluca, Mexico. His research interests are in the area of robust stability of multivariate polynomials.

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V. L. Kharitonov received the M.A. degree in 1973, the Candidate of Science degree in automatic control in 1977, the Doctor of Science degree in 1990, all from the Leningrad State University, Leningrad, Russia. He is a Professor in the Department of Applied mathematics and Control Theory St. Petersburg State University, St Petersburg, Russia, and is currently also with the Center of Research and Advanced Studies (CINVESTAV-IPN), Mexico City, Mexico. His scientific interest includes control, stability, and robust stability.