Open Problems Related to the Hurwitz Stability of Polynomials Segments

Abstract

In the framework of robust stability analysis of linear systems, the development of techniques and methods that help to obtain necessary and sufficient conditions to determine stability of convex combinations of polynomials is paramount. In this paper, knowing that Hurwitz polynomials set is not a convex set, a brief overview of some results and open problems concerning the stability of the convex combinations of Hurwitz polynomials is then provided.

1. Introduction

The design of control systems arises from the need of human beings to manipulate real plants with some degree of confidence and accuracy. However, this design requires mathematical models which are often very complex if all the dynamics that involve a real plant are taken into account. Thus, a complex model requires a complex controller design. Robust control theory intends to analyze a complex model by studying its linear approximation and always assuming that this approximation will incur with some degree of modeling errors. This error is regarded as uncertainty which is modeled and bounded to determine stability conditions of a system and control laws are obtained.

Problems that arise from robust control theory containing one uncertain parameter, such as gain or constants time, and the system stability should be determined for the entire uncertainty range. The study of criteria for deciding whether a convex combination of polynomials is stable can be applied to resolution of a wide variety of uncertainty systems. The problem to be raised is as follows: find conditions about the stable polynomials and such that the convex combination described by is stable (Hurwitz), for all .

The first result, where necessary and sufficient conditions were presented, is known as the Bialas Theorem [1–3]. A different approach in terms of the frequency domain is known as the Segment Lemma and it was found by Bhattacharyya et al. [4, 5]. This result shows that the stability of is equivalent to certain conditions that must be met by even and odd part of the polynomials and . In complex polynomials case, Bose developed a method to determine the stability of the segment of complex polynomials [6]. In regard to sufficient conditions, in [7, 8], the well-known Rantzer’s conditions were obtained.

Based on the above criteria computer algorithms have been developed to resolve this problem. The Segment Lemma has been used to design an algorithm in [9]. On this same path, a method to verify the stability of convex combinations of polynomials in a finite number of calculus is obtained in [10]. In [11] an algorithm to determine the stability of segment of complex polynomials is proposed; it is based on the work of Bose [12]. In the present paper, we will discuss some of these criteria and propose some open problems in these topics. Applications of Hurwitz stability criterion to Engineering problems are huge, for example, for the stability of wave propagation in strain gradient [13] or mixture [14].

The paper is organized as follows. In Section 2, the problem statement is presented. Some results with necessary and sufficient condition on stability of segment of Hurwitz polynomials and its open problems are presented in Section 3. In Section 4, some open problems with sufficient condition and using Rantzer’s conditions are postulated. In Section 5, a related problem with segments of stable polynomials is the calculating of the Minimum Left Extreme, which was studied by Bialas. In Section 6 some results and open problems related to Hadamardized Hurwitz polynomials are postulated. Finally, in Section 7, some concluding remarks are provided.

2. Problem Statement

In this section some definitions and one motivation example in the framework of the stability of polynomials are given.

Definition 1. A polynomial with real coefficients, , is Hurwitz if all its roots have negative real part.

Definition 2. Suppose that and are polynomials (real or complex) of degree . Letand consider the following one parameter family of polynomials:This family will be referred to as a segment of polynomials. We shall say that the segment is stable if and only if every polynomial on the segment is stable.

The following example shows that a segment of polynomials is not necessarily a stable segment although the two extremes are Hurwitz polynomials.

Example 3. Let and be Hurwitz; neverthelessis not Hurwitz for all :(i)for it is Hurwitz;(ii)for it is not Hurwitz;(iii)for it is Hurwitz.

Now, there are several criteria to determine the stability of the segment of polynomials; we present some of them below.

3. Necessary and Sufficient Conditions

In this section, we present two of the most known necessary and sufficient conditions on the stability of segment of Hurwitz polynomials.

3.1. The Bialas Theorem

Theorem 4 (see [2]). If is Hurwitz and , then is Hurwitz for all , if and only if the matrix has eigenvalues not in , where and are matrices Hurwitz of and , respectively.

Open Problem 1. Is it possible to find necessary and sufficient conditions in a similar way of the Bialas Theorem for case?
In order to establish the Segment Lemma we give some definitions.
Given a polynomial , we define then and can be written as , where In the Bialas Theorem, it is necessary that the two extremes of the segment of polynomials have different degree. Now we present the Segment Lemma, where the two extremes of the segment of polynomials have equal degrees.

3.2. The Segment Lemma

Lemma 5 (see [4, 5]). Let and    be -degree Hurwitz polynomials with leading coefficients of the same sign. Then the segment of polynomials is Hurwitz, if and only if there are not that satisfies the three conditions: (1),(2),(3).

Open Problem 2. Is it possible to get a result, based on the approach of the Segment Lemma, to be applicable when polynomials and have different degrees?
After we have discussed two necessary and sufficient conditions, now in the next section, we expose two sufficient conditions: Rantzer’s conditions and one approach of matrix inequalities.

4. Sufficient Conditions

In this section, we present two sufficient conditions for the stability of segment that are known in the literature: Rantzer’s conditions and one interesting approach with matrix inequalities.

4.1. Rantzer’s Conditions

Supposing that is a Hurwitz polynomial and is semistable (their roots have real part ), then the segment of polynomials consists of Hurwitz polynomials if you have one of the following four conditions.

(i) The difference satisfies

(ii) Each of the polynomials and has at least one root in and

(iii) Each of the polynomials and has at least one root in and

(iv) Each of the polynomials and has at least two roots in andThe previous conditions are known as Rantzer’s conditions; see [8].

Open Problem 3. Is there a mechanism to describe Rantzer’s conditions in terms of the polynomial coefficients?

Open Problem 4. What should we add to Rantzer’s conditions in order to obtain necessary and sufficient conditions?

In the next subsection, we present an interesting sufficient condition for checking the stability of segment of polynomials based on matrix inequalities.

4.2. A Matrix Inequality

Given a polynomial , we define the matrix as

Theorem 6. Let be a polynomial of degree . If the polynomials and are Hurwitz and vector satisfies the system of linear inequalities,then is Hurwitz for all Here the symbol () means that all components of a given vector are () and the symbol means that all components of a given vector are but there is at least one component which is >0. See [15, 16].

Open Problem 5. With these approach matrix inequalities, what should we add to obtain necessary and sufficient conditions?

4.2.1. Case for Grade

A similar case is obtained if . In this case, the matrix is defined byand the corresponding inequality is

There is a relation between segments and rays of polynomials, which let us give applications in control theory. We explain it in the next two subsections.

4.3. Relationship between Rays and Segment of Polynomials

If is Hurwitz for all , then is Hurwitz for all , from which if is Hurwitz, then, the stability of ray is equivalent to the stability of the segment . An analysis in terms of ray polynomials was made in [15].

For , is the matrix:for , is the matrix: for , is the matrix: Rewriting the results shown in [15, 16] in terms of segment polynomials yields the following result.

Theorem 7. Consider the Hurwitz polynomial . If is Hurwitz with or and the vector of coefficients satisfies the system of linear inequalities then the polynomial is Hurwitz for all ; where the matrix is determined by the degree of and is one of the matrices , or .

Example 8. Consider the polynomial Hurwitz . The vector of coefficients of polynomial is solution of system of linear inequalities (11):therefore, the segment polynomial is Hurwitz. However, is not a solution of (17):and Rantzer’s conditions are not verified: to the difference has

Open Problem 6. Will there be a stable segment of polynomial that satisfies Rantzer’s conditions but does not satisfy (11) or (17)?
The importance of the segments and rays of Hurwitz polynomials with the design of stabilizing controls is pointed out in the next subsection.

4.4. Applications to Control Systems

Consider the system , whereFor the uncontrolled system , the characteristic polynomial is . If into the controlled system we choose a control as , where and , then the controlled system takes the form:and its characteristic polynomial is that is,where .

Example 9. Consider the systemIn this case, the matrix is given bythen we have the systemTherefore, the system is stable . But this cannot be verified with the results presented by Aguirre et al. in [15].

The Minimum Left Extreme is a subject very related to the study of segment of Hurwitz polynomials; hence in the next sections we present some ideas about the Minimum Left Extreme in order to propose some open problems.

5. The Minimum Left Extreme

A related problem with segment of Hurwitz polynomials is the calculating of the Minimum Left Extreme, which was studied by Bialas. We begin with some definitions.

Let be a Hurwitz polynomial that generates a family of polynomials given as follows:where and is a polynomial with coefficient vector . If is the matrix defined in the previous section and is a solution of the system of linear inequalities , then is a Hurwitz polynomial .

What is the value of , so that is Hurwitz ? From Bialas (see [2]),where and are Hurwitz matrices of and , respectively, and is the smallest negative eigenvalue of the matrix

5.1. Estimation Technique Proposed

Consider the polynomial whereby the following matrix is defined:with the th row of the matrix and .

Theorem 10. Let and be a Hurwitz polynomial and a matrix defined above, respectively. If the vector is a solution to the system of linear inequalities and , then is Hurwitz , where ; that is, .

Example 11. Given and , we have thatThen . We have that , .

In the above example, the two approaches lead to being Hurwitz .

Example 12. Given and , we have thatThen . Finally we have , .

Now, in Example 12 with the approach proposed in Theorem 10, we obtain that is Hurwitz and applying the Bialas approach we obtain that is Hurwitz . That is, .

Example 13. Consider the following system:Here and . Then it holds Therefore, in addition, we can ensure that the system is stable for all (the control is a stabilizing control) and we can also say more, the system is stable for all because is Hurwitz .

Open Problem 7. Describe the cases when .

Remark 14. An advantage of our approach, in relation to the Bialas approach, is that we can estimate the Minimum Left Extreme when the polynomials have the same degree. Given the polynomial , the matrices and are defined byIf the vector is a solution of inequalities system and the polynomial is given by , then is a Hurwitz polynomial , where ; that is, .

Open Problem 8. With this approach find conditions for calculating as or as .
If we consider the set of the Hadamardized Hurwitz polynomials, then we can plant some open problems also. It is the motivation to include the next section.

6. Hadamardized Hurwitz Polynomials

In the present section, some results and open problems related to Hadamardized Hurwitz polynomials are postulated.

Another property of Hurwitz polynomials is related to Hadamard product. Consider two real polynomials and : The Hadamard product of and , denoted by , is defined byIn [17], it was proved that the set of Hurwitz polynomials is closed under Hadamard product. However in [18] it was shown that there are Hurwitz polynomials of degree 4 that do not have a Hadamard factorization in two Hurwitz polynomials. In [19, 20], some conditions to Hadamard factorization existence were presented. The set of Hurwitz polynomials that admits a Hadamard factorization, we called Hadamardized Hurwitz polynomials, is denoted by . Recently, in [21] some topological properties about this set were shown. But regarding the convex combination, we can propose the following problems.

Open Problem 9. Given , what conditions are required to guarantee that ?

Open Problem 10. Is there any property in set that does not share with set?

7. Conclusions

In this paper, a brief overview of some results concerning the stability of the convex combination of Hurwitz polynomials is provided. Also some criteria for deciding if a segment of polynomials is a Hurwitz polynomial were presented. Furthermore some open problems about these polynomials in order to motivate the study of them were given. As mentioned above, the study and solution of these problems also have their field of opportunity in the framework of robust systems.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is part of the sabbatical year by Faustino Ricardo García-Sosa realized at Universidad Autónoma del Estado de Hidalgo, Mexico. The author Carlos Arturo Loredo-Villalobos would like to thank the support of CONACYT through the postdoctoral fellowship.