# On Robust Stability for Hurwitz Polynomials via Recurrence Relations and Linear Combinations of Orthogonal Polynomials

Authors: Alejandro Arceo; Héctor F. Flores; Lino G. Garza; Luis E. Garza; Gerardo Romero
Journal: Complexity
 (2022)

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## Abstract

In this contribution, we use the connection between stable polynomials and orthogonal polynomials on the real line to construct sequences of Hurwitz polynomials that are robustly stable in terms of several uncertain parameters. These sequences are constructed by using properties of orthogonal polynomials, such as the well-known three-term recurrence relation, as well as by considering linear combinations of two orthogonal polynomials with consecutive degree. Some examples are presented.


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## Body

## 1. Introduction and Background

### 1.1. Hurwitz Polynomials and Robust Stability

A polynomial with real coefficients is called a Hurwitz polynomial if and only if all of its roots have negative real part. The study of Hurwitz polynomials is motivated by the role they play in the analysis of the stability of linear systems of the form(1)y˙t=Ayt,y∈ℝn,A∈ℝn×n,where A is a fixed matrix that represents some process and y˙t denotes the derivative of the vector yt componentwise. In fact, it is very well-known that Hurwitz polynomials characterize the stability of linear invariant-time continuous systems. More precisely, such a linear system is asymptotically stable (i.e., its solutions tend to zero as t⟶∞) if and only if the characteristic polynomial detA−xI, where I denotes the identity n×n matrix, is a Hurwitz polynomial [1–3]. As a consequence, many algebraic properties of Hurwitz (also called stable) polynomials have been studied in the literature, including many characterizations in order to determine whether or not a given real polynomial is Hurwitz without computing its roots explicitly. Among the most widely known criteria for determining the stability of a given polynomial, we can mention the Hurwitz–Routh test [4] and the stability test [5]. Other important characterizations are the Hermite–Biehler theorem [6] and Markov’s parameters theorem [2]. The latter two characterizations play a central role in the relation between Hurwitz polynomials and orthogonal polynomials, as it will be discussed in the sequel.On the other hand, in most of real-life applications, the physical process represented by equation (1) has uncertainty. This uncertainty can be caused by the inherent nature of the process under analysis, by the difficulty of taking precise measurements on the variables of the process, or a combination of both phenomena. As a consequence, it is useful to introduce an uncertain parameter on the mathematical representation of the linear system. In this situation, the system is represented by(2)y˙t=Aqyt,where q is a d−dimensional vector with d parameters defined in some domain D⊂ℝd, and the system is said to be robustly stable if detAq−xI is a degree invariant Hurwitz polynomial for every q∈D. Different types of uncertainty are considered in the literature. For instance, the uncertain structure defined by(3)px,q=qnxn+qn−1xn−1+…+q0,qi∈ai,bi,ai<bi,i=0,…,n,where q is a n+1−dimensional vector in the rectangular domain D=a0,b0×…×an,bn∈ℝn+1, is referred to as independent interval uncertainty. Here, the main tool is the celebrated Kharitonov’s theorem [7] that establishes that the robust stability of the whole family is equivalent to the stability of just four polynomials, the so-called Kharitonov polynomials. The case where the coefficients of the characteristic polynomial are linear combinations of the parameters qi is called affine linear uncertainty [5] and the fundamental result to determine the robust stability is called Edge’s theorem [8]. Polynomic uncertainty occurs when the coefficients of the characteristic polynomial are multivariate polynomials in the parameters qi. Finally, nonlinear uncertainty structure is obtained when the coefficients of the polynomial are more general nonlinear functions of the parameters. It is worth mentioning that there are not general theorems to determine the robust stability for polynomic and nonlinear uncertainty, and therefore most of the robustness analysis is carried out by using numerical approaches [5, 9, 10].


### 1.2. Orthogonal Polynomials on the Real Line

Given a positive Borel measureσ supported on some interval E⊂ℝ, an inner product on the linear space of polynomials can be defined by(4)p,qσ=∫Epxqxdσx.IfE is unbounded, we require that the moments cn=∫Exndσx are finite for n≥0. The Gram-Schmidt orthogonalization process guarantees the existence of a sequence of real polynomials pnn≥0 satisfying the orthogonality conditions(5)pn,pmσ=γnδm,n,γn≠0,n,m≥0,where δm,n is Kronecker’s delta. Such a sequence becomes unique once a suitable normalization is chosen. In this contribution, we will consider a monic sequence denoted by Pnn≥0. The second kind polynomials Qnn≥0 are defined by(6)Qnx=∫EPny−Pnxy−xdσy,and play a central role in Padé approximation [11]. It is not difficult to see that Qn is a polynomial of degree n−1.Orthogonal polynomials are a topic of study that dates back to the late 17th century, although a general theory started to flourish in the 18th century with the contributions of Stieltjes [12], Markov [13], and Chebyshev [14], among many others. Nowadays, it constitutes a very active area of research due to their numerous applications in approximation theory [11], numerical integration [15], mathematical physics [16], digital image processing [17, 18], stochastic processes [19, 20], signal theory [21, 22], and control theory [23], among many others. They have a number of nice properties, including the following [24, 25]:(1)
For anyn≥1, the roots of each Pn are real, simple, and located in the interior of the convex hull of E.(2)
For anyn≥1, the roots of Pn and Pn−1 interlace. That is, between any two consecutive roots of Pn, there is exactly one root of Pn−1.(3)
For anyn≥1, the roots of Pn and Qn interlace.Notice that the first property above means thatPn will have real, simple, and positive roots if E⊂ℝ+. On the other hand, Pnn≥0 and Qnn≥0 also satisfy the three-term recurrence relations(7)Pn+1x=x−bnPnx−anPn−1x,n≥0,(8)Qn+1x=x−bnQnx−anQn−1x,n≥1,with initial conditions P0=1, P−1=0, Q0=0, Qx=1, where ann≥0 and bnn≥0 are a positive and a real sequence of numbers, respectively. Notice that a0 is arbitrary and it is usually defined as a0=c0=∫Edσx. A converse result also holds: any sequence Pnn≥0 constructed by using equation (7) with arbitrary sequences ann≥0 (positive numbers) and bnn≥0 (real numbers) will be orthogonal with respect to some positive measure σ supported on ℝ. This fact is known as Favard’s theorem [24]. Moreover, it is possible to impose conditions on the coefficients an and bn in order to guarantee that the associated orthogonality measure is supported on ℝ. Namely, this will be the case if and only if bn>0 for every n≥0 and there exists a sequence gnn≥0 satisfying 0<gn<1 for n≥1, 0≤g0<1, and an/bnbn−1=gn1−gn−1 for n≥1. Any pair of sequences ann≥0, bnn≥0 satisfying those conditions are said to generate the chain sequence an/bnbn−1n≥1. As a particular case, this occurs when ann≥0 and bnn≥0 are chosen in such a way that an/bnbn−1=k for some fixed k satisfying 0<k≤1/4 (see [24, 26]). Notice that in this case we have gn=1/2 for every n≥0.


### 1.3. Robust Stability via Orthogonality

The relation between Hurwitz polynomials and orthogonal polynomials has been known in the literature for quite some time. They are both closely related with the theory of continued fractions and the moment problem, as pointed out in [2]. Also, in the early nineties, Genin (see [27]) describes a relation between the Euclid’s algorithm, the Routh–Hurwitz algorithm, and orthogonal polynomials and establishes that any Hurwitz polynomial can be identified with a (finite) sequence of orthogonal polynomials. This relation has been recently analyzed with more detail in [28] (in the more general framework of matrix orthogonal polynomials) and [26]. Specifically, if f2nx=x2n+a2n−1x2n−1+…+a1x+a0 is a (monic) polynomial of even degree 2n and we consider the decomposition(9)f2nx=hx2+xgx2,then h is a polynomial of degree n and g is a polynomial of degree at most n−1. Indeed, they are the even and odd parts of f2n. Hermite–Biehler theorem states that a polynomial fx=hx2+xgx2 is Hurwitz if and only if h and g have negative, simple, and interlacing roots, with the rightmost root being that of h, and h0g0>0 [4]. In [26] the authors show that, up to a change a variable and multiplication by a constant, h is an orthogonal polynomial with respect to some measure σ supported on ℝ+, and g is the associated second kind polynomial. More precisely, we have(10)Pnx=−1nh−x,Qnx=−1n−1g−x.A similar decomposition holds for odd degree Hurwitz polynomials, but now the odd part has the form(11)βPnx−Qnx=−1ng−x,where β is the constant term that appears in the power expansion of the rational function gx/hx. It is important to mention that the coefficients on this power expansion are precisely the moments of the orthogonality measure σ. In other words, to every Hurwitz polynomial f2n (or f2n+1), we can associate a finite sequence of orthogonal polynomials Pkk=0n and of course their corresponding second kind polynomials Qkk=0n. More importantly, as a straightforward consequence of the Hermite–Biehler theorem, the following converse result holds.Theorem 1 (see [26]).
Letσ be a positive measure supported on ℝ+ and denote by Pnn≥0 and Qnn≥0 its associated (monic) orthogonal polynomials and second kind polynomials, respectively. Define Gnx=Qnx+βnPnx with βn>0 such that the zeros of Pn and Gn are positive and interlaced. Then,(12)f2n+1x=−1nxPn−x2+−1nGn−x2,n≥0,f2nx=−1nPn−x2+−1n−1xQn−x2,n≥1,are Hurwitz polynomials.Notice that the construction of the odd degree Hurwitz polynomialf2n+1 in Theorem 1 requires the choice of a parameter βn in such a way that the roots of Pn and Gn interlace. In Proposition 2.2 in [26], it was shown that it is possible to choose βn=λ (independent of n) in such a way that the roots of Pn and Gn interlace for every n≥1. Indeed, it suffices to take λ>M=limn⟶∞Qn0/Pn0, provided such a limit exists. On the other hand, if we fix m>0 and define M=Qm0/Pm0, then Pk and Gk will have interlaced roots for all k=1,…,m if λ>M.Observe that the previous theorem implies that we can construct a sequence of Hurwitz polynomials by using sequencesPnn≥0 and Qnn≥0, when the orthogonality measure is supported on ℝ+; i.e., the zeros of each Pn are simple and positive. This property was used in [29] to construct robustly stable sequences of Hurwitz polynomials by using the Laguerre and Jacobi classical orthogonality measures supported on ℝ+ (see [25]). The main idea was to introduce a parameter t on the measure, in such a way that it remains positive. As a consequence, the corresponding inner product, the associated orthogonal sequence, and the second kind polynomials will depend on t. That is, Pnx,tn≥0 will be orthogonal for every value of t, and thus the sequence fnx,tn≥1 constructed by using Theorem 1 will be Hurwitz for every value of t. In other words, if we consider t as an uncertain parameter, the sequence fnx,tn≥1 will be robustly stable. It is worth mentioning that robustly stable families constructed in such a way have been already used in the design of a compensator that robustly stabilizes an interval plant with uncertain time-delay (see [30]) and can also be used for applications in control design.In this contribution, we propose two alternative ways to construct robustly stable sequences of Hurwitz polynomials by using other properties of orthogonal polynomials. The structure of the manuscript is as follows. In Section2, we introduce an uncertain parameter t on the coefficients of the three-term recurrence relation and then use the Favard theorem described in the previous subsection and Theorem 1 to construct a sequence of Hurwitz polynomials that is robustly stable for every value of t within an specified range. In Section 3, we generate robustly stable sequences by using linear combinations of orthogonal polynomials. Some examples are presented in Section 4. Finally, some conclusions and open problems are discussed in Section 5.


## 1.1. Hurwitz Polynomials and Robust Stability

A polynomial with real coefficients is called a Hurwitz polynomial if and only if all of its roots have negative real part. The study of Hurwitz polynomials is motivated by the role they play in the analysis of the stability of linear systems of the form(1)y˙t=Ayt,y∈ℝn,A∈ℝn×n,where A is a fixed matrix that represents some process and y˙t denotes the derivative of the vector yt componentwise. In fact, it is very well-known that Hurwitz polynomials characterize the stability of linear invariant-time continuous systems. More precisely, such a linear system is asymptotically stable (i.e., its solutions tend to zero as t⟶∞) if and only if the characteristic polynomial detA−xI, where I denotes the identity n×n matrix, is a Hurwitz polynomial [1–3]. As a consequence, many algebraic properties of Hurwitz (also called stable) polynomials have been studied in the literature, including many characterizations in order to determine whether or not a given real polynomial is Hurwitz without computing its roots explicitly. Among the most widely known criteria for determining the stability of a given polynomial, we can mention the Hurwitz–Routh test [4] and the stability test [5]. Other important characterizations are the Hermite–Biehler theorem [6] and Markov’s parameters theorem [2]. The latter two characterizations play a central role in the relation between Hurwitz polynomials and orthogonal polynomials, as it will be discussed in the sequel.On the other hand, in most of real-life applications, the physical process represented by equation (1) has uncertainty. This uncertainty can be caused by the inherent nature of the process under analysis, by the difficulty of taking precise measurements on the variables of the process, or a combination of both phenomena. As a consequence, it is useful to introduce an uncertain parameter on the mathematical representation of the linear system. In this situation, the system is represented by(2)y˙t=Aqyt,where q is a d−dimensional vector with d parameters defined in some domain D⊂ℝd, and the system is said to be robustly stable if detAq−xI is a degree invariant Hurwitz polynomial for every q∈D. Different types of uncertainty are considered in the literature. For instance, the uncertain structure defined by(3)px,q=qnxn+qn−1xn−1+…+q0,qi∈ai,bi,ai<bi,i=0,…,n,where q is a n+1−dimensional vector in the rectangular domain D=a0,b0×…×an,bn∈ℝn+1, is referred to as independent interval uncertainty. Here, the main tool is the celebrated Kharitonov’s theorem [7] that establishes that the robust stability of the whole family is equivalent to the stability of just four polynomials, the so-called Kharitonov polynomials. The case where the coefficients of the characteristic polynomial are linear combinations of the parameters qi is called affine linear uncertainty [5] and the fundamental result to determine the robust stability is called Edge’s theorem [8]. Polynomic uncertainty occurs when the coefficients of the characteristic polynomial are multivariate polynomials in the parameters qi. Finally, nonlinear uncertainty structure is obtained when the coefficients of the polynomial are more general nonlinear functions of the parameters. It is worth mentioning that there are not general theorems to determine the robust stability for polynomic and nonlinear uncertainty, and therefore most of the robustness analysis is carried out by using numerical approaches [5, 9, 10].


## 1.2. Orthogonal Polynomials on the Real Line

Given a positive Borel measureσ supported on some interval E⊂ℝ, an inner product on the linear space of polynomials can be defined by(4)p,qσ=∫Epxqxdσx.IfE is unbounded, we require that the moments cn=∫Exndσx are finite for n≥0. The Gram-Schmidt orthogonalization process guarantees the existence of a sequence of real polynomials pnn≥0 satisfying the orthogonality conditions(5)pn,pmσ=γnδm,n,γn≠0,n,m≥0,where δm,n is Kronecker’s delta. Such a sequence becomes unique once a suitable normalization is chosen. In this contribution, we will consider a monic sequence denoted by Pnn≥0. The second kind polynomials Qnn≥0 are defined by(6)Qnx=∫EPny−Pnxy−xdσy,and play a central role in Padé approximation [11]. It is not difficult to see that Qn is a polynomial of degree n−1.Orthogonal polynomials are a topic of study that dates back to the late 17th century, although a general theory started to flourish in the 18th century with the contributions of Stieltjes [12], Markov [13], and Chebyshev [14], among many others. Nowadays, it constitutes a very active area of research due to their numerous applications in approximation theory [11], numerical integration [15], mathematical physics [16], digital image processing [17, 18], stochastic processes [19, 20], signal theory [21, 22], and control theory [23], among many others. They have a number of nice properties, including the following [24, 25]:(1)
For anyn≥1, the roots of each Pn are real, simple, and located in the interior of the convex hull of E.(2)
For anyn≥1, the roots of Pn and Pn−1 interlace. That is, between any two consecutive roots of Pn, there is exactly one root of Pn−1.(3)
For anyn≥1, the roots of Pn and Qn interlace.Notice that the first property above means thatPn will have real, simple, and positive roots if E⊂ℝ+. On the other hand, Pnn≥0 and Qnn≥0 also satisfy the three-term recurrence relations(7)Pn+1x=x−bnPnx−anPn−1x,n≥0,(8)Qn+1x=x−bnQnx−anQn−1x,n≥1,with initial conditions P0=1, P−1=0, Q0=0, Qx=1, where ann≥0 and bnn≥0 are a positive and a real sequence of numbers, respectively. Notice that a0 is arbitrary and it is usually defined as a0=c0=∫Edσx. A converse result also holds: any sequence Pnn≥0 constructed by using equation (7) with arbitrary sequences ann≥0 (positive numbers) and bnn≥0 (real numbers) will be orthogonal with respect to some positive measure σ supported on ℝ. This fact is known as Favard’s theorem [24]. Moreover, it is possible to impose conditions on the coefficients an and bn in order to guarantee that the associated orthogonality measure is supported on ℝ. Namely, this will be the case if and only if bn>0 for every n≥0 and there exists a sequence gnn≥0 satisfying 0<gn<1 for n≥1, 0≤g0<1, and an/bnbn−1=gn1−gn−1 for n≥1. Any pair of sequences ann≥0, bnn≥0 satisfying those conditions are said to generate the chain sequence an/bnbn−1n≥1. As a particular case, this occurs when ann≥0 and bnn≥0 are chosen in such a way that an/bnbn−1=k for some fixed k satisfying 0<k≤1/4 (see [24, 26]). Notice that in this case we have gn=1/2 for every n≥0.


## 1.3. Robust Stability via Orthogonality

The relation between Hurwitz polynomials and orthogonal polynomials has been known in the literature for quite some time. They are both closely related with the theory of continued fractions and the moment problem, as pointed out in [2]. Also, in the early nineties, Genin (see [27]) describes a relation between the Euclid’s algorithm, the Routh–Hurwitz algorithm, and orthogonal polynomials and establishes that any Hurwitz polynomial can be identified with a (finite) sequence of orthogonal polynomials. This relation has been recently analyzed with more detail in [28] (in the more general framework of matrix orthogonal polynomials) and [26]. Specifically, if f2nx=x2n+a2n−1x2n−1+…+a1x+a0 is a (monic) polynomial of even degree 2n and we consider the decomposition(9)f2nx=hx2+xgx2,then h is a polynomial of degree n and g is a polynomial of degree at most n−1. Indeed, they are the even and odd parts of f2n. Hermite–Biehler theorem states that a polynomial fx=hx2+xgx2 is Hurwitz if and only if h and g have negative, simple, and interlacing roots, with the rightmost root being that of h, and h0g0>0 [4]. In [26] the authors show that, up to a change a variable and multiplication by a constant, h is an orthogonal polynomial with respect to some measure σ supported on ℝ+, and g is the associated second kind polynomial. More precisely, we have(10)Pnx=−1nh−x,Qnx=−1n−1g−x.A similar decomposition holds for odd degree Hurwitz polynomials, but now the odd part has the form(11)βPnx−Qnx=−1ng−x,where β is the constant term that appears in the power expansion of the rational function gx/hx. It is important to mention that the coefficients on this power expansion are precisely the moments of the orthogonality measure σ. In other words, to every Hurwitz polynomial f2n (or f2n+1), we can associate a finite sequence of orthogonal polynomials Pkk=0n and of course their corresponding second kind polynomials Qkk=0n. More importantly, as a straightforward consequence of the Hermite–Biehler theorem, the following converse result holds.Theorem 1 (see [26]).
Letσ be a positive measure supported on ℝ+ and denote by Pnn≥0 and Qnn≥0 its associated (monic) orthogonal polynomials and second kind polynomials, respectively. Define Gnx=Qnx+βnPnx with βn>0 such that the zeros of Pn and Gn are positive and interlaced. Then,(12)f2n+1x=−1nxPn−x2+−1nGn−x2,n≥0,f2nx=−1nPn−x2+−1n−1xQn−x2,n≥1,are Hurwitz polynomials.Notice that the construction of the odd degree Hurwitz polynomialf2n+1 in Theorem 1 requires the choice of a parameter βn in such a way that the roots of Pn and Gn interlace. In Proposition 2.2 in [26], it was shown that it is possible to choose βn=λ (independent of n) in such a way that the roots of Pn and Gn interlace for every n≥1. Indeed, it suffices to take λ>M=limn⟶∞Qn0/Pn0, provided such a limit exists. On the other hand, if we fix m>0 and define M=Qm0/Pm0, then Pk and Gk will have interlaced roots for all k=1,…,m if λ>M.Observe that the previous theorem implies that we can construct a sequence of Hurwitz polynomials by using sequencesPnn≥0 and Qnn≥0, when the orthogonality measure is supported on ℝ+; i.e., the zeros of each Pn are simple and positive. This property was used in [29] to construct robustly stable sequences of Hurwitz polynomials by using the Laguerre and Jacobi classical orthogonality measures supported on ℝ+ (see [25]). The main idea was to introduce a parameter t on the measure, in such a way that it remains positive. As a consequence, the corresponding inner product, the associated orthogonal sequence, and the second kind polynomials will depend on t. That is, Pnx,tn≥0 will be orthogonal for every value of t, and thus the sequence fnx,tn≥1 constructed by using Theorem 1 will be Hurwitz for every value of t. In other words, if we consider t as an uncertain parameter, the sequence fnx,tn≥1 will be robustly stable. It is worth mentioning that robustly stable families constructed in such a way have been already used in the design of a compensator that robustly stabilizes an interval plant with uncertain time-delay (see [30]) and can also be used for applications in control design.In this contribution, we propose two alternative ways to construct robustly stable sequences of Hurwitz polynomials by using other properties of orthogonal polynomials. The structure of the manuscript is as follows. In Section2, we introduce an uncertain parameter t on the coefficients of the three-term recurrence relation and then use the Favard theorem described in the previous subsection and Theorem 1 to construct a sequence of Hurwitz polynomials that is robustly stable for every value of t within an specified range. In Section 3, we generate robustly stable sequences by using linear combinations of orthogonal polynomials. Some examples are presented in Section 4. Finally, some conclusions and open problems are discussed in Section 5.


## 2. Stable Polynomials Generated by a Recurrence Relation

In this section, we consider the recurrence relation(13)Pn+1x,t=x−bntPnx,t−antPn−1x,t,n≥0,with initial conditions P−1x,t=0 and P0x,t=1, and ant,bnt are functions of an uncertain parameter t. If antn≥0,bntn≥0 generate a chain sequence for all t∈I, where I is some interval, then Favard’s theorem guarantees that the constructed monic sequence Pnx,tn≥0 is orthogonal with respect to some positive measure σ supported on ℝ+. Thus, the roots of each Pnx,t will be real, simple, and positive for every value of t∈I. Clearly, the associated second kind polynomials Qnx,tn≥0 will also be t-dependent, and for each n≥2 the roots of Pnx,t and Qnx,t will satisfy an interlacing property for every value of t∈I. As a consequence, we can use Theorem 1 to construct a sequence of polynomials fnx,tn≥1 that will be stable for every value of t∈I.Let us start with a simple example. Forn≥0, consider ant=t and(14)bnα,t=tifnis odd,αifnis even,forn≥0,where t∈I with I=0,∞ and α≥4. It is easy to check that antn≥0 and bnα,tn≥0 satisfy(15)0<antbn−1α,tbnα,t⩽14,n≥1.As pointed out in the previous section, they constitute a chain sequence. Notice that we have introduced two variablest and α that can be considered as uncertain parameters. From our previous discussion, the sequence Pnx,α,tn≥0 defined by(16)Pn+1x,α,t=x−bnα,tPnx,α,t−antPn−1x,α,t,n≥0,with P−1x,α,t=0, P0x,α,t=1, is an orthogonal sequence for all t>0 and all α≥4. Moreover, we can obtain the following closed expressions for these polynomials.Proposition 1.
Letantn≥0 and bntn≥0 be defined as before. For n≥1, t>0, and α≥4, we have(17)P2nx,α,t=∑j=0n−1j2n−j2n−jtjx−tn−jx−αn−j,(18)P2n−1x,α,t=∑j=0n−1−1j2n−1−j2n−j−1tjx−tn−jx−αn−1−j.Proof.
We proceed by induction onn. Since we clearly have P1x,α,t=x−t and P2x,α,t=x−tx−α−t, assuming equations (17) and (18) hold for n≥1, we will prove both of them for n+1. Taking into account the properties of the binomial coefficients and equation (16), with yt=x−t and yα=x−α, we obtain(19)P2n+1x,α,t=ytP2nx,α,t−tP2n−1x,α,t,=ytn+1yαn+∑j=1n−1j2n−j2n−jtjytn−j+1yαn−j+∑j=1n−1j2n−j2n−j+1tjytn−j+1yαn−j,=2n+12n+1ytn+1yαn+∑j=1n−1j2n+1−j2n−j+1tjytn−j+1yαn−j,=∑j=0n−1j2n+1−1−j2n+1−j−1tjytn+1−jyαn−j,
and equation (17) is proved. In a similar way, we get(20)P2n+2x,α,t=yαP2n+1x,α,t−tP2nx,α,t,=ytn+1yαn+1+∑j=1n−1j2n+1−j2n+1−j−1tjytn+1−jyαn+1−j+∑j=1n+1−1j2n+1−j2n+1−jtjytn+1−jyαn+1−j,=ytn+1yαn+1+∑j=1n−1j2n+2−j2n+1−jtjytn+1−jyαn+1−j+−1n+1tn+1,which yields equation (18) when n is replaced by n+1.
Of course, we can obtain similar expressions for the second kind polynomialQnx,α,t, since they satisfy the same recurrence relation, with different initial conditions. We state the following result without proof, since it is very similar to the proof of the previous proposition.Proposition 2.
Forn≥1, t>0, and α≥4,(21)Q2n+1x,α,t=∑j=0n−1j2n−j2n−jtjx−αn−jx−tn−j,Q2nx,α,t=∑j=0n−1−1j2n−1−j2n−j−1tjx−αn−jx−tn−1−j,with Q1x,α,t=1.Several observations are in order. First, notice that if we considert and α as uncertain parameters, the structure of the coefficients in Pnx,α,t and Qnx,α,t indicates that the Hurwitz polynomial constructed via Theorem 1 will have polynomic uncertainty. Second, observe that we can introduce some generality in the previous example. In fact, we can replace t by a more general continuous function gt such that gt>0 for every t∈I. For instance, we can consider the case when ant=g2t and bnα,t=αgt with α≥2. It is easily seen that antn≥0 and bnα,tn≥0 satisfy equation (15) and, as a consequence, they generate a chain sequence. Thus, again by using equation (16), we can construct a sequence of orthogonal polynomials that depend on t and α. As before, we can get explicit expressions for their coefficients.Proposition 3.
Forn≥0, let ant=g2t and bnα,t=αgt, where g is a continuous and positive function for all t∈I, and α≥2. Then,(22)P2n−1x,α,t=∑j=0n−1−1j2n−1−j2n−j−1g2jtx−αgt2n−j−1,(23)P2nx,α,t=∑j=0n−1j2n−j2n−jg2jtx−αgt2n−j.Proof.
The casen=0 is trivial. The proof is by induction on n≥1. Again, writing yα,t=x−αgt and using equation (16), we obtain(24)P2n+1x,α,t=yα,tP2nx,α,t−g2tP2n−1x,α,t,=yα,t2n+1+∑j=1n−1j2n−j2n−jg2jtyα,t2n−j+1+∑j=1n−1j−12n−j2n−j+1g2jtyα,t2n−j+1,=2n+12n+1yα,t2n+1+∑j=1n−1j2n+1−j2n−j+1g2jtyα,t2n−j+1,=∑j=0n−1j2n+1−1−j2n+1−j−1g2jtyα.t2n−j+1,which is equation (22) when n is replaced by n+1. Equation (23) can be obtained in a similar way.
Moreover, sinceantn≥0 and bnα,tn≥0 are constant sequences, if α,t are fixed, and both the orthogonal and the second kind polynomials satisfy the same recurrence relation (with different initial conditions), it follows that Qnx,α,t=Pn−1x,α,t for every n≥1. Hence, it is convenient to expand equations (22) and (23) in terms of x.Proposition 4.
Forn≥0, let ant=g2t, bnα,t=αgt, where g is a continuous and positive function for all t∈I, and α≥2. Then,(25)P2nx,α,t=∑k=0n∑j=1n+1−kcn,k+j−12k+j−12k−αgt2j−1x2k+∑k=0n−1∑j=1n−kcn,k+j2k+j2k+1−αgt2j−1x2k+1,P2n+1x,α,t=∑k=0n∑j=1n+1−kdn,k+j−12k+j−12k−αgt2j−1x2k+∑k=0n∑j=1n−kdn,k+j2k+j+12k+1−αgt2jx2k+1,where cn,k=−1n+kn+k2kg2n−kt and dn,k=−1n+kn+1+k2k+1g2n−kt.Proof.
The result follows easily by using the binomial expansion theorem on equations (22) and (23).
As a consequence, for this particular choice ofant and bnα,t, using Theorem 1 and the subsequent discussion, we can state the following result.Proposition 5.
LetPnx,α,tn≥0 be the sequence of orthogonal polynomials as in Proposition 4. For n≥0, t>0, and α≥2,(26)f2nx,α,t=−1nPn−x2,α,t+−1n−1xPn−1−x2,α,t.
is a stable polynomial of degree2n, where P−1x,α,t=0. On the other hand, fix n and choose M such that Pn−10,α,t/Pn0,α,t<M for t>0 and α≥2. Then,(27)f2k+1x,α,t=−1kxPk−x2,α,t+−1kGk−x2,α,t.is a stable polynomial with degree 2k+1 for all k=1,…,n. Here, Gkx,α,t=λPkx,α,t+Pk−1x,α,t with λ>M, where the coefficients of the even and odd powers of x in fnx,α,t are given in Proposition 4.We point out that since the only conditions on the functiong in Proposition 3 are positivity and continuity, the procedure to construct sequences of stable Hurwitz polynomials allows us to obtain different types of uncertainty. As an example, we can set gt=sint for t∈I=0,π and the obtained uncertainty structure will be nonlinear. Finally, it is also important to notice that there are many different choices for antn≥0 and bntn≥0. The only condition is that they generate a chain sequence, as pointed out in the previous section. In the general case, given both sequences, Pnx,tn≥0 and Qnx,tn≥0 can be computed by using equations (7) and (8), respectively, and the robustly stable Hurwitz polynomials can be obtained by using Theorem 1. Nevertheless, if both sequences are chosen in a simple way, as the cases considered in this section, then it is possible to get explicit expressions for the polynomials.


## 3. Stable Polynomials Associated with Linear Combinations of Orthogonal Polynomials

The particular choice of the constant sequencesantn≥0 and bnα,tn≥0 in the previous section led to the situation where Qn=Pn−1, and thus in the construction of robustly stable polynomials given in Proposition 5 a linear combination of Pn and Pn−1 appears in the polynomial Gn used for the odd degree polynomials. In this section, we further explore the idea of using linear combinations of orthogonal polynomials to construct robustly stable sequences of Hurwitz polynomials. Recall that if Pnn≥0 is a sequence of orthogonal polynomials, then the roots of Pn and Pn−1 are real, simple, and interlaced. The following result establishes an interlacing property for linear combinations of orthogonal polynomials.Theorem 2.
(see [31]). For n≥1 and λ1,λ2∈ℝ and both not zero, λ1Pnx+λ2Pn−1x has real and simple roots. Moreover, if μ1,μ2∈ℝ, then λ1Pnx+λ2Pn−1x and μ1Pnx+μ2Pn−1x have no common roots and their roots satisfy an interlacing property, unless one combination is a scalar multiple of the other.Set(28)Fnx;λ1,λ2=λ1Pnx+λ2Pn−1x,Gnx;μ1,μ2=μ1Pnx+μ2Pn−1x.λ1,λ2,μ1,μ2∈ℝ such that λ1μ2−μ1λ2≠0. Then, according to the previous theorem, the roots of Fnx;λ1,λ2 and Gnx;μ1,μ2 are real, simple, and interlaced. As a consequence, we can use the Hermite–Biehler theorem to construct Hurwitz polynomials as in Theorem 1. Namely,(29)f2n+1x=−1nxFn−x2;λ1,λ2+−1nGn−x2;μ1,μ2will be a Hurwitz polynomial of degree 2n+1 if the choice of the four parameters λ1,λ2,μ1,μ2∈ℝ guarantees the interlacing of the roots of Fn and Gn and the largest root of Fn is greater than the largest root of Gn. Notice that the latter condition implies we must have λ1≠0 and μ1≠0. A similar approach can be taken to construct Hurwitz polynomials of even degree, but it will be necessary to set μ1=0 so that Gn is a polynomial of degree n−1, and to write(30)f2nx=−1nFn−x2;λ1,λ2+−1nxGn−x2;0,μ2,where we again have the requirement λ1≠0 (which implies μ2≠0 so that λ1μ2−μ1λ2≠0). Observe that we can consider λ1,λ2,μ1,μ2∈ℝ as uncertain parameters, and by deducing appropriate ranges of values for each one of them such that the interlacing property of the roots of Fn and Gn is satisfied, the Hermite–Biehler theorem guarantees that the constructed polynomials will be robustly stable for all values of the parameters within an specified range. In this sense, we will determine such conditions on the four parameters. Observe that the even and odd degree cases are independent and thus will be analyzed separately. We begin with the even degree case. Recall that the sign function is denoted by(31)sgnx=1ifx>0,0ifx=0,−1ifx<0.Proposition 6.
LetPnxn≥0 be a sequence of monic polynomials orthogonal with respect to some positive measure supported in some interval E⊂ℝ+, and let Fnx;λ1,λ2 and Gnx;0,μ2 (i.e. μ1=0) be defined as above. Then, for n≥1,(32)f2nx=−1nsgnλ1Fn−x2;λ1,λ2+−1n−1sgnμ2xGn−x2;0,μ2.is a Hurwitz polynomial of degree 2n for every λ1≠0,μ2≠0, and λ2∈ℝ such that(33)λ2<−λ1sgnλ1Pn0Pn−10.Proof.
Sinceλ1,μ2 are nonzero and n≥1, sgnλ1Fnx;λ1,0=λ1Pnx and Pn−1x have positive, simple, and interlacing zeros, and thus their coefficients have alternating signs. Hence, both −1n−1μ2Pn−1−x and(34)−1nλ1Pn−x+−1nsgnλ1λ2Pn−1−x,have negative, simple, and interlacing zeros and positive coefficients, provided that λ2 satisfies sgnλ1λ2<−λ1Pn0/Pn−10. The result follows from the Hermite–Biehler theorem.
Now we consider the odd degree case. Recall that in this situation we must haveλ1≠0 and μ1≠0. Given n≥1, let us denote by xn,1λ1,λ2<…<xn,nλ1,λ2 and yn,1μ1,μ2<…<yn,nμ1,μ2 the roots of Fnx;λ1,λ2 and Gnx;μ1,μ2, respectively, in increasing order.
Assume firstλ2=0. As stated in Section 1, if the orthogonality measure is supported in E⊂ℝ+, then the roots of Fnx;λ1,0=λ1Pnx belong to the interior of the convex hull of E and interlace with the roots of Gnx;μ1,μ2 when λ1μ2≠0, from Theorem 2. Hence, our next goal is to determine when the largest root of Fnx;λ1,0 is greater than the largest root of Gnx;μ1,μ2 in terms of the parameters λ1,μ1,μ2,∈ℝ. Furthermore, in this situation the roots of Fnx;λ1,0 also interlace with the roots of Pn−1, and therefore sgnPn−1xn,nλ1,0=1. Thus, for all μ1,μ2∈ℝ,(35)sgnGnxn,nλ1,0;μ1,μ2=sgnμ2sgnPn−1xn,nλ1,0=sgnμ2.
As a consequence, forλ1μ2≠0, we get the following.(1)
Ifλ2=0, μ1>0 then(36)sgnμ2yn,kμ1,μ2<sgnμ2xn,kλ1,0.(2)
Ifλ2=0, μ1<0 then(37)sgnμ2xn,kλ1,0<sgnμ2yn,kμ1,μ2.We now turn to the caseλ2≠0. Notice that the continuity of the determinant λ1μ2−μ1λ2≠0 together with Theorem 2 implies that equations (36) and (37) still hold when λ2≠0, as long as the determinant is nonzero. Thus, we consider the following four cases:(3)
Ifλ2>0 and μ1>0, then(38)sgnμ1λ2λ1−μ2xn,kλ1,λ2<sgnμ1λ2λ1−μ2yn,kμ1,μ2.(4)
Ifλ2>0 and μ1<0, then(39)sgnμ1λ2λ1−μ2yn,kμ1,μ2<sgnμ1λ2λ1−μ2xn,kλ1,λ2.(5)
Ifλ2<0 and μ1>0, then(40)sgnμ1λ2λ1−μ2xn,kλ1,λ2<sgnμ1λ2λ1−μ2yn,kμ1,μ2.(6)
Ifλ2<0 and μ1<0, then(41)sgnμ1λ2λ1−μ2yn,kμ1,μ2<sgnμ1λ2λ1−μ2xn,kλ1,λ2.
Clearly, equations (36), (38), and (40) (resp. equations (37), (39), and (41)) imply that the inequality for the case μ1>0 (resp. μ1<0) is independent of value of λ2. As a consequence, we can now proceed with the construction of sequences of Hurwitz polynomials, by using the Hermite–Biehler theorem and the interlacing property of the zeros of Fnx;λ1,λ2 and Gnx;μ1,μ2. Recall that, for the odd degree case, the construction of a Hurwitz polynomial depends of the relative position of the zeros of Fnx;λ1,λ2 and Gnx;μ1,μ2 and therefore of the values of parameters λ1,λ2,μ2,μ1.Proposition 7.
LetPnxn≥0 be a sequence of monic polynomials orthogonal with respect to some positive measure supported in some interval E⊂ℝ+, and let Fnx;λ1,λ2 and Gnx;μ1,μ2 be defined as above, with λ1≠0 and μ1≠0. Then, for n≥1,(i)
ifμ1>0 and λ2∈ℝ such that sgnλ1λ2<−λ1Pn0/Pn−10, then(42)−1nsgnλ1xFn−x2;λ1,λ2+−1nGn−x2;μ1,μ2.is a Hurwitz polynomial of degree2n+1 if and only if(43)μ1λ2λ1<μ2<−μ1Pn0Pn−10.Also,(44)−1nxGn−x2;μ1,μ2+−1nsgnλ1Fn−x2;λ1,λ2.is a Hurwitz polynomial of degree2n+1 if and only if(45)μ2<minμ1λ2λ1,−μ1Pn0Pn−10.(ii)
ifμ1<0 and λ2∈ℝ such that sgnλ1λ2<−λ1Pn0/Pn−10, then(46)−1nsgnλ1xFn−x2;λ1,λ2+−1nsgnμ1Gn−x2;μ1,μ2.is a Hurwitz polynomial of degree2n+1 if and only if(47)−μ1Pn0Pn−10<μ2<μ1λ2λ1.Also,(48)−1nsgnμ1xGn−x2;μ1,μ2+−1nsgnλ1Fn−x2;λ1,λ2.is a Hurwitz polynomial of degree2n+1 if and only if(49)μ2>maxμ1λ2λ1,−μ1Pn0Pn−10.Proof.
Assumeμ1>0. The proof is based on the observation that both −1nsgnλ1Fn−x;λ1,λ2 and −1nGn−x;μ1,μ2 have negative, simple, and interlacing zeros, and thus their coefficients are positive if and only if sgnλ1λ2<−λ1Pn0/Pn−10 and μ1λ2/λ1≠μ2<−μ1Pn0/Pn−10, respectively. Indeed, if it were true that sgnλ1λ2<−λ1Pn0/Pn−10 and μ2=−μ1Pn0/Pn−10, we would have Fn0;λ1,λ2=0 and Gn0;μ1,μ2=0. Moreover, the fact that both Fnx;λ1,λ2 and Gnx;μ1,μ2 have the same roots follows from Theorem 2 if μ2=μ1λ2/λ1. Therefore, since E⊂ℝ+, we conclude that 0<yn,kμ1,μ2<xn,kλ1,λ2 (resp. 0<xn,kλ1,λ2<yn,kμ1,μ2) for all k=1,…,n if μ1λ2/λ1<μ2 (resp. μ2<μ1λ2/λ1). Hence i follows from the Hermite–Biehler theorem. A similar argument can be applied to the case when μ1<0.


## 4. Numerical Examples

In this section we provide some numerical examples, computed using Mathematica ®, to illustrate the behavior of the roots of stable polynomials generated by the three-term relation equation (13) and stable polynomials associated with linear combinations of orthogonal polynomials. More specifically, we will plot the location of the roots of some polynomials given by Propositions 5 and 7 for some values of the parameters.First, consider the case whengt=t1/2, t∈0,∞, and α≥2 in Proposition 5. Then, the first even degree polynomials are(50)f2x,α,t=x2+x+αt1/2,f4x,α,t=x4+x3+2αt1/2x2+αt1/2x+α2−1t1/2,f6x,α,t=x6+x5+3αt1/2x4+2αt1/2x3+3α2−2tx2+α2−1tx+αα2−2t3/2.And we obtain polynomic uncertainty in the parameterq1=α and nonlinear uncertainty in the parameter q2=t.Example 1.
For our first example, Figure1 illustrates the motion of the roots of f4x,α,t when α=2 and t varies from 0.1 to 90 with a step size of 0.1 (panel (a)) and when α=8 and t varies from 0.1 to 200 with a step size of 0.5 (panel (b)). For both cases, notice that the real part of all the complex roots tends approximately to −1/4 when t⟶90 and t⟶200, respectively.Figure 1
(a) The location of the roots off4x,2,t for several values of t. (b) The location of the roots of f4x,8,t for several values of t.
(a)(b)Example 2.
Proceeding in a similar way, Figure2 shows the location of the roots of f6x,α,t when α=2 and t varies from 0.1 to 500 with a step size of 0.1 (panel (a)) and when α=4 and t varies from 0.1 to 1000 with a step size of 0.2 (panel (b)). Analogously, the real part of each pair of conjugate complex roots tends to two distinct negative numbers when t⟶500 and t⟶1000, which approximately are −1/4 and −1/8, respectively.Figure 2
(a) Location of the roots off6x,2,t for several values of t. (b) Location of the roots of f6x,4,t for several values of t.
(a)(b)Example 3.
In Figure3 we present a comparison between the evolution of the roots of f4x,α,t and f6x,α,t with respect to α (with t fixed). The panel (a) corresponds to the case when t=1 and α varies from 2 to 200 with a step size of 2, while the graphic on the right corresponds to the case when t=0.1 and α varies from 2 to 200 with a step size of 0.5. In both cases, notice that the real part of all the complex roots approximately tends to −1/2 or zero when α⟶200.
On the other hand, to construct stable polynomials of degrees 3 and 5, we must compute−Pn−10,α,t/Pn0,α,t when n=2. In Proposition 5, take gt=t, t∈0,∞, and α≥2. Since P1x,α,t=x−αt and P2x,α,t=x2−2αtx+α2−1t2, the corresponding polynomials are(51)f1x,α,t=x+λf3x,α,t=x3+λx2+αtx+λαt−1,f5x,α,t=x5+λx4+2αtx3+2λαt−1x2+α2−1t2x+α2−1λt2−αt,where λ>αα2−1−1t−1.Figure 3
(a) The location of the roots off4x,α,1 for several values of α. (b) The location of the roots of f6x,α,0.1 for several values of α.
(a)(b)Example 4.
We chooseλ=λα,t=0.5+αα2−1−1t−1; then panel (a) in Figure 4 illustrates the location of the roots of f3x,6,t when t varies from 0.1 to 5 with a step size of 0.05 and panel (b) in Figure 4 shows the motion of the roots of f5x,2.2,t when t varies from 0.4 to 7 with a step size of 0.2. For both cases, the single real root tends to limt⟶∞−λα,t=−0.5 and the real part of all the complex roots tends approximately to 0 when t⟶5 and t⟶7, respectively.
Finally, we now show an example of robustly stable polynomials constructed by using linear combinations of Laguerre polynomials, which are orthogonal with respect to the measurexαe−x with support in ℝ+ and defined for α>−1. In particular, the first Laguerre monic polynomials are(52)L1αx=x−α+1,L2αx=x2−2α+2x+α2+3α+2,L3αx=x3−3α+3x2+3α2+5α+6x−α3+6α2+11α+6.Figure 4
Motion of the complex roots of two stable polynomials of degrees three (a) and five (b) constructed via Proposition5 for several values of t.
(a)(b)Example 5.
Taking into account these polynomials, Figure5 illustrates the motion of the roots of a stable polynomial of degree 3 constructed with α=2, λ1=1, λ2=0.7, and μ1=0.5 (panel (a)). According to Proposition 7, μ2 can vary from −2.95 to 0.05, and the roots are plotted using a step size of 0.3. Similarly, panel (b) in Figure 5 corresponds to the case when α=1/2, λ1=1, λ2=−0.5, μ1=−0.4, and μ2 varies from −0.925 to 0.125 with a step size of 0.15.Figure 5
Motion of the complex roots of a stable polynomial of degrees three (a) and five (b) constructed via Proposition7 for several values of μ2.
(a)(b)


## 5. Discussion and Further Comments

We have used algebraic properties of orthogonal polynomials to construct families of Hurwitz polynomials that are robustly stable in terms of a set of parameters. First, we use Favard’s theorem to introducet and α as parameters in the coefficients of the recurrence relation, which are now continuous functions of those parameters. Notice that there is a great deal of generality on the choice of ant and bnα,t. Indeed, the only requirement is that they generate a chain sequence, as mentioned in Section 2. Of course, one can introduce an arbitrary set of parameters, say ant1,…,tk, bnt˜1,…,t˜l, as long as the corresponding sequences generate a chain sequence for all values of all parameters within a specified range.On the other hand, we have also shown that the Hermite–Biehler theorem can be used to construct Hurwitz polynomials from linear combinations of two consecutive degree orthogonal polynomials, whose roots satisfy an interlacing property. Here, four real parameters are considered and Hurwitz polynomials of even and odd degree can be constructed if the parameters satisfy some inequalities.As a consequence of the results presented here, together with those of the recent contribution [29], it is possible to use orthogonal polynomials to construct sequences of robustly stable Hurwitz polynomials in three different ways: by introducing an uncertain parameter on the orthogonality measure, by introducing uncertain parameters on the three-term recurrence relation, and by using linear combinations of orthogonal polynomials. The results of this work can be extended in the following directions:(1)
It is possible to use Hurwitz polynomials constructed by using orthogonal polynomials in pole-placement design: to develop a controller that yields a closed-loop system with a prescribed characteristic polynomial [32, 33]. In this situation, an interesting open problem is to analyze the capabilities of the resulting system, such as closed-loop frequency response, disturbance rejection, and sensitivity [34]. It is worth noticing that properties of orthogonal polynomials have been used recently to propose a stabilization method for interval plants with uncertain time-delay by using the set value concept [30].(2)
Although the proposed Hurwitz polynomials are robustly stable for the prescribed values of the parameters, an interesting open problem is to deduce the equations of motion of the roots in terms of the parameters (i.e., their behavior in terms of the involved parameters or, equivalently, the algebraic expressions for the curves they describe on the complex plane), as well as to determine the regions of the complex plane where the roots will be located for every value of the parameters. This information would be useful in the design of controllers in pole-placement design previously mentioned.



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