mat.izt.uam.mxmat.izt.uam.mx/mat/documentos/produccion_academica/2016/2/articulo... · bol. soc....

1 23

Boletín de la Sociedad MatemáticaMexicanaThird Series ISSN 1405-213X Bol. Soc. Mat. Mex.DOI 10.1007/s40590-016-0086-x

Stable polynomial curves and someproperties with application in control

J. A. López-Renteria, B. Aguirre-Hernández & F. Verduzco

1 23

Your article is protected by copyright and

all rights are held exclusively by Sociedad

Matemática Mexicana. This e-offprint is for

personal use only and shall not be self-

archived in electronic repositories. If you wish

to self-archive your article, please use the

accepted manuscript version for posting on

your own website. You may further deposit

the accepted manuscript version in any

repository, provided it is only made publicly

available 12 months after official publication

or later and provided acknowledgement is

given to the original source of publication

and a link is inserted to the published article

on Springer's website. The link must be

accompanied by the following text: "The final

publication is available at link.springer.com”.

Bol. Soc. Mat. Mex.DOI 10.1007/s40590-016-0086-x

ORIGINAL ARTICLE

Stable polynomial curves and some properties withapplication in control

J. A. López-Renteria1 · B. Aguirre-Hernández1 ·F. Verduzco2

Received: 28 January 2015 / Accepted: 22 January 2016© Sociedad Matemática Mexicana 2016

Abstract The aim of this work is to give a Hurwitz path (which is a family of poly-nomials) joining any two arbitrary stable polynomials in the set of monic Hurwitzpolynomials with positive coefficients and fixed degree n, H+

n . This and the homo-topy of paths allow to prove the existence of a dense trajectory inH+

n . It implies, by theMöbius transform and Viète’s map, that we can find a connecting-path in the set of theSchur polynomials,Sn . Due to the formof the stable connecting-paths, a feedback con-trol is designedwhose structure can be used to stabilize continuous or discrete systems.

Keywords Control theory · Stable polynomials · Poles placement · Stabilizingfeedback control

Mathematics Subject Classification 93C05 · 93D09 · 93D20

1 Introduction

Let us to consider the dynamical system

ξ : f (ξ), ξ ∈ Rn,

Research partially supported by CONACyT with scholarship number 234083.

B B. Aguirre-Herná[email protected]

1 División de Ciencias Básicas e Ingeniería, Departamento de Matemáticas, UniversidadAutónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco #186, Col. Vicentina, 09340Mexico, DF, Mexico

2 División de Ciencias Exactas, Departamento de Matemáticas, Universidad de Sonora, BoulevardRosales y Luis Encinas s/n, Col. Centro, 83000 Hermosillo, Sonora, Mexico

Author's personal copy

http://crossmark.crossref.org/dialog/?doi=10.1007/s40590-016-0086-x&domain=pdf

J. A. López-Renteria et al.

with f a smooth vector field. In continuous time case the system iswritten as ξ = f (ξ),while in discrete time the system is written as ξ �→ f (ξ). The class of dynamicalsystems to dealwith in thiswork are the linear control systems in discrete or continuoustime of the form

ξ : �ξ + �u, (1)

where ξ ∈ Rn is the vector of states, � ∈ R

n×n , � ∈ Rn×m and u ∈ R

m is thevector of control signal which is an actuator for the system. A great deal of qualitativeinformation about the behaviors of its solutions is determined by the linear operator�,specifically its eigenvalues set {z1, . . . , zn}. Namely, if the system (1) is continuouswith u = 0, it is said to be asymptotically stable at the origin if and only if thecharacteristic polynomial of the matrix � has all of its roots with negative real part(roots in C−). The system is said to be unstable if it has at least one root in iR ∪C

+;while for the discrete case, the system is said to be stable if and only if the characteristicpolynomial of� has all of its roots with modulus less than one, that is, inside the openunit disk D = {z ∈ C : |z| < 1}. The discrete system is said to be unstable if it has atleast one root in the complement DC = {z ∈ C : |z| ≥ 1}. In both of these cases thecharacteristic polynomial is monic, that is, the coefficient of the leading term is equalto one. In this way we define the objects of study: we say that a polynomial isHurwitzor Hurwitz stable if all of its roots have negative real part, which is the continuouscase; and for the discrete case, a polynomial is Schur or Schur stable if all of its rootshave modulus less than one.The problem of obtaining (Hurwitz or Schur) stability for the control system (1), ifu �= 0, has been one of the most investigated problems in Engineering and AppliedMathematics; a way to achieve it is by mean of a suitable design of u. In order todesign the controller u it is convenient to transform the system (1) to its canonicalcontrollable form, that depends on the coefficients of its characteristic polynomial:let p�(t) = tn + a1tn−1 + · · · + an be the characteristic polynomial of �. It is wellknown that if the controllability matrix C = [� �� · · · �n−1�] has rank n, then thesystem is completely controllable and the transformation matrix Q = CM , with

M =

⎡⎢⎢⎢⎢⎢⎣

a1 a2 · · · an−1 1a2 a3 · · · 1 0...

......

...

an−1 1 · · · 0 01 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎦

, (2)

defines the change of variable x = Q−1ξ such that the system (1) becomes in thecanonical controllable form:

x : Ax + bu, (3)


Stable polynomial curves and some properties...

where

A = Q−1�Q =

⎡⎢⎢⎢⎢⎢⎣

0 1 0 · · · 00 0 1 · · · 0

... · · · ...

0 0 0 · · · 1−an −an−1 −an−2 · · · −a1

⎤⎥⎥⎥⎥⎥⎦

and b = Q−1� =

⎡⎢⎢⎢⎢⎢⎣

00...

01

⎤⎥⎥⎥⎥⎥⎦

where a j , j = 1, . . . , n − 1 are the coefficients of p�(t) = pA(t). We design ascalar linear feedback state control of the form u(k, x) = c(k)Tx , with c(k)T =(c1(k), . . . , cn(k)) and k a real parameter. Finally, the system (3) with the control u =u(k, x) is expressed as x : (A + bc(k)T)x , with characteristic polynomial P(k, t) =pA(t) + q(k, t), where q(k, t) = c1(k)tn−1 + c2(k)tn−2 + · · · + cn(k). In references[3,7,16] one can see the design of a scalar controller by taking c(k)T = −kcT, withthe constant vector cT = (c1, . . . , cn). Thus, the scalar control u(k, t) = −kcTx arisesthe family of polynomials P(k, t) = pA(t) + kq(t), which is a ray of polynomials,with vertex in pA(t) and direction q(t) = c1tn−1 + · · · + cn . Different techniques inthe study of the ray pA(t)+kq(t) are developed for finding the values of the parameterk to still keeping stability in the system if pA(t) is Hurwitz. The system is said to besimply stable if it is stable for a single value of k0; and it is said to be robustly stableif it is stable for all k in a parameter set K . If k− and k+ are the minimum and themaximum values of k for robust stability, respectively, then K is called the maximalstability interval. The approach can be changed by taking a direction λ = k−−k

k−−k+ ,for k ∈ [k−, k+], and define the linear convex combination (1 − λ)p0(t) + λp1(t),where if k = k− then λ = 0 and p0(t) = pA(t) + k−q(t); if k = k+ then λ = 1 andp1(t) = pA(t) + k+q(t). Thus, in this sense the ray pA(t) + kq(t) is Hurwitz stablefor all k ∈ [k−, k+] if and only if the segment of polynomials (1 − λ)p0(t) + λp1(t)is Hurwitz stable for all λ ∈ [0, 1]. In [6–8], techniques for determining the wholeHurwitz stability of segments of polynomials are developed.In the last years a lot of people have studied the sets of Hurwitz and Schur polynomialsin order to know their properties and apply them in the design of the controller u(k, x):the well known Hurwitz polynomials set, denoted by H+

n , described as the set ofmonic Hurwitz polynomials with positive coefficients; and the Schur polynomials set,denoted as Sn , formed by the monic Schur polynomials. In [9] one can see that H+

nis an open set, nevertheless it is not convex [20], thus not every segment or ray ofpolynomials consists of stable polynomials. In the case of study of rays, even whenthe polynomial extreme is stable, we do not warrant the entire stability of the ray,however, we can find a segment of such a ray which is stable, such problem is todeterminate the maximal stability interval (see [6,8,16] for instance). In [3], authorsgive a technique for determine stability of rays and cones of polynomials. In segmentscase, two stable extremes do not guarantee the entire stability of the segment due tothe lack of convexity ofH+

n . Particularly, in [8], is boarded the problem of stability ofa linear convex combination of stable polynomials. One can see a test for determinethe stability of segments of polynomials in [6]. In [4] we can see thatH+

n has structureof smooth trivial vector bundle over H+

n−l of rank l as well as generation of rays of



a Hurwitz polynomial. Some topological and differential properties of H+n are also

given.In relation of families of Schur polynomials, recently an interval of stability wasdetermined in order to obtain the stability in sampled-data systems [12]. In the paper[5] topological approaches have also been used to study the sets of Hurwitz and Schurpolynomials. In [11,14] the authors show that Sn is contractible to the polynomial tn

and as a consequence H+n is contractible to the polynomial (t + 1)n via the Möbius

transformation; an improved proof has been given in [2]. As a consequence, H+n

and Sn are path-connected. However, there is no work in control theory showing aparticular stable curve of polynomials joining any pair of arbitrary (Schur or Hurwitz)stable polynomials. Our main goal in this work is exhibit explicitly a robust Hurwitzstable curve joining any two Hurwitz stable polynomials and apply these results tothe design of a stabilizing feedback control. Respective results for Schur stability arealso given. The rest of paper is organized as follows: in Sect. 2 we present the toolto prove the existence of the connecting-curves. Basic concepts about homotopy ofpaths are presented which will be used to prove the existence of a connecting-curvein H+

n , as well as some properties of the Viète’s map and Möbius transformation.In the Sect. 3 we prove that for any two given polynomials p0(t) and p1(t) in H+

nthere exist homotopic Hurwitz curves joining both polynomials using linear convexcombinations. A dense curve in H+

n using homotopic paths product is also found.Next, other versions of the Möbius transformation are presented and used to prove theexistence of a connecting-curve and a dense trajectory in Sn is proved and exhibited.In Sect. 4, a stabilizing feedback control via the stable connecting-curves is proposedas well as an illustrative example is provided.

2 Preliminaries

In this sectionwepresent the needed tool for finding theHurwitz andSchur connecting-curves.

2.1 Homotopy of paths

A continuous map f : [0, 1] → X is called path in X . The points f (0) and f (1) arecalled initial and final points, respectively. Thus, f is a path joining to f (0) and f (1).A space X is said to be path-connected if for two arbitrary points x0 and x1 in X thereexists a path in X joining x0 to x1. If f and g are two paths in X , with f (1) = g(0)the product of f with g, f ∗ g, is given by

( f ∗ g)(λ) ={f (φ1(λ)) if λ ∈ [0, r ]g(φ2(λ)) if λ ∈ [r, 1] , (4)

where φ1(λ) and φ2(λ) are continuous reparameterizations on the variable λ such thatφ1(0) = 0, φ1(r) = 1 and φ2(r) = 0, φ2(1) = 1, and r defines a partition of theinterval I = [0, 1]. That is, if f is a path from x0 to x1 and g is a path from x1 to x2,then f ∗ g is a path from x0 to x2. Two paths f and g are homotopically equivalents,



f ∼ g, if there exists a continuous map F : I × I → I such that for λ ∈ I , τ ∈ I , wehave

F(λ, 0) = f (λ) and F(λ, 1) = g(λ);F(0, τ ) = f (0) and F(1, τ ) = g(1).

F(λ, τ ) is called the homotopy between f and g. If f ∼ g and g is a constant map,we say that f is nulhomotopic. Notice that∼ is an equivalence relation and we denoteby [ f ] the class of all the paths g related with f . So if f ∼ g then [ f ] = [g]. Thus,the path product defined above induces a well-defined operation on path-homotopyclasses [ f ] ∗ [g] = [ f ∗ g], and we have the following lemma which one can see aproof in [18].

Lemma 1 Let f be a path in X. Let 0 = a0 < a1 < · · · < an−1 < an = 1 be apartition of the interval [0, 1] and define fi = f |[ai−1,ai ] as the restriction of f to[ai−1, ai ]. Then

[ f ] = [ f1] ∗ [ f2] ∗ · · · ∗ [ fn].Even ∗ is not defined for all paths in X , it satisfies the groupoid properties. A

space X is said to be contractible if the identity map iX : X → X is nulhomotopic.Intuitively, a space X is contractible if it can be continuously deformed to a point.

2.2 Viète’s map

Denote by Cn[z] to the set of complex polynomials of degree less or equal to n inthe complex variable z, and let a0 + a1z + · · · + anzn be a polynomial in Cn[z], withan �= 0. Define the map

ϕ : Cn[z] → Cn+1 (5)

a0 + a1z + · · · + anzn �→ (a0, a1, . . . , an),

which is a homeomorphism between Cn[z] and Cn+1, with Cn[z] endowed (via ϕ) ofthe correspondingHermitian inner product and its induced topology. Similarly, let us todenote by Rn[t] to the set of real variable polynomials with real coefficients of degreeless or equal to n, endowed with the Euclidean inner product and its induced topologyvia the isomorphism ϕ. Let us to call Pn[t] ⊂ Rn[t] to the real monic polynomial setof fixed degree n. In the following, we shall see the relation between the roots of apolynomial in Cn[z] and its coefficients. The restraint for Pn[t] will be given.

Let us to consider the extended plane or also called Riemann sphere, which consistson the complex plane with the infinity point joined, denoted by S2. Consider the n-symmetric product of the Riemann sphere consisting on n-tuples without order of(not necessary distinct) elements of S2, and it is denoted by Symn(S2). Alternatively,this space is the quotient of the n−cartesian product (S2)×n with the symmetric groupn , endowed with the quotient topology. Given a n-tuple without order (z1, . . . , zn) ∈Symn(S2), we associate the polynomial



p(z) = an

n∏j=1

(z − z j ) = a0 + a1z + · · · + anzn (6)

with the coefficients vector (a0, a1, . . . , an) ∈ CPn , where CPn is the complex pro-

jective space of dimension n. This correspondence defines a homeomorphism betweenSymn(S2) and the projective space CP

n , and the complex structure on CPn carries

over to one on Symn(S2) under this correspondence. In [14], subsection 4.1.1, or [17]we can also see related theory with this correspondence. That such correspondence iscalled the Viète’s projective map and one can see it in [1,10,15]. In the following, wedescribe a version of the Viète’s map.

Let us to consider the elementary symmetric polynomials in n complex variablesz1, . . . , zn defined as the continuous functions σk : Symn(S2) → CP given by

σ0 = σ0(z1, . . . , zn) = 1,

σk = σk(z1, . . . , zn) =∑

1≤i1<···<ik≤n

zi1 · · · zik , k > 0.

Consider the complex polynomial p(z) given in (6), with the n-tuple of its roots(z1, . . . , zn) ∈ Symn(S2). The Viète’s map vp : Symn(S2) → CP

n is given by

vp(z1, . . . , zn) = ((−1)nσn(z1, . . . , zn), . . . , σ2(z1, . . . , zn),

− σ1(z1, . . . , zn), σ0(z1, . . . , zn)),

where for each k = 1, 2, . . . , n we have that (−1)kσk(z1, . . . , zn)an = an−k , whichare the inputs of the coefficient vector corresponding to the polynomial p(z) of (6).

Recall that the n-tuple (z1, . . . , zn) in Symn(S2) is unordered, hence the (n + 1)-vector of coefficients corresponding to the n-tuple of roots above is uniquelydetermined by the Viète’s projective map up unit multipliers. Henceforth we shall bedealing with monic polynomials and omit the n-th entry with the value 1 to identify thepolynomial setPn[t] directly withRn by assigning the vector (a0, a1, . . . , an−1) ∈ R

n

to the monic polynomial a0+a1t +· · ·+an−1tn−1+ tn . Furthermore, that consideredpolynomials are real polynomials with no roots at infinity whence the correspondingn-vector of coefficients of p(t) lies inRn (via the Viète’s map), with the restraint that ifz j is an entry of (z1, . . . , zn) and if z j = zk for some k �= j , then zk is also an entry of(z1, . . . , zn). With the aforementioned and since C ⊂ S2, we can to relax the Viète’smap to Symn(C) instead Symn(S2) and we simply write vp : Symn(C) → R

n .

2.3 Möbius transformations

For the complexnumbersa, b, c, d such thatad−bc �= 0, define themapm : S2 → S2

given by

m(z) = az + b

cz + d



for all z ∈ C such that cz + d �= 0. If c = 0, define m(∞) = ∞ and if c �= 0,define m(∞) = a

c and m(− dc ) = ∞. Such applications are called Möbius maps with

the following properties: Möbius maps are bijections from S2 onto itself, where theinverse is m−1(z) = dz−b

−cz+a ; they are homeomorphisms of S2. We shall consider themap m such that a = b = c = 1 and d = −1,

m(z) = z + 1

z − 1(7)

which is a biholomorphism of S2 onto itself with inverse m−1(·) = m(·), and trans-forms C− onto the open unit disk D and vice versa (see [14], p. 343, Lemma 3.4.79).We can see thatm(0) = −1 andm(∞) = 1. TheMöbius transformation for a complexpolynomial p(z) of the form p(z) = a0 + · · · + an−1zn−1 + anzn induced by Möbiusmap (7) is

p(z) = (z − 1)n p

(z + 1

z − 1

)(8)

= an(z + 1)n + an−1(z + 1)n−1(z − 1) + · · · + a0(z − 1)n

=n∑j=0

a j p j (z).

It is a vector space isomorphism ofCn[z] onto itself and it is involutive modulo a non-zero constant: ˜p = 2n p.Moreover, a polynomial p(z) is Schur stable if and only if p(z)is Hurwitz stable (see [14], p. 344, Lemma 3.4.81). Clearly, the transformation (·) isalso defined for real polynomials andwe shallwrite p(t) instead p(z)making referenceto the dealing with a real polynomial. When dealing with complex polynomials wesimply write p(z).

Remark 1 Sincewe shallworkwith fixed n-degree polynomialswe only have complexroots distinct to∞, therefore we can relax our domain from S2 to eitherC− orD. Thisrestraint does not affect Viète’s map and since the Möbius map is a homeomorphismwhich maps C− onto D and vice versa, all our results are true with this restraint.

3 Main results

3.1 A Hurwitz connecting-curve

As we have mentioned above, the fact that the Hurwitz polynomials set is contractibleto the polynomial (s + 1)n (see [11,14] for instance), via the Möbius transformationgiven in (8), it carries us to path-connectivity of H+

n . In the following theorem wegive one of these particular curves in explicit form using linear convex combinationsbetween the coefficients of the irreducible factors of two polynomials in H+

n . Weshall use the fact that in the field of real polynomials of fixed degree n, Rn[t], theirreducible factors are of the form t2 + At + B and t + C , with A, B,C ∈ R, andtherefore t2 + At + B and t +C are Hurwitz if and only if A, B and B are positive [9].



Theorem 1 Given any pair of polynomials p0 and p1 inH+n there exists a continuous

function Pt : [0, 1] → H+n such that Pt (0) = p0 and Pt (1) = p1. That is, H+

n ispath-connected.

Proof The proof will be based on a technical construction of Pt . Let

p0(t) =m1∏j=1

(t + δ j )

m1+n1∏j=m1+1

(t2 + α j t + β j ), t ∈ R, (9)

and

p1(t) =m2∏j=1

(t + ρ j )

m2+n2∏j=m2+1

(t2 + γ j t + η j ), t ∈ R, (10)

be the decomposition as irreducible factors of the Hurwitz polynomials p0 and p1,where δ j , α j , β j , ρ j , γ j , η j are positive for all j . The before arrays indicates that p0has m1 real and 2n1 complex roots (including multiplicities), and p1 has m2 real and2n2 complex roots (includingmultiplicities). It is clear thatm1+2n1 = m2+2n2 = n.Without loss of generality we can suppose thatm1 > m2. Then, there are n1 (n2 resp.)quadratic factors and m1 (resp. m2) linear factors of p0 (p1 resp.). Thus, let us callN1 = n2 − n1 and rewrite p0 as

p0(t) =m2∏j=1

(t + δ j )

m2+N1∏j=m2+1

(t2 + a j t + b j )

m2+n2∏j=m2+N1+1

(t2 + αN1+ j t + βN1+ j ),

t ∈ R, where a j = δ j + δN1+ j and b j = δ jδN1+ j , j = m2 + 1, . . . ,m2 + N1. Onother hand, by separating quadratic factors of p1 we may rewrite it as

p1(t) =m2∏j=1

(t + ρ j )

m2+N1∏j=m2+1

(t2 + γ j t + η j )

m2+n2∏j=m2+N1+1

(t2 + γ j t + η j ), t ∈ R.

Finally, define the function Pt : [0, 1] → H+n given by

Pt (λ) =m2∏j=1

(t + λ(ρ j − δ j ) + δ j )

×m2+N1∏j=m2+1

(t2 + [λ(γ j − a j ) + a j ]t + λ(η j − b j ) + b j )

×m2+n2∏

j=m2+N1+1

(t2 + [λ(γ j − αN1+ j ) + αN1+ j ]t + λ(η j − βN1+ j )+βN1+ j ),

(11)



t ∈ R, which is clearly continuous respect the variable λ ∈ [0, 1], where Pt (0) = p0and Pt (1) = p1. Moreover, the coefficients of each factor of Pt (λ), given by linearconvex combinations, are positive for all λ ∈ [0, 1]. Therefore, Pt (λ) is composedby Hurwitz factors for all λ ∈ [0, 1]. Consequently, Pt is a path between p0 and p1totally contained inH+

n .

Remark 2 In the construction of Pt , we collapsed quadratic factors with real rootst2 + a j t + b j onto quadratic factors with complex roots t2 + γ j t + η j to ensurestill having a real monoparametrica family of polynomials. In other words, if we taket2 + a j t + b j = (t + δ j )(t + δ j+1) and t

2 + γ j t + η j = (t + z j )(t + z j+1), and theparameterization [t + λ(z j − δ j ) + δ j ][t + λ(z j+1 − δ j+1) + δ j+1] is considered asa factor of the path Pt , it have no real coefficients for δ j �= δ j+1. That is, we mightcollapse real roots into complex roots and losing realness of Pt .

Any path contained in H+n is also called Hurwitz path, Hurwitz curve, or Hurwitz

polynomial curve, which is a Hurwitz stable polynomial family.TheHurwitz path Pt joins p0 and p1 in a particular array of their roots. The question

is, if we change the ordering of the factors, we still having a Hurwitz path or at leastan equivalent curve of Hurwitz polynomials? The answer is yes and such equivalenceis given in terms of homotopy between paths formed with distinct rearrangements ofthe irreducible factors of the polynomials p0 and p1.

Theorem 2 Any pair of Hurwitz paths from p0 to p1 are homotopically equivalents.

Proof Suppose that m1 > m2 (m1 < m2 case is analogous). For the polynomials p0and p1 given in (9) and (10), respectively, define for λ ∈ [0, 1] the linear functions

R j (λ) = λ(ρ j − δ j ) + δ j , j = 1, . . . ,m2,

Bj (λ) ={

λ(γ j − a j ) + a j if j = m2 + 1, . . . ,m2 + N1λ(γ j − αN1+ j ) + αN1+ j if j = m2 + N1 + 1, . . . ,m2 + n2,

C j (λ) ={

λ(η j − b j ) + b j if j = m2 + 1, . . . ,m2 + N1λ(η j − βN1+ j ) + βN1+ j if j = m2 + N1 + 1, . . . ,m2 + n2.

We can rewrite the Hurwitz path Pt given in (11) as

Pt (λ) =m2∏j=1

[t + R j (λ)]m2+n2∏j=m2+1

[t2 + Bj (λ)t + C j (λ)], λ ∈ [0, 1], t ∈ R.

Analogously, if we change the order of the factors of p0 and/or p1 we obtain anotherdistinct Hurwitz path:

Qt (λ) =m2∏j=1

[t + R′j (λ)]

m2+n2∏j=m2+1

[t2 + B ′j (λ)t + C ′

j (λ)], λ ∈ [0, 1], t ∈ R,



such that Qt (0) = p0 and Qt (1) = p1. Define the continuous function Ht : [0, 1] ×[0, 1] → H+

n given by

Ht (λ, τ ) =m2∏j=1

[t + Wj (λ, τ )]

×m2+n2∏j=m2+1

[t2 + X j (λ, τ )t + Y j (λ, τ )], τ ∈ [0, 1], t ∈ R,

where

Wj (λ, τ ) = τ [R′j (λ) − R j (λ)] + R j (λ),

X j (λ, τ ) = τ [B ′j (λ) − Bj (λ)] + Bj (λ),

Y j (λ, τ ) = τ [C ′j (λ) − C j (λ)] + C j (λ).

It is not hard to see that

Ht (λ, 0) = Pt (λ) and Ht (λ, 1) = Qt (λ);Ht (0, τ ) = Pt (0) and Ht (1, τ ) = Qt (1).

Therefore, Ht is the homotopy required. ��Remark 3 It can happen that Pt and Qt have different configurations on their factors,that is,

Pt (λ) =m1∏j=1

[t + R j (λ)]m1+n1∏j=m1+1

[t2 + Bj (λ)t + C j (λ)], λ ∈ [0, 1], t ∈ R,

and

Qt (λ) =m2∏j=1

[t + R′j (λ)]

m2+n2∏j=m2+1

[t2 + B ′j (λ)t + C ′

j (λ)], λ ∈ [0, 1], t ∈ R,

with m1 �= m2. If this is the case, for each fixed λ ∈ [0, 1] apply the technique usedin the Proof of Theorem 1 and connect continuously Pt to Qt obtaining then thehomotopy.

Now, due to we can join any two arbitrary Hurwitz polynomials with a completelystable curve, it is possible to obtain a dense trajectory inH+

n , in the topological senseof closeness. In addition to the tool presented in Sect. 2.1, let us present the followinglemma which can be viewed in [13].

Lemma 2 The polynomial set of fixed degree n with rational coefficients, Qn[t], is acountable set.



From the homeomorphism ϕ given by the correspondence (5) we have thatQn+1 ishomeomorphic to Qn[t] and by the countability and density of Q we have that Qn[t]is countable and dense in Rn[t]. Thus, the subset H+Q

n = Qn[t] ∩ H+n , which is the

Hurwitz polynomials set having rational positive coefficients, is countable and densein H+

n . Moreover, by numerability we can label each element of H+Qn with a natural

number and make a list of them, say,{p0, p1, . . . , p j , . . .

}and from Theorem 1, we

obtain the following result.

Theorem 3 There exists a dense path in H+n .

Proof Let{p0, p1, . . . , p j , . . .

}be an enumeration of the elements of H+Q

n . Let usto take r ∈ Q∩ (0, 1) and define the sequence r j = 1− r j . Thereupon,

{r j

}∞j=0 is an

increasing sequence in [0, 1] such that r0 = 0 and r j → 1 whenever j → ∞. Thence,the elements of the sequence r0 < r1 < · · · < r j < . . . define a partition of [0, 1].Define the parameterizations

φ j (λ) = λ − r j−1

r j − r j−1, λ ∈ [r j−1, r j ], j = 1, 2, . . . , (12)

where φ j (r j−1) = 0 and φ j (r j ) = 1. Now, by Theorem 1, there exists a sta-ble path Pt

j (φ j ) joining each pair of Hurwitz polynomials p j−1 and p j , such thatPtj (φ j (r j−1)) = p j−1 and Pt

j (φ j (r j )) = p j , j = 1, 2, . . .. Hence,

Ptj (φ j (λ)) ∗ Pt

j+1(φ j+1(λ)), λ ∈ [r j−1, r j+1],

is a Hurwitz path from p j−1 to p j+1, where ∗ is the path product defined in (4). Finally,like it is done in the Lemma 1, we define the continuous trajectory Ft : [0, 1] → H+

nas

Ft (λ) = Pt1(φ1(λ)) ∗ Pt

2(φ2(λ)) ∗ · · · ∗ Ptj (φ j (λ)) ∗ · · · , λ ∈ [0, 1],

where, Ptj (φ j ) = Ft |[r j−1,r j ], j = 1, 2, . . .. We have that the trajectory Ft reaches

every element of H+Qn in [0, 1]. Therefore, Ft is dense inH+

n . ��

Remark 4 Notice that Ft reaches and joins every polynomial H+Qn but it is not

contained since λ ∈ [0, 1]. Therefore, Ft does not have always (by linear convexcombinations) variable rational coefficients. However at the contrary, if we considerthe polynomial family

Ftλ := {Ft (λ) : λ ∈ [0, 1]},

then we have that H+Qn ⊂ Ft

λ ⊂ H+n .



3.2 A Schur connecting-curve

Let p(t) = a0 + a1t + · · · + an−1tn−1 + tn be a Schur polynomial (or Hurwitz).In according with the homeomorphism ϕ of correspondence between the coefficientsof p and its coefficients vector θp = (a0, . . . , an−1), we shall say that the vectorθp ∈ R

n is a Schur vector (Hurwitz vector resp.) if its corresponding polynomial isSchur (Hurwitz resp.). Denote by SVn ⊂ R

n (HV+n ⊂ R

n+ for Hurwitz case) to the setof Schur vectors or points. This correspondence and the Möbius transformation givenin (8) arise a natural homeomorphism h from HV+

n to HVn such that the followingdiagram commutes

HV+n

h�� SVnϕ−1 ↓ ↑ ϕ

H+n

∼−→ Sn

. (13)

Since the Möbius map m : C− → D is a homeomorphism, we can extend to thecorresponding symmetric, namely,

mn : Symn(C−) → Symn(D)

(z1, . . . , zn) �→ (m(z1), . . . ,m(zn)).

Clearly mn is also a homeomorphism. Note that m(z) = m(z) for all z ∈ C−.

This carries us to establish another homeomorphism h′ : HV+n → SVn given by the

correspondence

((−1)nσn, . . . , σ2,−σ1) �→ ((−1)nσn(mn), . . . , σ2(mn),−σ1(mn)),

where

σk = σk(z1, . . . , zn) =∑

1≤i1<···<ik≤n

zi1 · · · zik ,

and

σk(mn) = σk(m(z1), . . . ,m(zn)) =∑

1≤i1<···<ik≤n

m(zi1) · · ·m(zik ),

making the following diagram commute

HV+n

h′�� SVn

v−1p ↓ ↑ vp

HRnmn−→ SRn

(14)



where HRn ⊂ Symn(C−) (SRn ⊂ Symn(D) for Schur case) is the set of n−tupleswith complex entries of C− (D resp.) such that if z j is an entry of (z1, . . . , zn), thenz j is also an entry of (z1, . . . , zn).

Remark 5 The fact that m(z) = m(z) for all z ∈ C− and zi and z j are entries of

(z1, . . . , zn) ∈ HRn , i �= j , with zi = z j , it implies that m(zi ) and m(zi ) = m(z j )are entries of (m(z1), . . . ,m(zn)) ∈ SRn and therefore, if the projections σk are realfor all k = 1, . . . , n, then σk(mn) are also real for all k = 1, . . . , n.

In addition, from the Möbius transformation · : H+n → Sn given by the correspon-

dence p(t) �→ p(t) = (t − 1)n p(m(t)), t ∈ R, we have achieved to establish itscorresponding relation θp �→ θ p, where θp ∈ HV+

n and θ p ∈ SVn are the vectors ofcoefficients associated to p and p, respectively. Alternatively, the Möbius transforma-tion h′ : HV+

n → SVn allows us to establish the natural map · : H+n → Sn , induced

by the projective maps σk �→ σk(mn), as the correspondence p �→ p, where

p(t) =n∏

i=1

(t + zi ) = tn − σ1tn−1 + · · · + (−1)n−1σn−1t + (−1)nσn, t ∈ R,

and

p(t) =n∏

i=1

(t + m(zi ))

= tn − σ1(mn)tn−1 + · · · + (−1)n−1σn−1(mn)t + (−1)nσn(mn), t ∈ R.

Clearly, · is a homeomorphism that makes to commute the diagram

H+n

∧�� Snϕ ↓ ↑ ϕ−1

HV+n

h′−→ SVn

. (15)

In this way we have that p is Hurwitz stable if and only if p is Schur stable.The homeomorphisms h and h′ (· and ·, resp.) come to be Möbius transformations

between the spaces HV+n and SVn (H+

n and Sn , resp.); the first one in terms of thecoefficients and the second one in terms of the roots.

Now, consider the Hurwitz curve Pt as in the Proof of Theorem 2:

Pt (λ) =m2∏j=1

[t + R j (λ)]m2+n2∏j=m2+1

[t2 + Bj (λ)t + C j (λ)], λ ∈ [0, 1], t ∈ R,

and the commuting diagram



H+n

ϕ−→ HV+n

v−1p−→ HRn

∧ ↓ ↓ h′ ↓ mn

Snϕ−1

←− SVnvp←− SRn

. (16)

If zm1+1, . . . , zm1+2n1 and wm2+1, . . . , wm2+2n2 are the complex roots (includingmultiplicities) of p0 and p1, respectively, we can suppose that zm1+ j = zm1+n1+ j ,j = 1, 2, . . . , n1 andwm2+ j = wm2+n2+ j , j = 1, 2, . . . , n2.We shall use the diagram(16) over Pt . First, if we apply v−1

p ◦ϕ to Pt we obtain the n-vector function inHRn ,

V (λ) = (Z1(λ), . . . , Zn(λ)), (17)

which it is continuous respect λ ∈ [0, 1], such that for λ = 0 we have the vectorof roots V (0) = (δ1, . . . , δm1 , zm1+1, . . . , zn) and for λ = 1 the vector of rootsis V (1) = (ρ1, . . . , ρm2 , wm2+1, . . . , wn). The transformation mn applied to V (λ)

arises the continuous n-tuple

(mn ◦ V )(λ) = (m(Z1(λ)), . . . ,m(Zn(λ))), (18)

contained in SRn for all λ ∈ [0, 1], where

(mn ◦ V )(0) = (m(δ1), . . . ,m(δm1),m(zm1+1), . . . ,m(zn))

and

(mn ◦ V )(1) = (m(ρ1), . . . ,m(ρm2),m(wm2+1), . . . ,m(wn)).

Now, define the projection functions

σk(λ) =∑

1≤ j1<···< jk≤n

Z j1(λ) · · · Z jk (λ), λ ∈ [0, 1], (19)

and

σk(mn)(λ) =∑

1≤ j1<···< jk≤n

m(Z j1(λ)) · · ·m(Z jk (λ)), λ ∈ [0, 1], (20)

With the above calculations it is possible establish the following theorem analogousto Theorem 1.

Theorem 4 For any pair of polynomials p0 and p1 in Sn there exists a continuousfunction Pt : [0, 1] → Sn such that Pt (0) = p0 and Pt (1) = p1. That is, Sn ispath-connected.

Proof Suppose that p0(t) =n∏j=1

(t+z j ) and p1(t) =n∏j=1

(t+w j ), t ∈ R. Ifm−1(z j ) =

z j and m−1(w j ) = w j then from the Theorem 1, for the corresponding Hurwitz



polynomials p0(t) =n∏j=1

(t + z j ) and p1(t) =n∏j=1

(t +w j ), t ∈ R, and the projections

(19) we can obtain a Hurwitz path of the form

Pt (λ) = tn − σ1(λ)tn−1 + · · · + (−1)n−1σn−1(λ)t + (−1)nσn(λ),

λ ∈ [0, 1], t ∈ R, where σk(0) = σk(z1, . . . , zn) and σk(1) = σk(w1, . . . , wn). Byapplying the Möbius transformation · to the Hurwitz path Pt we get a Schur path

P t (λ) = tn − σ1(mn)(λ)tn−1 + · · · + (−1)n−1σn−1(mn)(λ)t + (−1)nσn(mn)(λ),

λ ∈ [0, 1], t ∈ R, where σk(mn) are the projections (20) such that

σk(mn)(0) = σk(m(z1), . . . ,m(zn))

= σk(z1, . . . , zn)

and

σk(mn)(1) = σk(m(w1), . . . ,m(wn))

= σk(w1, . . . , wn).

Thus P t is a Schur path from p0 to p1, as we claim. ��Taking the ideas of connecting given in the before theorem and by invoking the

Theorem 2, the following result is immediate.

Corollary 1 Any pair of Schur paths joining p0 and p1 are homotopically equivalents.

Analogous to the made for Theorem 3, we have that the numerability of the setSQn = Qn[t]∩Sn leaves us its enumeration

{p0, p1, p2, . . .

}, and as immediate result

from Theorems 3 and 4 we obtain the following corollary.

Corollary 2 There exists a dense trajectory in Sn.

Proof Let{p0, p1, p2, . . .

}be an enumeration of SQ

n . Consider the parameteriza-

tions φ j given in (12) and by Theorems 3 and 4 we obtain the paths P tj (φ j )|[r j−1,r j ],

with P tj (φ j (r j−1)) = p j−1 and P t

j (φ j (r j )) = p j , j = 1, 2, . . .. Finally, define thetrajectory

F t (λ) = P t1(φ1(λ)) ∗ P t

2(φ2(λ)) ∗ · · · ∗ P tj (φ j (λ)) ∗ · · · , λ ∈ [0, 1],

which is dense in Sn . ��The remark corresponding to Theorem 4 is analogous to Remark 2 made for Theo-

rem 1. As well as the Remark 4 made for Theorem 3, can be made it analogously forTheorem 2 by defining the family

F tλ := {F t (λ) : λ ∈ [0, 1]},



and then to get SQn ⊂ F t

λ ⊂ Sn .Similarly, for any pair of polynomials p0, p1 ∈ H+

n and their associated p0, p1 ∈Sn , let Pt and P t be their connecting paths, respectively. Recall that ϕ : Pn[t] → R

n

is a homeomorphism. Let us call ϕ(p j ) = θp j and ϕ( p j ) = θ p j , j = 0, 1, their vector

of coefficients associated. Then, clearly ϕ(Pt ( j)) = ϕ(p j ) = θp j and ϕ(P t ( j)) =ϕ( p j ) = θ p j , j = 0, 1. The following results are immediate.

Corollary 3 Given θp0 , θp1 ∈ HV+n (θ p0 , θ p1 ∈ SVn resp.), there exists a continuous

function Pϕ : [0, 1] → HV+n (Pϕ : [0, 1] → SVn resp.) with Pϕ( j) = θp j (Pϕ( j) =

θ p j resp.), j = 0, 1. That isHV+n (SVn resp.) is path connected.

Corollary 4 Any pair of Hurwitz (Schur resp.) vector paths joining θp0 to θp1 (θ p0 toθ p1 resp.) are homotopically equivalent.

Corollary 5 There exists a dense path in HV+n (resp. SVn).

4 An application in the design of a stabilizing feedback control

Consider the linear control (continuous or discrete) system

x : Ax + bu, x ∈ Rn (21)

where A ∈ Rn×n , b ∈ R

n and u ∈ R. In order to design a stabilizing control, let us tosuppose that the system is given in its canonical controllable form:

A =

⎡⎢⎢⎢⎢⎢⎣

0 1 0 · · · 00 0 1 · · · 0

... · · · ...

0 0 0 · · · 1−an −an−1 −an−2 · · · −a1

⎤⎥⎥⎥⎥⎥⎦

and b =

⎡⎢⎢⎢⎢⎢⎣

00...

01

⎤⎥⎥⎥⎥⎥⎦

,

with characteristic polynomial pA(t) = tn + a1tn−1 + · · · + an , where ak =(−1)kσk(z1, . . . , zn) are the Viète’s projections for which {z1, . . . , zn} ⊂ C is thespectrum of A. In the following we design a stabilizing feedback control.

4.1 Design and implementation of the control

We are interested on the stabilization of the system (21) and to achieve it we shalldesign a linear feedback state control of the form u(x, λ) = −c(λ)Tx , where c(λ)T =(cn(λ), . . . , c1(λ)), λ ∈ [0, 1]. The technique is a kind of poles assignment: Choosea suitable vector � = (ζ1, . . . , ζn) of complex numbers in HRn or SRn , dependingon the nature of the system (22), with associate monic polynomial p�(t) = tn +w1tn−1 + · · · + wn , t ∈ R, where wk = (−1)kσk(ζ1, . . . , ζn). Construct the stableconnecting path



Qt (λ) = tn + A1(λ)tn−1 + · · · + An(λ), λ ∈ [0, 1], (22)

which the functions Ak(λ), k = 1, . . . , n, λ ∈ [0, 1], are the continuous coefficientsof the polynomial family Qt such that Ak(0) = ak and Ak(1) = wk . Therefore,Qt (0) = pA is the characteristic polynomial of A and Qt (1) = p�. It is clearthat the vectors θpA = (an, . . . , a1)T, θp� = (wn, . . . , w1)

T and the vector pathθQt (λ) = (An(λ), . . . , A1(λ))T, λ ∈ [0, 1], are stable vectors, being θpA , θp� and θQt

the coefficients vectors of pA, p� and Qt , respectively. Finally, for λ ∈ [0, 1], definethe vector c(λ) as

c(λ) = θQt (λ) − θpA = (An(λ) − an, . . . , A1(λ) − a1)T. (23)

Thus, the closed-loop system (21) has the form

x : (A − bc(λ)T)x = Ac(λ)x, λ ∈ [0, 1].

where

Ac(λ) =

⎡⎢⎢⎢⎢⎢⎣

0 1 0 · · · 00 0 1 · · · 0

... · · · ...

0 0 0 · · · 1−An(λ) −An−1(λ) −An−2(λ) · · · −A1(λ)

⎤⎥⎥⎥⎥⎥⎦

.

The before discussion allows us to establish the following result.

Theorem 5 Consider the control system (21) and suppose that it is completely con-trollable. If the characteristic polynomial pA is Hurwitz stable (Schur stable resp.)and � ∈ HRn (SRn, resp.), then the control u(x, λ) = −cT(λ)x is a stabilizingfeedback control for the system (21) for all λ ∈ [0, 1].

It is clear that if the spectrum of A, {z1, . . . , zn}, forms an unordered n−tuplacontained inHRn (SRn , resp.) then the elements of � must also be the entries of anunordered n−tupla inHRn (SRn , resp.) to ensure the Hurwitz (Schur, resp.) stabilityof the vector R(λ) ∈ HV+

n (SVn , resp.), for all λ ∈ [0, 1].Note that if the system is not stable, it has at least one root with real part greater

than or equal to zero. The following result is held.

Corollary 6 Consider the control system (21) and suppose that it is completely con-trollable. If the characteristic polynomial pA is not Hurwitz stable (Schur stable resp.)and � ∈ HRn (SRn, resp.), then there exists a value λ0 ∈ [0, 1) such that the con-trol u(x, λ) = −cT(λ)x is a stabilizing feedback control for the system (21) for allλ ∈ (λ0, 1].

The following examples illustrate the control design technique.



Example 1 (Servo-motor System Model [19]) The servo-motor is armature controlledby a constant field whose model is given by the second order differential equation:

J θ (t) + BRα + KT Kb

Rα

θ(t) = KT

Rα

e(t) (24)

where θ(t) is the shaft position respect the time t ∈ R, Rα is the resistance armature,Kb is the electromotive force constant, J is the total moment of inertia connectedto the motor shaft and B is the total viscous friction. Moreover, KT in a parameterof the motor related with the torque and e(t) is the voltage for the armature circuit,continuous respect the time t . The state-variable model is derived by setting

ξ1 = θ,

ξ2 = θ ,

u = e.

Then, from Eq. (24)

ξ2 = θ = − BRα + KT Kb

J Rα

ξ2 + KT

J Rα

u.

Hence, the system of the servo-motor in state equations is described by

[ξ1ξ2

]=

[0 10 −a1

] [ξ1ξ2

]+

[0b2

]u (25)

where, a1 = BRα+KT KbJ Rα

> 0 and b2 = KTJ Rα

. Besides, the open-loop system charac-

teristic polynomial is pA(t) = t2 + a1t . Therefore the system is not stable, since theroots of pA are z1 = 0, and z2 = −a1. Moreover, it is not in canonical controllableform (3). However, the change of coordinates

[ξ1ξ2

]= CM

[x1x2

],

with

C =[0 b2b2 −a1b2

]and M =

[a1 11 0

](26)

transforms the system (25) into

[x1x2

]=

[0 10 −a1

] [x1x2

]+

[01

]u. (27)

We are now ready for the design of a stabilizing feedback. Suppose that we desire toassign the eigenvalues ζ1,2 = −0.3± 0.45i ∈ HR2. The associate monic polynomialis p�(t) = t2 + 0.6t + 0.2925, t ∈ R, which is Hurwitz stable. The Hurwitz path



is given by Pt (λ) = t2 + [λ(0.6 − a1) + a1]t + 0.2925λ, λ ∈ [0, 1], t ∈ R. Notethat the continuous coefficients A1(λ) = λ(0.6 − a1) + a1, A2(λ) = 0.2925λ, arepositive for λ ∈ (0, 1]. Therefore, Pt is Hurwitz stable for each λ ∈ (0, 1]. Takec(λ) = θQt (λ) − θpA = (λ(0.6 − a1), 0.2925λ), λ ∈ [0, 1]. Define the feedbacku = −c(λ)Tx for λ ∈ [0, 1] and x = (x1, x2)T. Thus, the design transforms thesystem (27) into

x =[

0 1−0.2925λ −λ(0.6 − a1) − a1

]x,

which is stable for all λ ∈ (0, 1].Finally, the stabilizing feedback in terms of the original coordinates is

u = − 1b (λ(0.6 − a1), 0.2925λ)ξ , λ ∈ (0, 1], ξ = (ξ1, ξ2)

T, that stabilizes system(25).

Example 2 Let us to consider the continuous-time control system

x = Ax + bu (28)

with

A =⎡⎣

0 1 00 0 1

−1 − 115 − 11

5

⎤⎦ and b =

⎡⎣001

⎤⎦ .

Then, the characteristic polynomial of the system is pA(t) = t3 + 115 t

2 + 115 t +1 with

set of roots{−1,− 3

5 ± i 45}and then pA(t) is Hurwitz. Suppose that the spectrum

desired for assigning is � = {−2,− 25 ,− 4

5

} ⊂ C−. Thus, in order to use the Hurwitz

connecting-path to design a stabilizing control, let us to make use of the Hurwitzpolynomial

p�(t) = (t + 2)

(t + 2

5

) (t + 4

5

)

= t3 + 16

5t2 + 68

25t + 16

25.

Since there is a pair of complex conjugate roots of the polynomial pA(t), it is conve-nient to connect the roots − 2

5 and − 45 with − 3

5 ± i 45 . To achieve it, let us to write thepolynomials pA(t) and p�(t) as

pA(t) = (t + 1)

(t2 + 6

5t + 1

),

p�(t) = (t + 2)

(t2 + 6

5t + 8

25

)



and a Hurwitz path is

Pt (λ) = (t + λ + 1)

(t2 + 6

5t + 1 − 17

25λ

),

for λ ∈ [0, 1], which has continuous roots given by

Z1(λ) = −(λ + 1),

Z2,3(λ) = −3

5± 1

5

√17λ − 16.

One can see that Pt (0) = pA(t) and Pt (1) = p�(t). Rewrite the polynomial familyPt (λ) as

Pt (λ) = t3 + A1(λ)t2 + A2(λ)t + A3(λ),

where

A1(λ) = λ + 11

5,

A2(λ) = 13

25λ + 11

5,

A3(λ) = λ

(8

25− 17

25λ

)+ 1.

Finally, define the feedback control u(x, λ) = − (λ

( 825 − 17

25λ), 1325λ, λ

)x and the

system (28) in closed-loop is expressed as

x =⎡⎣

0 1 00 0 1

−λ( 825 − 17

25λ) − 1 − 1325λ − 11

5 − 1325λ − 11

5

⎤⎦ x,

which is stable for all λ ∈ [0, 1].

5 Conclusions

In summary, we have obtained explicitly, by linear convex combinations, the pathPt , which connects any two elements in H+

n and moreover, it is robustly stable.The application of the stable connecting paths in the design of stabilizing feedbackcontrollers is a technique whose advantage is that the polynomial family involveddoes not need tests of stability due to the construction of the control. Furthermore, thesimplicity and structure of the control designed, allows implement the controller evenin continuous as discrete systems. Moreover, the design of the stabilizing controllercan be carried out of the stable spaces and replacing continuously any eigenvalues setby another desired spectrum for provoking and controlling bifurcations and chaos inmonoparametric nonlinear family of systems.



Acknowledgments Authors want to thank anonymous reviewers for their valuable corrections and com-ments.

References

1. Aguilar, M., Gitler, S., Prieto S.: Algebraic topology from a homotopical viewpoint, pp. 9–119. Uni-versitext. Springer, New York (2002)

2. Aguirre-Hernández, B., Cisneros-Molina, J., Frías-Armenta, M.: Polynomials in control theory para-metrized by their roots. Int. J. Math. Sci. 2012, 595076 (2012). doi:10.1155/2012/595076

3. Aguirre, B., Ibarra, C., Suárez, R.: Sufficient algebraic conditions for stability of cones of polynomials.Syst. Control Lett. 46, 255–263 (2002)

4. Aguirre-Hernández, B., Frías-Armenta, M., Verduzco, F.: Smooth trivial vector bundle structure of thespace of Hurwitz polynomials. Automatica 45, 2864–2868 (2009)

5. Aguirre-Hernández, B., Frías-Armenta, M., Verduzco, F.: On differential structures of polynomialspaces in control theory. J. Syst. Sci. Syst. Eng. 21(3), 372–382 (2012)

6. Aguirre, B., Suárez, R.: Algebraic test for the Hurwitz stability of a given segment of polynomials.Bol. Soc. Mat. Mex. (3rd series) 12(2), 261–275 (2006)

7. Barmish, B.R.: New tools for robustness of linear systems, vol. xvi, pp. 1–394. Macmillan, New York(1994)

8. Białas, S.: A necessary and sufficient condition for the stability of convex combinations of stablepolynomials or matrices. Bull. Polish Acad. Sci. Tech. Sci. 33, 473–480 (1985)

9. Bhattacharyya, S.P., Chapellat, H., Keel, L.H.: Robust control: the parametric approach, pp. 30–163.Prentice Hall, Englewood Cliffs (1995)

10. Dold, A., Thom, R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. Math. 67(2),239–281 (1958)

11. Fam, A.T., Meditch, J.: A canonical parameter space for linear systems design. IEEE Trans. Autom.Control 23(3), 454–458 (1978)

12. García, R., Aguirre, B., Suárez, R.: Stabilization of linear sampled-data systems by a time-delayfeedback control. Math. Probl. Eng. 2008, 270518 (2008). doi:10.1155/2008/270518

13. Herstein, I.N., Kaplansky, I.: Matters Mathematical, pp. 221–242. Chelsea Publishing Company, NewYork (1978)

14. Hinrichsen, D., Pritchard, J.: Mathematical systems theory I: modeling, state space analysis, stabilityand robustness. Texts in Applied Mathematics, vol. 48, pp. 369–430. Springer, Berlin (2005)

15. Kosniowski, C.: A first course in algebraic topology, pp. 11–118. Cambridge University Press, Cam-bridge (1980)

16. López-Rentería, J.A., Aguirre-Hernández, B., Verduzco, F.: The boundary crossing theorem and themaximal stability interval. Math. Probl. Eng. 2011, 123403 (2011). doi:10.1155/2011/123403

17. Mrowka, T.S., Ozsváth, P.S.: LowDimensional Topology, pp. 205–209. AmericanMathematical Soci-ety, Institute of Advanced Study, PCMS 15 (2009)

18. Munkres, J.R.: Topology, pp. 147–162. Prentice-Hall, Massachusetts (2000)19. Phillips, C.L., Nagle, H.T.: Digital control system analysis and design, pp. 235–261. 3rd ed. Prentice-

Hall, Englewood Cliffs (1995)20. Zeheb, E.: Necessary and sufficient conditions for robust stability of a continuous system—the con-

tinuous dependency case illustrated via multilinear dependency. IEEE Trans. Circ. Syst. 37, 47–53(1990)


http://dx.doi.org/10.1155/2012/595076

http://dx.doi.org/10.1155/2008/270518

http://dx.doi.org/10.1155/2011/123403

mat.izt.uam.mxmat.izt.uam.mx/mat/documentos/produccion_academica/2016/2/articulo... · bol. soc....

Documents