The Boundary Crossing Theorem and the Maximal Stability Interval

(1)

Volume 2011, Article ID 123403,13pages doi:10.1155/2011/123403

Research Article

The Boundary Crossing Theorem and the Maximal Stability Interval

Jorge-Antonio L ´opez-Renteria,

¹

Baltazar Aguirre-Hern ´andez,

¹

and Fernando Verduzco

²

1División de Ciencias Básicas e Ingenier´ıa, Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Avenida San Rafael Atlixco no. 186, Col. Vicentina, 09340 México, DF, Mexico

2Divisi´on de Ciencias Exactas y Naturales, Departamento de Matem´aticas, Universidad de Sonora, Boulevard Rosales y Luis Encinas s/n, Col. Centro, 83000, Hermosillo, Son, Mexico

Correspondence should be addressed to Baltazar Aguirre-Hern´andez,[email protected] Received 9 October 2010; Accepted 11 March 2011

Academic Editor: J. Jiang

Copyrightq2011 Jorge-Antonio L ´opez-Renteria et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The boundary crossing theorem and the zero exclusion principle are very useful tools in the study of the stability of family of polynomials. Although both of these theorem seem intuitively obvious, they can be used for proving important results. In this paper, we give generalizations of these two theorems and we apply such generalizations for finding the maximal stability interval.

1. Introduction

Consider the linear system

x˙Ax, 1.1

where Ais an n×n matrix with constant coeﬃcients andx ∈ ⁿ. It is known that if the characteristic polynomial,pAtis a Hurwitz polynomial, that is, all of its roots have negative real part, then the origin is an asymptoticly stable equilibrium. Great amount of information has been published about these polynomials since Maxwell proposed the problem of finding conditions for verifying if a given polynomial has all of its roots with negative real part 1. At first the researchers focused in the problem proposed for Maxwell and the Routh- Hurwitz criterion, the Hermite-Biehler theorem and other criteria were obtained. For reading the proofs of these results the books2–4can be consulted. After the scientists began to study other related problems, for instance, the problem of giving conditions for a given family

(2)

of polynomials consist of Hurwitz polynomials alone. This problem has its motivation in the applications because when a physical phenomenon is modeled, families of polynomials must be considered due the presence of uncertainties. Maybe the most famous results about families of Hurwitz polynomials are the Kharitonov’s theorem5and the Edge theorem6 which consider interval polynomials and polytopes of polynomials, respectively. However, other families have been studied also, for example, the cones and rays of polynomails see 7, 8 or the segments of polynomails 9–11. Besides in the study of Hurwitz polynomials, topological approaches have been used recentlysee12,13. Good references about Hurwitz polynomilas which were reported during 1987–1991 can be found in 14.

The books2,15,16are very recommendable works for consulting questions about families of stable polynomilas. Many proofs of results about the stability of families of polynomials are based in the boundary crossing theorem17and in the zero exclusion principlesee2 for a proof. In this way the importance of these two results has been appreciated. Now, in this paper we give generalizations as the boundary crossing theorem as the zero exclusion principle and we apply these generalizations for giving an alternative method to calculate the maximal interval for robust stability, which was studied by Białas. We will explain the diﬀerences between both methods.

2. Main Results

We have divided this section in two parts: in the first subsection we present generalizations of the mentioned theorems and in the second subsection we apply such generalizations for calculating the maximal stability interval for robust stability.

2.1. Generalizations of the Boundary Crossing Theorem and Zero Exclusion Principle

First, we give the known boundary crossing theorem. Consider a family of polynomials Pλ, tsatisfying the following assumption.

Assumption 2.1. Pλ, tis a family of polynomials of 1fixed degreen,

2coeﬃcients are continuous respectλin a fixed intervalI a, b.

Also let us to consider the complex plane and letS ⊂ be any given open set and denote the boundary ofSby∂Sand the complement asU − S. The following results are presented in2, page 34.

Theorem 2.2boundary crossing theorem. Under Assumption2.1, ifPa, thas all its roots in SwhereasPb, thas at least one root inU. Then there exists at least oneρina, bsuch that

iPρ, thas all its roots inS ∪∂S, iiPρ, thas at least one root in∂S.

(3)

Now suppose thatδt, pdenotes a polynomial whose coeﬃcients depend continuously on the parameter vectorp ∈ ^l which varies in a setΩ ⊂ ^l and thus generates the family of polynomials

Δt: δ

t, p

|p∈Ω

. 2.1

Theorem 2.3zero exclusion principle. Assume that the polynomial family2.1 has constant degree, contains at least one stable polynomial, andΩis pathwise connected. Then the entire family is stable if and only if

0∈/Δt^∗, ∀t^∗∈∂S. 2.2

Now we present our generalizations. In them we will considerS⁻.

Theorem 2.4generalization 1 of Theorem2.2. Under Assumption2.1, suppose thatPa, thas n1roots in⁻andn−n1roots in, andPb, thas at mostn1−1 roots in⁻ and at leastn−n11 roots in. Then there exists at least oneρina, bsuch that

iPρ, thas at leastn1roots in⁻∪i , iiPρ, thas at least one root ini .

Theorem 2.5generalization 2 of Theorem2.2. Under Assumption2.1, suppose thatPa, thas n1 roots in⁻ andn−n1 roots in, andPb, thasm1 roots in⁻ and n−m1 roots in. If n1/m1, Then there exists at least oneρina, bsuch that

iPρ, thas at leastn1roots in⁻∪i , iiPρ, thas at leastn−n1roots in ∪i , iiiPρ, thas at least one root ini .

Theorem 2.6 generalization of Theorem 2.3. Consider the polynomial family Pλ, t with constant degree, where λ ∈ Ω and Ω ⊂ ^l is a pathwise connected set. Suppose there exists an element of the family withn1 roots in⁻ andn−n1roots in. Then the entire family still having n1roots in⁻ andn−n1roots in if and only if

Pλ, iω/0 ∀λ∈Ω,∀ω∈ . 2.3

2.2. Application: An Alternative Method for Calculating the Maximal Stability Interval

We begin this subsection with three important definitions that can be seen in pages 50 to 51 of16.

(4)

Definition 2.7. Consider the uncertain polynomialPt, k p0t kp1twithp0tassumed Hurwitz stable and the uncertainty bounding setK k⁻, kwithk⁻ ≤ 0 andk ≥ 0. We define the subfamilies

Pk

p·, k|0≤k≤k , P

k⁻

p·, k|k⁻ ≤k≤0

. 2.4

Definition 2.8maximal stability interval. Associated with the subfamilyPkis the right- sided robustness margin

k_max sup

k:Pkis robustly stable

, 2.5

and asociated with the subfamilyPk⁻is the left-sided robustness margin k_min⁻ inf

k⁻:P k⁻

is robustly stable

. 2.6

Subsequently we callKmax k⁻_min, k_max the maximal interval for robust stability.

Definition 2.9. Given ann×nmatrixM, we defineλ_maxMto be the maximum positive real eigenvalue ofM. WhenMdoes not have any positive real eigenvalue, we takeλ_maxM 0. Similarly, we defineλ⁻_minMto be the minimum negative real eigenvalue ofM. When M does no have any negative real eigenvalue, we takeλ⁻_minM 0⁻.

Białas10proved the following theorem.

Theorem 2.10eigenvalue criterion. Consider the uncertain polynomialPt, k p0t kp1t withPt,0 p0tHurwitz stable and having positive coeﬃcients and degp0t>degp1t. Then the maximal interval for robust stability is described by

k_max 1

λmax

−H⁻¹ p0

H p1

,

k_min⁻ 1

λ⁻_min

−H⁻¹ p0

H p1

,

2.7

whereHp0andHp1are the corresponding matrices of Hurwitz ofp0tandp1t, respectively, and for purpose of conformability of matrix multiplication, Hp1 is an n×nmatrix obtained by treatingp1tas ann-degree polynomial.

Now we will explain our approach for calculating the maximal stability interval. Let p0tbe ann-degree polynomial andp1ta polynomial such thatn >degp1t. Consider the family of polynomialspct p0t kp1t. By evaluatingp0−tandpctiniωwe get

p0−iω P ω²

−iωQ ω²

, p_ciω P

ω² kp

ω² iω

Q ω²

kq

ω² , 2.8

(5)

wherep, q, P, Qare polynomials. Then pciωp0−iω P²

ω²

ω²Q² ω² k

p ω²

P ω²

ω²q ω²

Q ω² ikω

q ω²

P ω²

−p ω²

Q ω² .

2.9

Define the polynomials

Fω p ω²

P ω²

ω²q ω²

Q ω²

, Gω P²

ω²

ω²Q² ω²

, Hω q

ω² P

ω²

−p ω²

Q ω²

.

2.10

Therefore, we can rewritep_ciωp0−iωasp_ciωp0−iω Gω kFω ikωHω.

Definition 2.11. For an arbitrary polynomialftwe define the set of its roots as R

f

ζ∈ |fζ 0

. 2.11

LetRfÊdenote the set of positive real elements ofRf. It is clear thatRfÊcould be an empty set.

Now letFω, Gω, andHωbe the polynomials defined above. Define the sets K{Fωl|ωl∈RHÊ∪ {0}, Fωl>0},

K⁻{Fω_l|ω_l∈RHÊ∪ {0}, Fωl<0}. 2.12

If there is no elements inRHÊ∪ {0}such thatFωl> 0 then we will defineK {0}.

Similarly, if there is no elements inRHÊ∪ {0}such thatFω_l<0 then defineK⁻ {0⁻}.

Note that only can happen eitherK⁻ {0⁻}orK {0}but both at the same time never since we can always evaluate inω0. That is, we always have an extreme.

Therefore we have the following alternative method for calculating the maximal stability interval.

Theorem 2.12. Consider the polynomial familypct p0t kp1twithp0tHurwitz stable and having positive coeﬃcients, and letFω, Gω, andHωbe the polynomials defined above. Then the maximal interval of stability forp_ctis described by

k⁻_minmax

−Gωl

Fωl |Fωl∈K

,

k_maxmin

−Gωl

Fωl |Fωl∈K⁻

.

2.13

(6)

Remark 2.13. A diﬀerence with the Białas method is that in our approach it is not necessary to calculate the inverse of any matrix. Other diﬀerence is that in the Białas method the roots of ann-degree polynomial must be found while in our approach we have that if degree of both nandm, resp., n > mp0tandp1tis either even or odd then degHω nm−2 and in the other cases degHω nm−1. Therefore, by symmetry ofHωwe have to find the roots of a polynomial with degreenm−2/2 ornm−1/2 both less than or equal to n−1.

Example 2.14. Consider the polynomialp0t t³6t²12t6 andp1t t²−2t1, for the polynomial familypt, k p0t kp1t. We will verify the maximal stability interval by Białas method. First we have

H p0t

⎛

⎜⎜

⎝ 6 6 0 1 12 0 0 6 6

⎞

⎟⎟

⎠, H p1t

⎛

⎜⎜

⎝ 1 1 0 0 −2 0 0 1 1

⎞

⎟⎟

⎠. 2.14

Next,

−H p0t₋₁

H p1t

−

⎛

⎜⎜

⎜⎝ 2 11 −1

11 0

− 1 66

1 11 0 1

66 −1 11

1 6

⎞

⎟⎟

⎟⎠

⎛

⎜⎜

⎝ 1 1 0 0 −2 0 0 1 1

⎞

⎟⎟

⎠

⎛

⎜⎜

⎜⎝

− 2 11 −4

11 0 1

66 13 66 0

− 1 66 −4

11 −1 6

⎞

⎟⎟

⎟⎠,

2.15

whose characteristic polynomial is λ³ −2/11λ²−1/36λ 1/198 which is a 3-degree polynomial and has as roots λ1 λ2 −1/6 andλ3 2/11. Thus λ_maxHp0⁻¹Hp1

2/11 and λ⁻_minHp0⁻¹Hp1 −1/6. Then pt, k p0t kp1t is robustly stable in k⁻_min, k_max −6,11/2.

Now with our proposed method. We see

p0iω

6−6ω² iω

12−ω² P

ω²

iωQ ω²

, p1iω

1−ω²

iω−2 p

ω² iωq

ω² .

2.16

(7)

Thus,

Fω 8ω⁴−36ω²6, Gω ω⁶12ω⁴72ω²36, Hω −ω⁴25ω²−24.

2.17

ThusRHÊ{1,√

24}. Now,

F1 −22<0, F√

24

3750>0, F0 6>0, G1 121, G√

24

22527, G0 36.

2.18

Therefore

k_max−G1 F1 11

2 , k⁻_minmax

⎧⎨

⎩−G√ 24 F√

24

,−G0 F0

⎫⎬

⎭−6.

2.19

Note that we just only need to compute a 2-degree polynomial root in our method, while in Białas one it is a 3-degree polynomial.

Example 2.15. Consider the linear control system

x˙

⎛

⎜⎜

⎜⎝

0 1 0 0

0 0 1 0

0 0 0 1

−1 −7 −2 −3

⎞

⎟⎟

⎟⎠x

⎛

⎜⎜

⎜⎝ 0 0 0 1

⎞

⎟⎟

⎟⎠−3k,−2k,−k,0x. 2.20

(8)

From this system we have thatp0t t⁴t³7t²2t3 andp1t t²2t3 and their Hurwitz matrices are

H p0t

⎛

⎜⎜

⎜⎝

1 2 0 0 1 7 3 0 0 1 2 0 0 1 7 3

⎞

⎟⎟

⎟⎠, H p1t

⎛

⎜⎜

⎜⎝

0 2 0 0 0 1 3 0 0 0 2 0 0 0 1 3

⎞

⎟⎟

⎟⎠, 2.21

where the inverse ofHp0is

H p0t₋₁

⎛

⎜⎜

⎜⎝ 11

2 −4 7

6

7 0

−2 7

2 7 −3

7 0 1

7 −1 7

5

7 0

− 5 21

5 21 −32

21 1 3

⎞

⎟⎟

⎟⎠

. 2.22

Hence, the 4-degree characteristic polynomial of the matrix

−H p0t₋₁

H p1t

−

⎛

⎜⎜

⎜⎝ 11

2 −4 7

6

7 0

−2 7

2 7 −3

7 0 1

7 −1 7

5

7 0

−5 21

5 21 −32

21 1 3

⎞

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎝

0 2 0 0 0 1 3 0 0 0 2 0 0 0 1 3

⎞

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎝ 0 −18

7 −24 7 0 0 2

7 12

7 0 0 −1

7 −13 7 0 0 5

21 24

7 1

⎞

⎟⎟

⎟⎠

2.23

ist⁴ 12/7t³ 3/7t² −2/7t and has as rootsλ1 0, λ2 2/7 and λ3,4 −1. Then λ_maxHp0⁻¹Hp1 2/7 andλ⁻_minHp0⁻¹Hp1 −1. Therefore, system2.20is robustly stable in−1,7/2.

(9)

Now with our method. By evaluatingp0tandp1tiniωwe have p0iω

ω⁴−7ω²3 iω

−ω²2 P

ω²

iωQ ω²

, p1iω

−ω²3

iω2 p

ω² iωq

ω² .

2.24

Thus,

Fω −ω⁶8ω⁴−20ω²9, Gω ω⁸−13ω⁶51ω⁴−38ω²9, Hω ω²

−9ω² .

2.25

It is not hard to see thatRHÊ {0,3}. Next,F0 9 >0,F3 −252<0, andG0 9, G3 882. Therefore,

k_max −G3 F3 7

2 k⁻_min−G0

F0 −1.

2.26

Note the easiness of roots finding for our polynomialHω.

As we see, in our method computations are easier operatively speaking since matrices inverse and roots of bigger degree polynomials have been found in Białas method.

3. Proofs of the Theorems

Before start with the proofs of the theorems we present the following lemma wich can be found in3.

Lemma 3.1 continuous root dependence. Consider the family of polynomials P described by Pλ, t _n

i0aiλtⁱandλ∈Ωunder Assumption2.1. Then the roots ofPλ, tvary continuously with respect toλ∈Ω. That is, there exist continuously mappingst_i :Ω → fori1,2, . . . , nsuch thatt1λ, . . . , t_nλare the roots ofPλ, t.

3.1. Proof of Theorem

2.4

SincePλ, tsatisfies Assumption2.1, by Lemma3.1there existncontinuous function roots ofPλ, t, sayt1λ, . . . , tnλ,λ ∈ a, b. Let us denote αjλ ^Êtjλas the real parts of the roots. Without loss of generality we can suppose that forj 1, . . . , n1,α_ja ∈ ⁻ and

(10)

for j n1 1, . . . , n,αja ∈ , while forλ bat most n1−1 αjb’s belong to ⁻ and at leastn−n11 belong to . Then there exists at least onet_jλsuch thatα_ja < 0 and αjb > 0. Letαj1λ, . . . , αjmλbe such functions. Then by continuity and the intermediate value theorem we have that for each 1≤r ≤mthere existsρr ∈a, bsuch thatαjrρr 0.

Defineρmin{ρ_r |r1, . . . , m}. Therefore, forλρat least oneα_j_rρ 0. ThusPρ, thas n1roots in⁻∪i with at least one root ini , as we claim.

The proof of Theorem2.5is similar.

3.2. Proof of Theorem

2.6

⇒If all of the elements of the family haven1roots in⁻ andn−n1roots in the it is clear thatPλ, iω/0 for allω∈ and for allλ∈Ω.

⇐Suppose thatPλ, iω/0 for allω ∈ and for allλ ∈ Ω. If there isλ0 ∈Ωsuch that the polynomialPλ0, thasm1roots in⁻ andn−m1roots in withn1/m1, the from Theorem2.5there exsistsρsuch thatPρ, iω 0 for someω∈ , which is a contradiction.

3.3. Proof of Theorem

2.12

By generalization of zero exclusion principle, the polynomialpctp0−thasnroots in ⁻ andnroots in if and only if

pciωp0−iω/0 3.1

for allω∈ . Thus, ifksatisfies

p_ciωp0−iω 0 3.2

for someω∈ , then

Gω kFω ikωHω 0 3.3

for someω∈ . Consequently

ωHω 0,

Gω kFω 0. 3.4

And this system is satisfied ifk−Gω_l/Fω_l, whereω_l 0 orω_l∈RH. SinceGω>0 for allω∈ and we want to know the minimumk >0 and the maximumk <0 where it does not happen, then

k⁻_minmax

−Gωl

Fωl |Fωl>0, ωl0 orωl∈RH

,

k_maxmin

−Gωl

Fωl |Fωl<0, ωl0 or ωl∈RH

.

3.5

(11)

Now by symmetry ofFωandHωwe will just consider real positive roots ofHω. Thus

k⁻_minmax

−Gωl

Fω_l |Fωl∈K

,

k_maxmin

−Gωl

Fω_l |Fωl∈K⁻

.

3.6

This ends the proof.

Remark 3.2. In the proof it is not necessary to consider cases whenFωl 0, since in other case in the system3.4we would haveGω_l 0, but

Gω_l p0iω_l² p0iωlp0−iωl 0,

3.7

which is impossible sincep0tis Hurwitz. However, if occurs eitherK{0}orK⁻{0⁻}, then we evaluate inω0 and depending on sign ofF0we will get either

k⁻_min lim

r→0−G0

r −∞ 3.8

or

k_max lim

r→0⁻−G0

r ∞. 3.9

The following example illustrates the second part of Remark3.2.

Example 3.3. Consider the linear control system

x˙

⎛

⎜⎜

⎝

0 1 0

0 0 1

−6 −11 −6

⎞

⎟⎟

⎠x

⎛

⎜⎜

⎝ 0 0 1

⎞

⎟⎟

⎠

−13

2 k,−11k,−5k

x. 3.10

Then we have thatp0t t³6t²11t6 andp1t 5t²11t 13/2. First, for Białas’s method we have that the Hurwitz matrices ofp0tandp1tare

H p0t

⎛

⎜⎜

⎝ 6 6 0 1 11 0 0 6 6

⎞

⎟⎟

⎠, H p1t

⎛

⎜⎜

⎝ 5 13

2 0

0 11 0 0 5 13

2

⎞

⎟⎟

⎠. 3.11

(12)

Hence,

−H p0t₋₁

H p1t

⎛

⎜⎜

⎜⎝ 11 60 −1

10 0

− 1 60

1 10 0 1

60 −1 10

1 6

⎞

⎟⎟

⎟⎠

⎛

⎜⎜

⎝ 5 13

2 0

0 11 0 0 5 13

2

⎞

⎟⎟

⎠

⎛

⎜⎜

⎜⎝

−11 12 −11

120 0 1

12 −119 120 0

− 1 12

19 120 −13

12

⎞

⎟⎟

⎟⎠,

3.12

whose characteristic polynomial isλ³−359/120λ² 4297/1140λ−143/144which is a 3-degree polynomial and its roots areλ1,2 −229±i√

359/240 and λ3 −13/12. Hence λ_max−Hp0⁻¹Hp1 0⁻andλ⁻_min−Hp0⁻¹Hp1 −13/12. Thenpt, k p0t kp1t is robustly stable forkin−12/13,∞.

With our proposed method we can see that p0iω

6−6ω² iω

11−ω² P

ω²

iωQ ω²

, p1iω

13 2 −5ω²

11iω p

ω² iωq

ω² .

3.13

Thus,

Fω 19ω⁴112ω²39, Gω ω⁶−94ω⁴49ω²36, Hω −5ω⁴−9ω²−11.

3.14

Note that Fω > 0 for all ω ∈ and by a single test for second-order equations, Hω have no real roots. Therefore by Remark3.2kmax ∞andkmin −G0/F0 −12/13.

Consequently,p0t kp1tis stable for allk∈−12/13,∞.

4. Conclusions

In this paper, first we obtain generalizations of the Boundary Crossing Theorem and the Zero Exclusion Principle, which are results that allow to obtain important results about stability of families of polynomials. Next we use such generalizations for calculating the maximal

(13)

interval of stability, which is a diﬀerent approach to the Białas method. Since in Białas method we have to find the inverse of the matrixHp0and the roots of then-degree characteristic polynomial of−Hp0⁻¹Hp1, we have found that in our approach easier computations have arisen due if degree of bothp0tandp1tis either even or odd then degHω nm−2 and in the other cases degHω nm−1 and by symmetry ofHωwe must to find the roots of a polynomial with degreenm−2/2 ornm−1/2 both less than or equal to n−1, respectively.

References

1 T. C. Maxwell, “On governors,” Proceedings of the Royal Society, vol. 16, pp. 270–283, 1868.

2 S. P. Bhattacharyya, H. Chapellat, and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, Upper Saddle River, NJ, USA, 1995.

3 F. R. Gantmacher, The Theory of Matrices. Vols. 1, 2, translated by K. A. Hirsch, Chelsea Publishing, New York, NY, USA, 1959.

4 P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, 2nd edition, 1985.

5 V. L. Kharitonov, “The asymptotic stability of the equilibrium state of a family of systems of linear diﬀerential equations,” Diﬀerentsial’nye Uravneniya, vol. 14, no. 11, pp. 2086–2088, 1978.

6 A. C. Bartlett, C. V. Hollot, and H. Lin, “Root locations of an entire polytope of polynomials: it suﬃces to check the edges,” Mathematics of Control, Signals, and Systems, vol. 1, no. 1, pp. 61–71, 1988.

7 D. Hinrichsen and V. L. Kharitonov, “Stability of polynomials with conic uncertainty,” Mathematics of Control, Signals, and Systems, vol. 8, no. 2, pp. 97–117, 1995.

8 B. Aguirre, C. Ibarra, and R. Su´arez, “Suﬃcient algebraic conditions for stability of cones of polynomials,” Systems & Control Letters, vol. 46, no. 4, pp. 255–263, 2002.

9 B. Aguirre and R. Su´arez, “Algebraic test for the Hurwitz stability of a given segment of polynomials,”

Bolet´ın de la Sociedad Matem´atica Mexicana, vol. 12, no. 2, pp. 261–275, 2006.

10 S. Białas, “A necessary and suﬃcient condition for the stability of convex combinations of stable polynomials or matrices,” Bulletin of the Polish Academy of Sciences. Technical Sciences, vol. 33, no. 9-10, pp. 473–480, 1985.

11 H. Chapellat and S. P. Bhattacharyya, “An alternative proof of Kharitonov’s theorem,” IEEE Transactions on Automatic Control, vol. 34, no. 4, pp. 448–450, 1989.

12 B. Aguirre-Hern´andez, M. E. Fr´ıas-Armenta, and F. Verduzco, “Smooth trivial vector bundle structure of the space of Hurwitz polynomials,” Automatica, vol. 45, no. 12, pp. 2864–2868, 2009.

13 D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory. I. Modeling, State Space Analysis, Stability and Robustness, vol. 48 of Texts in Applied Mathematics, Springer, Berlin, Germany, 2005.

14 P. Dorato, R. Tempo, and G. Muscato, “Bibliography on robust control,” Automatica, vol. 29, no. 1, pp.

201–213, 1993.

15 J. Ackermann, Robust Control. The Parameter Space Approach, Springer, New York, NY, USA, 2002.

16 B. R. Barmish, New Tools for Robustness of Linear Systems, Macmillan, New York, NY, USA, 1994.

17 H. Chapellat, M. Mansour, and S. P. Bhattacharyya, “Elementary proofs of some classical stability criteria,” IEEE Transactions on Education, vol. 33, no. 3, pp. 232–239, 1990.

(14)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Probability and Statistics

Journal of

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Operations Research

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

The Boundary Crossing Theorem and the Maximal Stability Interval

Research Article