Research Article The Boundary Crossing Theorem and the Maximal Stability Interval

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1 Mathematical Problems in Engineering Volume, Article ID 343, 3 pages doi:.55//343 Research Article The Boundary Crossing Theorem and the Maximal Stability Interval Jorge-Antonio López-Renteria, Baltazar Aguirre-Hernández, and Fernando Verduzco División de Ciencias Básicas e Ingeniería, Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Avenida San Rafael Atlixco no. 86, Col. Vicentina, 934 México, DF, Mexico División de Ciencias Exactas y Naturales, Departamento de Matemáticas, Universidad de Sonora, Boulevard Rosales y Luis Encinas s/n, Col. Centro, 83, Hermosillo, Son, Mexico Correspondence should be addressed to Baltazar Aguirre-Hernández, bahe@xanum.uam.mx Received 9 October ; Accepted March Academic Editor: J. Jiang Copyright q Jorge-Antonio López-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.. Introduction Consider the linear system ẋ Ax,. where A is an n n matrix with constant coefficients and x R n. It is known that if the characteristic polynomial, p A t is 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. At first the researchers focused in the problem proposed for Maxwell and the Routh- Hurwitz criterion, the Hermite-Biehlertheoremand other criteria were obtained. For reading the proofs of these results the books 4 can be consulted. After the scientists began to study other related problems, for instance, the problem of giving conditions for a given family

2 Mathematical Problems in Engineering 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 theorem 5 and the Edge theorem 6 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, 8 or the segments of polynomails 9. Besides in the study of Hurwitz polynomials, topological approaches have been used recently see, 3. Good references about Hurwitz polynomilas which were reported during can be found in 4. The books, 5, 6 are 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 theorem and in the zero exclusion principle see 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.. 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... Generalizations of the Boundary Crossing Theorem and Zero Exclusion Principle First, we give the known boundary crossing theorem. Consider a family of polynomials P λ, t satisfying the following assumption. Assumption.. P λ, t is a family of polynomials of fixed degree n, coefficients are continuous respect λ in a fixed interval I a, b. Also let us to consider the complex plane C and let S C be any given open set and denote the boundary of S by S and the complement as U C S. The following results are presented in, page34. Theorem. boundary crossing theorem. Under Assumption., ifp a, t has all its roots in S whereas P b, t has at least one root in U. Then there exists at least one ρ in a, b such that i P ρ, t has all its roots in S S, ii P ρ, t has at least one root in S.

3 Mathematical Problems in Engineering 3 Now suppose that δ t, p denotes a polynomial whose coefficients depend continuously on the parameter vector p R l which varies in a set Ω R l and thus generates the family of polynomials Δ t : { δ t, p ) p Ω }.. Theorem.3 zero exclusion principle. Assume that the polynomial family. has constant degree, contains at least one stable polynomial, and Ω is pathwise connected. Then the entire family is stable if and only if / Δ t, t S.. Now we present our generalizations. In them we will consider S C. Theorem.4 generalization of Theorem.. Under Assumption., suppose that P a, t has n roots in C and n n roots in C,andP b, t has at most n roots in C and at least n n roots in C. Then there exists at least one ρ in a, b such that i P ρ, t has at least n roots in C ir, ii P ρ, t has at least one root in ir. Theorem.5 generalization of Theorem.. Under Assumption., suppose that P a, t has n roots in C and n n roots in C,andP b, t has m roots in C and n m roots in C.If n / m, Then there exists at least one ρ in a, b such that i P ρ, t has at least n roots in C ir, ii P ρ, t has at least n n roots in C ir, iii P ρ, t has at least one root in ir. Theorem.6 generalization of Theorem.3. Consider the polynomial family P λ, t with constant degree, where λ Ω and Ω R l is a pathwise connected set. Suppose there exists an element of the family with n roots in C and n n roots in C. Then the entire family still having n roots in C and n n roots in C if and only if P λ, iω / λ Ω, ω R..3.. Application: An Alternative Method for Calculating the Maximal Stability Interval We begin this subsection with three important definitions that can be seen in pages 5 to 5 of 6.

4 4 Mathematical Problems in Engineering Definition.. Consider the uncertain polynomial P t, k p t kp t with p t assumed Hurwitz stable and the uncertainty bounding set K k,k with k andk. We define the subfamilies P k { p,k k k }, P k ) { p,k k k }..4 Definition.8 maximal stability interval. Associated with the subfamily P k is the rightsided robustness margin k max sup{ k : P k is robustly stable },.5 and asociated with the subfamily P k is the left-sided robustness margin k min inf{ k : P k ) is robustly stable }..6 Subsequently we call K max k min,k max the maximal interval for robust stability. Definition.9. Given an n n matrix M, wedefineλ max M to be the maximum positive real eigenvalue of M.WhenMdoes not have any positive real eigenvalue, we take λ max M. Similarly, we define λ min M to be the minimum negative real eigenvalue of M. WhenM does no have any negative real eigenvalue, we take λ min M. Białas proved the following theorem. Theorem. eigenvalue criterion. Consider the uncertain polynomial P t, k p t kp t with P t, p t Hurwitz stable and having positive coefficients and deg p t > deg p t. Then the maximal interval for robust stability is described by k max λ max ) )), H p H p k min λ ) )), min H p H p. where H p and H p are the corresponding matrices of Hurwitz of p t and p t, respectively, and for purpose of conformability of matrix multiplication, H p is an n n matrix obtained by treating p t as an n-degree polynomial. Now we will explain our approach for calculating the maximal stability interval. Let p t be an n-degree polynomial and p t a polynomial such that n>deg p t.considerthe family of polynomials p c t p t kp t. By evaluating p t and p c t in iω we get p iω P ω ) iωq ω ), p c iω P ω ) kp ω ) [ iω Q ω ) kq ω )],.8

5 Mathematical Problems in Engineering 5 where p, q, P, Q are polynomials. Then p c iω p iω P ω ) ω Q ω ) [ k p ω ) P ω ) ω q ω ) Q ω )] [ ikω q ω ) P ω ) p ω ) Q ω )]..9 Define the polynomials F ω p ω ) P ω ) ω q ω ) Q ω ), G ω P ω ) ω Q ω ), H ω q ω ) P ω ) p ω ) Q ω ).. Therefore, we can rewrite p c iω p iω as p c iω p iω G ω kf ω ikωh ω. Definition.. For an arbitrary polynomial f t we define the set of its roots as R f ) { ζ C f ζ }.. Let R f R denote the set of positive real elements of R f. It is clear that R f R could be an empty set. Now let F ω, G ω, andh ω be the polynomials defined above. Define the sets K {F ω l ω l R H R {}, F ω l > }, K {F ω l ω l R H R {}, F ω l < }.. If there is no elements in R H R {} such that F ω l > then we will define K { }. Similarly, if there is no elements in R H R {} such that F ω l < then define K { }. Note that only can happen either K { } or K { } but both at the same time never since we can always evaluate in ω. That is, we always have an extreme. Therefore we have the following alternative method for calculating the maximal stability interval. Theorem.. Consider the polynomial family p c t p t kp t with p t Hurwitz stable and having positive coefficients, and let F ω,g ω, and H ω be the polynomials defined above. Then the maximal interval of stability for p c t is described by { k min max G ω } l F ω l F ω l K, k max min { G ω l F ω l F ω l K }..3

6 6 Mathematical Problems in Engineering Remark.3. A differencewith 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 an n-degree polynomial must be found while in our approach we have that if degree of both n and m, resp., n>m p t and p t is either even or odd then deg H ω n m and in the other cases deg H ω n m. Therefore, by symmetry of H ω we have to find the roots of a polynomial with degree n m / or n m / bothlessthanorequalto n. Example.4. Consider the polynomial p t t 3 6t t 6andp t t t, for the polynomial family p t, k p t kp t. We will verify the maximal stability interval by Białas method. First we have H p t ) 6 6, 6 6 H p t )..4 Next, H p t ) H p t ) , whose characteristic polynomial is λ 3 / λ /36 λ /98 which is a 3-degree polynomial and has as roots λ λ /6 andλ 3 /. Thus λ max H p H p / and λ min H p H p /6. Then p t, k p t kp t is robustly stable in k min,k max 6, /. Now with our proposed method. We see p iω 6 6ω ) iω ω ) P ω ) iωq ω ), p iω ω ) iω p ω ) iωq ω )..6

7 Mathematical Problems in Engineering Thus, F ω 8ω 4 36ω 6, G ω ω 6 ω 4 ω 36,. H ω ω 4 5ω 4. Thus R H R {, 4}.Now, F <, ) F 4 35 >, F 6 >, G, ) G 4 5,.8 G 36. Therefore kmax G F, ) G 4 k min max ), G 6. F 4 F.9 Note that we just only need to compute a -degree polynomial root in our method, while in Białas one it is a 3-degree polynomial. Example.5. Consider the linear control system ẋ x 3k, k, k, x. 3.

8 8 Mathematical Problems in Engineering From this system we have that p t t 4 t 3 t t 3andp t t t 3andtheir Hurwitz matrices are H p t ) 3, H p t ) 3,. 3 3 where the inverse of H p is H p t ) Hence, the 4-degree characteristic polynomial of the matrix H p t ) H p t ) is t 4 / t 3 3/ t / t and has as roots λ, λ / andλ 3,4. Then λ max H p H p /andλ min H p H p. Therefore, system. is robustly stable in, /.

9 Mathematical Problems in Engineering 9 Now with our method. By evaluating p t and p t in iω we have p iω ) ) ω 4 ω 3 iω ω P ω ) iωq ω ), ) p iω ω 3 iω p ω ) iωq ω )..4 Thus, F ω ω 6 8ω 4 ω 9, G ω ω 8 3ω 6 5ω 4 38ω 9,.5 H ω ω 9 ω ). It is not hard to see that R H R {, 3}. Next,F 9 >, F 3 5 <, and G 9, G Therefore, k max G 3 F 3 k min G. F.6 Note the easiness of roots finding for our polynomial H ω. 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 in 3. Lemma 3. continuous root dependence. Consider the family of polynomials P described by P λ, t n i a i λ t i and λ Ω under Assumption.. Then the roots of P λ, t vary continuously with respect to λ Ω. That is, there exist continuously mappings t i : Ω C for i,,...,nsuch that t λ,...,t n λ are the roots of P λ, t. 3.. Proof of Theorem.4 Since P λ, t satisfies Assumption., by Lemma 3. there exist n continuous function roots of P λ, t, sayt λ,...,t n λ, λ a, b. Letusdenoteα j λ R t j λ as the real parts of the roots. Without loss of generality we can suppose that for j,...,n, α j a R and

10 Mathematical Problems in Engineering for j n,...,n, α j a R, while for λ b at most n α j b s belong to R and at least n n belongtor. Then there exists at least one t j λ such that α j a < and α j b >. Let α j λ,...,α jm λ be such functions. Then by continuity and the intermediate value theorem we have that for each r m there exists ρ r a, b such that α jr ρ r. Define ρ min{ρ r r,...,m}. Therefore, for λ ρ at least one α jr ρ. Thus P ρ, t has n roots in C ir with at least one root in ir, as we claim. The proof of Theorem.5 is similar. 3.. Proof of Theorem.6 If all of the elements of the family have n roots in C and n n roots in C the it is clear that P λ, iω / forallω R and for all λ Ω. Suppose that P λ, iω / forallω R and for all λ Ω. Ifthereisλ Ω such that the polynomial P λ,t has m roots in C and n m roots in C with n / m,thefrom Theorem.5 there exsists ρ such that P ρ, iω forsomeω R, which is a contradiction Proof of Theorem. By generalization of zero exclusion principle, the polynomial p c t p t has n roots in C and n roots in C if and only if p c iω p iω / 3. for all ω R. Thus,ifk satisfies p c iω p iω 3. for some ω R, then G ω kf ω ikωh ω 3.3 for some ω R. Consequently ωh ω, G ω kf ω. 3.4 Andthissystemissatisfiedifk G ω l /F ω l,whereω l orω l R H. SinceG ω > for all ω R and we want to know the minimum k>andthemaximumk<whereitdoes not happen, then { k min max G ω } l F ω l F ω l >, ω l orω l R H, k max min { G ω l F ω l F ω l <, ω l orω l R H }. 3.5

11 Mathematical Problems in Engineering NowbysymmetryofF ω and H ω we will just consider real positive roots of H ω.thus { k min max G ω } l F ω l F ω l K, { kmax min G ω } l F ω l F ω l K. 3.6 This ends the proof. Remark 3.. In the proof it is not necessary to consider cases when F ω l, since in other case in the system 3.4 we would have G ω l, but G ω l p iω l p iω l p iω l 3., which is impossible since p t is Hurwitz. However, if occurs either K { } or K { }, then we evaluate in ω and depending on sign of F we will get either k min G lim r r 3.8 or kmax lim G. r r 3.9 The following example illustrates the second part of Remark 3.. Example 3.3. Consider the linear control system ẋ x ) k, k, 5k x. 3. Then we have that p t t 3 6t t 6andp t 5t t 3/. First, for Białas s method we have that the Hurwitz matrices of p t and p t are H p t ) 6 6, H p t )

12 Mathematical Problems in Engineering Hence, H p t ) H p t ) , 3 3. whose characteristic polynomial is λ 3 359/ λ 49/4 λ 43/44 which is a 3-degree polynomial and its roots are λ, 9 ± i 359 /4 and λ 3 3/. Hence λ max H p H p and λ min H p H p 3/. Then p t, k p t kp t is robustly stable for k in /3,. With our proposed method we can see that p iω 6 6ω ) iω ω ) P ω ) iωq ω ), ) 3 p iω 5ω iω p ω ) iωq ω ). 3.3 Thus, F ω 9ω 4 ω 39, G ω ω 6 94ω 4 49ω 36, 3.4 H ω 5ω 4 9ω. Note that F ω > forallω R and by a single test for second-order equations, H ω have no real roots. Therefore by Remark 3. k max and k min G /F /3. Consequently, p t kp t is stable for all k /3,. 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 Mathematical Problems in Engineering 3 interval of stability, which is a differentapproachto the Białas method. Since in Białasmethod we have to find the inverse of the matrix H p and the roots of the n-degree characteristic polynomial of H p H p, we have found that in our approach easier computations have arisen due if degree of both p t and p t is either even or odd then deg H ω n m and in the other cases deg H ω n m and by symmetry of H ω we must to find the roots of a polynomial with degree n m / or n m / bothlessthanorequalto n, respectively. References T. C. Maxwell, On governors, Proceedings of the Royal Society, vol. 6, pp. 83, 868. S.P.Bhattacharyya,H.Chapellat,andL.H.Keel,Robust Control: The Parametric Approach, Prentice Hall, Upper Saddle River, NJ, USA, F. R. Gantmacher, The Theory of Matrices. Vols.,, translated by K. A. Hirsch, Chelsea Publishing, New York, NY, USA, P. Lancaster and M. Tismenetsky, The Theory of Matrices, Computer Science and Applied Mathematics, Academic Press, Orlando, Fla, USA, nd edition, V. L. Kharitonov, The asymptotic stability of the equilibrium state of a family of systems of linear differential equations, Differentsial nye Uravneniya, vol. 4, no., pp , 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., no., pp. 6, 988. D. Hinrichsen and V. L. Kharitonov, Stability of polynomials with conic uncertainty, Mathematics of Control, Signals, and Systems, vol. 8, no., pp. 9, B. Aguirre, C. Ibarra, and R. Suárez, Sufficient algebraic conditions for stability of cones of polynomials, Systems & Control Letters, vol. 46, no. 4, pp ,. 9 B. Aguirre and R. Suárez, Algebraic test for the Hurwitz stability of a given segment of polynomials, Boletín delasociedadmatemática Mexicana, vol., no., pp. 6 5, 6. 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-, pp , 985. H. Chapellat and S. P. Bhattacharyya, An alternative proof of Kharitonov s theorem, IEEE Transactions on Automatic Control, vol. 34, no. 4, pp , 989. B. Aguirre-Hernández, M. E. Frías-Armenta, and F. Verduzco, Smooth trivial vector bundle structure of the space of Hurwitz polynomials, Automatica, vol. 45, no., pp , 9. 3 D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory. I. Modeling, State Space Analysis, Stability and Robustness, vol.48oftexts in Applied Mathematics, Springer, Berlin, Germany, 5. 4 P. Dorato, R. Tempo, and G. Muscato, Bibliography on robust control, Automatica, vol. 9, no., pp. 3, J. Ackermann, Robust Control. The Parameter Space Approach, Springer, New York, NY, USA,. 6 B. R. Barmish, New Tools for Robustness of Linear Systems, Macmillan, New York, NY, USA, 994. H. Chapellat, M. Mansour, and S. P. Bhattacharyya, Elementary proofs of some classical stability criteria, IEEE Transactions on Education, vol. 33, no. 3, pp. 3 39, 99.

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