The Boundary Crossing Theorem and the Maximal Stability Interval

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 systemẋ where A is an n × n matrix with constant coefficients and x ∈ Ê 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 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 books 2-4 can be consulted. After the scientists began to study other related problems, for instance, the problem of giving conditions for a given family 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 7,8 or the segments of polynomails 9-11 . Besides in the study of Hurwitz polynomials, topological approaches have been used recently see 12,13 . Good references about Hurwitz polynomilas which were reported during 1987-1991 can be found in 14 .
The books 2, 15, 16 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 17 and in the zero exclusion principle see 2 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. ii P ρ, t has at least one root in ∂S.

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Now suppose that δ t, p denotes a polynomial whose coefficients depend continuously on the parameter vector p ∈ Ê l which varies in a set Ω ⊂ Ê l and thus generates the family of polynomials Now we present our generalizations. In them we will consider S − . ii P ρ, t has at least n − n 1 roots in ∪ iÊ, iii P ρ, t has at least one root in iÊ.

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 of 16 .

Mathematical Problems in Engineering
Definition 2.7. Consider the uncertain polynomial P t, k p 0 t kp 1 t with p 0 t assumed Hurwitz stable and the uncertainty bounding set K k − , k with k − ≤ 0 and k ≥ 0. We define the subfamilies Definition 2.8 maximal stability interval . Associated with the subfamily P k is the rightsided robustness margin k max sup k : P k is robustly stable , 2.5 and asociated with the subfamily P k − is the left-sided robustness margin Subsequently we call K max k − min , k max the maximal interval for robust stability. Theorem 2.10 eigenvalue criterion . Consider the uncertain polynomial P t, k p 0 t kp 1 t with P t, 0 p 0 t Hurwitz stable and having positive coefficients and deg p 0 t > deg p 1 t . Then the maximal interval for robust stability is described by where H p 0 and H p 1 are the corresponding matrices of Hurwitz of p 0 t and p 1 t , respectively, and for purpose of conformability of matrix multiplication, H p 1 is an n × n matrix obtained by treating p 1 t as an n-degree polynomial.
Now we will explain our approach for calculating the maximal stability interval. Let p 0 t be an n-degree polynomial and p 1 t a polynomial such that n > deg p 1 t . Consider the family of polynomials p c t p 0 t kp 1 t . By evaluating p 0 −t and p c t in iω we get Mathematical Problems in Engineering   5 where p, q, P, Q are polynomials. Then

2.9
Define the polynomials
Definition 2.11. For an arbitrary polynomial f t we define the set of its roots as Let R f Ê denote the set of positive real elements of R f . It is clear that R f Ê could be an empty set. Now let F ω , G ω , and H ω be the polynomials defined above. Define the sets

2.12
If there is no elements in R H Ê ∪ {0} such that F ω l > 0 then we will define K
Note that only can happen either K − {0 − } or K {0 } but both at the same time never since we can always evaluate in ω 0. 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 family p c t p 0 t kp 1 t with p 0 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

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Mathematical Problems in Engineering 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 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 0 t and p 1 t is either even or odd then deg H ω n m − 2 and in the other cases deg H ω n m − 1. Therefore, by symmetry of H ω we have to find the roots of a polynomial with degree n m − 2 /2 or n m − 1 /2 both less than or equal to n − 1.
Example 2.14. Consider the polynomial p 0 t t 3 6t 2 12t 6 and p 1 t t 2 − 2t 1, for the polynomial family p t, k p 0 t kp 1 t . We will verify the maximal stability interval by Białas method. First we have

2.16
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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.

2.20
From this system we have that p 0 t t 4 t 3 7t 2 2t 3 and p 1 t t 2 2t 3 and their Hurwitz matrices are where the inverse of H p 0 is

2.22
Hence, the 4-degree characteristic polynomial of the matrix Now with our method. By evaluating p 0 t and p 1 t in iω we have

2.26
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.

Proofs of the Theorems
Before start with the proofs of the theorems we present the following lemma wich can be found in 3 .

Lemma 3.1 continuous root dependence .
Consider the family of polynomials P described by P λ, t n i 0 a i λ t i and λ ∈ Ω under Assumption 2.1. Then the roots of P λ, t vary continuously with respect to λ ∈ Ω. That is, there exist continuously mappings t i : Ω → for i 1, 2, . . . , n such that t 1 λ , . . . , t n λ are the roots of P λ, t .

Proof of Theorem 2.4
Since P λ, t satisfies Assumption 2.1, by Lemma 3.1 there exist n continuous function roots of P λ, t , say t 1 λ , . . . , t n λ , λ ∈ a, b . Let us denote α j λ Ê t j λ as the real parts of the roots. Without loss of generality we can suppose that for j 1, . . . , n 1 , α j a ∈ Ê − and 10 Mathematical Problems in Engineering for j n 1 1, . . . , n, α j a ∈ Ê , while for λ b at most n 1 − 1 α j b 's belong to Ê − and at least n − n 1 1 belong to Ê . Then there exists at least one t j λ such that α j a < 0 and α j b > 0. Let α j 1 λ , . . . , α j m λ be such functions. Then by continuity and the intermediate value theorem we have that for each 1 ≤ r ≤ m there exists ρ r ∈ a, b such that α j r ρ r 0. Define ρ min{ρ r | r 1, . . . , m}. Therefore, for λ ρ at least one α j r ρ 0. Thus P ρ, t has n 1 roots in − ∪ iÊ with at least one root in iÊ, as we claim.
The proof of Theorem 2.5 is similar.

Proof of Theorem 2.6
⇒ If all of the elements of the family have n 1 roots in − and n − n 1 roots in the it is clear that P λ, iω / 0 for all ω ∈ Ê and for all λ ∈ Ω. ⇐ Suppose that P λ, iω / 0 for all ω ∈ Ê and for all λ ∈ Ω. If there is λ 0 ∈ Ω such that the polynomial P λ 0 , t has m 1 roots in − and n − m 1 roots in with n 1 / m 1 , the from Theorem 2.5 there exsists ρ such that P ρ, iω 0 for some ω ∈ Ê, which is a contradiction.

Proof of Theorem 2.12
By 3.4 And this system is satisfied if k −G ω l /F ω l , where ω l 0 or ω l ∈ R H . Since G ω > 0 for all ω ∈ Ê and we want to know the minimum k > 0 and the maximum k < 0 where it does not happen, then k max min − G ω l F ω l | F ω l < 0, ω l 0 or ω l ∈ R H .

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Now by symmetry of F ω and H ω we will just consider real positive roots of H ω . Thus

3.6
This ends the proof.

Remark 3.2.
In the proof it is not necessary to consider cases when F ω l 0, since in other case in the system 3.4 we would have G ω l 0, but which is impossible since p 0 t is Hurwitz. However, if occurs either K {0 } or K − {0 − }, then we evaluate in ω 0 and depending on sign of F 0 we will get either The following example illustrates the second part of Remark 3.2.
Then we have that p 0 t t 3 6t 2 11t 6 and p 1 t 5t 2 11t 13/2 . First, for Białas's method we have that the Hurwitz matrices of p 0 t and p 1 t are 3.14 Note that F ω > 0 for all ω ∈ Ê and by a single test for second-order equations, H ω have no real roots. Therefore by Remark 3.2 k max ∞ and k min −G 0 /F 0 −12/13. Consequently, p 0 t kp 1 t is stable for all k ∈ −12/13, ∞ .

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 interval of stability, which is a different approach to the Białas method. Since in Białas method we have to find the inverse of the matrix H p 0 and the roots of the n-degree characteristic polynomial of −H p 0 −1 H p 1 , we have found that in our approach easier computations have arisen due if degree of both p 0 t and p 1 t is either even or odd then deg H ω n m − 2 and in the other cases deg H ω n m − 1 and by symmetry of H ω we must to find the roots of a polynomial with degree n m − 2 /2 or n m − 1 /2 both less than or equal to n − 1, respectively.