Elsevier

Automatica

Volume 37, Issue 2, February 2001, Pages 291-296
Automatica

Technical Communique
The use of Routh array for testing the Hurwitz property of a segment of polynomials

https://doi.org/10.1016/S0005-1098(00)00142-4Get rights and content

Abstract

In this paper we show that the test of Hurwitz property of a segment of polynomials (1−λ)p0(s)+λp1(s), where λ∈[0,1], p0(s) and p1(s) are nth-degree polynomials of real coefficients, can be achieved via the approach of constructing a fraction-free Routh array and using Sturm's theorem. We also establish the connection between the proposed approach and the finite-step methods based on the resultant theory and the boundary crossing theorem. In a certain sense, the proposed approach provides an efficient numerical implementation of the later two methods and, therefore, by which the robust Hurwitz stability of convex combinations of polynomials can be checked in a definitely finite number of arithmetic operations without having to invoke any root-finding procedure.

Introduction

Consider a linear system which has undergone parametric variations in a hyperbox B. If the coefficients of the characteristic polynomial depend affine linearly on parameters, the resulting parametric polynomial family is a polytope, which is the convex hull of the polynomials obtained by setting parameters to the vertices of the hyperbox B. Since the emergence of Edge Theorem (Bartlett, Hollot & Huang, 1988), which states that the robust Hurwitz stability analysis of a polytope of polynomials can be reduced to that of checking a finite number of segments of polynomials of the formA(s;λ)≔(1−λ)p0(s)+λp1(s),λ∈[0,1],where pi(s)=pi,e(s2)+spi,o(s2),i=0,1, are polynomials of the same degree; several authors (Bialas, 1985; Bose, 1989; Gutman, 1992; Bougerra, Chang, Yeh & Banda, 1990; Ozturk, 1992; Bollepalli & Pujara, 1994; Blondel, 1996; Barmish, 1994; Chapellat & Bhattacharyya, 1989; Bhattacharyya, Chapellat & Keel, 1995) have presented methods for testing the Hurwitzness of the polynomial family A(s;[0,1]).

Bialas (1985) was among the first to propose a necessary and sufficient condition for the Hurwitz stability of the polynomial family A(s;[0,1]). He proved that the parametric polynomial A(s;λ) is Hurwitz stable for λ∈[0,1] if and only if the real eigenvalues of the matrix −H1H0−1 are negative, where H0 and H1 are the Hurwitz matrices associated with the strict Hurwitz polynomials p0(s) and p1(s), respectively. All the methods subsequently presented in the literature for testing the Hurwitz property of a segment of polynomials are essentially based on verifying if there exists a λ∈[0,1] such that A(jω;λ)=Ar(ω,λ)+jAi(ω,λ)=0 for some real ω, where j=−1. According to the approaches adopted to check the stability boundary crossing as λ varying from 0 to 1, the methods can be grouped into two categories. In the first category (Bougerra et al., 1990; Ozturk, 1992; Bollepalli & Pujara, 1994; Blondel, 1996; Chapellat & Bhattacharyya, 1989; Bhattacharyya et al., 1995), the robust Hurwitz stability of the polynomial family A(s;[0,1]) is tested based on the following zero exclusion principle (Barmish, 1994). Suppose that the polynomial family A(s;λ),λ∈[0,1] has invariant degree. Then the members of A(s;[0,1]) all have the same number of zeros in the right-half plane if and only if A(jω;λ)≠0 for all λ∈[0,1] and ω∈[0,∞). For checking the absence of zeros in the domain (λ,ω)∈[0,1]×[0,∞) of the two-variable polynomials Ar(ω,λ) and Ai(ω,λ), it is required first to utilize Sturm's theorem (Marden, 1949). However, it has to invoke a root-finding procedure if indeterminacy occurs. In the second category (Bose, 1989; Gutman, 1992), the stability boundary crossing of the λ polynomial A(s;λ)=Ae(s2;λ)+sAo(s2;λ) is checked by application of the resultant theory to test the coprimeness of λ-polynomials Ae(x;λ) and Ao(x;λ). It involves the symbolic computation of the determinant of the (n−1) by (n−1) resultant matrixR(Ae(x;λ),Ao(x;λ))=R(p0,e(x),p0,o(x))+λR(p1,e(x)−p0,e(x),p1,o(x)−p0,o(x))and the check for the absence of zero in the interval (0,1) of the λ-polynomial |R(Ae(x;λ),Ao(x;λ))| using Sturm's theorem (Marden, 1949). In (2), R(f(x),g(x)) denotes the resultant matrix generated with the two polynomials f(x) and g(x). For the two polynomialsf(x)=anxn+an−1xn−1+⋯+a1x+a0,g(x)=bmxm+bm−1xm−1+⋯+b1x+b0,the resultant matrix R(f(x),g(x)) is defined asR(f(x),g(x))=anan−1a00m−10ana00m−20m−1ana00n−1bmb00bmb00n−2bmbm−1b00n−1,where 0k=[0⋯0]1×k.

In this paper, we apply the fraction-free Routh array (Jeltsch, 1979) with polynomial entries and Sturm's theorem (Marden, 1949) to test the robust Hurwitz stability of a segment of real-coefficient polynomials. Since the parametric array is a natural result of applying Euclidean algorithm to extract the greatest common divisor of the even part Ae(s2;λ) and the odd part Ao(s2;λ) of the parametric polynomial A(s;λ), the proposed approach for the robust Hurwitz stability test has a close connection with the ones based on the resultant theory (Bose, 1989; Gutman, 1992) and on the boundary crossing theorem (Ackermann, 1993). Hence, in a certain sense, the approach of using fraction-free Routh array for the robust Hurwitz stability test of a segment of polynomials can be viewed as an efficient numerical implementation of the later two methods.

Section snippets

Main results

Let p0(s) and p1(s) be nth-degree real-coefficient polynomials given bypi(s)=pi,0+pi,1s+⋯+pi,nsn,pi,n>0,i=0,1.Hence, the parametric polynomial in (1) can be written asA(s;λ)=a0(λ)+a1(λ)s+⋯+an(λ)sn,where the coefficients ak(λ) are first-degree polynomials in λ:ak(λ)=(1−λ)p0,k+λp1,k=p0,k+λ(p1,k−p0,k),k=0,1,…,n.Following the development given in Jeltsch (1979), we can construct the fraction-free parametric Routh array as shown in Table 1 for testing the Hurwitz stability of the parametric

A numerical example

To illustrate the proposed algorithm, we consider the real-coefficient segment of polynomials A(s;λ) given in Bollepalli & Pujara (1994):A(s;λ)=(1−λ)p0(s)+λp1(s),p0(s)=s5+4.6s4+9.13s3+10.52s2+6.78s+2.92,p1(s)=s5+6s4+21.58s3+62.244s2+96.112s+67.8.Using the algorithm in (9), we obtained polynomial entries of the 2nd, 3rd, and 4th rows of the parametric Routh array for A(s;λ) as follows:Row2:a2,0(λ)=3812.85237λ2+661.75296λ+44.666,a2,1(λ)=125.0648λ2+355.5392λ+28.268.Row3:a3,0(λ)=36297.72327λ3

Conclusion

In this paper, we have proposed an efficient procedure of using rational arithmetic operations to test the robust Hurwitz stability of a convex combination of two nth-degree real polynomials. The procedure is essentially based on constructing a fraction-free parametric Routh array and invoking Sturm's theorem for the checking of absence of real zeros in the interval [0,1], of a polynomial of degree n−1. From the computational point of view, the proposed procedure can be viewed as an efficient

Acknowledgements

This work was supported by the National Science Council of the Republic of China under Grant NSC88-2214-E-194-001.

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This paper was presented in the IFAC Conference Control Systems Design in Bratislava, Slovak Republic. This paper was recommended for publication in revised form by Associate Editor R. Middleton under the direction of Editor Paul Van den Hof.

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