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1.
The necessary conditions for a complex polynomial to be strictly Hurwitz are reviewed and rigorously proved. Both necessary conditions have been extended to cover nonmonic polynomials instead of monic polynomials. Also, based on these two results, some necessary conditions for an interval polynomial to be stable in terms of being strictly Hurwitz are obtained. They can be used to quickly determine the instability of a complex interval polynomial family. Finally, their application to the study of robust stability, in the case where coefficient perturbation intervals are functions of a single parameter, is briefly discussed  相似文献   

2.
Network realzability theory provides the basis for a unified approach to the stability of a polynomial or a family of polynomials. In this paper conditions are given, in terms of certain decompositions of a given polynomial, that are necessary and sufficient for the given polynomial to be Hurwitz. These conditions facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. This approach provides a simple interpretation of recent results for polynomials with real coefficients and also leads to the formulation of corresponding results for the case of polynomials with complex coefficients.  相似文献   

3.
We introduce the concept of the Hadamard power of a polynomial formed by real powers of its coefficients. We show that the Hadamard power of a Hurwitz polynomial remains Hurwitz.  相似文献   

4.
A discrete version of Foster's reactance theorem is developed and, subsequently, used to delineate necessary and sufficient conditions for a given polynomial with complex or real coefficients to be of the Schur type. These conditions, obtained from the decomposition of a polynomial into its circularly symmetric and anti-circularly symmetric components, facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. These results provide the complete discrete counterpart of recent results for a family of polynomials which are required to be tested for the Hurwitz property.  相似文献   

5.
Several sufficient conditions for the Hurwitz property of polynomials are derived by combining the existing sufficient criteria for the Schur property with bilinear mapping. The conditions obtained are linear or piecewise linear inequalities with respect to the polynomial coefficients. Making the most of this feature, the results are applied to the Hurwitz stability test for a polytope of polynomials. It turns out that checking the sufficient conditions at every generating extreme polynomial suffices to guarantee the stability of any member of the polytope, yielding thus extreme point results on the Hurwitz stability of the polytope. This brings about considerable computational economy in such a test as a preliminary check before going to the exact method, the edge theorem and stability test of segment polynomials.  相似文献   

6.
In this paper we give an efficient algorithm to find symbolically correct zeros of a polynomial f ∈ R[X] which can be represented by square roots. R can be any domain if a factorization algorithm over R[X] is given, including finite rings or fields, integers, rational numbers, and finite algebraic or transcendental extensions of those. Asymptotically, the algorithm needs O(Tf(d2)) operations in R, where Tf(d) are the operations for the factorization algorithm over R[X] for a polynomial of degree d. Thus, the algorithm has polynomial running time for instance for polynomials over finite fields or the rationals. We also present a quick test for deciding whether a given polynomial has zeros expressible by square roots and describe some additional methods for special cases.  相似文献   

7.
It is proved in this paper that all Hurwitz polynomials of order not less than n form two simply connected Borel cones in the polynomial parameter space. Based on this result, edge theorems for Hurwitz stability of general polyhedrons of polynomials and boundary theorems for Hurwitz stability of compact sets of polynomials are obtained. Both cases of families of polynomials with dependent and independent coefficients are considered. Different from the previous ones, our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials. Moreover, our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine  相似文献   

8.
In this paper we describe a conic approach to the stability theory of uncertain polynomials. We present necessary and sufficient conditions for a conic setp 0+K of polynomials to be Hurwitz stable (K is a convex cone of polynomials of degree n and degp 0=n). As analytical tools we derive an edge theorem and Rantzer-type conditions for marginal stability (semistability). The results are applied to prove an extremal-ray result for conic sets whose cone of directions is given by an interval polynomial.The second author would like to thank the Deutsche Forschungsgemeinschaft (DFG) for its support during the writing of this paper.  相似文献   

9.
This paper considers the robust stability verification of polynomials with coefficients depending polynomially on parameters varying in given intervals. Two algorithms are presented, both rely on the expansion of a multivariate polynomial into Bernstein polynomials. The first one is an improvement of the so-called Bernstein algorithm and checks the Hurwitz determinant for positivity over the parameter set. The second one is based on the analysis of the value set of the family of polynomials and profits from the convex hull property of the Bernstein polynomials. Numerical results to real-world control problems are presented showing the efficiency of both algorithms  相似文献   

10.
In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H -norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.  相似文献   

11.
An extension to a well-known theorem due to Kharitonov is presented, Kharitonov's theorem gives a necessary and sufficient condition for all polynomials in a given family to be Hurwitz stable. In Kharitonov's theorem, the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval. Kharitonov's theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials. Kharitonov's theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials  相似文献   

12.
A heuristic factorization scheme that uses learning and other heuristic programming techniques to improve the efficiency of determining the symbolic factorization of multivariate polynomials with integer coefficients and an arbitrary number of variables and terms is described. The learning program, POLYFACT, in which the factorization scheme is implemented is also described. POLYFACT uses learning through the dynamic construction and manipulation of first-order predicate calculus heuristics to reduce the amount of searching for the irreducible factors of a polynomial.Tables containing the results of factoring randomly generated multivariate polynomials are presented: (1) to demonstrate that learning does improve considerably the efficiency of factoring polynomials, and (2) to show that POLYFACT does learn from previous experience.The factorization times of polynomials factored by both the scheme implemented in POLYFACT and Wang's implementation of Berlekamp's algorithm are given. The two algorithms are compared, and two situations where POLYFACT'S algorithm can be used to improve the efficiency of Wang's algorithm are discussed.  相似文献   

13.
The theorem of Kharitonov on the Hurwitz property of interval families of polynomials cannot be extended, in genera, to obtain sufficient conditions for the stability of families of characteristic polynomials of discrete-time systems. Necessary and sufficient conditions for this stability problem are given. Such conditions naturally give rise to a computationally efficient stability test which requires the solution of a one-parameter optimization problem and which can be considered as a counterpart to the Kharitonov test for continuous-time systems. At the same time, the method used to derive the stability conditions provides a procedure for solving another stability robustness problem, i.e. the estimation of the largest domain of stability with a rectangular box shape around given nominal values of the polynomial coefficients  相似文献   

14.
Gives an elementary proof of Kharitonov's theorem using simple complex plane geometry without invoking the Hermite-Bieler theorem. Kharitonov's theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval. The result states that the whole class is Hurwitz if and only if four special, well-defined polynomials are Hurwitz. The paper also gives elementary proofs of two previously known extensions: for polynomials of degree less than six, the requirement is reduced to fewer than four polynomials; and the theorem is generalized to polynomials with complex coefficients  相似文献   

15.
Symbolic numeric algorithms for polynomials are very important, especially for practical computations since we have to operate with empirical polynomials having numerical errors on their coefficients. Recently, for those polynomials, a number of algorithms have been introduced, such as approximate univariate GCD and approximate multivariate factorization for example. However, for polynomials over integers having coefficients rounded from empirical data, changing their coefficients over reals does not remain them in the polynomial ring over integers; hence we need several approximate operations over integers. In this paper, we discuss computing a polynomial GCD of univariate or multivariate polynomials over integers approximately. Here, “approximately” means that we compute a polynomial GCD over integers by changing their coefficients slightly over integers so that the input polynomials still remain over integers.  相似文献   

16.
A state feedback is proposed to analyse the stability of a matrix polynomial in closed loop. First, it is shown that a matrix polynomial is stable if and only if a state space realisation of a ladder form of certain transfer matrix is stable. Following the ideas of the Routh–Hurwitz stability procedure for scalar polynomials, certain continued-fraction expansions of polynomial matrices are carrying out by unimodular matrices to achieve the Euclid’s division algorithm which leads to an extension of the well-known Routh–Hurwitz stability criteria but this time in terms of matrix coefficients. After that, stability of the closed-loop matrix polynomial is guaranteed based on a Corollary of a Lyapunov Theorem. The sufficient stability conditions are: (i) The matrices of one column of the presented array must be symmetric and positive definite and (ii) the matrices of the cascade realisation must satisfy a commutative condition. These stability conditions are also necessary for matrix polynomial of second order. The results are illustrated through examples.  相似文献   

17.
Sufficient conditions are developed for root clustering of a polynomial in a transformable region when each of the polynomial coefficients takes an arbitrary but fixed value within a specified closed interval. The conditions are in terms of the Kronecker products or the bialternate products. The sufficient conditions are then applied to different subregions in the complex plane. The work of Barmish (1984) on the invariance of the strict Hurwitz property for interval polynomials with perturbed coefficients is extended to root clustering in a region.  相似文献   

18.
New constructions of convex directions for Hurwitz stable polynomials are obtained. The technique is based on interpretations of the phase-derivative conditions in terms of the sensitivity of the root-locus associated with the even and odd parts of a polynomial  相似文献   

19.
Numerical algorithms for the canonical J-factorization of polynomial matrices with respect to the imaginary axis are given. The factorization problems for the non-regular polynomial matrices and for polynomials, whose determinants have zero roots, are considered. The constructed algorithms are not connected to the finding of roots of the polynomial. The basic calculation procedure is the construction of the stabilizing solution of the matrix algebraic Riccati equation.  相似文献   

20.
The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm.  相似文献   

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