A weak kharitonov theorem holds if and only if the stability region and its reciprocal are convex

We give necessary and sufficient conditions on the stability region for the validity of a weak version of Kharitonov's theorem, stating that stability of an ‘interval family’ of complex polynomials is implied by stability of the corner polynomials. Furthermore, we define a class of stability regions which do not satisfy the conditions, but for which the implication still holds in the case of real polynomials.     

A constructive approach for finding arbitrary roots of polynomials by neural networks

This paper proposes a constructive approach for finding arbitrary (real or complex) roots of arbitrary (real or complex) polynomials by multilayer perceptron network (MLPN) using constrained learning algorithm (CLA), which encodes the a priori information of constraint relations between root moments and coefficients of a polynomial into the usual BP algorithm (BPA). Moreover, the root moment method (RMM) is also simplified into a recursive version so that the computational complexity can be further decreased, which leads the roots of those higher order polynomials to be readily found. In addition, an adaptive learning parameter with the CLA is also proposed in this paper; an initial weight selection method is also given. Finally, several experimental results show that our proposed neural connectionism approaches, with respect to the nonneural ones, are more efficient and feasible in finding the arbitrary roots of arbitrary polynomials.     

Schur stability and stability domain construction

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.     

On the stability of a general polynomial in the space of Markov parameters

Markov parameters and the associated stability criterion were first introduced for continuous-time real polynomials. Recently, robust stability of such polynomials was considered in Markov parameters space, where efficient robust stability tests were obtained based on the Markov theorem. This has motivated the authors to extend the above idea to more general types of polynomials, and develop Markov parameters and the associated stability criterion for complex continuous-time as well as real and complex discrete-time polynomials. Moreover, for each polynomial type, we present compact relations in order to recover the coefficients of a polynomial corresponding to a given set of Markov parameters. The stability results presented here may be useful for 1-D and 2-D filtering and control applications.     

Damping margins of interval polynomials

The zero locations of interval polynomials are examined. In particular, it is shown that a family of interval polynomials will have zeros only in the left sector if the real and imaginary parts of four specially constructed complex polynomials have an interlacing real zero property. This is significant for the analysis of uncertain systems, as the computation cost associated with checking the zero locations of interval polynomials will be greatly reduced. The results presented can be readily extended to more general stability regions where the real and imaginary parts of the polar plot are polynomial functions     

Real algebraic numbers and polynomial systems of small degree

Based on precomputed Sturm–Habicht sequences, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. In particular, we express all isolating points as rational functions of the input polynomial coefficients. Although the roots are algebraic numbers and can be expressed by radicals, such representation involves some roots of complex numbers. This is inefficient, and hard to handle in applications in geometric computing and quantifier elimination. We also define rational isolating points between the roots of the quintic. We combine these results with a simple version of Rational Univariate Representation to isolate all common real roots of a bivariate system of rational polynomials of total degree ≤2 and to compute the multiplicity of these roots. We present our software within library synaps and perform experiments and comparisons with several public-domain implementations. Our package is 2–10 times faster than numerical methods and exact subdivision-based methods, including software with intrinsic filtering.     

A new tabular form for determining root distribution of a complexpolynomial with respect to the imaginary axis

The original Routh table dealing with real polynomials is further investigated for complex polynomials. A tabular form for determining root distribution of a complex polynomial with respect to the imaginary axis is developed, and modified procedures for directly treating singularities in the array are proposed. Also, procedures are developed for determining the respective orders of simple and/or repeated roots lying on the imaginary axis. Conditional stability and instability can be distinguished from each other by the criteria developed here     

Characterization of robust root loci of polytopes of polynomials

This paper deals with the robust root locus problem of a polytope of real polynomials. First, a simple and efficient algorithm is presented for testing if the value set of a polytopic family of polynomials includes the origin of the complex plane. This zero-inclusion test algorithm is then applied along with a pivoting procedure to construct the smallest set of regions in the complex plane which characterizes the robust root loci of a polytope of polynomials.     

Robust stability of polynomials with multilinear parameter dependence

The problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way. We show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials (obtained by setting parameters at their extreme values), while checking for unstable complex roots normally involves examining the real solutions of up to m + 1 simultaneous polynomial equations, where m is the number of parameters. When m = 2, this is an especially simple task.     

On multivariate interpolation by generalized polynomials on subsets of grids

This note may be regarded as a complement to a paper of H. Werner [17] who has carried over Newton's classical interpolation formula to Hermite interpolation by algebraic polynomials of several real variables on certain subsets of grids. Here generalized polynomials of several real or complex variables are treated. Recursive procedures are presented showing that interpolation by generalized multivariate polynomials is performed nearly as simply as interpolation by algebraic polynomials. Having in general the same approximation power, generalized polynomials may be better adapted to special situations. In particular, the results of this note can be used for constructing nonpolynomial finite elements since in that case the interpolation points usually are rather regular subsystems of grids. Though the frame is more general than in [17] some of our proofs are simpler. As an alternative method to evaluate multivariate generalized interpolation polynomials for rectangular grids a Neville-Aitken algorithm is presented.