Real and complex polynomial stability and stability domain construction via network realizability theory

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.     

A circular stability test for general polynomials

We extend a new stability test proposed recently for discrete system polynomials [1] to polynomials with complex coefficients. The method is based on a three-term recursion of a conjugate symmetric sequence of polynomials. The complex version has the same relative improved efficiency as the real version in comparison to the classical Schur-Cohn formulation for counting the number of zeros of a polynomial with respect to the unit circle. Furthermore, if desirable, the complex text can be carried out using only real polynomials and arithmetic.     

Two necessary conditions for a complex polynomial to be strictlyHurwitz and their applications in robust stability analysis

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     

Necessary and sufficient conditions for robust oscillatory stability

The problem of robust oscillatory stability of uncertain systems is investigated in this article. For the uncertain systems, whose characteristic polynomial sets belong to the interval polynomial family or diamond polynomial family, sufficient and necessary conditions are given based on the stability and/or oscillation properties of some special extreme point polynomials. A systematic approach exploiting Yang's complete discrmination system is proposed to check the robust oscillatory stability of such uncertain systems. The proposed method is efficient in computation and can be easily implemented.     

New stability conditions via reflection coefficients of polynomials

The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Starting from so-called barycentric simplex some necessary stability conditions in terms of unions of polytopes are obtained by splitting the unit hypercube of reflection coefficients. Sufficient stability conditions in terms of linear covers of reflection vectors of a family of stable polynomials improve the Cohn stability criterion.     

Robust stability of a class of polynomials with coefficientsdepending multilinearly on perturbations

Necessary and sufficient conditions are given for robust stability of a family of polynomials. Each polynomial is obtained by a multilinearity perturbation structure. Restrictions on the multilinearity are involved, but, in contrast to existing literature, these restrictions are derived from physical considerations stemming from analysis of a closed-loop interval feedback system. The main result indicates that all polynomials in the family of polynomials have their zeros in the strict left half-plane if and only if two requirements are satisfied at each frequency. The first requirement is the zero exclusion condition involving four Kharitonov rectangles. The second requirement is that a specially constructed &thetas;0-parameterized set of 16 intervals must cover the positive reals for each &thetas; ε[0,2π]     

Wronskian-based tests for stability of polynomial combinations

In this paper, a stability criterion based on counting the real roots of two specific polynomials is formulated. To establish this result, it is shown that a hyperbolicity condition and a strict positivity of a polynomial Wronskian are necessary and sufficient for the stability of any real polynomial. This result is extended to the stability study of some linear combinations of polynomials. Necessary and sufficient conditions of stability are obtained for polynomial segments and planes.     

On the stability of uncertain polynomials with dependentcoefficients

A sufficient condition is given for reducing the conservatism of the stability bounds for a family of polynomials with dependent coefficients, including nonlinear coefficients. It is also proved that if a finite family of stable polynomials has the same even part, then the polynomial with the even part and the odd part formed by adding any positive multiple of the even parts and odd parts, respectively, of the given family is also stable. Similar results holds if the given family of polynomials has the same odd part. A numerical example with nonlinear coefficients is given to illustrate the technique, and it is observed that the stability bounds obtained are larger than those acquired by Kharitonov's theorem     

Stability analysis of polynomials with coefficients in disks

The aim of this note is to report results on the stability of a class of polynomials from the small gain theorem point of view. The authors consider families of polynomials whose coefficients lie in closed circular disks around their nominal values. Various measures of variation of polynomial coefficients around their nominal value are considered and in each case necessary and sufficient conditions are presented for stability of the resulting family of polynomials. The stability region could be any closed region of the complex plane. Based on similar ideas of small gain, the authors also provide sufficient conditions for testing the stability of systems with commensurate time delays, and for two-dimensional type systems. These conditions become both necessary and sufficient in some special cases. All tests are easy to implement and require checking the stability of a matrix (or equivalently checking the stability of the central polynomial) and evaluation of a norm     

Discrete-sample curve fitting using chebyshev polynomials and the approximate determination of optimal trajectories via dynamic programming

Some useful properties of the Chebyshev polynomials are derived. By virtue of their discrete orthogonality, a truncated Chebyshev polynomials series is used to approximate a function whose discrete samples are the only available data. If minimization of the sum of the discrete squared error is used as the criterion, subject to some constraints on initial conditions and/or terminal conditions, the coefficients of the polynomials are easy to obtain. The simplicity of computing the coefficients of the polynomials from the discrete values of the function to be approximated is utilized to the approximate determination of optimal trajectories via dynamic programming using the technique of polynomial approximation. This allows use of the functional equation approach to solve multi-dimensional variational problems.     

Hurwitz stability of weighted diamond polynomials

In this note we consider real diamond polynomials of degree n whose roots are required to lie in the open left half plane. A minimal number of exposed edges whose stability guarantees the stability of the entire polynomial family is selected. A condition under which the stability of the vertices of the diamond guarantees the stability of the entire polynomial family is provided. Some illustrative examples are given.     

基于对称多项式理论及吴方法求解逆变器选择性消谐多项式

提出了一种利用对称多项式简化求解逆变器选择性消谐多项式的方法.基于余式理论求解逆变器选择性消谐多项式方程组,会出现当要求解多个开关角时,多项式方程的次数较高、计算工作量大的问题.为此,本文首先利用对称多项式理论降低该多项式方程组的次数,然后利用吴方法及置换法求解多项式方程组,结果表明,最后只需计算一些代数表达式就可得到选择性消谐多项式方程组的所有解,大大减少了计算量,提高了在线计算的速度.     

To stabilize an interval plant family it suffices to simultaneouslystabilize sixty-four polynomials

It is shown that stability of three specific polynomial families can be deduced from the stability of a finite number of polynomials. These polynomial families are the characteristic polynomials of unity feedback loops with the controller in the forward path, and where the plant includes a specific form of parameter uncertainty. For the first polynomial family, the plant has parameter uncertainty in the even or odd terms of the numerator or denominator polynomial. For the second polynomial family the plant has a numerator or denominator which is an interval polynomial. For the third polynomial family, the plant is interval. Because of the structure of these results it is shown that they lead to robust stabilization results. Two examples are included. The approach employed here was developed for plants with affine uncertainty. It is demonstrated that considerable simplification results if the plants under investigation are interval     

Robust stability of polynomials with multilinearly dependentcoefficient perturbations

Analysis of the robust stability of a polynomial with multilinearly dependent coefficient perturbations is presented in this note. Some sufficient conditions for forming a convex polygon with the value set of the polynomials with multilinearly dependent coefficient perturbations are obtained. A zero-exclusion algorithm is then given to determine the D-stability of such polynomials. The well known Kharitonov's theorem and the edge theorem for stability analysis can be included as special eases of the authors' conclusions     

Some results on robust stability of discrete-time systems

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     

Stabilisation of matrix polynomials

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.     

Sufficient conditions for Hurwitz polynomials and their application to the stability of a polytope of polynomials

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.     

Frequency domain conditions for the stability of perturbed polynomials

New necessary and sufficient conditions for the stability of perturbed polynomials of continuous systems are given in the frequency domain. The conditions are equivalent and in some respects more powerful than the well-known Kharitonov conditions. The new conditions allow considerable freedom in distributing the available uncertainty margin among the different coefficients of a polynomial and provide an indication as to whether the maximum allowable margin of uncertainty for a given polynomial has been reached.     

Solution of discrete scaled systems via general discrete orthogonal polynomials

General discrete orthogonal polynomials are introduced to analyse and approximate the solution of a class of discrete scaled systems. Using the general discrete shift transformation matrix, together with the general discrete scale matrix, the discrete scaled system can be reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of general discrete orthogonal polynomials can be determined simply by the derived algorithm. Examples are included to show the applicability of the general discrete orthogonal polynomial approximations.     

Interpolation type problems in the class of positive trigonometric polynomials of fixed order

This paper is devoted to a family of interpolation type problems for positive trigonometric polynomials of a given ordern. Via the Riesz-Fejér factorization theorem, they can be viewed as natural generalizations of the partial autocorrelation problem for discrete time signals of lengthn+1. The relevant variables for a specific problem are well-defined linear combinations of the coefficients of the underlying trigonometric polynomial. An efficient method is obtained to characterize the feasibility region of the problem, defined as the set of points having these variables as coordinates. It allows us to determine the boundary of that region by computing the extreme eigen values and the corresponding eigenvectors of certain well-defined Hermitian Toeplitz matrices of ordern+1. The method is an extension of one proposed by Steinhardt to solve the coefficient problem for positive cosine polynomials (which belongs to the family). Other interesting applications are the Nevanlinna-Pick interpolation problem for polynomial functions, and the simple interpolation problem for positive trigonometric polynomials. The close connection between the generalized Steinhardt method and classical techniques based on the polarity theorem for convex cones and on the Hahn-Banach extension theorem are established.