Damping margins of polynomials with perturbed coefficients

Considers the polynomial P(s)=t₀ Sⁿ+t₁ S^n-1 +···+t_n where 0<a _j⩽t_j⩽b_j. Recently, V.L. Kharitonov (1978) derived a necessary and sufficient condition for this polynomial to have only zeros in the open left-half plane. Two lemmas are derived to investigate the existence of theorems similar to the theorem of Kharitonov. Using these lemmas, the theorem of Kharitonov is generalized for P(s) to have only zeros within a sector in the complex plane. The aperiodic case is also considered     

On damping ratio of polynomials with perturbed coefficients

C.B. Soh and C.S. Berger (1988) derived a sufficient condition for a family of interval polynomials to have a damping ratio of φ using Kharitonov's theorem for complex polynomials. This paper points out that the transformations used by Soh and Berger to obtain the sufficient conditions also guarantee a simplification of Kharitonov's theorem for complex polynomials. That is, the number of required polynomials to be Hurwitz is half the number specified by Soh and Berger     

On the stability properties of polynomials with perturbed coefficients

Given a polynomialP_{c}(S) = S^{n} + t_{1}S^{n-1} + ... t_{n} = 0which is Hurwitz orP_{d}(Z) = Z^{n} + t_{1}Z^{n-1} + ... t_{n} = 0which has zeros only within or on the unit circle, it is of interest to know how much the coefficients tjcan be perturbed while preserving the stability properties. In this note, a method is presented for obtaining the largest hypersphere centered att^{T} = [t_{1} ... t_{n}]containing only polynomials which are stable.     

Invariance of the strict Hurwitz property for polynomials with perturbed coefficients

Given a strictly Hurwitz polynomialf(lambda) = lambda^{n} + a_{n-1} lambda^{n-1} + a_{n-2}lambda^{n-2}+...+ a_{1}lambda + a_{0}, it is of interest to know how much the coefficients aican be perturbed while simultaneously preserving the strict Hurwitz property. For systems withn leq 4, maximal intervals of the aiare given in a recent paper by Guiver and Bose [1]. In this note, a theorem of Kharitonov is exploited to obtain a general result for polynomials of any degree.     

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     

Lower bounds for polynomials with algebraic coefficients

We give a method, based on algebraic geometry, to show lower bounds for the complexity of polynomials with algebraic coefficients. Typical examples are polynomials with coefficients which are roots of unity, such as

$\text{Σ}\text{j=1}\text{d}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>i</mtext>}}\text{X}{}^{\mathrm{i}}$

and

$\text{Σ}\text{j=i}\text{d}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>pi</mtext>}}\text{X}{}^{\mathrm{j}}$

where p_j is the jth prime number.We apply the method also to systems of linear equations.     

Semi-numerical absolute factorization of polynomials with integer coefficients

In this paper, we propose a semi-numerical algorithm for computing absolute factorization of multivariate polynomials. It is based on some properties appearing after a generic change of coordinate. Using numerical computation, Galois group action and rational approximation, this method provides an efficient probabilistic algorithm for medium degrees. Two implementations are presented and compared to other algorithms.     

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     

On the stability of low-order perturbed polynomials

It is shown that for low-order perturbed continuous system polynomials (N⩽6), stability can be guaranteed by checking very simple conditions based on the Hermite-Biehler theorem. For N ⩽5 no numerical computation of the roots is required to check stability. For N=6, one root of a third-order polynomial needs to be found, with the rest of the conditions reducing to simple algorithmic relationships. The result is illustrated by numerical examples     

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.     

Stability of a family of polynomials with coefficients bounded in adiamond

The robust stability property is examined for family of nth-order real polynomials where the coefficients are bounded within a diamond in the (n+1)-dimensional space. It is shown that such a family of polynomials is Hurwitz if and only if four specially selected edge polynomials are Hurwitz     

‘Stability’ criterion for ARMA systems with perturbed coefficients†

A sufficient condition for certain internal stability of a perturbed parameter ARMA model is proved. It is argued heuristically that BIBO stability of the model follows under this condition, at least in some weaker probabilistic sense, so that the output of the perturbed model will not diverge away from that of its simplified averaged coefficient model asymptotically. This result is complementary to earlier work in which such approximations were studied.     

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 pole assignment via reflection coefficients of polynomials

A robust version of the output controller design for discrete-time systems is introduced. Instead of a single stable point a stable polytope (or simplex) is preselected in the closed-loop characteristic polynomial coefficients space. A constructive procedure for generating stable polytopes is given starting from the unit hypercube of reflection coefficients of monic polynomials. This procedure is quite straightforward because for a special family of polynomials the linear cover of so-called reflection vectors is stable. The roots placement of reflection vectors is studied. If a stable target simplex is preselected then the robust output controller design task is solved by quadratic programming approach.     

Invariance of a strict Hurwitz property for uncertain polynomials with dependent coefficients

This note is concerned with the invariance of strict Hurwitz property for uncertain polynomials having dependent coefficients. A feasible test criterion is provided which reduces the conservatism of the Kharitonov theorem.     

Sliding-mode boundary control for perturbed time fractional parabolic systems with spatially varying coefficients using backstepping

In this paper, we consider the boundary stabilization of an uncertain time fractional parabolic systems governed by time fractional parabolic partial differential equations (PDEs) with a boundary input disturbance and spatially varying coefficients (nonconstant coefficients) using a fractional-order sliding-mode controller. For this, the backstepping approach is used to transform an original system into a target system with a new manipulable input and perturbation. Then, the fractional-order sliding-mode algorithm is employed to design this new discontinuous boundary input to achieve the asymptotical stabilization of the target system (and, therefore, of the original system as well) by the fractional Lyapunov method. Apart from this, the well-posedness of the fractional parabolic system is analyzed theoretically. Fractional-order numerical simulations are provided to validate the developed technique.     

Frequency domain criteria for robust stability of perturbed systems with uncoupled variations in odd and even coefficients

Sets of (characteristic) polynomials with uncoupled variations in odd and even coefficients arise when systems experience parameter variations. This paper derives frequency domain criteria for robust stability of polynomial sets with uncoupled variations in odd and even coefficients. The frequency domain criteria only require one frequency domain plot to check the robustness of polynomial sets with uncoupled variations in odd and even coefficients. Furthermore, the largest stable polynomial set can be graphically estimated from the same frequency domain plot