An elementary proof of Kharitonov's stability theorem withextensions

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     

On the boundedness of the set of stabilizing controllers

This paper shows that the set of rational, strictly proper, robustly stabilizing controllers for single‐input single‐output linear‐time invariant plants will form a bounded (can even be empty) set in the controller parameter space if and only if the order of the stabilizing controller cannot be reduced any further; if the set of proper stabilizing controllers of order r is not empty and the set of strictly proper controllers of order r is bounded, then r is the minimal order of stabilization. The paper also extends this result to characterize the set of controllers that guarantee some pre‐specified performance specifications. In particular, it is shown here that the minimal order of a controller that guarantees specified performance is l iff (1) there is a controller of order l guaranteeing the specified performance and (2) the set of strictly proper, robustly stabilizing controllers of order l and guaranteeing the performance is bounded. Moreover, if the order of the controller is increased, the set of higher‐order controllers which satisfies the specified performance will necessarily be unbounded. This characterization is provided for performance specifications, such as gain margin and robust stability, which require a one‐parameter family of real polynomials to be Hurwitz, where the parameter is in a closed interval. Other performance specifications, such as phase margin and ℋ︁_∞ norm, can be reduced to the problem of determining a set of stabilizing controllers that renders a family of complex polynomials Hurwitz. The characterization of the set of controllers for the stabilization of complex polynomials is provided and is used to show the boundedness properties for the set of controllers that guarantee a given phase margin or an upper bound on the ℋ︁_∞ norm. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability of polynomials with conic uncertainty

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.     

Properties of the entire set of Hurwitz polynomials and stabilityanalysis of polynomial families

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     

Necessary and sufficient condition for the existence of Kharitonov-like theorems

Kharitonov has shown that a family of interval polynomials is Hurwitz if and only if the vertex polynomials are Hurwitz. A necessary and sufficient condition to check the existence of Kharitonov-like theorems for other regions of interest in the complex plane is presented.     

On the complete instability of interval polynomials

In this paper, we study “complete instability” of interval polynomials, which is the counterpart of classical robust stability. That is, the objective is to check if all polynomials in the family are unstable. If not, a subsequent goal is to find a stable polynomial. To this end, we first propose a randomized algorithm which is based on a (recursive) necessary condition for Hurwitz stability. The second contribution of this paper is to provide a probability-one estimate of the volume of stable polynomials. These results are based on a combination of deterministic and randomized methods. Finally, we present two numerical examples and simulations showing the efficiency of the proposed methodology for small and medium-size problems.     

Hurwitz testing sets for parallel polytopes of polynomials

In considering robustness of linear systems with uncertain paramenters, one is lead to consider simultaneous stability of families of polynomials. Efficient Hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis. This paper gives a stability criterion for parallel polytopes in terms of Hurwitz stability of a number of corners and edges. The ‘testing set’ of edges and corners depends entirely on the edge directions of the polytope, hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions.It follows as a consequence, that Kharitonov's four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials, when the stability region is a sector Ω = { e^iv | > 0, rπ/q < | v | ≤ π } and r/q is a rational number.     

A simple procedure for the exact stability robustness computation of polynomials with affine coefficient perturbations

This paper considers the problem of the stability robustness computation of polynomials with coefficients which are affine functions of the parameter perturbations. A polynomial is said to be stable if its roots are contained in an arbitrarily pre-specified open set in the complex plane, and its stability robustness is then measured by the norm of the smallest parameter perturbation which destabilizes the polynomial. A simple and numerically effective procedure, which is based on the Hahn-Banach theorem of convex analysis and which is applicable for any arbitrary norm, is obtained to compute the stability robustness. The computation is then further simplified for the case when the norm used is the Hölder ∞-norm, 2-norm or 1-norm.     

Inverse Routh table construction and stability of delay equations

Based on an inversion of the Routh table construction, a unimodular characterization of all Hurwitz polynomials is obtained. In parameter space, the Hurwitz polynomials of degree n correspond to the positive 2ⁿ-tant. The method is then used to construct classes of stable continuous-time delay-difference equations and delay differential equations of neutral type, by a suitable limiting process.     

Sufficient algebraic conditions for stability of cones of polynomials

In this paper a sufficient condition for a cone of polynomials to be Hurwitz is established. Such condition is a matrix inequality, which gives a simple algebraic test for the stability of rays of polynomials. As an application to stable open-loop systems, a cone of gains c such that the function u=−kc^Tx is a stabilizing control feedback for all k>0 is shown to exist.     

A sufficient condition for the stability of interval matrixpolynomials

The root location of sets of scalar polynomials whose coefficients are confined to intervals and the associated extension to eigenvalues of sets of constant matrices whose coefficients are contained in intervals are reviewed. A central result for complex scalar interval polynomials is a theorem developed by V.L. Kharatonov (1978), which states that each member of a set of such polynomials is stable if and only if eight special polynomials from the set are stable. The case of interval matrix polynomials is examined, and a Kharitonov-like result for their strong stability is provided. This in turn yields a sufficient condition for stability of a set of interval matrix polynomials     

On the spectrum of convex sets of matrices

Let K be an arbitrary compact convex set of square matrices. This paper presents necessary and sufficient conditions for the spectrum of K to have no eigenvalues in a prespecified closed convex subset of the complex plane. The obtained result implies different criteria for analysis of the spectral set of K. In particular, we have formulated criteria for nonsingularity, inertia and Hurwitz stability of K which can be used in the robustness analysis of linear control systems with uncertain parameters     

A Kharitonov‐like theorem for robust stability independent of delay of interval quasipolynomials

In this paper, a Kharitonov‐like theorem is proved for testing robust stability independent of delay of interval quasipolynomials, p(s)+∑eq_k(s), where p and q_k's are interval polynomials with uncertain coefficients. It is shown that the robust stability test of the quasipolynomial basically reduces to the stability test of a set of Kharitonov‐like vertex quasipolynomials, where stability is interpreted as stability independent of delay. As discovered in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), the well‐known vertex‐type robust stability result reported in (IMA J. Math. Contr. Info. 1988; 5 :117–123) (See also (IEEE Trans. Circ. Syst. 1990; 37 (7):969–972; Proc. 34th IEEE Conf. Decision Contr., New Orleans, LA, December 1995; 392–394) does contain a flaw. An alternative approach is proposed in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), and both frequency sweeping and vertex type robust stability tests are developed for quasipolynomials with polytopic coefficient uncertainties. Under a specific assumption, it is shown in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov‐like vertex quasipolynomials. In this paper, we show that the assumption made in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is redundant, and the Kharitonov‐like result reported in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is true without any additional assumption, and can be applied to all quasipolynomials. The key idea used in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) was the equivalence of Hurwitz stability and ?_‐o‐stability for interval polynomials with constant term never equal to zero. This simple observation implies that the well‐known Kharitonov theorem for Hurwitz stability can be applied for ?_‐o‐stability, provided that the constant term of the interval polynomial never vanishes. However, this line of approach is based on a specific assumption, which we call the CNF‐assumption. In this paper, we follow a different approach: First, robust ?_‐o‐stability problem is studied in a more general framework, including the cases where degree drop is allowed, and the constant term as well as other higher‐orders terms can vanish. Then, generalized Kharitonov‐like theorems are proved for ?_‐o‐stability, and inspired by the techniques used in (IEEE Trans. Autom. Control 2008; 53 :1219–1234), it is shown that robust stability independent of delay of an interval quasipolynomial can be reduced to stability independent of delay of a set of Kharitonov‐like vertex quasipolynomials, even if the assumption adopted in (IEEE Trans. Autom. Control 2008; 53 :1219–1234) is not satisfied. Copyright © 2009 John Wiley & Sons, Ltd.     

A generalisation of Jury’s conjecture to arbitrary dimensions and its proof

In 1986 E. I. Jury conjectured by analogy to the theory of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function is of the form H = 1/P, where P is a very strict Hurwitz polynomial (VSHP). In this article we prove a generalisation of Jury’s conjecture to r-dimensional analog filters (r ≥ 2) with proper transfer function H = Q/P, where the denominator P is a robustly stable polynomial, i.e., a strict Hurwitz polynomial which retains this property under small variations of its coefficients. In the bivariate case these polynomials are the VSHPs. Financial support by the Austrian FWF via project 18974 is appreciated.     

Robust Schur-stability of control systems with interval plants

The main contribution of this paper is a generalization of the Box Theorem, which was originally introduced for the Hurwitz case, for determining the robust Schur-stability of linear time-invariant discrete-time control systems containing an interval plant. This generalization provides necessary and sufficient conditions for the stability of a family of closed-loop characteristic polynomials <(z) = Q₁,(z)P₁(z) +... + Q_m(z)P_m(z), where the Q_is are fixed (controller) and then P_is are interval (plant) polynomials whose coefficients vary within a prescribed interval. This method requires checking a set of prescribed segment polynomials whose number is considerably less than that of the edges required by the edge theorem. A summary of the robust Schur-stability of discrete-time interval polynomials is presented.     

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     

On stability of quasipolynomials with annular uncertainties and interval commensurate delays

The paper concerns the Hurwitz stability of a family of quasipolynomials with commensurate delays. Each coefficient of the quasipolynomials belongs to a prescribed annulus in the complex plane, and the delay belongs to a prescribed real interval. A computationally tractable robust stability criterion is the main result of the paper.     

Multidimensional BIBO stability and Jury’s conjecture

Twenty years ago E. I. Jury conjectured by analogy to the case of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function has the form H = 1/P where P is a very strict Hurwitz polynomial (VSHP). In more detail he conjectured that the impulse response of the filter is an absolutely integrable function. However, he did not specify the exact equations of these filters and did not prove the existence of the impulse response. In the present paper we generalise Jury’s conjecture to arbitrary proper transfer functions H = Q/P where P is a bivariate VSHP and prove this generalisation. In particular, we show the existence of a suitable impulse response or fundamental solution for any multivariate proper rational function. However, this impulse response is a measure and not a function. We have not succeeded to prove an analogue of Jury’s conjecture in higher dimensions than two yet, but we propose a new conjecture in context with the robustly stable multivariate polynomials investigated by Kharitonov et. al. For the discrete case we prove that the structurally stable rational functions after Bose, Lin et al. coincide with the stable rational functions discussed in context with the stabilisation of discrete input/output systems. These rational functions are BIBO stable, but the converse is not true as established by several authors. Financial support of M. Scheicher through the Austrian FWF-project P18974 is gratefully acknowledged.     

Computation of the stability radius of a Hurwitz polynomial with diamond-like uncertainties

This paper considers the stability radius problem of Hurwitz polynomials whose coefficients have Hölder 1-norm-bounded uncertainties. We show that the solution to this problem demands the computation of the minimum of a piece-wise real-rational function ρ(λ), called the stability radius function. It is then shown that the calculations of ρ(λ) at the intersection points where ρ(λ) changes its representation and at the stationary points where ρ′(λ)=0 can be reduced to two sets of eigenvalue problems of matrices of the form H_β⁻¹H_γ, where both H_β and H_γ are frequency-independent Hurwitz matrices. Using root locus technique, we analyze this function further and prove that, in some special cases, the minimum of this function can be achieved only at the intersection points. Extensions of the eigenvalue approach to cover other robust stability problems are also discussed.     

Robust D-decomposition under l p-bounded parametric uncertainties

Consideration was given to stability of an affine family of uncertain polynomials defined by two real or one complex parameter, the rest of the parameters characterizing indeterminacy. On the plane of family parameters, a domain was established where the uncertain polynomials are stable. The method of robust D-decomposition was used. For the cases where the uncertain parameters are real and bounded in the Euclidean norm or are complex and bounded in the l _p norm, expressions for the boundary of these domains were obtained.