Edge Theorem Research Articles

Domain Stability Test of Polytopes of 2-D Polynomials

The necessary and sufficient conditions of domain stability of polytopes of 2-D polynomials have been established. The conditions are based on edge theorem and convex directions of 2-D polynomials. The number of edges to be tested for the domain stability of a polytope of 2-D polynomials can be reduced by a testing set constructed by us. We introduce how to construct the testing set of polytopes of 2-D polynomials by using the phase property of 2-D polynomials.

Stability of Multivariate Polynomial Conic Sets

We present necessary and sufficient conditions for a conic set of bivariate polynomials p0+ K to be stable (where K is a convex cone). We derive from edge theorem for semistability, the main coefficients implication are obtained from the analysis of p(s1, s2).

The monadic second-order logic of graphs XIII: Graph drawings with edge crossings

We introduce finite relational structures called sketches, that represent edge crossings in drawings of finite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges that may cross one another and that may cross the edges of the frame. We prove the 3- Edge Theorem stating that a structure of appropriate type with a frame is a sketch if and only if every induced substructure representing the frame and at most 3 edges not in the frame is a sketch.

On Stability and Robust Stability of Multivariate Polynomials

The aim is to summarize recent results on stability and robust stability of multivariate polynomials. The largets class of polynomials preserving stability when thay are subjected to small coefficient variations is recalled. Then adaptation of the zero exlcusion principle for this class is introduced. Applications of this theory are shown for the edge theorem, stability of interval families and complex structured radius, all of them for the so-called class of stable multivariate polynomials.

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.

Stability analysis of Hopfield-type neural networks

The paper applies several concepts in robust control research such as linear matrix inequalities, edge theorem, parameter-dependent Lyapunov function, and Popov criteria to investigate the stability property of Hopfield-type neural networks. The existence and uniqueness of an equilibrium is formulated as a matrix determinant problem. An induction scheme is used to find the equilibrium. To verify whether the determinant is nozero for a class of matrix, a numerical range test is proposed. Several robust control techniques in particular linear matrix inequalities are used to characterize the local stability of the neural networks around the equilibrium. The global stability of the Hopfield neural networks is then addressed using a parameter-dependent Lyapunov function technique. All these results are shown to generalize existing results in verifying the existence/uniqueness of the equilibrium and local/global stability of Hopfield-type neural networks.

The uniform distribution: A rigorous justification for its use in robustness analysis

Consider a control system with admissible uncertain parameter values which exceed the bounds specified by classical robustness theory. The article concerns trade-offs between performance degradation risk and uncertainty tolerance. If a large increase in the uncertainty bound can be established, an acceptably small risk may be justified. Since robustness formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances for any distribution. The paper concentrates on problems associated with Kharitonov's theorem and the edge theorem. If p(s, q) denote the uncertain polynomial under consideration and /spl Pscr/(/spl omega/) a frequency dependent convex target set for the uncertain values p(j/spl omega/, q), /spl Pscr/(/spl omega/) symmetric with respect to the nominal p(j/spl omega/, 0), the uncertain parameters q/sub i/ zero-mean independent random variables with known support interval, and for each uncertainty /spl Fscr/ consists of symmetric nonincreasing density functions, then, for fixed frequency /spl omega/, the first theorem indicates that the probability that p(j/spl omega/, q) is in /spl Pscr/(/spl omega/) is minimized by the uniform distribution for q. The second theorem, a generalization of the first, indicates that the same result holds uniformly with respect to frequency. Then, probabilistic guarantees for robust stability are given in the third theorem. In many cases, one can far exceed classical robustness margins while keeping the risk of instability small.

A New Algorithm for Testing the Stability of a Polytope: A Geometric Approach for Simplification

Considered in this paper is the robust stability of a class of systems in which a relevant characteristic equation is a family of polynomials F: f(s, q) = a0(q) + a1(q)s + … + an(q)sn with its coefficients ai(q) depending linearly on q unknown-but-bounded parameters, q = (p1, p2, …, pq)T. It is known that a necessary and sufficient condition for determining the stability of such a family of polynomials is that polynomials at all the exposed edges of the polytope of F in the coefficient space be stable (the edge theorem of Bartlett et al., 1988). The geometric structure of such a family of polynomials is investigated and an approach is given, by which the number of edges of the polytope that need to be checked for stability can be reduced considerably. An example is included to illustrate the benefit of this geometric interpretation.

Robust stability of convex and compact sets of complex polynomials

A necessary and sufficient condition for a convex and compact set of complex polynomials to be strictly Hurwitz is given. The result implies and generalizes several results on strict Hurwitz property of polynomials. In particular, our result covers the “edge theorem” for polytopes of strictly Hurwitz polynomials and requires weaker conditions than the “edge theorem”. It is also shown that previously established results on the robust strict Hurwitzness of diamond polynomials can be implied by our result. Finally, applying the result, we derive a new result on the robust positivity of a complex diamond rational function.

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 setp0+K of polynomials to be Hurwitz stable (K is a convex cone of polynomials of degree ≤n and degp0=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.

A class of multilinear uncertain polynomials to which the edge theorem is applicable

This paper gives a class of multilinear uncertain polynomials to which the edge theorem is applicable. In the paper, a notion of uncertainty is introduced. The main result is as follows: if a multilinear uncertain polynomial has a symmetric uncertainty structure and the uncertain parameters are nonoverlapping, then the corresponding value sets will be convex polygons. This means that the use of the zero exclusion condition to test the robust stability of such uncertain polynomials is feasible since we need only to calculate the extreme points of the convex polygons. Furthermore, a version of the edge theorem can be used to test the robust stability of such uncertain polynomials.

Robust stability of neutral delay differential systems

In this paper we consider the robust stability problem for a neutral delay-differential system where the characteristic equations involve a polytope of quasi-polynomials. First, the ‘Edge theorem’ is developed for robust stability analysis of a polytope of neutral quasi-polynomials. Based on the Edge theorem, the robust stability of neutral systems with commensurate delays will be reduced to the checking of the H (Hurwitz)-stability of all the edges of a polytope of quasi-polynomials and the Schur stability of all the edges of the subpolytope of neutral terms. Then, an effective graphical test is made for checking the robust stability of a polytope of neutral quasi-polynomials. Finally, we show that checking vertic quasi-polynomials is sufficient for the asymptotic stability of the entire family of first-order quasi-polynomials of the neutral type.

Robust stability of polynomials with multilinearly dependent coefficient 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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust stability of time-delay systems

There are two fundamental results available when we study stability of a polynomial family that is described by convex polytope in the coefficient space: the edge theorem and the theory based on the concept of convex directions. Many known results can be explained with these two results. This paper deals with a generalization of this line of research to the case of quasipolynomials that are entire functions which include both degree of the independent variable and exponential functions. The main objects of the paper are the developing of the concept of convex directions for quasipolynomials and exploiting this concept for construction of testing sets for quasipolynomial families. One of the primary sources of motivation for the class of problems considered in this paper is derived from process control. A typical problem formulation almost always includes a delay element in each subsystem process block. When we interconnect a number of such blocks in a feedback system, the study of robust stability involves quasipolynomials of the sort considered in this paper.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Properties of the entire set of Hurwitz polynomials and stability analysis 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. >

Boundary Theorem and Edge Theorem of Complex Polynomial Families

The value mapping, boundary theorem and parameterization as a new idea has afforded approaches and theory for the robust stability analysis of real polynomial families. In this note the problems of robust stability analysis of complex polynomial families, which is of significance in several engineering applications, are discussed via the above approaches. It will be shown that value mapping, boundary theorem and edge theorem can be successfully applied to the robust stability analysis of complex polynomial families.

Economic Impact of the Structural Changes of Production Technology: A Robust Approach

In this article, edge theorem and boundary theorem, one of the most important results obtained recently in the field of robust control theory, is applied to analyze the economic impact of structural change of production technology in one economic sector, by means of the famous dynamic Leontief input–output model. Two theorems, parallel to edge theorem and boundary theorem, are given in terms of economics in this paper. It is argued that in a sense the complex analysis about the economic impact of production technological changes is equivalent to a one-dimensional computation problem, which is easy to solve. A case study is given in the last section of this paper.

A survey of extreme point results for robustness of control systems

This paper surveys a subset of the body of research which was sparked by Kharitonov's Theorem. The focal point is extreme point results for robust stability and robust performance. That is, we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family. The results which are surveyed apply mainly to systems with structured real parametric uncertainty. In addition, a number of counterexamples are given to illustrate cases for which an extreme point result does not hold. For such cases, a solution via the so-called Edge Theorem is often possible.

Characterization of zero locations of polytopes of real polynomials

Given a stability region of a polytope of polynomials, the edge theorem can be used to verify certain root-clustering properties of the polytope. The authors consider the inverse problem; starting with a polytope of real polynomials, they present a result that allows the construction of the smallest set of regions in the complex plane which characterizes its zero location.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The edge theorem and graphical tests for robust stability of neutral time-delay systems

We consider the robust stability problem for a class of uncertain neutral time-delay systems where the characteristic equations involve a polytope p of quasipolynomials of neutral type. Given a stability region D in the complex plane our goal is to find a constructive technique to verify the D-stability of p (i.e. to verify whether the roots of every quasipolynomial in p all belong to D). We first show that, under a certain assumption on the stability region D, p is D-stable if and only if the edges of p are D-stable. Hence, the D-stability problem of a higher dimensional polytope is reduced to the D-stability problem of a finite number of pairwise convex combinations of vertices. Based on this result, we then give an effective graphical test for checking the D-stability of a polytope of quasipolynomials of neutral type.