Hurwitz Determinants Research Articles

Robust PID control of a tuned mass damper system and its stability analysis validation by Kharitonov and Edge theorems

ABSTRACT A robust control and stability analysis of a tuned mass damper attached to benchmark the model of mass-spring-damper system via different Kharitonov-related approaches, including the post-Kharitonov Edge Theorem, is presented in this paper. The control of uncertain parameters of the given system is demonstrated via robust Proportional-Integral-Derivative (PID) system tuning techniques in MATLAB. Usually, the Kharitonov theorem is sufficient to check the stability analysis of a controlled interval system. A detailed stability analysis of the given system by forming the Kharitonov rectangular parameter boxes and testing the positivity of Hurwitz determinants of the enclosed segments connecting opposite corners of the box is explained here. Stability checking is validated by the analysis based on the Hurwitz matrices of two Kharitonov polynomials. The multi-linear affine system model in the overall system plant is controlled and analysed via Edge theorem.

A Note on Lienard-Chipart Criteria and its Application to Epidemic Models

An important part of the study of epidemic models is the local stability analysis of the equilibrium points. The linear algebra method which is commonly employed is the well-known Routh-Hurwitz criteria. The criteria give necessary and sufficient conditions for all of the roots of the characteristic polynomial to be negative or have negative real parts. To date, there are no epidemic models in the literature which employ Lienard-Chipart criteria. This note recommends an alternative linear algebra method namely Lienard-Chipart criteria, to significantly simplify the local stability analysis of epidemic models. Although Routh-Hurwitz criteria are a correct method for local stability analysis, Lienard-Chipart criteria have advantages over Routh-Hurwitz criteria. Using Lienard-Chipart criteria, only about half of the Hurwitz determinants inequalities are required, with the remaining conditions of each set concern with only the sign of the alternate coefficients of the characteristic polynomial. The Lienard-Chipart criteria are especially useful for polynomials with symbolic coefficients, as the determinants are usually significantly more complicated than original coefficients as degree of the polynomial increases. Lienard-Chipart criteria and Routh-Hurwitz criteria have similar performance for systems of dimension five or less. Theoretically, for systems of dimension higher than five, verifying Lienard-Chipart criteria should be much easier than verifying Routh-Hurwitz criteria and the advantage of Lienard-Chipart criteria may become clear. Examples of local stability analysis using Lienard-Chipart criteria for two recently proposed models are demonstrated to show the advantages of simplified Lienard-Chipart criteria over Routh-Hurwitz criteria.

Adaptive emergency control system with an algorithm for rapid assessment of sustainability united energy system

In the present study described the methodology for determination of sustainable modes of long equivalent of the interconnected power systems of various structures with simplified representation of excitation controllers of generators. According to the results of numerous studies, the criterion of oscillatory disturbance of static stability based on the control of the partial derivative ∂AN/∂δ is formulated. To confirm the formulated criterion of violation of the static vibrational stability performed quantitative mapping of polynomial coefficients of haracteristics equations and additional applications of the matrix equations is redundant points of the equivalent circuit of the Electrical Power Systems (EPS) obtained under the criterion Hurwitz. Quantitative comparison and qualitative analysis of the nature of the change in the roots of the characteristic equation and the Hurwitz determinants confirm that the transition across the boundary of the stability region, found by the formulated criterion on the basis of the control of the partial derivative ∂AN/∂δ taking into account the proposed simplified model of generators at the constant excitation voltage Uf = const, is accompanied by the appearance of reciprocal values of the Hurwitz determinants. Taking into account the generalizations and conclusions obtained, a modified method of studying the static oscillatory stability of the EPS is proposed. The efficiency of the developed algorithm for determining the boundary of the region of potentially stable regimes is estimated. It is shown that the proposed method has the best performance indicators and can reduce the computational cost.

Oscillations and bistability in a model of ERK regulation.

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust-this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network-namely, the fully processive dual-site network-which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.

Implications of network topology on stability.

In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network’s “influence topology”, a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network’s steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network’s stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

Hurwitz rational functions

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establish an analog of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.

Non-Linear Analysis of Disc Brake-Induced Vibrations for Railway Vehicles

In this article, a two-degree-of-freedom model for the disc brake system is set up by considering the vertical motion of the brake unit and the pitch motion of the bogie frame, and the sliding friction interface between the brake lining and disc is included in the model. Using the Hurwitz determinants, an algebraic criterion for determining the Hopf bifurcation point of the system is proposed. The vehicle speed at the Hopf bifurcation point is defined as the critical speed where the equilibrium position of the disc brake system loses stability and the limit cycle emerges. In order to investigate the stability of the disc brake system, parametric studies are undertaken. It is shown that the mass of brake unit, vertical damping of primary suspension, friction coefficient, and brake normal force between the lining and disc have significant influences on the critical speed. The instability of the system may not occur if the parameters are properly designed. The limit cycle arisen due to the loss of stability of the equilibrium position is studied using the numerical integration method. It is found that once the brake system loses its stability at a certain speed, the stick—slip motion occurs and the limit cycle oscillation will last until the vehicle comes to a standstill. It is also known that the frequency of limit cycle is significantly affected by the mass and suspended stiffness of the brake unit but is not related to the vehicle speed in the braking process.

The Hurwitz determinants and the signatures of irreducible representations of simple real Lie algebras

Abstract The paper deals with the real classical Lie algebras and their finite dimensional irreducible representations. Signature formulae for Hermitian forms invariant relative to these representations are considered. It is possible to associate with the irreducible representation a Hurwitz matrix of special kind. So the calculation of the signatures is reduced to the calculation of Hurwitz determinants. Hence it is possible to use the Routh algorithm for the calculation.

Ullemar's formula for the jacobian of the complex moment mapping

The complex moment sequence μ ( P ) is assigned to a univalent polynomial P ( z ) by the Cauchy transform of the domain , where is the unit disk. We establish the representation of the Jacobian in terms of roots of the derivative . Combining this result with the special decomposition for the Hurwitz determinants, we prove a formula for , which was previously conjectured by Ullemar. As a consequence, we show that the boundary of the class of all locally univalent polynomials in is contained in the union of three irreducible algebraic surfaces.

Deciding Hopf Bifurcations by Quantifier Elimination in a Software-component Architecture

In this paper we give a semi-algebraic description of Hopf bifurcation fixed points for a given parameterized polynomial vector field. The description is carried out by use of the Hurwitz determinants, and produces a first-order formula which is transformed into a quantifier-free formula by the use of usual-quantifier elimination algorithms. We apply techniques from the theory of sub-resultant sequences and of Gröbner bases to come up with efficient reductions, which lead to quantifier elimination questions that can often be handled by existing quantifier elimination packages. We could implement the algorithms for the conditions on Hopf bifurcations by combining the computer algebra system Maple with packages for quantifier elimination using a Java-based component architecture recently developed by the second author. In addition to some textbook examples we applied our software system to an example discussed in a recent research paper.

A comment on ‘ Routh-Hurwitz. determinants and integral square errors ’

The recently published results of Allwright (1980) on-integral square errors and Hurwitz determinants are shown not to be new, but to. have a long history.

A note on Routh-Hurwitz determinants and integral square errors

Let p(s) be a polynomial, all of whose roots lie in the left half plane. The purpose of this note is to show that when the transfer function p(0)/p(s) tracks unit step input, the integral square error (ISE) of the response has a simple expression in terms of the Routh—Hurwitz determinants for p. This enables us to generalize a result of Hall : he pointed out that for a quadratic polynomial with fixed natural frequency, the minimum ISE is attained for a damping factor of ½. We find the corresponding result for polynomials of any degree.

An optimal fraction free Routh array†

The classical Routh's algorithm has the drawback that it involves divisions. Hence if one starts for example with integer or polynomial entries one ends up with rational numbers or rational functions respectively. Using Hurwitz determinants instead one can avoid this drawback in principle. However an efficient way to compute these determinants without introducing fractions has been missing. An algorithm to compute these determinants in a fraction free manner is presented. It is shown that this algorithm is optimal with respect to the growth of the entries.

Generalized theorems of Li�nard and Shepherd

The paper considers a real polynomial p(x)=α0+α1x+...+αnxn(α0 > 0) for which there hold inequalities Δ1>0, Δ3>0, ... or Δ2>0, Δ4>0, ..., where Δ1, Δ2, ..., Δjn are the Hurwitz determinants for polynomial p(x). It is proven that polynomial p(x) can have, in the right half-plane, only real roots, where the quantity of positive roots of polynomial p(x) equals the quantity of changes of sign in the system of coefficients a0, a2, ..., an−2, an, when n is even, and α0, a2,..., an−1, an, when n is odd. From the proven theorem, in particular, there follows the Lienard and Shepherd criterion of stability.

New reductions of Hurwitz determinants

Two lesser-known classical elimination techniques are applied in matrix form to the linear system stability problem of determining when the roots of a real nth degree polynomial all have negative real parts. In particular, for calculation of the critical Hurwitz determinant Δ n−1 a new tabular scheme is obtained which is superior to the Routh array for parametric stability problems. An extension leads to new half-order expressions for the other Δt Another reduction of Δ−1 in terms of minors of order about ¼n is also given.

A new formulation of the Liénard–Chipart stability criterion

AbstractThe Liénard–Chipart criterion for determining whether all the zeros of a real nth degree polynomial a(λ) have negative real parts involves calculation of only about half the Hurwitz determinants, the order of the largest being n − 1. It is shown, by using the companion matrix of a polynomial formed from a(λ), that the required determinants are equal to minors of a matrix whose order is ½ n or ½(n−1) according as n is even or odd. In either case this matrix is very easy to obtain A simple application of the result provides a criterion for a(λ) to be stable and aperiodic.

A New Formulation of the Theorems of Hurwitz, Routh and Sturm

It is shown that the Hurwitz determinants associated with a real polynomial of degree n can be obtained from minors of matrices having orders n/2 or (« —1)/2 according as n is even or odd. The method used is based on forming companion matrices of appropriate polynomials, and is extended to calculation of Routh arrays, Sturm sequences and to complex polynomials, thus providing a new formulation of a number of classical theorems.

Design curves for linear systems using Hurwitz determinants

By using Hurwitz determinants, boundary curves (α and ζ diagrams) are constructed according to specified stability constraints. From these diagrams, the dominant roots to be used for design purposes can be located.

A New Representation of Hurwitz's Determinants in the Expansion of Certain Ladder Filters

Given a real strictly Hurwitz polynomial H_{n}(s) = a_{0}\prod_{\upsilon = 1}^{n} (s - s_{\upsilon}), n = 3, 4, \cdots , the standard method of calculating the continued fraction expansion of \{[odd H_{n}(s)]/ [even H_{n}(s)]\}^{\pm 1} about its pole at infinity uses Routh's scheme or Hurwitz's determinants \Delta_{r}, r = 1, 2, \cdots , n , in the coefficients of H_{n}(s) (on the equivalence of the two, see [2]). In filter theory, cases are often encountered where knowledge of the zeros of H_{n}(s) precedes that of its coefficients, and one would then prefer to have formulas for the coefficients in the above continued fraction expansion directly in terms of the former rather than the latter. This is achieved by expressing \Delta_{r} as bialternants in the zeros of H_{n}(s) and reads \Delta_{r} = (-)^{r(r+1)/2} a_{0}^{r}A(0, 1, \cdots , n - r - 1, n - r + 1, \cdots , n + r - 1)/A(0, 1, \cdots , n - 1) , where the alternant in the denominator is the Vandermonde in s_{1}, s_{2}, \cdots , s_{n} , whereas the alternant in the numerator is obtained from it on replacing the exponents 0, 1, \cdots , n - 1 by 0, 1, \cdots, n - r - 1, n - r + 1, \cdots , n + r - 1 . Examples include H_{n}(s) = \prod_{\upsilon = 1}^{n} [s - j \exp (2 \upsilon - 1)j \pi /2n] and H_{n}(s) = (s + 1)^{n} .

A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov

ABSTRACTThe second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.