Routh-Hurwitz Conditions Research Articles

On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems

Some Routh–Hurwitz stability conditions are generalized to the fractional order case. The results agree with those obtained numerically for Lorenz, Rössler, Chua and Chen fractional order equations. The case of coupled map lattice is briefly discussed.

Collection of master–slave synchronized chaotic systems

In this work the open-plus-closed-loop (OPCL) method of synchronization is used in order to synchronize the systems from the Sprott's collection of the simplest chaotic systems. The method is general and we were looking for the simplest coupling between master and slave system. The interval of parameters were synchronization is achieved are obtained analytically using Routh-Hurwitz conditions. Detailed calculations and numerical simulation are given for the system I from the Sprott's collection. Working in the same manner for non-linear systems based on ordinary differential equations the method can be adopted for the teaching of the topic.

Characterizations and stable tests for the Routh–Hurwitz conditions and for total positivity

Given a polynomial of degree n, a test of O(n 2) elementary operations and growth factor 1 is presented in order to check the Routh–Hurwitz conditions. This optimal growth factor guarantees that the test presents better stability properties than other known tests. We also present a test of O(n 3) elementary operations and growth factor 1 in order to check if a matrix is strictly totally positive. Finally, totally positive matrices are characterized by their symmetric-triangular decompositions.

STABILITY SWITCHES OF TIME-DELAYED DYNAMIC SYSTEMS WITH UNKNOWN PARAMETERS

The paper presents a systematic method of stability analysis for high-dimensional dynamic systems involving a time delay and some unknown parameters. Here, the term “unknown” means that the parameters are constants but yet to be determined. The analysis focuses on the stability switches of those systems with increase of the time delay from zero to infinity. On the basis of the generalized Sturm criterion, the parameter space of concern is divided into several regions determined by a discrimination sequence and the Routh–Hurwitz conditions. It is found, as the time delay increases, that the system may undergo no stability switch, exactly one stability switch, or more than one stability switches when the parameters are chosen from different regions. To demonstrate the approach, a detailed analysis of the stability switches is made in the paper for an active vehicle suspension equipped with a delayed “sky-hook” damper and a four-wheel steering vehicle with time delay in driver's response, respectively.

Finance in a Classical and Harrodian Cyclical Growth Model

This paper is an extension of an earlier working paper (Finance and the Macroeconomic Process in a Classical Growth and Cycles Model, Levy Institute Working Paper No. 253). The basic structure of the model remains unchanged in that it is based on a social accounting matrix (SAM) with endogenous money. Investment in circulating capital adds to output and investment in fixed capital adds to potential output. Driving the model's fast adjustment process, which describes the disequilibrium adjustment between aggregate demand and supply, is the dual disequilibria relationship in which the excess of monetary injections over desired money holdings fuels spending in the markets for goods and services. This excess also spills over into the bond market and lowers the interest rate. The model's slow adjustment process entails adjustments in fixed investment so that actual and normal (desired) capacity utilization fluctuate around each other. Over the long run investment is internally financed and regulated by the rate of profit. The current paper has three innovations. First, inventory investment is treated explicitly. Second, the SAM itself has been split into a current and capital account, thereby making it easier to derive the balance sheet counterpart of the flow matrix. Third, the paper discusses the stability properties of the 4 x 4 nonlinear differential equation system that describes the fast adjustment process. The key to stability is the negative feedback effect of business debt on investment. In the 4 x 4 case, a necessary condition for stability is that the reaction coefficient h2 on the debt term in the circulating investment equation be positive; a necessary and sufficient condition is that h2 ?h2* where h2* is some critical value. In crossing this critical value, the system undergoes a Hopf bifurcation. Finally, if the model is reduced to a 3 x 3 system by considering a budget deficit that is wholly bond financed, then necessary and sufficient conditions for stability can be derived using the modified Routh-Hurwitz conditions. These stability conditions, in this case, imply that h2 > 0.

Asymptotic 2p-moment stability of stochastic linear systems

The exponential p-moment stability of dynamical systems governed by a system of linear Ito stochastic differential equations is revisited. It is well-known that the system of equations governing the evolution of these p-moments is linear and, therefore, available results for asymptotic stability of the linear systems of deterministic first-order homogeneous differential equations are applicable [2,4,7,10,11]. Specifically, the necessary and sufficient conditions for asymptotic stability of a system of deterministic linear equations is that the real parts of all the eigenvalues of the system matrix are negative. The search for stability boundaries involves repeated solutions of eigenvalue problems of dimension equal to the dimension of the system of moments. Alternatively, the well-known Routh-Hurwitz procedure provides conditions for stability in the form of inequalities which involve the system and parametric excitation characteristics. However, these conditions are quite cumbersome since they involve computation of a large number of determinants of orders up to the order of the system of moments. Therefore, Routh-Hurwitz conditions are practical for obtaining stability boundaries only for low order systems. For the special case of an n-th order linear Ito stochastic differential equation, Khasminskii [9] has derived simplified conditions for exponential mean-square (p = 2) stability which mainly depend on the conditions of stability of the first moments, supplemented by only an extra condition involving the evaluation of an n-th order determinant. In this study, new simplified 2p-moment stability conditions are developed which provide significant advantages for the analytical and numerical estimation of the 2p-moment stability border and stability regions. Specifically, it is shown that there exists a real eigenvalue of the system of 2p-moments which is an upper bound of the real parts of all other eigenvalues. Thus, the stability of the system of moments can be examined by computing only the maximum real eigenvalue of the state matrix describing the evolution of the 2p moments. In particular, at the border of stability, one eigenvalue of the system of 2p-moments is equal to zero. Thus, a necessary condition for the system configuration to correspond to a point on the 2p-moment stability boundary is that the determinant of the matrix describing the system of 2p-moments be zero. This condition is a generalization of the mean-square stability boundary condition obtained in [8]. It provides a single algebraic expression for computing all candidate stability boundaries. It is also shown in this study that all candidate 2p-moment stability boundaries can be obtained by computing the real eigenvalues of a matrix of dimension equal to the dimension describing the system of 2p-moments.

A Criterion for Stability of Matrices

A necessary and sufficient condition for the stability ofn×nmatrices with real entries is proved. Applications to asymptotic stability of equilibria for vector fields are considered. The results offer an alternative to the well-known Routh–Hurwitz conditions.

Evaluation of operating procedures based on stationary-state stability

The safe and reliable operation of chemical plants depends on high-integrity operating procedures. Such procedures often involve process transition through stationary states. The stability of these states is an important part of the integrity evaluation. The methodology presented qualitatively identifies families of inherently stable chemical operations and establishes design rules for the synthesis of stable stationary states. These rules have been developed using the Routh-Hurwitz conditions for three qualitative levels of dynamic system models. These levels represent the gains of the dynamic system by the signs of the gains (+,0,-), the signs and their equality(+,0,-,=), and finally the gains represented by their order of magnitude values (+,0,-,=, ,>>). Example design rules have been developed during the synthesis of operating procedures for a series of vessels with composition control and temperature control of an exothermic chemical reaction. The most robust design rules apply to systems that can be proven stable by the sign level of gain representation. Less robust rules with a wider range of applicability to dynamic systems have been developed using the equality and order of magnitude representations

Use of the parametric representation method in revealing the root structure and hopf bifurcation

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =uo +u1x +g(x), whereuo andu1 are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the “stable region” on the parameter plane, where the real parts of the roots off are negative. The D-curve (along which the discriminant off is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.

Stability analysis of a model of infectious diseases with two transmission routes

Programs of symbolic manipulation are used to analyze stability of equilibria of a model of infectious diseases with two transmission routes. The condition of the transition from disease-persistent state to disease-free state is obtained. The listed MACSYMA programs can also be used to verify the Routh-Hurwitz conditions for the stability analysis of other systems.

On estimating the attraction regions of the equilibrium states of dynamic systems by the direct lyapunov method

The method of non-local reduction /1, 2/ is proposed for estimating the attraction regions of asymptotically stable equilibriums states of dynamic systems linked to the construction of Lyapunov functions in which the thoroughly investigated trajectories of two-dimensional systems are contained. The class of functions described below widens the class of functions obtained on the basis of Lur'e-Postnikov functions /3, 4/. That region entirely belongs to the attraction region of the system equilibrium state and is wider than the estimate of the attraction region estimated by the Lur'e-Postnikov function basis. The attraction region may exceed the limits of the interval in which the non-linearity satisfies the generalized Routh-Hurwitz conditions /5/. The algorithm proposed below contains the logarithms for specific conditions based on Lyapunov functions of the Lur'e type and the special condition of V.M. Popov.

A “Simplest Possible” Property of the Generalized Routh–Hurwitz Conditions

To decide whether a prescribed complex polynomial has all its zeros with negative real parts, there are available many tests involving the checking of rational or polynomial inequalities in the coefficients. It is shown that the generalized Routh–Hurwitz conditions are from a certain point of view not replaceable by simpler conditions.

Development of lyapunov functions for freely spinning dual-spin spacecraft

Lyapunov functions for linear multivariable time-invariant systems are developed by use of construction techniques which are based on Lyapunov's second method. Two methods of construction are studied. In the first method (direct method) the Lyapunov function is obtained by directly solving the Lyapunov matrix equation while in the second method (indirect method) it is generated by reducing the system matrix by two successive similarity transformations to the Schwarz canonical form. The conditions of stability on the resulting matrix from Lyapunov's second method are the same as the Routh-Hurwitz conditions. Both construction techniques are applied to a freely-spinning dual-spin system, with viscous rate damping assumed about both principal transverse axes. It is seen that when the rotor is spinning relative to the main part, damping about only one of the principal transverse axes is required for stability. The Lyapunov functions constructed are quadratic forms whose different coefficients, collectively, involve all of the system parameters including the damping coefficients.

On the boundedness of solutions of some non-autonomous differential equations of the fourth-order†

The boundedness of solutions of a fourth-order, non-linear, non-autonomous differential equation is investigated. The result is a further extension of Sinha and Hoft (1971). The Routh-Hurwitz conditions are satisfied for the linear autonomous system.

New algebraic criteria for positive realness

We present new algebraic criteria for positive realness of real rational functions and matrices, which are formulated entirely in terms of the Routh-Hurwitz conditions. It is first shown how the Routh algorithm can be modified to serve as a criterion for positive realness. Due to the outstanding algebraic simplicity of the Routh algorithm, this criterion yields an efficient numerical procedure for testing positive realness. Once the Routh algorithm is successfully reformulated, the Hurwitz version of positive realness is almost automatic. Hurwitz-like determinants are obtained, which provide explicit conditions for positive realness often desired in the theory of networks and systems. The matrix generalization of the proposed Routh-Hurwitz criteria lead to numerical procedures for testing the positive real character of real rational matrices with desirable simplicity and suitability for machine calculations.

On stability of a fifth-order non-linear differential equation

The stability of an isolated equilibrium or critical point of a fifth-order nonlinear differential equation is investigated. The result is a further extension of Ku [1], Skidmore [2], and Leighton [3]. The Routh-Hurwitz conditions are satisfied for the linear system.

A modification of the energy metric algorithm to include the Routh-Hurwitz criteria

A simple extension of the energy metric algorithm is presented which modifies the system equation so that the resultant form of the Lyapunov function generates the Routh-Hurwitz conditions for the reduced linear system. A third-order example is included.

Routh-Hurwitz conditions and Lyapunov methods for the transient-stability problem

In this paper, Lyapunov functions are generated to determine the regions of asymptotic stability of power systems under transient disturbances. With suitable assumptions, the swing equations of the synchronous machines connected to a power system are second- or third-order nonlinear autonomous differential equations. The Lyapunov functions V, employed to determine the domains of asymptotic stability of these nonlinear differential equations, are simple quadratic forms, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. For a synchronous machine swinging against an infinite bus, three typical Lyapunov functions are generated, taking transient saliency and positive-sequence damping into account. The application of Aizerman's method to the same problem leads to a fourth Lyapunov function. The domains of stability given by these Lyapunov functions are compared with the actual stability region obtained by numerical integration. Consideration of one time constant of the prime-mover governor leads to a third-order differential equation. A Lyapunov function is generated, and the stability surface obtained is verified by computing the swing curves numerically for various initial conditions. Finally, it is explained how some of these techniques could be extended for generating Lyapunov functions for multimachine systems.

Routh-Hurwitz conditions and lyapunov methods for the transient-stability problem

Routh-Hurwitz conditions and lyapunov methods for the transient-stability problem

Stability of linear time-varying systems

It is shown that Lyapunov functions similar to the Ralston-Parks and Kalman-Bertram forms (which wore employed to derive the Routh-Hurwitz conditions through Lynpunov theory) can be formed from the time-varying coefficients of time-varying differential equations for the study of stability. These Lyapunov functions are used to conclude asymptotic stability of solutions of differential equations whoso time-varying coefficients approach constant values as time tends to infinity.