Routh-Hurwitz Stability Research Articles

Design of a combined self-stabilizing electrodynamic passive magnetic bearing support for the automotive turbocharger rotor

The purpose of this study is to create a concept for what would be a structurally simple and operationally robust support for the automotive turbocharger rotor in electrodynamic passive magnetic bearings. Because this kind of magnetic suspension—in its fundamental version—is dynamically unstable, to avoid the disadvantages contained therein, what is being proposed is the addition of external damping through the employment of the newly designed combined self-stabilizing electrodynamic passive magnetic bearing. The electromagnetic stiffness and damping characteristics of combined electrodynamic passive magnetic bearings have been determined for various shaft rotational speeds by means of the advanced 3D finite element method. In this study, a dynamic interaction between the turbocharger rotor shaft and the passive magnetic suspension is proposed as a support for both the fundamental electrodynamic passive magnetic bearings and the suggested combined self-stabilizing passive magnetic bearings. Here, the main attention is focused on the asymptotic stability of both the rotor shaft suspension variants. The additional damping magnitudes required to stabilize the most sensitive lateral eigenmodes of the object under consideration have been determined by means of the Routh–Hurwitz stability criterion.

Chaos and Coexisting Bifurcations in a Novel 3D Autonomous System with a Non-Hyperbolic Fixed Point: Theoretical Analysis and Electronic Circuit Implementation

A 3D autonomous chaotic system with the distinguishing feature of having a couple of fixed points, one of which is non-hyperbolic, is proposed. Interestingly, these fixed points become all hyperbolic in other parameters’ regions yielding diverse modes of oscillations in the system. The stability of the equilibria is discussed based on the Routh–Hurwitz stability criterion. The complex dynamics of the proposed system is numerically explored by using phase space trajectory plots, bifurcation diagrams, graphs of Lyapunov exponents, and basins of attraction. It is found that the system experiences period-doubling bifurcation, coexisting bifurcations, periodic windows, and chaos when monitoring its parameters. When the system is tuned to develop non-hyperbolic chaos, the basin of attraction of the strange attractor (coexisting with one of the fixed points) intersects with neighborhood of equilibria which is typical of self-excited oscillations. The coexistence between periodic and chaotic behaviors is found for specific parameter values. The analysis of the basins of attraction for the coexisting attractors reveals extremely complex structures. PSpice simulations based on a suitably designed electronic analog of the system confirm the results of theoretical analysis. The model proposed in this work shows “elegant” mathematical simplicity (i.e., only quadratic nonlinearities) and extremely rich modes of oscillations, and thus may be regarded as a prototypal member of the recently discovered and very restricted class of nonlinear systems developing non-hyperbolic chaos.

Stability analysis of mathematical model (sirb) in the spread of cholera with vaccination and disinfection

Cholera is a disease as a kind of acute diarrhea caused by bacteria V. Cholerae. The spread of cholera can be modeled in the form of nonlinear differential equation systems with 4 variables SIRB. This paper aims to research to analyze the local stability of the equilibrium point of the dynamical population in the spread of cholera by the Routh-Hurwitz stability criterion and bifurcation method. Next Generation Matrix (NGM) method is used to get the basic reproductive numbers ( ℜ0 ) to find the local stability at the equilibrium point of the. The disease-free equilibrium point is locally asymptotically stable if ℜ0 < 1, while the endemic equilibrium point is locally asymptotically stable if ℜ0 > 1 the results of numerical simulations obtained ℜ0 = 0 0.87 indicated that the disease-free equilibrium point is locally asymptotically stable. In endemic condition ( ℜ0 > 1 ) show that increasing the rate of vaccination and disinfection can reduce the population of susceptible, infected and bacteria of V. Cholerae.

Chatter model with structure-process-control coupled and stability analyses in the cold rolling system

In the cold tandem rolling process of high-strength thin plates, the chatter usually brings significant influences on equipment safety, production efficiency and product quality. The higher the rolling speed is, the more frequently and seriously the chatter occurs. Considering the equivalent elastic-plastic stiffness within the rolling gap, an eight Degree-of-Freedom (DOF) centralized mass model in the vertical direction for a six-roller mill is proposed, and a dynamic model of rolling process is established based on the Slab method. Then the structure-process coupling model is constructed and validated by calculating the equivalent stiffness and the rolling force. Subsequently, both the feed-forward control of entry thickness and the feed-back control of exit thickness are considered, and a hydraulic Auto Gauge Control (AGC) unit is modeled, which links with the established structure-process model to form a complete structure-process-control coupling model. Furthermore, by applying the Routh stability criterion to the presented structure-process-control coupling model, a threshold calculation model of rolling speed is derived, which is determined by the strip specification, technique condition and control parameter. Additionally, it is found that the experimental signals monitored from the actual rolling process are almost entirely consistent with the theoretical calculation, and more details involved in the vibration signals are demonstrated. Therefore the newly proposed structure-process-control coupling model is expected to be valuable for instability prediction and chatter suppression of the cold tandem rolling.

A hybrid analytical scheme for the numerical computation of time fractional computer virus propagation model and its stability analysis

Abstract In this paper, we present an application of the homotopy perturbation transform method to compute the approximate analytical solution of the nonlinear fractional order computer virus propagation (CVP) model. The fractional derivatives are used in Caputo sense. The proposed approximate method generates the numerical solution in the shape of a rapid convergent series by utilizing the provided initial conditions. The main purpose of the paper is to analyze the effect of variation of fractional order α on the meeting time of susceptible, infected and recovered computers. Moreover, the local stability analysis of the fractional order computer virus model is also presented using Routh–Hurwitz stability criterion.

On some extended Routh–Hurwitz conditions for fractional-order autonomous systems of order α ∈ (0, 2) and their applications to some population dynamic models

Abstract The Routh–Hurwitz stability criterion is a useful tool for investigating the stability property of linear and nonlinear dynamical systems by analyzing the coefficients of the corresponding characteristic polynomial without calculating the eigenvalues of its Jacobian matrix. Recently some of these conditions have been generalized to fractional systems of order α ∈ [0, 1). In this paper we extend these results to fractional systems of order α ∈ [0, 2). Stability diagram and phase portraits classification in the (τ, Δ)-plane for planer fractional-order system are reported. Finally some numerical examples from population dynamics are employed to illustrate our theoretical results.

Stability and Adaptive Control with Sychronization of 3-D Dynamical System

A three-dimensional system is presented with unknown parameters that employs two nonlinearities terms. The basic characteristics of the system are studied. The stability is measured by Characteristic equation roots, Routh stability criteria, Hurwitz stability criteria and Lapiynov function, all show that the system unstable. Then, Chaoticity is measured by maximum Lapiynov exponent of (Lmax=2.509426) and “Kaplan-Yorke” dimension (DL=2.22349544). The system is controlled effectively and synchronized by designed adaptive controllers. Furthermore, the theoretical and graphic results of the system before and after control are compared.

The Impact of Immigration on a Stability Analysis of Lotka-Volterra System

The aim of this paper is to investigate three species Lotka-Volterra system with the immigration and harvesting effect. This work verifies the parametric conditions for the existence of the non-negative equilibrium points and its asymptotic stability. First, the sign stability of the system is analyzed through signed digraphs associated with the Jacobian matrix. Second, the asymptotic stability of the system is obtained by attaining the suitable parameters which are satisfying the Routh-Hurwitz stability criterion. In this model, the focus is mainly on the role of the immigration effect of every species. The paper is concluded with the immigration effect that supports to stabilize the unstable system and how the numerical result provides the guarantee to achieve the theoretical results.

Proving Routh’s Theorem using the Euclidean Algorithm and Cauchy’s Theorem

This paper presents a proof of Routh’s theorem for polynomials with real coefficients, determining the number of roots in the right half plane (RHP). The proof exploits the relationship of the Routh array to the Euclidean algorithm and applies Cauchy’s theorem in an analogous way to that of applying the Nyquist criterion to investigate the stability of a control system. While a number of papers have been published over the years with different proofs of Routh’s stability criterion or Routh’s theorem, the aim in this paper is to present a proof that may offer most insight to undergraduate students of engineering. Routh’s theorem and his array are introduced without any proof in most undergraduate texts on control theory, whereas the Nyquist criterion is typically treated quite extensively in such texts. As well as presenting a proof for the regular case when all the coefficients in the first column of the Routh array are non-zero, analogous proofs are given for the singular cases when some of the leading coefficients in a row, or the coefficients of the entire row, become zero. In the first case, these result in a statement on the number of roots in the RHP, more explicit than those typically presented in papers on Routh’s theorem. In the second case, the only case where there may be roots on the imaginary axis, use is made of the modified array introduced by Routh, often referred to as the Q-method, to determine the number of such roots, differentiating between simple and multiple roots. One can thus distinguish between exponential stability, marginal stability and polynomial instability, when there are no roots in the RHP, with these results.

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

This article presents a general six-step discrete-time Zhang neural network (ZNN) for time-varying tensor absolute value equations. Firstly, based on the Taylor expansion theory, we derive a general Zhang et al. discretization (ZeaD) formula, i.e., a general Taylor-type 1-step-ahead numerical differentiation rule for the first-order derivative approximation, which contains two free parameters. Based on the bilinear transform and the Routh–Hurwitz stability criterion, the effective domain of the two free parameters is analyzed, which can ensure the convergence of the general ZeaD formula. Secondly, based on the general ZeaD formula, we design a general six-step discrete-time ZNN (DTZNN) for time-varying tensor absolute value equations (TVTAVEs), whose steady-state residual error changes in a higher order manner than those presented in the literature. Meanwhile, the feasible region of its step size, which determines its convergence, is also studied. Finally, experiment results corroborate that the general six-step DTZNN model is quite efficient for TVTAVE solving.

Porous panels with fixed ends lose stability by divergence

The aeroelastic stability of one-dimensional porous panels with a Darcy boundary condition on its surface is examined theoretically. Analytical and numerical analyses demonstrate that a porous panel in a uniform, single-sided, incompressible flow becomes aeroelastically unstable via divergence. This primary route of instability is identical to the well-known mechanism for non-porous panels. However, the divergence speed of a porous panel is always greater than the non-porous limit and increases with a dimensionless porosity parameter formed by the aeroelastic system. Various chordwise porosity distributions along the panel are also investigated, where the uniformly-porous panel is shown to be the most stable configuration. The generality and robustness of the primary divergence instability for porous panels is established analytically using a simple but general flutter analysis approach based on the Routh–Hurwitz stability criterion.

Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order

In this paper, we propose and analyze a fractional order model of viral kinetics for primary infection of HIV-1 in presence of immune control with treatment, as the classical model of target-cell-limited model is unable to predict long term viral kinetics unless a delayed immune effect is assumed. Further, the reverse transcriptase treatment can be incorporated by reducing the target cell infection rate. The existence of equilibria and their asymptotical stability results using fractional Routh–Hurwitz stability criterion will be discussed. A threshold value ′R0′ known as basic reproduction number has been found to ensure the extinction or persistence of the infection. The numerical solution using generalized Adams–Bashforth–Moulton method to the proposed HIV-1 fractional model is obtained. Also, the sufficient conditions that guarantee the asymptotic stability of endemic equilibrium point is presented. Meanwhile, global asymptotic stability of the endemic equilibrium point is investigated by constructing a suitable Lyapunov functions. The fractional derivative is described in Caputo sense. The obtained numerical results of the proposed model show the effectiveness and strength of the Adams–Bashforth–Moulton method. Some numerical simulations are given to illustrate the analytical results.

Stability of Non-Axisymmetric Rotor and Bearing Systems Modeled With Three-Dimensional-Solid Finite Elements

Abstract Although rotors are simplified to be axisymmetric in rotordynamic models, many rotors in the industry are actually non-axisymmetric. Several authors have proposed methods using 3D finite element, rotordynamic models, but more efficient approaches for handling a large number of degrees-of-freedom (DOF) are needed. This task becomes particularly acute when considering parametric excitation that results from asymmetry in the rotating frame. This paper presents an efficient rotordynamic stability approach for non-axisymmetric rotor-bearing systems with complex shapes using three-dimensional solid finite elements. The 10-node quadratic tetrahedron element is used for the finite element formulation of the rotor. A rotor-bearing system, matrix differential equation is derived in the rotor-fixed coordinate system. The system matrices are reduced by using Guyan reduction. The current study utilizes the Floquet theory to determine the stability of solutions for parametrically excited rotor-bearing systems. Computational efficiency is improved by discretization and parallelization, taking advantage of the discretized monodromy matrix of Hsu's method. The method is verified by an analytical model with the Routh–Hurwitz stability criteria, and by direct time-transient, numerical integration for large order models. The proposed and Hill's methods are compared with respect to accuracy and computational efficiency, and the results indicate the limitations of Hill's method when applied to 3D solid rotor-bearing systems. A parametric investigation is performed for an asymmetric Root's blower type shaft, varying bearing asymmetry and bearing damping.

Low-Voltage Flipped Voltage Follower Cell Based Current Mirrors for High Frequency Applications

The paper proposes low-voltage current mirrors namely low-voltage current mirror (LVCM) and flipped voltage follower based low-voltage current mirror (FVFLVCM) with high bandwidth. The stability of the proposed circuits has been performed by time-domain approach and frequency domain approach using Routh–Hurwitz stability criteria and phase margin calculations, respectively. The bandwidth of the proposed FVFLVCM has also been enhanced using resistive compensation technique. The proposed LVCM and FVFLVCM have wide input current range of 0–100 µA and 0–150 µA, low DC error of 5.9% and 4.9%, low DC power dissipation of 110.39 µW and 99 µW, high bandwidth of 367 MHz and 624 MHz, low input impedance of 16.3 KΩ and 12.3 KΩ, low THD of 0.26% and 1% and low output noise of 2.25 nA and 2.83 nA over 100 MHz, respectively. The results have been simulated using SPICE in the TSMC 0.18 µm CMOS technology and are presented to validate the effectiveness of the proposed current mirrors.

Identification of open quantum system: Via closed feedback control

For stochastic quantum systems with given dissipations, the identification method of Hamiltonian is given in this paper. First, the stability of the system is realized through designing a real-time feedback control method. Second, by using Routh–Hurwitz stability criterion, the intervals of the element values of the system Hamiltonians are given. Finally, the identification of Hamiltonian is realized by selecting the equilibrium state of the system equal to the desired target state. For two typical categories of noise: Purely dephasing decoherence and amplitude damping decoherence, we check the validity of the proposed estimation scheme.

Numerical Investigation of the Aerodynamic Characteristics and Attitude Stability of a Bio-Inspired Corrugated Airfoil for MAV or UAV Applications

In this study, two-dimensional (2D) and three-dimensional (3D) numerical calculations were conducted to investigate the aerodynamic characteristics, especially the unsteady aerodynamic characteristics and attitude stability of a bio-inspired corrugated airfoil compared with a smooth-surfaced airfoil (NACA2408 airfoil) at the chord Reynolds number of 4000 to explore the potential applications of non-traditional, corrugated dragonfly airfoils for micro air vehicles (MAVs) or micro-sized unmanned aerial vehicles (UAVs) designs. Two problem settings were applied to our numerical calculations. First, the airfoil was fixed at a constant angle of attack to analyze the aerodynamic characteristics and the hydrodynamic moment. Second, the angle of attack of airfoils was passively changed by the fluid force to analyze the attitude stability. The current numerical solver for the flow field around an unsteady rotating airfoil was validated against the published numerical data. It was confirmed that the corrugated airfoil performs (in terms of the lift-to-drag ratio) much better than the profiled NACA2408 airfoil at low Reynolds number R e = 4000 in low angle of attack range of 0 ∘ – 6 ∘ , and performs as well at the angle of attack of 6 ∘ or more. At these low angles of attack, the corrugated airfoil experiences an increase in the pressure drag and decrease in shear drag due to recirculation zones inside the cavities formed by the pleats. Furthermore, the increase in the lift for the corrugated airfoil is due to the negative pressure produced at the valleys. It was found that the lift and drag in the 2D numerical calculation are strong fluctuating at a high angle of attacks. However, in 3D simulation, especially for a 3D corrugated airfoil with unevenness in the spanwise direction, smaller fluctuations and the smaller average value in the lift and drag were obtained than the results in 2D calculations. It was found that a 3D wing with irregularities in the spanwise direction could promote three-dimensional flow and can suppress lift fluctuations even at high angles of attack. For the attitude stability, the corrugated airfoil is statically more unstable near the angle of attack of 0 ∘ , has a narrower static stable range of the angle of attack, and has a larger amplitude of fluctuations of the angle of attack compared with the profiled NACA2408 airfoil. Based on the Routh–Hurwitz stability criterion, it was confirmed that the control systems of the angle of attack passively changed by the fluid force for both two airfoils are unstable systems.

Load Capacity Analysis of Self-Excited Induction Generators Based on Routh Criterion

In this paper, the Routh criterion has been used to analyze the stability of a self-excited induction generator-based isolated system which is regarded as an autonomous system. Special focus has been given to the load capacity of the self-excited induction generator. The state matrix of self-excited induction generators with resistor-inductor load has been established based on transient equivalent circuits in the stator stationary reference-frame. The recursive Routh table of self-excited induction generators is established by the characteristic polynomial coefficients of the state matrix. According to the Routh stability criterion, the necessary and sufficient condition to predict the critical loads of self-excited induction generators is deduced, from which the critical load impedance can be calculated. A simple self-excited induction generator-based isolated power system has been built up with a 2.2 kW self-excited induction generator. The theoretical analysis and experiments were all carried out based on this platform. In the range determined by the minimum excitation capacitance (Cmin) and the maximum excitation capacitance (Cmax), the critical loads under various power factors have been calculated. The agreement of the calculated theoretical results and experimental results demonstrate the effectiveness and accuracy of the proposed analysis method. The conclusions achieved lay a foundation for further application of Routh stability criterion in self-excited induction generator-based power systems analysis.

Modeling and stability analysis methods of neutrosophic transfer functions

Uncertainty is inherent property in actual control systems because parameters in actual control systems are no constants and changeable under some environments. Therefore, actual systems imply their indeterminate parameters, which can affect the control behavior and performance. Then, a neutrosophic number (NN) presented by Smarandache is very easy expressing determinate and/or indeterminate information because a NN p = c + dI is composed of its determinate term c and its indeterminate term dI for c, d ∈ R (R is all real numbers), where the symbol “I” denotes indeterminacy. Unfortunately, all uncertain modeling and analysis of practical control systems in existing literature do not provide any concept of NN models and analysis methods till now. Hence, this study firstly proposes a neutrosophic modeling method and defines a neutrosophic transfer function and a neutrosophic characteristic equation. Then, two stability analysis methods of neutrosophic linear systems are established based on the bounded range of all possible characteristic roots and the neutrosophic Routh stability criterion. Finally, the proposed methods are used for two practical examples on the RLC circuit and mass–spring–damper systems with NN parameters. The analysis results demonstrate the effectiveness and feasibility of the proposed methods.

Investigating Pole behavior of Reduced Order Model

In this paper, pole placement and stability behavior of reduced order model using Pade approximation method and Routh Stability Criterion by V. Krishnamurthy Method has been investigated. Higher order systems are reduced to lower order systems using Reduced Order Modelling (ROM) Techniques.In this reduction technique we here conclude that the shifting of the poles of original system and reduced system gives some important results about the Pade approximation unstability problem.. The results are compared with the Routh Stability Criterion by V. Krishnamurthy Method. Pade approximation and Routh stability criterion by V.Krishnamurthy methods are very good and simple methods of reducing a higher order system.

Simulation and experiment of algorithm and circuit design for UPQC

ABSTRACTPower quality issues have become one of the most important issue for researchers to concern. In this paper, simulation and experiment of algorithm and circuit design of Unified Power Quality Conditioner (UPQC) are provided. Control algorithm and topology design of one UPQC which includes active power filter (APF) and dynamic voltage restorer (DVR) are introduced. Stability condition of the filter unit is deduced and proved by Routh stability criterion. Simulation for APF and DVR is carried out in PSCAD to show the proposed control strategy. Experiments such as current tracking, harmonic detection and compensation and voltage drop compensation are provided in details. Experimental results show that the proposed control method and the designed topology are effective and practical.