Hurwitz Stability Research Articles

Approximate analytical solution of fractional order biochemical reaction model and its stability analysis

Approximate analytical solution of the system of coupled nonlinear Ordinary Differential Equations (ODEs) of a biochemical reaction model is much relevant due to its practical significance to biochemists. In this paper, an effective and powerful mathematical technique, viz. fractional homotopy analysis transform method (FHATM), is employed to get the numerical solutions of biochemical reaction model with time fractional derivatives. The adopted scheme is the beautiful copulation of homotopy analysis technique and Laplace transform algorithm. This paper shows that the adopted scheme is quite easy as well as computationally attractive in the context of a solution procedure. The Caputo-type fractional derivatives are considered in the present paper. Approximate results of the probability density functions of the time fractional biochemical reaction model are computed for miscellaneous fractional Brownian motions as well as for classical motion and are presented graphically. The time fractional biochemical reaction model with respect to stability analysis for various values of fractional order [Formula: see text] is also analyzed. In the context of stability discussion, we have used the fractional Routh–Hurwitz stability criterion to establish the local stability of the biochemical reaction model of fractional order.

The generalized Hadamard product of polynomials and its stability

For two polynomials of degrees n and m () we define a set of polynomials , where for , and call it a generalized Hadamard product of f and g . We give sufficient conditions for the Hurwitz stability of . The obtained results show that the famous Garloff–Wagner theorem on the Hurwitz stability of the Hadamard product of polynomials is a special case of a more general fact. We also show that for every polynomial with positive coefficients (even not necessarily stable) one can find a polynomial such that their generalized Hadamard product is stable. Some connections with polynomials admitting the Hadamard factorization are also given. Numerical examples complete and illustrate the considerations.

Infinite polytopes in Hurwitz stability region

We consider Hurwitz stability region in the parameter space of monic polynomials. Firstly, a multilinear map from the positive first octant of the cartesian coordinate system to the stability region is defined. Based on this map a necessary and sufficient condition for stability is obtained. Starting from stable points, polytopes with infinite edge lengths are defined. Stability of the edges of these polytopes is equivalent to the stability of the convex combinations of third and fourth order factor polynomials. If all edges of polytope are stable then by the edge theorem the polytope is robust stable. The obtained result is important for understanding the geometry of stability domain, and can be used for generation of convex subsets of stable polynomials and for stabilization by fixed order controllers and other related problems.

General Five-Step Discrete-Time Zhang Neural Network for Time-Varying Nonlinear Optimization

In this paper, a general five-step discrete-time Zhang neural network is proposed for time-varying nonlinear optimization (TVNO). First, based on the bilinear transform and Routh–Hurwitz stability criterion, we propose a general five-step third-order Zhang dynamic (ZeaD) formula, which is shown to be convergent with the truncation error $$O(\tau ^3)$$ with $$\tau >0$$ the sampling gap. Second, based on the designed ZeaD formula and the continuous-time Zhang dynamic design formula, a general five-step discrete-time Zhang neural network (DTZNN) model with quartic steady-state error pattern is presented for TVNO, which includes many multi-step DTZNN models as special cases. Third, based on the Jury criterion, we derive the effective domain of step size in the DTZNN model, which determines its convergence and convergence speed. Fourth, a nonlinear programming is established to determine the optimal step size. Finally, simulation results and discussions are given to validate the theoretical results obtained in this paper.

Analisis Kestabilan dan Usaha Pemanenan Model Predator Prey Tipe Holling III dengan Keuntungan Maksimum

In this paper discussed Stability Analysis and Harvesting Effort at second Predator Prey Populations model Holling Type III with Maximum Profit. The step this research is to determine the equilibrium point, linearize the model, stability analysis of the equilibrium point, and numerical simulation. Result shows that obtained an interior point T that asymptotic stable based on Hurwitz stability test then obtained maximum profit from exploitation harvesting effort of second predator prey populations. This second populations will always exist, even though exploited with harvesting effort done by humans. Harvesting effort of second predator-prey populations given maximum profit that occur on critical points of surface profit function

Mathematical modelling of Zika virus as a mosquito-borne and sexually transmitted disease with diffusion effects

Zika virus disease is an infection transmitted primarily by Aedes mosquitoes, and can be a serious epidemic if not contained and controlled in its early stages. The purpose of this study is to investigate the transmission dynamics of Zika virus epidemic. A mathematical model of Zika virus with the inclusion of diffusion in the system is formulated, using a system of ordinary differential equations. Mathematical models assume transmission of the virus by mosquitoes, with and without transmission through sexual contact by humans. The basic properties of the model in the absence and presence of diffusion are determined including, points of the disease-free equilibrium (DFE) and the endemic equilibrium (EE). Stability of the disease-free equilibrium using the reproduction number, R0 is established. The stability of the endemic equilibrium is established using the Routh–Hurwitz stability conditions. Bifurcation values for transmission rates and human recovery rate are also calculated. Numerical simulations based on different population distributions help to understand how the diffusion of human and mosquitoes affects the transmission of the disease.

Stability Region of Integrating Process based on Routh- Hurwitz Stability Criteria

This paper presents a new tuning method for P and PID controller in a double-loop feedback structures. The stability region is first determined by Routh-Hurtwitz stability criteria. Simple tuning rules are established to determine the optimal P/PID parameter values within the stability regions, to meet desired performance-robustness criteria. The effectiveness of the newly established tuning rules are evaluated via numerical study and compared with those based on the same existing PID tuning methods.

Nonlinear PID controller design for a 6-DOF UAV quadrotor system

A Nonlinear PID (NLPID) controller is proposed to stabilize the translational and rotational motion of a 6-Degree of Freedom (DOF) unmanned aerial vehicles (UAV) quadrotor system and enforce it to track a given trajectory with minimum energy and error. The complete nonlinear model of the 6-DOF quadrotor system is obtained using Euler-Newton formalism and used in the design process, taking into account the velocity and acceleration vectors resulting in a more accurate 6-DOF quadrotor model and closer to the actual system. Six NLPID controllers are designed, each for Roll, Pitch, Yaw, Altitude, and the Position subsystems, where their parameters are tuned using Genetic Algorithm (GA) to minimize a multi-objective Output Performance Index (OPI). The stability of the 6-DOF UAV subsystems has been analyzed in the sense of Hurwitz stability theorem under certain conditions on the gains of the NLPID controllers. The simulations have been accomplished under MATLAB/SIMULINK environment and included three different trajectories, i.e., circular, helical, and square. The proposed NLPID controller for each of the six subsystems of the 6-DOF UAV quadrotor system has been compared with the Linear PID (LPID) one and the simulations showed the effectiveness of the proposed NLPID controller in terms of speed, control energy, and steady-state error.

Model experiments and study on regulating stability of isolated hydropower plants

Hydropower shoulders a large portion of the regulation and balancing duty in many power systems all over the world. The stable regulation of hydropower plants (HPPs) is a conventional and crucial research topic, and many meaningful studies have been carried out by theoretical analysis and numerical simulations; however seldom experimental works on this issue are reported. This paper investigates stability of isolated HPPs based on model experiments on an integral experiment platform for transient processes of pumped storage plants in Wuhan University. Theoretical stability analysis is conducted based on Routh – Hurwitz stability criterion, and the stability region of the model power plant is obtained. The experiment results validate the theoretical analysis on the system stability, and the instability of the hydropower system due to poor settings of speed governor parameters are captured and demonstrated. This work presents initial results of experimental study on oscillation characteristics and regulation performance of HPPs for power system stability.

Eliminating the Fluid‐Induced Vibration and Improving the Stability of the Rotor/Seal System Using the Inerter‐Based Dynamic Vibration Absorber

A theoretical research on eliminating the instability vibration and improving the stability of the rotor/seal system using the inerter‐based dynamic vibration absorber (IDVA) is presented in this paper. The modified Jeffcott rotor and Muszynska nonlinear seal force models are employed. The proposed IDVA is the damping element of the traditional dynamic vibration absorber (DVA) replaced by one of the six configurations of the inerter. The instability threshold speed of the system is obtained by applying the Routh–Hurwitz stability criterion. The quantum particle swarm optimization (QPSO) method is utilized to optimize the parameters of the IDVA. The numerical method is applied to investigate the nonlinear dynamic responses and stability. The results show that the IDVA can effectively improve the stability and reduce the instability vibration of the rotor/seal system. Furthermore, the performance of the IDVA is more effective than that of the traditional DVA.

Consensus-Based Formation of Second-Order Multi-Agent Systems via Linear-Transformation-Based Partial Stability Approach

The paper studies the time-invariant formation problem of second-order multi-agent systems under a time-invariant directed communication topology. Extensions of the consensus protocol are introduced in the formation control. By choosing appropriate consensus states, the state-linear-transformation approach and the partial stability theory are adopted to analyze the formation problem. Sufficient and necessary algebraic criteria are derived for the formation regulation problem with or without velocity constrains and the formation tracking problem. They are expressed in terms of Hurwitz stability of matrices which are constructed from the gain matrices of formation control protocols.

Synthesis of Hurwitz Polynomial Families Using Root Locus Portraits

The paper deals with the problem of synthesis of a stable characteristic polynomial families describing the control systems' dynamics in conditions of the interval uncertainty. Investigation is based on the system mathematical model in the form of its root locus portrait generated by the polynomial free term variation that is named in the paper as the "free root locus portrait". The root loci of the Kharitonov's polynomials family (subfamily) is picked out of the whole polynomial family and is considered for carrying out the investigation. Specific regularities of the interval root locus portrait have been discovered. On the basis of these regularities main properties of the system root locus portrait have been defined. A stability condition has been formulated that allows to calculate the polynomial free term variation interval ensuring the polynomial family hurwitz stability. This stability condition is applicable to the class of polynomials having their free root locus poles lying within the left half-plane of roots or, in other words, being stable when their free term is equal to zero. The stable family is being synthesized by setting up (adjusting) the given initial family that is supposed to be unstable, i.e. the proposed method of synthesis allows to turn stable (hurwitz) the given nonhurwitz interval polynomial family. The setting up criterion is specified in terms of proximity i.e. as the nearest distance from the "unstable" system roots to the "stable" ones as measured along the root trajectories. The stable polynomial could be selected as the nearest to the given unstable one with or without consideration of the system quality requirements. In the course of the setting up procedure new boundaries of only the polynomial free term variation interval (stability interval) are calculated that allows to ensure system stability without modification of its root locus portrait configuration. A numerical example of the polynomial setting up procedure has been given.

New Versions of the Hermite Bieler Theorem in Stability Contexts

The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.

Nonlinear Torsional Vibration Analysis and Control of Semidirect Electromechanical Coupling Transmission System in Shearer

The nonlinear torsional vibration and instability oscillation caused by nonlinear damping in the shearer electromechanical coupling cutting transmission system in shearer driven by the permanent magnet synchronous motor (PMSM) is investigated in this paper. The electromechanical coupling transmission system in the shearer is equivalent to a concentrated mass model for the purpose of establishing the system dynamic model by the Lagrange–Maxwell equation. Then, the Routh–Hurwitz criterion is used to determine the torsional vibration critical point and stability region for the Hopf bifurcation for the cutting transmission system. According to the Routh–Hurwitz stability criterion, the Hopf bifurcation type and the effect of the supercritical Hopf bifurcation in the torsional vibration of the cutting transmission system are analyzed. Furthermore, based on the washout filter, the Hopf bifurcation controller is designed for suppressing the transmission system’s large vibration amplitude and unstable oscillation. In addition, the influences of the linear gain and nonlinear gain on the bifurcation point and the limit cycle amplitude are discussed. Finally, the numerical simulation results indicate the effectiveness of the designed controller. The research achievements can provide a theoretical basis for design or optimize the cutting transmission system of high‐reliability shearer driven by PMSM.

Decoupled linear model and S-shaped curve motion trajectory for load sway reduction control in overhead cranes with double-pendulum effect

Overhead cranes with double-pendulum effect seem more practical than those with single-pendulum effect. However, in this case, it is difficult to find an applicable controller for such systems. Hence, a linearized and decoupled double-pendulum overhead crane dynamic model is derived by adopting a disturbance observer and modal analysis technique. The S-shaped trajectory is planned by solving algebraic equations, and the stability of the system is confirmed by the Routh–Hurwitz stability theory. Experimental results and simulations demonstrate the effectiveness of the proposed method. It could be realized to operate the crane accurately without sensor systems for measuring load sways by using the proposed method.

Control of Memristor-Based Simplest Chaotic Circuit with One-State Controllers

This paper deals with the control of the memristor-based simplest chaotic circuit by means of only one-state controller. Three distinct control techniques, namely linear feedback control, sliding mode control, and nonlinear control, are examined for the control. Routh–Hurwitz stability criteria are used for constructing the linear feedback and sliding mode gains. Lyapunov function is used for ensuring the global asymptotic stability of the system with the nonlinear controller. Numerical simulations are demonstrated not only to validate the theoretical analyses, but also to compare the control results. They have shown that even if one-state controller is used, all the methods are effective for controlling the memristor-based simplest chaotic circuit. However, the control is observed in a better time period with the sliding mode controller.

Linear Minimax Filtering of a Stationary Random Process under the Condition of the Interval Fuzziness in the State Matrix of the System with a Restricted Variance

The equations for the transition functions of optimal minimax filters are derived under the condition of the interval fuzziness of the linear dynamic system with the parametric uncertain assignment of only the state matrix in the context of Hurwitz stability. The method by which the required filter is synthesized as the suboptimal minimax filter of the stable and unstable first-order systems with the given degree of stability is discussed.

Charge and Discharge Control Method for Grid-Connected Electric Vehicle

A inverter with LCL filter was adopted as a main interface between electric vehicle and power grid. The mathematical model of the LCL filter was built, and then a charge and discharge control method for grid-connected electric vehicle was proposed. This control method employed an outer grid-side current loop and an inner inverter-side current loop. And this control method can protect power switches effectively, but also ensure the system stability and high in-grid power factor. Based on the Hurwitz stability criterion and Lienard Chipart stability criterion, the design method of controller parameters in inner and outer loops is derived. Simulations and experiments of electric vehicle in grid-connected mode have been conducted. Simulation and experiment results validate the correctness and feasibility of the proposed method.

Aggregation and the economy prototyping of municipality in the conditions of lability and robust changes

Purpose of the study . The purpose of the scientific paper is the formation of a model of managing the economy of municipality through aggregation and prototyping in conditions of lability and robustness of changes. As part of the stated goal, the author is supposed to conduct the economy prototyping of the municipality based on built-in management platforms; to form an adapted risk assessment model in the implementation of smart-project; to propose an aggregated model for assessing the development of the economy. Materials and methods . While writing a scientific article, the methods of situational and simulation modeling, approximation and grouping of specific elements of the structural object, and the process of algorithmization of functional processes are used. Particular emphasis in this paper is made on the methodological apparatus, which allows drawing conclusions based on subjective author’s opinion, to take into account external changes in the system under study. These methodological approaches include entropy of scientific knowledge, aberration, approximation, robustness. Results. Within the framework of the conducted research the following conclusions are formed: the municipal economy is in the process of transformation, it is possible to obtain positive changes with the help of prototyping the model of municipality; smart- projects as the basis of the economy of the municipality are subject to robustness and lability of changes, provoking negative risks; game theory is a rather important tool, used in the context of the definition of the riskiness of the smart-project the aggregated model for assessing the development of the economy of the municipality makes it possible to formulate certain premises of the Smart City concept in municipalities. Conclusion. For the effective development of municipalities, it is necessary to transform the economies towards the formation of municipality based on using prototyping and aggregation tools in conditions of lability and robustness of the changes. Presented in the scientific paper, the aspect of aggregation and prototyping of the economy of the municipal formation is the initial stage in the development of the concept of management of the Smart Economy of the municipal formation. The transformation of municipal economies regulates the need for a new tool for the development of these territories. The author of the scientific article proposes to use the adapted model of risk assessment of Smart-design based on game theory and the aggregated model for assessing the development of the economy of a municipal formation. The adapted model of risk assessment of Smart-design on the ground of the theory of games is based on the use of the Bayes criterion, Wald’s maximin criterion, the Savage decision-making criterion, the Hurwitz stability criterion, the Hodges-Lehmann criterion. The selected criteria contribute to the definition of the least risky Smart- project with the aim of its implementation in the municipal economy. The aggregated model for assessing the economy development of the municipal entity projects the regression parameter data algorithm – Constructive Coste Model (COCOMO) on the basis of which a conclusion is given about the level of the economy development of the municipal formation. In the future, simulation models of economic management can supplement this study in conditions of the prevalence of knowledge and intelligence, as well as methods for assessing the quality of urban technological infrastructure.

The Modeling Dynamics of HIV and CD4 $$^{+}$$ + T-cells During Primary Infection in Fractional Order: Numerical Simulation

This article is devoted to introduce a numerical treatment using Adams–Bashforth–Moulton method of the fractional model of HIV-1 infection of CD4 $$^{+}$$ T-cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. The fractional derivative is described in Caputo sense. Special attention is given to present the local stability of the proposed model using fractional Routh–Hurwitz stability criterion. Qualitative results show that the model has two equilibria: the disease-free equilibrium and the endemic equilibrium points. We compare our numerical solutions with those numerical solutions using fourth-order Runge–Kutta method (RK4). The obtained numerical results of the proposed model show the simplicity and the efficiency of the proposed method.