Nonlinear Autonomous Systems Research Articles

Finite-time stability results for fractional damped dynamical systems with time delays

This paper is explored with the stability procedure for linear nonautonomous multiterm fractional damped systems involving time delay. Finite-time stability (FTS) criteria have been developed based on the extended form of Gronwall inequality. Also, the result is deduced to a linear autonomous case. Two examples of applications of stability analysis in numerical formulation are described showing the expertise of theoretical prediction.

Coordinate-wise Prescribed-time Stabilization in Block Control Form

Dynamic systems with finite time convergence have been studied intensively in numerous publications on sliding mode control, systems with terminal control, systems with homogeneity properties. Recently research has been initiated for a new class of such systems with remarkable property -finite time convergence does not depend on initial conditions. The main attention was initially paid to SISO systems in canonical form (chain of n integrators). The paper demonstrates that the method can be generalized for MIMO systems in the block control form, consisting of a set of lower order systems with dimension equal to the dimension of intermediate controls. First it is shown that the design procedure let the designer decouple systems in canonical form into set of problems of the first order. Dealing with non-autonomous linear systems is another novelty of the presented investigation.

Automatic Rescheduling of Generator for Rotor Speed Regulation by KRASOVSKII’s Theorem

Emergency control of the power system is performed via generator rescheduling or load shedding depending upon the availability of generation and demand. This paper presents a generator rescheduling method based on Krasovskii’s theorem for enhancing rotor speed regulation during multiple contingency conditions. The speed governor in automatic generation control (AGC) and automatic voltage regulator (AVR) are indispensable components of the control mechanism for deploying any control scheme. Contingencies may lead the security state of the power system towards a preventive, emergency, and restorative state. Generally, there is a violation of equality constraints of active and reactive power in the corrective security state. However, in this paper, the corrective security state considers a significant imbalance in active power with a generation capacity always greater than load. For rotor angle stability, active power delivery by the generator met the demands, whilst the turbine and automatic speed governor together balance it by changing the set point of the turbine inlet. This paper investigates the rotor stability by triggering a new set point by the rescheduling of the generator. The strategy proposed encompasses the broad domain of active power imbalance using Krasovskii’s theorem, which is an extensive form of the Lyapunov stability theorem for the nonlinear multivariable autonomous system. The proposed control scheme is deployed on an IEEE-10 machine 39-bus system for validation and feasibility detection. It is found that the proposed scheme is more accurate than the conventional turbine automatic speed governor close loop system. Moreover, there is turbine power-saving up to 400 MW. In addition, the rotor attains steady-state speed quicker up to 5-10 sec and attains synchronous speed in all the cases considered, i.e., single load outage; multiple load outage; area outage; and load increment. The MATLAB® and the CVX platform are used to validate the proposed concept.

Studying the Stability of the ψ‐Hilfer Fractional Differential System

This paper devotes to the study on the stability and decay of solution to fractional differential system involving the ψ‐Hilfer fractional derivative of order α ∈ (0,1) and type β ∈ [0,1]. We first derive the solution of linear system by using the generalized Laplace transform, which can be represented by the form of Mittag‐Leffler function. Then, in terms of the asymptotic expansion of the Mittag‐Leffler function, stability properties of linear system are analyzed in more detail. Finally, we construct a linearization theorem and determine the stability near the equilibrium for the autonomous nonlinear differential system with the ψ‐Hilfer derivative.

On the admissibility of observation for perturbed non-autonomous linear systems

In this paper, we are concerned with unbounded observation operators for perturbed non-autonomous evolution equations. First, we briefly survey the necessary background about the concept of maximal regularity. Second, we establish the admissibility invariance of observation operators under non-autonomous Miyadera-Voigt perturbations. Finally, we illustrate our abstract result with an example.

On the Use of the Time-Integrals of the Output in Observer Design for Nonlinear Autonomous Systems

The goal of this article is to show that the observation scheme recently proposed for linear systems by Menini <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2019, achieving good performance in the presence of a high-frequency measurement disturbance, can be adapted to be used for nonlinear autonomous systems. The design is based on the use of the time-integrals of the measured output and requires the availability of a state observer able to provide a rough estimate of the state; the proposed scheme can be then understood as an observer postprocessing algorithm. For the implementation of the proposed scheme, the time-integrals of the output are required to express as functions of the state of the system. Such functions are called directional antiderivatives; in the article, conditions to guarantee their existence and methods for their computation, in closed-form or by approximation, are resumed. The performance of the proposed observation scheme is illustrated through an example.

Fuzzy extended state observer for the fault detection and identification

This paper introduces a novel methodology to detect and identify faults for a class of autonomous nonlinear systems. In the proposed design, a fuzzy extended system observer (FESO) based on the Mandami-type fuzzy system is used to estimate the fault that is considered to be the extended system state. In this method, the Mamdani-type fuzzy system is based on a single-input single-output (SISO) where the observer error is considered as the fuzzy input variable. Additionally, the stability analysis under Lyapunov criteria verifies that the solutions of proposed FESO are ultimately bounded. Finally, simulation examples are given to corroborate the feasibility of the proposed FESO.

Multistability and period-adding in a logarithmic Lorenz system

In this paper, we design and explore a novel 3D autonomous nonlinear system with logarithmic nonlinearity. It is framed from the Lorenz model by replacing the variable [Formula: see text] in the second equation by the nonlinear logarithmic term [Formula: see text]. The bi-parametric dynamics of the modeled system is presented. Studies reveal the emergence of shrimp-shaped periodic structures in chaotic regime with period-adding phenomenon, and coexisting periodic-chaotic attractors, which do not occur in the Lorenz model. The stability, bifurcation patterns, and chaotic behaviors of the system are explored with the aid of bifurcation diagrams, phase portraits and Lyapunov exponents. The complexity of the basin sets for the coexisting attractors is also studied.

Dynamical Analysis and Parameter Estimation of Hepatitis B Disease Model in Malang

In this article, a model representing the spread of Hepatitis B disease is constructed as a nonlinear autonomous system. The model divides the considered human population into three classes, namely susceptible, infected, and recovered class. The dynamical analysis shows that there are two equilibrium points in the model, namely a disease-free equilibrium point and an endemic equilibrium point. The existence and stability of the equilibrium points depend on the basic reproduction number (R_0). The disease-free equilibrium point is local asymptotically stable when R_0&lt;1Â while the endemic equilibrium point exists and is local asymptotically stable if R_0&gt;1. The five parameters of the model are estimated by applying Downhill Simplex (Nelder-Mead) Algorithm and by using the infected data cases taken from such a hospital in Malang. The estimated parameters are the transmission of infection rate, the saturation rate, the vaccination rate, the recovery rate, and the immunity loss rate. The resulting parameter estimation supports the analytical result and is used to illustrate the analytical results numerically. Based on the considered model and the result of the parameters estimation, it can be concluded that the Hepatitis B spread in Malang is controllable.Keywords: downhill simplex (Nelder-mead) algorithm, dynamical analysis, hepatitis B model, parameter estimation.Â

Invariants of Linear Control Systems with Analytic Matrices and the Linearizability Problem

The paper continues the authors’ study of the linearizability problem for nonlinear control systems. In the recent work (Sklyar, Syst Control Lett 134:104572, 2019), conditions on mappability of a nonlinear control system to a preassigned linear system with analytic matrices were obtained. In the present paper, we solve more general problem on linearizability conditions without indicating a target linear system. To this end, we give a description of invariants for linear nonautonomous single-input controllable systems with analytic matrices, which allow classifying such systems up to transformations of coordinates. This study leads to one problem from the theory of linear ordinary differential equations with meromorphic coefficients. As a result, we obtain a criterion for mappability of nonlinear control systems to linear control systems with analytic matrices.

Performance comparison of estimation-based control schemes for hybrid systems

In this paper, the authors have proposed an estimation-based control scheme. Artificial neural network and unscented Kalman filter are used for state estimation and inverse dynamics controller is implemented that utilizes the estimation of unmeasured state variables. The benchmark example taken for the analysis and implementation is nonlinear autonomous three-tank hybrid system and non-autonomous switched mode non-isothermal continuous stirred tank reactor, which are subjected to state and measurement noise. The performance comparison of both proposed estimation-based control scheme has been carried out and results are presented. Further, the proposed scheme is validated with three-tank hybrid system experimental setup.

Counting equilibria of large complex systems by instability index

We consider a nonlinear autonomous system of [Formula: see text] degrees of freedom randomly coupled by both relaxational ("gradient") and nonrelaxational ("solenoidal") random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of "absolute instability" where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.

A Discussion on the Existence of Smooth Lyapunov Functions for Continuous Stable Systems

Lyapunov's theorem is the basic criteria to establish the stability properties of the nonlinear dynamical systems. In this method, it is a necessity to find the positive definite functions with negative definite or negative semi-definite derivative. These functions that named Lyapunov functions, form the core of this criterion. The existence of the Lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse Lyapunov theorems. On the other hand, for the cases where the equilibrium point is stable in the sense of Lyapunov, converse Lyapunov theorems only ensure non-smooth Lyapunov functions. In this paper, it is proved that there exist some autonomous nonlinear systems with stable equilibrium points that despite stability don’t admit convex Lyapunov functions. In addition, it is also shown that there exist some nonlinear systems that despite the fact that they are stable at the origin, but do not admit smooth Lyapunov functions in the form of V(x) or V(t,x) even locally. Finally, a class of non-autonomous dynamical systems with uniform stable equilibrium points, is introduced. It is also proven that this class do not admit any continuous Lyapunov functions in the form of V(x) to establish stability.

Computing tensor Z-eigenvalues via shifted inverse power method

The positive definiteness of an even degree homogeneous polynomial plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control, the detection of P or P0 tensor in tensor complementarity problems and spectral hypergraph theory, and more. Owing to the positive definiteness of an even degree homogeneous polynomial is equivalent to that of an even order symmetric tensor. In this paper, we propose a shifted inverse power method for computing tensor Z-eigenpairs, which can be viewed as a generalization of the inverse power method for matrices case. We also formulate it as a fixed point iteration form, and reveal that the relationship between the fixed points and the Z-eigenvectors of symmetric tensors. The advantages of the proposed method are simple operations and readily comprehensible convergence analysis. An efficient initialization strategy is also developed, which makes the proposed method converges to a better solution compared to not using the initialization strategy. Finally, we present applications of the proposed method in nonlinear autonomous systems, the detection of P or P0 tensor and symmetric tensor Z-eigenproblems, some numerical results are reported to illustrate the effectiveness of the proposed method.

Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network

Identification of autonomous nonlinear dynamical system based on discrete-time multiscale wavelet neural network

Finite-time stability of a class of nonautonomous systems

In this paper, the finite-time stability of nonautonomous systems is studied by using Lyapunov Krasovskii method and Gronwall inequality. Firstly, a new lemma for the finite-time stability for nonautonomous systems is given by using Gronwall inequality. which make the derivative of Lyapunov function not to be negative all the time. Then, based on the new lemma and Lyapunov Krasovskii method, some finite-time stability criteria about nonautonomous linear systems and nonlinear systems are proposed. Based on this, by designing controllers, the finite-time stabilizable theorem is obtained. In addition, it is pointed out that the conclusion of this paper is more general than the existing literature in the remarks. Finally, several examples are simulated with MATLAB to verify the effectiveness of the results.

Dynamics of mathematical model interaction of living populations in making biogas as an alternative of renewable energy

Renewable energy which is often referred to alternative energy is needed as an alternative or substitute for conventional energy. Biogas is a type of energy source that comes from living things (plants and animals). Biogas is the most energy-efficient and environmentally friendly. Biogas as alternative energy has considerable potential to help reduce the use of fossil energy. One way to determine the effect of each aspect is to model it in mathematical language. Mathematical models represent the production of biogas by considering the interactions between living populations in making alternative renewable energy alternatives. The model is constructed as a nonlinear autonomous system with three dependent variables. After forming a mathematical model, dynamical behavior such as investigate the equilibrium point, a local stability analysis process is then carried out on the model. The numerical simulation will represent an illustration analysis results of the constructed model. The simulation results show the same stability analysis.

Dynamical complexity of the teleparallel gravity cosmology

The exploration of teleparallel gravity has been done from a dynamical systems point of view in order to be tested against the cosmological evolution currently observed. So far, the proposed autonomous systems have been restrictive over a constant dynamical variable, which contains information related to the dynamics on the $H_0$ value. It is therefore that in this paper we consider a generalization of the dynamical system by imposing a nonconstant degree of freedom over it which allows us to rewrite a generic autonomous dynamical analysis. We describe the treatment of our nonlinear autonomous system by studying the hyperbolic critical points and discuss an interesting phenomenological feature in regards to $H_0$: the possibility to obtain a best-fit value for this parameter in a cosmologically viable $f(T,B)$ model, a mixed power law. This result allows us to present a generic scenario in which it is possible to fix constraints to solve the $H_0$ tension at late times where its linearized solutions are considered.

Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.

Reduction of a Tri-Modal Lorenz Model of Ferrofluid Convection to a Cubic–Quintic Ginzburg–Landau Equation Using the Center Manifold Theorem

The differential geometric method of the center manifold theorem is applied to the magnetic-Lorenz model of ferrofluid convection in an electrically non-conducting ferrofluid. The analytically intractable tri-modal nonlinear autonomous system (magnetic-Lorenz model) is reduced to an analytically tractable uni-modal cubic–quintic Ginzburg–Landau equation. The inadequacy of the cubic Ginzburg–Landau equation and the need for the cubic–quintic one is shown in the paper. The heat transport is quantified using the solution of the cubic–quintic equation and the effect of ferrofluid parameters on it is demonstrated. The stable and unstable regions in the conductive regime and the conductive-convective regime is depicted using a bifurcation diagram. The noticeable discrepancy between the results of the two models is highlighted and the quintic non-linearity effects are delineated.