Fourth-order Nonlinear System Research Articles

INFLUENCE OF EXTERNAL CUBIC VISCOSITY ON THE DYNAMICS OF A CANTILEVER ROD LOADED WITH A TRACKING FORCE

The analysis of the influence of external nonlinear cubic viscosity on the behavior of a cantilever rod loaded with a compressive tracking force is performed. The Bernoulli – Euler beam model is used for the describing of the rod bending deformations. After the transition to dimensionless variables, we obtained the form of notation in which the coefficients of the equation do not depend on the coefficient of cubic viscous friction. This indicates the independence of the critical load and system frequency characteristics from the level of nonlinear attenuation. Using the standard procedure of the Bubnov – Galerkin method, taking into account the first two forms of eigen oscillations, the problem is reduced to the analysis of a fourth order nonlinear system of ordinary differential equations with respect to generalized coordinates and generalized velocities. For the linear version of the equations with low attenuation, the values of the critical load are obtained in dependence on the ratio of the damping coefficients of the eigenforms. The behavior of the system is analyzed taking into account the considered cubic nonlinearity using the method of nonlinear normal forms. The problem is reduced to the analysis of a nonlinear dynamical system with one degree of freedom. There are constructive analytical methods of qualitative and approximate quantitative research for this class of problems. The critical value of the compressive force was determined by the analysis of the dependence of the first Lyapunov magnitude on the load parameter. It turned out to be less than the value corresponding to the system without damping (the destabilizing effect of external nonlinear friction). A bifurcation occurs in the system at the critical load value. Therefore, the zero equilibrium becomes an unstable complex focus of the first order and a limit cycle is born. The function of the self-oscillations amplitude of the load magnitude is constructed using the harmonic linearization method.

Robust control of a single-ended primary inductor converter using adrc technique

Control of a single-ended primary inductor converter (SEPIC) is a challenging task as it is a fourth-order non-linear system. In this paper, an active disturbance rejection control (ADRC) scheme is proposed for the SEPIC converter with a high switching frequency. Selection of the observers and controller bandwidth is done according to the maximum sensitivity requirement to achieve robust performance towards varying input voltage and load. The effectiveness of the suggested ADRC is shown by calculating power and efficiencies for different loading conditions.

Control Oriented Modeling, Experimentation, and Stability Analysis of an Autorotating Samara

Abstract This paper presents a control-oriented model for describing the steady-state and dynamic behavior of a single-winged samara seed-pod in autorotative descent. A negligible lateral center of mass motion and constant, prescribed roll-angle to develop a simplified and compact model. Spanwise aerodynamic dependence is exchanged for an independent blade element representation with two tuned parameters to account for the effects of leading-edge vortex phenomena. The resulting model is a fourth-order nonlinear dynamical system. The accuracy of this model is established by validating it against our own experimental data as well as against those reported in the literature by other researchers. The validation exercise reveals that zero roll-angle is a viable assumption that significantly reduces model complexity while retaining accuracy. A necessary condition is derived for the existence of steady autorotation of the samara under free descent. Furthermore, a stability analysis is conducted suggesting that the eigenvalues of the fourth-order system, linearized about the autorotational equilibrium, can be well-represented by those of two decoupled two-dimensional systems. The analysis reveals the critical parameters that determine stability of sustained autorotation. Such stability analysis provides a platform for similar analytical exploration of future model improvements. The validity of this compact model suggests the plausibility of designing and controlling simple autorotative mechanisms based on these dynamics.

Wind-driven transport on the rotating spherical Earth

The dynamics of a water column at the surface of the ocean on the rotating spherical Earth forced by zonal wind stress is analyzed by substituting the angular momentum for the zonal velocity as one of the system's dependent variables. This substitution results in a model of the column's trajectory as a quasiparticle in a time dependent potential well. Explicit solutions are derived for the temporal changes in the angular momentum and the associated minima of the potential well as well as for the oscillations about these minima. The analytic results are confirmed by numerical solutions of the fourth-order nonlinear system of ordinary differential equations. For the eastward directed wind stress, our results provide exact formulas for the time it takes a column to reach the equator, where the dynamics is trivially described by the non-rotating paradigm of a particle subject to a constant force. In mid-latitudes, the analysis underscores the pivotal role played by the latitude where the wind-stress changes sign. Columns originating north or south of this latitude either converge to it or diverge away from it depending on whether the latitudinal change of the wind stress at this latitude is positive or negative. The oscillatory motion about this latitude is linearly unstable, and the growth rate of the amplitude is proportional to the gradient of the wind stress at that latitude.

Coupled non-oscillatory Duffing oscillators: Multistability, multiscroll chaos generation and circuit realization

We consider the dynamics of two coupled double-well non-oscillatory cubic Duffing oscillators. The coupling is realized by perturbing each one’s amplitude with a signal proportional to the amplitude of the other one. We prove that the coupling induces in the resulting fourth-order self-driven nonlinear system up to nine equilibrium points and engendered extremely complex dynamic features amongst which the coexistence of several bifurcation branches, multiple Hopf bifurcations (when a single parameter is considered), multistable behaviors (e.g. a pair of coexisting double scroll chaos, four coexisting mono-scroll chaos), and four-scroll chaotic attractor. A detailed investigation of these latter features is carried out by resorting to both analytical (i.e. Hopf bifurcation theorem, Routh–Hurwitz criterion) and numerical tools (e.g. Lyapunov exponent diagrams, 1D bifurcation diagrams, frequency spectrum, phase space trajectory plots, time series and attraction basins). An analog circuit implementing the coupled Duffing oscillators is performed by using simple electronic components (i.e. capacitors, resistors, quadruple operational amplifiers and analog multipliers chips). Experimentally captured phase portraits support the conclusions of the theoretical study. One of the important conclusions of this work is that the coupling of non-oscillatory Duffing type oscillators can be considered as an alternative approach to the generation of multiscroll chaos.

Sliding Mode Control of Underactuated Nonlinear Systems Based on Piecewise Double Power Reaching Law

A sliding mode control (SMC) strategy using novel piecewise double power reaching law (PDPRL) for a class of fourth-order nonlinear underactuated systems (USs) is proposed to realize simultaneous control of the unactuated and the actuated part. A novel PDPRL is designed in the form of a piecewise function to reduce the chattering produced by SMC. The fixed-time convergence characteristic and the existence of maximum convergence time independent of the initial value of the sliding mode surface for the PDPRL are analyzed. Considering the uncertainties of model parameters and external disturbance in the USs, a supertwisting disturbance observer (STDO) is provided to accurately estimate the disturbance in real time, and the estimated value is compensated to the controller. The underactuated inverted pendulum system is taken to verify the effectiveness of the devised strategy, and the simulation results show that the proposed strategy can reduce chattering, suppress the disturbance, and enhance the robustness of the system.

Swing up and stabilization control of rotary inverted pendulum based on energy balance, fuzzy logic, and LQR controllers

Rotary Inverted Pendulum (RIP) mimics the behavior of many practical control systems like crane mechanism, segway, unicycle robot, traction control in vehicles, rocket stabilization, and launching. RIP is a fourth-order nonlinear open-loop unstable dynamical system and is widely used for testing the effectiveness of the newly developed control algorithms. In this paper, a Hybrid Control Scheme (HCS) based on energy balance and fuzzy logic controllers is proposed to implement the swing up and stabilization control of RIP. In the proposed control scheme, the fuzzy logic-based state feedback gains are dynamically tuned in real-time by minimizing the absolute error between the desired and actual states to get robust control performance. The proposed HCS is also compared with the conventional Linear Quadratic Controller (LQR) for this application. The comparative results show that the proposed fuzzy logic-based hybrid control scheme gives the optimal control performance in terms of achieving satisfactory transient, steady-state, and robust responses from a given RIP system, as compared to the conventional LQR based control scheme. The proposed control scheme is also relatively less complex with a low computational cost and provides desired response characteristics as compared to the existing ones in the literature.

On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm to study the stability of the found regime and confirm the type of attractor.

Fuzzy logic and gradient descent-based optimal adaptive robust controller with inverted pendulum verification

This paper develops an adaptive robust combination of feedback linearization (FL) and sliding mode controller (SMC) based on fuzzy rules and gradient descent laws. The new suggested control algorithm is tested to stabilize a fourth-order under-actuated nonlinear inverted pendulum system. More precisely, the reliable feedback linearization approach and the robust SMC controller are combined to design a stable control effort. In order to enhance the performance of the suggested controller, an adaptation technique as long as fuzzy rules are applied to update the control gains and the boundary layer parameter. Then, a novel evolutionary algorithm termed multi-objective ant lion optimizer (MOALO) is implemented to determine the control coefficients. The analysis and results conducted on the inverted pendulum system demonstrate the desired performance of the proposed control scheme by providing an optimal smooth control input, suitable tracking performance, and proper time responses.

Orders & Contracts

This paper develops an adaptive robust combination of feedback linearization (FL) and sliding mode controller (SMC) based on fuzzy rules and gradient descent laws. The new suggested control algorithm is tested to stabilize a fourth-order under-actuated nonlinear inverted pendulum system. More precisely, the reliable feedback linearization approach and the robust SMC controller are combined to design a stable control effort. In order to enhance the performance of the suggested controller, an adaptation technique as long as fuzzy rules are applied to update the control gains and the boundary layer parameter. Then, a novel evolutionary algorithm termed multi-objective ant lion optimizer (MOALO) is implemented to determine the control coefficients. The analysis and results conducted on the inverted pendulum system demonstrate the desired performance of the proposed control scheme by providing an optimal smooth control input, suitable tracking performance, and proper time responses.

Computational intelligence approach using Levenberg–Marquardt backpropagation neural networks to solve the fourth-order nonlinear system of Emden–Fowler model

Computational intelligence approach using Levenberg–Marquardt backpropagation neural networks to solve the fourth-order nonlinear system of Emden–Fowler model

Study of a thermoacoustic-Stirling engine connected to a piston-crank-flywheel assembly.

This paper deals with the theoretical description of self-sustained oscillations resulting from the coupling of a piston-crank-flywheel assembly with a thermoacoustic-Stirling prime mover. The governing equations of the piston-flywheel motion are coupled to those of the thermoacoustic system, which is described in the time domain through a rational differential operator relating acoustic pressure fluctuations inside the cavity to the piston's velocity. As a result, the complete device is described by means of a fourth-order nonlinear dynamical system and solved numerically. The dynamical behavior of the system is studied as a function of the temperature difference along the thermoacoustic unit, and it is shown that the regime of stable rotations of the flywheel appears through a saddle-node bifurcation above a threshold value of the temperature difference. Moreover, the simulation results show good agreement with experiments.

Extended state observer based robust sliding mode control for fourth order nonlinear systems with experimental validation

It is commonly known that the design of control is difficult for the systems with non-minimum phase behavior, the coupling between the controlled variables and integral instability. The paper presents such a typical class of nonlinear system and propose a new technique generalized extended state observer-based sliding mode decoupling control. The proposed control algorithm is also able to compensate for the matched and mismatched uncertainties encountered in such nonlinear non-minimum phase systems. The decoupling control approach is used to decouple the entire system framework into subsystems. The sliding surface of the individual subsystem was composed using the estimated states of GESO. The proposed technique can nullify the effect of external disturbances, parametric uncertainties and unmodelled dynamics due to the inclusion of disturbance compensation gain in the control law. The proposed methodology enables the system to accomplish asymptotic stability and better dynamic response with lesser IAE and ITAE values than existing decoupling control strategies. An experimental application of the proposed technique to Quanser’s rotary pendulum module is investigated.

Optimal Design of a Fuzzy Controller Based on the Imperialist Competitive Algorithm for Fourth-order Nonlinear Systems

Nowadays, control and optimization of systems are two important and noticeable topics in all fields of science and engineering. One of the control methods, widely used because of its simplicity and accuracy, is the fuzzy control, which is based on the fuzzy logic theory. In the uncertainty conditions, this theory is able to formulate many inaccurate and vague implications, variables and systems in mathematical language, and provides requirements for reasoning, inference, control and decision making. This paper studies the stabilization of a Translation Oscillations with a Rotational Actuator (TORA) nonlinear fourth-order system by using a new fuzzy control technique which utilizes the small number of membership functions and rules. Besides, to determine the proper values for the controller parameters, the Imperialist Competitive Algorithm (ICA) is employed. Unlike the other optimization algorithms, ICA is not originated from evolutionary behavior of nature, but it is inspired by a human natural phenomenon called imperialism. In the optimization procedure, the objective function is defined as the sum of integrals of the rotor angle and the cart position. Finally, the comparison between the results obtained by the proposed strategy and some other researchers’ methods is provided, which shows the superiority of this work.

On the existence of periodic solutions of a fourth-order nonlinear system of parabolic equations

On the existence of periodic solutions of a fourth-order nonlinear system of parabolic equations

Nonsingular decoupled terminal sliding-mode control for a class of fourth-order under-actuated nonlinear systems with unknown external disturbance

This study is concerned with the design of a nonsingular decoupled terminal sliding mode controller for a class of fourth-order under-actuated uncertain nonlinear systems with unknown external disturbance. For the unmeasured disturbance, a disturbance observer with finite-time convergence of estimation error to zero is proposed. The nonsingular decoupled terminal sliding mode controller is designed by utilizing the output of the proposed disturbance observer. Also, an input saturation constraint and control singularity are considered in the controller design. The finite-time stability and convergence of the disturbance observer are proved for the closed-loop system. In addition, the control of an electrostatically actuated Timoshenko nanobeam subjected to Casimir force is simulated to demonstrate the effectiveness and performance of the proposed control scheme.

Characteristics of higher-order vector rogue waves to a coupled fourth-order nonlinear Schrödinger system in a two-mode optical fiber

Optical fiber communication serves a significant role in recent communication. Investigated in the work is a coupled fourth-order nonlinear Schrodinger system for the ultrashort optical pulses in a two-mode optical fiber. For the complex envelopes of two field polarization components, we work out the Darboux transformation and higher-order vector rogue-wave solutions. The second-order vector rogue waves with the triangle structure, each component of which is composed of three four-petaled rogue waves, are shown. With the strength of the higher-order linear and nonlinear effects increasing, width of the triangle structure along the distance axis decreases, while width of the triangle structure along the time axis increases. We obtain the second-order vector rogue waves with the merged structure and the third-order vector rogue waves with the triangle and pentagon structures. We expect that our results could be helpful in studying the rogue-wave phenomena in the nonlinear optics.

POWER SYSTEM STABILIZER DESIGN USING NOTCH-FILTER – BARKA II (OMAN) - CASE STUDY

The use of power system stabilizers (PSSs) to damp power system swing mode oscillations is of extreme practical importance. This manuscript presents an approach to the stabilization of a single machine infinite bus system (SMIB). The proposed control is based on the notch-filter approach to cancel the poles near to the imaginary axes. The approach is based on the root locus method. Application to Barka II power station connected to the main interconnected system of Oman is presented. The peak load at summer is considered as the system is near to instability. The Barka SMIB is modeled as a fourth order non-linear system. A linearized model is then obtained using MATLAB/Simulink. The simulation results validate the proposed design for the PSS.

Vector breathers for the coupled fourth-order nonlinear Schrödinger system in a birefringent optical fiber

Abstract Optical fiber communication is widely applied in the information transmission and capacity handling of information. In this paper, coupled fourth-order nonlinear Schrodinger system, which describes the propagation of ultrashort optical pulses in a birefringent optical fiber, is studied. The Nth-order Darboux transformation and corresponding vector breather solutions are constructed, where N is a positive integer. Based on such solutions, we graphically show four different types of breathers: (1) the one with the one component containing the anti-eye-shaped breather and the other component containing the eye-shaped breather; (2) the one with each component containing the four-petaled breather; (3) the one with each component containing the eye-shaped breather; (4) the one with each component containing the Y-shaped breather. Moreover, we find that the range of the breather along the time axis decreases, and the angle between the breather and the time axis increases as the value of |γ| decreases, where γ represents the strength of higher-order linear and nonlinear effects.

An optimal adaptive hybrid controller for a fourth-order under-actuated nonlinear inverted pendulum system

In this paper, a combination of approximate feedback linearization and sliding mode control approaches is applied to stabilize a class of fourth-order nonlinear systems. In order to improve the performance of the proposed controller, design parameters of the sliding surface are adjusted using the adaptation laws, based on the gradient descent technique. Eventually, a new version of particle swarm optimization is implemented to optimize the control gains with respect to the integral of time-weighted absolute errors as the objective function, and a constraint on the maximum control effort value. The introduced idea is put into practice for the under-actuated inverted pendulum system, and the procedure is described with details step by step. The time responses of cart position, pendulum angle, and control effort of the proposed method is then compared with those of several recently published references so as to evaluate the effectiveness of this scenario.