Nonlinear Pendulum Research Articles

Existence of a period two solution of a delay differential equation

We consider the existence of a symmetric periodic solution for the following distributed delay differential equation \begin{document}$ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $\end{document} where \begin{document}$ f(x) = r\sin x $\end{document} with \begin{document}$ r>0 $\end{document} . It is shown that the well studied second order ordinary differential equation, known as the nonlinear pendulum equation, derives a symmetric periodic solution of period \begin{document}$ 2 $\end{document} , expressed in terms of the Jacobi elliptic functions, for the delay differential equation. We here apply the approach in Kaplan and Yorke (1974) for a differential equation with discrete delay to the distributed delay differential equation.

Robust adaptive anti-synchronization control of multiple uncertain chaotic systems of different orders

The precise anti-synchronization control of uncertain chaotic systems has always remained an interesting problem. The anti-synchronization control of multiple different orders uncertain chaotic systems increases the complexity and enhances the security of the information signal in secure communications. Hence, it confines the hacking in digital communication systems. This paper proposes a novel adaptive control technique and studies the double combination anti-synchronization of multiple different orders uncertain chaotic systems. The proposed adaptive feedback control technique consists of three fundamental nonlinear components. Each component accomplishes a different objective; (i) stability of the closed-loop, (ii) smooth and fast convergence behaviour of the anti-synchronization error, and (iii) disturbance rejection. The theoretical analysis in (i) to (iii) uses the Lyapunov stability theory. This paper also provides parameters adaptation laws that stabilize the uncertain parameters to some constants. The paper discusses the simulation results of two representative examples of four different orders uncertain chaotic systems. These examples demonstrate anti-synchronization among hyperchaotic Lü, uncertain chaotic Shimizu Morioka, uncertain second-order nonlinear duffing, and uncertain parametrically excited second-order nonlinear pendulum systems. The computer-based simulation results certify the efficiency and performance of the proposed anti-synchronization control approach and compare them with peer works.

Experimental Verification of two Theoretical Solutions of the Pendulum for Large Angles in Frequency Domain for Teaching Support

During pendulum analysis, the approximation for small angles is usually performed as a simple harmonic motion. However, for large angles, this approximation is not convenient so exact solutions are proposed by differ-ent methods. This paper presents the comparison of two solutions for the displacement of the pendulum in the domain of time and frequency, the solution by Jacobi elliptic function and the solution by the numerical method Dormand-Prince with the results of measurements obtained by means of a physical prototype designed for the teaching of physics. We consider that this comparative study allows a better understanding of the phenomenon of the non-linear pendulum in the students of undergraduate careers in the physics of waves as well as a previous training for the course of analysis of signals being in good exercise of teaching

The interplay of damping and amplitude in the nonlinear pendulum

A highly accurate solution of the nonlinear pendulum with small initial amplitude and damping proportional to the velocity is derived using multi-time scale perturbation methods. The nonlinear pendulum has been studied extensively, but its formulation has rarely incorporated damping. This paper extends the body of traditional results by using a mathematical framework that incorporates damping to get a clear, robust solution in which the interplay of the damping and amplitude yields new physical insights into the pendulum's motion.

Gait modification for improving walking stability of exoskeleton assisted paraplegic patient

The currently available exoskeleton for assisting the paraplegic patient in walking usually adapts a pre-programmed gait that involves the patient following an exoskeleton lead. The system allows the patient to hold a pair of canes in order to keep balance, and does not contribute to keeping balance without the patient’s action. This paper proposes an algorithm based on the zero moment point (ZMP) to modify the gait generated through human walking synergy for paraplegic patients who make use of the exoskeleton system and hold their canes. The proposed ZMP will enable the paraplegic patient to keep balance during walking and also reduce the burden in maintaining balance. First, a pair of cane is used as an interface to control the user’s walking and then, the synergy between legs and canes is used to synchronize the user’s walking intention during the exoskeleton movement. The walking synergy is extracted from the able-bodied subject walking with a pair of canes and analyzed using principal component analysis (PCA). In order to improve the walking stability, the hip joint angle on stance leg during walking was modified based on ZMP. Furthermore, a nonlinear inverted pendulum (NIP) model was utilized in order to generate a gait with a fully stretched knee joint angle that is similar to human gait. The proposed method was verified via the Gazebo simulation using a walking robot to simulate a patient wearing an exoskeleton. The experiment results show that the walking stability was highly improved after gait modification.

Oscillation free robust adaptive synchronization of chaotic systems with parametric uncertainties

The complexity of the closed-loop system, short transient response time, and fast synchronization error convergence rates are the three basic parameters that limit hacking in the data encryption and secure the communication systems. This paper addresses the following two challenges: The full-order synchronization (FOS) of two parametrically excited second-order nonlinear pendulum (PENP) chaotic systems with uncertain parameters. The reduced-order synchronization (ROS) between the canonical projection part of an uncertain third-order chaotic Rossler and the uncertain PENP systems. This article designs a new robust adaptive synchronization control (RASC) algorithm to address the above two challenges. The proposed controller achieves the FOS and ROS in a shorter transient time, and the synchronization error signals converge to the origin with faster rates in the presence of bounded unknown state-dependent and time-dependent disturbances. The Lyapunov direct method verifies this convergence behavior. The paper provides parameters updated laws that confirm the convergence of the uncertain parameters to some fixed values. The controller does not cancel the nonlinear terms of the plant; this property of the controller keeps the nonlinear terms in the closed-loop that results in the enhanced complexity of the dynamical system. The proposed RASC strategy is successful in synthesizing oscillation free convergence of the synchronization error signals to the origin for reducing the transient time and guarantees the asymptotic stability at the origin. The simulation results endorse the theoretical findings.

An improved implicit method with dissipation control capability: The simple generalized composite time integration algorithm

In this paper, a simple and accurate implicit composite time integration scheme is newly developed for more efficient transient analyses of structural problems. The new scheme does not require the acceleration vector of the previous time step, and the initial acceleration vector is not needed, either. As a result, the factorization of the mass matrix is unnecessary, while factorizations of the effective coefficient matrix are still needed. Besides, the new method is designed to possess controllable algorithmic dissipation. Despite the absence of the acceleration vector of the previous time step, the numerical performance of the new method is equivalent to those of the existing methods. The excitation of an elastic bar problem and the simple nonlinear pendulum problem are numerically analyzed for the test of the new scheme. In the numerical experiments, the effects of consistent and lumped mass matrices are also discussed.

Adaptive stabilization by delay with biased measurements

The problem of output robust adaptive stabilization for a class of Lipschitz nonlinear systems is studied under assumption that the measurements are available with a constant bias. The state reconstruction is avoided by using delayed values of the output in the feedback and adaptation laws. The control and adaptation gains can be selected as a solution of the proposed linear matrix inequalities (LMIs). The efficiency of the presented approach is demonstrated on a nonlinear pendulum through simulations.

Tuning of parameters of PID controller using Grey Wolf Optimizer

In this paper, the Grey Wolf Optimizer (GWO) is used for tuning the parameters of PID (Proportional, Integral and Derivative) controller. The GWO is a meta-heuristic approach inspired by grey wolves (Canis lupus). The GWO algorithm consists of four types of grey wolves known as alpha (∝) as a leader or decision maker, beta (𝛽) as to assist alpha in decision making, delta (𝛿) as a subordinate wolf and omegas (𝜔) the followers. The hunting behavior of GWO continues with the three main following steps and they are searching for prey, encircling the prey and attacking the prey. Firstly, the parameters of the PID controller are tuned using the Ziegler-Nichols (Z-N) technique. Then these tuned parameters along with some randomly assumed coefficients are used as the first iteration step for the GWO algorithm. The tuning principle of GWO ensures the minimization of performance index. The performance index used in this paper is Integral Square error (ISE). The GWO algorithm for tuning the parameters of PID controller is validated by controlling the highly nonlinear in-verted pendulum system. Simulation results showed that the GWO algorithm minimizes the ISE of the system and stabilized the system within the stipulated time.

How A PID Controlling A Nonlinear Plant

This paper investigates the weakness of feedback controller to find the best parameter of PID controller of a nonlinear inverted pendulum system (NL-IPS). Stability and performance analysis of a NL-IPS are carried out with combination of feedback and PID controllers. It is found that the NL-IPS which was stabilized by feedback controller provides some weakness. By addition of PID controller on closed loop NL-IPS will provide better tracking and response on zero steady state error.

Adaptive and Robust Second Order Sliding Mode Controller Dedicated to Nonlinear Uncertain Pendulum Application System

The research work presented within this paper deals with an innovative second-order sliding mode control (SOSMC) allocated to adaptive gain and associated with nonlinear systems subject to unknown but bounded uncertainties. The derived controller guarantees the control gain dynamical adaptation for the sake of counteracting the system’s uncertainties and to mitigate the chattering phenomenon. The Lyapunov method is also used to analyses the stability of any closed loop system (CLS) within a finite-time under bounded uncertainties assumptions. To assess how effective is the approach considered within this paper, the adaptive controller has been carefully studied on a benchmark of nonlinear systems on a damped overturned pendulum.

Adaptive and Robust Second Order Sliding Mode Controller Dedicated to Nonlinear Uncertain Pendulum Application System

The research work presented within this paper deals with an innovative second-order sliding mode control (SOSMC) allocated to adaptive gain and associated with nonlinear systems subject to unknown but bounded uncertainties. The derived controller guarantees the control gain dynamical adaptation for the sake of counteracting the system’s uncertainties and to mitigate the chattering phenomenon. The Lyapunov method is also used to analyses the stability of any closed loop system (CLS) within a finite-time under bounded uncertainties assumptions. To assess how effective is the approach considered within this paper, the adaptive controller has been carefully studied on a benchmark of nonlinear systems on a damped overturned pendulum.

Designing, implementing and analysing optimal controllers on a non-linear reaction wheel pendulum

This paper discusses a reaction wheel control system. A reaction wheel pendulum is good example of a non-linear and underactuated system, which attracts control system researchers to develop many control algorithms. The reaction wheel plant is usually used for studying advanced control system courses. In this paper, a mathematical model of the state space is discussing. A proposed LQR control algorithm is explained. Simulation and real time experiments have seen carried out to verify the performance of the proposed closed loop system. The LQR optimisation algorithm was able to find the optimum feedback gains. The simulation and real time experiments show that the reaction wheel pendulum could stabilize the pendulum at upright position.

Enhanced studies on the composite sub-step algorithm for structural dynamics: The Bathe-like algorithm

The Bathe algorithm is superior to the trapezoidal rule in solving nonlinear problems involving large deformations and long-time durations. Generally, the parameter γ=2−2 is highly recommended due to its optimal numerical properties. This paper further studies this implicit composite sub-step algorithm and thus presents a class of the Bathe-like algorithm. It not only gives a novel family of composite algorithms whose numerical properties are the exactly same as the original Bathe algorithm with γ=2−2, but also provides the generalized alternative to the original Bathe algorithm with any γ. In this study, it has been shown that the Bathe-like algorithm, including the original Bathe algorithm, can reduce to two common single-step algorithms: the trapezoidal rule and the backward Euler formula. Besides, a new parameter called the algorithmic mode truncation factor is firstly defined to describe the numerical property of the Bathe-like algorithm and it can estimate which modes to be damped out. Finally, numerical experiments are provided to show the superiority of the Bathe-like algorithm over some existing methods. For example, the novel Bathe-like algorithms are superior to the original Bathe algorithm when solving the highly nonlinear pendulum.

Anomalies in the switching dynamics of C -type antiferromagnets and antiferromagnetic nanowires

Antiferromagnets (AFMs) are widely believed to be superior than ferromagnets in spintronics because of their high stability due to the vanishingly small stray field. It is thus expected that the order parameter of AFM should always align along the easy-axis of the crystalline anisotropy. In contrast to this conventional wisdom, we find that the AFM order parameter switches away from the easy-axis below a critical anisotropy strength when an AFM is properly tailored into a nano-structure. The switching time first decreases and then increases with the damping. Above the critical anisotropy, the AFM order parameter is stable and precesses under a microwave excitation. However, the absorption peak is not at resonance frequency even for magnetic damping as low as 0.01. To resolve these anomalies, we first ascertain the hidden role of dipolar interaction that reconstructs the energy landscape of the nano-system and propose a model of damped non-linear pendulum to explain the switching behavior. In this framework, the second anomaly appears when an AFM is close to the boundary between underdamped and overdamped phases, where the observed absorption lineshape has small quality factor and thus is not reliable any longer. Our results should be significant to extract the magnetic parameters through resonance techniques.

Adaptive fault-tolerant consensus for a class of leader-following systems using neural network learning strategy

In this paper, the leader-following consensus problem of a class of nonlinearly multi-dimensional multi-agent systems with actuator faults is addressed by developing a novel neural network learning strategy. In order to achieve the desirable consensus results, a neural network learning algorithm composed of adaptive technique is proposed to on-line approximate the unknown nonlinear functions and estimate the unknown bounds of actuator faults. Then, on the basis of the approximations and estimations, a robust adaptive distributed fault-tolerant consensus control scheme is investigated so that the bounded results of all signals of the resulting closed-loop leader-following system can be achieved by using Lyapunov stability theorem. Finally, efficiency of the proposed adaptive neural network learning strategy-based consensus control strategies is demonstrated by a coupled nonlinear forced pendulums system.

Toward an artificial intelligence physicist for unsupervised learning.

We investigate opportunities and challenges for improving unsupervised machine learning using four common strategies with a long history in physics: divide and conquer, Occam's razor, unification, and lifelong learning. Instead of using one model to learn everything, we propose a paradigm centered around the learning and manipulation of theories, which parsimoniously predict both aspects of the future (from past observations) and the domain in which these predictions are accurate. Specifically, we propose a generalized mean loss to encourage each theory to specialize in its comparatively advantageous domain, and a differentiable description length objective to downweight bad data and "snap" learned theories into simple symbolic formulas. Theories are stored in a "theory hub," which continuously unifies learned theories and can propose theories when encountering new environments. We test our implementation, the toy "artificial intelligence physicist" learning agent, on a suite of increasingly complex physics environments. From unsupervised observation of trajectories through worlds involving random combinations of gravity, electromagnetism, harmonic motion, and elastic bounces, our agent typically learns faster and produces mean-squared prediction errors about a billion times smaller than a standard feedforward neural net of comparable complexity, typically recovering integer and rational theory parameters exactly. Our agent successfully identifies domains with different laws of motion also for a nonlinear chaotic double pendulum in a piecewise constant force field.

A Novel Robust Control of Uncertain Furuta Pendulum Based on a General Lyapunov Function

AbstractA novel robust approach for the obedience control of Furuta pendulum with uncertainty is proposed. The uncertainty considered in this paper is (possibly fast) time-varying and bounded, which may exist in any stage of the pendulum subsystem. By the Lagrangian formulation of the nonlinear pendulum system, a robust control, based on a general Lyapunov function, is designed to render the Furuta pendulum a position obedience. As a consequence of the Lyapunov approach, the control design is not restricted to linearize the pendulum system. The system performance under the proposed control is guaranteed as uniform boundedness and uniform ultimate boundedness. The salient features of this new control are demonstrated both analytically and numerically. The experiment is conducted in the Furuta pendulum system to prove the validity and effectiveness of the control design.

Broadband energy harvesting by nonlinear magnetic rolling pendulum with subharmonic resonance

Nonlinear systems may exhibit secondary resonances, which can provide an additional and thus broadened bandwidth for energy harvesting. However, the secondary resonances of nonlinear energy harvesters reported in the literature suffer from low-power output and limited bandwidth. This work proposes a novel magnetic rolling pendulum (MRP) with a large bandwidth and high power output in both primary and secondary resonances for energy harvesting. The MRP employs the rolling motion of a magnetically levitated permanent magnet with minimal mechanical damping. A prototype was fabricated and characterised. An analytical model combined with finite element analysis was developed and validated by experiment. Both experiment and simulation show that the MRP has a linear resonance frequency of 4.6 Hz and peak power of 3.7 mW. It exhibits strong nonlinear behaviours and broadband characteristics with excitation amplitude as low as 2 m/s2 in the primary resonance. As the excitation amplitude is larger than 5 m/s2, the secondary resonance (1/2 order subharmonics) is excited. The responses of the MRP at the subharmonic resonance take the same form as the primary resonance in terms of displacement and power outputs. This helps the subharmonic resonance to produce the same power level as the primary resonance but with a larger bandwidth. When excited at 14 m/s2, the MRP shows 1-mW-bandwidth of 9.7 Hz, 2/3 of which is attributed to the subharmonic resonance.

Fixed-Time Synchronization Control for a Class of Master–Slave Systems Based on Homogeneous Method

This brief investigates the fixed-time synchronization problem for a class of master–slave systems with second-order nonlinear dynamics via output feedback. Based on the homogeneous system theory, a new fixed-time synchronization control algorithm and a fixed-time observer are designed such that the synchronization of master–slave coupling systems can be achieved in a fixed time. It is theoretically shown that the synchronization time can be pre-determined which is independent of any initial conditions. Numerical simulations on synchronization of master–slave nonlinear forced pendulum systems are finally performed to illustrate the effectiveness of the proposed method.