Unstable Equilibrium Point Research Articles

Generalised Proportional Integral Control for Magnetic Levitation Systems Using a Tangent Linearisation Approach

This paper applies a robust generalised proportional integral (GPI) controller to address the problems of stabilisation and position tracking in voltage-controlled magnetic levitation systems, with consideration of the system’s physical parameters, non-linearities and exogenous disturbance signals. The controller has been developed using as a basis a model of the tangent linearised system around an arbitrary unstable equilibrium point. Since the approximate linearised system is differentially flat, it is therefore controllable. This flatness gives the resulting linearised system a relevant cascade characteristic, thus allowing simplification of the control scheme design. The performance of the proposed GPI controller has been analysed by means of numerical simulations and compared with two controllers: (i) a standard proportional integral derivative (PID) control, and (ii) a previously designed exact feedforward-GPI controller. Simulation results show that the proposed GPI control has a better dynamic response than the other two controllers, along with a better performance in terms of the integral squared tracking error (ISE), the integral absolute tracking error (IAE), and the integral time absolute tracking error (ITAE). Finally, experimental results have been included to illustrate the effectiveness of the proposed controller in terms of position stabilisation and tracking performance when appreciable non-linearities and uncertainties exist in the underlying system. Comparative graphs and metrics have shown a superior performance of the proposed GPI scheme to control the magnetic levitation platform.

Estimation of the spawning stock and recruitment relationship of Octopus vulgaris in Asturias (Bay of Biscay) with generalized depletion models: implications for the applicability of MSY

Abstract We developed a new type of generalized depletion model adapted to the rapid life cycle of Octopus vulgaris and the data collection framework put in place in its co-managed small-scale fishery (SSF) in Asturias, northwest Spain. The model uses weekly data from 19 seasons to estimate annual recruitment and female spawners emigration out of the vulnerable stock to attend to their broods. The matched annual spawners-recruitment (SR) magnitudes are then used as observations in a non-Bayesian hierarchical inference framework. The Shepherd SR model was selected among three alternatives. The stock has a high degree of density-dependence, leading to overcompensation. The estimated SR model has an unstable equilibrium point, driving the stock to deterministic cyclic fluctuations under small disturbances. Under those conditions the maximum sustainable yield (MSY) is not applicable, lending further credence to Sidney Holt’s opposition to the MSY as a basis for the elaboration of fishery management objectives, with potentially wide relevance to the management of SSF and cephalopod fisheries. We argue that harvest rates based on mean latent productivity, a concept that includes the MSY as a special case, are more adequate and sustainable for fluctuating stocks.

Angular dependence of vertical force and torque when magnetic dipole moves vertically above flat high-temperature superconductor* *Projects supported by the National Natural Science Foundation of China (Grant No. 11572232) and the China Three Gorges Corporation Research Project (Grant No.202103407).

The interaction between a permanent magnet (PM) assumed as a magnetic dipole and a flat high-temperature superconductor (HTS) is calculated by the advanced frozen-image model. When the dipole vertically moves above the semi-infinite HTS, the general analytical expression of vertical force and that of torque are obtained for an arbitrary angle between the magnetization direction of the PM and the c axis of the HTS. The variations of the force and torque are analyzed for angle and vertical movements in both zero-field cooling (ZFC) condition and field cooling (FC) condition. It is found that the maximum vertical repulsive or attractive force has the positive or negative cosine relation with the angle. However, the maximum torque has the positive or negative sine relation. From the viewpoint of the rotational equilibrium, the orientation of the magnetic dipole with zero angle is deemed to be an unstable equilibrium point in ZFC, but the same orientation is considered as a stable equilibrium point in FC. In addition, both of the variation cycles of the maximum force and torque with the angle are π.

Model‐free control of mechatronic systems based on algebraic estimation

AbstractThis paper presents the practical application of a class of model‐free control considering the algebraic estimation approach. Two mechatronic systems are considered: a reaction wheel inverted pendulum (RWIP) and a two degree‐of‐freedom (2‐DOF) helicopter. For the RWIP, since it is under‐actuated, it is proposed a modified ultra‐local model including a feedback of the reaction wheel velocity, and then, the algebraic estimator obtains the parameters of this ultra‐local model around the unstable equilibrium point. For the 2‐DOF helicopter, the ultra‐local model is estimated during a tracking task in a wide range of operation. Some insights on how to choose the fixed parameters of controller are presented. Simulation and practical results corroborate the efficiency of the method, without needing any information on the systems' dynamics.

Self-excited oscillations in an inverted cart–pendulum based on the two-relay approach

Solving the orbital stabilization of a class of underactuated systems near its open-loop unstable equilibrium point is a challenging task but useful in repetitive tasks. This paper addresses the control problem of an inverted cart–pendulum’s motion driven by a two-relay controller. This controller is tuned to set the desired amplitude and frequency using the locus of the perturbed relay system, which is a frequency-domain method. We solved the periodic motion of the pendulum occurring in its open-loop unstable equilibrium point. The experimental results showed that the proposed two-relay controller forced the pendulum into a periodic motion around the upright position.

A Signal Compensation-Based Robust Swing-Up and Balance Control Method for the Pendubot

This article proposes a novel robust swing-up and balance control method using data-driven compensation signals for the Pendubot system, which is underactuated and subject to dynamic friction, backlash, and modeling uncertainty. The method involves three major developments. First, the uncertainty of the system is described by a previous sampled unknown nonlinear term and its changing rate. Compensation signals are then designed to eliminate the influence of the previous sampled unknown nonlinear term and its changing rate on the outputs of the controlled plant. Second, based on the Lyapunov stability theory, a robust swing-up control method is proposed. Third, in order to eliminate the limit cycle phenomenon caused by the uncertainty near the unstable equilibrium point, a proportional-derivative balance controller based on compensation signals is proposed. The stability and tracking error analysis show that the compensation signals can effectively eliminate the influence of dynamic friction, backlash, and modeling uncertainty on the system. Experimental results verify the effectiveness of the proposed method by showing significant improvements in production rate, accuracy, and cost.

Design of a Fuzzy Optimization Control Structure for Nonlinear Systems: A Disturbance-Rejection Method

This paper tackles the control problem of nonlinear disturbed polynomial systems using the formalism of output feedback linearization and a subsequent sliding mode control design. This aims to ensure the asymptotic stability of an unstable equilibrium point. The class of systems under investigation has an equivalent Byrnes–Isidori normal form, which reveals stable zero dynamics. For the case of modeling uncertainties and/or process dynamic disturbances, conventional feedback linearizing control strategies may fail to be efficient. To design a robust control strategy, meta-heuristic techniques are synthesized with feedback linearization and sliding mode control. The resulting control design guarantees the decoupling of the system output from disturbances and achieves the desired output trajectory tracking with asymptotically stable dynamic behavior. The effectiveness and efficiency of the designed technique were assessed based on a benchmark model of a continuous stirred tank reactor (CSTR) through numerical simulation analysis.

Impulsive control of a class of multiple unstable neural networks

This paper addresses the issue of stability of a class of multiple unstable Cohen–Grossberg neural networks(CGNNs) under impulsive control. Some novel sufficient conditions are given to make the unstable equilibrium points of the model locally μ-stable. An example is offered to demonstrate the effectiveness of the control strategy by comprehensive computer simulations.

On the Existence and Stability of Periodic Solutions of Airy’s Equation with Elastic Coefficients

In this paper, the existence and stability of periodic solutions of a certain second order differential equation with elastic coefficient were investigated using power series method, eigenvalue approach and lyapunov direct method. Existence of analytical solution which is independent of time was achieved using the power series method. Eigenvalue approach and Lyapunov direct method were used to investigate the stability of the resulting solution. Periodic solution was obtained using the eigenvalues of the resulting matrix. The first stability method further examined stability of the equilibrium point by considering the intervals around the origin and it’s discriminate. The equilibrium points for the intervals and the discriminate were unstable because the real part of the characteristics root is zero. Unstable equilibrium point was also obtained for the second stability method using the energy function and time derivative around the equilibrium point. The two unstable results indicated that there were highly instability regions with a strictly positive elastic coefficient. The highly instability regions were confirmed by the presence of elastic coefficient which reduces oscillation with an increase in amplitude. Furthermore, numerical simulations for existence and stability of Airy’s equation at different values of the elastic coefficient were illustrated in order to demonstrate the behaviour of the solutions which extends some results in literature.

Block Cipher Nonlinear Confusion Components Based on New 5-D Hyperchaotic System

The security strengths of block ciphers greatly rely on the confusion components which have the tendency to transform the data nonlinearly into the perplexed form. This paper proposes to put forward a novel scheme of generating cryptographically strong nonlinear confusion components of block ciphers, usually termed as substitution-boxes (S-boxes). The anticipated S-box design scheme is based on a novel five-dimensional (5-D) chaotic system analyzed in this paper. The proposed 5-D dynamical system consists of hyperchaotic phenomenon, KY dimension, conservativity, unstable equilibrium point, and complex phase attractors which are suited for cryptographic applications. The S-box based on hyperchaotic system is made to evolve in order to generate an optimized S-box for high nonlinearity score to make it robust against many linear attacks. The performance analysis of proposed S-box demonstrates that it has bijectivity, high nonlinearity; satisfied strict avalanche criterion and bits independent criterion; low differential and linear probabilities. Moreover, performance appraisal of proposed S-box justifies its better strength and features over many recently investigated S-boxes.

A Novel Megastable Oscillator with a Strange Structure of Coexisting Attractors: Design, Analysis, and FPGA Implementation

Megastable chaotic systems are somehow the newest in the family of special chaotic systems. In this paper, a new megastable two‐dimensional system is proposed. In this system, coexisting attractors are in some islands, interestingly covered by megalimit cycles. The introduced two‐dimensional system has no defined equilibrium point. However, it seems that the origin plays the role of an unstable equilibrium point. Therefore, the attractors are determined as hidden attractors. Adding a forcing term to the system, we can obtain chaotic solutions and coexisting strange attractors. Moreover, the effect of three different values of the forcing term’s amplitude is studied. The dynamical properties of the designed system are investigated using attractor plots, bifurcation diagrams, and Lyapunov Exponents diagram. Phase portraits of the novel megastable oscillator are presented by FPGA design. Xilinx system generator block diagrams of the proposed system and trigonometric functions are also presented.

Vehicle drifting: mathematical theory and dynamic analysis for stabilised drifting of RWD vehicles

This paper introduces a mathematical definition for drifting of vehicles. A necessary kinematic condition for enabling negative body side-slip angle at centre of the front wheel during a left-hand turn is identified as drifting indicator. Dynamics of the drifting motion is investigated by means of a simplified three-wheel vehicle model in steady-state, focusing on rear wheel drive (RWD) vehicles. Drifting point is identified as an unstable equilibrium point of the nonlinear system. The equilibrium point is made stable by means of direct control only over yaw velocity, as a single key stabilising objective, and using the steady-state model as feedforward. A four-wheel planar dynamic simulation model is used together with a combined slip tyre model to investigate the accuracy of the proposed analysis. Steady-state RWD drifting motion is achieved by using the proposed method, validating the proposed dynamics analysis. Finally, the drifting metric is used to measure the amount of drifting achieved during a manoeuvre and its effectiveness is observed.

Vehicle drifting: mathematical theory and dynamic analysis for stabilised drifting of RWD vehicles

This paper introduces a mathematical definition for drifting of vehicles. A necessary kinematic condition for enabling negative body side-slip angle at centre of the front wheel during a left-hand turn is identified as drifting indicator. Dynamics of the drifting motion is investigated by means of a simplified three-wheel vehicle model in steady-state, focusing on rear wheel drive (RWD) vehicles. Drifting point is identified as an unstable equilibrium point of the nonlinear system. The equilibrium point is made stable by means of direct control only over yaw velocity, as a single key stabilising objective, and using the steady-state model as feedforward. A four-wheel planar dynamic simulation model is used together with a combined slip tyre model to investigate the accuracy of the proposed analysis. Steady-state RWD drifting motion is achieved by using the proposed method, validating the proposed dynamics analysis. Finally, the drifting metric is used to measure the amount of drifting achieved during a manoeuvre and its effectiveness is observed.

Design and Analysis of a Novel Six-Dimensional Hyper Chaotic System

The chaotic system has been widely studied. A new six-dimension hyper chaotic system is introduced in this paper. We used a new chaotic system based on a six-dimension for the purpose of increasing chaos in the system, where the new system has eleven positive parameters, complicated chaotic dynamics behaviors and gives an analysis of the new systems. The basic characteristics and dynamic behavior of this system are investigated with a presence of chaotic attractor, Dissipativity, symmetry, equilibrium points, Lyapunov Exponents, Kaplan-Yorke dimension, waveform analysis and sensitivity toward initial conditions. The results of the analysis exhibit that the new system contains three unstable equilibrium points and the six Lyapunov exponents. Maxim non-negative Lyapunov Exponent (MLE) is obtained as 4.72625, and Kaplan-Yorke are obtained as 3.92566, and the new system characteristics with, unstable, high complexity, and unpredictability, the new system dynamics is simulated utilizing MATHEMATICA program. The phase portraits and the qualitative properties of the new hyper chaotic system have been described at the detail.

Nonintegrability of the problem of a spherical top rolling on a vibrating plane

This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.

Design and experimental research of haw core removal machine based on buoyancy orientation mechanism

AbstractIn order to solve the problems of the labor cost and the low mechanization level in the intensive processing of the haw products, a haw core removal machine based on the buoyancy orientation was designed, which solved the two key technical problems including the oriented conveying and the core removal in its coring process. The effects of different cutting edge structures on the force on the pulp and the shape of the final pulp were studied, and the effects of the haw's own shape and the medium flow velocity on the orientation righting rate were analyzed. The results showed that the ratio of height to diameter had a direct effect on the orientation righting rate of hawthorn, and the two unstable equilibrium points could be improved by increasing the flow velocity of the medium. The prototype experiment showed that the buoyancy oriented conveying was a feasible and effective method in the processing of the small‐scale fruit with the uniaxial symmetry such as the haw. In the same scale of the haw core removal processing, its production capacity could be increased by about 30%. The research results provided a basis for the follow‐up improvement of the automatic haw core removal machine.Practical applicationsThe traditional haw deep processing enterprises mostly use manual or semi‐automatic equipment for the haw core removal, which greatly restricts the production efficiency of enterprises. The present study provides an automatic haw core removal processing equipment based on the buoyancy orientation mechanism and gives the optimal cutting edge structure. The experimental results show that the main factors affecting the haw orientation and centering are the shape of the haw and the flow velocity. Furthermore, on the basis of orthogonal experiment, the process parameter surface of the orientation righting rate is established, which can provide reference for the subsequent design of the haw core removal machine and the optimization of the process parameters in the production process.

Unraveling the combined actions of a Holling type III predator–prey model incorporating Allee response and memory effects

In this article, we have studied a memory dependent prey–predator model with logistic type prey growth subject to the Allee effect and Holling type-III functional response. To include the memory effect, we have used here a fractional order system. The model contains at most four equilibrium points among which one is trivial, two are axial, and the last one is interior. Here we have studied behavior of these four equilibrium points using eigen analysis method. Our investigation shows that the trivial equilibrium point is stable for strong Allee effect, and unstable for weak Allee effect. Between the two axial equilibrium points, one is stable under certain condition but the other is always unstable. This unstable equilibrium point exists only for strong Allee case. The interior equilibrium point arises when the axial equilibrium point is unstable. Stability condition of this interior equilibrium point depends on the parametric relationship. The schematic diagram divides the a-p plane into five sub-regions in both the cases of memory dependent and memory less system. The number and nature of the equilibrium points in different sub-regions are different. Finally, we have drawn the bifurcation diagrams considering all the equilibrium points in different sub-regions and the results are concluded at the end.

WITHDRAWN: Hybrid synchronization for a novel class of 6D system with unstable equilibrium points

Abstract In recent years, the classification of attractor’s systems witnessed a lot of interest because of its important applications in various fields. Via utilizing a state feedback control in the 5D hyperchaotic Lorenz model, a new three unstable equilibrium points hyperchaotic model is reported. Despite the newly constructed system is simple, six-dimensional (6D) with eighteen terms, including three quadratic nonlinearities, it can possess self-excited attractors. The complicated self-excited attractor's dynamic various characteristics of this model are discovered using Lyapunov exponents, stability, and so on. In addition, the corresponding implementation hybrid synchronization (HS) between two identical for a new system by linearization and Lyapunov stability methods via four different controllers. Finally, the theoretical results are compatible with numerical simulations.

A novel dual memristor hyperchaotic system and its application for secure communication based on three‐fold function projection synchronization

AbstractA novel simplest four‐dimensional hyperchaotic system with dual memristors is designed by using a charge‐controlled memristor and a generalized voltage‐controlled memristor. Based on the conventional dynamic analysis method and MATLAB simulation software, the dynamic characteristics of the dual memristor hyperchaotic system such as Lyapunov exponent spectrum, phase diagram, and bifurcation diagram are studied. The novel system is found to exhibit singular nonlinear phenomena, such as limit cycle, state transition and multiple scroll attractor with certain parameter values. Being different from most known chaotic systems, which contain only two isolated unstable equilibrium points, the novel system has an infinite number of unstable equilibriums points and a larger Lyapunov exponent. What is more, the three‐fold function projection synchronization of the dual memristor hyperchaotic system and its application for secure communication is studied. Based on Lyapunov stabilization theory and adaptive sliding mode control method, three driving systems and three response systems are synchronized according to the corresponding scaling function matrix, respectively. In the secure communication system, the message signal is separated into three parts loaded in the different drive systems with different initial conditions to increase the reliability. The corresponding numerical simulations demonstrate the effectiveness of the results obtained.

Wolbachia

Ary Hoffman introduces Wolbachia.