Second Method Of Lyapunov Research Articles

A New Predefined Time Zeroing Neural Network With Drop Conservatism for Matrix Flows Inversion and Its Application.

Zeroing neural network (ZNN) can effectively solve the matrix flows inversion problem. Nevertheless, quite a few related research works focus on the improvement of the convergence and robustness performance of the ZNN models and ignore the conservatism of their predefined time. Therefore, this article adopts a polymorphous activation function (PAF) to construct a new predefined time ZNN (NPTZNN) model. The second method of Lyapunov is utilized to analyze the stability, convergence, and robustness of the NPTZNN model. The Beta function is dexterously employed in the process of calculating the predefined time of the NPTZNN model, reducing its conservatism. Furthermore, the correctness of the theoretical analyses is verified by numerous experiments. Finally, the NPTZNN model is applied to robot manipulator control and can improve the tracking speed, extending the applicability of the model.

Robust Approximate Dynamic Programming for Nonlinear Systems With Both Model Error and External Disturbance.

Model error and external disturbance have been separately addressed by optimizing the definite H∞ performance in standard linear H∞ control problems. However, the concurrent handling of both introduces uncertainty and nonconvexity into the H∞ performance, posing a huge challenge for solving nonlinear problems. This article introduces an additional cost function in the augmented Hamilton-Jacobi-Isaacs (HJI) equation of zero-sum games to simultaneously manage the model error and external disturbance in nonlinear robust performance problems. For satisfying the Hamilton-Jacobi inequality in nonlinear robust control theory under all considered model errors, the relationship between the additional cost function and model uncertainty is revealed. A critic online learning algorithm, applying Lyapunov stabilizing terms and historical states to reinforce training stability and achieve persistent learning, is proposed to approximate the solution of the augmented HJI equation. By constructing a joint Lyapunov candidate about the critic weight and system state, both stability and convergence are proved by the second method of Lyapunov. Theoretical results also show that introducing historical data reduces the ultimate bounds of system state and critic error. Three numerical examples are conducted to demonstrate the effectiveness of the proposed method.

New Qualitative Outcomes for Ordinary Differential Systems of Second Order

This paper deals with a nonlinear ordinary differential system of second order. In the paper, qualitative properties of solutions of the system called asymptotic stability (AS), uniform stability (US), boundedness, ultimately boundedness (UB) and integrability of solutions, are investigated by using the second method of Lyapunov. We give four new qualitative results and an example as a numerical application of the results. The results of this article extend and improve some earlier ones in the literature.

On uniform-ultimate boundedness and periodicity results of solutions to certain second order non-linear vector differential equations

In this paper, we employ the second method of Lyapunov to examine sufficient conditions for the uniform-ultimate boundedness of solutions and existence of at least one periodic solution to the following second order vector differential equation: Ẍ+ F(X, Ẋ ) Ẋ + H(X) = P(t, X, Ẋ ), when the non-linear term H(X) is: (i) differentiable, (ii) non-necessarily differentiable. The results contain in this paper are new and complement related ones in the literature.

On the Stability and Boundedness of Solutions to Certain Second Order Differential Equation

In this paper, we investigate by means of second method of Lyapunov, sufficient conditions that guarantee uniform-asymptotic stability of the trivial solution and ultimate boundedness of all solutions to a certain second order differential equation. We construct a complete Lyapunov function in order to discuss the qualitative properties mentioned earlier. The boundedness result in this paper is new and also complement some boundedness results in literature obtained by using an incomplete Lyapunov function together with a signum function. Finally, we demonstrate the correctness of our results with two numerical examples and graphical representation of the trajectories of solutions to the examples using Maple software.

Operating characteristics of a 15 W free-piston Stirling generator

Free-piston Stirling generators (FPSGs) show great potential in portable power sources for the advantages of high reliability, high efficiency, low noise, compact structure, long lifetime, etc. An FPSG aimed at 15 W electric power output is numerically and experimentally studied. By adopting the second method of Lyapunov, the onset characteristics of the FPSG are analyzed based on a thermodynamic model. The effects on the onset conditions of the interaction between the displacer and power piston are explored. It is shown that the optimal operating pressure and spring stiffness corresponding to the lowest onset heating temperature are 6.2 MPa and 5500 N/m, and the FPSG is easier to start when the damping of the displacer and power piston is smaller and the mass of the displacer and power piston is well matched. The experimental results of the effects of the pressure and spring stiffness on the onset temperature agree well with the calculated results. The FPSG can generate 15.5 W electric power with a thermal-to-electric efficiency of 16.0% in the experiment when the heating and chilling water temperatures are 820 K and 300 K, respectively. And the relative errors between the calculated and experimental PV power and their efficiencies are about 2.1%–6.8% and 24.4%–26.9%, respectively. This work can lay a good foundation for studying the onset characteristics and stable operating characteristics of FPSGs.

Distributed control of nonlinear conductivity heat transfer equation in a thick functionally graded plate

In this paper, distributed control of nonhomogeneous nonlinear conductivity heat transfer equation on a single plate is studied. The inhomogeneity of the equation is due to the use of functionally graded materials. In the proposed method, the governing equation of the system, which is a nonlinear partial differential equation, is represented as some linear partial differential equation using the Takagi-Sugeno linearization technique; so that their fuzzy combination accurately retrieve the original nonlinear equation. This process is done to prevent the phenomenon of frequency disorder or aliasing. Then, the modified wavenumber method is utilized to convert the obtained linear partial differential equations to a set of ordinary differential equations. In this way, a combination of fast Fourier transform and finite difference methods is implemented. Consequently, by combining Takagi-Sugeno and the modified wavenumber strategies, several ordinary differential equations are obtained for each grid point. In this regard, the parallel distributed compensation method is used to stabilize each grid point. The second method of Lyapunov is implemented to study the stability of the proposed control strategy. Numerical Simulations have been used to demonstrate the efficacy of the proposed control scheme and also its relative robustness with respect to measurement noises and parametric uncertainties.

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

Controlling the PVTOL Aircraft System with an Inverted Pendular Load by means of Nested Saturation Functions and GPI Controller

This paper presents a control scheme that allows height position regulation and stabilization for an unmanned planar vertical takeoff and landing aircraft system with an inverted pendular load. The proposed controller consists of nested saturations and a generalized proportional integral (GPI). The GPI controls the aircraft height and the roll attitude; the latter is used as the fictitious input control. Next, the system is reduced through linear transformations, expressing it as an integrator chain with a nonlinear perturbation. Finally, the nested saturation function-based controller stabilizes the aircraft’s horizontal position and the pendulum’s angle. Obtaining the control approach was a challenging task due to the underactuated nature of the aircraft, particularly ensuring the pendulum’s upright position. The stability analysis was based on the second method of Lyapunov using a simple candidate function. The numerical simulation confirmed the control strategy’s effectiveness and performance. Additionally, the numerical simulation included a comparison against a PD controller, where its corresponding performance indexes were estimated, revealing that our controller had a better response in the presence of unknown disturbances.

Adaptive Active Disturbance Rejection Control of Solar Tracking Systems with Partially Known Model

In this article, the trajectory tracking control of a solar tracking system is tackled by means of an adaptive active disturbance rejection control scheme. The state and disturbance estimation system is based on the combination of a time varying identification system and an adaptive observer. The stability and robustness of the controller is mathematically tested by means of the second method of Lyapunov, and its effectiveness is experimentally tested in a robotic test bed, achieving both lower energy consumption and better tracking results with respect to a PID-based controller.

Comparative Study Between Integrator Backstepping and Fuzzy Logic Control Applied to an Electric Powered Wheelchair

The aim of this paper is the control of electric powered wheelchairs (EPW) which was made for people suffering of temporary or permanent disabilities due to illnesses or accidents. The EPW is powered by two Permanent Magnet Synchronous Motors (PMSM) that are characterized by high efficiency, high torque, low noise and robustness; hence the dynamic model of the both EPW-motors is presented in the first. After that, a comparative study is made between two nonlinear command theory; Integrator Backstepping based on the second method of Lyapunov which combine the choice of the energy function with the laws control, and, fuzzy logic introduced to approach human reasoning with the help of an adequate representation of knowledge. To evaluate the performance of the two controls, numerical simulations are presented to show the evolution of electrical and mechanical quantities, the energy consumed and the squared error of the displacement and velocity. However, the reference trajectory used is that generated by the fifth-degree polynomial interpolation, which ensures a regular trajectory that is continuous in positions, velocities and accelerations.

Boundedness and Stability Properties of Solutions of Mathematical Model of Measles.

In this paper, asymptotic stability and global asymptotic stability of solutions to a deterministic and compartmental mathematical model of measles infection is considered using the ideas of the Jacobian determinant as well as the second method of Lyapunov, criteria/conditions that guaranteed asymptotic stability of disease free equilibrium and endemic equilibrium were established. Also the basic reproductive number $R_0$ was obtained. The results in this work compliments existing work and provided further information in controlling the disease in an open population.

Output based bilateral adaptive control of partially known robotic systems

This paper describes the design and implementation of a robust adaptive bilateral observer based control to solve the trajectory tracking problem for a class of master–slave robotic systems. The output-based bilateral control uses an adaptive observer to estimate the force and the unmeasurable velocity state considering only the master–slave system order as well as the position measurements. The stability of the observer and controller is simultaneously proved via the second method of Lyapunov. A set of experimental tests proved the performance of the presented algorithm for a manipulation task in a bilateral teleoperated system based on two delta robots.

Novel Lyapunov-Based Autonomous Controllers for Quadrotors

In this paper, we look into the dynamic motion planning and control of an unmanned aerial vehicle, namely, the quadrotor, governed by its dynamical equations. It is shown for the first time that the Direct or the Second Method of Lyapunov is an effective tool to derive a set of continuous nonlinear control laws that not only provide smooth trajectories from a designated initial position to a designated target, but also continuously minimise the roll and pitch of the quadrotor en route to its targets. The latter successfully addresses the challenging problem of a quadrotor autonomously transporting valuable and fragile payloads safely to the designated target. Computer simulations are used to illustrate the effectiveness of the proposed control laws.

The synthesis of the algorithms for adaptive control by nonlinear dynamic objects on the basis of the neural network

The present work is devoted to a problem of Synthesis of the algorithms for adaptive control by nonlinear dynamic objects with the incomplete mathematical description. The method of synthesis adaptive neural networks is considered on the basis of application of some positions of a method of analytical designing. The law of adaptation is defined on a condition of maintenance of stability of the closed system with the help of the second method of Lyapunov. The resulting control systems can operate in uncertain conditions caused by external and internal disturbances. The designed parameter adaptation law of the controllers admits simple implementation, thereby facilitating the on-line adaptation process.

A Novel Low Voltage Ride-Through Technique of Three-Phase Grid-Connected Inverters Based on a Nonlinear Phase-Locked Loop

In this paper, a novel low voltage ride-through (LVRT) technique for three-phase grid-connected inverters is proposed. The proposed technique consists of two parts: a nonlinear phase locked loop based on complex-coefficient filters (NLCCF-PLL) and an LVRT control scheme. Generally, the synchronization process of three-phase grid-connected inverters is performed via PLL with a relatively low bandwidth, which delays the detection of voltage sag and recovery during the LVRT process. To accelerate the synchronization process, the NLCCF-PLL with adaptive controller gains is proposed to improve both the filtering capability and dynamic performance of PLL at the same time. The stability of the NLCCF-PLL is validated by the second method of Lyapunov in the nonlinear model, and the superiority of its operating performance is verified. The proposed LVRT control scheme consists of a reference current calculation block to effectively suppress the power ripples and an inner loop controller with strong robustness as well as fast dynamic response. By comparing the proposed LVRT technique with the existing LVRT technique on the basis of experimental results, the superiority of the proposed LVRT technique is confirmed.

New Qualitative Results for Solutions of Functional Differential Equations of Second Order

In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.

Overflow oscillation-free realization of digital filters with saturation

PurposeThe purpose of this paper is to establish a criterion for the global asymptotic stability of fixed-point state–space digital filters using saturation overflow arithmetic.Design/methodology/approachThe method of stability analysis used in this paper is the second method of Lyapunov. The approach in this paper makes use of a precise upper bound of the state vector of the system and a novel passivity property associated with the saturation nonlinearities.FindingsThe presented criterion leads to an enhanced stability region in the parameter-space as compared to several existing criteria.Practical implicationsWhen dealing with the design of fixed-point state–space digital filters, it is desirable to have a criterion for selecting the filter coefficients so that the designed filter becomes free of overflow oscillations. The criterion presented in this paper provides enhanced saturation overflow stability region and therefore facilitates the designer greater flexibility in selecting filter parameters for overflow oscillation-free realization of digital filters.Originality/valueThe approach uses the structural properties of the saturation nonlinearities in a greater detail. The exploitation of upper bound of the system state vector together with a new passivity property of saturation nonlinearities is a unique feature of the present approach. The presented approach may lead to results not covered by several existing approaches.

New qualitative criteria for solutions of Volterra integro-differential equations

In this paper, we consider certain non-linear scalar Volterra integro-differential equations and Volterra integro-differential systems of first order. We investigate the boundedness, stability, uniformly asymptotic stability, integrability and square integrability of solutions to the scalar equations and the system considered. The technique used to prove the results of the paper is based on the second method of Lyapunov. From the obtained results, we extend and improve some related results that can be found in the literature.

Điều khiển robot n bậc tự do

A hybrid control scheme is presented in this paper for the control of manipulators which have uncertainties in parameters and disturbances. The robustness is achieved by applying the model reference Adaptive Approach in order to obtain the nonlinear compensation laws which do not require the exact knowledge of parameters and disturbances. At first modeling of discrete control system of n degree of freedom robot is analysed. Then using the second method of Lyapunov a control scheme is proposed which results in asymptotically stable system performance.