Derivative Of Lyapunov Function Research Articles

Higher-order derivatives of Lyapunov functions and partial boundedness of solutions with partially controllable initial conditions

Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives of the Lyapunov functions.

The control and synchronization of fractional-order Genesio–Tesi system

In this paper, we prove an elementary lemma which estimates fractional derivatives of Lyapunov function in the sense Caputo when $$\alpha \in (0,1)$$ . Moreover, by using this result, we study the control and synchronization of fractional-order Genesio–Tesi system. The chaotic system is assumed that only the output variable is available and the output variable is the first state variable of Genesio–Tesi system. Based on Lyapunov stability theorem, some novel criteria for the control and synchronization of Genesio–Tesi system are proposed via the output variable. Numerical simulations are shown to further verify the feasibility of the presented control schemes.

Dwell time-based stabilisation of switched delay systems using free-weighting matrices

ABSTRACTIn this paper, we present a quasi-convex optimisation method to minimise an upper bound of the dwell time for stability of switched delay systems. Piecewise Lyapunov–Krasovskii functionals are introduced and the upper bound for the derivative of Lyapunov functionals is estimated by free-weighting matrices method to investigate non-switching stability of each candidate subsystems. Then, a sufficient condition for the dwell time is derived to guarantee the asymptotic stability of the switched delay system. Once these conditions are represented by a set of linear matrix inequalities , dwell time optimisation problem can be formulated as a standard quasi-convex optimisation problem. Numerical examples are given to illustrate the improvements over previously obtained dwell time bounds. Using the results obtained in the stability case, we present a nonlinear minimisation algorithm to synthesise the dwell time minimiser controllers. The algorithm solves the problem with successive linearisation of nonlinear conditions.

Higher-Order Derivatives of Lyapunov Functions and Partial Boundedness of Solutions with Partially Controllable Initial Conditions

В терминах высших производных функций Ляпунова получены достаточные признаки видов частичной ограниченности решений с частично контролируемыми начальными условиями. Библиография: 16 названий.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

Global finite-time stabilization for a class of high-order nonlinear systems with multiple unknown control directions

This paper considers the problem of the global finite-time stabilization for the high-order nonlinear systems with unknown control directions. Due to the uncertainty of control directions, the paper analyzes all possible conditions of the directions. The Lyapunov-based logic switching rule ensures that we can find the correct control directions. The adaptive switching controller with a switching parameter which is to be tuned online guarantees that the derivative of Lyapunov function is less than a negative definite function and the closed-loop system is globally finite-time stable. The effectiveness of the proposed method is illustrated by an example.

Delay-dependent stability and dissipativity analysis of generalized neural networks with Markovian jump parameters and two delay components

This paper focuses on the problem of delay-dependent stability and dissipativity analysis of generalized neural networks (GNNs) with Markovian jump parameters and two delay components. By constructing novel augmented Lyapunov–Krasovskii functional (LKF), using free-matrix-based inequality to estimate the derivative of Lyapunov function and employing the reciprocally convex approach to consider the relationship between the time-varying delay and its interval, some improved delay-dependent stability criteria and dissipativity criteria are established in terms of linear matrix inequalities. Moreover, the obtained criteria is extended to analyze the stability analysis of GNNs with two delay components and the passivity analysis of GNNs with one delay. Numerical examples are given to show the effectiveness and the significant improvement of the proposed methods.

Decentralized adaptive coupling synchronization of fractional-order complex-variable dynamical networks

In this paper, we combine decentralized adaptive control with the fractional-order techniques to investigate the synchronization of fractional-order complex-variable dynamical networks. A new lemma is proposed for estimating the Caputo fractional derivatives of Hermitian quadrtic Lyapunov functions. Based on local information among neighboring nodes, an effective fractional-order decentralized adaptive strategy to tune the coupling gains among network nodes is designed. This analysis is further extended to the case where only a small fraction of coupling gains are choosen to be adjusted. By constructing suitable Lyapunov functions and utilizing the proposed lemma, two sufficient criteria are derived to guarantee the network synchronization by using the proposed adaptive laws. Finally, numerical examples are given to validate the theoretical results.

Control and Synchronization with Known and Unknown Parameters

In this paper, we consider the chaos control for 4D hyperchaotic system by two cases, known & unknown parameters based on Lyapunov stability theory via nonlinear control. We find that there are two cofactors that have an effect on determining any case to achieve the control, the two cofactors are proposed in the control and the matrix that produce from the time derivative of Lyapunov function. In adding, we find some weakness cases in Lyapunov stability theory. For this reason, we design with only one controller and perform a simple change in this control in order to recognize the difference between these cases although all of the controllers are almost similar.

Speed estimation of an induction machine based on designed Lyapunov candidate functions

This paper describes a new approach to estimate speed of the induction machine from measured terminal voltages and currents. In this approach, the speed is assumed to be unknown constants, because it changes slowly compared to electrical variables such as currents and fluxes. Based on this assumption, a state observer is defined to eliminate the flux information of the machine. Then, a Lyapunov function and a new candidate expression for the time derivative of Lyapunov function that guaranties the stability conditions of the system dynamics are developed from which the speed of the machine is derived. The proposed system is analyzed and verified by both simulation and experiment.

Stability of Caputo fractional differential equations by Lyapunov functions

The stability of the zero solution of a nonlinear nonautonomous Caputo fractional differential equation is studied using Lyapunov-like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov-like function along the given fractional equation. Comparison results using this definition for scalar fractional differential equations are presented. Several sufficient conditions for stability, uniform stability and asymptotic uniform stability, based on the new definition of the derivative of Lyapunov functions and the new comparison result, are established.

Controller Design for Tracking Control of an Under-Actuated Surface Ship

 Abstract—In this paper an output feedback controller for tracking control of surface ships based on Euler-Lagrange equations has been proposed. It has been assumed that a surface ship is moving in a horizontal plane and under-actuated in sway direction. The change of coordinate's method is applied to overcome the third order component that arises in the Lyapunov function derivatives due to Coriolis and centripetal forces term. The design of the controller is based on the backstepping control technique and Lyapunov stability theory. Firstly, the observer is derived using the change of coordinate method. Next, backstepping control technique is employed to derive the control law. Finally, a global asymptotic convergence is proven using Lyapunov stability theorems. Simulations are provided to demonstrate the performance of the designed controller and prove tracking error of the controller convergence.

Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays

Some comparison principles for fractional-order linear systems with multiple time delays are established, after Mittag–Leffler functions are showed to be positive. Then by the stability theory of fractional linear delayed systems, the comparison system with multiple time delays is showed to be asymptotically stable under some conditions. Based on the comparison results, the asymptotical stability of the original systems follows from that of the comparison system. Then the obtained results are applied to investigate the asymptotical stability of nonlinear fractional-order cellular neural networks with multiple time delays. In terms of the inequality satisfied by the fractional derivative of Lyapunov function, some criteria ensuring asymptotical stability of fractional neural models are derived. Numerical simulations are presented to demonstrate the validity and feasibility of the proposed stability criteria.

UPFCs control design for avoiding generator trip of electric power grid with barrier function

Abstract A novel transient control scheme for unified power flow controller (UPFC) devices is proposed for avoiding generator trip in electrical power grid. The control scheme is derived by an proposed control lyapunov function, which consists of energy function and barrier functions. Undesirable generator trips are prevented thanks to the barrier functions penalty imposed by UPFCs, because the value of the barrier function approaches infinity as generator frequency deviation approaches pre-set threshold. The control effects of the UPFC devices make the time derivative of control lyapunov function be negative for asymptotic stability. The attraction region of the system is reshaped by control system with the desired stable equilibrium point, and the generator trips are prevented in this region. Meanwhile, the attraction region and its design are discussed. Two numerical simulations are carried out using 3 bus and IEEE 162 bus test system, to demonstrate the effectiveness of the proposed method.

Volterra-type Lyapunov functions for fractional-order epidemic systems

In this paper we prove an elementary lemma which estimates fractional derivatives of Volterra-type Lyapunov functions in the sense Caputo when α∈(0,1). Moreover, by using this result, we study the uniform asymptotic stability of some Caputo-type epidemic systems with a pair of fractional-order differential equations. These epidemic systems are the Susceptible–Infected–Susceptible (SIS), Susceptible–Infected–Recovered (SIR) and Susceptible–Infected–Recovered–Susceptible (SIRS) models and Ross–Macdonald model for vector-borne diseases. We show that the unique endemic equilibrium is uniformly asymptotically stable if the basic reproductive number is greater than one. We illustrate our theoretical results with numerical simulations using the Adams–Bashforth–Moulton scheme implemented in the fde12 Matlab function.

Fixed-order Linear Parameter Varying Controller Design for a 2DOF Gyroscope

This paper presents an approach for fixed-order Linear Parameter Varying (LPV) controller design with application to a 2 Degree-of-Freedom (2DOF) gyroscope experimental setup. Inner convex approximation of the non-convex set of all stabilizing fixed-order LPV controllers is characterized through a set of Linear Matrix Inequalities (LMIs). This is achieved through the use of two slack matrices which enable decoupling of the controller and Lyapunov matrix parameters in the derivative of Lyapunov function. The LPV model obtained by the approximation of the nonlinear model of the 2DOF gyroscope is used for the design of a second-order LPV controller. Experimental results show good tracking performance in the presence of scheduling parameter variations.

Guaranteed estimates of the domain of attraction for a class of hybrid systems

SummaryThis paper addresses the estimation of the domain of attraction for a class of hybrid nonlinear systems where the state space is partitioned into several regions. Each region is described by polynomial inequalities, and the union of all the regions is a complete cover of the state space. The system dynamics are defined on each region independently from the others by polynomial functions. First, the problem of computing the largest sublevel set of a Lyapunov function included in the domain of attraction is considered. An approach is proposed for addressing this problem based on linear matrix inequalities, which provides a lower bound of the sought estimate by establishing negativity of the Lyapunov function derivative on each region. Second, a sufficient and necessary condition is provided for establishing optimality of the found lower bound, which requires to solve linear algebra operations in typical cases. Third, the problem of looking for Lyapunov functions that maximize the estimate of the domain of attraction is considered, describing several strategies where the proposed approach can be readily adopted. Copyright © 2014 John Wiley & Sons, Ltd.

General Convex Integral Control

In this paper, a fire-new general integral control, named general convex integral control, is proposed. It is derived by defining a nonlinear function set to form the integral control action and educe a new convex function gain integrator, introducing the partial derivative of Lyapunov function into the integrator and resorting to a general strategy to transform ordinary control into general integral control. By using Lyapunov method along with the LaSalle’s invariance principle, the theorem to ensure regionally as well as semi-globally asymptotic stability is established only by some bounded information. Moreover, the lemma to ensure the integrator output to be bounded in the time domain is proposed. The highlight point of this integral control strategy is that the integral control action seems to be infinity, but it factually is finite in the time domain. Therefore, a simple and ingenious method to design the general integral control is founded. Simulation results showed that under the normal and perturbed cases, the optimum response in the whole control domain of interest can all be achieved by a set of control gains, even under the case that the payload is changed abruptly.

A note on global stability of the virose equilibrium for network-based computer viruses epidemics

It has been recently conjectured in Yang et al. (2013) [27] that the unique virose equilibrium of a network-based model is globally asymptotically stable in the feasible region. The aim of this paper is to investigate the global stability of the virose equilibrium by using the direct Lyapunov method and appropriate grouping techniques for Lyapunov function derivatives. Some sufficient conditions are established, which improves the relevant results. Moreover, we provide a new brief proof for global stability of the virus-free equilibrium by using comparison method, and discuss quantitatively the effect of various immunization schemes, including uniform immunization, targeted immunization and high-risk immunization, on the viruses spread, which suggests promising strategy to halt viruses spread in computer networks.

Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.