Time Derivative Of Lyapunov Function Research Articles

Adaptive control of a class of uncertain nonlinear systems using brain emotional learning and Legendre polynomials

In this paper, an adaptive controller for a class of uncertain nonlinear systems is presented using a combination of Legendre polynomials and brain emotional learning-based intelligent controller (BELBIC). Recently, some versions of BELBIC have been presented with the aim of satisfying the universal approximation property using Gaussian basis function. However, the size of regressor vector is too large that imposes a heavy computational load to the processor. The novelty of this paper is presenting a new version of BELBIC with less computational burden using Legendre polynomials. Moreover, there are very few tuning parameters in Legendre polynomials. Another contribution of this paper is editing the stability analysis presented in recent related works. Due to the intrinsic non-differentiability of the adaptation rules of BELBIC, the second time derivative of Lyapunov function is undefined and thus, the Barbalat’s lemma cannot be applied to verify the asymptotic convergence of the error function. Therefore, bounded-input-bounded-output (BIBO) stability can only be claimed for this controller. Simulation results on different case studies show that Legendre polynomials can improve the universal approximation property of BELBIC with less tuning parameters. Moreover, in the absence of the robust control term in the control law, the performance Legendre polynomials will not deteriorate, while the performance degrade in Gaussian basis function is quite considerable.

Exponential stability for nonlinear time-varying systems on time scales

In this paper, some less conservative sufficient conditions for uniform exponential stability, local uniform exponential stability and uniform practical exponential stability of nonlinear time-varying systems on time scales are proposed. Compared with the traditional exponential stability results, the negative definiteness of the time derivatives of Lyapunov functions is weakened, that is, the time derivatives of related Lyapunov functions in this paper are allowed to be indefinite. The main tools used are some Gronwall's inequalities on time scales. Two examples are also included to verify the effectiveness of the results obtained.

Asymptotic stability for nonlinear time-varying systems on time scales

This paper is concerned with nonlinear time-varying systems on time scales, which can cover not only trivial continuous and discrete dynamical systems, but also some other nontrivial cases. Some less conservative sufficient conditions for asymptotic stability of nonlinear systems are derived. Compared with the traditional stability results, the negative definiteness of the time derivatives of Lyapunov functions is weakened, that is, the time derivatives of related Lyapunov functions in this paper are allowed to be indefinite. Two examples about trivial and nontrivial systems are included to illustrate the effectiveness of the results obtained.

Finite-Time Stabilization of Switched Systems Under Mode-Dependent Event-Triggered Impulsive Control

This article studies the event-triggered finite-time stabilization problem of nonlinear switched systems in which the time derivatives of Lyapunov functions of modes are indefinite. Based on a mode-dependent average dwell time constraint and event-triggered impulsive control (ETIC) strategy, a new type of the mode-dependent event-triggered mechanism (MDETM) which can efficiently avoid Zeno behavior for switched systems is presented. Some Lyapunov-based criteria for finite-time stability (FTS) and finite-time contractive stability (FTCS) are obtained, respectively, where a relationship between the prescribed bound and event-triggered mechanism is established. Then, we apply these proposed ETIC strategies to impulsive switched systems and design a class of LMI-based MDETMs to guarantee the FTS/FTCS property. Finally, the effectiveness of presented ETIC strategies is illustrated by two examples.

Impulsive systems with hybrid delayed impulses: Input-to-state stability

This paper investigates input-to-state stability (ISS) and integral-input-to-state stability (iISS) of nonlinear impulsive systems with hybrid delayed impulses. Based on Lyapunov method, some sufficient conditions ensuring ISS and iISS of impulsive systems are obtained, where the time derivative of Lyapunov function is indefinite, and the hybrid effects of delayed impulses are also fully considered. It is shown that the impulsive system is ISS provided that the combined action of time delay existing in impulses, continuous dynamic, and the cumulative strength of hybrid impulses satisfies some conditions, even if the hybrid delayed impulses play a destabilizing effect on ISS. Examples and their simulations are presented to illustrate the applicability of the proposed results.

Fixed‐time nonsingular adaptive containment control for uncertain nonlinear multi‐agent systems with predefined accuracy

SummaryThis article studies the containment control problem for uncertain nonlinear multi‐agent systems. To solve this problem, a performance‐oriented fuzzy adaptive algorithm is proposed. By constructing a new Lyapunov function and using the predefined accuracy functions, the algorithm ensures the negative‐definite property of the time derivative of the Lyapunov function while guaranteeing that the containment errors converge to a user‐defined interval. Different from most existing results, the control algorithm proposed in this article realizes the negative‐definite property of the time derivative of Lyapunov function without adjusting parameters. Then, an event‐triggered mechanism is introduced to reduce the communication burden between the controller and the actuator. Moreover, by using the singularity‐avoidance functions, a fixed‐time fuzzy nonsingular controller is constructed to avoid the possible “singularity” problem in the backstepping recursive design. Based on the fixed‐time stability criterion, it is proved that all output signals converge to a predefined interval in a fixed time. Finally, several simulation results further demonstrate the effectiveness of the presented control strategy.

Direct Adaptive Fuzzy Control Scheme With Guaranteed Tracking Performances for Uncertain Canonical Nonlinear Systems

In our recent work, we propose an indirect adaptive fuzzy control scheme for uncertain unparametrizable nonlinear systems, which ensures that the number of adaptive laws does not increase with the number of fuzzy rules, and the time derivative of the chosen Lyapunov function is negative semidefinite. However, the scheme involves a class of high-order smooth functions and their time derivatives, which can make the controller structure become sophisticated especially when the relative degree of system is quite large. To overcome this problem, in the article, we propose a new direct adaptive fuzzy control scheme based on a class of reduced-order smooth functions. With the scheme, no partial-derivative term is involved in controller and virtual controllers, only one adaptive law is used regardless of the increase of fuzzy rules, and also the time derivative of Lyapunov function can be ensured negative semidefinite. It is proved that all closed-loop signals are bounded, and the output tracking error converges to a prescribed interval asymptotically. The transient tracking performance and robustness of the proposed control scheme are also considered. The effectiveness of the obtained results is illustrated by two practical control systems.

Extensions of Razumikhin-type stability theorems for nonlinear time-delay systems on time scales

This paper focuses on the stability analysis of nonlinear time-delay systems on time scales, which are of generality as they can include not only the traditional continuous and discrete ones, but also some other cases, such as systems on general uniform or non-uniform time domains. One existing classical method for analyzing stability of such systems is the Razumikhin-type theorem with requiring the time derivative of relating Lyapunov function to be non-positive for uniform stability and negative for uniform asymptotic or exponential stability, which are difficult to be satisfied. To relax these restrictions, by introducing the time-scale type uniformly stable function and uniformly asymptotically stable function, this paper presents several less conservative stability criteria, in which the time-scale time derivative of Lyapunov function can be positive or non-negative. To demonstrate the effectiveness of the theoretic results, a numerical example about non-continuous and non-discrete time-delay systems is given.

Predicting onset conditions of a free piston Stirling engine

This research predicts the onset conditions of the free piston Stirling engines using analytical solution of the governing equations and averaging-based Lyapunov technique. In this regard, firstly, the linear dynamic equations of the free piston Stirling engines are extracted. Then, the analytical solutions of the motion equations of the displacer and the power pistons are obtained. Afterwards, these analytical solutions are put into the mean value of the time derivative of Lyapunov function to obtain a convenient relation which provides meaningful information regarding the onset condition of free piston Stirling engines. The mentioned procedure has been verified by the results of multiple numerical simulations and the experimental data (B10-B and SUTech-SR-1). In addition, the results of this method have been validated by the outcomes obtained from the state-space technique. Lastly, by comparing the results of the simulation works and the experiments, it has been confirmed that this method is capable of precisely predicting the onset condition of free piston Stirling engines. Besides, the proposed approach could be used as a powerful tool to predict the behavior of various free piston Stirling engines regardless of whether they meet the onset condition or not.

New stability results for impulsive neural networks with time delays

This paper investigates the stability of impulsive neural networks with time delays. Based on a new tool called as uniformly exponentially convergent functions, an improved Razumikhin method leads to new, more permissive stability results. By comparison with the existing results, the rigorous restrictions on impulses, which are presented in the previous Razumikhin stability theorems, are removed. Moreover, the obtained results do not restrict that the time derivative of Lyapunov function is negative definite or positive definite under the Razumikhin condition. The effectiveness of the proposed results is demonstrated by three simple numerical examples.

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.

Lyapunov-Based Decentralized Excitation Control for Global Asymptotic Stability and Voltage Regulation of Multi-Machine Power Systems

This paper proposes a novel Lyapunov-based excitation control (LEC) technique to deal with both global asymptotic stability and voltage regulation for the multi-machine power systems. Lyapunov function is built as a quadratic form of the control objectives consisted of active power offset, rotor speed offset and terminal voltage offset. A completely controllable linear system is constructed to design the trajectories of the time-derivatives of the control objectives. Furthermore, time-derivative of Lyapunov function (TLF) is also expressed as a quadratic form of the control objectives. Feedback gains are chosen to guarantee the negative definition of the TLF for any real number except for the equilibrium point. Therefore, the global asymptotic stability is achieved. The design flexibility of the proposed method provides convenience to introduce the voltage feedback signal for maintaining acceptable voltage level. The system voltage is forced to converge to its reference accompanying with the energy attenuation of Lyapunov function. As a result, voltage deviation is eliminated when Lyapunov function is equal to zero. Finally, a Lyapunov-based decentralized excitation controller is developed for the multi-machine power systems, and it is only related to local measurements. Simulations on a six-machine power system have illustrated better performances in comparison to the existing controllers.

Chaos synchronization by variable strength linear coupling and Lyapunov function derivative in series form

A new general strategy to achieve chaos synchronization by variable strength linear coupling without another active control is proposed. They give the criteria of chaos synchronization for two identical chaotic systems and two different chaotic dynamic systems with variable strength linear coupling. In this method, the time derivative of Lyapunov function in series form is firstly used. Lorenz system, Duffing system, Rössler system and Hyper-Rössler system are presented as simulated examples.

Lyapunov 강인 안정성 조건을 이용한 강인 최적 뱅뱅 제어기

There are mainly two types of bang-bang controllers for nominal linear time-invariant (LTI) system. Optimal bang-bang controller is designed based on optimal control theory and suboptimal bang-bang controller is obtained by using Lyapunov stability condition. In this paper, the suboptimal bang-bang control method is extended to LTI system involving both control input saturation and structured real parameter uncertainties by using Lyapunov robust stability condition. Two robust optimal bang-bang controllers are derived by minimizing the time derivative of Lyapunov function subjected to the limit of control input. The one is developed based on the classical quadratic stability(QS), and the other is developed based on the affine quadratic stability(AQS). And characteristics of the two controllers are compared. Especially, bounds of parameter uncertainties which theoretically guarantee robust stability of the two controllers are compared quantitatively for 1DOF vibrating system. Moreover, the validity of robust optimal bang-bang controller based on the AQS is shown through numerical simulations for this system.