Derivative Of Lyapunov Function Research Articles

Nonlinear adaptive robust control with continuous error reduction mapping

Abstract Achieving high stability and high-precision control by traditional control algorithms is difficult due to the strong nonlinearity, parameter uncertainty, and random disturbance characteristics of electro-hydraulic proportional servo systems. Herein, we developed a nonlinear adaptive control algorithm based on Lyapunov functions. By estimating the parameters and converging their errors, this algorithm solved the parameter adaptation problem. Additionally, based on convergence inference from the nonnegativity of the derivative of Lyapunov functions, the algorithm designed a disturbance mapping through the derived function. The random disturbance was reduced through the mapping, improving the stability of the system under random disturbance. Finally, we carried out continuous derivation and moved the mapping relationship out of the intermediate temporary controller. This reduced the complexity of the calculation during controller design, prevented performance degradation of the error convergence algorithm after complex calculation, and enhanced the robustness of the algorithm. The algorithm was verified through the simulation model established by Simulink and experience. Results showed that the nonlinear adaptive control algorithm demonstrates fast response, high accuracy, and strong anti-interference ability compared with the high-gain robust control algorithm. It realizes the adaptation of unknown parameters and reduces the control error introduced by random disturbance.

Fixed-Time Control and Estimation of Discontinuous Fuzzy Neural Networks: Novel Lyapunov Method of Fixed-Time Stability.

This article considers the fixed-time stability (FXTS) issue for discontinuous system described by differential equation (DE) with time-varying parameters. Using the tool of differential inclusion (DI), several improved FXTS criteria and estimation formulas of settling time (SET) are derived by employing a relaxed Lyapunov method. The special novelty of the developed Lyapunov method is that the derivative of Lyapunov function possesses indefiniteness. As an important application, the established FXTS theorems are utilized to deal with fixed-time (FXT) synchronization issue for fuzzy neural networks (FNNs) possessing state discontinuity, where the fuzzy operation is used in the synaptic law computing, and one typical time-varying switching control protocol is designed. Moreover, a series of estimations of SET for FXT synchronization are given out. Finally, the simulation examples are present to substantiate the validity of the obtained results.

On linear stabilization of a class of nonlinear systems in a critical case

In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

A Passivity-based Control Combined with Sliding Mode Control for a DC-DC Boost Power Converter

In this paper, a passivity-based control combined with sliding mode control for a DC-DC boost power converter is proposed. Moreover, a passivity-based control for a DC-DC boost power converter is also proposed. Using a co-ordinate transformation of state variables and control input, a DC-DC boost power converter is passive. A new plant is zero-state observable and the equilibrium point at origin of this plant is asymptotically stable. Then, a passivity-based control is applied to this plant such that the capacitor voltage is equal to the desired voltage. Additionally, the sliding mode control law is chosen such that the derivative of Lyapunov function is negative semidefinite. Finally, a passivity-based control combined with sliding mode control law is applied to this plant such that the capacitor voltage is equal to the desired voltage. The simulation results of the passivity-based control, the sliding mode control and the passivity-based control combined with sliding mode control demonstrate the effectiveness and show that the capacitor voltage is kept at the desired voltage when the desired voltage, the input voltage E and the load resistor R are changed. The results show that compared with the passivity-based control, the passivity-based control combined with sliding mode control has better performance such as shorter settling time, 8.5 ms when R changes and it has smaller steady-state error, which is indicated by the value of integral absolute error (IAE), 0.0679 when the desired voltage changes. The paper has limitations such as the assumed circuit parameters.

New predefined-time stability results of impulsive systems with time-varying impulse strength and its application to synchronization of delayed BAM neural networks

In this paper, the predefined-time synchronization of BAM neural networks with hybrid impulsive effects and time-varying delays is explored. First, based on the improved Lyapunov function method, two novel predefined-time stability lemmas of impulsive dynamical systems are given, in which the derivative of Lyapunov function can be negative definite or indefinite. Second, different from previous studies on constant impulse strength, we consider the impulse strength as a time-varying function, which can take different values at different moments and allow for the coexistence of both stabilizing and destabilizing impulses simultaneously. Then, two novel controllers are designed, and some sufficient conditions are derived to ensure predefined-time synchronization of the established system. Finally, the effectiveness of the obtained criteria is verified by a numerical simulation. The results show that the synchronized time can be adjusted according to actual needs and does not depend on the initial values and parameters of the system.

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.

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

Stability and adaptive control-based synchronization of delayed uncertain fractional-order gene regulatory networks

This paper is devoted to investigating stability and synchronization issues for delayed uncertain fractional-order gene regulatory networks (DUFOGRNs). First, the existence and uniqueness of the equilibrium point of the DUFOGRNs are explored via constructing contraction map, whereafter, a stability criterion is studied by means of the fractional-order Lyapunov–Razumikhin approach and property of Mittag-Leffler function. Second, a novel fractional-order Lemma is established by using the theory of fractional calculus showing that the derivative of Lyapunov functions in the fractional-order systems (FOSs) satisfy certain form of convergence, which can be able to discuss the problems about finite-time stability and synchronization of FOSs. Third, adaptive control strategies are designed based on the established Lemma, which are different control protocols anent the synchronization of gene regulatory networks proposed in previous work. Furthermore, sufficient synchronization criteria are derived by probing fractional calculus theory and inequality analysis techniques for guaranteeing the complete synchronization (CS) and finite-time synchronization (F-TS) of DUFOGRNs. Finally, three numerical examples are provided to verify the availability of the proposed method.

Fixed-time synchronization of large-scale systems via aperiodically intermittent control

This paper studies the fixed-time synchronization of large-scale systems. An aperiodically intermittent control (AIC) based on the average control rate is introduced to achieve fixed-time synchronization for the first time. Compared with previous work, the AIC based on the average control rate reduces the limitation of control width and rest width, thus, the conservativeness is reduced. Moreover, the designed control uses the hyperbolic sine function instead of the sign function, which can avoid the chattering phenomena. A concession is made between rest and control intervals by constructing the auxiliary timers, such that the derivative of Lyapunov function (ignoring the exogenous disturbance) built is always non-positive. Then by combining the graph theory and Lyapunov method, some sufficient conditions are given. Finally, a numerical example is presented to verify the effectiveness of the results developed.

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.

Fixed-time synchronization of delayed memristive neural networks with impulsive effects via novel fixed-time stability theorem

In this study, the fixed-time synchronization (FXTS) of delayed memristive neural networks (MNNs) with hybrid impulsive effects is explored. To investigate the FXTS mechanism, we first propose a novel theorem about the fixed-time stability (FTS) of impulsive dynamical systems, where the coefficients are extended to functions and the derivatives of Lyapunov function (LF) are allowed to be indefinite. After that, we obtain some new sufficient conditions for achieving FXTS of the system within a settling-time using three different controllers. At last, to verify the correctness and effectiveness of our results, a numerical simulation was conducted. Significantly, the impulse strength studied in this paper can take different values at different points, so it can be regarded as a time-varying function, unlike those in previous studies (the impulse strength takes the same value at different points). Hence, the mechanisms in this article are of more practical applicability.

Finite-time stability of solutions for non-instantaneous impulsive systems and application to neural networks

This work extends the finite-time stability (FTS) results to non-instantaneous impulsive time-varying systems on the basis of general impulsive systems via Lyapunov theory. The impacts of impulse numbers, impulse time sequences and impulse amplitudes on FTS and settling-time estimation are discussed. The effects of different types of impulses on settling-time are discussed when the impulse number is known or pre-given. The average impulsive interval is introduced to extend the above results to the case when the impulse number is unknown, and the derivative of Lyapunov function is allowed to be indefinite instead of being negative definite or semi-negative definite. Moreover, the theoretical results are applied to non-instantaneous impulsive neural networks. Finally, we provide two numerical examples to demonstrate the effectiveness and feasibility of the obtained results.

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 passivity analysis for Caputo fractional BAM reaction–diffusion delayed neural networks

The issues of finite time passivity are explored for BAM reaction–diffusion neural networks including discrete delayed and Caputo fractional partial differential operator. With the help of fractional Lyapunov function and existing passivity definition (integral form or fractional integral form), four finite time passivity definitions are given out in Caputo-type fractional derivative form under the double-layer network structure. In this way, the fractional derivative of Lyapunov function can be directly obtained to judge whether the system is passive. Constructing a suitable mixed controller with sign function, several passivity conditions are established as matrix inequality form by utilizing the Green’s theorem, Jensen’s integral inequality, Lyapunov functional approach and four novel definitions. At last, one example is appeared to better illustrate the obtained conditions.

Exponential stability analysis for nonlinear time-varying perturbed systems on time scales

<abstract><p>This paper is concerned with the stability of nonlinear time-varying perturbed system on time scales under the assumption that the corresponding linear time-varying nominal system is uniformly exponentially stable. Some less conservative sufficient conditions for uniform exponential stability and uniform practical exponential stability are proposed by imposing different assumptions on the perturbation term. Compared with the traditional exponential stability results of perturbed systems, the time derivatives of related Lyapunov functions in this paper are not required to be negative definite for all time. The main tools employed are two Gronwall's inequalities on time scales. Some examples are also given to illustrate the effectiveness of the theoretical results.</p></abstract>

New Control Design via Switching Approach with Higher Order Derivatives of Lyapunov Function for Takagi-Sugeno Fuzzy Systems

This paper proposes a new control design method via two techniques to decrease the conservativeness in control design conditions for Takagi-Sugeno fuzzy systems that can represent a wide class of nonlinear systems. The first technique we employ is to use higher order derivatives of Lyapunov function that is shown to reduce the conservativeness in the literature. The second one is a switching approach that takes advantages of a property of the membership function. In this paper, we provide control design methods and their less conservative conditions based on the above-mentioned techniques. The proposed control design conditions are independent of the membership function and are written in a linear matrix inequality form. Finally, Numerical examples illustrate the effectiveness of the proposed control methods by comparing with the existing methods.

Adaptive Predefined-Time Control for Lü Chaotic Systems via Backstepping Approach

This brief focuses on the adaptive predefined-time backstepping control for Lü chaotic systems with an unknown parameter. Different from the existing research on convergence-time control for chaotic systems, based on the backstepping technique, an improved predefined-time stable lemma for strict-feedback systems is proposed, which can achieve precise control about convergence time. Moreover, a predefined-time control scheme is designed using the proposed lemma to suppress the chaotic phenomenon and stabilize the Lü chaotic systems with predefined time. And, a novel adaptive updating law is designed to make the derivative of Lyapunov function satisfy the predefined-time stable form. The stability analysis proves that the signals in the Lü chaotic system are predefined-time stable and the convergence time can be specified by the user. Finally, a simulation example is provided to prove the effectiveness of the presented predefined-time control scheme.

LaSalle–Yoshizawa Theorem for nonlinear systems with external inputs: A counter-example

LaSalle–Yoshizawa Theorem is an important tool in guaranteeing convergence of a nonlinear time-varying adaptive system. It claims boundedness and convergence when the derivative of a Lyapunov function is negative semidefinite. For a nonlinear system with an external input, the input-to-state stability (ISS) Lyapunov theorem reveals boundedness of system solutions when the derivative of an ISS Lyapunov function is negative definite with an input term. It is interesting to seek a LaSalle–Yoshizawa like criterion for a nonlinear system with an external input. An intuitive question is whether a certain boundedness property is guaranteed when the derivative of an ISS Lyapunov function is negative semidefinite with an input term. This technical communique gives a negative answer with a counter-example.

A Robust Fractional-Order Control Technique for Stable Performance of Multilevel Converter-Based Grid-Tied DG Units

In this article, we propose a sliding fractional-order (FO) control strategy to provide robustness feature for integrating a multilevel converter into the power grid under the system parameter uncertainties, nonlinear load alterations, and grid voltage sag. All dynamics achieved from the input capacitors and output inductances of the multilevel converter are utilized to design the proposed sliding fractional surfaces (SFSs) such that a robust operation of the grid-tied converter-based system is achieved. Then, an SFS-based Lyapunov function is built to assess the proposed controller stability. A thorough analysis of the Lyapunov function is carried out for distinguishing suitable boundaries of the controller coefficients when the error components of the system state variables are changed, accordingly. Using the derivative of Lyapunov function and SFS, a comprehensive evaluation is executed to detail the regulation procedure for the robustness and fractional PI controller coefficients. Finally, a closed-loop system using the converter current dynamics aims to discern the differences between different values of FO. Experimental tests using dSPACE-1202 along with MATLAB/Simulink simulations are employed to verify the effectiveness of the proposed control technique for providing proper harmonic compensation, enhanced power quality, unity power factor, voltage sag tolerance, and very low total harmonic distortion grid current.

Stability analysis for fractional‐order neural networks with time‐varying delay

SummaryThis paper focuses on the stability analysis for fractional‐order neural networks with time‐varying delay. A novel Lyapunov's asymptotic stability determination theorem is proved, which can be used for fractional‐order systems directly. Different from the classical Lyapunov stability theorem, constraint condition on the derivative of Lyapunov function is revised as an uniformly continuous class‐K function in the fractional‐order case. Based on this novel Lyapunov stability theorem and free weight matrix method, a new sufficient condition on Lyapunov asymptotic stability of fractional‐order Hopfield neural networks is derived by constructing a suitable Lyapunov function. Moreover, two numerical examples are provided to illustrate the effectiveness of these criteria.