Lyapunov Energy Function Research Articles

Logic-Based Reverse Analysis: A Covid-19 Surveillance Data Set Classification Problem

This study focuses on the application of formal logic systems to real-world problem-solving, specifically in the classification of the COVID-19 Surveillance Data Set (CSDS). The research introduces the integration of a random three satisfiability problem of Boolean logic into a Hopfield Neural Network (HNN) to obtain an optimal representation of Random kSatisfiability for CSDS classification. The primary goal is to utilize the optimization capabilities of the Lyapunov energy function in the HNN to extract logical relationships and identify significant features contributing to COVID-19 detection. The CSDS used in this study is sourced from the reputable UCI dataset, and the HNN's energy minimization mechanism is employed for logical mining. Computational simulations are performed with varying numbers of clauses to validate the efficacy of the proposed model in training the CSDS for classification purposes. The results showcase the efficiency and robustness of employing reverse analysis using k-satisfiability in conjunction with a Hopfield Neural Network. This approach successfully extracts dominant features related to the logical framework underlying the CSDS. By combining formal logic systems with the power of neural networks, this research offers insights into the correlation between logical rules and COVID-19 detection. The findings contribute to our understanding of how the HNN can effectively learn and classify data, opening avenues for enhanced classification techniques in the healthcare sector and other domains.

Approaching the Transient Stability Boundary of a Power System: Theory and Applications

Estimating the stability boundary is a fundamental and challenging problem in transient stability studies. It is known that a proper level set of a Lyapunov function or an energy function can provide an inner approximation of the stability boundary, and the estimation can be expanded by trajectory reversing methods. In this paper, we streamline the theoretical foundation of the expansion methodology, and generalize it by relaxing the request that the initial guess should be a subset of the stability region. We investigate topological characteristics of the expanded boundary, showing how an initial guess can approach the exact stability boundary locally or globally. We apply the theory to power system transient stability assessment, and propose expansion algorithms to improve the well-known Potential Energy Boundary Surface (PEBS) and Boundary of stability region based Controlling Unstable equilibrium point (BCU) methods. Case studies on the IEEE 39-bus system well verify our results and demonstrate that estimations of the stability boundary and the critical clearing time can be significantly improved with modest computational cost. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper was motivated by the problem of determining whether a power system can retain transient stability subject to large disturbances such as short-circuit fault and generator tripping. The most well-known approaches to such a problem are the direct methods that can avoid time-costly simulations and “directly” determine the transient stability by estimating the stability boundary. Nevertheless, such estimations are generally too conservative. This paper proposes a new expansion method to improve the direct methods, which allows the initial estimation to lie partially outside the real boundary and hence enables applications to both global and local direct methods. Our method can expand the initial estimation towards the real stability boundary, thus reducing the conservativeness and providing a more accurate stability assessment such as the critical clearing time (CCT) of a fault. In this paper, we first mathematically establish the expansion theory and then propose numerical algorithms for power system applications. Simulations on the IEEE 39-bus benchmark verify our approach and show that the conservativeness can be significantly reduced with modest computational cost. Our approach can be further extended to scenarios where a valid Lyapunov (energy) function is unavailable.

Enhanced Operation of a Virtual Synchronous Machine Under Unbalanced Grid Voltage Condition Based on Lyapunov Energy Function

Three-phase grid connected inverters integrated with renewable energy sources like photovoltaic systems require to provide inertia to damp out oscillations in the frequency. However, during grid voltage sag or unbalance classical virtual synchronous machines (VSMs) cannot maintain oscillation free power supplied to the grid. Therefore, the maximum power point tracking of the photovoltaic system cannot be maintained, and optimal usage of the available photovoltaic power cannot be accomplished. To cater this problem, in this article a Lyapunov energy function-based augmented controller architecture has been proposed alongside a VSM during unbalanced grid voltage to effectively suppress these double frequency oscillations in the power. Identifying the dynamics of these oscillations followed by detailed stability analyses for the proposed architecture have been presented. In comparison to classical VSMs, the augmented system provides better transient response as well as double frequency power oscillation damping during grid voltage sag or unbalance. To verify the efficacy and understand the challenges of practical implementation of the proposed architecture, experimental verification on a reduced scale laboratory prototype is performed and various important case study results are obtained and presented in this article.

Power Feasible Region Ensuring Transient Stability of Droop-Based Multiconverters DC System

This article systematically focuses on the transient stability problem of droop-based multiconverters dc (MCdc) system caused by large power disturbances. Therefore, a new and effective transient stability analysis method is proposed. In the present method, it starts with reduced-order nonlinear mathematical modeling, followed by Lyapunov energy function building, and then the inscription of the largest estimated domain of attraction (LEDA) and the power feasible region (PFR). First, a state-variable equivalent transformation method is proposed. With this method, the droop-based MCdc system can be modeled as an equivalent single-converter (ESC) model without ignoring any coupling effects between the control state-variables. Second, a Takagi–Sugeno fuzzy model of the ESC is developed, which is able to construct the LEDA. Third, to ensure the transient stability of the MCdc system under large power disturbance, the concept of PFR is proposed. Furthermore, a PFR inscription algorithm is given based on the LEDA. In addition, the influence of control parameters on the PFR is analyzed, which will provide guidelines for the optimization of the PFR. Finally, the time domain experiments are carried out on the RT-BOX hardware-in-the-loop experimental platform to verify the effectiveness of the above theoretical analysis.

Orbit quantization in a retarded harmonic oscillator

We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential with state-dependent time-delayed feedback. In the limit of small time-delays, we show that the oscillator is equivalent to a Liénard system. This allows us to analytically predict the value of the first Hopf bifurcation, unleashing a self-oscillatory motion. We compute bifurcation diagrams for several model parameter values and analyze multistable domains in detail. Using the Lyapunov energy function, two well-resolved energy levels represented by two coexisting stable limit cycles are discerned. Further exploration of the parameter space reveals the existence of a superposition limit cycle, encompassing two degenerate coexisting limit cycles at the fundamental energy level. When the system is driven very far from equilibrium, a multiscale strange attractor displaying intrinsic and robust intermittency is uncovered.

Angle Instability and Oscillations Control using SVC: A Deep Reinforcement Learning Enhanced Local Controller

An advanced control scheme to deal with transient angle instability and low-frequency angle oscillations in power systems is proposed in this paper. The control combines an existing controller (derived from the concept of Lyapunov energy functions) and deep Q-network (a deep reinforcement learning algorithm) to control a static VAr compensator. This control is modified in this paper in a way to directly control thyristor controller reactor part of the compensator, offering an easier implementation. The aim is to improve performances of existing control through the use of deep Q-network while retaining its basic system stabilization characteristics. Advantages of the proposed control scheme are illustrated by implementation of a model of four-machine test power system (this system is considered as the benchmark when studying the phenomena dealt with in this paper) in MATLAB/Simulink environment and using TensorFlow toolkit for deep Q-network usage.

Optimization of Hopfield Neural Network (HNN) using multiconnection and Lyapunov Energy Function (LEF) for storage and recall of handwritten images

Optimization of Hopfield Neural Network (HNN) using multiconnection and Lyapunov Energy Function (LEF) for storage and recall of handwritten images

A Lyapunov-Based Model Predictive Control Approach for Photovoltaic Microgrid Integration via Multilevel Flying Capacitor Inverter

In this study, we introduce a Model Predictive Control (MPC) approach based on a Lyapunov energy function, ensuring global asymptotic stability for a single-phase multilevel flying capacitor inverter (FCI) that interfaces with photovoltaic systems and microgrids. The defined cost function is derived from the Lyapunov energy function, harnessing the stored energy within the capacitor and inductor. This choice is rooted in the principle that as long as total energy is continuously dissipated, the system’s states will eventually reach the equilibrium point. Moreover, this cost function eliminates the need for tuning the weighting factors, a process that can be cumbersome and time-consuming due to the lack of clear guidelines. To assess the efficacy of the proposed control strategy, we conducted simulations using MATLAB/Simulink under varying weather conditions. The results obtained demonstrate that the MPC strategy not only ensures overall stability but also delivers high-quality sinusoidal current with minimal total harmonic distortion (THD), practical low steady-state error in the grid current, and rapid dynamic response, even in the face of changing weather conditions.

Model Predictive Control Strategy for Induction Motor Drive Using Lyapunov Stability Objective

This article presents a novel Lyapunov-based model predictive control strategy for squirrel-cage induction motor fed by a voltage source inverter. The model predictive control method has received enormous attention thanks to its rapid response to load perturbations. However, the traditional model predictive control method may suffer from instability due to the poor choice of the objective function, weighting factors, or other design parameters. In particular, the selection of the objective function may not be sufficient to ensure global stability. Since the performance of the model predictive control method highly relies on the explicit model of the system, a simple penalization term in the objective function can lead to system instability. As a result, the transient and steady-state performances are negatively affected. The objective function is reformulated as the Lyapunov energy function to deal with this ambiguous situation in this article. The control input constraints are defined in the formulated optimal control problem, and the optimal solution is explored by assessing the Lyapunov stability criterion. The proposed method ensures asymptotic stability since the control action that satisfies the Lyapunov stability condition is selected. The experimental work proves the theoretical concepts. Moreover, the proposed method does not require the weighting factors to control the multiple control goals. The experimental results demonstrate that the proposed control strategy improves the steady-state performance. The machine torque and speed are well regulated, and the quality of the stator current is improved.

Design of a Hybrid Transformer based on an Indirect Matrix Converter for Distribution Networks

This paper presents a novel hybrid transformer (HT) to regulate voltage sags/swells and to provide a unity power factor at the connection point to the medium voltage grid. The HT is a promising instrument for constructing an active distribution grid in the future. To overcome the applied design limitations concerning the converter and control system of the existing HTs, the HT structure and the control design have been proposed based on the indirect matrix converter (IMC) and Lyapunov energy function, respectively. The proposed HT topology consists of a direct power converter and a line frequency transformer. The proposed HT employs an IMC with space vector modulation that is connected to the distribution network via a series transformer. This power electronic converter increases the reliability, controllability and intelligence of the distribution substation. The Lyapunov direct control method is used to design the IMC controller that ensures tracking the input reference current and voltage. A systematic experimental and numerical investigation is performed in order to validate the effectiveness of the proposed topology. The proposed HT can compensate voltage sags/swells (more than 20% of nominal voltage). In addition, it performs properly in power factor correction and reduces the capacitor bank in the station.

Lyapunov-based model predictive control of dual-induction motors fed by a nine-switch inverter to improve the closed-loop stability

This study presents a novel Lyapunov-based model predictive control strategy for a nine-switch inverter-based multi-drive system. The drive system includes two ac motors and a single nine-switch inverter (NSI) that is capable of feeding the multiple induction motors. The model predictive control (MPC) is a popular feedback strategy in ac-drive applications due to its fast dynamic characteristics and the skill of handling multiple control goals. Although the MPC offers a rapid dynamic response to load variations, it suffers from closed-loop instability when the cost function is poorly designed. The unwise choice of the tuning parameters, cost function terms, and control constraints cause instability resulting in poor system performance. In particular, the formulation of the selected objective function greatly impacts the closed-loop stability since it tailors the closed-loop characteristics. A simple penalization of the control goal error in the cost function may not ensure asymptotic stability. Motivated by the MPC stability problem, the cost function is reformulated as a Lyapunov energy function to strengthen the closed-loop stability in this paper. The proposed method selects the control input that meets the Lyapunov stability criterion and applies it to the multi-motor driver. The proposed strategy ensures asymptotic stability and provides a robust motor operation by introducing the Lyapunov control constraints. The proposed feedback strategy is experimentally verified using a lab-scaled hardware test platform. The experimental results demonstrate that the suggested control routine offers an improved steady-state characteristic compared to the conventional MPC method. © 2017 Elsevier Inc. All rights reserved.

Decentralized Mutual Damping Control of Cascaded-Type VSGs for Power and Frequency Oscillation Suppression

This paper reveals the power and frequency oscillation mechanism of the islanded cascaded-type virtual synchronous generators (VSGs), and proposes a decentralized mutual damping suppression method. In the proposed scheme, the mutual damping term is constructed via the decentralized manner with transmission line current. It contributes to increasing damping and accelerating the frequency convergence of cascaded-type VSGs in the dynamic process. As a result, the power and frequency oscillations are restrained, and the system dynamic performance is improved. Further, the system stability based on the Lyapunov energy function method is proved. Finally, the effectiveness of the proposed decentralized mutual damping control is verified by simulations and experimental results.

AILC for Rigid-Flexible Coupled Manipulator System in Three-Dimensional Space with Time-Varying Disturbances and Input Constraints

In this paper, an adaptive iterative learning control (AILC) law is developed for two-link rigid-flexible coupled manipulator system in three-dimensional (3D) space with time-varying disturbances and input constraints. Based on the Hamilton’s principle, a dynamic model of a manipulator system is established. The conditional equation that is coupled by ordinary differential equations and partial differential equations is derived. In order to achieve high-precision tracking of the revolving angles and vibration suppression of the elastic part, the iterative learning control law based on the disturbance observer is considered in the process of the design controller. The composite Lyapunov energy function is proposed to prove that the angle errors and elastic deformation can eventually converge to zero with the increase of the number of iterations. Ultimately, the simulation results to rigid-flexible coupled manipulator system are given to prove the convergence of the control objectives under the adaptive iterative learning control law.

Digital Sliding Mode Based Model-Free PWM Current Control of Switched Reluctance Machines

This article presents a digital sliding mode based robust model-free pulsewidth mode current control method for switched reluctance machine (SRM) drives. The model-free nature of the proposed controller allows complete elimination of the phase inductance or flux-linkage identification process. From the controller point of view, this exclusion yields excellent computational efficiency and low memory usage. In the proposed control method, the model information required to achieve a consistent closed-loop dynamic response is treated as an extended state and it is estimated online via unit time-step delayed approximation. The effect of this delay becomes significant as the operating speed increases. The estimation accuracy of the extended state is also affected by the noise in the current measurement. The effect of these factors on the control performance is compensated by using an auxiliary control action derived from a Lyapunov energy function analysis in the discrete-time domain. The resultant controller achieves accurate tracking of the reference phase current profile at fixed switching frequency modulation. It demonstrates a robust tracking performance throughout the controllable operating range of the SRM drive amid low signal-to-noise ratio.

Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method

Abstract The paper calculates the nonlinear dynamic differential equation model based on the stiffness of the gear teeth and gives the calculation method of the spring stiffness of the transmission system. Choose the Lyapunov energy function and derive the adaptive law that can make the system asymptotically stable globally. At the same time, we discussed the influence of the phase combination of the coupling shaft’s torsional stiffness and the gears’ meshing stiffness in the multi-stage gear transmission system on the system dynamics. The example calculation shows that the asymptotic method has higher solution accuracy and higher calculation efficiency. This algorithm is a highly versatile analytical solution method.

Optimization of Hopfield Neural Network for Improved Pattern Recall and Storage Using Lyapunov Energy Function and Hamming Distance

In this paper, we propose a multiconnection-based Hopfield neural network (MC-HNN) based on the hamming distance and Lyapunov energy function to address the limited storage and inadequate recalling capability problems of Hopfield Neural Network (HNN). This study uses the Lyapunov energy function and Hamming Distance to recall correct stored patterns corresponding to noisy test patterns during the convergence phase. The proposed method also extends the traditional HNN storage capacity by storing the individual patterns in the form of etalon arrays through the unique connections among neurons. Hence, the storage capacity now depends on the number of connections and is independent of the total number of neurons in the network. The proposed method achieved the average recall success rate of 100% for bit map images with a noise level of 0, 2, 4, 6 bits, which is a better recall success rate than traditional and genetic algorithm-based HNN methods, respectively. The proposed method also shows quite encouraging results on hand-written images compared with some latest state of art methods.

Mean-square pinning control of fractional stochastic discrete-time complex networks

This paper considers the mean-square pinning control problem of fractional stochastic discrete-time complex networks. First, a new fractional stochastic discrete-time complex networks model with stochastic noise is established. Then, some pinning controllers and sufficient conditions are developed for the complex networks. By adopting Lyapunov energy function theory and matrix analysis theory, it proved that the synchronization of the fractional stochastic discrete-time complex networks can be achieved in a mean-square sense via pinning control. In addition, these results are extended to solve the synchronization problem of general fractional discrete-time complex networks without noise. Finally, several numerical examples are given to verify the correctness of the obtained theoretical results.

An optimal representation of Random Maximum kSatisfiability on a Hopfield Neural Network for High order logic(k 3)

This paper proposed a new logical rule by incorporating Random maximum kSatsifiability (RANMAXkSAT) in the Hopfield neural network (HNN) as a single network model (HNN-RANMAXkSAT). The purpose is to combine the optimization capacity of the Hopfield neural network (HNN) for optimal representation to random maximum kSatsifiability (MAXRANkSAT). The energy function of a Hopfield neural network has been considered as a programming language for dynamics minimization mechanism. Several optimization and search problems associated with machine learning (ML), decision Science (DS) and artificial intelligence (AI) have been expressed on the Hopfield neural network(HNN) optimally by modelling the problem into variables to minimize the objective function that corresponds to Lyapunov energy function. The computer simulation has been developed based on RANMAXkSAT to explore the feasibility of a Hopfield neural network as a neuro-symbolic integration model in carrying out RANMAXkSAT logic programming optimally. The proposed model has been compared with the existing models published in the literature in term of Global minimum ratio (zM), Fitness energy landscapes (FEL), Root Means square error (RMSE), Mean absolute errors and computation time (CPU). Hence, based on the experimental simulation results, it revealed that the RANMAXkSAT can optimally and effectively represented in the Hopfield neural network (HNN) with 85.1 % classification accuracy.

Novel stability and passivity analysis for three types of nonlinear LRC circuits

In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

Sufficient conditions in terms of linear matrix inequalities for guaranteed ultimately boundedness of solutions of switched Takagi-Sugeno fuzzy systems using the S-procedure

• LMIs sucient conditions guarantee ultimate boundedness of the solutions of switched TS fuzzy systems. • A constrained switching law plays the role of the controller to assure a bounded solution of a class of non-linear systems. • The derivative of the used Lyapunov-like energy function can assume positive values in a bounded set described as level sets. • The LMIs explore the S-procedure to reduce the conservativeness of the results. • Estimates of the upper bounds on the derivative of the membership functions are used to obtain bounded sets. In this paper, sufficient conditions to ensure the existence of a switching law that makes the solutions of switched Takagi-Sugeno (TS) fuzzy systems ultimately bounded are developed by means of linear matrix inequalities (LMIs). These LMIs are based on the existence of a scalar function, which plays a role similar to Lyapunov energy functions for an auxiliary system formed by a convex combination of all subsystems of the switched system. A feature of the developed results is that the derivatives of the scalar function can assume positive values in a bounded set described as level sets. The LMIs explore the S-procedure to obtain low levels of conservativeness and do not require the calculation of the derivative of the membership functions, which facilitates their application to switched TS fuzzy systems with many rules. Exploring the proposed conditions, we estimated the attractor and basin of attraction of some examples of switched TS fuzzy systems under a measurable switching law. These numerical examples showed the effectiveness of the proposed approach in maximizing the estimation of the bounded attraction domain.