Lyapunov Stability Criterion Research Articles

Projective synchronization for a new class of chaotic/hyperchaotic systems with and without parametric uncertainty

The term “chaotic system” refers to a kind of dynamical system that is characterized by its extreme sensitivity to its initial conditions. The ability of such dynamical systems to maintain synchronization is one of the extremely crucial features of these systems. There is an increasing fascination in chaotic systems with real-state variables, and they are being discovered as well as explored in greater depth in a variety of areas of nonlinear science. In this manuscript, a mixture of active nonlinear control scheme as well as adaptive control scheme has been recommended to investigate the full-order and reduced-order projective synchronization for a unique class of chaotic and hyperchaotic systems with as well as without uncertain parameters. Active nonlinear control technique is an effective method, which has been extensively used to synchronize two chaotic systems in master–slave configuration when parameters of the systems under consideration are fully known in advance. However, in practical scenarios, system parameters are not exactly known in advance. This often occurs due to the factors such as limitations associated with the experimental conditions, mutual interference among system components or due to the influence of external environmental factors. Therefore, in such cases, the designer has to go for the adaptive control technique to develop the adaptation control laws for unknown system parameters. Adaptive active nonlinear controller and parameter adaptation laws have been introduced here using Lyapunov stability criteria to confirm that the error dynamics of the synchronizing chaotic systems become asymptotic stable. Furthermore, numerical simulations on Lorenz hyperchaotic fourth-order system and Pehlivan chaotic third-order system are shown to demonstrate that the proposed synchronization scheme is working effectively.

Nonlinear Path Generation and Tracking Control Design for a Mobile Robot*

This paper proposes trajectory generation and tracking control methods for a differential drive wheel mobile robot. The reference trajectory is generated as a polynomial function in time variable while the tracking control algorithm is constructed using Lyapunov's stability criteria. Experimental results are reported to show the good performance of the proposed trajectory generation and tracking control algorithms.

Adaptive Fractional‐Order Super‐Twisting Sliding Mode Controller for Lower Limb Rehabilitation Exoskeleton in Constraint Circumstances Based on the Grey Wolf Optimization Algorithm

In this work, a lower limb exoskeleton with uncertainties and external disturbances under constrained motion has been controlled by developing a model‐free adaptive optimal fractional‐ordersuper‐twisting sliding mode (AOFSTSM) controller. Contrary to popular existing modeling methods, the modeling of the lower limb exoskeleton has been accomplished in the case of contact with the ground. A fractional‐order operator and super‐twisting sliding mode control have been combined to achieve an excellent tracking performance, chatter‐free control inputs, and robustness to external disturbances and uncertainties. The controller structure is model‐independent which has been derived without needing to know the dynamics of the system. An adaptive control strategy has been adopted as an approximating method to evaluate the uncertain dynamics of the system without the necessity for knowing the prior knowledge of the upper bounds. A grey wolf optimization technique has been used to find the optimal values of controller parameters. The stability of the overall system has been investigated and derived from the Lyapunov stability criterion. To validate the effectiveness of our proposed controller structure, a series of comparative simulations have been conducted with optimal fractional‐order super‐twisting sliding mode (OFSTSM) controller and fractional‐order super‐twisting sliding mode (FSTSM) controller. The comparative simulations have been also carried out with other types of recently existing control methods and recently existing optimization algorithms. The results of numerical simulations indicate the superiority of the AOFSTSM controller over other types of recently existing optimization algorithms and controllers in terms of tracking error and robustness towards the uncertainties and external disturbances.

A Feedforward Quadrotor Disturbance Rejection Method for Visually Identified Gust Sources Based on Transfer Reinforcement Learning

With the rapid development of urbanization, quadrotors are flying more and more frequently between highdensity buildings. In urban areas, there are many gust sources with obvious visual characteristics, such as ventilation outlet of buildings or subways, outdoor unit of HVAC (heating ventilating and air conditioning), or other artificial gust sources, which can cause sudden gust disturbances to the passing quadrotors. Aiming at reducing the effect of visually identified gust disturbances, this article presents a quadrotor feedforward compensator framework based on transfer reinforcement learning (TRL). In this structure, the Discriminative Model Prediction (DiMP) algorithm is used to extract the visual features of the gust source and achieve tracking of such sources. Then, the TRL algorithm is used to transfer the obtained tracking network to a compensation network, which completes a DiMP-TRL structure. The output of the trained DiMP-TRL compensation network is added to the quadrotor controller as a feedforward part, which will help generate preventive action when encountering the identified gust disturbance. Only a front-facing camera is needed to collect gust information, no additional anemometer or accurate wind model is required. The ultimately uniformly bounding of the quadrotor system can be guaranteed using the Lyapunov stability criteria. The proposed method is compared with Cascade PID and a novel feedback compensation-based method called DCF in a realistic simulation environment built by a physics engine. Two sets of results demonstrate that the proposed method can reject gust disturbance more efficiency.

Neuro-Based Higher Order Sliding Mode Control for Perturbed Nonlinear Systems

One of the great concerns when tackling nonlinear systems is how to design a robust controller that is able to deal with uncertainty. Many researchers have been working on developing such type of controllers. One of the most efficient techniques employed to develop such controllers is sliding mode control (SMC). However, the low order SMC suffers from chattering problem which harm the actuators of the control system and thus unsuitable to be used in many practical applications. In this paper, the drawbacks of low order traditional sliding mode control (FOTSMC) are resolved by presenting a novel adaptive radial basis function neural network–based generalized rth order sliding mode control strategy for nth order uncertain nonlinear systems. The proposed solution adopts neural networks for their excellent capability in function approximation and thus used to approximate the nonlinearities and uncertainties for systems under consideration. The approximation errors are completely considered in the developed approach. The proposed approach can be used with any order of sliding mode and thus can be generally used with various types of applications. The global stability of the proposed control approach is proved through Lyapunov stability criterion. The proposed approach is validated and assessed through simulations on the nonlinear inverted pendulum system with severe modeling uncertainties. The simulations results show that the proposed approach provide superior performance compared with other approaches in the literature.

On the stability of the diffusive and non-diffusive predator-prey system with consuming resources and disease in prey species.

This research deals with formulating a multi-species eco-epidemiological mathematical model when the interacting species compete for the same food sources and the prey species have some infection. It is assumed that infection does not spread vertically. Infectious diseases severely affect the population dynamics of prey and predator. One of the most important factors in population dynamics is the movement of species in the habitat in search of resources or protection. The ecological influences of diffusion on the population density of both species are studied. The study also deals with the analysis of the effects of diffusion on the fixed points of the proposed model. The fixed points of the model are sorted out. The Lyapunov function is constructed for the proposed model. The fixed points of the proposed model are analyzed through the use of the Lyapunov stability criterion. It is proved that coexisting fixed points remain stable under the effects of self-diffusion, whereas, in the case of cross-diffusion, Turing instability exists conditionally. Moreover, a two-stage explicit numerical scheme is constructed, and the stability of the said scheme is found by using von Neumann stability analysis. Simulations are performed by using the constructed scheme to discuss the model's phase portraits and time-series solution. Many scenarios are discussed to display the present study's significance. The impacts of the transmission parameter 𝛾 and food resource f on the population density of species are presented in plots. It is verified that the availability of common food resources greatly influences the dynamics of such models. It is shown that all three classes, i.e., the predator, susceptible prey and infected prey, can coexist in the habitat, and this coexistence has a stable nature. Hence, in the realistic scenarios of predator-prey ecology, the results of the study show the importance of food availability for the interacting species.

Structural Vibration Suppression Using a Reduced-Order Extended State Observer-Based Nonsingular Terminal Sliding Mode Controller with an Inertial Actuator

In this paper, we mainly aimed to design a reduced-order extended state observer-based active vibration controller for a structural vibration control system with total disturbances, i.e., model uncertainties, higher harmonics, and external excitations. A reduced-order extended state observer (RESO)-based nonsingular terminal sliding mode vibration control (RESO–NTSMVC) method is proposed for the vibration suppression of an all-clamped plate structure with an inertial actuator. First, a second-order state space model of the thin plate, with an inertial actuator, was established by solving the dynamic partial differential equation and analyzing the physical model. Second, the total disturbances, i.e., model uncertainties, higher harmonics, and external excitations, were estimated and compensated for by using a RESO via a feedforward part. Third, a NTSMVC based on an estimated value was designed to obtain a fast-tracking rate and effective vibration suppression performance. In addition, the stability of the closed-loop system was proven by using a Lyapunov stability criterion. Finally, a semi-physical experimental instrument was built based on the MATLAB/Simulink real-time environment and the NI-PCIE6343 acquisition card to verify strong anti-disturbance performance and effective vibration control performance of the designed method. The experimental comparison results showed that the vibration amplitudes of the proposed method could be reduced by 11.7 dB, when the traditional extended state observer-based nonsingular terminal sliding mode vibration control (ESO–NTSMVC) method achieved a control effect of only 6.5 dB. The comparative experimental results showed that the proposed method possessed better vibration suppression performance and anti-disturbance performance.

The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication

A novel approach for the synchronization of a class of chaotic systems with uncertainty, unknown time delays, and external disturbances is presented. The control method given here is expressed by combining sliding mode control approaches with adaptive rules. A sliding surface of fractional order has been developed to construct the control strategy of the abovementioned sliding mode by employing the structure of nonlinear fractional PID (NLPID) controllers. The suggested control mechanism using Lyapunov’s theorem developed robust adaptive rules in such a way that the estimation error of the system’s unknown parameters and time delays tends to be zero. Furthermore, the proposed robust control approach’s stability has been demonstrated using Lyapunov stability criteria and Lipschitz conditions. Then, in order to assess the performance of the proposed mechanism, the presented control approach was used to simulate the synchronization of two chaotic jerk systems with uncertainty, unknown time delays, and external distortion. The results of the simulation confirm the robust and desirable synchronization performance. Finally, a secure communications mechanism based on the proposed technique is shown as a practical implementation of the introduced control strategy, in which the message signal is disguised in the transmitter with high security and well recovered in the receiver with high quality, according to the mean squared error (MES) criteria.

Artificial intelligence based nonlinear control of hybrid DC microgrid for dynamic stability and bidirectional power flow

Conventional power generation resources are depleting rapidly and world power sector has been moving towards renewable energy sources (RESs). In this paper nonlinear control for renewable energy and hybrid energy storage system (HESS) based DC microgrid (DCMG) has been presented. PV and wind energy being the renewable sources whereas fuel cell, battery and ultracapacitor constitute the HESS. The global mathematical model of the said system has been presented. Datasets of varying solar irradiance and temperature have been trained by Artificial Neural Network for the reference voltage generation of PV. Integral terminal sliding mode controller (ITSMC) has been proposed for the output DC bus voltage regulation. Lyapunov stability criterion has ensured the overall stability of the system. A comparison of ITSMC with SMC and Lyapunov redesign controller has also been presented. Grid connected battery electric vehicle charger with grid to vehicle (G2V) and vehicle to grid modes (V2G) has been presented being an application of DCMG. The proposed system has been validated by using MATLAB/Simulink (2020b). Moreover, the hardware in loop setup has been used to observe the real time applicability of the proposed controller.

Robust MTPA Control for Novel EV-WFSMs Based on Pure SM Observer Based Multistep Inductance Identification Strategy

Wound field synchronous motors (WFSMs) without installing slip rings and brushes are drawing increasing attention in the electric vehicle (EV) propulsion systems. To ensure high control performance of the EV-WFSMs, this article proposes a robust maximum torque per ampere (MTPA) control strategy based on a series of new sliding mode (SM) inductance observers. First, after establishing the model of a WFSM based on capacitive coupling, the current control scheme is designed for the MTPA control. Second, to eliminate the impacts of the inductance uncertainties on the MTPA control, the SM mutual inductance observer, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -axis inductance observer, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> -axis inductance observer are designed, which need to be implemented one by one. Then, the Lyapunov stability criterion is used to analyze the stability conditions for the observers. Moreover, considering that the observers are achieved by using the offline inductance information provided by the suppliers, the robustness against parameter mismatch is innovatively discussed at length, and an analytical method that can avoid estimation errors is developed. The proposed inductance identification techniques and MTPA control method are verified by both simulation and experiments, which are conducted on a 580-W three-phase WFSM drive.

Sensorless Control of Permanent Magnet Synchronous Motor Based on New Sliding Mode Observer with Single Resistor Current Reconstruction

To solve the chattering problem caused by discontinuous switching function in traditional sliding mode observer, a piecewise square root switching function with continuously varying characteristics is designed, and its stability is analyzed by using Lyapunov stability criterion. Secondly, according to the relationship among bus current, switching state and phase current, a single bus resistance sampling current reconstruction scheme without current sensors is adopted, which effectively reduces the cost of motor system. Finally, the feasibility and effectiveness of the proposed scheme are verified by simulation.

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.

Saturated Adaptive-Law-Based Backstepping and Its Applications to a Quadrotor Hover

This article presents a saturated adaptive-law-based backstepping control method for a class of uncertain nonlinear systems with external dynamic disturbances. In this approach, a novel adaptive backstepping control method with a new command filter is primarily proposed to design the feedback controllers. In order to address the system unknown nonlinearities caused by external dynamic disturbances, a method named “saturated adaptive law approach” is used to estimate the system unknown nonlinearities online. Meanwhile, a filter called “dual command filter” is applied to estimate the derivative of certain signal such that the saturated adaptive law approach can be well implemented. With the help of Lyapunov stability criteria, it is proved that the target signal can be tracked by the output of the system with small error. Finally, our control method is applied to a quadrotor hover system, and two sets of comparison experiments are given to show the effectiveness of the proposed control method.

Active Control of Stick-Slip Torsional Vibrations of Drill-Strings using Non-Switching Sliding Mode Controller

Torsional vibrations of drill-strings are actively controlled using a non-switching sliding mode strategy. The controller is intended also to reject the effect of the interaction between the drill-bit and rock formation which induces non-linear stick-slip friction torque while tracking a desirable constant angular velocity of the drill-string. In order to demonstrate the merits of the proposed control algorithm, it is integrated with a simple two degrees-of-freedom model of the drill-string which is considered as a benchmark model for studying stick-slip induced torsional vibrations. Details of the development of the individual components making up the entire control action of the sliding mode controller (SMC) are presented. These components aim at rejecting the effect of the external disturbance of the stick-slip disturbance and tracking a reference angular velocity of the drill-string. Such development of the control action components is established using Lyapunov stability criterion. Numerical examples are presented to demonstrate the effectiveness of the SMC controller as influenced by the weight-on-bit (WOB), the drill-string rotational speed, and the main parameters of the controller. The stick-slip sensitivity (SSS) maps are developed to illustrate the performance characteristics of the active system for different operating conditions and important design parameters of the SMC. The presented active control approach is envisioned to present an invaluable and practical means for effectively mitigating the undesirable effects of stick-slip frictional disturbances on drill-stings.

Fast reaching law based integral terminal sliding mode controller for photovoltaic-fuel cell-battery-super capacitor based direct-current microgrid

In this paper, a fast reaching law based integral terminal sliding mode controller has been designed for photovoltaic based DC microgrid system. The proposed microgrid system comprised of photovoltaic system as main energy source and fuel cell, battery and supercapacitor as auxiliary energy sources. To avoid the stress on individual energy sources, an energy management system has been devised to allocate the load among these energy sources. Using the Lyapunov stability criteria, the stability of proposed framework has been validated. The performance of the proposed controller has been verified by simulating the dc microgrid under varying environmental conditions and load demands in MATLAB/Simulink platform. The comparison of the proposed control laws against PID and lyapunov controllers has been provided in order to assess its accuracy and robustness. The hardware-in-loop experiments have been performed to verify the real-time efficacy of the proposed microgrid system.

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.

Finite-time robust control of robotic manipulator with input deadzone and time-varying disturbance compensation

In this paper, the problem of trajectory tracking for robotic manipulator with input deadzone and time-varying disturbances is investigated using a continuous non-singular terminal sliding mode control (SMC). The robotic manipulator plays a vital role in industrial as well as many other applications. However, it faces numerous problems for accurate and fast trajectory tracking response in the presence of unknown uncertainties and external disturbances, that always have adverse affects on system closed-loop performance. This paper proposes a valid control design scheme to address the aforementioned practical issue. First, a non-singular terminal sliding mode surface is designed. Then, a robust control law is developed to guarantee the finite-time converges of system states to the origin. In addition, the sign function is substituted by saturation function, which can effectively reduce the chattering driven by the high-frequency switching item in the traditional SMC method. Moreover, the Lyapunov stability criterion is used to prove the stability of the closed-loop system. The efficiency of the designed control scheme is validated with numerical simulations.

Finite-Time Neuro-Sliding-Mode Controller Design for Quadrotor UAVs Carrying Suspended Payload

Due to the quadrotor’s underactuated nature, suspended payload dynamics, parametric uncertainties, and external disturbances, designing a controller for tracking the desired trajectories for a quadrotor that carries a suspended payload is a challenging task. Furthermore, one of the most significant disadvantages of designing a controller for nonlinear systems is the infinite-time convergence to the desired trajectory. In this paper, a finite-time neuro-sliding mode controller (FTNSMC) for a quadrotor with a suspended payload that is subject to parametric uncertainties and external disturbances is designed. By constructing a finite-time sliding mode controller, the quadrotor can follow the reference trajectories in finite time. Furthermore, despite time-varying nonlinear dynamics, parametric uncertainties, and external disturbances, a neural network structure is added to the controller to effectively reduce chattering phenomena caused by high switching gains, and significantly reduce the size of the control signals. Following the completion of the controller design, the system’s stability is demonstrated using the Lyapunov stability criterion. Extensive numerical simulations with various scenarios are run to demonstrate the effectiveness of the proposed controller.

An Optimization-based Adaptive Sliding Mode Control for an Inverted Pendulum

The work proposes an optimization-based robust adaptive gain sliding mode control technique to handle bounded behavior of uncertainties/perturbations without having the exact knowledge of the bound a priori. The constant of the sliding surface follows dynamical adaptive nature using three optimization techniques, i.e. Harmony Search Algorithm, Bacterial Foraging Optimization Algorithm and Gradient Search Algorithm to extenuate the extent of chattering. The gain of the switching control law, which is a part of the sliding mode control design, is calculated using the gain adaption technique. Lyapunov Stability Criterion is applied to assure the stability of the closed-loop system with adaptable gain and the optimized value of the sliding constant. An inverted Pendulum system has been taken to validate the proposition both in the simulation environment and the hardware platform.

Improved adaptive backstepping control of MPCVD reactor systems with non-parametric uncertainties

This paper presents an improved adaptive backstepping control strategy for a microwave plasma chemical vapor deposition (MPCVD) reactor system. To describe the dynamics of the reactor system, a non-parametric uncertain nonlinear system model with unknown control direction functions is built up. In order to deal with the non-parametric uncertainties of the system, direct adaptive laws are primarily proposed to estimate such uncertainties, which is superior to traditional adaptive backstepping that can only address parametric uncertainties. Moreover, an equivalence transformation of the system is given to address the system unknown control direction functions. By using Lyapunov stability criterion, it can be proved that the target signals (desired temperature, desired pressure, and desired gas flow rates) are tracked by the reactor system outputs with small errors. Finally, simulation results are given to demonstrate the effectiveness of the proposed control method.