Lyapunov Criterion Research Articles

Mitigation of Uncertainty in an Islanded Microgrid Using Robust Voltage Controller

In microgrid, a severe problem occurs in terms of voltage oscillations due to mismatch between synchronizing and damping torque. In this study, a centralized nonlinear sliding mode voltage controller proposes to minimize the rotor and DC voltage oscillations issue in an islanded microgrid. The linear matrix inequality (LMI) technique has been applied for bounding the state error. Lyapunov criteria is utilized for assurance of asymptotic convergence and LMI optimization approach is used for obtaining the controller parameters. The proposed controller is able to tackle the parametric uncertainties of diesel generator, oscillations of rotor and voltage fluctuation as well as the closed-loop system responses also improves both transient responses and steady state responses simultaneously. The simulation results authenticate that the proposed controller confirms speedy recovery of nominal frequency without any oscillations and reduced the limitation of chattering notably without any loss in control accuracy. The performance and robustness of the proposed controller is also compared with conventional controllers.

Data-Driven Fault Compensation Tracking Control for Coupled Wastewater Treatment Process

Dear Editor, This letter is concerned with the data-driven fault compensation tracking control for a coupled wastewater treatment process (WWTP) subject to sensor faults. Invariant set theory is introduced to eliminate the completely bounded and differentiable conditions of coupled non-affine dynamics and to explicitly express the control inputs. An adaptive fault compensation mechanism is constructed to accommodate the effects of sensor faults. By employing a cubic absolute-value Lyapunov criteria, it is shown that all the signals are bounded and the tracking error converges to an adjustable neighborhood near the origin. Experiment studies are executed on a standardized platform of WWTP to illustrate the effectiveness of the proposed strategy.

Variants of Two-Measure Input-to-State Stability

For attracting closed sets that are not compact we consider different formulations of input-to-state stability properties and related Lyapunov criteria. In the compact case, there exists a well-established hierarchy of such properties and some formulations have been quickly discarded as equivalent to input-to-state stability. In this paper we show for the noncompact case, several new phenomena appear. In particular, input-to-state stability (ISS) does not imply integral input-to-state stability, and ISS is not equivalent to integral-input-to-integral-state stability. The criteria are formulated in terms of measurement functions, which allows a uniform presentation of a number of related results.

Adaptive Terminal Sliding Mode Control of Picking Manipulator Based on Uncertainty Estimation

In this paper, a robust nonsingular fast terminal sliding mode control scheme for the picking manipulator under the condition of load change and nonlinear friction disturbance is presented. Firstly, the dynamic equation of the picking manipulator under the condition of load change and nonlinear friction disturbance is established. Then, in order to avoid the singularity problem existing in the terminal sliding mode and improve the convergence time, a new nonsingular fast terminal sliding mode control strategy is adopted to design the control law of the picking manipulator, which can guarantee the finite time convergence. The adaptive law is used to estimate the uncertainties of the system, and the finite time convergence of the system state is proved by the Lyapunov criterion. In addition, the genetic algorithm is used to identify the friction parameters to realize the nonlinear friction compensation control of the system. Finally, the simulation results of the picking manipulator under different load conditions show that the controller designed in this paper realizes the fast and accurate positioning of the picking manipulator under load change and nonlinear friction, and the control strategy is reasonable and effective.

Iterative Learning Tracking Control of High-Speed Trains With Nonlinearly Parameterized Uncertainties and Multiple Time-Varying Delays

The precise operation control of high-speed trains is pivotal to maintain the safety and efficiency of trains, while the inevitable state delays will seriously attenuate the performance of control system. In this paper, an adaptive iterative learning control (ILC) approach for high-speed trains is presented in the presence of the nonlinearly parameterized uncertainties and multiple unknown state delays, aiming to drive that the displacements and velocities of trains can track the desired reference trajectories. To describe the operational dynamics of trains more realistically, the multi-particle model of trains involving multiple time-varying delays is established by analyzing the aerodynamic resistance, mechanical resistance, and coupler force acting on different cars. The proposed adaptive ILC scheme fully leverages various techniques, e.g., the hyperbolic tangent function, the parameter separation, to cope with the inherent nonlinearities, uncertainties and couplings of system. Specially, to eliminate the negative influence of unknown delays, an appropriate Krasovskii function is integrated into the Lyapunov criterion to devise the learning controller and check the stability of control systems. The novelties of our work lie in that the refinement model and periodical characteristic are simultaneously utilized to improve the practicability and performance of control scheme for the high-speed trains with multiple state delays.

Distributed Kalman Filtering for Interconnected Dynamic Systems.

This article is concerned with the distributed Kalman filtering problem for interconnected dynamic systems, where the local estimator of each subsystem is designed only by its own information and neighboring information. A decoupling strategy is developed to minimize the impact of interconnected terms on the estimation performance, and then the recursive and distributed Kalman filter is derived in the minimum mean-squared error sense. Moreover, by using Lyapunov criterion for linear time-varying systems, stability conditions are presented such that the designed estimator is bounded. Finally, a heavy duty vehicle platoon system is employed to show the effectiveness and advantages of the proposed methods.

Stability analysis of quantum systems: A Lyapunov criterion and an invariance principle

In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In analog to the classical probability measure for Markovian processes, we show that the set of invariant density operators is both closed and convex. We then show how to analyze the stability of this set via a candidate Lyapunov operator. We complete our analysis of the set of invariant density operators by introducing an analog of the Barbashin–Krasovskii–LaSalle theorem on the dynamics of quantum systems.

Propagation of periodic and solitary waves in a highly dispersive cubic–quintic medium with self-frequency shift and self-steepening nonlinearity

We study the dynamics of femtosecond light pulse propagation in a cubic–quintic medium exhibiting dispersive effect up to the fourth order as well as self-frequency shift and self-steepening nonlinearity. A rich variety of periodic and solitary wave solutions are derived for the governing generalized higher-order nonlinear Schrödinger equation in the presence of self-frequency shift and self-steepening effects. It is found that the frequency shift, inverse velocity, amplitude and wave number of both periodic and solitary waves depend on dispersion coefficients and nonlinearity parameters as well. The conditions on optical fiber parameters for the existence of these structures are presented. The stability of these periodic and solitary wave solutions is studied numerically by adding white noise. It is proved by using the numerical split-step Fourier method that the derived exact solutions are stable by Lyapunov criteria.

Design and Analysis of Rate Predictive Fractional-Order Sliding Mode Controller (RP-FOSMC) for MPPT and Power Regulation of DFIG-based Wind Energy Conversion System (WECS)

Now a days, maximum power point tracking (MPPT), power regulation and transient load reduction are the major issues in the WECSs. To overcome these challenges and to cope with WECS nonlinearity and MIMO nature, nonlinear predictive control algorithms based on Tip-Speed Ratio MPPT (TSR-MPPT) and Rate Predictive Control (RPC) strategies are proposed for the variable speed DFIG-based WECS. The designed control system consists of a FOSMC that is based on fractional-order PID sliding surface model and super-twisting control algorithm (STA) for MPPT through Optimal Torque Control (OTC) algorithm and a Nonlinear-RPC (N-RPC) that is designed based on STA and RPC strategies for power regulation via pitch angle control. FOSMC based on STA improve the dynamics of the controlled system by increasing the degrees of freedom of the sliding surface, minimizing the effects caused by imperfect modelling of nonlinear systems and provide chattering free response of the system and the stability of the designed STA based FOSMC law was proven by Lyapunov criterion. N-RPC is a novel adaptive model-less feedback control algorithm that addresses the limitations of MPC and PID controllers. It limits the power production above rated wind speed and ensure the transient load reduction. The proposed cascaded control algorithms are implemented in a MATLAB/Simulink environment. In this work, the system parameters of Sany SE7715 wind turbine that is installed in Adama-II wind farm, Ethiopia with generating capacity of 1.5 MW is used. The simulation result confirms that the proposed control algorithms achieve optimum power extraction as compared to second-order SMC (2-SMC) and excellent stress reduction in the drive train under variable wind speed. Moreover, they are effective and robust regarding to the application of different external disturbance and sensor noise (i.e., Gaussian & non-Gaussian) and perfectly achieved the required objectives of our investigation.

Extended State Observer Based-Backstepping Control for Virtual Synchronous Generator

The penetration of distributed generators (DGs)-based power electronic devices leads to low inertia and damping properties of the modern power grid. As a result, the system becomes more susceptible to disruption and instability, particularly when the power demand changes during critical loads or the system needs to switch from standalone to a grid-connected operation mode or vice versa. Developing a robust controller to deal with these transient cases is a real challenge. The inverter control method via the virtual synchronous generator (VSG) control method is a better way to supply the system’s inertia and damping features to boost system stability. Therefore, a nonlinear control strategy for VSG with uncertain disturbance is proposed in this paper to enhance the system stability in the islanded, grid-connected, and transition modes. Firstly, the mechanical equations for a VSG’s rotor, which include virtual inertia and damping coefficient, are presented, and the matching mathematical model is produced. Then, the nonlinear backstepping controller (BSC) method combined with the extended state observer (ESO) is constructed to compensate for the uncertainty. The Lyapunov criteria were used to prove the method’s stability. Considering the issue of uncertain items, a second-order ESO is built to estimate uncertainty and external disruption. Finally, the suggested control strategy is validated through three simulation experiments; the findings reveal that the proposed control method has an excellent performance with fast response and tracking under various operating situations.

Adaptive Fuzzy Controller Design for Uncertain Robotic Manipulators Subject to Nonlinear Dead Zone Inputs.

This paper comes up with the adaptive fuzzy scheme for multi-degree robotic manipulators with dead zone. Fuzzy logic system is employed to approximate unknown nonlinear functions and input nonlinear functions with dead zones that are introduced. The adaptive fuzzy technique is utilized to deal with the problems of multi-joint robotic manipulators with unknown nonlinear functions in dead zones. Based on Lyapunov criterion, all the states and signals involved in the system are maintained in a certain bounded region and the tracking error converges to a small domain of origin. Finally, a simulation example of robotic manipulators with two joints is structured to confirm the practicability of the raised scheme.

Improved event-triggered robust adaptive control for marine vehicle with the fault compensating mechanism

This paper investigates an improved event-triggered robust adaptive control algorithm for the marine surface vehicle with the fault compensating mechanism. A dynamic event-triggered mechanism is designed with a concise form to reduce the updating frequencies of the system state. Different from the exiting results, the proposed algorithm can ensure that both the controller and the state transmitter are activated synchronously only at the triggering instants. That can ameliorate the communication obstruction in both the channels of sensor-to-controller and controller-to-actuator. To improve the fault-tolerant capability of the theoretical algorithm, gain-related adaptive parameters are derived to compensate for gain uncertainties and possible actuator faults. In addition, a compensating term via the robust neural damping technique is introduced to stabilize the unpredictable perturbation from the model uncertainties and the external disturbance. Through the Lyapunov criterion, the corresponding closed-loop system is with the semi-global uniformly ultimately bounded (SGUUB) stability. The simulation results are illustrated to verify the effectiveness of the algorithm.

Wavelet neural network sliding mode control of two rigid joint robot manipulator

To solve the problems of low accuracy and poor stability due to uncertainties, external disturbances and unknown load, which exist in the position control of rigid joint robot manipulator, this article is to propose Non-Singular Fast Terminal Sliding Mode Control strategy with Wavelet neural networks observer (NSFTSMCW). The wavelet observer is designed using the online approximation capability of the neural network, which is used to online estimate the modeling error, external disturbances and uncertainties generated by the dynamic surface control of the joint robot online. Combining the above strategies, the robot manipulators position controller is designed. The stability of this control strategy is demonstrated by stability analysis using the Lyapunov criterion. Simulations on the 2-Link Rigid Joint (2LRJ) robot show that the control strategy can overcome the chattering phenomena ensures the accuracy and stability of the joint robot position control.

High-Precision Speed Control for Low-Speed Gimbal Systems Using Discrete Sliding Mode Observer and Controller

The high-precision control of angular speed in low-speed gimbal servo systems is an essential topic in the operation of the single gimbal control moment gyro (SGCMG). This article proposes a discrete control algorithm to suppress the influence of various disturbances on the low-speed gimbal servo system and the constraint between controller parameters and sampling time is established to ensure the stability of the system. To begin with, the disturbance analysis of the low-speed gimbal servo system is presented, including the torque ripple in the permanent magnet synchronous motor (PMSM), nonlinear friction, coupling torque caused by the spacecraft motion, and the parameter uncertainties. Then, a composite control method combining the sliding mode controller (SMC) with sliding mode observer (SMO) is applied to the speed loop and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -axis current loop. The interferences are estimated by two SMOs, while SMCs are designed to eliminate the influence brought by the residual disturbances. The stability of the overall system is proven based on the Lyapunov criterion. Finally, simulation and experimental results verify the effectiveness of the proposed composite control method.

New algorithm based active method to eliminate stick-slip vibrations in drill string systems

The drill string is always exposed to various types of vibrations among which, stick slip is one of the most important types. It is a severe state of torsional vibrations. This phenomenon can decrease the rate of penetration of drilling, wear of expensive equipment prematurely and cause catastrophic failures. In this paper, a novel adaptive sliding mode (SM) controller is proposed to eliminate stick slip in drill string systems. This proposed algorithm has a more robust capacity than existing 1st-order SM schemes in the literature regarding the robustness to parametric uncertainties, variations in weight on bit (WOB), variations in reference velocity and measurement noise. Moreover, the proposed controller does not require a priori knowledge of the upper bounds of parametric uncertainties, external disturbances and can be easily applied for any operating mode of the drill rig. A proof of stability based on the Lyapunov criterion of the system is given. Simulation results show that the proposed algorithm suppresses the stick-slip while keeping good performances compared to other SM controllers. A comparative study between the proposed controller and classic SM controllers and other adaptive SM scheme is performed in order to assess the advantages of the proposed algorithm and illustrate the overall performance improvements. The obtained results show that the proposed controller succeeded to eliminate the stick-slip phenomenon with the best performance compared to the classic SM controllers. In fact, the proposed controller presented a reduction of nearly 26 in terms of overshoot and 1.6 times better settling time values while having the smoother input signal.

Spectral criteria to stability and observability of mean-field stochastic periodic systems

This paper addresses stability and observability of discrete-time mean-field stochastic systems with periodic coefficients and multiplicative noise. By constructing a monodromy operator associated to the periodic coefficients of the considered dynamics, spectral criterion and Lyapunov-type criterion are presented for asymptotic mean square stability and weak stability, respectively. Based on the Lyapunov criterion, a necessary and sufficient condition is supplied for regional stabilizability. Furthermore, Popov–Belevitch–Hautus (PBH) criterion is obtained for observability of mean-field stochastic periodic systems. By the proposed spectral criterion of stability/observability, the intrinsic relationships are clarified for stability/observability among deterministic periodic systems, stochastic periodic systems and mean-field stochastic periodic systems. Finally, as an application of PBH criterion, a Barbashin–Krasovskii stability theorem is established to indicate the connection between observability and stability of the considered systems.

TDE Based Model-free Control for Rigid Robotic Manipulators under Nonlinear Friction

This paper establishes a model-free finite-time tracking control of nonlinear robotic manipulator systems. The proposed controller incorporates both time delay estimation (TDE) and an enhanced terminal sliding mode control (TSMC). The improved TSMC scheme is devised using fractional-order TSMC (FOTSMC) and proportional-integral-derivative (PID) control to obtain robust tracking and high control performance. The TDE is designed to estimate the unknown nonlinear dynamics of robotic manipulators including the Stribeck friction and the external disturbances. Due to Stribeck friction, the effect of TDE error may fail to obtain the desired error performance; thus, another TDE loop is devised to compensate for TDE error generated by non-smooth frictions. The Lyapunov criterion is used to investigate the finite-time stability to analyze the behavior of the designed approach. Finally, computer simulations of the proposed method on PUMA 560 robotic manipulators are performed in contrast with FOTSMC and adaptive fractional-order nonsingular terminal sliding mode control (AFONTSM).

ZOH-based composite neural event-triggered control of underactuated cable-laying vehicle in presence of varying depth

To achieve the autonomous cable-laying operation under the varying depth, this note investigates the path-following problem of underactuated cable-laying vehicles with constraints of the unknown actuator gain. A double-stage logical (DSL) guidance principle is developed to realize the practical laying path programming for the cable-laying vehicle. In the guidance law, the catenary model is incorporated to derive the towing distance, i.e., the horizontal projection distance between the cable landing point and the vehicle’s gravity center. That can guarantee the cable-laying operation is solidly implemented in the condition of the varying depth. As for the control part, the composite neural event-triggered control algorithm is proposed by fusing the zero-order holder (ZOH) to stabilize the track error between the target position and the actual one. The event-triggered mechanism is employed to effectively reduce the occupancy of the communication channel between sensors and controllers. Moreover, for the merits of robust neural damping and composite learning techniques, the gain uncertainty of actuators can be compensated by updating two adaptive laws online. The stability analysis of the closed-loop system is rigorously performed via Lyapunov criterion and simulation results are provided to verify the performance of the proposed scheme.

Internal Model Control-Based Observer for the Sideslip Angle of an Unmanned Surface Vehicle

Since the sideslip angle is often ignored or simplified in the process of path following of unmanned surface vehicle by using the line-of-sight (LOS) guidance law because of its fast change and the difficulty of measurement, an observer was proposed by internal model control (IMC) to quickly estimate the sideslip angle in the LOS guidance law. First, a prediction model was established for the tracking error, and a state space model for prediction errors was constructed as an internal model. With the introduction of the auxiliary variables, a new augmented system was set up for a state space model of the prediction errors. Then, the sideslip angle observer was designed by the theory of state feedback with the feature of the control law of a proportional-integral type. Theoretically, the stability of the system was proved based on the Lyapunov criteria. A simulation and experiment verified the effectiveness of the proposed sideslip angle observer in improving the path-following accuracy. The results show that the IMC-based observer introduces a proportional term of tracking error that is not considered by other observers, which is easier to implement and adjust, and has a faster response speed and a smaller steady-state error for the sideslip angle. In addition, the assumption of a small sideslip angle is not introduced in the design process, so the proposed observer provides an accurate estimation method for a large sideslip angle.

On Appearance of Fast or Late Self-Synchronization between Non-Ideal Sources Mounted on a Rectangular Plate Due to Time Delay

The present paper aims to present the effects of late switching on (time delay) between two or three DC electrical machines characterized by limited power supplies on their fast or late self-synchronization when mounted on a rectangular plate with simply supported edges. The DC electrical machines are considered here as non-ideal oscillators, rotating in the same direction and acting as an external excitation on a specific surface of the plate. The stability analysis of the whole studied system (with two machines) around the obtained fixed point is done through analytical and numerical approaches by using the generalized Lyapunov and Routh-Hurwitz criterion. The existence conditions of the fixed point and the stability conditions are derived and presented. Great attention is put on the incidence of such study on the vibrations amplitude of the plate, which are considerably reduced in some cases. It appears that the time delay induces a rapid or late synchronization observed between the DC sources. This has been observed by numerically exploring the dynamics of the system for various possibilities that could occur. Moreover, in the modelling of the system, the positions on the plate occupied by DC electrical machines are taken into account by using the Heaviside function. It is shown that, in the case of three DC electrical machines, these positions influence the time to obtain a synchronous state between the DC electrical machines.