Lyapunov Theorem Research Articles

Third-Order Robust Fuzzy Sliding Mode Tracking Control of a Double-Acting Electrohydraulic Actuator

In the industrial sector, an electrohydraulic actuator (EHA) system is a common technology. This system is often used in applications that demand high force, such as the steel, automotive, and aerospace industries. Furthermore, since most mechanical actuators' performance changes with time, it is considerably more difficult to assure its robustness over time. Therefore, this paper proposed a robust fuzzy sliding mode proportional derivative (FSMCPD) controller. The sliding mode controller (SMC) is accomplished by utilizing the exponential law and the Lyapunov theorem to ensure closed loop stability. By replacing the fuzzy logic control (FLC) function over the signum function, the chattering in the SMC controller has been considerably reduced. By using the sum of absolute errors as the objective function, particle swarm optimization (PSO) was used to optimize the controller parameter gain. The experiment results for trajectory tracking and the robustness test were compared with the sliding mode proportional derivative (SMCPD) controller to demonstrate the performance of the FSMCPD controller. According to the findings of the thorough study, the FSMCPD controller outperforms the SMCPD controller in terms of mean square error (MSE) and robustness index (RI). Keywords—tracking control, electrohydraulic actuator (EHA), robust control design, fuzzy sliding mode proportional derivative (FSMCPD), particle swarm optimization (PSO).

Passivity-Based Control for Interleaved Double Dual Boost Converters in DC Microgrids Supplying Constant Power Loads

The interleaved double dual boost converter (IDDBC) is considered a promising and reliable approach to interface the low-voltage renewable energy sources (RESs), such as battery packs, fuel cells and photovoltaic (PV) panels, to the DC bus of the DC microgrids. It could give a higher voltage step-up ratio and reduce voltage and current stresses on the power devices. However, the connection of various tightly regulated power converters, which are known as constant power loads (CPLs), would cause negative incremental impedance. Meanwhile, the unmodeled dynamics and load perturbation disturbances are unavoidable in the DC microgrid. These issues would deteriorate the system stability and even lead to system failure. To address these problems, this paper proposes an adaptive passivity-based control (PBC) to handle the DC bus’s large-scale stability issues of the N-phase IDDBC feeding CPL system. A reduced and practical N-phase IDDBC model supplying CPL considering unmodeled dynamics and load perturbation disturbances is proposed to facilitate the controller design. Besides, a nonlinear disturbance observer (NDO) is incorporated with the proposed PBC to improve the robustness against parameter uncertainties and external disturbances. And the global stability of the closed-loop system is proved through the Lyapunov theorem. Finally, simulation and experimental results are demonstrated to verify the feasibility and effectiveness of the proposed controller.

Decentralized formation cooperative control for homing of a swarm of underactuated AUVs via singular perturbation

This paper addresses a decentralized control mechanism for the homing process of underactuated AUVs. The mechanism can be decoupled into foraging and tracking. During the homing process, the mother submarine is regarded as a Gaussian type of nutrient profile and underactuated AUVs are regarded as agents in swarm. The problem of homing is transformed into the social foraging. In the swarm, each individual has a double-integrator dynamic. Under the proposed artificial potential field, the motion of each individual is determined by repulsion or attraction from other individuals, attraction from nutrient profile and repulsion from toxicant profile. Therefore, each agent generates a collision-free trajectory tend to the nutrient profile. The trajectories can be used as reference trajectories for the underactuated AUVs. For the trajectory tracking, the speed of attitude dynamics is much faster than that of position dynamics. Such characteristic can be naturally treated by time scale decomposition method and singular perturbation theory which allows independent analysis of the dynamics in each time scale. In addition, the stability of the collective behavior of the swarm and the trajectory tracking controller are proposed by Lyapunov theorem. Finally, the effectiveness of the proposed control mechanism is demonstrated via a set of simulation results.

Robust Fuzzy-Fractional-Order Nonsingular Terminal Sliding-Mode Control of LCL-Type Grid-Connected Converters

Sliding-mode control (SMC) has been widely used in grid-connected converter system (GCC) systems because of its robustness to parameter variations and external disturbances. However, chattering in SMC may deteriorate the tracking accuracy and can easily excite high-frequency unmodeled dynamics. To solve this problem, this article presents a fuzzy-fractional-order nonsingular terminal sliding-mode controller (Fuzzy-FONTSMC) for the grid current control of LCL–GCCs. First, the system modeling, design of the integer-order NTSMC controller, and state estimation based on the Kalman filter to minimize the sampling sensors are described. Second, the Fuzzy-FONTSMC controller is introduced for optimal fraction-order selection and chattering mitigation, this controller exhibits fast convergence with high tracking accuracy and strong robustness. Finally, the Lyapunov theorem is used to analyze the system stability. Experimental comparisons on a 10-kVA laboratory prototype validate the superior performance and effectiveness of the proposed method under many scenarios.

Adaptive backstepping sliding mode control of permanent magnet spherical motor based on disturbance observer.

This paper proposes an adaptive backstepping sliding mode control method based on a nonlinear disturbance observer (NDO) to solve the problem that the tracking performance of the motion system of a permanent magnet spherical motor (PMSpM) is degraded due to the influence of external nonlinear disturbance. First, the dynamic model of the PMSpM under the compound interference is established. Second, an NDO and an adaptive backstepping sliding mode controller are designed to compensate for the external disturbances and modeling uncertainties, and the stability of the closed-loop system using the proposed method is confirmed through the Lyapunov theorem. Then, compared with the results of proportion differentiation control and conventional sliding mode control, the simulation results show that the proposed method can significantly reduce input signal chattering and improve the trajectory tracking performance of the PMSpM. Finally, experimental results are provided to validate the effectiveness of the proposed method.

Fixed-Time Lyapunov Criteria and State-Feedback Controller Design for Stochastic Nonlinear Systems

This paper investigates the fixed-time stability theorem and state-feedback controller design for stochastic nonlinear systems. We propose an improved fixed-time Lyapunov theorem with a more rigorous and reasonable proof procedure. In particular, an important corollary is obtained, which can give a less conservative upper-bound estimate of the settling time. Based on the backstepping technique and the addition of a power integrator method, a state-feedback controller is skillfully designed for a class of stochastic nonlinear systems. It is proved that the proposed controller can render the closed-loop system fixed-time stable in probability with the help of the proposed fixed-time stability criteria. Finally, the effectiveness of the proposed controller is demonstrated by simulation examples and comparisons.

Lyapunov theorems for stability and semistability of discrete-time stochastic systems

This paper develops Lyapunov and converse Lyapunov theorems for discrete-time stochastic semistable nonlinear dynamical systems expressed by Itô-type difference equations possessing a continuum of equilibria. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose difference operator involves a discrete-time analog of the infinitesimal generator for continuous-time Itô dynamical systems and decreases along the dynamical system sample solution sequences satisfying an inequality involving the average distance to the set of the system equilibria.

Collision-free formation-containment control for a group of UAVs with unknown disturbances

This paper focuses on the formation-containment control problem for a group of fixed-wing unmanned aerial vehicles (UAVs) subjected to unknown disturbances. To solve this problem, a two-layer control framework is developed by decoupling the problem into a trajectory generation problem at the group level and a trajectory tracking problem at the individual level. Firstly, based on the directed interaction topology and the required configuration of the UAVs, two types of finite-time estimators are constructed to generate the desired trajectories of the leading UAVs and the following UAVs, respectively. Meanwhile, to guarantee the safety of each UAV, the smooth collision avoidance potential function and obstacle avoidance potential function are designed. After that, using the dynamic surface control (DSC) technique and combined with the established potential functions, a distributed control law with the ability of collision avoidance is presented for each UAV to track the generated trajectory without any collision, where the homogeneous extended state observer (HESO) is developed to compensate for the mismatched and matched disturbances. The stability of the whole closed-loop system is provided using the Lyapunov theorem. Finally, simulation results demonstrate the effectiveness of the proposed control algorithm.

Chaos synchronization control for stochastic nonlinear systems of interior PMSMs based on fixed-time stability theorem

This paper proposes an enhanced chaos synchronization control scheme of stochastic nonlinear systems for the permanent magnet synchronous motors(PMSMs) applications for dynamic operating mechanism under different working environment. Firstly, a novel fixed-time stochastic stability theorem and convergence for the stochastic nonlinear system is valid through formulating a novel Lyapunov function. Different from what has been achieved on the fixed-time stability, it’s not the usual range but it’s more precise for settling time which improving control performance effectually. Secondly, the chaos synchronization issue of driven-response PMSMs with stochastic noise is firstly investigated based on the proposed Lyapunov theorem synchronizing the complex dynamic systems considering practical significance. Finally, the fixed-time synchronization control scheme is developed via applying the adaptive algorithms, furthermore, through constructing a reasonable synchronization dynamic error system achieving stability in probability of fixed time. A simulation analysis is demonstrated to verify the availability of the proposed control strategy.

On the simultaneous stabilization of stochastic nonlinear systems

SummaryIn this paper, the problem of simultaneous stabilization in probability by state feedback is investigated for a class of stochastic nonlinear systems whose drift and diffusion terms are dependent on the control and for which classical methods are not applicable. Under the assumption that a collection of stochastic control Lyapunov functions (SCLFs) is known and based on the generalized stochastic Lyapunov theorem, we derive sufficient conditions for the simultaneous stabilization in probability by a continuous state feedback controller that we explicitly compute. We also derive a necessary condition when the system coefficients satisfy some regularity conditions. This work generalizes previous results on the simultaneous stabilization of stochastic nonlinear systems. The obtained results are illustrated by a numerical example.

Continuous non‐singular terminal sliding‐mode control of permanent magnet spherical actuator for trajectory tracking based on a modified nonlinear disturbance observer

Multi-degree-of-freedom permanent magnet spherical actuators (PMSAs) are highly nonlinear, coupled, and multivariable systems that are subject to different types of disturbances including unknown payloads, unmodelled dynamics, frictions, and other external disturbances, which adversely affect the trajectory tracking performance of a system. In this paper, a disturbance-observer-based continuous non-singular terminal sliding-mode control (NDOB-CNTSMC) strategy is developed for a PMSA to enhance its trajectory tracking performance. Firstly, a nonlinear disturbance observer (NDOB) with a parameter design method is proposed to suppress uncertainty and reject disturbance in a PMSA trajectory tracking control system. Secondly, a continuous non-singular terminal sliding-mode controller (CNTSMC) is presented to compensate for the effect of the disturbance observation error and unobservable disturbances. The stability of the proposed control strategy is proved by the Lyapunov theorem. Both simulation and experimental results have shown that the control strategy proposed in this research can effectively address the issues of the inaccuracy of modelling and various disturbances and improve the trajectory tracking precision of the PMSA.

Finite-time synchronisation of delayed fractional-order coupled neural networks

This paper considers the global synchronisation and finite-time synchronisation for a class of delayed fractional-order complex neural networks (DFOCNNs). Based on the properties of fractional-order calculus and the Razumikhin-type Lyapunov theorem of a fractional-order system, two new lemmas are proved. These lemmas are employed to formulate a couple of novel criteria for both finite-time synchronisation and global synchronisation of DFOCNNs. Moreover, the upper bound of the setting time for synchronisation is given. Three examples are provided to verify the effectiveness of the obtained results.

Adaptive Friction Compensation for a Novel Two-Axis Differential Microfeed System

The conventional drive feed system inevitably enters the nonlinear creep region at very low speed. This seriously affects its low-speed performance and tracking accuracy. The new two-axis differential microfeed system (TDMS) can overcome the low-speed crawling and creep phenomenon caused by the inherent characteristics of the traditional electromechanical servo system structure. In this study, the dynamic model of TDMS is first established, and the LuGre friction model is improved according to its motion characteristics. The friction model of the screw nut whole assembly of TDMS is established. On this basis, two nonlinear observers are designed to estimate the internal state of the friction model, and the TDMS adaptive friction compensation controller is designed to estimate the uncertain parameters in TDMS through the parameter adaptive law. The global asymptotic stability of the TDMS controller is proved by Lyapunov theorem. Finally, the experimental verification is carried out by using dSPACE system. The results showed that the proposed TDMS adaptive friction compensation controller has good control accuracy. And it can effectively compensate the friction torque of lead screw nut differential drive. Compared with the application of traditional PID controller in TDMS, the tracking accuracy is improved by an order of magnitude.

Estimating the Region of Attraction for Power Systems Using Gaussian Process and Converse Lyapunov Function

The significant penetration of renewable energy resources into power systems introduces intermittent disturbances that drive the system away from its stable operating point (i.e., equilibrium). Nevertheless, as long as the system states are still contained inside a region of attraction (ROA), the system will finally return to the normal operating equilibrium. Thus, the ROA plays a crucial role in the stability assessment of dynamical power systems under uncertainties. This brief introduces a novel framework to construct the ROA of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytic Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need for an analytic Lyapunov function and learning unknown Lyapunov functions with the Gaussian process (GP) approach. In addition, a GP upper confidence bound (GP-UCB)-based sampling algorithm is designed to reconcile the tradeoff between the exploitation for enlarging the ROA and the exploration for enhancing the confidence level of the sample region. Within the constructed ROA, it is guaranteed in the probability that the system state will converge to the stable equilibrium with a confidence level. Extensive simulations and experimental validations are conducted to substantiate the assessment approach for the ROA of IEEE test systems and a real smartgrid using phasor measurement unit (PMU) data. It is demonstrated that the proposed approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Development of a method for calculation of information protection from the clustering coefficient and information flow in social networks

The object of research is the system of information protection of the social network. The article investigates the dynamic models of the information protection system in social networks taking into account the clustering coefficient, and also analyzes the stability of the protection system. In graph theory, the clustering factor is a measure of the degree to which nodes in a graph tend to group together. The available data suggest that in most real networks, and in particular in social networks, nodes tend to form closely related groups with a relatively high density of connections. It is probability is greater than the average probability of a random connection between two nodes. There are two variants of this term: global and local. The global version was created for a general idea of network clustering, while the local one describes the nesting of individual nodes. There is a practical interest in studying the behavior of the system of protection of social networks from the value of the clustering factor. Dynamic systems of information protection in social networks in the mathematical sense of this term are considered. A dynamic system is understood as any object or process for which the concept of state as a set of some quantities at a given moment of time is unambiguously defined and a given law is described that describes the change (evolution) of the initial state over time. This law allows the initial state to predict the future state of a dynamic system. It is called the law of evolution. The study is based on the nonlinearity of the social network protection system. To solve the system of nonlinear equations used: the method of exceptions, the joint solution of the corresponding homogeneous characteristic equation. Since the differential of the protection function has a positive value in some data domains (the requirement of Lyapunov's theorem for this domain is not fulfilled), an additional study of the stability of the protection system within the operating parameters is required. Phase portraits of the data protection system in MatLab/Multisim are determined, which indicate the stability of the protection system in the operating range of parameters even at the maximum value of influences.

Quaternion-based finite-time fault-tolerant trajectory tracking control for autonomous underwater vehicle without unwinding

This brief disposes the finite-time anti-unwinding trajectory tracking control problem of the autonomous underwater vehicle (AUV) encountering model uncertainties, ocean disturbances and actuator failures. Primarily, the kinematic model of translational and rotational motions is depicted by unit quaternion in lieu of classical Euler angle such that the AUV’s dynamics could be globally and uniquely formulated. Subsequently, two finite-time control strategies are presented here to leave the state variables of AUV can converge to an adjustable region. Additionally, the adaptive laws could be applicable to the existence of actuator faults by utilizing a passive fault-tolerant technology. By integrating the initial value of the scalar quaternion into the sliding mode surface, the proposed controllers are characterized with anti-unwinding property. Then, with the application of hyperbolic tangent function, finite-time stability will be achieved for tracking errors without singularity. Stability of the closed-loop system is verified via the Lyapunov theorem. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.

Tracking control strategy for space flexible manipulator considering nonlinear friction torque based on adaptive fuzzy compensation sliding mode controller

Space flexible manipulator (SFM) is one of the most promising development directions of space mechanism, which is attributed to more and more spacecraft on-orbit using it to perform maintenance, capture, and assembly operations. The SFM is a complex multi-output dynamic system with nonlinear friction torque. Nonlinear friction torque makes it difficult for precise tracking control of space flexible manipulators. In this paper, nonlinear friction torque in dynamics equations of the SFM is identified by the fuzzy rules based on the universal approximation property. The nonlinear friction torque is identified to improve the preciseness of the control torque in the SFM servo system. Thus, the accuracy of the rotation angle is improved, and the tracking errors of end-effectors are reduced. Firstly, the dynamic equations of the SFM considering nonlinear friction torque are established by the Hamilton principle. Then the sliding mode control strategy is proposed to control the SFM based on adaptive fuzzy compensation. Besides, the stability of the closed-loop servo system in the SFM is proved by the Lyapunov theorem. Finally, numerical simulation analysis and ground control experiments of the SFM are carried out. The results of control experiments show that the tracking error can be reduced in this control strategy. Therefore, the control strategy can improve the control accuracy of the SFM.

Adaptive fuzzy sliding-mode consensus control of nonlinear under-actuated agents in a near-optimal reinforcement learning framework

This study presents a new framework for merging the Adaptive Fuzzy Sliding-Mode Control (AFSMC) with an off-policy Reinforcement Learning (RL) algorithm to control nonlinear under-actuated agents. In particular, a near-optimal leader-follower consensus is considered, and a new method is proposed using the framework of graphical games. In the proposed technique, the sliding variables’ coefficients are considered adaptively tuned policies to achieve an optimal compromise between the satisfactory tracking performance and the allowable control efforts. Contrary to the conventional off-policy RL algorithms for consensus control of multi-agent systems, the proposed method does not require partial knowledge of the system dynamics to initialize the RL process. Furthermore, an actor-critic fuzzy methodology is employed to approximate optimal policies using the measured input/output data. Therefore, using the tuned sliding vector, the control input for each agent is generated which includes a fuzzy term, a robust term, and a saturation compensating term. In particular, the fuzzy system approximates a nonlinear function, and the robust part of the input compensates for any possible mismatches. Furthermore, the saturation compensating gain prevents instability due to any possible actuator saturation. Based on the local sliding variables, the fuzzy singletons, the bounds of the approximation errors, and the compensating gains are adaptively tuned. Closed-loop asymptotic stability is proved using the second Lyapunov theorem and Barbalat's lemma. The method's efficacy is verified by consensus control of multiple REMUS AUVs in the vertical plane.

Adaptive Formation Control of Unmanned Underwater Vehicles with Collision Avoidance under Unknown Disturbances

In this paper, the formation control problem for a group of unmanned underwater vehicles (UUVs) is investigated considering collision avoidance and environment disturbances. To address the external force effect of the environment, such as waves and currents, a sliding mode disturbance observer is designed to compensate for the unknown dynamic disturbances in finite time. A bounded artificial potential field is incorporated into the control law to ensure collision avoidance among UUVs. The form of an artificial potential function is much simpler and convenient for engineering applications. A controller is devised to guarantee all the error signals are bounded, and the formation pattern can be achieved in finite time after collision avoidance. The stability of UUV formation with collision avoidance is proven by using the Lyapunov theorem, and the scheme has been shown to be convergent using Barbalat’s lemma. Comparative simulations are presented to demonstrate the effectiveness of the proposed method in 2-D and 3-D environments.

Improved neural network adaptive control for compound helicopter with uncertain cross-coupling in multimodal maneuver

The main goal of this study is to create a robust control system that could guide or replace pilots in the tracking of the commanded velocity and attitude in multimodal maneuver, while complex dynamics and uncertain aerodynamic cross-coupling among control surfaces of compound helicopter are considered. To this end, a Pi-Sigma neural network (PSNN) adaptive controller is proposed based upon the certainty-equivalence (CE) principle, where a novel Lyapunov-based weight self-tuning algorithm augmented with e-modification is designed to realize efficient uncertainty approximation and guarantee robustness of convergence process. Compared with traditional neural networks in control field, stronger generalization ability of PSNN must be balanced against weaker stability, which leads to inevitable parameters perturbation. Therefore, an incremental nonlinear dynamic inversion (INDI) framework is established to decouple original overactuated system and reject parameters perturbation in PSNN. Meanwhile, by incorporating Lagrange multiplier method into allocation, an original incremental allocation method is designed to get globally ideal control input according to time-varying working capability of each surface. In terms of Lyapunov theorem, it is demonstrated that the closed-loop augmented system driven by the proposed control scheme is semi-global uniformly ultimately bounded. Finally, by comparing with existing methods, the simulation validates the effectiveness of proposed control scheme.