Lyapunov Function Approach Research Articles

Stability analysis for highly nonlinear switched stochastic systems with time-varying delays

In this paper, we examine the stability of highly nonlinear switched stochastic systems (SSSs) with time-varying delays, where the switching time instants are deterministic rather than stochastic. Herein, the boundedness of the global solution is first proven for highly nonlinear SSSs via the average dwell time (ADT) method and multiple Lyapunov function (MLF) approach. Then, the stability criteria for qth moment exponential stability and almost surely exponential stability are presented. The main difficulty lies in the presence of switching and time-varying delay terms, which prevents the validation of existing methods. New inequality techniques have been developed to counteract the effects of switching signals and time-varying delays. Finally, an example is provided to verify the effectiveness of the results.

Disturbance observer-based constrained attitude control for flexible spacecraft

This paper investigates the difficult issue of fixed-time constrained attitude control for a flexible spacecraft under the influence of the system uncertainties, environmental disturbance and abrupt actuator faults. The contributions of the proposed control framework are twofold. Firstly, an observer is presented to precisely reconstruct the uncertain dynamics within a fixed time regardless of the initial estimation error while the settling time is given as a specific parameter. Secondly, using a combination of prescribed performance control (PPC) and barrier Lyapunov function (BLF) approaches, a simple structure constrained attitude control for flexible spacecraft is proposed while desired performance specifications for both quaternion and rotation velocity are indirectly achieved by constraining the sliding manifold. A distinctive feature of the suggested control framework is that the settling time of the closed-loop system is finite and explicitly expressed as two tunable gains even when abrupt actuators faults happen. Numerical simulations substantiate the ability of the offered control to effectively accomplish favorable attitude maneuver under the negative effect of the inertia matrix uncertainty, external disturbances, flexible structures vibration and actuators faults.

Active fault-tolerant tracking control for discrete-time switched LPV system based on virtual actuator method

In this paper, the active fault-tolerant tracking control issue is studied for a class of discrete-time switched linear parameter varying system by using the virtual actuator approach. First, a switched gain-scheduled virtual actuator is proposed to improve the faulttolerance performance. Second, in order to obtain better tracking performance, a sub-optimal trajectory recovery target is determined by minimizing the H∞-norm of the virtual actuator's relevant transmission channel. By using the multiple parameter-dependent Lyapunov function approach, sufficient conditions for the tracking closed-loop system and virtual actuator system to be globally uniformly exponentially stable are derived, which also guarantee the desired H∞ performance indices requirement. Finally, effectiveness and applicability of the developed approaches are validated via a GE-90 turbofan-engine model system

A Multiple Lyapunov Functions Approach for Formation Tracking Control

This paper proposes a leader-follower formation tracking control problem of autonomous unmanned systems with switching directed topologies, where the leader is dynamic. It is considered that the followers have only access to the leader's position. The distributed observer is designed to estimate the velocity of the leader for each follower. Based on the estimated states, this paper presents a distributed formation tracking control algorithm. Using multiple Lyapunov functions (MLFs) stability is ensured. For dealing with switching topologies, the average dwell time (ADT) concept has been used, which relaxes the traditional dwell time restriction for switching topologies. Finally, a numerical simulation demonstrates the theory's effectiveness.

Robust synchronization analysis of delayed fractional order neural networks with uncertain parameters

<abstract><p>This paper is concerned with the robust synchronization analysis of delayed fractional order neural networks with uncertain parameters (DFNNUPs). Firstly, the DFNNUPs drive system model and response system model are established. Secondly, using multiple matrix quadratic Lyapunov function approach and inequality analysis technique, the robust synchronization conditions are derived in the form of the matrix inequalities. Finally, the correctness of the theoretical results is verified by an example.</p></abstract>

Quantized-State-Feedback-Based Neural Control for a Class of Switched Nonlinear Systems With Unknown Control Directions

This paper investigates the problem of unknown virtual control directions in a state-quantized adaptive recursive control design for a class of arbitrarily switched uncertain pure-feedback nonlinear systems in a band-limited network. State quantization is considered for state feedback control in a band-limited network. The primary contribution of this study is to provide a quantized state feedback adaptive control strategy to address the unknown control direction and arbitrarily switched nonaffine nonlinearities. Herein, a coupling problem between Nussbaum functions and quantization errors caused by quantized state feedback control laws is considered in the Lyapunov-based design and stability analysis. A state-quantized adaptive recursive control scheme using the function approximation is constructed without a priori knowledge of the signs of the control gain functions, where the estimated parameters and Nussbaum-type functions are adaptively updated via quantized states. Theoretical lemmas are derived to show that the adaptive parameters and quantization errors of the closed-loop signals are bounded using the proposed control scheme. The boundedness of the closed-loop signals and the convergence of tracking error to a neighborhood of the origin are proved using the common Lyapunov function approach. Two simulation examples are shown to illustrate the effectiveness of the proposed theoretical result.

Finite Time Annular Domain Robust Stability Analysis and Controller Design for T-S Fuzzy Interval Positive Systems

This paper is concerned with the finite time annular domain robust stability (FTADRS) analysis and controller design for T-S fuzzy positive systems with interval uncertainties. The concept of finite time annular domain stability is first introduced for positive systems. Based on this and using the copositive Lyapunov function approach, some sufficient conditions for FTADRS are derived. Subsequently, the finite time annular domain robust controller is designed via the linear programming technique. Finally, two numerical examples and an application example are employed to show the effectiveness of our results.

Resilient adaptive event-triggered dissipative control for networked control systems with DoS attacks

This paper focuses the design of resilient control for networked control systems with an adaptive event-triggered mechanism in the presence of denial-of-service attacks, actuator saturation and actuator faults. The model for denial-of-service attacks (DoS-As) in networked control systems, which mainly destroy the control channel and occur aperiodically with an unknown attack strategy, is constructed. An adaptive event-triggered mechanism (AETM) is introduced to reduce unimportant communication transmissions in a network provided adaptive varying threshold while maintaining satisfactory system performance. By using the Lyapunov function approach, linear matrix inequality (LMI)-based sufficient conditions are obtained to ensure the exponential stability and satisfying the prescribed performance index of the closed-loop systems under DoS-As. Moreover, the feasibility of the jointly developed schemes for the control gain and event-triggered parameters is calculated using the presented LMIs. The potentiality of the proposed scheme is validated using a numerical example.

Asymptotic stabilization with group‐wise sparse input based on control Lyapunov function approach

AbstractThis study proposes a novel stabilizing controller for nonlinear systems using group‐wise sparse inputs. The input variables are divided into several groups. In the situations when the input constraints can be ignored, one input becomes active for each group at each moment. Our method improves energy efficiency, as sparse input vectors often reduce the standby power of inactive actuators. Large‐scale systems, such as those consisting of multiple subsystems, often require the manipulation of multiple inputs simultaneously to be controlled. Our method can be applied to such systems due to the group‐wise sparsity of the inputs. The proposed controller is based on the control Lyapunov function approach and includes Sontag's universal formula as a special case. The controllers designed in our method have best‐effort property, which means even when a restriction for the decreasing rate of the Lyapunov function cannot be fulfilled, the controller minimizes the time derivative of the Lyapunov function within the input constraint. The effectiveness of the proposed method can be confirmed through simulations.

A Lyapunov-Based Optimal Integral Finite-Time Tracking Control Approach for Asymmetric Nonholonomic Robotic Systems

This study suggests a control Lyapunov-based optimal integral terminal sliding mode control (ITSMC) technique for tracker design of asymmetric nonholonomic robotic systems in the existence of external disturbances. The design procedure is based on the control Lyapunov function (CLF) approach. Hence, the output tracking problem is solved by combining the ITSMC with optimal control. The CLF synthesizes a nonlinear optimal control input for the nominal system. Once the control system’s states lie far away from the operating point, it is activated to drive them toward the equilibrium point optimally. However, on the condition that the system perturbations are the main factor, the ITSMC would be designed to take over in the vicinity of the equilibrium point. Accordingly, the control goals, such as robustness and precise control, are warranted in the perturbed system. The usefulness of the suggested method is demonstrated with a wheeled mobile robot via a simulation study.

Observer-Based Sliding Mode Control for Stabilization of Mismatched Disturbance Systems With or Without Time Delays

In this article, the stabilization problem of mismatched disturbance systems with or without time delays is studied via observed-based sliding mode control (SMC). Two kinds of SMC schemes for systems with time delay and without time delay are considered, respectively. When time delay is addressed in the system, an SMC strategy is structured via disturbance observer, where the unknown external disturbances are supposed to be generated by an exogenous dynamic. When time delay is not addressed, an SMC approach is designed in which the unknown external disturbances are assumed to tend to a constant steady state in infinite time. Sufficient conditions for stability of the corresponding sliding motion are derived by using Lyapunov–Krasovskii functional and Lyapunov function approach, respectively. Our results can be applied when the bound of the disturbances are unmeasured or unknown. Two simulation examples are shown to illustrate the proposed methods.

Distributed resilient state estimation over sensor networks with random nonlinearities, fading measurements, and stochastic gain variations

AbstractThe distributed resilient state estimation problem of nonlinear discrete systems in sensor networks is investigated in this article. The system model under consideration involves three phenomena of incomplete information: randomly occurring nonlinearities, fading measurements, and random gain variations. The probabilistic characteristics of the above phenomena are depicted by three sets of independent random variables subject to more general random distribution. Based on the above model, by applying Lyapunov functional approach and random distribution solution method, the asymptotic stability in the mean square sense of the estimation error system with a given attenuation level is proved. Further, the estimator parameters are solved by introducing a novel linearization method. Finally, a numerical simulation is given to illustrate the validity of the theoretical results.

Small-Gain Technique-Based Adaptive Neural Output-Feedback Fault-Tolerant Control of Switched Nonlinear Systems With Unmodeled Dynamics

In this article, the issue of adaptive neural fault-tolerant control (FTC) is addressed for a class of uncertain switched nonstrict-feedback nonlinear systems with unmodeled dynamics and unmeasurable states. In such a system, the uncertain nonlinear parts are identified by radial basis function (RBF) neural networks (NNs). Also, with the help of the structural characteristics of RBF NNs, the violation between the nontsrict-feedback form and backstepping method is tackled. Then, based on the small-gain technique, input-to-state practical stability (ISpS) theory, and common Lyapunov function (CLF) approach, an adaptive fault-tolerant tracking controller with only three adaptive laws is developed by designing an observer. It is shown that the designed controller can ensure that all the closed-loop signals are bounded under arbitrary switching, while the tracking error can converge to a small area of the origin. Finally, two simulation examples are provided to demonstrate the feasibility of the suggested control approach.

Analysis of a Tuberculosis Infection Model considering the Influence of Saturated Recovery (Treatment)

Tuberculosis (TB) is a serious global health threat that is caused by the bacterium Mycobacterium tuberculosis, is extremely infectious, and has a high mortality rate. In this paper, we developed a model of TB infection to predict the impact of saturated recovery. The existence of equilibrium and its stability has been investigated based on the effective reproduction number R C . The model displays interesting dynamics, including backward bifurcation and Hopf bifurcation, which further results in the stable disease-free and stable endemic equilibria to be coexisting. Bifurcation analysis demonstrates that the saturation parameter is accountable for the phenomenon of backward bifurcation. We derive a condition that guarantees that the model is globally asymptotically stable using the Lyapunov function approach to global stability. The numerical simulation also reveals that the extent of saturation of TB infection is the mechanism that is fuelling TB disease in the population.

Observer-based [formula omitted] control for persistent dwell-time switched networked nonlinear systems under packet dropout

This paper focuses on the observer-based controller design issue for a class of discrete-time networked nonlinear systems. Due to the stochastic effects, the switching between subsystems occurs. The persistent dwell-time switching mechanism, which is more general than dwell-time switching and average dwell-time switching mechanisms, can well describe these switching cases with both fast and slow switchings. Furthermore, there is packet dropout during data transmission, which is described by Bernoulli white sequence, and a corresponding mechanism is used to compensate for the impact of packet dropout. Then based on the Lyapunov function approach, some sufficient conditions for mean-square exponentially stable and an expected H∞ performance are derived. Finally, an illustrative example that demonstrates the superiority and feasibility of the proposed method is given.

Adaptive tracking control of switched cyber-physical systems with cyberattacks

This study investigates the problem of adaptive tracking control for switched cyber-physical systems against cyberattacks. A tracking signal is generated using an exogenous system. An adaptive tracking controller is designed to compensate for the attacks and then stabilize the studied systems. A switching signal is constructed using the mode-dependent average dwell time (MDADT) method. By using the multiple Lyapunov function approach, which is associated with the adaptive estimated signal, the closed-loop error systems with cyberattacks are asymptotically stable under the designed adaptive tracking controller and MDADT switching signal, and the system states can asymptotically track the reference signal. Finally, two examples are provided to validate the theoretical results.

Non‐fragile state estimation for second‐order memristive neural networks with unbounded time‐varying delays

SummaryWithout converting the second‐order memristive neural networks (SMNNs) into the usual first‐order systems, this article works on the non‐fragile state estimation of SMNNs with unbounded time‐varying delays. The memristor is treated as uncertainty by recommending some measurable functions. According to Lyapunov functional approach, inequality techniques, and combining Barbalat lemma, a new Lyapunov functional is proposed to straightway discuss the asymptotic stability for the error system, and several sufficient conditions are obtained to assure the existence of the preset estimators. The nonfragile estimators are designed and several explicit descriptions of the estimators are given by solving linear matrix inequalities. To overcome the uncertainty of state variables, the problem of parameter mismatch is as well considered. Finally, two examples are given to verify the availability of the results.

Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission

The coronavirus infectious disease (COVID-19) is a novel respiratory disease reported in 2019 in China. The infection is very destructive to human lives and caused millions of deaths. Various approaches have been made recently to understand the complex dynamics of COVID-19. The mathematical modeling approach is one of the considerable tools to study the disease spreading pattern. In this article, we develop a fractional order epidemic model for COVID-19 in the sense of Caputo operator. The model is based on the effective contacts among the population and environmental impact to analyze the disease dynamics. The fractional models are comparatively better in understanding the disease outbreak and providing deeper insights into the infectious disease dynamics. We first consider the classical integer model studied in recent literature and then we generalize it by introducing the Caputo fractional derivative. Furthermore, we explore some fundamental mathematical analysis of the fractional model, including the basic reproductive number R0 and equilibria stability utilizing the Routh-Hurwitz and the Lyapunov function approaches. Besides theoretical analysis, we also focused on the numerical solution. To simulate the model, we use the well-known generalized Adams–Bashforth Moulton Scheme. Finally, the influence of some of the model essential parameters on the dynamics of the disease is demonstrated graphically.

Adaptive robust fault-tolerant control for a two-slider synchronization system with a flexible beam

In this paper a novel adaptive robust fault-tolerant sync control method is proposed for a two-slider system where two sliders are constrained by a flexible beam. At first the dynamic models of sync motion system subject to external disturbances and actuator faults are derived. In order to avoid the shortcomings of truncated model, the model of flexible beam is described by using infinite dimensional equation. Then based on the models a novel disturbance observer and an adaptive fault-tolerant control law are designed. The disturbance observer is used to estimate and cancel external disturbances. The adaptive fault-tolerant control is used to deal with the partial loss of effectiveness faults. Lyapunov functional approach is used to prove that the closed-loop system with the proposed control laws is uniformly bounded stable. Finally, some simulation results display that the proposed control laws can obtain excellent sync performance in the present of external disturbances and actuator partial loss of effectiveness faults.

Fully distributed event-triggered time-varying formation control of multi-agent systems subject to mode-switching denial-of-service attacks

In this study, the time-varying state formation control problem for general linear multi-agent systems (MASs) subject to mode-switching denial-of-service (MSDoS) attacks is considered. Based on only the sampled state information of itself and neighboring agents at event-triggered instants, a novel fully distributed event-triggered secure control strategy without continuous communication between agents is delicately designed. Note that the control strategy is implemented in a fully distributed manner, which means that it does not require any global network information. By using the multiple Lyapunov function approach based on edge-dependent average dwell time, this study presents the MASs subject to MSDoS attacks with limited attack frequency and attack width can achieve the specified time-varying state formation structure under the designed control strategy. Furthermore, this study presents that in any finite time, the control strategy does not exhibit Zeno behavior. Finally, the effectiveness and performance of the designed control strategy are validated by a numerical simulation.