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This paper investigates the stability problem of fractional-order periodic piecewise nonlinear systems. As a preparation, the existence and uniqueness of solutions of the system are derived under the Lipschitz condition and expressed in an equivalent integral expression based on the memorability of fractional derivative. We further propose an upper bound of the system states by using the Laplace transform technique. With the help of piecewise Lyapunov functions and class- $\mathcal {K}$ functions, some sufficient conditions are given to verify the Mittag-Leffler and asymptotic stability of the investigated system mixed with stable and unstable subsystems. Furthermore, asymptotic stability criteria are proposed for a general case, i.e., all subsystems are unstable. Three numerical simulation examples are finally provided to illustrate the effectiveness of the proposed results.

The asymptotic stability and extended dissipativity performance of T-S fuzzy generalized neural networks (GNNs) with mixed interval time-varying delays are investigated in this paper. It is noted that this is the first time that extended dissipativity performance in the T-S fuzzy GNNs has been studied. To obtain the improved results, we construct the Lyapunov-Krasovskii functional (LKF), which consists of single, double, triple, and quadruple integral terms containing full information of the delays and a state variable. Moreover, an improved Wirtinger inequality, a new triple integral inequality, and zero equation, along with a convex combination approach, are used to deal with the derivative of the LKF. By using Matlab’s LMI toolbox and the above methods, the new asymptotic stability and extended dissipativity conditions are gained in the form of linear matrix inequalities (LMIs), which include passivity, $L_{2}-L_{\infty }$ , $H_{\infty }$ , and dissipativity performance. Finally, numerical examples that are less conservative than previous results are presented. Furthermore, we give numerical examples to demonstrate the correctness and efficacy of the proposed method for asymptotic stability and extended dissipativity performance of the T-S fuzzy GNNs, including a particular case of the T-S fuzzy GNNs.

This paper is concerned with the stability analysis of fixed-point state-space digital filters with generalized overflow arithmetic and a time-varying delay. This paper aims to derive a delay and nonlinear function bound dependent asymptotical stability criterion with less conservatism. Firstly, a new Lyapunov functional with several augmented terms, including extra free matrices and overflow nonlinear function, is constructed such that it has a relaxed positive condition. Then, for bounding the summation term arising in the forward difference of Lyapunov functional, a new lemma is developed to introduce the terms for linking the delayed states and the overflow nonlinear function, the Wirtinger-based summation inequality and several zero-value terms are applied to add more cross terms. As a result, a stability criterion with less conservatism is established and its conservatism. Finally, several numerical examples are given to illustrate the advantages of the proposed method.

Active magnetic bearing (AMB) system has been recently employed widely as an ideal equipment for high-speed rotating machines. The inherent challenges to control the system include instability, nonlinearity and constricted range of operation. Therefore, advanced control technology is essential to optimize AMB system performance. This paper presents an application of model predictive control (MPC) based on linear parameter-varying (LPV) models to control an AMB system subject to input and state constraints. For this purpose, an LPV model representation is derived from the nonlinear dynamic model of the AMB system. In order to provide stability guarantees and since the obtained LPV model has a large number of scheduling parameters, the parameter set mapping (PSM) technique is used to reduce their number. Based on the reduced model, a terminal cost and an ellipsoidal terminal set are determined offline and included into the MPC optimization problem which are the essential ingredients for guaranteeing the closed-loop asymptotic stability. Moreover, for recursive feasibility of the MPC optimization problem, a slack variable is included into its cost function. The goal of the proposed feedback control system is twofold. First is to demonstrate high performance by achieving stable levitation of the rotor shaft as well as high capability of reference tracking without violating input and state constraints, which increases the overall safety of the system under disturbances effects. Second is to provide a computationally tractable LPVMPC algorithm, which is a substantial requirement in practice for operating the AMB system with high performance over its full range. Therefore, we propose an LPVMPC scheme with frozen scheduling parameter over the prediction horizon of the MPC. Furthermore, we demonstrate in simulation that such frozen LPVMPC can achieve comparable performance to a more sophisticated LPVMPC scheme developed recently and a standard NL MPC (NMPC) approach. Moreover, to verify the performance of the proposed frozen LPVMPC, a comparison with a classical controller, which is commonly applied to the system in practice, is provided.

In this paper, the problem of distributed event-based control of large scale power systems in presence of denial-of-service (DoS) cyber attacks is addressed. Towards this end, a direct current (DC) microgrid composed of multiple interconnected distributed generation units (DGUs) is considered. Voltage stability is guaranteed by utilizing decentralized local controllers for each DGU. A distributed discrete-time event-triggered (ET) consensus-based control strategy is then designed for current sharing in the DGUs. Through this mechanism, transmissions occur while a specified event is triggered to prevent unessential utilization of communication resources. The asymptotic stability of the ET-based controller is shown formally by using Lyapunov stability via linear matrix inequality (LMI) conditions. The behavior of the DGUs subject to DoS cyber attacks are also investigated and sufficient conditions for secure current sharing are obtained. Towards this end, a switching framework is considered between the communication and attack intervals in order to derive sufficient conditions on frequency and duration of DoS cyber attacks to reach the secure current sharing. The validity and capabilities of the presented approach is confirmed through a simulation case study.

At present, the theory and application of fractional-order neural networks remain in the exploratory stage. We study the asymptotic stability of fractional-order neural networks with Riemann-Liouville (R-L) derivatives. For non-delayed and delayed systems, we propose an asymptotic stability criterion based on the combination of the Lyapunov method and linear matrix inequality (LMI) method. The highlights include the following: (1) for fractional-order neural networks with time delay, the existence and uniqueness of solutions are proven by using matrix analysis theory and contraction mapping theorem, and (2) based on the unique solution, a suitable Lyapunov functional is constructed. Based on the inequality theorem and LMI method, two sets of asymptotic stability criteria for fractional-order neural networks are proven, which avoids the difficulty of solving the fractional derivative by the Leibniz law. Finally, the results are verified using numerical simulations.

An artificial Earth satellite (AES) with three different principal central moments of inertia is under consideration. The AES moves along a Keplerian circular equatorial near-Earth orbit. The AES is equipped with electrodynamic attitude control system that simultaneously generates Lorentz and magnetic control torques. The possibility of using an electrodynamic attitude control system for monoaxial attitude stabilization of AES in the orbital coordinate system is analyzed. The development of the concept of electrodynamic attitude control, including the use of a restoring torque with a distributed delay (integral term), is proposed. The conditions are found under which the electromagnetic attitude control system with distributed delay solves the problem of AES monoaxial stabilization in the presence of the disturbing gravitational torque. In a nonlinear formulation, sufficient conditions for the asymptotic stability of the AES equilibrium position are obtained. A theorem on the asymptotic stability of the AES programmed attitude motion is proved. The effectiveness of the constructed attitude control with a distributed delay is confirmed by numerical modeling.

The stability of grid-connected inverters is very sensitive to varying grid impedance, especially in the weak grid condition. In this paper, based on the Lyapunov method and with the micro-grid impedance identification, an active damping adaptive control strategy is proposed to guarantee local asymptotic stability for an LCL inverter system. First, using the backstepping design and taking the state variable errors as new variables, an appropriate Lyapunov function is constructed, so that the asymptotic stability control law can be obtained. Then, with this asymptotic stability controller, taking the varying impedance into account, an adaptive controller is developed using the Lyapunov direct method. Hence, the controller presented here can assure asymptotic stability, the system performance can also be improved instantly according to the adaptive control, and the expected stability and performance can also be achieved. Finally, the feasibility of the proposed control strategy was verified via a simulation and an experiment.

In this paper, the sampled-data control problem is studied for asynchronously switched linear systems (SLSs) without minimum dwell time (MDT) constraints. The asynchronous phenomenon exists due to that the information of system mode can be acquired only at the sampling instant. First, a sufficient condition of global asymptotic stability (GAS) is presented for sampled-date switched control systems with a novel class of switching signals, which allows the switching number to be a non-affine function of time, and does not involve any point-wise bound on the switching number. Moreover, unlike the existing literature concerned with sampled-data control problem of switched systems, the MDT constraints are removed. We allow that no sampling happens between two adjacent switching instants, which makes the results more applicable to practice. Then, a sufficient condition checking the existence of sampled-data controllers is presented in terms of linear matrix inequalities (LMIs). Finally, it is shown by a boost converter circuit system that the designed sampled-data controller and switching signals can stabilize the system cooperatively.

In this paper, a theorem with very weak conditions on vector field functions for asymptotic stability of Caputo fractional order switching systems is proposed. Based on Vainikko’s lemmas, the Caputo fractional derivative of a continuously differentiable and convex Lyapunov function, along trajectories of any Caputo fractional order switching systems with vector field function being only continuous, proves to be continuous and have a very useful estimation. This weakens the smoothness requirement on the vector field functions for fractional stability analysis from differentiability as in existing results to only continuity. Finally, the numerical implementation of a Caputo fractional order switching system with only continuous vector field function illustrates the effectiveness of the proposed theorem.

This paper investigates distributed adaptive fault-tolerant control schemes for spacecraft formation subject to external disturbances, model uncertainties and communication delays. Two novel adaptive fault-tolerant control approaches are presented based on directed communication interaction. The developed adaptive laws are used to estimate the actuator effectiveness, the spacecraft masses and the upper bound on external disturbances. Then, an adaptive fault-tolerant control algorithm with time-varying communication delays is proposed. In particular, the conditions on the controller parameters and the communication delays necessary to assure the asymptotic stability of the closed-loop system are given. Finally, numerical simulations are presented to validate the effectiveness of the two proposed fault-tolerant control schemes for the spacecraft formation system.

This technical note focus on the tracking control problem of uncertain nonlinear systems with multiple states constraints. Based on the Barrier Lyapunov function and backstepping technology, a new continuous smooth control solution can be ultimately synthesized to realize asymptotic tracking control in presence of multiple states constraints and modeling uncertainties. Firstly, the modeling uncertainties are divided into periodic and un-periodic components and Fourier expansion technology is employed to translate the periodic disturbance into the form which can be easily compensated. Then the Barrier Lyapunov function are flexibly utilized to design the virtual control law of every step and the final controller, which can guarantee the specified states within certain bounds regardless of the amplitude of system output. Meanwhile, a novel nonlinear control technology is introduced to each design step to realize the final asymptotic tracking control despite the matched and mismatched uncertainties. By analyzing the choice of the control parameters, the backstepping cross-term is skillfully dealt with and the stability of the whole system is proved rigorously. Finally, the simulation results on a three-order nonlinear hydraulic system demonstrate the satisfactory performance of the proposed control method.

In this article, a novel method to design the observer for a class of uncertain Lipschitz nonlinear parabolic partial differential equations (PDE) systems is investigated. First, the observer and the dynamic errors with undetermined parameters for the parabolic PDE systems subject to appropriate boundary conditions are presented. The conditions of the designed observer are involved. Then the analysis of asymptotic stability and H∞ performance conditions for the observer design of uncertain nonlinear parabolic PDE systems are studied and presented in terms of matrix inequalities based on the Lyapunov stability theory. Finally, the effectiveness of the proposed method is validated by a numerical parabolic PDE system.

Successful treatment of COVID-19 that outbroke worldwide since the beginning of 2020 has demonstrated the importance of effective isolation, which is aimed at asymptomatic and symptomatic infected persons in the incubation period. In this paper, to further analyze the transmission dynamics behavior of epidemics with the latent state, we construct a class of health state - latent state - infection - recovery state (SEIR) infectious disease model with heterogeneity and time delay characteristic based on considering the nonlinear incidence rate formed by psychological inhibition factors. Also, the dynamics of the epidemic, the threshold condition, and stability are studied by creating Lyapunov functions reasonably, applying LaSalle's Invariance Principle and mean-field equation theory. The research shows that, the basic reproduction number R0 of the system depends on birth rate, death rate, recovery rate, disease transmission rate, and network topology. If R0 0, and if R0 > 1, the system is sound at the endemic equilibrium point E*. Moreover, it is also proved that latent delay and psychological inhibitory factors can influence the peak and rate of the infected nodes in the system before their convergence to the equilibrium point, but not the system's global stability. Meanwhile, the theoretical results are verified by numerical simulation finally.

This paper considers the stabilization problem for under-actuated rotary inverted pendulum systems (RotIPS) via a fuzzy-based continuous sliding mode control approach. Various sliding mode control (SMC) methods have been proposed for stabilizing the under-actuated RotIPS. However, there are two main drawbacks of these SMC approaches. First, the existing SMCs have a discontinuous structure; therefore, their control systems suffer from the chattering problem. Second, a complete proof of closed-loop system stability has not been provided. To address these two limitations, we propose a fuzzy-based (continuous) super-twisting stabilization algorithm (FBSTSA) for the under-actuated RotIPS. We first introduce a new sliding surface, which is designed to resolve the under-actuation problem, by combining the fully-actuated (rotary arm) and the under-actuated (pendulum) variables to define one sliding surface. Then, together with the proposed sliding surface, we develop the FBSTSA, where the corresponding control gains are adjusted based on a fuzzy logic scheme. Note that the proposed FBSTSA is continuous owing to the modified super-twisting approach, which can reduce the chattering and enhance the control performance. With the proposed FBSTSA, we show that the sliding variable can reach zero in finite time and then the closed-loop system state converges to zero asymptotically. Various simulation and experimental results are provided to demonstrate the effectiveness of the proposed FBSTSA. In particular, (i) compared with the existing SMC approaches, chattering is alleviated and better stabilization is achieved; and (ii) the robustness of the closed-loop system (with the proposed FBSTSA) is guaranteed under system uncertainties and external disturbances.

This paper proposes an improved model predictive control (MPC) scheme with a robust prediction and stability-constrained finite states for three-phase voltage source inverters (3Φ-VSIs) with an LC filter. In this paper, the stability-constrained finite states are selected via the asymptotic stability conditions as a key factor to reduce the total harmonic distortions (THDs) and steady-state errors. Meanwhile, the Kalman filter-based observers improve the overall robustness against model mismatches and noises via a robust prediction. To select the stabilized finite states, the stability conditions are derived by the equivalent feedback-gains and constrained in the exhausting search of the proposed finite-set (FS) MPC. Unlike conventional FS-MPC methods, three control targets (i.e., robustness, stability, and optimality) are simultaneously achieved as the new contributions to remarkably enhance the voltage control performance of the 3Φ-VSI, which are also the challenge of the conventional control methods. To verify the superiority of the proposed FS-MPC, comparative studies are conducted on a prototype three-wire 3Φ-VSI system with a TI TMS320F28335 DSP under practical conditions (i.e., parameter mismatches, linear/nonlinear-load step-changes). The experimental results confirm that the performance of the proposed FS-MPC has been significantly improved in terms of lower THDs, smaller steady-state errors, faster dynamic response, and more robustness under critical system changes as compared with the conventional FS-MPC.

In this paper, we present a novel event-triggered control mechanism for nonlinear systems with hysteresis quantization. To describe the tendency of the dynamic quantizer parameter, an auxiliary parameter is created and the relationship between the quantizer parameter and the auxiliary parameter has been discussed. Via thoroughly analyzing the structure of the hysteresis quantizer, the hysteresis quantizer based event-triggered control mechanism is designed to appropriately decide the accurate signal transmission instants with simple calculation process. Moreover, under the proposed method, the nonlinear system could be stabilized and Zeno behavior could be intrinsically avoided. Furthermore, a numerical example is given to demonstrate the efficiency of the proposed mechanism.

Web-winding systems are generally large-scale interconnected systems with many motor-driven subsystems. If the centralized fault diagnosis methods are adopted, the information exchanges among the motor driven subsystems will be required. However, the information exchanges are not always available due to the communication restriction, high system cost and so on. Moreover, it will lead to the high computational cost. To solve this problem well, the web-winding system is considered as a synthetic system with several dynamic subsystems subjected to multiple disturbances and actuator faults. Then, the methods of disturbance attenuation based distributed fault diagnosis method and disturbance compensation based distributed fault diagnosis method are developed to estimate the actuator faults. The objective of fault detection, fault isolation and fault estimation can also be realized via these methods. Meanwhile, sufficient conditions of asymptotic stability of the estimation error system are derived based on the Lyapunov theory. Observer gain matrices are obtained by solving the linear matrix inequalities (LMIs). Finally, simulations and analysis are performed on the three-motor web-winding system to verify the effectiveness of the proposed two fault diagnosis methods.

This paper studies the adaptive asymptotic tracking problem for a class of unknown nonlinear systems in pure-feedback form. Different from the traditional literatures which only tackle the bounded tracking problem for pure-feedback systems, this paper investigates the asymptotic tracking problem by developing a novel controller design method. Moreover, the differentiable assumption on nonaffine functions is canceled, and only a mild semi-bounded assumption is required as the controllability condition. By utilizing Lyapunov theorem, it is proved that all the variables of the resulting closed-loop system are semi-globally uniformly ultimately bounded, and the output tracking error can converge to zero asymptotically by choosing design parameters appropriately. Finally, a simulation result is presented to verify the effectiveness of the proposed control scheme.

This paper addresses the asymptotic tracking problem of adaptive neural control for a class of uncertain strict-feedback nonlinear systems. As a universal approximator, the neural network is widely utilized to solve the tracking control problem of unknown continuous nonlinear systems. Due to the existence of neural network approximation errors, previous neural network-based control approaches can only achieve the bounded tracking rather than the asymptotic tracking. This paper designs an asymptotic error eliminating term to achieve the adaptive neural asymptotic tracking. By utilizing the Lyapunov stability theory, all the variables of the resulting closed-loop system are proven to be semi-globally uniformly ultimately bounded, and the tracking error can converge to zero asymptotically by choosing design parameters appropriately. A simulation example is presented to show the effectiveness of the proposed control approach.

The guaranteed cost control problem for a class of nonlinear discrete time-delay systems is investigated. Based on the Lyapunov matrix, a complete-type Lyapunov-Krasovskii functional is constructed. Thereby, the Lyapunov stability theory is employed to design the exact form of the controller to ensure that the resultant closed-loop system is asymptotically stable and the cost function is bounded. A numerical example is presented to illustrate the usefulness of the theoretical results.

This paper introduces new theoretical insights into Nash equilibrium-based asymptotic stability (NEAS) of two-group and three-group asymmetric evolutionary games in typical scenarios of electricity market (EM). EM competition has become a complex dynamic evolution process accomplished by more complex characteristics of market economy behavior. Replicator dynamics in evolutionary game theory, as well as Lyapunov stability theory, are employed to solve incomplete-information and bounded-rationality game issues in EM, so as to overcome theoretical demerits of classical game theory in solving multi-group games in EM. First, the NEAS of a unilateral two-group asymmetric evolutionary game (AEG) is investigated. Then, this is expanded to a complicated 2 × 2 × 2 trilateral multi-group AEG, and the NEAS of it under different game situations in EM is thoroughly discussed. Finally, a practical case study is conducted for verification. The case illustrates how the factors affect the payoff matrix which will change ultimate evolutionary stable state of the multi-group AEG in EM. One main finding demonstrates the complete dynamics behavior and multi-group evolutionary stable strategy (MESS) of the AEG system in 3-D mixed strategy space. The other one reveals that EM policies formulated by government and other factors can gradually influence the MESS via changing the payoff distribution matrix.

In this paper, we propose a novel controller design technique which is applicable to nonlinear systems. Based on the classical power shaping methodology, we establish some sufficient conditions which are able to guarantee the stability of the resulting closed-loop system. By relaxing some preliminary conditions, the particular expression of the nonlinear controller is given and the asymptotic stability of desired equilibria can be proved according to the Lyapunov theory. To motivate the application of the new proposed controller design technique and illustrate its feasibility, we verify the proposed methodology with a magnetic levitated system. Moreover, simulation results are presented to show the performance of the designed controller.

In this paper, a composite control scheme is presented to asymptotically stabilize a class of port-controlled Hamiltonian systems under nonvanishing disturbances. First, based on the damping injection method and the nonlinear disturbance observer (NDOB) technique, the robust composite control strategy is designed. The NDOB, as an effective observation tool, is developed to estimate the disturbances, and furthermore, the disturbances can be feedforward compensated using the estimates of disturbances. Then, for the augmented system, an asymptotic stability theorem is proposed via an input-to-state stability technique and Lyapunov stability theorems. The proposed control approach exhibits not only good robustness and disturbance rejection performances but also the property of nominal performance recovery. Finally, a simulation example on a circuit system is given to show the feasibility and advantage of the composite control method.

This paper addresses the stability analysis problem for planar periodic switching systems. We characterize the stability margin in the space constituted by the dwell times of the subsystems, by which we can assess the asymptotic stability of the overall system in the necessary and sufficient sense. The mutual constraint conditions on the dwell times in nature depend on the type of equilibrium point of each subsystem. The stability conditions are expressed in terms of a family of transcendental inequalities, which can be numerically solved and precisely depicted in the time-domain space. An example is worked out in detail to illustrate the theoretical results.

In this paper, a nonlinear controller with a current-limiting property is proposed to guarantee accurate dc output voltage regulation and unity power factor operation for single-phase pulse-width-modulating rectifiers without the need of a phase-locked-loop (PLL). The proposed current-limiting controller is fully independent of the system parameters and can guarantee asymptotic stability and convergence to a unique solution for the closed-loop system using nonlinear control theory. Without requiring the instantaneous measurement of the grid voltage, a PLL, an external limiter, or a saturation unit, the proposed strategy guarantees that the input current of the rectifier will always remain below a given value. An analytic framework for selecting the controller parameters is also presented to provide a complete controller design procedure and it is also proved that the current-limiting property is maintained even when the grid voltage drops. Extensive experimental results are presented to verify the proposed controller when the load changes, the reference dc output voltage changes, and the grid voltage drops.