Lyapunov Conditions Research Articles

Soft-Reset Control With Max-of-Quadratics Lyapunov Certificates

We further develop a novel control paradigm encoding the core behavior of a reset control system within a continuous-time feedback, described by a differential inclusion. A tuning knob allows adjusting how close the continuous-time feedback resembles the behavior of the underlying reset controller. The Lyapunov conditions that we derive naturally lead to a sum-of-squares formulation allowing for the design of max-of-quadratics Lyapunov certificates associated with polynomial multipliers. A few application examples confirm the practical effectiveness of the proposed solutions.

Distributed finite‐time control with event triggered mechanism for multi‐agent systems under fixed and switching topologies

AbstractThis paper addresses the problem of distributed finite‐time (FT) event‐triggered consensus of multi‐agent systems (MASs) under fixed and switching topologies. To conserve the limited communication resources, a new distributed event‐triggered control protocol is proposed, in which the agents exchange information via discontinuous communication among the follower agents. Then, to solve the FT consensus problem, norm‐normalized sign functions are employed, and sufficient Lyapunov conditions are provided. On this basis, the MASs proves capable of reaching consensus within finite time under fixed and switching topologies. Finally, the effectiveness of the developed algorithm is verified via numerical simulation.

Fixed-time supper twisting disturbance observer and sliding mode control for a secure communication of fractional-order chaotic systems

This paper proposed a new disturbance observer (DO) based on the fixed-time supper-twisting algorithm (FTSTA) for rejecting perturbations of secure communication of the fractional-order chaotic systems (FOCSs). First, the FOCSs are remodeled into the Takagi–Sugeno fuzzy (TSF) system with the expectation of softening the calculations of the design of the controller and observer. Second, the synchronization control was designed based on the FTSTA sliding mode control (SMC). Third, a new DO for the secure communication system (SCS) of FOCSs was proposed to estimate the disturbance on public channels and uncertainty variations. Fourth, the stability of the proposed synchronization control was obtained by using the Lyapunov condition. Finally, to verify the correction of the proposed method, the MATLAB simulation with the use of FONCOM toolbox was used to meet the desired expectations of theoretically proposed method. There are, small reaching times, small steady-states, and most tested disturbances and parameters’ variations were compensated by the proposed DO.

Time-averaging principle for G-SDEs based on Lyapunov condition

In this paper, we tame the uncertainty about the volatility in time-averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs) based on the Lyapunov condition. That means we treat the time-averaging principle for stochastic differential equations based on the Lyapunov condition in the presence of a family of probability measures, each corresponding to a different scenario for the volatility. The main tool for the mathematical analysis is the G-stochastic calculus, which is introduced in the book by Peng (Nonlinear Expectations and Stochastic Calculus Under Uncertainty. Springer, Berlin, 2019). We show that the solution of a standard equation converges to the solution of the corresponding averaging equation in the sense of sublinear expectation with the help of some properties of G-stochastic calculus. Numerical results obtained using PYTHON illustrate the efficiency of the averaging method.

Exponential Stabilization of Nonlinear Impulsive Systems via Output-Based Event-Triggered Control

This article applies output-based event-triggered control (ETC) strategies to nonlinear systems involving impulses and investigates the problem for globally exponential stability (GES). The impulses are not directly related to events and two kinds of control schemes (static and dynamic ETC) are fully considered, respectively. Moreover, the stabilizing and destabilizing impulses can coexist (i.e., hybrid impulses). By employing properties of impulsive system theory and Lyapunov conditions, some theoretical conditions for GES and non-Zeno behavior are derived under static and dynamic event-triggered mechanism, respectively. And the results for stability are different but complementary to each other. Based on the linear matrix inequality, the design criteria of expected triggered mechanism and control gains are derived. Finally, two numerical examples containing simulations are given to demonstrate the effectiveness of the proposed control strategy.

Exponential ergodicity for non-dissipative McKean-Vlasov SDEs

Under Lyapunov and monotone conditions, the exponential ergodicity in the induced Wasserstein quasi-distance is proved for a class of non-dissipative McKean-Vlasov SDEs, which strengthen some recent results established under dissipative conditions in long distance. Moreover, when the SDE is order-preserving, the exponential ergodicity is derived in the Wasserstein distance induced by one-dimensional increasing functions chosen according to the coefficients of the equation.

Delay tolerance for stochastic complex networks

In this paper, we investigate the delay tolerance for stochastic complex networks (SCNs, in short) under Lyapunov conditions. Based on global Lipschitz coefficients, we show that when the SCN without delay is pth moment exponentially stable, the system with small delays is still pth moment exponentially stable. In addition, we also give the explicit delay bounds. An example is provided to demonstrate the obtained results.

Adaptive finite-time fault-tolerant control for nano-satellite attitude tracking under actuator constraints

In this paper, a finite-time fault-tolerant control method is developed for attitude tracking of a nano-satellite with three magnetorquers and one reaction wheel in the presence of actuator faults, inertia uncertainties, external distribution, and actuator constraints. For this purpose, the modified non-singular fast terminal sliding mode control is used due to the ability of robustness against uncertainties and finite-time convergence. In this approach, the modified sliding surface variable is constructed based on angular velocity and attitude error to converge the attitude error to zero in a finite time. In addition, three adaptive parameters are employed to reduce the adverse effect of uncertain expressions and to track the attitude commands with high control accuracy. The adaptive parameters and sliding surface variable are used in the reaching phase of the controller to decrease the chattering phenomenon. The dynamics of adaptive parameters, with regard to the angular velocities of the nano-satellite and reaction wheel and the sliding surface variable, are obtained during the stability proof of the closed-loop system. Also, the finite-time convergence of sliding surface and attitude variables are proved by the extended Lyapunov condition, and the convergence regions of attitude tracking errors are obtained. The simulations are accomplished and compared with an existing adaptive sliding mode control approach to evaluate the performance of the proposed controller. The results verify that the proposed controller performs better than the mentioned controller in terms of finite-time convergence, the accuracy of attitude tracking, and the reduction of the chattering phenomenon.

Optimal consensus control for multi-satellite assembly in elliptic orbit with input saturation

This paper presents an optimal control design method for multi-satellite assembly in an elliptic orbit in close proximity operations considering input saturation constraints. Tschauner–Hempel (T–H) equations are utilized to model the relative motion dynamics of the satellites in the Local Vertical Local Horizontal (LVLH) frame of the target satellite, where every satellite in the assembly is modeled as a point mass. The communication topology between the satellites in the assembly is modeled using graph theory, where the graph has several nodes representing the satellites and several edges representing the data exchange direction. It is assumed that every satellite exchanges its states with a few neighbors in the team, where the neighbors of every satellite are prespecified and do not change during the mission. This graph is utilized to construct the global dynamic model of the assembly, in which every node is represented by the dynamic model of a single satellite. The local control law, which is applied to every satellite in the assembly, consists of a state feedback gain and a coupling gain. The state feedback gain is designed using the discrete-time periodic Riccati equation to guarantee the local stability of every satellite separately, while the coupling gain is selected to satisfy the Lyapunov condition to guarantee the global stability of the assembly. The invariant-set method is utilized to prove the stability of the global dynamic system that is subjected to symmetric input saturation. The precision and optimality of the proposed control law are successfully demonstrated by numerical nonlinear simulations in a MATLAB environment for a cubic formation satellite assembly.

Deterministic and Stochastic Fixed-Time Stability of Discrete-time Autonomous Systems

This paper studies deterministic and stochastic fixed-time stability of autonomous nonlinear discrete-time (DT) systems. Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT systems is certified. Extensions to systems under deterministic perturbations as well as stochastic noise are then considered. For the former, sensitivity to perturbations for fixed-time stable DT systems is analyzed, and it is shown that fixed-time attractiveness results from the presented Lyapunov conditions. For the latter, sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented. The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed-time attractive systems, and a stochastic settling-time function fixed upper bound is derived for stochastic DT systems. Illustrative examples are given along with simulation results to verify the introduced results.

Reset control and ℒ2-gain analysis of piecewise-affine systems

This paper is concerned with reset controller design and L 2 -gain stability of piecewise-affine systems under the framework of hybrid systems. Firstly, stability conditions of piecewise-affine systems under dynamic state-feedback control are established through bilinear matrix inequality conditions. Secondly, a reset controller with reset rules under the hybrid systems framework is proposed in the sense of Lyapunov and sufficient conditions for exponential and L 2 -gain stability of the closed-loop systems are provided. Different from the piecewise-affine systems with the dynamic state-feedback controller, the reset controller is designed such that the L 2 -gain performance of piecewise-affine systems can be enhanced. Thirdly, an LMI approach is proposed to avoid the difficulty of solving the bilinear matrix inequalities. Furthermore, robustness to inflations of the flow and jump sets is established, and robustness to the norm-bounded uncertainties is proposed. Finally, numerical simulations including a robot arm system, an inverted pendulum system and the Chua's circuit system are provided to illustrate the results.

Disturbance Observer Based on Fixed Time Sliding Mode Control and Optimal State Observer for Three-Phase Three-Level T-Type Inverters

This paper presents the advanced control method together with the disturbance rejection method with the aims to obtain the low of total harmonic distortion (THD). The expectation of the paper is that the output voltage is satisfied the requirements of the EN 62040 with the THD is less than 5%. Therefore, this paper proposes a new disturbance observer (DOB) without the requirement of the information of the first derivative disturbance for the three phase three level (TPTL) T-type inverter system. The proposed DOB is based on the information of the measurable states, which are used such an inversed model of T-type inverter to find the information of real disturbance. First, the DOB was designed based on the given measured and estimated states. However, to design the DOB, the state observer is required. Therefore, second, the state observer (SOB) was designed based on linear matrix inequality (LMI) to identify the optimal poles of the state-tracking error function. Third, the estimated states were used to construct the fixed time (FT) sliding mode control (SMC) to control the T-type inverter system. Fourth, the Lyapunov condition was used to verify the correction of the proposed method. The performance of the proposed control strategy is validated by simulations and experiments during steady-state, transients caused by load change, and unbalanced grid conditions. The estimated states precisely tracked the measured states. The output signals precisely converged to the predefined trajectories in a predefined time and the tracking errors are small. The obtained results showed that the proposed control method provided excellent steady-state and good performances with low THD in the line currents, zero steady-state error in the output voltage, and a very fast dynamic response.

On Input-to-State Stability of Linear Difference Equations and Its Characterization with a Lyapunov Functional

This paper investigates necessary and sufficient Lyapunov conditions for Input-to-State Stability (ISS) of Linear Difference Equations with pointwise delays and an additive exogenous signal. Grounding on recent works in the literature on necessary conditions for the exponential stability of such Difference Equations, we propose a quadratic Lyapunov functional involving the derivative of the so-called delay Lyapunov matrix of the corresponding homogeneous Difference Equation. We prove that the ISS of Linear Difference Equations is equivalent to the existence of an ISS Lyapunov functional. We apply this result to the stability and ISS analysis of hyperbolic Partial Differential Equations of conservation laws.

Lyapunov Conditions for Differentiability of Markov Chain Expectations

We consider a family of Markov chains whose transition dynamics are affected by model parameters. Understanding the parametric dependence of (complex) performance measures of such Markov chains is often of significant interest. The derivatives and their continuity of the performance measures w.r.t. the parameters play important roles, for example, in numerical optimization of the performance measures, and quantification of the uncertainties in the performance measures when there are uncertainties in the parameters from the statistical estimation procedures. In this paper, we establish conditions that guarantee the smoothness of various types of intractable performance measures—such as the stationary and random horizon discounted performance measures—of general state space Markov chains and provide probabilistic representations for the derivatives. Funding: C.-H. Rhee is supported by the National Science Foundation [Grant CMMI-2146530].

Lyapunov-Malkin type approach of equilibrium stability in a critical case applied to a switched model of a servomechanism with state delay

A general theorem on equilibrium stability in a critical case is applied to switched strong nonlinear differential equations with time delay on state, characterising electrohydraulic servomechanisms dynamics. Basically, its proof involves the use of the Lyapunov-Malkin approach to stability and multiple complete Lyapunov-Krasovskii functionals. The fulfilment of the equilibrium stability condition of the nonlinear system returns to the asymptotic stability condition of the linearised equations and to the so-called Lyapunov conditions for the latter. The transformation of the nonlinear system into the canonical form specific to the Lyapunov-Malkin theorem, and the verification of the two conditions mentioned above require analytical developments doubled by numerical simulations, since the mathematical models are too complex to be approached only analytically. As a consequence, an important result is obtained, for the first time, regarding the thresholds of admissible delay in preserving the stability of a real world object, vital for the safety of the aircraft.

Ergodic Stationary Distribution and Threshold Dynamics of a Stochastic Nonautonomous SIAM Epidemic Model with Media Coverage and Markov Chain

A stochastic nonautonomous SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic model with Markov chain and nonlinear noise perturbations has been constructed, which is used to research the hybrid dynamic impacts of media coverage and Lévy jumps on infectious disease transmission. The uniform upper bound and lower bound of the positive solution are studied. Based on defining suitable random Lyapunov functions, we researched the existence of a nontrival positive T-periodic solution. Sufficient conditions are derived to discuss the exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, the persistence in the average sense and extinction of infectious disease are investigated using stochastic analysis techniques. Finally, numerical simulations are utilized to provide evidence for the dynamical properties of the stochastic nonautonomous SIAM.

Lyapunov conditions for finite-time stability of disturbed nonlinear impulsive systems

What we concern in this paper is finite-time control of nonlinear impulsive systems involving external disturbance, where practical finite-time and finite-time stabilization are studied with respect to nonvanishing and vanishing disturbance, respectively. A relationship between the finite settling time and the impulsive frequency is presented to show the stabilizing effect of impulses. It is shown that systems subject to nonvanishing disturbance can enter a disturbance-dependent ultimate bound in a finite-time sense, and a relatively smaller bound of settling time is obtained by utilizing stabilizing impulses. Meanwhile, systems subject to vanishing disturbance can achieve finite-time stabilization at the origin. Moreover, compared with the situation without impulses, the corresponding bound of settling time is also smaller. For the sake of illustrating the validity of proposed results, some examples and their simulations are provided.

A Novel Disturbance Rejection Method Based on Robust Sliding Mode Control for the Secure Communication of Chaos-Based System

This paper mainly proposes a new disturbance observer (DO) for a secure communication system (SCS) of the chaos-based system (CBS). First, the fractional-order (FO) Chen chaotic system is remodeled by a Takagi–Sugeno (T–S) fuzzy system with the aim of softening in calculation. Second, the robust fixed-time was designed to synchronize two nonidentical chaotic systems. Third, a new disturbance observer was proposed to compensate for the disturbance and uncertainty of the secure communication system. Fourth, the proof of the proposed method based on Lyapunov condition together with simulation are given to illustrate the correctness and effectiveness of the proposed theory. The tested disturbance on the public channel was mostly compensated by the appropriately estimated disturbance value. The states of master and slave systems (MSSs) were closed to each other in fixed-time. These factors are used to confirm that the symmetry of two chaotic systems were obtained by the proposed control methods.

Existence, uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with Markovian switching

The paper is dedicated to studying the problem of existence and uniqueness of solutions as well as existence of and exponential convergence to invariant measures for McKean-Vlasov stochastic differential equations with Markovian switching. Since the coefficients are only locally Lipschitz, we need to truncate them both in space and distribution variables simultaneously to get the global existence of solutions under the Lyapunov condition. Furthermore, if the Lyapunov condition is strengthened, we establish the exponential convergence of solutions' distributions to the unique invariant measure in Wasserstein quasi-distance and total variation distance, respectively. Finally, we give two applications to illustrate our theoretical results.

"Lyapunov Conditions for One-Sided Discrete-Time Random Dynamical Systems"

"This paper considers nonuniform exponential stability and nonuniform exponential instability concepts for one-sided discrete-time random dynamical systems. These concepts are generalizations from the deterministic case. Using this, characterizations in terms of Lyapunov functions respectively Lyapunov norms are presented. Also, an approach in terms of considered concepts for the inverse and adjoint random discrete- time systems is derived."