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In this paper, we shall prove a stochastic averaging principle for two-time-scale jump-diffusion SDEs under the non-Lipschitz coefficients.

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Ito formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Ito formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random “noise”, under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process.

In this paper, we address the group controllability issue for discrete-time multi-agent systems (MASs) with two-time-scale feature. Since the existence of singular perturbation parameter in modeling the two-time-scale MASs will cause ill-posedness problems, we decouple the MASs into fast and slow subsystems by using the iterative and approximate methods, and then eliminate the singular perturbation parameter. By using matrix theory, we propose a lot of necessary and/or sufficient conditions of group controllability, where some easy-to-use criteria are determined only by eigenvalues of matrices. Lastly, a simulation example is given to illustrate the effectiveness of the proposed theoretical results.

In this paper, a feedback control law based on two time‐scale assignment and state extension is implemented for the nonlinear non‐minimum phase dynamics of three degree‐of‐freedom autonomo...

A two-time-scale system involves both fast and slow dynamics. This paper studies the design of nonlinear observers for general nonlinear two-time-scale systems and presents a reduced-order observer design approach. The reduced-order observer is derived based on a lower dimensional model to reconstruct the slow states, along with the algebraic slow-motion invariant manifold function to reconstruct the fast states. Through an error analysis, it is shown that even though the observer is designed based on the reduced model by neglecting the fast dynamics, it is capable of providing accurate estimation of the states of the original detailed system. It will render a vanishing estimation error, with exponential convergence rate governed by the subsystem of fast dynamics and the chosen observer design parameters.

In this article, modeling and control of a rotating hub-beam system are studied. The system consists of a solid rotating cylinder and an attached flexible arm with a payload at the end. The rotation is supposed to be in the presence of gravity and the flexible arm is assumed to be a Euler-Bernoulli beam. To derive the equations of motion of the system, Lagrange’s method is applied. Moreover, Galerkin’s technique is employed to discretize the equations of motion. Furthermore, designing an appropriate two-time (slow and fast) scale controller in the presence of uncertainties is considered in order to track the desired hub angular position and suppress vibrations of the arm simultaneously. For the so-called slow subsystem, a novel controller design is proposed as two different cases, with and without the presence of uncertainties in system dynamics are considered; and accordingly, a control law for tracking the desired path is introduced based on the idea of using the cross-term constructed Lyapunov function. For the fast subsystem, a pole placement technique is used to suppress vibration of the beam. The simulation results indicate notable effectiveness of the proposed controller.

In this study, the controllability of multi-agent systems with second-order dynamic under discrete-time model, which evolve on two time scales. The authors first characterise the systems by difference equations with a singular perturbation parameter. Then, in order to get rid of the singular perturbation parameter, they split the systems into a slow subsystem and fast subsystem based on the iterative method and approximate approach. Moreover, according to matrix theory, they deduce a lot of necessary and/or sufficient controllability criteria of the systems. In particular, a second-order controllability criterion only depending on the eigenvalues of matrices is proposed, which is more easy-to-use and efficient. A simulation example is lastly given to clarify the validity of all theoretical results.

This paper studies the distributed consensus problem of a new class of multi-agent systems, where each agent is represented by the two-time-scale system. Two fully distributed, ϵ-dependent adaptive...

This paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stab...

This work is concerned with competitive Lotka–Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the pap...

This paper addresses the controllability for an interconnected two-time-scale second-order multi-agent system. Firstly, to eliminate the singular perturbation parameter, we separate the multi-agent system into slow subsystem and fast subsystem by using singular perturbation methods. Then, based on matrix theory, some necessary and/or sufficient criteria are derived for second-order controllability of two-time-scale multi-agent systems with multiple leaders. Moreover, we propose some easy-to-use second-order controllability criteria determined only by eigenvalues of system matrices. Lastly, the effectiveness of the proposed theoretical results is illustrated by a simulation example.

This paper investigates the controllability of discrete-time leader-follower multiagent systems (MASs) with two-time-scale and heterogeneous features, motivated by the fact that many real systems are operating in discrete-time. In this study, singularly perturbed difference systems are used to model the two-time-scale heterogeneous discrete-time MASs. To avoid the ill-posedness problem caused by the singular perturbation parameter when using the classical control theory to study the model, the singular perturbation method was first applied to decompose the system into two subsystems with slow-time-scale and fast-time-scale feature. Then, from the perspective of algebra and graph theory, several easier-to-use controllability criteria for the related MASs are proposed. Finally, the effectiveness of the main results is verified by simulation.

Motivated by broad applications within engineering and sciences, we study distributed consensus-based gradient methods for solving optimization problems over a network of nodes. A fundamental challenge for solving this problem is the impact of finite communication bandwidth, so information that is exchanged between the nodes must be quantized. In this paper, we utilize the dithered (random) quantization and study the distributed variant of the well-known two-time-scale methods for solving the underlying optimization problems under the constraint of finite bandwidths. In addition, we provide more insight and an explicit formula of how to design the step sizes of these two-time-scale methods and their impacts on the performance of the algorithms.

The key issue in on-orbit manipulation is to transport large flexible modules to pre-assembly position accurately. However, the cooperative manipulation by a team of space robots can induce large vibration in the robot-beam system. This paper presents a two-time scale control scheme for vibration minimization in trajectory tracking of the flexible module. The control scheme is composed of a slow controller ensuring precise tracking of the beam's rigid motion and a fast controller to minimize the systemic vibration. Our approach to the slow controller design is based on non-singular terminal sliding mode control, with which fast and finite-time convergence of tracking errors against disturbances is obtained. The fast controller guarantees minimum vibration induced during the manipulation by applying optimal control technique. Numerical results by comparison show that the two-time scale control scheme for cooperative manipulation demonstrates higher precision and finite-time convergence in tracking the rigid motion with robustness and minimizing the vibration simultaneously.

This paper deals with averaging principle for two-time-scale stochastic differential equations (SDEs) with non-Lipschitz coefficients, which extends the existing results: from Lipschitz to non-Lipschitz case. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for coupled system is established, and as a result, the system can be reduced to a single SDEs with a modified coefficient which is also non-Lipschitz. Moreover, it is shown that the slow variable strongly converges to the solution of the corresponding averaging equation.

As a basic concept in modern control theory, controllability reveals the fundamental structural characteristics of a dynamic system, and it also plays an important part in the analysis and control of a dynamic system. With the increasing complexity of multi-agent systems, the multi-agent networks can be divided into some subnetworks in terms of time scales. This paper concentrates on the group controllability of two-time-scale multi-agent networks, establishes the necessary and sufficient criterion of group controllability based on singular perturbation methods, and deduces easy-to-use group controllability criteria by using matrix theory and graph theory. Lastly, a simulation example is presented to illustrate the effectiveness of the proposed methods.

This paper presents a novel multivariable two-time scale control method for flexible joint robots. The fundamental limitations of the conventional two-time scale control are clearly explained based on a singular perturbation analysis with a linearized dynamic model. Unlike the conventional methods, the bandwidth of the slow control can be made high and the derivatives of the rigid control are not required. The closed-form solution of a slow model that is more accurate than the rigid model is presented. It is shown that the modeling error of the proposed two-time scale control can be made arbitrarily small by the control gain only. A real-world experiment with a six degrees-of-freedom commercial industrial robot was performed to validate the presented approach.

In this paper, the global asymptotic stability of two-time-scale competitive neural networks(CNNs) with multiple time-varying delays is investigated. By constructing a new e-dependent Lyapunov functional, sufficient conditions for the global asymptotic stability of the concerned systems are established, and an optimization problem is formulated to get the best estimate of the e-bound. Compared with the existing results, the proposed results are more general and less conservative in the sense of determining an upper bound for the time-scale parameter e. Finally, three examples are given to illustrate the advantages of the obtained results.

The paper investigates the impact of jumps in forecasting co-volatility, accommodating leverage effects. We modify the jump-robust two time scale covariance estimator of Boudt and Zhang (2013) such that the estimated matrix is positive definite. Using this approach we can disentangle the estimates of the integrated co-volatility matrix and jump variations from the quadratic covariation matrix. Empirical results for three stocks traded on the New York Stock Exchange indicate that the co-jumps of two assets have a significant impact on future co-volatility, but that the impact is negligible for forecasting weekly and monthly horizons.

We analyze a time-varying Mperi/Geo2timeScale/N queueing system. The arrival process is periodic Poisson. The service time of a customer has components in different time scales: length of stay (LOS) in days and departure time (hdis) in hours. This queueing system has been used to study patient flows from the emergency department (ED) to hospital inpatient wards. In that setting, the LOS of a patient is simply the number of days she spends in a ward, and her departure time hdis is the discharge hour on the day of her discharge. We develop a new analytical framework that can perform exact analysis on this novel queueing system. This framework has two steps: first analyze the midnight customer count process and obtain its stationary distribution, then analyze the time-dependent customer count process to compute various performance measures. We also develop approximation tools that can significantly reduce the computational time. In particular, via Stein’s method, we derive explicit expressions to approximate the stationary distribution of the midnight count. We provide error bounds for these approximations and numerically demonstrate that they are remarkably accurate for systems with various sizes and load conditions. Our theoretical and numerical analysis have produced a number of insights that can be used to improve hospital inpatient flow management. We find that the LOS term affects the overnight wait caused by the mismatch between daily arrivals and discharges, whereas the hdis term affects the intraday wait caused by the nonsynchronization between the arrival and discharge time patterns. Thus, reducing LOS or increasing capacity can impact the daily average performance significantly; shifting the discharge timing to earlier times of a day can alleviate the peak congestion in the morning and mainly affects the time-dependent performance. The e-companion is available at https://doi.org/10.1287/opre.2016.1566 .

This paper presents a two-time-scale neurodynamic approach to constrained minimax optimization using two coupled neural networks. One of the recurrent neural networks is used for minimizing the objective function and another is used for maximization. It is shown that the coupled neurodynamic systems operating in two different time scales work well for minimax optimization. The effectiveness and characteristics of the proposed approach are illustrated using several examples. Furthermore, the proposed approach is applied for $H_\infty $ model predictive control.

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied, and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method., 32 pages

Motivated by their broad applications in reinforcement learning, we study the linear two-time-scale stochastic approximation, an iterative method using two different step sizes for finding the solutions of a system of two equations. Our main focus is to characterize the finite-time complexity of this method under time-varying step sizes and Markovian noise. In particular, we show that the mean square errors of the variables generated by the method converge to zero at a sublinear rate $\Ocal(k^{2/3})$, where $k$ is the number of iterations. We then improve the performance of this method by considering the restarting scheme, where we restart the algorithm after every predetermined number of iterations. We show that using this restarting method the complexity of the algorithm under time-varying step sizes is as good as the one using constant step sizes, but still achieving an exact converge to the desired solution. Moreover, the restarting scheme also helps to prevent the step sizes from getting too small, which is useful for the practical implementation of the linear two-time-scale stochastic approximation.

This work focuses on stochastic bifurcation for a slow–fast dynamical system driven by non-Gaussian α-stable Lévy noise. We prove the main result for the stochastic equilibrium states for the original system and the reduced system based on the random slow manifold. Then, it is verified that the slow reduced system bears the stochastic bifurcation phenomenon inherited from the original system. Furthermore, we investigate the number and stability type of stochastic equilibrium states for dynamical systems through numerical simulations, and it is illustrated that the slow reduced system captures the stochastic bifurcation of the original system.

In transient stability analysis, the fifth-order generator model neglects the stator transients and may result in different simulation results as compared with the seventh-order model. Based on the two-time scale property of generators, a dynamic correction method is proposed to offset the difference. Using the singular perturbation theory, fast components in the stator flux linkage is captured to restore stator dynamics and correct the electromagnetic torque. Simulations on a single-machine-infinite-bus (SMIB) system and a 3-machine system show that, the difference between the two generator models can be remarkably reduced by the proposed method.

The convergence of pedestrian flow in a unidirectional corridor is studied based on the two-time-scale hybrid traffic models for pedestrian crowds. In the two-time-scale framework, the corridor is divided into a set of virtual lanes. The width of each lane is equal to the diameter of a pedestrian which is modeled as a disk. Being situated in a corridor, the pedestrians can either walk in a lane following each other or change to the adequate lanes. The in-lane movements of pedestrians are viewed as a one-line multi-agent dynamic system in continuous time. The lane changing behavior of pedestrians is modeled as a Markov decision process. Pedestrians will change lanes according to the transition probabilities which are functions depending on pedestrian velocities in the related lanes. Furthermore, the quantized consensus state of pedestrian flow is defined. We show that when pedestrians wish to change to faster lanes, the numbers of pedestrians in each lane converge to a balanced distribution (quantized consensus state) as shown typically in pedestrian flows. Numerical simulations and theoretical analysis are used to demonstrate the convergence properties of this method. Simulation results show that the model can capture the main features of unidirectional pedestrian flows.

In this paper, a two time-scale based controller is developed for the problem of a quadrotor in the presence of wind-gust disturbances. A backstepping control and a model based PD control method are proposed to perform position (slow model) and attitude (fast model) control of the quadrotor, respectively. The wind-gust disturbances are regard as external forces and torques acting on quadrotor, which are estimated by two Square-Root Unscented Kalman filter (SR-UKF) in the fast and slow modes. Simulated results are presented to demonstrate the good performance of the controller even under wind-gust disturbances which are assumed as a mixture of multiple unknown sinusoidal signals.

This paper reviews some results from the theory of two-time scale dynamical systems (also called fast-slow systems) and presents the detailed analysis of some systems of interest in electrical engineering and fusion plasma physics. The aim of this work is to introduce engineers to singular perturbation methods as a tool for providing additional insights in modeling and understanding systems whose components evolve on different time scales.

A nonlinear process with input multiplicity has two or more input values for a given output at the steady state, and the process steady state gain changes its sign as the operating point changes. A control system with integral action will be unstable when both signs of the process gain and the controller integral gain are different, and its stability region will be limited to the boundary where the process steady state gain is zero. Unlike processes with output multiplicities, feedback controllers cannot be used to correct the sign changes of process gain. To remove such stability limitation, a simple control system with parallel compensator is proposed. The parallel compensator can be easily designed based on the process steady state gain information and tuned in the field. Using the two time scale method, the stability of proposed control systems for processes with input multiplicities can be checked.

A Freidlin-Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [3]., Comment: 30 pages

In this paper, event-triggered strategies for control of discrete-time plants with two-time scale property are developed and analyzed. The plant is assumed input-to-state stable with respect to measurement errors and the control law is updated once a triggering condition involving the norm of a measurement error is violated. The results are also extended to a self-triggered formulation, where the next control updates are decided at the previous ones, thus relaxing the need for continuous monitoring of the measurement error. The results are illustrated through simulated examples.

The problem of exponential mean-square stability of two-time scale, linear stochastic hybrid systems has been studied in this paper. To obtain the sufficient conditions of stability, two basic approaches of stability analysis: for one-time hybrid systems with a Markovian switching rule as well as switching rule and singularly perturbed nonhybrid systems, were combined. The Lyapunov techniques were used in both approaches.

In this paper, we investigate the controllability problems for heterogeneous multiagent systems (MASs) with two-time-scale feature under fixed topology. Firstly, the heterogeneous two-time-scale MASs are modeled by singular perturbation system with a singular perturbation parameter, which distinguishes fast and slow subsystems evolving on two different time scales. Due to the ill-posedness problems caused by the singular perturbation parameter, we analyze the two-time-scale MASs via the singular perturbation method, instead of the general methods. Then, we split the heterogeneous two-time-scale MASs into slow and fast subsystems to eliminate the singular perturbation parameter. Subsequently, according to the matrix theory and the graph theory, we propose some necessary/sufficient criteria for the controllability of the heterogeneous two-time-scale MASs. Lastly, we give some simulation and numerical examples to demonstrate the effectiveness of the proposed theoretical results.

This short communication is a discussion of the paper entitled “Adaptive sliding mode observer for induction motor using two-time-scale approach” by A. Mezouar, M.K. Fellah and S. Hadjeri published in the Electric Power Systems Research 77 (2007) pp. 604–618. In the discussed paper the authors present a current and flux sliding mode observer for the induction motor that also incorporates an adaptive law in order to estimate the inverse of the rotor time constant. However the proposed observer design uses the real value of the rotor time constant which is unknown and therefore cannot be used in the observer design.

This short communication is a discussion of the paper entitled “Adaptive sliding-mode-observer for sensorless induction motor drive using two-time-scale approach” by A. Mezouar, M.K. Fellah and S. Hadjeri published in the Simulation Modelling Practice and Theory 16 (2008) 1323–1336. In the discussed paper the authors present a current and flux sliding mode observer for the induction motor that also incorporates an adaptive laws in order to estimate the rotor speed and the inverse of the rotor time constant. However the proposed design for the observer and for the adaptive laws, employs the real value of the rotor time constant and the real value of the rotor speed, which are unknown, and therefore cannot be used in the observer design nor in the adaptive laws design.

We examine the initial layer problem and the infinite Prandtl number limit of the thermosolutal convection, which is applicable to magma chambers. We derive the effective approximating system of the Boussinesq system at large Prandtl number using two time scale approach [M. Holmes, Introduction to Perturbation Methods, Springer, New York, 1995, A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York, American Mathematical Society, Providence, RI, 2003]. We show that the effective approximating system is nothing but the infinite Prandtl number system with initial layer terms. We also show that the solutions of the Boussinesq system converge to solutions of the effective approximating system with the convergence rate of O(ϵ).

The problem of exponential mean-square stability of two-time scale, linear stochastic hybrid systems has been studied in this paper. To obtain the sufficient conditions of stability two basic approaches of stability analysis for one-time hybrid systems with a Markovian switching rule and any switching rule and singularly perturbed nonhybrid systems were combined. In both approaches the Lyapunov techniques were used.