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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 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 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.

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.

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 .

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.

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.

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.

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 paper presents a two-time-scale neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems. The problem is formulated as a bi-convex optimization problem with spectral or Frobenious condition number as robustness measure. Coupled recurrent neural networks are applied for solving the formulated problem in different time scales. Simulation results of the proposed neurodynamic approach for benchmark problems and control of autonomous underwater gliders are reported to demonstrate its superiority.

This paper presents a model predictive control (MPC) method for a single supply chain (SC) unit. In order to make the most profit, the prediction of demand and the policy of inventory are the key factors for an SC unit. The ordering is a manipulate variable for an SC unit, and the pricing is introduced as another manipulate variable in this paper. The supposed price has determinate effect on the demand. The demand then can be regarded as a determinate part and a stochastic part. The inventory is regarded as a probability problem because of the stochastic demand. The two manipulate variables bring different dynamics into the SC unit. An MPC is designed, and the two-time scale characteristic of the two manipulate variables is considered simultaneously for the SC unit. By comparing with MPC without dynamic pricing policy and order-up-to-level policy, we found that the simulation results approve the validity of the MPC with dynamic pricing policy. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

This paper considers the problem of robust multi-objective

In this paper, a non-linear controller for linear switched reluctance motors (LSRMs) is developed by utilizing their properties and the energy dissipation theory. As the electrical time constant is much smaller than the mechanical time constant, the whole LSRM driving system can be treated as a two-time-scale system. It is then decomposed into an electrical subsystem and a mechanical subsystem, which are interconnected by negative feedbacks. Controllers for these two subsystems are designed to guarantee that both systems are passive. Because a system consisting of two passive subsystems connected through negative feedbacks is still passive as a whole, stability of an LSRM driving system can be achieved at system level. The proposed control strategy is characterized by a simple structure and easy implementation. Experimental results are also provided to prove the effectiveness and robustness of the proposed position control for LSRM.

A new uniform methodology to study the fast subsystem stability of general two-time scale nonlinear systems is developed. It consists of a direct method that provides estimates of the stability region of the fast subsystem that are uniform with respect to the slow variables, which are treated as uncertainties. The methodology is illustrated on small power systems leading to much improved results in estimating the stability region, critical clearing times as compared to traditional methods. As a by-product, it gives the required theoretical support to justify and to correct the heuristic approaches used in power system stability analysis literature.

This comment points out several mathematical errors in the proof of Therorem 3, and gives the correct expression of $B_{3}$ .

In this paper a reduced-order model for control design applications of a seventh-order two-time-scale discrete-time model of synchronous machines is developed. The numerical simulation is performed on synchronous machines, and the simulation results obtained using the iterative separation of time-scales provide improved accuracy of the reduced-order models, compared with that of the same obtained via the classical quasi-steady-state (qss) technique.

In this work we focus on a stochastic optimization based approach to make distributed routing and server management decisions in the context of large-scale, geographically distributed data centers, which offers significant potential for exploring power cost reductions. Our approach considers such decisions at different time scales and offers provable power cost and delay characteristics. The utility of our approach and its robustness are also illustrated through simulation-based experiments under delay tolerant workloads.

A design method to achieve robust stability and performance criteria for discrete two‐time‐scale systems controlled by low‐order observer‐based compensators is proposed. Sufficient conditions for stability and performance robustness are established by an easy extension of the small gain theorem. The theoretical analysis is illustrated by a numerical example.

This article deals with the trajectory tracking of linear plants with sampled and delayed outputs. A class of sampled-data controllers with two time-scales is proposed which allows sampled tracking with a delay twice as much as that of the plant output. Numerical simulation results are presented to demonstrate the effectiveness of the proposed approach.

This paper examines the problem of designing a robust H1 state-feedback controller for a class of nonlinear two-time scale systems with Markovian Jumps described by a Takagi-Sugeno (TS) fuzzy model. Based on a linear matrix inequality (LMI) approach, LMI-based sufficient conditions for the uncertain Markovian jump nonlinear two-time scale systems to have an H1 performance are derived. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard nonlinear two-time scale systems. A numerical example is provided to illustrate the design developed in this paper. Keywords—TS fuzzy, Markovian jumps, LMI, Two-time scale systems.

This chapter is concerned with jump diffusions involving Markovian switching regimes. In the models, there are a finite set of regimes or configurations and a switching process that dictates which regime to take at any given instance. At each time t, once the configuration is determined by the switching process, the dynamics of the system follow a jump-diffusion process. It evolves until the next jump takes place. Then the post-jump location is determined and the process sojourns in the new location following the evolution of another jump-diffusion process and so on. The entire system consists of random switches and jump-diffusive motions.

A decentralized model predictive controller applicable for some systems which exhibit different dynamic characteristics in different channels was presented in this paper. These systems can be regarded as combinations of a fast model and a slow model, the response speeds of which are in two-time scale. Because most practical models used for control are obtained in the form of transfer function matrix by plant tests, a singular perturbation method was firstly used to separate the original transfer function matrix into two models in two-time scale. Then a decentralized model predictive controller was designed based on the two models derived from the original system. And the stability of the control method was proved. Simulations showed that the method was effective.

Increasing demands on the fidelity of simulations for real-time and high-fidelity simulations are stressing the capacity of modern processors. New integration techniques are required that provide maximum efficiency for systems that are parallelizable. However many current techniques make assumptions that are at odds with non-cascadable systems. A new serial multi-step/multi-rate integration algorithm for dual-timescale continuous state systems is presented which applies to these systems, and is extended to a parallel multi-step/multi-rate algorithm. The superior performance of both algorithms is demonstrated through a representative example.

Theobjective is thedevelopmentandevaluationofa fast and reliable,multiple-timescalealgorithmfor the inverse simulation of rotorcraft maneuvering tasks. A recent, two-timescale approach to the solution of inverse problems of aircraft motion represents the background for devising a technique that accounts for speciŽ c issues of rotorcraft dynamics such as the large effects of the fast, primarymomentgenerating controls on the slow dynamicsassociated to the vehicle trajectory and the system being frequently nonminimumphase. Accurate solutions are obtained for the inverse simulations of the fast and slow reduced-order systems because the quasi-steady-state values of the fast controls are considered in the slow timescale. High-amplitudeoscillations in the control inputs, revealed in previous applications of helicopter inverse simulation, are interpreted as due to the presence of nonobservablemotions and are ruled out by the multiple timescale approach. The results show that the expected computer time reduction is realized, that the well-known difŽ culties of inverse methods for Ž nding feasible solutions at convergence are practically eliminated, and, Ž nally, that steady-state  ight conditions are accurately recovered at the end of the prescribed maneuvers.

This paper will present controller designs for two—time scale systems with aunified approach using delta operators that guarantee stability with a knownH°° norm bound and also allow the incorporation of reliability with respect tofailures in some control channels. Here, sensor (or measurement) outages casewill be studied. By such a unified approach, the procedure and result for thesolution are shown simpler and more accurate than those of discrete systemusing shift operators.Key Words: Reliable Control, two-time scale, H°° norm, delta operator, shiftoperator, unified approach, finite word length. 1. INTRODUCTION To facilitate solving the two-time scale problem, technique of matrix diagonalization, which can be found in [1], is essential.Kokotovic et al.[2}, made pioneering contributions to solve the problemsfor the singularly perturbed systems. Also extended works of this field havebeen done by Luse[14] and Naidu[3]. For the H°° control in the frequency domain has been pioneered by

The first SLAM II network simulation model for Juvenile Detention Management in Washington State was built in 1989 by the Washington State Office of the Administrator for the Courts. The juvenile detention process can be viewed as a complicated queuing system from the juvenile's arrival at the detention facility to final release from the facility. The detention process is simulated through the use of RESOURCEs, AWAITs and GATEs that control the movement of the juveniles entering and leaving the system. The goal of this research is to help judges and court administrators apply Operations Research to court management and to examine how parameters as well as structure in the simulation model influence the detention system operation. For example, how does varying the arrival rate, the service rate, or service style affect bed usage and judge time in a detention center? Or, how do changes in legal procedures effect the structure of the model and detention facilities' operations.

Two-time-scale decoupling is adopted to obtain slow and fast subsystems. Based on these subsystems, a multirate composite control strategy is derived. It is shown how a recently presented multirate scheme can be modified to handle slow, nonzero mean disturbances, by introducing time incremental models. The improvements with the given multirate control law are illustrated by an example. >

This paper deals with two time scale systems which contain products of fast variables in the slow sub-system. A nonlinear transformation to reduce the fast product terms to order epsilon is presented. A synchronous machine example with this two time scale property is also given.