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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.

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.

We consider a dynamic average consensus problem where a group of agents is required to track the average of their time-varying inputs. We assume that the inputs are sinusoidal with a single unknown frequency. We develop a distributed two-time-scale estimator that estimates the unknown frequency and achieves average consensus of the inputs. We establish input-to-state (ISS) properties of the estimator using two-time-scale averaging theory. We also explore benefits of fusing the agents' individual frequency estimates. Using a simulation example, we demonstrate that the average consensus performance is significantly improved by incorporating the fused frequency estimates into the estimator.

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 deals with fractional robust control of a two time scale system. The CRONE design methodology is applied on a singularly perturbed system. The system is decomposed on two subsystems: slow and fast. Then, the CRONE method is applied to obtain a fractional controller ensuring robustness towards gain variations. A numerical example is given to show the efficiency of the proposed method.

This paper discusses the foundations and proposes a general framework to decompose the stability analysis of electrical power systems in time-scales. Assuming that the electrical power system model possesses two-time-scale features, the stability analysis of the original two-time-scale power system model is decomposed into the stability analysis of two simpler models: the fast and slow subsystems. Conditions to guarantee that stability of these simpler subsystems implies stability of the complete model are given. The general framework is suitable for stability analysis of power systems in a time frame that can range from a few seconds, which is typically the scenario of transient stability analysis, to a few minutes, which is typical of mid-term stability analysis, in which voltage stability problems are commonly encountered.

This paper proposes a new Two-Time Scale (TTS) BCU method, and reports the first ever known use of the TTS-CUEP method in a multi-machine power system. The proposed TTS-BCU method is a numerical algorithm to correctly compute the slow and fast CUEPs of the TTS-CUEP method. It also provides a more robust algorithm to compute the CUEP of the original system.

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.

In this paper, the problem of robust multi-objective control design with an H ∞ constrain is studied for a class of linear two-time scale systems. The design is based on a new modelling approach under the assumption of norm-boundedness of the fast dynamics. In this method, a portion of the fast dynamics is treated as a norm-bounded perturbation in the design by its maximum possible gain. In this view, the problem of robust multi-objective control design is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. One illustrative example is used to demonstrate the validity of the proposed approach.

This paper presents the foundations of controlling unstable equilibrium point (CUEP) theory for stability assessment of two-time-scale power system models. A conceptual two-time-scale CUEP method is developed. The two-time scale CUEP method is faster and more robust as compared to the traditional CUEP method. Furthermore, more insight into power system dynamics and less conservative estimates of the stability region and critical clearing time are obtained.

In this paper, the problem of H ∞ model reduction for singularly perturbed control systems is investigated. A new algorithm based on the slow-fast decomposition of the original system is suggested to obtain the reduced-order model. The associated model reduction error meets a prescribed H ∞ norm bound constraint and the two-time-scale property of the original singularly perturbed control system is retained in the reduced-order model. A numerical example is presented to show the effectiveness of the model reduction method.

We investigate the effect of the parameter mismatches on the oscillation frequency of a popular voltage controlled oscillator, whose behavior is schematized by a system of two mutually coupled oscillators. A perturbation analysis of this system is carried out using the method of two time scale. The existence of synchronization phenomena leading to a locking frequency is shown, which significantly differs from the natural frequencies of the tanks. Circuit simulations confirm the presence of a synchronized oscillation which is consistent with the prediction of the presented analysis.

In this paper we propose a robust redesign technique which recovers the trajectories of a nominal control design in the presence of additive input uncertainties. We design a high-gain filter and employ the fast variables arising from this filter in the feedback control law to cancel the effect of the uncertainties in the plant. Using singular perturbation theory we prove that the trajectories of the redesigned system approach those of the nominal system when the filter gain is increased. We illustrate the redesign by applying it to two physically motivated examples.

A new order model reduction technique for two-time-scale systems is presented in this paper. This reduction technique provides the advantage of forcing the dominant poles of the original system to be the dominant poles of the reduced order model. The reduction technique is performed based on the two-time-scale system reduction technique, while the dominant eigenvalue preservation is achieved by the implementation of a neural network and the use of the matrix reducibility concept. The eigenvalues of the reduced order model are selected as a subset of the full order model eigenvalues. Simulation and comparison with other techniques for a third order system along with its results are presented as part of this paper.

Active pneumatic vibration isolators have been widely studied as the use of precision equipment increases. This paper focuses on the robust two-time-scale control design of an active pneumatic vibration isolator using the singular perturbation method and time delay control (TDC). The singular perturbation method, or more generally, the two-time-scale approach to feedback design is used to cope with the ill-conditioning of the pneumatic vibration isolator; TDC is used to overcome inevitable modeling errors in more accurate physical model of pneumatic vibration isolator and unknown disturbance such as base excitation. Because the proposed controller in this paper is based on more accurate physical model than rough model, accurate control is feasible. Numerical experiment demonstrates the effectiveness of the proposed robust two-time-scale control design.

The nonlinearity and coupling of the aerospace vehicle dynamics can be linearized and decoupled by well known feedback linearization techniques. But, this technique fails to handle the uncertainties in the system due to aerodynamic coefficients. This problem becomes acute when an aerospace vehicle tries to attain high angles of attack during flight. Significant research works are devoted in accordance to this. In this paper, a new robust nonlinear controller structure using feedback linearization technique is presented. The design is carried out in two time scale separation. The fast dynamics (inner body rate loop) and slow dynamics (outer latax loop) are controlled separately by two nonlinear controllers. The performances of the design in terms of robustness, minor coupling between the longitudinal motion and lateral motion, stability and guaranteed tracking are shown in a very realistic 6-DOF simulation.

The singular perturbation approach to the analysis and design of two-time-scale systems is applied to the H/sub /spl infin// filtering problem. A sequential procedure is described to decompose the problem into slow and fast subproblems. In this procedure, a fast filter is designed first using the fast model; then a slow filter is designed under a condition on the value of the filter at infinity. A composite filter is formed as the parallel connection of the fast with the strictly proper part of the slow filter. The asymptotic validity of the composite filter is established.

Motivated by the problem of the asymptotic behavior of a class of actor-critic learning algorithms recently proposed by Konda and Tsitsiklis, the almost sure asymptotic properties of two time-scale stochastic approximation are analyzed under violated Kushner-Clark noise conditions and very general stability conditions. The analysis covers the algorithms with additive noise, as well as those with nonadditive noise. The algorithms with additive noise are analyzed for the case where the noise is state-dependent. The analysis of the algorithms with non-additive state-dependent noise is carried out for the case where the noise is a Markov chain controlled by the algorithm states, while the algorithms with non-additive exogenous noise are analyzed for the case where the noise is correlated and satisfies uniform or strong mixing conditions. The obtained results characterize the robustness of two time-scale stochastic approximation towards the violation of the Kushner-Clark noise condition. Moreover, they cover a fairly broad class of highly non-linear two time-scale stochastic approximation algorithms, including the actor-critic learning algorithms proposed by Konda and Tsitsiklis.

The problem of designing a robust controller to solve a tracking control problem for improving plasma characteristics in Reactive Ion Etching Systam is studied. The presented design methodology is based on the construction of a two-time scale motions of the closed-loop system. It has been shown that a sufficiently small perturbation parameter associated with the dynamical controller consisting of high order derivatives of the output signal, induces a two time-scale separation of the fast and slow modes in the closed-loop system. Stability conditions imposed on the fast and slow models can ensure that the full-order closed-loop system achieves the desired properties so that the output transient performances are insensitive to parameter variations and external disturbances.

A nodel aggregation algorithm has been proposed to compute the slow aggregate of linearized power system networks. The algorithm decomposes the aggregation process into recursive two-node aggregations. Recursive formulas for aggregation are derived, replacing the inversion of a possibly large, dense matrix. To reduce the amount of computation and storage, the algorithm can utilize sparsity in choosing the nodes to be aggregated at each step. The algorithm reduces to the nodal elimination algorithm when one of the nodes to be aggregated is static. >

The multi-time-scale systems are subject of a great number of research works concerning their reduction and simplification, in order to elaborate the control. The singular perturbations theory is perfectly adapted to systems presenting multi-time-scale. The multi-time-scale property is due to the presence of slow and fast dynamic phenomena. The use of singular perturbations permits the decomposition of the global system in two sub-systems giving prominence to slow and fast parts. So, the determination of the control from these systems is faster than its calculation from the global one. >

Two-time scale (TTS) distributions are introduced. For a class of stable systems, it is shown that every TTS distribution has a two-frequency-scale (TFS) Laplace transform. Conversely, it is shown that the impulse response of any stable TFS transfer function, and hence any stable (standard) singularly perturbed system, can be characterized in terms of a stable TTS distribution. A time domain decomposition for TTS distributions is obtained which parallels the slow and fast decompositions of singularly perturbed systems and also the frequency domain decompositions of TFS transfer functions. It is shown that every stable TTS distribution can be decomposed in terms of two simpler distributions represented in two different time scales. A composite distribution is constructed from these two which approximates the TTS distribution arbitrarily closely in the L/sub 1/ norm. >

We deal with the design of robust controllers for a linear time invariant uncertain system exhibiting two time scale properties for the entire range of parameter variations. The uncertain system has been decomposed into slow and fast uncertain subsystems. The stabilizing controllers for the lower order subsystems are designed and a composite controller is obtained from them. This composite controller is shown to stabilize the higher order uncertain system, if applied to it. A numerical example illustrates the procedure. >

This paper deals with the synthesis of continuous state feedback laws for a class of uncertain two-time-scale systems to induce a prespecified input/output response in the nominal system, achieve arbitrary degree of asymptotic attenuation of the effect of uncertainty on the output and guarantee boundedness of the trajectories of the closed-loop system. >

Sampled data systems having two frequency scale behavior are introduced. Properties of these systems are characterized in terms of their transfer matrices. An approximation theorem is given by which a two frequency scale sampled data (TFS-SD) system can be approximated in terms of its low and high frequency parts. The relationship between a TFS-SD system and its time domain counterpart, a two time scale sampled data (TTS-SD) system, is established. >

The objective of this paper is to design a compensator by a singular perturbation approach for flexible structures subjected to moving vibratory system (e.g. bridge vibration system with moving vehicle). The proposed design of the controller includes a composite control strategy derived by the singular perturbation approach. In order to overcome the problem of lack of full state availability, a compensator is applied to the controller. Using this compensator, the vibration amplitude of the elastic beams can be reduced and the performance of the dynamic system improved. Although the research idea originated from exploration of the dynamic characteristics of the bridge system, the findings of this paper can be applied to various mechanical systems such as space structures, ballistic systems, gantry crane, and other dynamic systems with moving loads.

The problem of pole placement in a specified circular region for discrete two-time scale system is considered. An algorithm for pole assignment in a specified circular region using a bilinear transformation onto the unit circle is presented for discrete two-time scale systems. We decompose the discrete regulator problem into two subsystem discrete regulator problem and then they are solved for optimal pole assignment in specified circular region. A fifth-order steam power system model is worked out to demonstrate the analytical development.

We present a linear quadratic design for singularly perturbed systems by a unified approach with /spl delta/ operators. We convert a two-time scale discrete system to a two-time-scale unified system and solve the problem and compare the results of the two systems.

Control of underactuated mechanical systems often leads to unstable internal dynamics, which can be handled by resorting to prediction when the system bandwidth is small. The paper considers systems with a high bandwidth and proposes a two-time-scale controller for decoupling the system while ensuring internal stability. A toy helicopter in which the speed of the propellers is manipulated to vary the aerodynamic force is taken up as a case study.

The paper concerns the reduced order modeling of nonlinear two-time-scale power systems by approximate slow subsystems. In particular, we investigate the issue if the full order power system model is conservative, whether the slow subsystems will also be conservative. For linearised models, the slow subsystems obtained from truncated asymptotic expansions are symmetrisable, implying that energy functions can be construted from a Lagrangian. For the nonlinear systems with linearized fast dynamics slow subsystems obtained from the first three terms of an asymptotic expansion shown to possess energy functions. For higher order approximate slow subsystems, this property does not seem to hold.

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.

This paper deals with the linear model-following control (LMFC) of a class of discrete systems which have the two-time-scale (TTS) property. It is shown that, under certain sufficient conditions, the proposed reduced-order LMFC can force the slow subsystem of the plant to follow the stable model asymptotically. Both state feedback and observer-based feedback control are considered. An example [8] is used to demonstrate the derived schemes. The design algorithms of the controllers are summarized in the conclusion.

Stabilization problem of discrete two-time-scale systems via block implementation is considered in this paper. Block dynamical equations for the original system and slow and fast subsystems are examined. It is shown that the eigenvalue assignment problem for subsystems, both in state and output feedback cases, can be exploited to stabilize the full system by adding certain extra compensators.

A formulation is presented for designing optimal output feedback regulators of fixed order. The formulation exploits an observer canonical form to represent the compensator dynamics. The formulation precludes the use of direct feedback of the plant output. The major contribution of the paper is to present how this approach to compensator design may be extended to output feedback of singularly perturbed systems. The result is that two compensators are designed that operate on separate time scales, and which may be implemented with different sampling rates.