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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 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 proposes a two-time scale simulation approach of both short-term and long-term dynamics in power systems. The short-term transient period is analyzed by a full-time scale simulation while a quasi-steady state model of the power system is used to simulate long-term dynamics. The proposed method is inspired of Singular Perturbation theory to model the interaction between the short-term and long-term dynamics. Based on this interaction, a suitable criterion to determine when the quasi-steady state model of a power system can be considered as a uniform approximation of the full-time scale model is proposed, which also determines the appropriate switching time between models. The proposed method is demonstrated on the 10-machine, 39-bus New England system and a 46-machine, 190-bus equivalent Mexican system.

We consider the problem of large deviations for a two-time-scale reflected diffusion process, possibly with delays in the dynamical terms. The Dupuis-Ellis weak convergence approach is used. It is perhaps the most intuitive and simplest for the problems of concern. The results have applications to the problem of approximating optimal controls for two-time-scale systems via use of the averaged equation.

International audience; A temporal multiscale modelling applied to fatigue damage evolution in cortical bone is presented. Microdamage accumulation in cortical bone, ensued from daily activities, leads to impaired mechanical properties, in particular by reducing the bone stiffness and inducing fatigue. However, bone damage is also known as a stimulus to bone remodelling, whose aim is to repair and generate new bone, adapted to its environment. This biological process by removing fatigue damage seems essential to the skeleton lifetime. As daily activities induce high frequency cycles (about 10,000 cycles a day), identifying two-time scale is very fruitful: a fast one connected with the high frequency cyclic loading and a slow one related to a quasi-static loading. A scaling parameter is defined between the intrinsic time (bone lifetime of several years) and the high frequency loading (few seconds). An asymptotic approach allows to decouple the two scales and to take into account history effects (Guennouni and Aubry in CR Acad Sci Paris Ser II 20:1765-1767, 1986). The method is here applied to a simple case of fatigue damage and a real cortical bone microstructure. A significant reduction in the amount of computation time in addition to a small computational error between time homogenized and non homogenized models are obtained. This method seems thus to give new perspectives to assess fatigue damage and, with regard to bone, to give a better understanding of bone remodelling.

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.

We use the two-time scale subordination in order to describe dynamical processes in continuous media with a long-term memory. Our consideration touches two physical examples in detail. First we study a temporal evolution of the species concentration for the trapping reaction in which a diffusing reactant is surrounded by a sea of randomly moving traps. The analysis is based on the random-variable formalism of anomalous diffusive processes. We find that the empirical trapping-reaction law, according to which the reactant concentration decreases in time as a product of an exponential and a stretched exponential function, can be explained by the two-time scale subordination of random processes. Another example is connected with a state equation for continuous media with memory. If the pressure and the density of a medium are subordinated in two different random processes, then the ordinary state equation becomes fractional with two time scales. This allows one to arrive at the state equation of Bagley-Torvik type., 15 pages

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.

We study a two-time-scale system of jump-diffusion stochastic differential equations. The main goal is to study the convergence rate of the slow components to the effective dynamics. The convergence established here is in the strong sense, i. e., uniformly in time. For the ergodicity assumptions, we use the existence of a Lyapunov function to control the return times. This assumption is weaker than the one- sided Lipschitz condition, frequently used for deriving rates.

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 deals with the problem of H ? control of linear two-time scale systems. The authors' attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain --- in the L 2 sense --- over different values of the inputs. One of the advantagesof this approach is that --- unlike in the conventional approaches of the order reduction --- the reduced-order system still keeps some information of the `deleted' subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (?) in the two-time scale systems by using linear fractional transformations and structured singular values (?) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard ?-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ?-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H ? controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.

The present paper focuses on the two time scale dynamics generated by 2D polynomial noninvertible maps T of (Z0 - Z2) and (Z1 - Z3 - Z1) types. This symbolism, specific to noninvertible maps, means that the phase plane is partitioned into zones Zk, where each point possesses the k real rank-one preimages. Of special interest here is the structure of slow and fast motion sets of such maps. The formation mechanism of a stable invariant close curve through the interaction of fast and slow dynamics, as well as its transformation into a canard are studied. A few among the plethora of chaotic attractors and chaotic transients produced by such maps are described as well.

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 decomposition of singularly perturbed systems and also the frequency domain decomposition 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 1 norm.

A new method to obtain approximate solutions by placing the only poles of the slow subsystem for the two-time-scale aircraft dynamic systems. The two kinds of approximate solutions are obtained by a matrix block diagonalization. One is called the uncorrected solution, and the other is called the corrected solution. The former has an error of , and the latter has an error of . Of course, both solutions are robust enough even though they are reduced solutions. The excellence of the proposed method is illustrated by an numerical example of an aircraft longitudinal dynamics.

This note is concerned with the robust stability problem of uncertain two-time-scale systems, where the only information available for system uncertainties is their norm upper bounds. The stability bound /spl epsiv//sup */ is derived via a state transformation and the constructive use of a Lyapunov function. Finally, examples are given to illustrate the applications of the proposed results.

Accepted: 2003-02-28, 資料番号: SA1003987000

We investigate the evolution of a system composed of N non-interacting point particles of mass m in a container divided into two chambers by a movable adiabatic piston of mass M≫m. Using a two-time-scale perturbation approach in terms of the small parameter α=2m/(M+m), we show that the evolution towards thermal equilibrium proceeds in two stages. The first stage is a fast, deterministic, adiabatic relaxation towards mechanical equilibrium. The second stage, which takes place at times $$\mathcal{O}$$ (M), is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A very simple equation is derived which shows that in the second stage, the position of the piston is given by X M (t)= L[1/2−ξ(αt)] where the function ξ is independent of M. Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.

The present paper has two main purposes. The first is to provide an explicit state-space solution of the robust stabilization problem with respect to the NLCF for two-time scale systems. In this construction only the solutions of the control and of the filtering Riccati equations associated with the slow and with the fast dynamics of the nominal plant are used. These solutions do not depend upon the singular perturbation and therefore all computations are well-conditioned. The second purpose of the paper is to investigate the case when an k -independent robust controller can be used for the robust stabilization of a two-time scale nominal system. Two constructions of such controllers are developed under some additional assumptions. Numerical examples illustrate the theoretical developments.

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.

The problem of sensor noise rejection in digital control systems with nonlinear plant is discussed. The design method based on the formation of two-time-scale motions in the closed loop system is used. It has been shown that if a sufficient time-scale separation between the fast and slow modes in the discussed closed loop system and stability of fast modes are provided then slow modes have the desired form. The relationships for the control law parameters calculation in accordance with the requirement to high-frequency noise rejection are obtained. Finally, an example with simulation results is presented.

In this paper, we address the dynamic modeling of reactive distillation processes, where the presence of fast reaction or momentum transport induces two time scale dynamics. We present a systematic modeling framework that allows identifying the source of time Scale multiplicity and deriving nonlinear non-stiff models of the slow dynamics.

The problems of the robust digital controller design and control accuracy analysis are discussed. The design method based on the formation of two-time scale motions in the closed loop system is used. If a sufficient time-scale separation between the fast and slow modes in the discussed closed loop system and stability of fast modes are provided then slow modes have the desired form. Thus the output transient performance indices are insensitive to external disturbances. The relationships for control accuracy analysis and choice of the sampling period are obtained. Finally, an example with simulation results is presented.

New regulations for emission control require the improvement of the system composed by spark ignition internal combustion engine and three-way catalytic converter (TWC). In particular, an important problem is to minimize harmful emissions during the transient warm-up phase where the TWC is not working yet and, hence, a large amount of pollutants are emitted in the air. Toward this goal we present a dynamical thermo-chemical TWC model simple enough for the design and test of warm-up control strategies. The model is obtained through an asymptotic approximation of a more detailed model, i.e., by letting the adsorption coefficient between gas and substrate tend to infinity. Further, we present a fast integration algorithm based partly on a “method of lines” space-discretization, partly on the “method of characteristics” for “quasi linear” hyperbolic partial differential equations, the separation being allowed by a two time scale analysis of the system. The model has been identified, through a purposely designed genetic algorithm, and validated on experimental data.

This note considers the problem of robustness of discrete two-time-scale systems. Robustness is first analyzed for a nominal system as the general e-bound problem which characterizes an upper bound of the singular perturbation parameter. The stability robustness is then extended to handle the problem of systems subject to parameter variations. For any singular perturbation parameter bounded above by that obtained from the e-bound problem, a stability robustness bound with respect to parameter variations is derived. The results include both unstructured and structured variations.

In this paper, the H 2 guaranteed cost output feedback control for uncertain singularly perturbed systems is addressed. The design procedure shows how the fullorder uncertain system can be stabilized for small value of G > 0 avoiding high dimensionality and ill-conditioning. It is shown that when necessary and sufficient conditions formulated for the slow and fast subsystems are satisified, a singularly perturbed controller which stabilizes the full-order system is obtained. The solution of the H 2 guaranteed cost problem is searched in the set of such two time scale controllers.

The problem of forming desired output transients for discrete-time system is discussed under assumption of incomplete information about varying parameters of the system and unknown external disturbances. The main result of the paper is the procedure to analyse of fast and slow motions in the discrete-time control system with small parameter where control system is designed by dynamic contraction method. It is shown, that if a sufficient time-scale separation between the fast and slow modes in the closed loop system and stability of fast motions are provided then in the closed loop system the output transient performance indices are insensitive to parameter variations of the plant and external disturbances. Finally, in order to guaranteed cost control an universal digital control law for a sampled-data system with small sampling period is received and numerical example is presented.

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

Under the effect of strong genetic drift, it is highly probable to observe gene fixation or gene loss in a population, shown by singular peaks on a potential landscape. The genetic drift-induced noise gives rise to two-time-scale diffusion dynamics on the bipeaked landscape. We find that the logarithmically divergent (singular) peaks do not necessarily imply infinite escape times or biological fixations by iterating the Wright-Fisher model and approximating the average escape time. Our analytical results under weak mutation and weak selection extend Kramers's escape time formula to models with B (Beta) function-like equilibrium distributions and overcome constraints in previous methods. The constructed landscape provides a coherent description for the bistable system, supports the quantitative analysis of bipeaked dynamics, and generates mathematical insights for understanding the boundary behaviors of the diffusion model.

This paper focuses on the synthesis of well-conditioned multivariable robust controllers for a broad class of multi-input multi-output two-time-scale nonlinear systems with time-varying uncertain variables, modeled within the framework of singular perturbations. The proposed controller stabilizes the fast dynamics, guarantees boundedness of trajectories, and ensures that the ultimate discrepancy between the outputs and the external reference inputs in the closed-loop system can be made arbitrarily small by an appropriate choice of controller parameters, provided that the singular perturbation parameter is sufficiently small. The developed control method is tested, through simulations, on a fluidized catalytic cracking reactor.

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.

The indirect method of solving optimal control problems requires the solution of a Hamiltonian boundary-value problem (HBVP). Geometric properties of Hamiltonian systems and the notion of a dichotomy guide the development of a solution approach for HBVPs that involve two distinct time scales and boundary layers.

This paper describes a fundamental study into modeling the effects of frequency variation on network and shunt impedances in long term power system stability simulations. The focus is on extreme post-fault situations where significant generation-load mismatch causes wide excursions in power system frequency. A novel application of singular perturbations is presented. This method is compared with the "constant reactance" technique, the "variable reactance" method in which the network reactances are varied with system frequency, and a method in which the network transients are fully simulated. Stability simulations for an interconnected two-area power system employing the different modeling techniques show that a two time scale analysis using the singular perturbation method is a feasible technique for accurate prediction of long term post fault system behaviour.

In this paper, we study an averaging principle for stochastic FitzHugh-Nagumo system with different time scales driven by cylindrical Wiener processes and Poisson jumps, where the slow equation is non-autonomous and the fast equation is autonomous case. Under suitable assumptions, we show that the slow component mean-square strongly converges to the solution of the corresponding averaging equation, and the rate of the convergence as a by-product is also affirmed. Finally, we give some open problems which are derived from this paper.

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.

A two-stage observer-based controller design of discrete two-time-scale (DTTS) systems is presented in this paper. The necessary and sufficient conditions for this kind of controlled system are all addressed. The robust slow-observer-based (SOB) controller is investigated in detail. For a fast sampling rate model, prediction type observer-based controller and SOB controller are proposed. Both prediction type and current estimation type observer-based controller and SOB controllers are developed for a slow sampling rate model of DTTS systems. A twin-engine aviation aircraft is utilized to illustrate the derived schemes.

International audience; The aim of this article is to derive a simplified sedimentation model thanks to an asymptotic analysis. We consider a two time scale erosion process due to tidal effects and we show that the approximation at the first order can model bed-load transport well. To this end, the simplified model is validated through numerical tests (evolution of a dune submitted to tidal effects in the ocean, run up near the coast) and compared to direct simulations that are very expensive in terms of computation time.

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.

This paper concerns two-time-scale single-input single-output nonlinear processes. For such processes, we address and solve the problem of synthesizing well-conditioned state feedback control laws that induce a well-characterized input/output behavior and guarantee stability of the closed-loop system.

International audience; This paper addresses the state estimation of two-time scale nonlinear systems with an unknown input multi-observer (UIMO). In order to design such an observer, the nonlinear system presented in a singular perturbed form is transformed into an equivalent multiple model with unmeasurable premise variables (UPV) affected by unknown inputs (UI). Then an observer is built and the stability analysis of the state reconstruction error is performed by using the Lyapunov method that leads to the resolution of linear matrix inequalities (LMIs). The performances of the proposed estimation method are highlighted through the application to a wastewater treatment plant (WWTP) model.

Ruppert [Handbook of Sequential Analysis, B. K. Ghosh and P. K. Sen, eds., Marcel Dekker, New York, 1991] and Polyak [Automat. Remote Control, 51 (1990), pp. 937–946] independently introduced the averaging principle for Robbins–Monro's algorithm in order to construct an algorithm that is asymptotically efficient, and that is quickly updated. Their averaging principle has been extended to other algorithms (including Kiefer–Wolfowitz's algorithm), leading to algorithms that are asymptotically efficient, but whose updatings are less straightforward. We introduce the use of two-time-scale stochastic approximation algorithms to construct a large class of asymptotically efficient algorithms. This method generalizes the averaging principle. Moreover, for the algorithms that do not behave as Robbins–Monro's algorithm, it allows one to provide algorithms whose updating is as straightforward as that of the averaged Robbins–Monro's algorithm.

In the quarter car model, the active suspension with nonlinear characteristics is investigated. An active suspension model is made by two time scale modeling. Two models have relation to handling and the ride comfort respectively. A new control method that is based on the singular perturbation approach is proposed. Our control law for the full model combines with control laws for slow and fast submodels. The simulation shows that the new design approach is very useful.

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.