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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.

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.

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

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.

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.

International audience; This paper deals with the problem of the state estimation of a nonlinear system. The system under study is characterized by a two-time scale model. The state estimation is performed by designing a proportional integral observer (PIO) with unknown inputs. In order to design such an observer, the nonlinear system is transformed into an equivalent multiple model form and the fast dynamics are considered as unknown inputs. An application to an ASM1 model of Wastewater Treatment Plants (WWTP) is considered and the obtained numerical results show the performances of the proposed approach.

Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$, or vice-versa (provided of course that $A$, or $B$, is not empty). The number of balls in $B$ after $n$ steps is of order $\sqrt n$, and this number remains essentially the same after $\sqrt n$ further steps. Observe that each ball in the urn $B$ after $n$ steps has a probability bounded away from $0$ and $1$ to be placed back in the urn $A$ after $\sqrt n$ additional steps. So, even though the number of balls in $B$ does not evolve significantly between $n$ and $n+\sqrt n$, the precise contain of urn $B$ does. This elementary observation is the source of an interesting two-time-scale phenomenon which we illustrate using a simple model of fragmentation-coagulation. Inspired by Pitman's construction of coalescing random forests, we consider for every $n\in \mathbb{N}$ a uniform random tree with $n$ vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits subaging in the sense that when it is observed after $k$ steps in the regime $k\sim tn+s\sqrt n$ with $t>0$ fixed, it seems to reach a statistical equilibrium as $n\to\infty$; but different values of $t$ yield distinct pseudo-stationary distributions.

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.

We investigate a general scenario for ``glassy'' or ``jammed'' systems driven by an external, non-conservative force, analogous to a shear force in a fluid. In this scenario, the drive results in the suppression of the usual aging process, and the correlation and response functions become time translation invariant. The relaxation time and the response functions are then dependent on the intensity of the drive and on temperature. We investigate this dependence within the framework of a dynamical closure approximation that becomes exact for disordered, fully-connected models. The relaxation time is shown to be a decreasing function of the drive (``shear thinning'' effect). The correlation functions below the glass transition temperature (Tc) display a two time scales relaxation pattern, similar to that observed at equilibrium slightly above Tc. We also study the violation of the fluctuation dissipation relationship in the driven system. This violation is very reminiscent of the one that takes place in a system aging below Tc at zero drive. It involves in particular the appearance of a two-temperatures regime, in the sense of an effective fluctuation dissipation temperature. Although our results are in principle limited to the closure relations that hold for mean-field models, we argue that a number of the salient features are not inherent to the approximation scheme, and may be tested in experiments and simulations., Version accepted for publication - Physical Review E

Asymptotic behaviour of a two time scale stochastic approximation algorithm is analysed in terms of a related singular ordinary differential equation.

Based on differences in dynamic response times in the crop production process, a decomposition of optimal greenhouse climate management is proposed and evaluated in simulations. Using the favourable results of this decomposition, a hierarchical concept for optimal greenhouse climate management is derived and discussed in view of application in horticultural practice.

International audience; The aim of this paper is to study two-time-scale nonlinear transient models and their associated parameter identification. When it is possible to consider two well-separated time scales, and when the fast component of the applied loading is periodic, a periodic time homogenization scheme, similar to what exists in space homogenization, can be used to derive a homogenized model. A parameter identification process for this latter is then proposed, and applied to an academic example, which allows to show the benefits of such a strategy.