Robust output stabilization: improving performance via supervisory control

We analyze robust stability, in an input-output sense, of switched stable systems. The primary goal (and contribution) of this paper is to design switching strategies to guarantee that input-output stable systems remain so under switching. We propose two types of {\em supervisors}: dwell-time and hysteresis based. While our results are stated as tools of analysis they serve a clear purpose in design: to improve performance. In that respect, we illustrate the utility of our findings by concisely addressing a problem of observer design for Lur'e-type systems; in particular, we design a hybrid observer that ensures ``fast'' convergence with ``low'' overshoots. As a second application of our main results we use hybrid control in the context of synchronization of chaotic oscillators with the goal of reducing control effort; an originality of the hybrid control in this context with respect to other contributions in the area is that it exploits the structure and chaotic behavior (boundedness of solutions) of Lorenz oscillators.Comment: Short version submitted to IEEE TA

Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter

We consider the problem of an ensemble Kalman filter when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. To limit the variance of the latter poorly resolved variables we derive a variance limiting Kalman filter (VLKF) in a variational setting. We analyze the variance limiting Kalman filter for a simple linear toy model and determine its range of optimal performance. We explore the variance limiting Kalman filter in an ensemble transform setting for the Lorenz-96 system, and show that incorporating the information of the variance of some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.Comment: 32 pages, 11 figure

Model error and sequential data assimilation. A deterministic formulation

Data assimilation schemes are confronted with the presence of model errors arising from the imperfect description of atmospheric dynamics. These errors are usually modeled on the basis of simple assumptions such as bias, white noise, first order Markov process. In the present work, a formulation of the sequential extended Kalman filter is proposed, based on recent findings on the universal deterministic behavior of model errors in deep contrast with previous approaches (Nicolis, 2004). This new scheme is applied in the context of a spatially distributed system proposed by Lorenz (1996). It is found that (i) for short times, the estimation error is accurately approximated by an evolution law in which the variance of the model error (assumed to be a deterministic process) evolves according to a quadratic law, in agreement with the theory. Moreover, the correlation with the initial condition error appears to play a secondary role in the short time dynamics of the estimation error covariance. (ii) The deterministic description of the model error evolution, incorporated into the classical extended Kalman filter equations, reveals that substantial improvements of the filter accuracy can be gained as compared with the classical white noise assumption. The universal, short time, quadratic law for the evolution of the model error covariance matrix seems very promising for modeling estimation error dynamics in sequential data assimilation

Aggressive shadowing of a low-dimensional model of atmospheric dynamics

Predictions of the future state of the Earth's atmosphere suffer from the consequences of chaos: numerical weather forecast models quickly diverge from observations as uncertainty in the initial state is amplified by nonlinearity. One measure of the utility of a forecast is its shadowing time, informally given by the period of time for which the forecast is a reasonable description of reality. The present work uses the Lorenz 096 coupled system, a simplified nonlinear model of atmospheric dynamics, to extend a recently developed technique for lengthening the shadowing time of a dynamical system. Ensemble forecasting is used to make forecasts with and without inflation, a method whereby the ensemble is regularly expanded artificially along dimensions whose uncertainty is contracting. The first goal of this work is to compare model forecasts, with and without inflation, to a true trajectory created by integrating a modified version of the same model. The second goal is to establish whether inflation can increase the maximum shadowing time for a single optimal member of the ensemble. In the second experiment the true trajectory is known a priori, and only the closest ensemble members are retained at each time step, a technique known as stalking. Finally, a targeted inflation is introduced to both techniques to reduce the number of instances in which inflation occurs in directions likely to be incommensurate with the true trajectory. Results varied for inflation, with success dependent upon the experimental design parameters (e.g. size of state space, inflation amount). However, a more targeted inflation successfully reduced the number of forecast degradations without significantly reducing the number of forecast improvements. Utilized appropriately, inflation has the potential to improve predictions of the future state of atmospheric phenomena, as well as other physical systems.Comment: 14 pages, 16 figure

Fast Ensemble Smoothing

Smoothing is essential to many oceanographic, meteorological and hydrological applications. The interval smoothing problem updates all desired states within a time interval using all available observations. The fixed-lag smoothing problem updates only a fixed number of states prior to the observation at current time. The fixed-lag smoothing problem is, in general, thought to be computationally faster than a fixed-interval smoother, and can be an appropriate approximation for long interval-smoothing problems. In this paper, we use an ensemble-based approach to fixed-interval and fixed-lag smoothing, and synthesize two algorithms. The first algorithm produces a linear time solution to the interval smoothing problem with a fixed factor, and the second one produces a fixed-lag solution that is independent of the lag length. Identical-twin experiments conducted with the Lorenz-95 model show that for lag lengths approximately equal to the error doubling time, or for long intervals the proposed methods can provide significant computational savings. These results suggest that ensemble methods yield both fixed-interval and fixed-lag smoothing solutions that cost little additional effort over filtering and model propagation, in the sense that in practical ensemble application the additional increment is a small fraction of either filtering or model propagation costs. We also show that fixed-interval smoothing can perform as fast as fixed-lag smoothing and may be advantageous when memory is not an issue

Multifractal characterization of stochastic resonance

We use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals. To characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence. We use wavelet transform modulus maxima method for the singularity spectrum computations. It is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic (stochastic or chaotic) input signals. We show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization. This fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also. Moreover, our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence. We use also the proposed approach to characterize the noise-enhanced dynamics of a coupled stochastic neurons model.Comment: 10 pages, 21 EPS-figures, RevTe

On the concept of complexity in random dynamical systems

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In random dynamical system, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent computed considering two nearby trajectories evolving under the same randomness. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models.Comment: 35 pages, LaTe