A late time accelerated FRW model with scalar and vector fields via Noether symmetry

We study the evolution of a three-dimensional minisuperspace cosmological model by the Noether symmetry approach. The phase space variables turn out to correspond to the scale factor of a flat Friedmann-Robertson-Walker (FRW) model, a scalar field with potential function $V(\phi)$ with which the gravity part of the action is minimally coupled and a vector field of its kinetic energy is coupled with the scalar field by a coupling function $f(\phi)$. Then, the Noether symmetry of such a cosmological model is investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generator of the desired symmetry. We explicitly calculate the form of the coupling function between the scalar and the vector fields and also the scalar field potential function for which such symmetry exists. Finally, by means of the corresponding Noether current, we integrate the equations of motion and obtain exact solutions for the scale factor, scalar and vector fields. It is shown that the resulting cosmology is an accelerated expansion universe for which its expansion is due to the presence of the vector field in the early times, while the scalar field is responsible of its late time expansion.Comment: 9 pages, 2 figures, typos corrected, Refs. adde

A Stieltjes transform approach for analyzing the RLS adaptive Filter

Although the RLS filter is well-known and various algorithms have been developed for its implementation, analyzing its performance when the regressors are random, as is often the case, has proven to be a formidable task. The reason is that the Riccati recursion, which propagates the error covariance matrix, becomes a random recursion. The existing results are approximations based on assumptions that are often not very realistic. In this paper we use ideas from the theory of large random matrices to find the asymptotic (in time) eigendistribution of the error covariance matrix of the RLS filter. Under the assumption of a large dimensional state vector (in most cases n = 10-20 is large enough to get quite accurate predictions) we find the asymptotic eigendistribution of the error covariance for temporally white regressors, shift structured regressors, and for the RLS filter with intermittent observations

A Stieltjes transform approach for studying the steady-state behavior of random Lyapunov and Riccati recursions

In this paper we study the asymptotic eigenvalue distribution of certain random Lyapunov and Riccati recursions that arise in signal processing and control. The analysis of such recursions has remained elusive when the system and/or covariance matrices are random. Here we use transform techniques (such as the Stieltjes transform and free probability) that have gained popularity in the study of large random matrices. While we have not yet developed a full theory, we do obtain explicit formula for the asymptotic eigendistribution of certain classes of Lyapunov and Riccati recursions, which well match simulation results. Generalizing the results to arbitrary classes of such recursions is currently under investigation

On the steady-state performance of Kalman filtering with intermittent observations for stable systems

Many recent problems in distributed estimation and control reduce to estimating the state of a dynamical system using sensor measurements that are transmitted across a lossy network. A framework for analyzing such systems was proposed in and called Kalman filtering with intermittent observations. The performance of such a system, i.e., the error covariance matrix, is governed by the solution of a matrix-valued random Riccati recursion. Unfortunately, to date, the tools for analyzing such recursions are woefully lacking, ostensibly because the recursions are both nonlinear and random, and hence intractable if one wants to analyze them exactly. In this paper, we extend some of the large random matrix techniques first introduced in to Kalman filtering with intermittent observations. For systems with a stable system matrix and i.i.d. time-varying measurement matrices, we obtain explicit equations that allow one to compute the asymptotic eigendistribution of the error covariance matrix. Simulations show excellent agreement between the theoretical and empirical results for systems with as low as n = 10, 20 states. Extending the results to unstable system matrices and time-invariant measurement matrices is currently under investigation