11 research outputs found
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
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
Inferring hidden states in Langevin dynamics on large networks: Average case performance
We present average performance results for dynamical inference problems in
large networks, where a set of nodes is hidden while the time trajectories of
the others are observed. Examples of this scenario can occur in signal
transduction and gene regulation networks. We focus on the linear stochastic
dynamics of continuous variables interacting via random Gaussian couplings of
generic symmetry. We analyze the inference error, given by the variance of the
posterior distribution over hidden paths, in the thermodynamic limit and as a
function of the system parameters and the ratio {\alpha} between the number of
hidden and observed nodes. By applying Kalman filter recursions we find that
the posterior dynamics is governed by an "effective" drift that incorporates
the effect of the observations. We present two approaches for characterizing
the posterior variance that allow us to tackle, respectively, equilibrium and
nonequilibrium dynamics. The first appeals to Random Matrix Theory and reveals
average spectral properties of the inference error and typical posterior
relaxation times, the second is based on dynamical functionals and yields the
inference error as the solution of an algebraic equation.Comment: 20 pages, 5 figure
Gossip and Distributed Kalman Filtering: Weak Consensus under Weak Detectability
The paper presents the gossip interactive Kalman filter (GIKF) for
distributed Kalman filtering for networked systems and sensor networks, where
inter-sensor communication and observations occur at the same time-scale. The
communication among sensors is random; each sensor occasionally exchanges its
filtering state information with a neighbor depending on the availability of
the appropriate network link. We show that under a weak distributed
detectability condition:
1. the GIKF error process remains stochastically bounded, irrespective of the
instability properties of the random process dynamics; and
2. the network achieves \emph{weak consensus}, i.e., the conditional
estimation error covariance at a (uniformly) randomly selected sensor converges
in distribution to a unique invariant measure on the space of positive
semi-definite matrices (independent of the initial state.)
To prove these results, we interpret the filtered states (estimates and error
covariances) at each node in the GIKF as stochastic particles with local
interactions. We analyze the asymptotic properties of the error process by
studying as a random dynamical system the associated switched (random) Riccati
equation, the switching being dictated by a non-stationary Markov chain on the
network graph.Comment: Submitted to the IEEE Transactions, 30 pages
Modelling and Adaptive Control; Proceedings of an IIASA Conference, Sopron, Hungary, July 1986
One of the main purposes of the workshop on Modelling and Adaptive Control at Sopron, Hungary, was to give an overview of both traditional and recent approaches to the twin theories of modelling and control which ultimately must incorporate some degree of uncertainty. The broad spectrum of processes for which solutions of some of these problems were proposed was itself a testament to the vitality of research on these fundamental issues. In particular, these proceedings contain new methods for the modelling and control of discrete event systems, linear systems, nonlinear dynamics and stochastic processes