Journals:
Y. Sokratous, F. Rezaei and C.D. Charalambous, “Nonlinear Estimation for a Class of Systems,” IEEE Transactions on Information Theory, Vol. 55, No.4, pp. 1930-1938, April 2009
Abstract
This paper considers nonlinear estimation problems for classes of models, and employs relative entropy to describe the uncertainty classes. Two optimization problems are formulated on general Banach spaces, and their solutions are sought: 1) when the transition probability between the signal to be estimated X and the measurement Y or stochastic kernel is unknown, and 2) when the joint probability induced by the random variables (RVs) X, Y is unknown. For both problems, the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution. The results include existence of the optimal measures using weak convergence techniques, and properties associated with the estimate of the true distribution. Classical examples are chosen to illustrate the applicability of the results.
N.U. Ahmed and C.D. Charalambous, “Measurement Strategy for Nonlinear Filtering,” SIAM Journal on Control and Optimization, Vol.45, No.2, pp.519-531, April 2006
Abstract
In this paper we consider the question of an optimal measurement strategy for nonlinear filtering. Given a set of measurement and observation options, the problem faced by a designer is to choose the option that gives the best nonlinear filter. We develop the necessary conditions of optimality whereby one can determine the best filter. Read More: https://epubs.siam.org/doi/10.1137/040610428
C.D. Charalambous, R.J. Elliott, and V. Krishnamurthy, “Conditional moment generating functions for integrals and stochastic integrals,’’ SIAM Journal on Control and Optimization, Vol. 42, No. 5, pages1578-1603, 2004
Abstract
In this paper we present two methods for computing filtered estimates for moments of integrals and stochastic integrals of continuous-time nonlinear systems. The first method utilizes recursive stochastic partial differential equations. The second method utilizes conditional moment generating functions. An application of these methods leads to the discovery of new classes of finite-dimensional filters. For the case of Gaussian systems the recursive computations involve integrations with respect to Gaussian densities, while the moment generating functions involve differentiations of parameter dependent ordinary stochastic differential equations. These filters can be used in Volterra or Wiener chaos expansions and the expectation-maximization algorithm. The latter yields maximum-likelihood estimates for identifying parameters in state space models. Read More: https://epubs.siam.org/doi/abs/10.1137/S036301299833327X?journalCode=sjcodc
N.U. Ahmed and C.D. Charalambous, “Filtering for linear systems driven by fractional Brownian motion,’’ SIAM Journal on Control and Optimization, Vol. 41, No.1, pages 313-330, 2002
Abstract
In this paper we study continuous time filtering for linear multidimensional systems driven by fractional Brownian motion processes. We present the derivation of the optimum linear filter equations which involve a pair of functional-differential equations giving the optimum error covariance (matrix-valued) functions and the optimum filter. These equations are the appropriate substitutes of the matrix-Riccati differential equation arising in classical Kalman filtering. However, the optimum filter has the classical appearance, and, as usual, it is driven by the increments of theobserved process. Read More: https://epubs.siam.org/doi/abs/10.1137/S0363012900368715?mobileUi=0
S. Dey and D. Charalambous, “Discrete-time risk-sensitive filters for non-Gaussian conditions and their ergodicity properties,” Asian Journal of Control, vol.3, no.4, pp.262-271, 2001
Abstract
In this paper, we study asymptotic stability properties of risk‐sensitive filters with respect to their initial conditions. In particular, we consider a linear time‐invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk‐sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk‐sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non‐Gaussian initial conditions, we derive the expression for the risk‐sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon‐Nikodym derivative, we show that a suboptimal risk‐sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk‐sensitive filter for non‐Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims.
S. Dey and D. Charalambous, “On asymptotic stability of continuous-time risk-sensitive filters with respect to their initial conditions,” Systems and Control Letters, vol.41, no.1, pp.9-18, 2000
Abstract
In this paper, we consider the problem of risk-sensitive filtering for continuous-time stochastic linear Gaussian time-invariant systems. In particular, we address the problem of forgetting of initial conditions. Our results show that suboptimal risk-sensitive filters initialized with arbitrary Gaussian initial conditions asymptotically approach the optimal risk-sensitive filter for a linear Gaussian system with Gaussian but unknown initial conditions in the mean square sense at an exponential rate, provided the arbitrary initial covariance matrix results in a stabilizing solution of the (H∞-like) Riccati equation associated with the risk-sensitive problem. More importantly, in the case of non-Gaussian initial conditions, a suboptimal risk-sensitive filter asymptotically approaches the optimal risk-sensitive filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non-Gaussian density and a slow growth condition satisfied by a certain Radon–Nikodym derivative.
C.D. Charalambous and R.J.Elliott, “New explicit filters and smoothers for diffusions with nonlinear drift and measurements,’’ Systems and Control Letters, vol. 33, no. 2, pages 89-103, March 1998
Abstract
The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 ⩽ τ ⩽ t} is given by the conditional mean E[x(t)|y(τ); 0 ⩽ τ ⩽ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫0tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation ϖ/ϖxf(x) + f(x)2 = f(x)2 = αx2 + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫0tx(s) ds + ∫0t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 ⩽ τ ⩽ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.
J.L. Hibey and C.D. Charalambous, “Performance analysis for a changepoint problem,’’ IEEE Transactions on Automatic Control, Vol. 44, Issue 8, pages 1628-1638, August 1999
Abstract
Nonlinear stochastic differential equations are used to model a version of the changepoint problem. State estimates of the minimum mean-square-error-type are used in likelihood-ratio tests to detect the time of change. System performance is evaluated by deriving theoretical expressions for the false alarm probability and a special case of the miss error probability. The approach uses a Fokker-Planck equation.
J.L. Hibey and C.D. Charalambous, “Conditional densities for continous-time nonlinear hybrid systems with applications to fault detection,’’ IEEE Transactions on Automatic Control, Vol. 44, Issue 11, pages. 2164-2169, November 1999
Abstract
Continuous-time nonlinear stochastic differential state and measurement equations, all of which have coefficients capable of abrupt changes at a random time, are considered; finite-state jump Markov chains are used to model the changes. Conditional probability densities, which are essential in obtaining filtered estimates for these hybrid systems, are then derived. They are governed by a coupled system of stochastic partial differential equations. When the Q matrix of the Markov chain is either lower or upper diagonal, it is shown that the system of conditional density equations is finite-dimensional computable. These findings are then applied to a fault detection problem to compute state estimates that include the failure time.
C.D. Charalambous, S. Dey, R.J. Elliott, “New finite-dimensional risk-sensitive filters: small-noise limits,” IEEE Transactions on Automatic Control, Vol. 43, Issue 10, pages 1424-1429, October 1998
Abstract
The paper is concerned with continuous-time nonlinear risk-sensitive filters. It is shown that for large classes of nonlinearities entering both the dynamics and measurements, these filters are finite-dimensional generalizations of the Benes filters. Specific examples are discussed. The small noise limiting analog is discussed using change of probability measures.
Conferences:
C. D Charalambous and P. Stavrou, “Optimization of Directed Information and Relations to Filtering Theory,” in Proceedings of the European Control Conference, Strasbourg France, July 24-27, 2014
Abstract
In this paper we generalize the relation between nonanticipative Rate Distortion Function (RDF) and filtering theory, to processes which are affected by the reproduction process, by utilizing the topology of weak convergence of probability measures. Specifically, this generalization is established via an optimization on the space of conditional distributions of the so-called directed information, subject to a fidelity constraint. Existence of the optimal reproduction distribution of the general nonanticipative RDF is shown, while the closed form expression of the optimal reproduction distribution is obtained for nonstationary processes. The expression of the optimal reproduction conditional distribution is recursively computed, backward in time. Finally, the realization procedure of the general nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described.
P.A. Stavrou and C.D. Charalambous, “On the Relation of Nonanticipative Rate Distortion Function and Filtering Theory,” in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, July 17-19, 2013
Abstract
In this paper the relation between nonanticipative rate distortion function (RDF) and Bayesian filtering theory is investigated using the topology of weak convergence of probability measures on Polish spaces. The relation is established via an optimization on the space of conditional distributions of the so-called directed information subject to fidelity constraints. Existence of the optimal reproduction distribution of the nonanticipative RDF is shown, while the optimal nonanticipative reproduction conditional distribution for stationary processes is derived in closed form. The realization procedure of nonanticipative RDF which is equivalent to joint-source channel matching for symbol-by-symbol transmission is described, while an example is introduced to illustrate the concepts.
Y. Sokratous, C.D. Charalambous and C. Georghiades “Robust Estimation with Applications to Phase and Envelop Estimation in Frequency Selective Wireless Fading Channels,” in 2008 International Symposium on Information Theory, Toronto, Canada, June 6-11, 2008
Abstract
This paper derives robust minimax estimators for a class of uncertain models. The uncertainty is described by a relative entropy constraint between the unknown joint distribution and a fixed nominal joint distribution. The maximization is addressed using variational methods, while the minimization is addressed using the concept of a sufficient statistic, which is an unnormalized version of the a posteriori density. The theory developed is applied to multipath fading wireless channels, to derive minimax envelope and phase estimates. Related results are also derived when the uncertainty shrinks to zero.
Y. Sokratous and C.D. Charalambous “Robust Noncoherent Estimation for a Class of Frequency Wireless Fading Channels,” in 2008 Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, December 9-11, 2008
Abstract
This paper is concerned with the control of a class of dynamical systems over finite capacity communication channels. Necessary conditions for reliable data reconstruction and stability of a class of dynamical systems are derived. The methodology is information theoretic. It introduces the notion of entropy for a class of sources, which is defined as a maximization of the Shannon entropy over a class of sources. It also introduces the Shannon information transmission theorem for a class of sources, which states that channel capacity should be greater or equal to the mini-max rate distortion (maximization is over the class of sources) for reliable communication. When the class of sources is described by a relative entropy constraint between a class of source densities, and a given nominal source density, the explicit solution to the maximum entropy, is given, and its connection to Rényi entropy is illustrated. Furthermore, this solution is applied to a class of controlled dynamical systems to address necessary conditions for reliable data reconstruction and stability of such systems.
A. Kyprianou and C. Mares and C.D. Charalambous and J.E. Mottershead, “Minimum Relative Entropy for Damage Detection and Location,” in 7th International Conference on Damage Assessment of Structures, Torino, Italy, June 25-27, 2007
Abstract
Relative entropy has been employed, as an alternative to other regularization methods, in solving ill-conditioned linear inverse problems. Damage detection when treated as structural modification imparted by the damage leads to a linear inverse problem involving frequency response functions. This problem is amenable to ill-conditioning issues that could arise from the low frequency response values and noisy experiments. This article formulates and solves using the minimum relative entropy method the damage detection and localization problem on a simulated cantilever beam.
C.D. Cralambous and S. Djouadi, “Stochastic Nonlinear Minimax Filtering in Continuous-Time,” in Proceedings of the 40th IEEE Conference on Decision and Control, Vol. 3, pp. 2520-2525, Florida, Orlando, U.S.A., December 4-7, 2001
Abstract
This paper discusses nonlinear stochastic minimax games in which the minimizing player is the state estimate while the maximizing players are square-integrable stochastic disturbances. A pathwise optimization method is considered, and an information state is introduced which is governed by a second-order Hamilton-Jacobi-Bellman (HJB) equation. The HJB equation is subsequently employed to characterize the dissipation properties of the estimator error with respect to the stochastic disturbances and to introduce a certainty equivalence estimator.
N.U. Ahmed and C.D. Charalambous, “Filtering for linear systems driven by fractional Brownian motion,’’ in Proceedings of the 39th IEEE Conference on Decision and Control, Vol.5, pp. 4259-4263, Sydney, Australia, 2000
Abstract
In this paper we study continuous time filtering for linear multidimensional systems driven by fractional Brownian motion processes. We present the derivation of the optimum linear filter equations which involve a pair of functional-differential equations giving the optimum error covariance (matrix-valued) functions and the optimum filter. These equations are the appropriate substitutes of the matrix-Riccati differential equation arising in classical Kalman filtering. However, the optimum filter has the classical appearance, and, as usual, it is driven by the increments of the observed process..
S. Dey and D. Charalambous, “Discrete-time risk sensitive filters with non-Gaussian initial conditions and their ergodic properties,” in Proceedings of the 1999 American Control Conference, Vol. 6, pages 4403-4407, San Diego, California, 1999
Abstract
We study asymptotic stability properties of risk-sensitive filters with respect to their initial conditions. In particular, we consider a linear time-invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk-sensitive filter asymptotically converges to any suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense, provided the incorrect initializing value for the covariance matrix results in a risk-sensitive filter that is asymptotically stable. For non-Gaussian initial conditions, we show that under certain conditions, a suboptimal risk-sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk-sensitive filter for non-Gaussian initial conditions in the mean square sense.
C.D. Charalambous and A. Logothetis, “New Finite-Dimensional Filters for Log-Likehood Gradient, Hessian and Fisher Information Matrices: The Discrete-Time Case,” in proceedings of the 37th IEEE Conference on Decision and Control, Tampa Florida,U.S.A., December 16-18,1998
C. D. Charalambous, A. Logothetis, R.J. Elliott, “Bank Filters for ML parameter estimation via the Expectation-Maximization algorithm: the continuous-time case,” in Proceedings of the 37th IEEE Conference on Decision and Control, Vol, 2, pp. 2317-2322, Tampa, Florida, U.S.A., December 16-18,1998
Abstract
In this paper we consider continuous-time partially observed systems in which the parameters are unknown. We employ conditional moment generating functions of integrals and stochastic integrals to derive new maximum-likelihood (ML) parameter estimates which are required in the implementation of the expectation-maximization algorithm. Each parameter is estimated by a bank of Kalman filters consisting of four statistics: two are the Kalman filter statistics while the remaining two have the structure of the Kalman filter driven by the innovations process.
C. D. Charalambous and R. J. Elliott, “Certain results concerning filtering and control diffusions in small white noice,” in Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, Vol. 3, pp. 2773-2778, December 1997 [invited paper]
Abstract
We examine in detail the binary hypothesis decision and/or estimation problem using a risk-sensitive cost criterion, when the state and observation processes are diffusion signals. We demonstrate the role played by a Feynman-Kac version of the Duncan-Mortensen-Zakai stochastic partial differential equation in making decisions. The question of performance is addressed by relating Chernoff bounds to this Feynman-Kac stochastic equation. We also examine in detail the behaviour of our calculations, in the limit as the covariances of the random inputs tend to zero. The procedure employs large deviations techniques. This approach enables us to establish relations between stochastic and deterministic methods in tackling the binary decision problem. The latter reveals a natural formulation of the binary decision problem in terms of an H/sup /spl infin//-disturbance attenuation framework.
C. D. Charalambous and R. J. Elliott and V. Krishnamurthy, “Conditional Moment generating functions for integrals and stochastic integrals,” in Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, Vol. 4, pp. 3944-3949, December 1997 [invited paper]
Abstract
We present two methods for computing filtered estimates for moments of integrals and stochastic integrals of continuous-time nonlinear systems. The first method utilizes recursive stochastic partial differential equations. The second method utilizes conditional moment generating functions. For the case of Gaussian systems the recursive computations involve integrations with respect to Gaussian densities, while the moment generating functions involve differentiations of parameter dependent ordinary stochastic differential equations. The second method is applied in the expectation maximization algorithm.
C. D. Charlambous, “Conditional Moment generating functions for integrals and stochastic integrals: Maximum likehood estimation,” in Proceedings of the 5th IEEE Mediterranean Conference on Control Systems, Paphos, Cyprus, Abstracts TA3-3, July 21-23, 1997 [invited paper]
C. D. Charlambous, S. Dey, R. J. Elliott, “New finite –dimensional risk-sensitive filters,” in Proceedings of the 1997 American Control Conference, Albuquerque, New Mexico, pp. 2830-2835, June, 1997
Abstract
This paper is concerned with continuous-time nonlinear risk-sensitive filters. It is shown that for large classes of nonlinearities entering both the dynamics and measurements, these filters are finite-dimensional, generalizations of the Benes filters. Examples are given for rational and exponential nonlinearities. The small-noise limiting analog is discussed.
C.D. Charalambous and J.L. Hibey, “Differential geometry and measured transformations applied to nonlinear estimation and control,” Presented in the SIAM Conference on control and its Applications, Minneapolis, Minnesota, Abstracts and authors Index, A36, 1992
