H 2 And H ∞ Filtering Design Subject To Implementation Uncertainty

This paper presents new filtering design procedures for discrete-time linear systems. It provides a solution to the problem of linear filtering design, assuming that the filter is subject to parametric uncertainty. The problem is relevant, since the proposed filter design incorporates real world implementation constraints that are always present in practice. The transfer function and the state space realization of the filter are simultaneously computed. The design procedure can also handle plant parametric uncertainty. In this case, the plant parameters are assumed not to be exactly known but belonging to a given convex and closed polyhedron. Robust performance is measured by the H 2 and H ∞ norms of the transfer function from the noisy input to the filtering error. The results are based on the determination of an upper bound on the performance objectives. All optimization problems are linear with constraint sets given in the form of LMI (linear matrix inequalities). Global optimal solutions to these problems can be readily computed. Numerical examples illustrate the theory. © 2005 Society for Industrial and Applied Mathematics.442515530Gevers, M., Li, G., (1993) Parametrizations in Control, Estimation and Filtering Problems, , Springer-Verlag, LondonWilliamson, D., Finite wordlength design of digital Kalman filters for state estimation (1985) IEEE Trans. Automat. Control, 30, pp. 930-939Williamson, D., Kadiman, K., Optimal finite wordlength linear quadratic regulators (1989) IEEE Trans. Automat. Control, 34, pp. 1218-1228Liu, K., Skelton, R.E., Grigoriadis, K., Optimal controllers for finite wordlength implementation (1992) IEEE Trans. Automat. Control, 37, pp. 1294-1304Hwang, S.Y., Minimum uncorrelated unit noise in state-space digital filtering (1977) IEEE Trans. Acoustics Speech Signal Process, 25, pp. 273-281Amit, G., Shaked, U., Minimization of roundoff errors in digital realizations of Kalman filters (1989) IEEE Trans. Acoustics Speech Signal Process, 37, pp. 1980-1982De Oliveira, M.C., Skelton, R.E., Synthesis of controllers with finite precision considerations (2001) Digital Controller Implementation and Fragility: A Modern Perspective, pp. 229-251. , R. S. H. Istepanian and J. F. Whidborne eds., Springer-Verlag, New YorkKeel, L.H., Bhattacharyya, S.P., Robust, fragile or optimal (1997) IEEE Trans. Automat. Control, 42, pp. 1098-1105Keel, L.H., Bhattacharyya, S.P., Authors' reply to: "Comments on 'Robust, fragile or optimal' " by P. M. Mäkilä (1998) IEEE Trans. Automat. Control, 43, p. 1268Dorato, P., Non-fragile controller design: An overview (1998) Proceedings of the 1998 American Control Conference, 5, pp. 2829-2831. , Philadelphia, IEEE, Piscataway, NJFamularo, D., Dorato, P., Abdallah, C.T., Haddad, W.H., Jadbabaie, A., Robust non-fragile LQ controllers: The static state feedback case (2000) Internat. J. Control, 73, pp. 159-165Yang, G.H., Wang, J.L., Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty (2001) IEEE Trans. Automat. Control, 46, pp. 343-348Haddad, W.M., Corrado, J.R., Robust resilient dynamic controllers for systems with parametric uncertainty and controller gain variations (2000) Internat. J. Control, 73, pp. 1405-1423Keel, L.H., Bhattacharyya, S.P., Stability margins and digital implementation of controllers (1998) Proceedings of the 1998 American Control Conference, 5, pp. 2852-2856. , (Philadelphia), IEEE, Piscataway, NJGeromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Trans. Signal Process, 47, pp. 168-175Nesterov, Y., Nemirovskii, A., (1994) Interior-Point Polynomial Algorithms in Convex Programming, , SIAM, PhiladelphiaGeromel, J.C., Bernussou, J., Garcia, G., De Oliveira, M.C., H 2 and H ∞ robust filtering for discrete-time linear systems (2000) SIAM J. Control Optim., 38, pp. 1353-1368Geromel, J.C., De Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM J. Control Optim., 41, pp. 700-711De Oliveira, M.C., Bernussou, J., Geromel, J.C., A new discrete-time robust stability condition (1999) Systems Control Lett., 37, pp. 261-265Sayed, A.H., A framework for state-space estimation with uncertain models (2001) IEEE Trans. Automat. Control, 46, pp. 998-1013Balakrishnan, V., Huang, Y., Packard, A., Doyle, J.C., Linear matrix inequalities in analysis with multipliers (1994) Proceedings of the 1994 American Control Conference, 2, pp. 1228-1232. , Baltimore, MD, IEEE, Piscataway, NJGeromel, J.C., Peres, P.L.D., Bernussou, J., On a convex parameter space method for linear control design of uncertain systems (1991) SIAM J. Control Optim., 29, pp. 381-40

Filtering and Smoothing with Score-Driven Models

We propose a methodology for filtering, smoothing and assessing parameter and filtering uncertainty in misspecified score-driven models. Our technique is based on a general representation of the well-known Kalman filter and smoother recursions for linear Gaussian models in terms of the score of the conditional log-likelihood. We prove that, when data are generated by a nonlinear non-Gaussian state-space model, the proposed methodology results from a first-order expansion of the true observation density around the optimal filter. The error made by such approximation is assessed analytically. As shown in extensive Monte Carlo analyses, our methodology performs very similarly to exact simulation-based methods, while remaining computationally extremely simple. We illustrate empirically the advantages in employing score-driven models as misspecified filters rather than purely predictive processes.Comment: 33 pages, 5 figures, 6 table

Robust filtering with randomly varying sensor delay: The finite-horizon case

Copyright [2009] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we consider the robust filtering problem for discrete time-varying systems with delayed sensor measurement subject to norm-bounded parameter uncertainties. The delayed sensor measurement is assumed to be a linear function of a stochastic variable that satisfies the Bernoulli random binary distribution law. An upper bound for the actual covariance of the uncertain stochastic parameter system is derived and used for estimation variance constraints. Such an upper bound is then minimized over the filter parameters for all stochastic sensor delays and admissible deterministic uncertainties. It is shown that the desired filter can be obtained in terms of solutions to two discrete Riccati difference equations of a form suitable for recursive computation in online applications. An illustrative example is presented to show the applicability of the proposed method

Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation

Copyright [2001] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.We investigate the robust filter design problem for a class of nonlinear time-delay stochastic systems. The system under study involves stochastics, unknown state time-delay, parameter uncertainties, and unknown nonlinear disturbances, which are all often encountered in practice and the sources of instability. The aim of this problem is to design a linear, delayless, uncertainty-independent state estimator such that for all admissible uncertainties as well as nonlinear disturbances, the dynamics of the estimation error is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are proposed to guarantee the existence of desired robust exponential filters, which are derived in terms of the solutions to algebraic Riccati inequalities. The developed theory is illustrated by numerical simulatio

Optimal investment and hedging under partial and inside information

This article concerns optimal investment and hedging for agents who must use trading strategies which are adapted to the filtration generated by asset prices, possibly augmented with some inside information related to the future evolution of an asset price. The price evolution and observations are taken to be continuous, so the partial (and, when applicable, inside) information scenario is characterised by asset price processes with an unknown drift parameter, which is to be filtered from price observations. We first give an exposition of filtering theory, leading to the Kalman-Bucy filter. We outline the dual approach to portfolio optimisation, which is then applied to the Merton optimal investment problem when the agent does not know the drift parameter of the underlying stock. This is taken to be a random variable with a Gaussian prior distribution, which is updated via the Kalman filter. This results in a model with a stochastic drift process adapted to the observation filtration, and which can be treated as a full information problem, and an explicit solution to the optimal investment problem is possible. We also consider the same problem when the agent has noisy knowledge at time $0$ of the terminal value of the Brownian motion driving the stock. Using techniques of enlargement of filtration to accommodate the insider's additional knowledge, followed by filtering the asset price drift, we are again able to obtain an explicit solution. Finally we treat an incomplete market hedging problem. A claim on a non-traded asset is hedged using a correlated traded asset. We summarise the full information case, then treat the partial information scenario in which the hedger is uncertain of the true values of the asset price drifts. After filtering, the resulting problem with random drifts is solved in the case that each asset's prior distribution has the same variance, resulting in analytic approximations for the optimal hedging strategy

Optimal investment and hedging under partial and inside information

This article concerns optimal investment and hedging for agents who must use trading strategies which are adapted to the filtration generated by asset prices, possibly augmented with some inside information related to the future evolution of an asset price. The price evolution and observations are taken to be continuous, so the partial (and, when applicable, inside) information scenario is characterised by asset price processes with an unknown drift parameter, which is to be filtered from price observations. We first give an exposition of filtering theory, leading to the Kalman-Bucy filter. We outline the dual approach to portfolio optimisation, which is then applied to the Merton optimal investment problem when the agent does not know the drift parameter of the underlying stock. This is taken to be a random variable with a Gaussian prior distribution, which is updated via the Kalman filter. This results in a model with a stochastic drift process adapted to the observation filtration, and which can be treated as a full information problem, and an explicit solution to the optimal investment problem is possible. We also consider the same problem when the agent has noisy knowledge at time $0$ of the terminal value of the Brownian motion driving the stock. Using techniques of enlargement of filtration to accommodate the insider's additional knowledge, followed by filtering the asset price drift, we are again able to obtain an explicit solution. Finally we treat an incomplete market hedging problem. A claim on a non-traded asset is hedged using a correlated traded asset. We summarise the full information case, then treat the partial information scenario in which the hedger is uncertain of the true values of the asset price drifts. After filtering, the resulting problem with random drifts is solved in the case that each asset's prior distribution has the same variance, resulting in analytic approximations for the optimal hedging strategy

Robust filtering for bilinear uncertain stochastic discrete-time systems

Copyright [2002] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper deals with the robust filtering problem for uncertain bilinear stochastic discrete-time systems with estimation error variance constraints. The uncertainties are allowed to be norm-bounded and enter into both the state and measurement matrices. We focus on the design of linear filters, such that for all admissible parameter uncertainties, the error state of the bilinear stochastic system is mean square bounded, and the steady-state variance of the estimation error of each state is not more than the individual prespecified value. It is shown that the design of the robust filters can be carried out by solving some algebraic quadratic matrix inequalities. In particular, we establish both the existence conditions and the explicit expression of desired robust filters. A numerical example is included to show the applicability of the present method

Robust filtering under randomly varying sensor delay with variance constraints

Copyright [2004] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper deals with a new filtering problem for linear uncertain discrete-time stochastic systems with randomly varying sensor delay. The norm-bounded parameter uncertainties enter into the system matrix of the state space model. The system measurements are subject to randomly varying sensor delays, which often occur in information transmissions through networks. The problem addressed is the design of a linear filter such that, for all admissible parameter uncertainties and all probabilistic sensor delays, the error state of the filtering process is mean square bounded, and the steady-state variance of the estimation error for each state is not more than the individual prescribed upper bound. We show that the filtering problem under consideration can effectively be solved if there are positive definite solutions to a couple of algebraic Riccati-like inequalities or linear matrix inequalities. We also characterize the set of desired robust filters in terms of some free parameters. An illustrative numerical example is used to demonstrate the usefulness and flexibility of the proposed design approach

Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noises

Copyright [2002] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, a robust finite-horizon Kalman filter is designed for discrete time-varying uncertain systems with both additive and multiplicative noises. The system under consideration is subject to both deterministic and stochastic uncertainties. Sufficient conditions for the filter to guarantee an optimized upper bound on the state estimation error variance for admissible uncertainties are established in terms of two discrete Riccati difference equations. A numerical example is given to show the applicability of the presented method