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This note develops robust filters for time-varying uncertain discrete-time systems. The developed filters are based on a data regularization solution and they enforce a minimum state-error variance propoerty. Simulation results confirm their superior performance over other robust filter designs. Index Terms--Convex optimization, least-squares, parametric uncertainty, regularization, robust filter.

This note addresses the problem of robust multiobjective filtering for discrete time-delay systems with mixed stochastic and deterministic uncertainties, in addition to unmodeled nonlinearities. A procedure is developed for the design of linear and exponentially stable tilters with a bounded error variance, exponential rate of decay, and robust performance for the error system. Index Terms--Exponential stability, Kalman tilter, linear matrix inequality, robust tilter, stochastic uncertainties, time-delay systems.

We show that several interesting problems in [H.sup.[infinity]] - filtering, quadratic game theory, and risk sensitive control and estimation follow as special cases of the Krein-space linear estimation theory developed in [1]. We show that all these problems can be cast into the problem of calculating the stationary point of certain second-order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein-space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns.

The authors develop a self-contained theory for linear estimation in Krein spaces. The derivation is based on simple concepts such as projections and matrix factorizations and leads to an interesting connection between Krein space projection and the recursive computation of the stationary points of certain second-order (or quadratic) forms. The authors use the innovations process to obtain a general recursive linear estimation algorithm. When specialized to a state-space structure, the algorithm yields a Krein space generalization of the celebrated Kalman filter with applications in several areas such as [H.sup.[infinity]]-filtering and control, game problems, risk sensitive control, and adaptive filtering.

We present a new approach to the Chandrasekhar recursions and some generalizations thereof. The derivation uses the generalized Schur recursions, which are O([N.sup.2]) recursions for the triangular factorization of N X N matrices having a certain Toeplitz-like displacement structure. It is shown that when the extra structure provided by an underlying state-space model is properly incorporated into the generalized Schur algorithm, it reduces to the Chandrasekhar recursions, which are O(N[n.sup.2]) recursions for estimating the n-dimensional state of a time-invariant (or constant-parameter) system from N measured outputs. It is further noted that the generalized Schur algorithm factors more general structured matrices, and this fact is readily used to extend the Chandrasekhar recursions to a class of time-variant state-space models, special cases of which often arise in adaptive filtering.

We describe a new solution to the four-block problem using the method of generalized Schur analysis. We first reduce the general problem to a simpler one by invoking a coprime factorization with a block-diagonal inner matrix. Then, using convenient spectral factorizations, we are able to parameterize the unknown entry in terms of a Schur-type matrix function, which is shown to satisfy a finite number of interpolation conditions of the Hermite-Fejer type. All possible interpolating functions are then determined via a simple recursive procedure that constructs a transmission-line (or lattice) cascade of elementary J-lossless sections. This also leads to a parameterization of all solutions of the four-block problem in terms of a linear fractional transformation.

We derive a new recursive solution for a general time-variant interpolation problem of the Hermite-Fejer type, based on a fast algorithm for the recursive triangular factorization of time-variant structured matrices. The solution follows from studying the properties of an associated cascade system and leads to a triangular array implementation of the recursive algorithm. The system can be drawn as a cascade of first-order lattice sections, where each section is composed of a rotation matrix followed by a storage element and a tapped-delay filter. Such cascades always have certain blocking properties, which can be made equivalent to the interpolation conditions. We also illustrate the application of the algorithm to problems in adaptive filtering, model validation, robust control, and analytic interpolation theory.

We extend the discrete-time Chandrasekhar recursions for least-squares estimation in constant parameter state-space models to a class of structured time-variant state-space models, special cases of which often arise in adaptive filtering. It can be shown that the much studied exponentially weighted recursive least-squares filtering problem can be reformulated as an estimation problem for a state-space model having this special time-variant structure. Other applications arise in the multichannel and multidimensional adaptive filtering context.

This paper develops a framework for state-space estimation when the parameters of the underlying linear model are subject to uncertainties. Compared with existing robust filters, the proposed filters perform regularization rather than deregularization. It is shown that, under certain stabilizability and detectability conditions, the steady-state filters are stable and that, for quadratically-stable models, the filters guarantee a bounded error variance. Moreover, the resulting filter structures are similar to various (time- and measurement-update, prediction, and information) forms of the Kalman filter, albeit ones that operate on corrected parameters rather than on the given nominal parameters. Simulation results and comparisons with [H.sub.[infinity]], guaranteed-cost, and set-valued state estimation filters are provided. Index Terms--Estimation, guaranteed-cost design, [H.sub.[infinity]]-filtering, Kalman filtering, parametric uncertainty, quadratic stability, regularized least-squares, set-valued estimation, steady-state filter.

In this paper we develop array algorithms for [H.sup.[infinity]] filtering. These algorithms can be regarded as the Krein space generalizations of [H.sup.2] array algorithms, which are currently the preferred method for implementing [H.sup.2] filters. The array algorithms considered include two main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of [H.sup.[infinity]] filters, as these conditions are built into the algorithms themselves. However, since [H.sup.[infinity]] square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the [H.sup.2] case, further investigation is needed to determine the numerical behavior of such algorithms. Index Terms--Array algorithms, estimation, fast algorithms, robustness, [H.sup.[infinity]].

An efficient recursive solution is presented for the one-sided unconstrained tangential interpolation problem. The method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data. The recursive procedure admits a physical interpretation in terms of discretized transmission lines. In this framework the generating system is constructed as a cascade of first-order sections. Singular steps occur only when the input data is contradictory, i.e., only when the interpolation problem does not have a solution. Various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants. The algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general Hermite matrix interpolation problem. A recursive method is proposed to compute a column-reduced generating system that can be used to solve the minimal tangential interpolation problem. Index Terms - Interpolation, matrix decomposition, numerical stability, polynomial matrices, rational functions, rational matrices.