linear filtering for bilinear stochastic differential systems with unknown inputs investigates the problem of state estimation for bilinear stochastic multivariable differential systems in the presence of an additional disturbance whose statistics are completely unknown a linear filter is proposed based on a suitable decomposition of the state of the bilinear system into two components the first one is a computable function of the observations while the second component is estimated via a suitable linear filtering algorithm no a priori information on the disturbance is required for the filter implementation the proposed filter is robust with respect to the unknown input in that the covariance of the estimation error is not affected by such input numerical simulations show the effectiveness of the proposed filter

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

embedded trees estimation of gaussian processes on graphs with cycles graphical models provide a powerful general framework for encoding the structure of large scale estimation problems however the graphs describing typical real world phenomena contain many cycles making direct estimation procedures prohibitively costly in this paper we develop an iterative inference algorithm for general gaussian graphical models it operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph when these subproblems are suitably chosen the algorithm converges to the correct conditional means moreover and in contrast to many other iterative methods the tree based procedures we propose can also be used to calculate exact error variances although the conditional mean iteration is effective for quite densely connected graphical models the error variance computation is most efficient for sparser graphs in this context we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree structured counterparts thanks both to the expressive power of these models and to the efficient inference algorithms developed herein the convergence properties of the proposed tree based iterations are characterized both analytically and experimentally in addition by using the basic tree based iteration to precondition the conjugate gradient method we develop an alternative accelerated iteration that is finitely convergent simulation results are presented that demonstrate this algorithm s effectiveness on several inference problems including a prototype distributed sensing application

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

the linear quadratic optimal regulator for descriptor systems discrete time case abstract in this paper the linear quadratic optimal regulator problem for the discrete time descriptor system ex k 1 ax k bu k where e is in general a singular matrix is investigated the continuous time version of this problem was treated elsewhere the discrete time theory is largely parallel to the continuous time theory but there are several important differences one is that in the discrete time case a fairly general problem can be solved with singular cost matrices the key to the solution as for the continuous time problem is a coordinate system which can be computed by performing a singular value decomposition of e this is called an svd coordinate system the optimization problem is solved by applying hamiltonian minimization to compute the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors in the general singular cost case this trajectory is computed by recursively solving a discrete time riccati equation from the optimal trajectory the optimal descriptor variable feedback gain relating the control to the descriptor variable can be computed several riccati difference equations are derived all of which have the same solution this solution can be used to compute the optimal trajectory and cost for non singular problems even in the finite horizon case the optimal feedback gain can be computed non recursively by solving an eigenvalue eigenvector problem formulated from the system parameters in the svd coordinate system the optimal gain for the infinite horizon steady state problem on the other hand can be computed from the solution of an eigenvector problem involving the untransformed system parameters

kalman filtering and riccati equations for descriptor systems a general formulation of a discrete time filtering problem for descriptor systems is considered it is shown that the nature of descriptor systems leads directly to the need to examine singular estimation problems using a dual approach to estimation the authors derive a so called 3 block form for the optimal filter and a corresponding 3 block riccati equation for a general class of time varying descriptor models which need not represent a well posed system in that the dynamics may be either over or under constrained specializing in the time invariant case they examine the asymptotic properties of the 3 block filter and in particular analyze in detail the resulting 3 block algebraic riccati equation the noncausal nature of discrete time descriptor dynamics implies that future dynamics may provide some information about the present state a modified form for the descriptor kalman filter that takes this information into account is presented

boundary recursion for descriptor variable systems it is shown that if an n dimensional descriptor variable system satisfies the properties of solvability and controllability the family of solutions is n dimensional and the set of two point boundary values that are consistent with the system can be expressed as the solutions to a linear system of equations involving only the boundary values this equation is termed a boundary mapping equation and it provides a compact representation of the system the boundary mapping generalizes the state transition matrix associated with state variable systems the author presents the boundary mapping theory describes the recursive processes required and applies both to some special cases including general linear time invariant systems and adjoint systems

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

internal stabilizability of some diffusive models abstract we consider a single species population dynamics model with age dependence spatial structure and a nonlocal birth process arising as a boundary condition we prove that under a suitable internal feedback control one can improve the stabilizability results given in kubo and langlais j math biol 29 1991 363 378 this result is optimal our proof relies on an identical stabilizability result of independent interest for the heat equation that we state and prove in section 3

optimal filtering for systems with unknown inputs an optimal filtering formula is derived for linear time varying discrete systems with unknown inputs by making use of the well known innovations filtering technique the derivation is an extension of a new observer design method for time invariant deterministic systems with unknown inputs the systems under consideration have the most general form the derived optimal filter has a similar form to the standard kalman filter with some modified covariance and gain matrices

brief paper robust filtering for uncertain linear discrete time descriptor systems this paper is concerned with the problem of robust filtering for uncertain linear discrete time descriptor systems the matrices of the system state space model are uncertain belonging to a given polytope a linear matrix inequality based method is proposed for designing a linear stationary filter that guarantees the asymptotic stability of the estimation error and gives an optimized upper bound on the asymptotic error variance irrespective of the parameter uncertainty the proposed robust filter design is based on a parameter dependent lyapunov function which is shown to outperform parameter independent ones

h_ infty unbiased filtering for linear descriptor systems via lmi this note concerns the optimal h infin unbiased filtering problem for linear descriptor systems it unifies the design of reduced order minimal order and full order filters for continuous and discrete time systems necessary and sufficient conditions for the solvability of the problem are obtained in terms of a set of linear matrix inequalities lmis the parametrization of all h infin unbiased filters is presented application to standard systems with or without unknown inputs is given a numerical example is given to show the applicability of the presented approach

on kalman filter solution of space time interpolation the approximate kalman filtering algorithm presented previously see ibid vol 3 p 773 88 nov 1994 for image sequence processing can introduce unacceptable negative eigenvalues in the information matrix and can have degraded performance in some applications the improved algorithm presented in this note guarantees a positive definite information matrix leading to more stable filter performance

state estimation for descriptor systems via the unknown input filtering method this paper addresses the optimal unbiased minimum variance state estimation of descriptor systems in the framework of unknown input filtering it is shown that any descriptor system can be equivalently transformed into a standard state space system with unknown inputs the existing globally optimal state estimators can then be readily used to facilitate the optimal filter design this research highlights the relationship between descriptor state estimation and unknown input filtering for standard state space systems with unknown inputs a direct application to the state estimation of descriptor systems with unknown inputs via the proposed results is also addressed an illustrative example is given to show the novelty of the proposed results

high level primitives for recursive maximum likelihood estimation this paper proposes a high level language constituted of a small number of primitives and macros for describing recursive maximum likelihood ml estimation algorithms this language is applicable to estimation problems involving linear gaussian models or processes taking values in a finite set the use of high level primitives allows the development of highly modular ml estimation algorithms based on simple numerical building blocks the primitives which correspond to the combination of different measurements the extraction of sufficient statistics and the conversion of the status of a variable from unknown to observed or vice versa are first defined for linear gaussian relations specifying mixed deterministic stochastic information about the system variables these primitives are used to define other macros and are illustrated by deriving new filtering and smoothing algorithms for linear descriptor systems the primitives are then extended to finite state processes and used to implement the viterbi ml state sequence estimator for a hidden markov model

h filter design for a class of nonlinear discrete time singular systems this paper deals with the h filter design problem for a class of discrete time lipschitz nonlinear singular systems the approach is based on the parameterisation of the obtained algebraic constraints from the estimation errors sufficient conditions for the existence of the filters which guarantee stability and the worst case filter error energy over all bounded energy disturbances is minimised are given the method also unifies the design for the full order reduced order minimal order filters for discrete time systems the presented results are less conservative than those existing in the literature this is due to the introduction of new slack variables application to systems with unknown input is presented via a numerical example

information filtering and array algorithms for descriptor systems subject to parameter uncertainties this paper presents information filters in riccati recursions and in array algorithms for descriptor systems subject to parameters uncertainties the filters are developed in filtered and predicted forms the inversion of the state matrix is avoided in the new information recursive formulas therefore it turns out clear that the invertibility of the state matrix usually considered in the state space information recursions is not necessary a numerical example is provided to illustrate the performance of the proposed robust array algorithms

a causal realization theory part i linear deterministic systems we study acausal linear systems their controllability and observability properties and the weighting patterns that they realize a complete classification is given of all minimal real analytic realizations of a given weighting pattern and of all minimal autonomous realizations of a stationary weighting pattern

existence uniqueness and stability of solutions to singular linear quadratic optimal control problems we consider general continuous and discrete time linear quadratic optimal control problems where the dynamics of the system is a linear differential or difference algebraic equation with constant coefficient matrices and the cost functionals are defined by infinite integrals or sums these problems occur in numerous applications in many different fields such as economics operations research electrical engineering model and system design biology and chemistry since they form two of the simplest optimal control problems and they are also used as linear approximations for many nonlinear problems which are much harder to solve we extend some of the known results to these general problems

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

optimal filtering for systems with unknown inputs via the descriptor kalman filtering method this paper addresses globally optimal unbiased minimum variance state estimation for systems with unknown inputs that affect both the system and the output with the descriptor kalman filtering method it is shown that directly applying the conventional descriptor kalman filter dkf to the considered problem may not yield the globally optimal solution because the unknown input vector may not be estimable to remedy this problem three approaches are proposed to facilitate optimal filter design the transformed approach uses some input and output transformations the untrammeled approach does not require any transformations and the augmented approach reconstructs the unknown input dynamics then three 5 block forms of the extended dkf 5 block edkf are derived as globally optimal state estimators in the sense that the first two filters are equivalent to the recently developed extended recursive three step filter and the third is equivalent to the conventional augmented state kalman filter the relationship between the proposed edkfs and the existing results in the literature is addressed simulation results are given to illustrate the usefulness of the proposed filters

kalman filtering for general discrete time linear systems recursive state estimation problems for explicit and implicit time invariant linear systems both for systems with and without unknown inputs can be formulated as a single problem usually referred to as descriptor kalman filtering solutions to this problem have been proposed in the literature however these solutions either neglect possible contributions of future dynamics to the current estimate or make unnecessary assumptions on the structure of the system in this paper the authors propose a solution to this problem which leads to a constructive method lifting these unnecessary assumptions this method uses a generalization of the shuffle algorithm

a robust null space method for linear equality constrained state estimation we present a robust null space method for linear equality constrained state space estimation exploiting a degeneracy in the estimator statistics an orthogonal factorization is used to decompose the problem into stochastic and deterministic components which are then solved separately the resulting dimension reduction algorithm has enhanced numerical stability solves the constrained problem completely and can reduce computational load by reducing the problem size the new method addresses deficiencies in commonly used pseudo observation or projection methods which either do not solve the constrained problem completely or have unstable numerical implementations due in part to the degeneracy in the estimator statistics we present a numerical example demonstrating the effectiveness of the new method compared to other current methods

h infinity kalman estimation for rectangular descriptor systems with unknown inputs this note considers h filtering and prediction for rectangular descriptor systems with unknown inputs that affect both the system and the output an optimal h descriptor kalman estimator hdke which can simultaneously solve the h filtering and prediction problems for rectangular descriptor systems with unknown inputs is developed based on the maximum likelihood descriptor kalman filtering method the hdke serves as a unified solution to solve h and kalman filtering for descriptor systems and standard systems with or without unknown inputs to reduce the computational complexity problem some efficient alternatives to the developed hdke are further proposed the relationship between the hdke and the existing literature results is also addressed an illustrative example is given to show the usefulness of the proposed results

score tests for dose effect in the presence of non responders when only a certain proportion of subjects respond to treatment responders or may never experience an event of interest thus cured mixture models often lead to increased understanding of the treatment or disease process this paper focuses on hypothesis testing in a dose response framework and shows that increased power is possible by using a mixture model where both the logit of the response rate and the response mean are linear functions of the dose level three score tests are developed for testing an overall effect and permutation methods are used to control the type i error extensive simulations establish the power properties of the tests and show that our proposed score test has the best performance the approach is illustrated by a multi country clinical trial of rapid acting intramuscular olanzapine copyright 2004 john wiley sons ltd

brief risk sensitive filtering prediction and smoothing for discrete time singular systems this paper is concerned with steady state risk sensitive filtering prediction and smoothing problems for discrete time singular systems it is shown that a risk sensitive estimator can be obtained by ensuring the minimum of an indefinite quadratic form to be maximum minimum when the risk sensitivity parameter q is negative positive an auxiliary state space signal model and an innovation sequence in krein space are introduced to simplify the derivation of the estimator the estimator is calculated based on one j spectral factorization for risk seeking q 0 a numerical example is given to demonstrate the applicability of the result

brief paper recursive linear estimation for general discrete time descriptor systems this paper considers the optimal linear estimates recursion problem for discrete time linear systems in its more general formulation the system is allowed to be in descriptor form rectangular time variant and with the dynamical and measurement noises correlated we propose a new expression for the filter recursive equations which presents an interesting simple and symmetric structure convergence of the associated riccati recursion and stability properties of the steady state filter are provided

multiple window moving horizon estimation long horizon lengths in moving horizon estimation are desirable to reach the performance limits of the full information estimator however the conventional mhe technique suffers from a number of deficiencies in this respect first the problem complexity scales at least linearly with the horizon length selected which restrains from selecting long horizons if computational limitations are present second there is no monitoring of constraint activity inactivity which results in conducting redundant constrained minimizations even when no constraints are active in this study we develop a multiple window moving horizon estimation strategy mw mhe that exploits constraint inactivity to reduce the problem size in long horizon estimation problems the arrival cost is approximated using the unconstrained full information estimator arrival cost to guarantee stability of the technique a new horizon length selection criterion is developed based on the sensitivity between remote states in time the development will be in terms of general causal descriptor systems which includes the standard state space representation as a special case the potential of the new estimation algorithm will be demonstrated with an example showing a significant reduction in both computation time and numerical errors compared to conventional mhe

filtering and lqg problems for discrete time stochastic singular systems discrete time stochastic singular systems are discussed filtering and linear quadratic gaussian lqg problems are considered the problems are first transformed into the normal form via state augmentation and then solved by utilizing standard results for nonsingular systems the linear unbiased least squares state estimation algorithm and the optimal control law for the lqg problem are given the order of the filtering algorithm obtained in this way is not much increased moreover this algorithm allows the presence of some kinds of control inputs

optimal unbiased reduced order filtering for discrete time descriptor systems via lmi abstract in this paper the reduced order unbiased optimal h filtering problem for discrete time descriptor linear systems is considered necessary and sufficient conditions for the existence of unbiased causal time invariant filters are obtained solutions of the filtering problems are given in terms of linear matrix inequalities lmis parametrization of all optimal causal unbiased filters is provided the proposed method generalizes the existing results in the standard case

technical communique descriptor wiener state estimators based on the modern time series analysis method a new time domain wiener filtering approach is presented asymptotically stable wiener state estimators are presented for discrete linear stochastic descriptor systems they can be implemented via the autoregressive moving average arma recursive filters they can handle the optimal state filtering smoothing and prediction problems in a unified framework and can simply be obtained based on the arma innovation model the solution of the diophantine equations and riccati equations is avoided so that the computational burden is reduced a simulation example shows the effectiveness of the new approach

multiscale systems kalman filters and riccati equations an algorithm analogous to the rauch tung striebel algorithm spl minus consisting of a fine to coarse kalman filter like sweep followed by a coarse to fine smoothing step spl minus was developed previously by the authors ibid vol 39 no 3 p 464 78 1994 in this paper they present a detailed system theoretic analysis of this filter and of the new scale recursive riccati equation associated with it while this analysis is similar in spirit to that for standard kalman filters the structure of the dyadic tree leads to several significant differences in particular the structure of the kalman filter error dynamics leads to the formulation of an ml version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the hamiltonian for the smoothing problem in addition the notion of stability for dynamics requires some care as do the concepts of reachability and observability using these system theoretic constructs the stability and steady state behavior of the fine to coarse kalman filter and its riccati equation are analysed

bounded constrained filtering for gps ins integration this paper considers estimation problems where inequality constraints are imposed on the outputs of linear systems and can be modeled by nonlinear functions in this case censoring functions can be designed to constrain measurements for use by filters and smoothers it is established that the filter and smoother output estimates are unbiased provided that the underlying probability density functions are even and the censoring functions are odd the bounded real lemma is employed to ensure that the output estimates satisfy a performance criterion a global positioning system gps and inertial navigation system ins integration application is discussed in which a developed solution exhibits improved performance during gps outages when a priori information is used to constrain the altitude and velocity measurements

brief paper on the global optimality of unbiased minimum variance state estimation for systems with unknown inputs in this paper a globally optimal filtering framework is developed for unbiased minimum variance state estimation for systems with unknown inputs that affect both the system state and the output the resulting optimal filters are globally optimal within the unbiased minimum variance filtering over all linear unbiased estimators globally optimal state estimators with or without output and or input transformations are derived through the global optimality evaluation of this research the performance degradation of the filter proposed by darouach zasadzinski and boutayeb darouach m zasadzinski m boutayeb m 2003 extension of minimum variance estimation for systems with unknown inputs automatica 39 867 876 is clearly illustrated and the global optimality of the filter proposed by gillijns and de moor gillijns s de moor b 2007b unbiased minimum variance input and state estimation for linear discrete time systems with direct feedthrough automatica 43 934 937 is further verified the relationship with the existing literature results is addressed a unified approach to design a specific globally optimal state estimator that is based on the desired form of the distribution matrix from the unknown input to the output is also presented a simulation example is given to illustrate the proposed results

deadbeat control and tracking of discrete time systems a new approach for forcing the state of a linear discrete time system to zero in a minimum number of steps is discussed the problem is formulated as the solution to a steady state optimal control problem with no cost on the control this problem in turn is set up as the solution to an associated eigenvalue problem no special assumption on the open loop system matrix and or the ratio of the number of states to controls is required stable numerical techniques are presented for solving for the feedback gain robust deadbeat tracking is also discussed