An exact solution of the time-invariant discrete Kalman filter

An exact solution is presented of the matrix Riccati difference equation associated with a time-invariant discrete Kalman filter. The time-varying solution is expressed by means of the corresponding steady-state algebraic solution. An exact solution of the closed-loop transition matrix is also presented.     

The location of the continuous-time stationary Kalman filter poles

It is shown that the closed-loop poles of the continuous-time Kalman filter reside in a region in the left half of the complex plane that is confined by two concentric circles whose radii depend on the system matrices and the signal-to-noise ratio. This region includes the system open-loop poles and excludes the imaginary axis. In the case where the system dynamic matrix has a simple eigenstructure, this region possesses an additional boundary, that is parallel to the imaginary axis at a distance that varies with the signal-to-noise ratio.     

Hybridizing discrete- and continuous-time models for batch sizing and scheduling problems

This paper proposes a new hybrid technique called “partial parameter uniformization” (hereafter PPU). The technique simplifies problems by ignoring the different values that certain problem parameters can take, which may facilitate the solution of some hard combinatorial optimization problems. PPU is applied to complex batch sizing and scheduling problems. Some information can be obtained from a discrete-time model in which job durations have been made uniform. This information is then exploited by a more detailed continuous-time model to generate feasible solutions and further improve these solutions. Good, or optimal solutions to the Westenberger and Kallrath Benchmark problems have been obtained in this way, at relatively low computational cost, as have solutions to the newer problems of Blömer and Günther.     

On the stability of the continuous-time Kalman filter subject to exponentially decaying perturbations

This paper details the stability analysis of the continuous-time Kalman filter dynamics for linear time-varying systems subject to exponentially decaying perturbations. It is assumed that estimates of the input, output, and matrices of the system are available, but subject to unknown perturbations which decay exponentially with time. It is shown that if the nominal system is uniformly completely observable and uniformly completely controllable, and if the state, input, and matrices of the system are bounded, then the Kalman filter built using the perturbed estimates is a suitable state observer for the nominal system, featuring exponentially convergent error dynamics.     

An approximation of the Kalman filter equations

This correspondence presents the results of the application of the matrix inversion lemma to the Kalman filter equation. This operation eliminates the inversion process in the Kalman filter and enables one to sequentially compute the optimum estimate of the state without the use of the inversion process.     

An interlaced extended Kalman filter

An estimation algorithm for a class of discrete time nonlinear systems is proposed. The system structure we deal with is partitionable into in subsystems, each affine w.r.t. the corresponding part of the state vector. The algorithm consists of a bank of m interlaced Kalman filters, and each of them estimates a part of the state, considering the remaining parts as known time-varying parameters whose values are evaluated by the other filters at the previous step. The procedure neglects the subsystem coupling terms in the covariance matrix of the state estimation error and counteracts the errors so introduced by suitably “increasing” the noise covariance matrices. Comparisons through numerical simulations with the extended Kalman filter and its modified versions proposed in the literature illustrate the good trade-off provided by the algorithm between the reduction of the computational load and the estimation accuracy     

An extension of the SRIF Kalman filter

We consider the problem of implementing the Kalman filter recursions in square root information filter form. We suggest a general linear, dynamical model which directly incorporates the fact that many of the unknowns are not time varying. The resulting implementation is widely applicable, numerically sound, and extends easily to smoothing problems.     

Relation between filter using covariance information and Kalman filter

This paper presents sequential algorithms for the optimal impulse function, Kalman gain and the error variance in linear least squares filtering problems, when the autocovariance function of the signal is given in the form of a semi-degenerate kernel, and the additive observation noise in white Gaussian. A digital simulation result indicates that the algorithms presented in this paper are feasible, and that the values of Kalman gain and the error variance calculated by these algorithms approach to those obtained by the Kalman filter theory, for time sufficiently large.     

Duality between the discrete-time Kalman filter and LQ control law

In the context of linear quadratic Gaussian (LQG) controller design for single-input, single-output, discrete-time, linear, time-invariant systems, a duality is demonstrated between the Kalman filter design phase and the linear quadratic (LQ) design phase. Furthermore, an asymptotic recovery design procedure is proposed. Through specific Kalman filter design choices, the LQG open-loop transfer function is forced to approach the LQ open-loop transfer function     

An analysis of the divergence problem in the Kalman filter

The effect of modeling errors in a linear discrete stochastic system upon the Kalman filter state estimates is investigated. Errors in both plant dynamics and noise covariances are permitted. The errors are characterized in such a manner that a linear recursion relation for the actual estimation error covariances can be derived. Conditions which guarantee that the covariance matrix remains bounded are described in terms of the asymptotic stability of the homogeneous part of the covariance equation and the boundedness of the forcing terms in the inhomogeneous equation.     

Cubature 卡尔曼滤波与Unscented 卡尔曼滤波估计精度比较

对于不同维数下非线性系统的估计问题,为从常用的Unscented卡尔曼滤波(UKF)和Cubature卡尔曼滤波(CKF)中选取合适的滤波方法,从函数泰勒展开式和数值稳定性上对其进行了分析和比较.由于不同维数下它们捕获函数泰勒展开式高阶项的程度和数值稳定性不同,两者滤波精度出现差异,从而得到了不同维数下滤波方法的选择途径.仿真结果验证了理论分析的正确性.     

Design of stabilizing switching control laws for discrete- and continuous-time linear systems using piecewise-linear Lyapunov functions

In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose 'global' Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.     

Kalman filter and quantization

We give an interpretation of the problem of filtering of diffusion processes as a quantization problem. Based on this, we show that the classical Kalman-Bucy linear filter describes a flow of automorphisms of the Heisenberg algebra. We obtain new formulas for the unnormalized conditional density in the linear case, a new interpretation of the Mehler formula for the fundamental solution of the Schrödinger operator for a harmonic oscillator, and formulas for a regularized determinant of a Sturm-Liouville operator.     

Divergence of the Kalman filter

The Kalman estimation technique is examined from the point of view of the asymptotic behavior of the errors in the estimates. It is shown that, under certain conditions, the mean-square errors may become unbounded with time, and that this divergence may or may not be correctable by increasing the intensity of process noise assumed in the filtering model General results are derived for multidimensional systems, and both "true" and "apparent" divergence are demonstrated by a simple scalar system. Divergence due to numerical inaccuracies is considered, and an example problem in orbital navigation is used to demonstrate divergence and its elimination.     

Optimal decentralized Kalman filter and Lainiotis filter

A method to implement the optimal decentralized Kalman filter and the optimal decentralized Lainiotis filter is proposed; the method is based on the a priori determination of the optimal distribution of measurements into parallel processors, minimizing the computation time. The resulting optimal Kalman filter and optimal Lainiotis filter require uniform distribution or near to uniform distribution of measurements into parallel processors. The optimal uniform distribution has the advantages of elimination of idle time for the local processors and of low hardware cost, but it is not always applicable. The optimal filters present high parallelism speedup; this is verified through simulation results and is very important due to the fact that, in most real-time applications, it is essential to obtain the estimate in the shortest possible time.     

导弹跟踪中的卡尔曼与不敏卡尔曼滤波

卡尔曼滤波是在线性高斯情况下利用最小均方误差准则获得目标的动态估计,但在实际系统中,许多情况下观测数据与目标动态参数间的关系是非线性的。对于非线性滤波问题,至今尚未得到完善的解法。本文采用了两种方法来进行滤波:一种是将观测变量进行坐标系变化,使量测方程线性化,然后直接进行线性卡尔曼滤波;另一种方法是直接采用非线性滤波方法的不敏卡尔曼滤波。对仿真导弹轨迹的仿真结果显示,第一种方法在本系统中优于第二种方法。     

An iterated fuzzy extended Kalman filter for nonlinear systems

The fuzzy extended Kalman filter (FEKF) for state estimation can be used to deal with fuzzy uncertainty effectively. However, the linearisation processing of the FEKF introduces truncation error, which degrades the estimation precision. In order to reduce the error, a new iterated fuzzy extended Kalman filter (IFEKF), based on the FEKF and the maximum a posteriori estimation, is proposed in this article. Compared with the FEKF, the proposed algorithm can be used not only to deal with the fuzzy uncertainty, but also to reduce the truncation error and to estimate the states more accurately. With an algebraic example and a passive location simulation, it is shown that the IFEKF has better estimation precision than that of the FEKF.     

Two-level form of the Kalman filter

Sequential estimation of the states of several high-order interconnected systems may be prohibitive on computer time and storage if the problem is formulated as for a single system. Therefore, multilevel systems theory has been applied to derive a coordination algorithm, with one-step convergence, for a number of subsystem Kalman estimators. The procedure may be computationally attractive for sparsely coupled subsystems with few stochastic inputs.     

Accuracy of the decoupled Kalman filter

An explicit analytical solution is obtained for the suboptimal covariance matrix of a decoupled Kalman filter. The result is used to determine when filter decoupling breaks down.     

Decomposition of the extended Kalman filter

The use of the extended Kalman filter as an approximate estimator for the states and parameters of nonlinear systems is well known. A decomposition is pointed out in this letter, which is possible with the usual augumentations of the state space by parameters, and which may lead to a more efficient computing algorithm.