Finite Impulse Response Smoother Research Articles

A continuous finite-time convergence fixed-lag FIR smoother using multiple IIR filters

In this paper, we propose a continuous finite-time convergence finite impulse response (FIR) fixed-lag smoother using multiple, or more than two, computationally efficient IIR filters. We describe the optimal design to improve and further optimize an existing scheme based on two IIR filters. Multiple IIR filters are utilized to minimize the estimation error variance of the proposed smoother under the condition that its estimate converges to a real state in a finite time. As the number of adopted IIR filters increases, the proposed smoother improves and its performance approaches that of the heavy computational fixed-lag minimum variance FIR smoother. By choosing the appropriate number of IIR filters, we can balance the trade-off between improved accuracy and increased implementation costs. To realize the optimal design of IIR filters with the limited number of IIR filters, their gains are determined using a particle swarm optimization scheme. Numerical examples are used to show that with an increasing number of IIR filters, the estimation error variance decreases monotonically while guaranteeing finite-time convergence.

Discrete Time $q$-Lag Maximum Likelihood FIR Smoothing and Iterative Recursive Algorithm

The finite impulse response (FIR) approach is known to be more robust than the Kalman approach. In this paper, we derive a batch q-lag maximum likelihood (ML) FIR smoother for full covariance matrices and represent it with an iterative algorithm using recursions for diagonal covariance matrices. It is shown that, under ideal conditions of fully known model, the ML FIR smoother occupies an intermediate place between the more accurate Rauch-Tung-Striebel (RTS) smoother and the less accurate unbiased FIR smoother. With uncertainties and errors in noise covariances, ML FIR smoothing is significantly superior to RTS smoothing. It is also shown experimentally that ML FIR smoothing is more robust than RTS smoothing against measurement outliers.

Non‐linear FIR smoothing filter for systems with a modelling error and its application to the DR/GPS integrated navigation

Infinite impulse response (IIR) filter generally used in estimation, positioning, etc. has problems in that, when there is a modeling error even if the system is completely observable, the estimates may converge to incorrect values or the varying values cannot be estimated quickly. The finite impulse response (FIR) filter, which has been investigated as a method to solve this problem, has faster estimation performance than the IIR filter in the presence of modelling error, but there is a limit to the convergence characteristic of the state variables. In this paper, a non-linear FIR smoothing (NFS) filter is proposed to overcome the limitation of the state variable convergence characteristic of the FIR filter. The proposed NFS filter adds the smoothing concept to the modified receding horizon Kalman FIR filter. In order to verify the performance of the NFS filter, this filter is applied to a delay-tolerant integrated navigation system using magnetic compass (MC) of which error is difficult to be modelled accurately. If an un-modeled jump or ramp error occurs in the MC measurement, it shows that the NFS filter can estimate the measurement error more accurately than the FIR filter as well as the IIR filter through a simulation.

이산 비선형 시스템에 대한 유한 임펄스 응답 고정 시간 지연 평활기

In this paper, a finite impulse response(FIR) fixed-lag smoother is proposed for discrete-time nonlinear systems. If the actual state trajectory is sufficiently close to the nominal state trajectory, the nonlinear system model can be divided into two parts: The error-state model and the nominal model. The error state can be estimated by adapting the optimal time-varying FIR smoother to the error-state model, and the nominal state can be obtained directly from the nominal trajectory model. Moreover, in order to obtain more robust estimates, the linearization errors are considered as a linear function of the estimation errors. Since the proposed estimator has an FIR structure, the proposed smoother can be expected to have better estimation performance than the IIR-structured estimators in terms of robustness and fast convergence. Additionally the proposed method can give a more general solution than the optimal FIR filtering approach, since the optimal FIR smoother is reduced to the optimal FIR filter by setting the fixed-lag size as zero. To illustrate the performance of the proposed method, simulation results are presented by comparing the method with an optimal FIR filtering approach and linearized Kalman filter.

이산 시변 상태공간 모델을 위한 최적 고정 시간 지연 FIR 평활기

In this paper, we propose an optimal fixed-lag FIR (Finite-Impulse-Response) smoother for a class of discrete time-varying state-space signal models. The proposed fixed-lag FIR smoother is linear with respect to inputs and outputs on the recent finite horizon and estimates the delayed state so that the variance of the estimation error is minimized with the unbiased constraint. Since the proposed smoother is derived with system inputs, it can be adapted to feedback control system. Additionally, the proposed smoother can give more general solution than the optimal FIR filter, because it reduced to the optimal FIR filter by setting the fixed-lag size as zero. A numerical example is presented to illustrate the performance of the proposed smoother by comparing with an optimal FIR filter and a conventional fixed-lag Kalman smoother.

FIR Smoothing of Discrete-Time Polynomial Signals in State Space

We address a smoothing finite impulse response (FIR) filtering solution for deterministic discrete-time signals represented in state space with finite-degree polynomials. The optimal smoothing FIR filter is derived in an exact matrix form requiring the initial state and the measurement noise covariance function. The relevant unbiased solution is represented both in the matrix and polynomial forms that do not involve any knowledge about measurement noise and initial state. The unique <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> -degree unbiased gain and the noise power gain are derived for a general case. The widely used low-degree gains are investigated in detail. As an example, the best linear fit is provided for a two-state clock error model.

A Low-Complexity Sliding-Window Kalman FIR Smoother for Discrete-Time Models

The information filter is a form of the Kalman filter that, in many of its realizations, allows optimal, unbiased, recursive state estimation without an initial state estimate. We review a number of forms of the information filter. We then derive the coefficients for the sliding-window Kalman finite impulse response (FIR) smoother (also known as a receding or moving horizon Kalman FIR smoother) starting from the equations for the information filter. The resulting FIR smoother has a simple, recursive form for calculating the coefficients, allowing them to be calculated with O(N) complexity versus the O(N 2) to O(N 3) complexity of previous approaches, where N is the length of the batch. It also allows for a control input, something not present in previous algorithms. This method is only limited in the assumption that the state transition matrix is invertible, which, however, is satisfied in most practical problems.

Minimax FIR smoothers for deterministic continuous-time state space models

In order to design a smoother for a deterministic continuous-time state space model, a new performance criterion is proposed, which is given by a ratio of the current estimation error to the weighted energy of the deterministic disturbance applied during the recent finite horizon. Among smoothers with the deadbeat property and finite impulse response (FIR) structure, a minimax FIR smoother (MFS) is obtained to optimize the proposed performance criterion. To begin with, the functional optimization problem is formulated with respect to kernel functions of the MFS and then its solution is explicitly presented. The MFS depends only on inputs and outputs on the finite recent horizon, and is independent of any a priori state information. The MFS is first represented in an integral form for simple representation and then a differential form is introduced for efficient numerical computation. As in H ∞ IIR smoothing and H 2 IIR filtering, it is shown that the proposed MFS for a deterministic system can be interpreted as the minimum variance unbiased FIR smoother for a stochastic system.

New Quasi-Deadbeat FIR Smoother for Discrete-Time State-Space Signal Models: An LMI Approach

In this letter, we propose a new H2 smoother (H2S) with a finite impulse response (FIR) structure for discrete-time state-space signal models. This smoother is called an H2 FIR smoother (H2FS). Constraints such as linearity, quasi-deadbeat property, FIR structure, and independence of the initial state information are required in advance to design H2FS that is optimal in the sense of H2 performance criterion. It is shown that H2FS design problem can be converted into the convex programming problem written in terms of a linear matrix inequality (LMI) with a linear equality constraint. Simulation study illustrates that the proposed H2FS is more robust against uncertainties and faster in convergence than the conventional H2S.

New H FIR Smoother for Linear Discrete-Time State-Space Models

This letter propose a new H∞ smoother (HIS) with a finite impulse response (FIR) structure for discrete-time state-space models. This smoother is called an H∞ FIR smoother (HIFS). Constraints such as linearity, quasi-deadbeat property, FIR structure, and independence of the initial state information are required in advance. Among smoothers with these requirements, we choose the HIFS to optimize H∞ performance criterion. The HIFS is obtained by solving the linear matrix inequality (LMI) problem with a parametrization of a linear equality constraint. It is shown through simulation that the proposed HIFS is more robust against uncertainties and faster in convergence than the conventional HIS.

Fixed-lag maximum likelihood FIR smoother for state-space models

In this paper, we propose a new fixed-lag maximum likelihood smoother with a finite impulse response (FIR) structure for discrete-time state-space models. This smoother is called a maximum likelihood FIR smoother (MLFS). The MLFS is linear with the most recent finite outputs and does not require a prior initial state information on a receding horizon. It is shown that the proposed MLFS possesses the unbiasedness property and the deadbeat property. Simulation study illustrates that the proposed MLFS is more robust against uncertainties and faster in convergence than the conventional fixed-lag Kalman smoother.

Optimal FIR filters for time-varying state-space models

An optimal FIR (finite impulse response) filter and smoother is introduced for the time-varying state-space model. The suggested filter has an FIR structure and utilizes finite observation. It is shown that the impulse response of the optimal FIR filter can be obtained by a simple Riccati-type matrix differential equation. Especially for time-invariant systems, this FIR filter reduces to previously known simple forms. For implementation, a recursive form of the optimal FIR filter and smoother is derived by using adjoint variables, and computational algorithms are suggested. It is also shown by sensitivity analysis that the proposed optimal FIR filter alleviates potential divergence characteristics of the standard Kalman filter.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

FIR filters and recursive forms for discrete-time state-space models

An FIR (finite impulse response) filter and an FIR smoother are introduced for discrete-time state-space models with system noise. The FIR structure not only guarantees both the BIBO stability and the robustness to some parameter changes, but also improves the filter divergence problem. It is shown in this paper that impulse responses of both the FIR filter and the FIR smoother are easily obtained from Riccati-type difference equations. Especially for time-invariant state-space models, they are also to be time-invariant and finally determined by simple equations on a finite interval. For the purpose of implementation, recursive forms of the FIR filter and smoother are derived using each other as adjoint variables. Improved characteristics of the FIR filter in the divergence problem are demonstrated via a simulation.

FIR Filters and Recursive Forms for Discrete-time State-space Models

Abstract An FIR (finite impulse response) filter and an FIR smoother are introduced for discrete-time state-space models with system noises, since the FIR structure not only guarantees both the BIBO (bounded input/bounded output) stability and the robustness to some parameter changes, but also improves the filter divergence problem. It is shown in this paper that impulse responses of both the FIR filter and the FIR smoother are given by a Riccati-type difference equation. Especially for time-invariant systems, they are also to be time-invariant and determined by simpler equations on a finite interval once and for all. For implementational purpose, recursive forms of the FIR filter and smoother are derived using each other as adjoint variables. The improved characteristics of the FIR filter in the divergence problem is demonstrated via a simulation.

FIR filters and recursive forms for continuous time-invariant state-space models

An FIR (finite impulse response) filter and an FIR smoother are introduced for continuous time-invariant state-space models. It is shown in this note that finite impulse responses of the FIR filter and smoother can be easily determined by solving a simple Riccati-type matrix differential equation on a finite interval. Especially for systems with stationary processes, finite impulse responses of the FIR filter and smoother become time-invariant and can be computed from simpler equations. For fast computational purposes, recursive forms of the FIR filter and smoother are derived by using adjoint variables. In this case all gains for recursive forms are shown to be constant.