Fixed-lag Smoothing Algorithm Research Articles

Adaptive Fixed-Lag Smoothing Algorithms Based on the Variational Bayesian Method

In this article, we address the adaptive fixed-lag smoothing (FLS) problem in the presence of unknown and slowly time-varying measurement noise covariance matrix (MNCM). Based on the variational Bayesian (VB) method, we use the VB inference to jointly estimate the system state and the unknown MNCM. According to different implementation methods, we propose two adaptive FLS algorithms: One is based on the augmented state-space models and another one is based on a sliding Rauch-Tung-Striebel smoothing. In addition, the evidence lower bound and the convergence criterion of the above two algorithms are given. The proposed adaptive algorithms are evaluated by a simulation example.

Flight Control Design using Incremental Nonlinear Dynamic Inversion with Fixed-lag Smoothing Estimation

In this study, a flight control design strategy based on incremental nonlinear dynamic inversion (INDI) and smoothing algorithm is presented. The INDI is an enhanced version of the nonlinear dynamic inversion technique with a better robust performance. It reduces the aircraft model dependence via the feedback information of the state derivative, that is, angular acceleration for the aircraft attitude motion (or angular velocity related to the wind axis parameters). However, the state derivatives cannot always be obtained by direct measurement, thus they need to be estimated on-line. Moreover, taking into account that INDI requires one-step delay of state derivative and the fact that the inertial measurement unit (IMU) operates faster than the flight control computer (FCC), there will be additional information that can be used to improve the estimation. Therefore, a fixed-lag smoothing algorithm based on a discrete Kalman filter is proposed for angular velocity and angular acceleration estimation. The smoother utilizes the state variable from the sensor measurement and thus dealing with noise and delay.

Semi-widely linear estimation algorithms of quaternion signals with missing observations and correlated noises

Abstract The paper deals with the estimation problem of a discrete-time vectorial quaternion signal which is observed through a linear dynamic system with intermittent observations and autocorrelated and cross-correlated noises. Under C η -properness conditions, a semi-widely linear processing is considered to provide filtering, fixed-point and fixed-lag smoothing algorithms for estimating the quaternion state. The proposed solutions give a substantial reduction in computational burden in relation to the widely linear estimation techniques, this benefit being impossible to be fully captured by using a real number framework. The feasibility and performance of the aforementioned algorithms are illustrated by means of simulation examples.

One Way to Obtain the Smoothed Estimate of Phase Vector in the Task of Motion Control of Convoy Robots

In this paper, we formulated and solved the problem of fixed-lag smoothing estimation for continuous linear dynamical systems. The problems of other fixed-lag smoothing algorithm for continuous linear dynamical systems are indicated. Firstly, sometimes the fixed-lag smoothing algorithm is not stable. Secondly, the knowledge of state transition matrix is needed to implement algorithm. It is different to get it, especially for time-vary system. An algorithm is developed for generating the optimal smoothed estimate x(t - T|t) of the state x(t) of a continuous linear system, where t is the most recent measurement and T is a positive real constant. The recurrent algorithm is obtained using the properties of state transition matrix Ф(t 1 , t 2 ) of a linear dynamical system. The developed method makes it possible to bypass the time-consuming procedure of finding state-transition matrix. The process of computing the fixed-lag smoothed estimate is discussed in terms of the algorithm's dependence on the solutions of the filtering and fixed-point smoothing problems. Therefore, the calculation of the estimation of fixed-lag smoothing depends on the value obtained during the filtering process. We give the simulation result of fixed-lag smoothing algorithms for a nonstationary linear system and compared it with the simulation result of filtering algorithms. It is showed that the estimation errors of fixed-lag smoothing are less than the estimation errors of filtering. We also used the fixed-lag smoothing algorithms to improve the localization accuracy of robots in the task of motion control of convoy robots. A node named fix_lag_smoothing is written using C++ in ROS

Least-squares finite impulse response fixed-lag smoother and filter in linear discrete-time stochastic systems

This paper proposes the least-squares (LS) finite impulse response (FIR) fixed-lag smoother and filter in linear discrete-time wide-sense stationary stochastic systems. The FIR fixed-lag smoothing estimate is given as a linear convolution sum of the impulse response function and the observed values. It is assumed that the signal is observed with additional white noise, which is uncorrelated with the signal process. By solving the simultaneous linear equations transformed from the Wiener–Hopf equation, the optimal impulse response function is obtained. The necessary information of the LS FIR fixed-lag smoothing algorithm is the auto-covariance function of the signal process and the variance of the observation noise process. In particular, this paper proposes the Levinson–Durbin algorithm, which needs less amount of arithmetic operations than the Gauss–Jordan elimination method in the inverse of the Toeplitz matrix, for the optimal impulse response function. From the numerical simulation example, the proposed LS FIR fixed-lag smoother and filter are superior in estimation accuracy to the RLS Wiener FIR estimators.

Estimation of UAV Position with Use of Smoothing Algorithms

Abstract The paper presents methods of on-line and off-line estimation of UAV position on the basis of measurements from its integrated navigation system. The navigation system installed on board UAV contains an INS and a GNSS receiver. The UAV position, as well as its velocity and orientation are estimated with the use of smoothing algorithms. For off-line estimation, a fixed-interval smoothing algorithm has been applied. On-line estimation has been accomplished with the use of a fixed-lag smoothing algorithm. The paper includes chosen results of simulations demonstrating improvements of accuracy of UAV position estimation with the use of smoothing algorithms in comparison with the use of a Kalman filter.

A robust fixed-lag smoothing algorithm for dynamic systems with correlated sensor malfunctions

Abstract A new robust fixed-lag smoothing algorithm for fault-tolerant signal processing in stochastic dynamic systems in the presence of correlated sensor malfunctions has been developed. The algorithm is developed using a state vector augmentation method and the Gaussian approximation of the current estimate probability density function. The algorithm can be used in the real-time fault-tolerant control systems as well as in radar tracking systems working in the complex interference environment. The performance of the developed algorithm are evaluated by simulations and compared with smoothing and nonlinear filtering algorithms.

Incorporating delayed and infrequent measurements in Extended Kalman Filter based nonlinear state estimation

Abstract This work deals with state estimation in the presence of delayed and infrequent measurements. While most measurements (referred to as secondary measurements) are available frequently and instantaneously, there might be a delay associated with acquiring other measurements ( primary measurements) due to long analysis times involved. The primary measurements are usually sampled at irregular intervals and the exact delay is also unknown. The traditional fixed-lag smoothing algorithm, which has been applied for a variety of chemical processes systems, can be computationally inefficient for such situations and alternate methods to handle delays are necessary. In this paper, we analyze several existing methods to incorporate measurement delays and reinterpret their results under a common unified framework (for Extended Kalman Filter). Extensions to handle time-varying and uncertain delays, as well as out of sequence measurement arrival are also presented. Simulation studies on a linear distillation column and a nonlinear polymerization reactor are used to compare the performance of these methods based on RMSE values and computation times. A large scale nonlinear reactive distillation column example is also used to illustrate the practicality of the suggested method.

On Approximate Maximum-Likelihood Methods for Blind Identification: How to Cope With the Curse of Dimensionality

We discuss approximate maximum-likelihood methods for blind identification and deconvolution. These algorithms are based on particle approximation versions of the expectation-maximization (EM) algorithm. We consider three different methods which differ in the way the posterior distribution of the symbols is computed. The first algorithm is a particle approximation method of the fixed-interval smoothing. The two-filter smoothing and the novel joined-two-filter smoothing involve an additional backward-information filter. Because the state space is finite, it is furthermore possible at each step to consider all the offsprings of any given particle. It is then required to construct a novel particle swarm by selecting, among all these offsprings, particle positions and computing appropriate weights. We propose here a novel unbiased selection scheme, which minimizes the expected loss with respect to general distance functions. We compare these smoothing algorithms and selection schemes in a Monte Carlo experiment. We show a significant performance increase compared to the expectation maximization Viterbi algorithm (EMVA), a fixed-lag smoothing algorithm and the Block constant modulus algorithm (CMA).

Design of fixed-lag smoother using covariance information based on innovations approach in linear discrete-time stochastic systems

This paper newly designs the recursive least-squares (RLS) Wiener fixed-lag smoother and filter using the covariance information in linear discrete-time stochastic systems. The estimators require the information of the observation matrix, the system matrix for the state variable, related with the signal, the variance of the state variable, the cross-variance function of the state variable with the observed value and the variance of the white observation noise. It is assumed that the signal is observed with additive white noise. The current fixed-lag smoothing algorithm has a characteristic, as shown in Theorem 1, that the fixed-lag smoothing estimate of the state vector is calculated in the reverse direction of time.

Fixed-Lag Smoothing Algorithm under Non-independent Uncertainty

This paper discusses the least-squares linear filtering and fixed-lag smoothing problems of discrete-time signals from uncertain observations when the random interruptions in the observation process are modelled by a sequence of not necessarily independent Bernoulli variables. It is assumed that the observations are perturbed by white noise and the autocovariance function of the signal is factorizable. Using an innovation approach we obtain the filtering and fixed-lag smoothing recursive algorithms, which do not require the knowledge of the state-space model generating the signal. Besides the observed values, they use only the matrix functions defining the factorizable autocovariance function of the signal, the noise autocovariance function, the marginal probabilities and the (2, 2)-element of the conditional probability matrices of the Bernoulli variables. The algorithms are applied to estimate a scalar signal which may be transmitted through one of two channels.

New design of fixed-lag smoother using covariance information in linear discrete-time stochastic systems

This paper proposes recursive least-squares fixed-lag smoothing and filtering algorithms in linear discrete-time stochastic systems. The estimators require the information of the factorized autocovariance function of the signal. Namely, the estimation algorithms use the observation matrix, the system matrix for the state variable and the crossvariance function between the state variable and the signal. Also, it is assumed that the variance of the white observation noise is known.

Tracking of multiple maneuvering targets in clutter using IMM/JPDA filtering and fixed-lag smoothing

We consider the problem of tracking multiple maneuvering targets in clutter using switching multiple target motion models. A suboptimal filtering algorithm is developed by applying the basic interacting multiple model (IMM) approach and the joint probabilistic data association (JPDA) technique to a Markovian switching system. A suboptimal fixed-lag smoothing algorithm is developed by applying the IMM and the JPDA approaches to a state-augmented system. The algorithms are illustrated via a simulation example involving tracking of two highly maneuvering targets.

Multisensor tracking of a maneuvering target in clutter using IMMPDA fixed-lag smoothing

We present a suboptimal fixed-lag smoothing algorithm for tracking a highly maneuvering target in a cluttered environment using multiple sensors. The fixed-lag smoothing algorithm is developed by applying the basic interacting multiple model (IMM) approach and the probabilistic data association (PDA) technique to a state-augmented system. In the past this approach had been restricted to Markovian switching systems with no uncertainty regarding the origin of the measurements (i.e., no clutter). The algorithm is illustrated via a highly maneuvering target tracking simulation example where two sensors, a radar and an infrared sensor, are used and are assumed to operate in a cluttered environment. Compared with an existing IMMPDA filtering algorithm, the proposed smoothing algorithm achieves significant improvement in the accuracy of track estimation by introducing a small time lag between the instants of estimation and latest measurements whereas the computational load increases linearly with lag. However, the tracking delay may lead to undesired effects on control loops in certain applications.

Interacting multiple model fixed-lag smoothing algorithm for Markovian switching systems

We investigate a suboptimal approach to the fixed-lag smoothing problem for Markovian switching systems. A fixed-lag smoothing algorithm is developed by applying the basic Interacting Multiple Model (IMM) approach to a state-augmented system. The computational load is roughly d (the fixed lag) times beyond that of filtering for the original system. In addition, an algorithm that approximates the "fixed-lag" mode probabilities given measurements up to current time is proposed. The algorithm is illustrated via a target tracking simulation example where a significant improvement over the filtering algorithm is achieved at the cost of a time delay (i.e., data up to time k are used to produce the smoothed state estimate at time k-d where the fixed large d>0). the IMM fixed-lag smoothing performance for the given example is comparable to that of an existing IMM fixed-interval smoother. Compared with fixed-interval smoothers, the fixed-lag smoothers can be implemented in real-time with a small delay.

Fixed-lag smoothing in the identification of time-varying systems with unknown dead time

This paper considers a new application of fixed-lag smoothing algorithms for identification. It could be used online in conjunction with the extended recursive least-squares identification algorithm. This offers a scheme for identifying time-varying process models with unknown dead time from sampled input-output records. Application to short records with time-varying dynamics, where recursive algorithms without smoothing may not give a clear indication of the time variation, is of particular interest. Optimal Kalman-filter and fixed-lag smoothing algorithms are reviewed, and their computational requirements and stability in an identification context are examined. The most attractive measures of model structure obtained from identification algorithms are tested on artificial and real records. It is concluded that the innovation and adjoint variables are feasible and effective indicators for identifying model structure goodness.

Improved Demodulation of Sampled FM Signals in High Noise

Simulation results are presented which are very convincing in favor of FM demodulators driven by in-phase and quadraturephase signals. Application of the extended Kalman filtering algorithms to the appropriate signal model directly yields demodulators in which the error covariance equations are uncoupled from the state estimate equations. These demodulators perform a little better than others derived using different nonlinear filtering techniques and the necessary approximations to achieve "decoupling." More significantly, these demodulators using in-phase and quadrature-phase sampled signals are readily augmented to achieve demodulation with delay by the application of fixed-lag smoothing algorithms. Simulations highlight the attractive trade-offs between demodulation complexity and performance results for this class of demodulator.

Smoothing algorithms for nonlinear distributed parameter systems

Fixed interval and fixed-lag smoothing algorithms are developed for a class of noisy nonlinear distributed parameter systems with unknown volume disturbances. Using a least-squares estimation criterion a two point boundary value problem is obtained. The solution of this boundary value problem is obtained by converting it into an initial value problem using a Riccati transformation. Using a Gaussian assumption for the disturbances, approximate error covariance expressions are developed. The possible instability of the fixed-lag smoothing algorithm is noted. An alternate fixed-interval algorithm is developed to obviate storage requirements. The applicability of these algorithms is illustrated with an example from nuclear reactor kinetics.

An efficient fixed-lag smoothing algorithm for discrete linear systems

The modified Bryson-Frazier fixed interval smoothing algorithm [6], is an addendem to the Kalman filter. This algorithm when applied to the problem of fixed-lag smoothing is computationally more efficient than the algorithms recently reported in refs. [1–3]. Features of the algorithm are ease of implentation, computational efficiency, reduced storage requirements, and stability.

Fixed-lag demodulation of discrete noisy measurements of FM signals

This paper considers an application of some recent results of fixed-lag smoothing algorithms to the demodulation of discrete noisy measurements of FM signals. The problem is attacked by first modelling the communications system in terms of a finite dimensional nonlinear stochastic system and relating the resulting model to an estimation problem for which approximate solutions can be obtained. Performance results are presented, using the modified truncated second order filtering algorithm. The results illustrate that if delay is allowed in arriving at an “on-line” optimum estimate, significant improvement can be obtained by even a very moderate fixed-lag.