Optimal and robust noncausal filter formulations

Einicke, G.A.

doi:10.1109/tsp.2005.863042

Cited by 29 publications

(21 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A proof appears in [7] and proceeds similarly to that within Lemma 1 of Chapter 2. The simplification D k = 0 is assumed below unless stated otherwise.…”

Section: N   Satisfies (3)mentioning

confidence: 93%

Discrete-Time Smoothing

Einicke¹

2012

Smoothing, Filtering and Prediction - Estimating the Past, Present and Future

View full text Add to dashboard Cite

1 IntroductionObservations are invariably accompanied by measurement noise and optimal filters are the usual solution of choice. Filter performances that fall short of user expectations motivate the pursuit of smoother solutions. Smoothers promise useful mean-square-error improvement at mid-range signal-to-noise ratios, provided that the assumed model parameters and noise statistics are correct.In general, discrete-time filters and smoothers are more practical than the continuous-time counterparts. Often a designer may be able to value-add by assuming low-order discretetime models which bear little or no resemblance to the underlying processes. Continuoustime approaches may be warranted only when application-specific performance considerations outweigh the higher overheads.This chapter canvasses the main discrete-time fixed-point, fixed-lag and fixed interval smoothing results [1] - [9]. Fixed-point smoothers [1] calculate an improved estimate at a prescribed past instant in time. Fixed-lag smoothers [2] -[3] find application where small end-to-end delays are tolerable, for example, in press-to-talk communications or receiving public broadcasts. Fixed-interval smoothers [4] - [9] dispense with the need to fine tune the time of interest or the smoothing lags. They are suited to applications where processes are staggered such as delayed control or off-line data analysis. For example, in underground coal mining, smoothed position estimates and control signals can be calculated while a longwall shearer is momentarily stationary at each end of the face [9]. Similarly, in exploration drilling, analyses are typically carried out post-data acquisition.The smoother descriptions are organised as follows. Section 7.2 sets out two prerequisites: time-varying adjoint systems and Riccati difference equation comparison theorems. Fixedpoint, fixed-lag and fixed-interval smoothers are discussed in Sections 7.3, 7.4 and 7.5, respectively. It turns out that the structures of the discrete-time smoothers are essentially the same as those of the previously-described continuous-time versions. Differences arise in the calculation of Riccati equation solutions and the gain matrices. Consequently, the treatment "An inventor is simply a person who doesn't take his education too seriously. You see, from the time a person is six years old until he graduates from college he has to take three or four examinations a year. If he flunks once, he is out. But an inventor is almost always failing. He tries and fails maybe a thousand times. If he succeeds once then he's in. These two things are diametrically opposite. We often say that the biggest job we have is to teach a newly hired employee how to fail intelligently. We have to train him to experiment over and over and to keep on trying and failing until he learns what will work." Charles Franklin Kettering www.intechopen.com is somewhat condensed. It is reaffirmed that the above-mentioned smoothers outperform the Kalman filter and the minimum-variance smoother provides the best performance. Prer...

show abstract

“…A proof appears in [7] and proceeds similarly to that within Lemma 1 of Chapter 2. The simplification D k = 0 is assumed below unless stated otherwise.…”

Section: N   Satisfies (3)mentioning

confidence: 93%

Discrete-Time Smoothing

Einicke¹

2012

Smoothing, Filtering and Prediction - Estimating the Past, Present and Future

View full text Add to dashboard Cite

show abstract

“…From the arguments of Chapters 1 -2 and [28], for single-input-single-output plants When the problem is stationary (or time-invariant), the filter gain is precalculated as…”

Section: Trading-off H ∞ Performancementioning

confidence: 99%

“…The following fixed-interval smoother for output estimation [28] employs the gain for the H ∞ predictor,…”

Section: H ∞ Solutionmentioning

confidence: 99%

See 1 more Smart Citation

Robust Prediction, Filtering and Smoothing

Einicke¹

2012

Smoothing, Filtering and Prediction - Estimating the Past, Present and Future

View full text Add to dashboard Cite

Unit 405, Office Block, Hotel Equatorial Shanghai This book describes the classical smoothing, filtering and prediction techniques together with some more recently developed embellishments for improving performance within applications. It aims to present the subject in an accessible way, so that it can serve as a practical guide for undergraduates and newcomers to the field. The material is organised as a ten-lecture course. The foundations are laid in Chapters 1 and 2, which explain minimum-mean-square-error solution construction and asymptotic behaviour. Chapters 3 and 4 introduce continuous-time and discrete-time minimum-variance filtering. Generalisations for missing data, deterministic inputs, correlated noises, direct feedthrough terms, output estimation and equalisation are described. Chapter 5 simplifies the minimum-variance filtering results for steady-state problems. Observability, Riccati equation solution convergence, asymptotic stability and Wiener filter equivalence are discussed. Chapters 6 and 7 cover the subject of continuous-time and discrete-time smoothing. The main fixed-lag, fixedpoint and fixed-interval smoother results are derived. It is shown that the minimum-variance fixed-interval smoother attains the best performance. Chapter 8 attends to parameter estimation. As the above-mentioned approaches all rely on knowledge of the underlying model parameters, maximum-likelihood techniques within expectation-maximisation algorithms for joint state and parameter estimation are described. Chapter 9 is concerned with robust techniques that accommodate uncertainties within problem specifications. An extra term within Riccati equations enables designers to trade-off average error and peak error performance. Chapter 10 rounds off the course by applying the afore-mentioned linear techniques to nonlinear estimation problems. It is demonstrated that step-wise linearisations can be used within predictors, filters and smoothers, albeit by forsaking optimal performance guarantees. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following:

show abstract

“…The Kalman-Bucy filter [4] extends the discrete-time sequence to the continuous-time domain. The fixed interval smoother [5][6][7] provides an optimum estimate of state variables using a fixed interval. The set-valued Kalman filter [8,9] has the filtering mechanism to handle uncertain initial values.…”

Section: Introductionmentioning

confidence: 99%

An Extended Kalman Filtering Mechanism Based on Generalized Interval Probability

Wang

Cheng

et al. 2015

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems Part B: Mechanical Engineering

View full text Add to dashboard Cite

Kalman filter has been widely applied for state identification in controllable systems. As a special case of the hidden Markov model, it is based on the assumption of linear dependency relationships and Gaussian noise. The classical Kalman filter does not differentiate systematic error from random error associated with observations. In this paper, we propose an extended Kalman filtering mechanism based on generalized interval probability, where state and observable variables are random intervals, and interval-valued Gaussian distributions model the noises. The prediction and update procedures in the new mechanism are derived. Two examples are used to illustrate the developed mechanism. It is shown that the method is an efficient alternative to sensitivity analysis for assessing the effect of systematic error.

show abstract

Optimal and robust noncausal filter formulations

Cited by 29 publications

References 21 publications

Discrete-Time Smoothing

Discrete-Time Smoothing

Robust Prediction, Filtering and Smoothing

An Extended Kalman Filtering Mechanism Based on Generalized Interval Probability

Contact Info

Product

Resources

About