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1999 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Introduction to Optimal Estimation

verfasst von: E. W. Kamen, PhD, J. K. Su, PhD

Verlag: Springer London

Buchreihe : Advanced Textbooks in Control and Signal Processing

Enthalten in: Professional Book Archive

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Über dieses Buch

The topics of control engineering and signal processing continue to flourish and develop. In common with general scientific investigation, new ideas, concepts and interpretations emerge quite spontaneously and these are then discussed, used, discarded or subsumed into the prevailing subject paradigm. Sometimes these innovative concepts coalesce into a new sub-discipline within the broad subject tapestry ofcontrol and signal processing. This preliminary batde between old and new usually takes place at conferences, through the Internet and in the journals of the discipline. After a litde more maturity has been acquiredhas been acquired by the new concepts then archival publication as ascientificorengineering monograph mayoccur. Anewconceptin control and signal processing is known to have arrived when sufficient material has developed for the topic to be taught as a specialised tutorial workshop or as a course to undergraduates, graduates or industrial engineers. The Advanced Textbooks in Control and Signal Processing Series is designed as a vehicle for the systematic presentation ofcourse material for both popular and innovative topics in the discipline. It is hoped that prospective authors will welcome the opportunity to publish a structured presentation of either existing subject areas or some of the newer emerging control and signal processing technologies.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
One of the most common problems in science and engineering is the estimation of various quantities based on a collection of measurements. This includes the estimation of a signal based on measurements that relate to the signal, the estimation of the state of a system based on noisy measurements of the state, and the estimation of parameters in some functional relationship. The use of estimation techniques occurs in a very wide range of technology areas such as aerospace systems, communications, manufacturing, and biomedical engineering. Specific examples include the estimation of an aircraft’s or spacecraft’s position and velocity based on radar measurements of position, the estimation of congestion in a computer communications network, the estimation of process parameters in a manufacturing production system, and the estimation of the heath of a person’s heart based on an electrocardiogram (ECG).
E. W. Kamen, J. K. Su
Chapter 2. Random Signals and Systems with Random Inputs
Abstract
In the problem of estimating a signal s(n) from the measurements z(n) = g(s(n), v(n), n), the noise term v(n) usually varies “randomly,” and thus modeling v(n) requires that we use a random signal formulation. The signal s(n) may also include some random variation, and thus it too must be modeled in general as a random signal. The random signal formulation is generated by taking v(n) and s(n) to be random variables for each value of the time index n. We begin by presenting the fundamentals of random variables in Section 2.1, and then in Section 2.2 we consider random discrete-time signals. In the last section of the chapter, we study linear time-varying and time-invariant discrete-time systems driven by random signal inputs. The treatment of random signals and systems with random inputs given in this chapter is presented in sufficient depth to allow the reader to then follow the development of optimal filtering given in this text.
E. W. Kamen, J. K. Su
Chapter 3. Optimal Estimation
Abstract
The preceding chapters provide the background necessary to introduce the optimal estimation problem. An “optimal estimate” is a best guess. However, we may express the “goodness” of an estimate in different ways, depending upon the particular engineering problem. After presenting the basic optimal estimation problem and some desirable properties of an estimate, we introduce three commonly-used optimality criterion: the maximum-likelihood, maximum a posteriori, and minimum mean-square error criteria. Each leads to a different estimate and a different form for the estimator. The estimators we discuss are typically implemented in digital systems, so we restrict ourselves to discrete-time signals and systems. Finally, we compare and contrast the different approaches.
E. W. Kamen, J. K. Su
Chapter 4. The Wiener Filter
Abstract
This chapter introduces the Wiener filter, which is used in many control and signal-processing applications. The Wiener filter is a LTI filter, and it may have different forms, depending upon the constraints imposed on the filter (e.g., finite or infinite impulse response, and causality). For the given constraints, the Wiener filter produces the LMMSE estimate of a signal s(n).
E. W. Kamen, J. K. Su
Chapter 5. Recursive Estimation and the Kalman Filter
Abstract
In this chapter we study estimation based on the causal, available past. Chapter 4 discussed the three Wiener filters: noncausal, causal, and FIR. Certainly, the noncausal Wiener filter cannot be employed for causal estimation. The causal Wiener filter requires all observations from the entire past: from time n = — ∞ to the present. Finally, the FIR filter uses only the N most-recent observations. At time N, the observation z(0) is discarded, at time N + 1, z(1) is discarded, and so on. In other words, past observations that might contain information about s(n) are abandoned. However, if we could store the entire past, we would expect our estimator to perform better.
E. W. Kamen, J. K. Su
Chapter 6. Further Development of the Kalman Filter
Abstract
The previous chapter took us from the idea of recursive estimation to the Kalman filter. This chapter contains several extensions of the Kalman filter. It begins with a discussion of the innovations, which are the estimation error for predicting the measurements. The innovations can be viewed as the “new information” about x(n) that is conveyed by z(n) and have a number of interesting and useful properties. An alternate derivation, based on the innovations, of the Kalman filter follows.
E. W. Kamen, J. K. Su
Chapter 7. Kalman Filter Applications
Abstract
At this point, we have derived the Kaiman filter, presented some of its important properties, and demonstrated some simple examples. In this chapter, we examine some applications employing the Kaiman filter. We first present the problem of tracking a single target based on noisy measurements. In this case, the SMM may be unstable, since the position of the target need not be zero-mean. We also consider three special cases of Kaiman filtering: the case of colored (non-white) process noise, the case of correlated process and measurement noises, and the case of colored measurement noise. The target tracking problem is revisited for the case of measurements in polar, rather than Cartesian, form. Finally, we show how the Kaiman filter can be used to estimate the parameters of a LTI system.
E. W. Kamen, J. K. Su
Chapter 8. Nonlinear Estimation
Abstract
The Kalman filter is based on the assumption of a linear SMM. However, many physical systems are described by nonlinear equations. In this chapter, we consider modifying the Kalman filter to address nonlinear systems. We derive the resulting estimator, known as the extended Kalman filter (EKF).
E. W. Kamen, J. K. Su
Backmatter
Metadaten
Titel
Introduction to Optimal Estimation
verfasst von
E. W. Kamen, PhD
J. K. Su, PhD
Copyright-Jahr
1999
Verlag
Springer London
Electronic ISBN
978-1-4471-0417-9
Print ISBN
978-1-85233-133-7
DOI
https://doi.org/10.1007/978-1-4471-0417-9