Preface

This text encompasses the basic idea of the model-based approach to signal processing by incorporating the often overlooked, but necessary, requirement of obtaining a model initially in order to perform the processing in the first place. Here we are focused on presenting the development of models for the design of model-based signal processors (MBSP) using subspace identification techniques to achieve a model-based identification (MBID) as well as incorporating validation and statistical analysis methods to evaluate their overall performance 1. It presents a different approach that incorporates the solution to the system identification problem as the integral part of the model-based signal processor (Kalman filter) that can be applied to a large number of applications, but with little success unless a reliable model is available or can be adapted to a changing environment 2. Here, using subspace approaches, it is possible to identify the model very rapidly and incorporate it into a variety of processing problems such as state estimation, tracking, detection, classification, controls and communications to mention a few 3,4. Models for the processor evolve in a variety of ways, either from first principles accompanied by estimating its inherent uncertain parameters as in parametrically adaptive schemes 5 or by extracting constrained model sets employing direct optimization methodologies 6, or by simply fitting a black-box structure to noisy data 7,8. Once the model is extracted from controlled experimental data, or a vast amount of measured data, or even synthesized from a highly complex truth model, the long-term processor can be developed for direct application 1 . Since many real-world applications seek a real-time solution, we concentrate primarily on the development of fast, reliable identification methods that enable such an implementation 9-11. Model extraction/development must be followed by validation and testing to ensure that the model reliably represents the underlying phenomenology - a bad model can only lead to failure!

System identification 6 provides solutions to the problem of extracting a model from measured data sequences either time series, frequency data or simply an ordered set of indexed values. Models can be of many varieties ranging from simple polynomials to highly complex constructs evolving from nonlinear distributed systems. The extraction of a model from data is critical for a large number of applications evolving from the detection of submarines in a varying ocean, to tumor localization in breast tissue, to pinpointing the epicenter of a highly destructive earthquake, or to simply monitoring the condition of a motor as it drives a critical system component 1 . Each of these applications require an aspect of modeling and fundamental understanding (when possible) of the underlying phenomenology governing the process as well as the measurement instrumentation extracting the data along with the accompanying uncertainties. Some of these problems can be solved simply with a "black-box" representation that faithfully reproduces the data in some manner without the need to capture the underlying dynamics (e.g. common check book entries) or a "gray-box" model that has been extracted, but has parameters of great interest (e.g. unknown mass of a toxic material). However, when the true need exists to obtain an accurate representation of the underlying phenomenology like the structural dynamics of an aircraft wing or the untimely vibrations of a turbine in a nuclear power plant, then more sophisticated representations of the system and uncertainties are clearly required. In cases such as these, models that capture the dynamics must be developed and "fit" to the data in order to perform applications such as condition monitoring of the structure or failure detection/prediction of a rotating machine. Here models can evolve from lumped characterizations governed by sets of ordinary differential equations, linear or nonlinear, or distributed representations evolved from sets of partial differential equations. All of these representations have one thing in common, when the need to perform a critical task is at hand - they are represented by a mathematical model that captures their underlying phenomenology that must somehow be extracted from noisy measurements. This is the fundamental problem that we address in this text, but we must restrict our attention to a more manageable set of representations, since many monographs have addressed problem sets targeting specific applications 12,13.

In fact, this concept of specialty solutions leads us to the generic state-space model of systems theory and controls. Here the basic idea is that all of the theoretical properties of a system are characterized by this fundamental set of models that enables the theory to be developed and then applied to any system that can be represented in the state-space. Many models naturally evolve in the state-space, since it is essentially the representation of a set of th-order differential equations (ordinary or partial, linear or nonlinear, time (space) invariant or time (space) varying, scalar or multivariable) that are converted into a set of first-order equations, each of which is a state. For example, a simple mechanical system consisting of a single mass, spring, damper construct is characterized by a set of second-order, linear, time-invariant, differential equations that can simply be represented in state-space form by a set of two first-order equations, each one representing a state: one for displacement and one for velocity 12 . We employ the state-space representation throughout this text and provide sufficient background in Chapters 2 and 3.

System identification is broad in the sense that it does not limit the problem to various classes of models directly. For instance, for an unknown system, a model set is selected with some perception that it is capable of representing the underlying phenomenology adequately, then this set is identified directly from the data and validated for its accuracy. There is clearly a well-defined procedure that captures this approach to solve the identification problem 6 -15. In some cases, the class structure of the model may be known a priori, but the order or equivalently the number of independent equations to capture its evolution is not (e.g. number of oceanic modes). Here, techniques to perform order estimation precede the fitting of model parameters first, then are followed by the parameter estimation to extract the desired model 14. In other cases, the order is known from prior information and parameter estimation follows directly (e.g. a designed mechanical structure). In any case, these constraints govern the approach to solving the identification problem and extracting the model for application. Many applications exist, where it is desired to monitor a process and track a variety of parameters as they evolve in time, (e.g. radiation detection), but in order to accomplish this on-line, the model-based processor must update the model parameters sequentially in order to accomplish its designated task. We develop these processors for both linear and nonlinear models in Chapters 4 and 5.

Although this proposed text is designed primarily as a graduate text, it will prove useful to practicing signal processing professionals and scientists, since a wide variety of case studies are included to demonstrate the applicability of the model-based subspace identification approach to real-world problems. The prerequisite for such a text is a melding of undergraduate work in linear algebra (especially matrix decomposition methods), random processes, linear systems, and some basic digital signal processing. It is somewhat unique in the sense that many texts cover some of its topics in piecemeal fashion. The underlying model-based approach of this text is the thread that is embedded throughout in the algorithms, examples, applications, and case studies. It is the model-based theme, together with the developed hierarchy of physics-based models, that contributes to its uniqueness coupled with the new robust, subspace model identification methods that even enable potential real-time methods to become a reality. This text has evolved from four previous texts, 1 , 5 and has been broadened by a wealth of practical applications to real-world, model-based problems. The introduction of robust subspace methods for model-building that have been available in the literature for quite a while, but require more of a systems theoretical background to comprehend. We introduce this approach to identification by first developing model-based processors that are the prime users of models evolving to the parametrically adaptive processors that jointly estimate the signals along with the embedded model parameters 1 , 5 . Next, we introduce the underlying theory evolving from systems theoretic realizations of state-space models along with unique representations (canonical forms) for multivariable structures 16,17. Subspace identification is introduced for these deterministic systems. With the theory and algorithms for these systems in hand, the algorithms are extended to the stochastic case, culminating with a combined solution for both model sets, that is, deterministic and stochastic.

In terms of the...