The book is devoted to the effects of finite arithmetic computations on the performance of digital filters, controllers and estimators. In contrast to the history and achievements of standard numerical analysis, this area is relatively new with first results obtained in the mid seventies. The basic approach adopted in the book is the formulation of optimal realization (or parametrization) problems in which the aim is to minimize a measure of the performance degradation due to numerical (roundoff in particular) errors. Indeed, although different realizations are equivalent in exact arithmetic (without roundoff errors), they have at the same time different sensitivities to parameter errors which leads to different propagation of errors.
The book has several objectives: to collect the existing results and present them in a coherent way, to present a number of new optimal filter realization results, to produce optimal and suboptimal realizations in the context of numerical errors, to investigate the optimal parametrization of controllers in terms of reducing the finite precision effects on the closed loop system, to compare the robustness to numerical errors of optimal shift operator state space realizations vs. optimal \(\delta\)-operator realizations and finally to investigate the role of pre-filtering the data in estimation, identification and filtering problems.
The book is divided into 12 chapters, includes 60 figures and 127 references. A number of numerical examples is also included.
A motivation and a general statement of objectives are given in the Introduction. Chapter 2 (Finite Word Length Errors and Computations) is devoted to the fundamentals of finite arithmetic computations (fixed-point computations in particular) in signal processing. We note that there are significant differences between floating- and fixed-point computations. Usually the first are considered in standard numerical analysis while the last are typical for the performance of digital filters. The third chapter (Parametrizations in Digital System Design) presents a review on the existing results on optimal digital filter design. It is shown here that the error properties of a digital filter depend significantly on the particular state space realization. In chapter 4 (Frequency Weighted Optimal Design), the authors give a generalization of the standard \(L_1/L_2\) sensitivity measure for the transfer function \(z\mapsto H(z)= c^{\top}(zI- A)^{-1} b+d\) of the system \(x(t+ 1)= Ax(t)+ bu(t)\), \(y(t)= c^{\top}x(t)+ d\) to a frequency weighted \(L_1/L_2\) measure. In chapter 5 (A New Transfer Function Sensitivity Measure) an \(L_2\) sensitivity measure is considered. It is shown that the corresponding optimal realization problem has a solution and a computational algorithm is described. Chapter 6 (Pole and Zero Sensitivity Minimization) deals with various ways to define pole and zero sensitivities and to solve the corresponding optimal filter design problems. In chapter 7 (A Synthetic Sensitivity-Roundoff Design) a combined sensitivity plus roundoff noise measure is introduced which is a linear combination of the classical roundoff noise gain and the \(L_2\) sensitivity measure introduced in chapter 5. A solution to the corresponding minimization problem is provided. Chapter 8 (Sparse Optimal and Suboptimal Realizations) deals with various techniques to find optimal and suboptimal realizations. The real Schur canonical form relative to orthogonal state transformations is implemented in particular. In this case a sparse and computationally reliable realization is applied. Other sparse forms of the system are also considered. Chapter 9 (Parametrizations in Control Problems) deals with optimal design of controllers in the presence of numerical errors. The linear-quadratic optimization problem and the pole assignment problem are considered in particular. A synthetic compensator design procedure is proposed in chapter 10 (Synthetic Finite Word Length Compensator Design). Chapter 11 (Parametrizations in the Delta Operator) deals with the comparison between the effects on sensitivity and roundoff noise gain of the shift operator parametrization and of a \(\delta\)-operator parametrization. Finally, generalized polynomial parametrizations are considered in chapter 12 (Generalized Transfer Function Parametrizations for Adaptive Estimation). They correspond to expressing the numerator \(p(z)\) and denominator \(q(z)\) of the transfer function \(H(z)= p(z)/q(z)\) in polynomial basis functions that are not limited to the powers of \(z\). Applications of these parametrizations are illustrated by several examples.
This book will be (and already is since it was published in 1993) of interest to graduate students, scientists and engineers working in signal processing and systems theory and especially in the area of adaptive and robust control.