Fundamental Limitations in Filtering and Control

This book deals with the issue of fundamental limitations in filtering and control system design. This issue lies at the very heart of feedback theory since it reveals what is achievable, and conversely what is not achievable, in feedback systems. The subject has a rich history beginning with the seminal work of Bode during the 1940's and as subsequently published in his well-known book Feedback Amplifier Design (Van Nostrand, 1945). An interesting fact is that, although Bode's book is now fifty years old, it is still extensively quoted. This is supported by a science citation count which remains comparable with the best contemporary texts on control theory. Interpretations of Bode's results in the context of control system design were provided by Horowitz in the 1960's. For example, it has been shown that, for single-input single-output stable open-loop systems having rela­ tive degree greater than one, the integral of the logarithmic sensitivity with respect to frequency is zero. This result implies, among other things, that a reduction in sensitivity in one frequency band is necessarily accompa­ nied by an increase of sensitivity in other frequency bands. Although the original results were restricted to open-loop stable systems, they have been subsequently extended to open-loop unstable systems and systems having nonminimum phase zeros.

Table of Contents:
I Introduction.- 1 A Chronicle of System Design Limitations.- II Limitations in Linear Control.- 2 Review of General Concepts.- 3 SISO Control.- 4 MIMO Control.- 5 Extensions to Periodic Systems.- 6 Extensions to Sampled-Data Systems.- III Limitations in Linear Filtering.- 7 General Concepts.- 8 SISO Filtering.- 9 MIMO Filtering.- 10 Extensions to SISO Prediction.- 11 Extensions to SISO Smoothing.- IV Limitations in Nonlinear Control and Filtering.- 12 Nonlinear Operators.- 13 Nonlinear Control.- 14 Nonlinear Filtering.- A Review of Complex Variable Theory.- A.1 Functions, Domains and Regions.- A.2 Complex Differentiation.- A.3 Analytic functions.- A.3.1 Harmonic Functions.- A.4 Complex Integration.- A.4.1 Curves.- A.4.2 Integrals.- A.5 Main Integral Theorems.- A.5.1 Green’s Theorem.- A.5.2 The Cauchy Integral Theorem.- A.5.3 Extensions of Cauchy’s Integral Theorem.- A.5.4 The Cauchy Integral Formula.- A.6 The Poisson Integral Formula.- A.6.1 Formula for the Half Plane.- A.6.2 Formula for the Disk.- A.7 Power Series.- A.7.1 Derivatives of Analytic Functions.- A.7.2 Taylor Series.- A.7.3 Laurent Series.- A.8 Singularities.- A.8.1 Isolated Singularities.- A.8.2 Branch Points.- A.9 Integration of Functions with Singularities.- A.9.1 Functions with Isolated Singularities.- A.9.2 Functions with Branch Points.- A. 10 The Maximum Modulus Principle.- A. 11 Entire Functions.- Notes and References.- B Proofs of Some Results in the Chapters.- B.1 Proofs for Chapter 4.- B.2 Proofs for Chapter 6.- B.2.1 Proof of Lemma 6.2.2.- B.2.2 Proof of Lemma 6.2.4.- B.2.3 Proof of Lemma 6.2.5.- C The Laplace Transform of the Prediction Error.- D Least Squares Smoother Sensitivities for Large ?.- References.