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.

I Introduction.- 1 A Chronicle of System Design Limitations.- 1.1 Introduction.- 1.2 Performance Limitations in Dynamical Systems.- 1.3 Time Domain Constraints.- 1.3.1 Integrals on the Step Response.- 1.3.2 Design Interpretations.- 1.3.3 Example: Inverted Pendulum.- 1.4 Frequency Domain Constraints.- 1.5 A Brief History.- 1.6 Summary.- Notes and References.- II Limitations in Linear Control.- 2 Review of General Concepts.- 2.1 Linear Time-Invariant Systems.- 2.1.1 Zeros and Poles.- 2.1.2 Singular Values.- 2.1.3 Frequency Response.- 2.1.4 Coprime Factorization.- 2.2 Feedback Control Systems.- 2.2.1 Closed-Loop Stability.- 2.2.2 Sensitivity Functions.- 2.2.3 Performance Considerations.- 2.2.4 Robustness Considerations.- 2.3 Two Applications of Complex Integration.- 2.3.1 Nyquist Stability Criterion.- 2.3.2 Bode Gain-Phase Relationships.- 2.4 Summary.- Notes and References.- 3 SISO Control.- 3.1 Bode Integral Formulae.- 3.1.1 Bode’s Attenuation Integral Theorem.- 3.1.2 Bode Integrals for S and T.- 3.1.3 Design Interpretations.- 3.2 The Water-Bed Effect.- 3.3 Poisson Integral Formulae.- 3.3.1 Poisson Integrals for S and T.- 3.3.2 Design Interpretations.- 3.3.3 Example: Inverted Pendulum.- 3.4 Discrete Systems.- 3.4.1 Poisson Integrals for S and T.- 3.4.2 Design Interpretations.- 3.4.3 Bode Integrals for S and T.- 3.4.4 Design Interpretations.- 3.5 Summary.- Notes and References.- 4 MIMO Control.- 4.1 Interpolation Constraints.- 4.2 Bode Integral Formulae.- 4.2.1 Preliminaries.- 4.2.2 Bode Integrals for S.- 4.2.3 Design Interpretations.- 4.3 Poisson Integral Formulae.- 4.3.1 Preliminaries.- 4.3.2 Poisson Integrals for S.- 4.3.3 Design Interpretations.- 4.3.4 The Cost of Decoupling.- 4.3.5 The Impact of Near Pole-Zero Cancelations.- 4.3.6 Examples.- 4.4 Discrete Systems.- 4.4.1 Poisson Integral for S.- 4.5 Summary.- Notes and References.- 5 Extensions to Periodic Systems.- 5.1 Periodic Discrete-Time Systems.- 5.1.1 Modulation Representation.- 5.2 Sensitivity Functions.- 5.3 Integral Constraints.- 5.4 Design Interpretations.- 5.4.1 Time-Invariant Map as a Design Objective.- 5.4.2 Periodic Control of Time-invariant Plant.- 5.5 Summary.- Notes and References.- 6 Extensions to Sampled-Data Systems.- 6.1 Preliminaries.- 6.1.1 Signals and System.- 6.1.2 Sampler, Hold and Discretized System.- 6.1.3 Closed-loop Stability.- 6.2 Sensitivity Functions.- 6.2.1 Frequency Response.- 6.2.2 Sensitivity and Robustness.- 6.3 Interpolation Constraints.- 6.4 Poisson Integral formulae.- 6.4.1 Poisson Integral for S°.- 6.4.2 Poisson Integral for T°.- 6.5 Example: Robustness of Discrete Zero Shifting.- 6.6 Summary.- Notes and References.- III Limitations in Linear Filtering.- 7 General Concepts.- 7.1 General Filtering Problem.- 7.2 Sensitivity Functions.- 7.2.1 Interpretation of the Sensitivities.- 7.2.2 Filtering and Control Complementarity.- 7.3 Bounded Error Estimators.- 7.3.1 Unbiased Estimators.- 7.4 Summary.- Notes and References.- 8 SISO Filtering.- 8.1 Interpolation Constraints.- 8.2 Integral Constraints.- 8.3 Design Interpretations.- 8.4 Examples: Kalman Filter.- 8.5 Example: Inverted Pendulum.- 8.6 Summary.- Notes and References.- 9 MIMO Filtering.- 9.1 Interpolation Constraints.- 9.2 Poisson Integral Constraints.- 9.3 The Cost of Diagonalization.- 9.4 Application to Fault Detection.- 9.5 Summary.- Notes and References.- 10 Extensions to SISO Prediction.- 10.1 General Prediction Problem.- 10.2 Sensitivity Functions.- 10.3 BEE Derived Predictors.- 10.4 Interpolation Constraints.- 10.5 Integral Constraints.- 10.6 Effect of the Prediction Horizon.- 10.6.1 Large Values of ?.- 10.6.2 Intermediate Values of ?.- 10.7 Summary.- Notes and References.- 11 Extensions to SISO Smoothing.- 11.1 General Smoothing Problem.- 11.2 Sensitivity Functions.- 11.3 BEE Derived Smoothers.- 11.4 Interpolation Constraints.- 11.5 Integral Constraints.- 11.5.1 Effect of the Smoothing Lag.- 11.6 Sensitivity Improvement of the Optimal Smoother.- 11.7 Summary.- Notes and References.- IV Limitations in Nonlinear Control and Filtering.- 12 Nonlinear Operators.- 12.1 Nonlinear Operators.- 12.1.1 Nonlinear Operators on a Linear Space.- 12.1.2 Nonlinear Operators on a Banach Space.- 12.1.3 Nonlinear Operators on a Hilbert Space.- 12.2 Nonlinear Cancelations.- 12.2.1 Nonlinear Operators on Extended Banach Spaces.- 12.3 Summary.- Notes and References.- 13 Nonlinear Control.- 13.1 Review of Linear Sensitivity Relations.- 13.2 A Complementarity Constraint.- 13.3 Sensitivity Limitations.- 13.4 The Water-Bed Effect.- 13.5 Sensitivity and Stability Robustness.- 13.6 Summary.- Notes and References.- 14 Nonlinear Filtering.- 14.1 A Complementarity Constraint.- 14.2 Bounded Error Nonlinear Estimation.- 14.3 Sensitivity Limitations.- 14.4 Summary.- Notes and References.- 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.