Fundamental Limitations in Filtering and Control

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.