Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects

1 Introduction.- 1.1 Motivation and general statement of objectives.- 1.2 A historical view and some motivating examples.- 1.3 Outline of the book.- 2 Finite Word Length errors and computations.- 2.1 Introduction.- 2.2 Representations of binary numbers.- 2.3 Overflow and quantization errors.- 2.4 Arithmetic computations and roundoff errors.- 2.5 Dynamic range and scaling.- 2.6 Conclusions.- 3 Parametrizations in digital system design.- 3.1 Introduction.- 3.2 State space realization set.- 3.3 Sensitivity measure of a state space realization.- 3.4 Optimal realizations with respect to a sensitivity measure.- 3.5 Roundoff noise gain of state space realizations.- 3.6 Minimal roundoff noise gain realizations.- 3.7 Relationship between sensitivity measure and roundoff noise gain.- 3.8 Examples and simulations.- 3.9 Alternative approaches.- 3.10 Conclusions.- Appendix 3: Proof of Theorem 3.2.- 4 Frequency weighted optimal design.- 4.1 Introduction.- 4.2 Minimization of a frequency weighted sensitivity measure.- 4.2.1 Weighted sensitivity measure of a realization.- 4.2.2 Optimal FWL realizations.- 4.2.3 Existence and uniqueness.- 4.3 Computation of the optimal realization set.- 4.4 Numerical example.- 4.5 Conclusions.- Appendix 4.A: Proof of existence of a minimum.- Appendix 4.B: Computation of weighted Gramians.- 5 A new transfer function sensitivity measure.- 5.1 Introduction.- 5.2 Minimization of an L2 sensitivity measure.- 5.2.1 The L2 sensitivity measure of a realization.- 5.2.2 Optimal L2 sensitivity realizations.- 5.2.3 Solution of the optimal realization problem.- 5.3 Relationship between L1/L2 and L2 sensitivity measures.- 5.4 An example.- 5.5 Conclusions.- 6 Pole and zero sensitivity minimization.- 6.1 Introduction.- 6.2 A pole-zero sensitivity measure.- 6.3 The eigenvalue sensitivity problem.- 6.4 Pole sensitivity minimization and normal matrices.- 6.5 Zero sensitivity measure.- 6.6 Pole-zero sensitivity coordinate dependence.- 6.7 Optimal realizations for pole-zero sensitivity minimization.- 6.8 Numerical example.- 6.9 Conclusions.- 7 A synthetic sensitivity - roundoff design.- 7.1 Introduction.- 7.2 A synthetic FWL noise gain.- 7.3 Optimizing the Total Noise Gain.- 7.4 A numerical example.- 7.5 Conclusions.- Appendix 7: Existence of a constrained minimum.- 8 Sparse optimal and suboptimal realizations.- 8.1 Introduction.- 8.2 Sparse optimal realizations.- 8.2.1 Hessenberg optimal realizations.- 8.2.2 Schur optimal realizations.- 8.2.3 Sparse block-balanced realizations.- 8.3 Theoretical versus actual sensitivity measure.- 8.3.1 Pole sensitivity of Schur realizations.- 8.3.2 A numerical example.- 8.4 Sparse quasi-optimal realizations.- 8.4.1 Constrained similarity transformations.- 8.4.2 Extensions of the Bomar and Hung algorithm.- 8.5 Sparse suboptimal realizations.- 8.6 Conclusion.- 9 Parametrizations in control problems.- 9.1 Introduction.- 9.2 Implementation of a pole placement controller.- 9.2.1 The ideal pole placement controller.- 9.2.2 Finite precision aspects in a closed loop compensator: problem formulation.- 9.2.3 Sensitivity analysis and optimal structures.- 9.2.4 Roundoff noise study and optimal realizations.- 9.2.5 Design example.- 9.3 FWL LQG controller design.- 9.3.1 The 'ideal' LQG controller.- 9.3.2 Roundoff noise study of an LQG controller.- 9.3.3 Optimal implementations of FWL LQG controllers.- 9.3.4 Optimal FWL LQG controllers.- 9.4 Conclusions.- Appendix 9: Sensitivity functions of closed loop system.- 10 Synthetic FWL compensator design.- 10.1 Introduction.- 10.2 State space description of a compensator.- 10.3 Analysis of FWL effects of a compensator.- 10.4 Optimal FWL compensator realizations.- 10.5 A design example.- 10.6 Conclusion.- 11 Parametrizations in the Delta operator.- 11.1 Introduction and motivation.- 11.2 Delta operator parametrizations.- 11.3 Sensitivity of delta parametrizations.- 11.3.1 Sensitivity measure.- 11.3.2 Optimal realization set.- 11.3.3 Numerical example.- 11.4 Roundoff noise analysis.- 11.4.1 Introduction.- 11.4.2 The roundoff noise gain of shift and delta operator realizations.- 11.4.3 Minimization of the roundoff noise gain.- 11.4.4 New conditions for superiority of ?-realizations.- 11.4.5 Numerical examples.- 11.5 Conclusions.- Appendix 11: Proof of Theorem 11.5.- 12 Generalized transfer function parametrizations for adaptive estimation.- 12.1 Introduction.- 12.2 Parameter estimation and the information matrix.- 12.3 Connection between parametrization and data filtering.- 12.4 Optimal and suboptimal design choices.- 12.4.1 Manipulating the information matrix with data information.- 12.4.2 A robustness property.- 12.5 ?-operator parametrizations.- 12.5.1 A transfer function in ?-operator.- 12.5.2 Computing the information matrix.- 12.5.3 Analysis of the information matrix.- 12.5.4 Comparison with ?-operator parametrizations.- 12.6 Applications in estimation and adaptive filtering.- 12.7 Conclusions.- References.