Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects

This book is all about finite wordlength errors in digital filters, con­ trollers and estimators, and how to minimize the deleterious effects of these errors on the performance of these devices. This does by no means imply that all about finite wordlength errors in filters, controllers and estimators is to be found in this book. We first ventured into the world of finite wordlength effects in 1987 when Gang Li began his PhD thesis in this area. Our more experienced readers might well say 'This shows', but we believe that the extent of our new contributions largely offsets our relative inexperience about the subject that might surface here and there in the book. Our naive view on the subject of finite wordlength errors in 1987 could probably be summarized as follows: • numerical errors due to finite wordlength encoding and roundoff are something that one has to live with, and there is probably not much that can be done about them except to increase the wordlength by improvements on the hardware; • these errors are as old as finite arithmetic and numerical analysis and they must therefore be well understood by now; • thus, if something can be done to minimize their effects, it must have been analysed and put into practice a long time ago. It is almost fair to say that we were wrong on all counts.

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.