Digital Controller Implementation and Fragility

In the usual process of control system design, the assumption is made that the controller is implemented exactly. This assumption is usually reasonable, since clearly, the plant uncertainty is the most significant source of uncertainty in the control system, while controllers are implemented with high-precision hardware. However, inevitably, there will be some amount of uncertainty in the controller, a fact that is largely ignored in existing modern advanced robust control techniques. If the controller is implemented by analogue means, there are some tolerances in the analogue components. More commonly, the controller will be implemented digitally, and consequently there will be uncertainty involved with the quantization in the analogue-digital conversion and rounding in the parameter representation and in the numerical computations. A failure to account for these uncertainties in the controller may result in a controller that is "fragile". A controller is fragile in the sense that very small perturbations in the coefficients of the designed controller destabilize the closed-loop control system.
This book collects a number of articles which consider the problems of finite-precision computing in digital controllers and filters. Written by leading researchers, topics that the book covers include:
- analysis of fragility and finite-precision effects;
- design of optimal controller realizations;
- design of non-fragile robust controllers;
- design of low-complexity digital controllers;
- analysis of quantization effects in fuzzy controllers.

1 Finite-precision Computing for Digital Control Systems: Current Status and Future Paradigms.- 1.1 Introduction.- 1.2 Finite-precision Control and Fragility.- 1.3 Hardware Issues and Development of Control System Process ing Structures.- 1.4 Future Paradigms and Relevant Research Problems.- References.- 2 Stability Margins and Digital Implementation of Controllers.- 2.1 Introduction.- 2.2 Digital Implementation.- 2.3 Simulation Setup.- 2.4 Examples.- 2.5 Concluding Remarks.- Acknowledgements.- References.- 3 Finite Word-length Effects in Systems with Fast Sampling.- 3.1 Introduction.- 3.2 The Case of Small Sampling Periods.- 3.3 A Reformulated Aström’s Theorem.- 3.4 Estimation of the Word-length.- 3.5 Examples.- 3.6 Remarks about Regulator Design and Implementation.- 3.7 Problems with Model Identification.- 3.8 Conclusions and Notes.- References.- 4 Implementation of a Class of Low Complexity, Low Sensitivity Digital Controllers Using Adaptive Fixed-point Arithmetic.- 4.1 Introduction.- 4.2 Digital Feedback Controller.- 4.3 Q-Parameterized Controller.- 4.4 Dynamically Scaled Controllers.- 5 Convexity and Diagonal Stability: an LMI Approach to Digital Filter Implementation.- 5.1 Introduction.- 5.2 Convex Approach to the Diagonal Stability Issue.- 5.3 Application to Digital Filter Implementations.- 5.4 Concluding Remarks.- 6 The Determination of Optimal Finite-precision Controller Realisations Using a Global Optimisation Strategy: a Pole-sensitivity Approach.- 6.1 Introduction.- 6.2 Problem Formulation.- 6.3 A New Pole-sensitivity Stability Related Measure.- 6.4 Optimisation Procedure.- 6.5 A Numerical Example.- 6.6 Conclusions.- References.- 6.A Appendix - Theorem Proof.- 6.B Appendix - Adaptive Simulated Annealing.- 7 Computational Algorithms For Sparse Optimal Digital Controller Realisations.- 7.1 Introduction.- 7.2 Digital Controller Coefficient Quantisation.- 7.3 Stability-optimal Controller Realisations.- 7.4 Numerical Issues.- 7.5 Concluding Remarks.- References.- 8 On the Structure of Digital Controllers in Sampled-data Systems with Stability Consideration.- 8.1 Introduction.- 8.2 Digital Controller State Space Implementation.- 8.3 A Stability Robustness Related Measure.- 8.4 Optimal Controller Structures.- 8.5 Sparse Structures.- 8.6 A Design Example.- References.- 9 An Evolutionary Algorithm Approach to the Design of Finite Word-length Controller Structures.- 9.1 Introduction.- 9.2 Multi-objective Optimisation.- 9.3 Evolutionary Algorithms and the Multi-objective Genetic Algorithm.- 9.4 A Linear System Equivalence Completion Problem.- 9.5 FWL Controller Structure Design using Evolutionary Computation.- 9.6 Application Example.- 9.7 Concluding Remarks.- References.- 10 Non-fragile Robust Controller Design.- 10.1 Introduction.- 10.2 Robustness and Fragility Analysis.- 10.3 Another View on Robustness and Fragility.- 10.4 Factored Controller Form.- 10.5 Partial Fraction Controller Form.- 10.6 Conclusions.- Acknowledgements.- References.- 11 Robust Resilient Controller Design.- 11.1 Introduction.- 11.2 Robust Stability and Performance.- 11.3 Sufficient Conditions for Robust Stability and Performance.- 11.4 Multiplicative Controller Uncertainty Structure and Guaranteed Cost Bound.- 11.5 Decentralised Static Output Feedback Formulation.- 11.6 Sufficient Conditions for Fixed-order Resilient Compensation with Multiplicative Uncertainty.- 11.7 Additive Controller Uncertainty Structure and Guaranteed Cost Bound.- 11.8 Decentralised Static Output Feedback Formulation.- 11.9 Sufficient Conditions for Fixed-order Resilient Compensation with Additive Uncertainty.- 11.10 Quasi-Newton Optimisation Algorithm.- 11.11 Illustrative Numerical Examples.- 11.12 Conclusion.- References.- 12 Robust Non-fragile Controller Design for Discrete Time Systems with FWL Consideration.- 12.1 Introduction.- 12.2 Problem Statement and Preliminaries.- 12.3 Robust Non-fragile H2 Control with Additive Controller Uncertainty.- 12.4 Robust Non-fragile H2 control with Multiplicative Controller Uncertainty.- 12.5 Example.- 12.6 Conclusion.- Acknowledgements.- References.- 13 Synthesis of Controllers with Finite-precision Considerations.- 13.1 Introduction.- 13.2 A Model for Finite-precision Controller Design.- 13.3 The Noise Model.- 13.4 Finite-precision Effects on Closed-loop Performance.- 13.5 Optimal Controller Coordinates.- 13.6 Optimal Controller Design.- 13.7 Skewed Sampling.- 13.8 A Numerical Example.- Acknowledgements.- References.- 13.A Proof of Lemma 13.1.- 13.B Proof of Lemma 13.2.- 13.C Proof of Lemma 13.3.- 14 Quantisation Errors in Digital Implementations of Fuzzy Controllers.- 14.1 Introduction.- 14.2 Fuzzy Systems.- 14.3 Sources of Quantisation Errors.- 14.4 Digitised FLCs.- 14.5 Consequences of the Digitisation in Feedback Fuzzy Systems.- 14.6 Conclusions.- References.