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Loop-shaping Robust Control

Loop-shaping Robust Control

Philippe Feyel

ISBN: 978-1-118-57475-1

Jun 2013, Wiley-ISTE

288 pages



The loop-shaping approach consists of obtaining a specification in relation to the open loop of the control from specifications regarding various closed loop transfers, because it is easier to work on a single transfer (in addition to the open loop) than on a multitude of transfers (various loopings such as set point/error, disturbance/error, disturbance/control, etc.). The simplicity and flexibility of the approach make it very well adapted to the industrial context.
This book presents the loop-shaping approach in its entirety, starting with the declension of high-level specifications into a loop-shaping specification. It then shows how it is possible to fully integrate this approach for the calculation of robust and efficient correctors with the help of existing techniques, which have already been industrially tried and tested, such as H-infinity synthesis. The concept of a gap metric (or distance between models) is also presented along with its connection with the prime factors of a set of systems shaping a ball of models, as well as its connections with robust synthesis by loop-shaping, in order to calculate efficient and robust correctors. As H-infinity loop-shaping is often demanding in terms of the order of correctors, the author also looks at loop-shaping synthesis under an ordering constraint. Two further promising lines of research are presented, one using stochastic optimization, and the other non-smooth optimization. Finally, the book introduces the concept of correction with two degrees of freedom via the formalism of prime factorization.
Avenues for future work are also opened up by the author as he discusses the main drawbacks to loop-shaping synthesis, and how these issues can be solved using modern optimization techniques in an increasingly competitive industrial context, in accordance with ever more complex sets of functional specifications, associated with increasingly broad conditions of usage.


1. The Loop-shaping Approach
2. Loop-shaping H-infinity Synthesis
3. Two Degrees-of-Freedom Controllers
4. Extensions and Optimizations
Appendix 1. Demonstrative Elements on the Optimization of Robust Stabilization with Order Constraint
Appendix 2. Establishment of Real LMIs for the Quasi-Convex Problem of Optimization of the Weighting Functions

About the Authors

Philippe Feyel is an R&D Engineer for the high-tech company Sagem Défense Sécurité, part of the defence and security business of the SAFRAN group, in Paris, France.

Introduction ix

Chapter 1 The Loop-shaping Approach 1

1.1 Principle of the method 1

1.2 Generalized phase and gain margins 14

1.3 Limitations inherent to bandwidth 17

1.4 Examples 18

1.5 Conclusion 30

Chapter 2 Loop-shaping H∞ Synthesis 33

2.1 The formalism of coprime factorizations 33

2.2 Robustness of normalized coprime factor plant descriptions 42

2.3 Explicit solution of the problem of robust stabilization of coprime factor plant descriptions 54

2.4 Robustness and υ-gap 77

2.5 Loop-shaping synthesis approach 82

2.6 Discrete approach 120

Chapter 3 Two Degrees-of-Freedom Controllers 135

31 Principle 135

3.2 Two-step approach 143

3.3 One-step approach 156

3.4 Comparison of the two approaches 165

3.5 Example 166

3.6 Compensation for a measurable disturbance at the model’s output 174

Chapter 4 Extensions and Optimizations 187

4.1 Introduction 187

4.2 Fixed-order synthesis 188

4.4 Towards a new approach to loop-shaping fixed-order controller synthesis, etc 242                               


Appendix 1 247

Appendix 2 251

Bibliography 255

Index 259