# ORGANIZATION

The book is organized as follows:

■ Chapter 1 (Part 1): the ц framework is presented. The general case of a closed loop system subject to both parametric uncertainties and neglected dynamics is considered. The first step is to transform this uncertain closed loop into a standard interconnection structure. The s. s.v. is then introduced as a tool for studying the robustness proper­ties of this interconnection structure, and thus equivalently those of the original uncertain closed loop system. Beyond its mathematical definition, a physical interpretation of the s. s.v. is especially given.

■ Chapter 2 (Part 1): the airplane and missile examples are explained in details. The plant model is described, as well as the method for synthesizing the control law. The way to introduce uncertainties in the model is also presented.

■ Chapters 3 and 4 (Part 2): the issue is to transform the ori­ginal uncertain closed loop into the standard interconnection struc­ture. Chapter 3 is devoted to the problem of parametric uncertainties entering the open loop plant model, since this appears as the key is­sue to be solved. The idea is to realize this parametrically uncertain system as an LFT transfer, in which the uncertainties appear as an internal feedback. To a large extent, this difficult problem remains open from a theoretical point of view. A simple solution is presented here for the case of physical systems. As an illustration of the tech­niques of chapter 3, the standard interconnection structures for the airplane and missile problems are obtained in chapter 4.

■ Chapter 5 (Part 3): different ways to compute (bounds of) the clas­sical s. s.v. n are presented. Are considered the particular case of real parametric uncertainties and the general case of mixed uncertainties (i. e. a model perturbation simultaneously containing real parametric uncertainties and neglected dynamics).

■ Chapter 6 (Part 3): the aim is twofold. On the one hand, the H tools developed in chapter 5 are evaluated on the aeronautical ex­amples. On the other hand, the resolution of the first physical prob­lem, which is considered in this book, is detailed (point (1) of the previous section).

■ Chapter 7 (Part 4): through the presentation of some of the phys­ically motivated problems, which are solved in this book, this chapter illustrates in a rather qualitative way that the skewed s. s.v. v can solve a large set of engineering problems. It is emphasized that many /r problems, which are encountered in practice, appear to be skewed /л problems.

■ Chapter 8 (Part 4): because of the practical importance of the skewed s. s.v. v (see chapter 7), it is interesting to develop specific tools for computing skewed fi bounds: this is done in chapter 8.

■ Chapter 9 (Part 4): the aim is twofold. The skewed v tools de­veloped in chapter 8 are evaluated on the missile example. On the other hand, the resolution of the physical problem, which corresponds to point (2) of the previous section, is detailed. Note that the skewed ц tools of chapter 8 will also be used in chapters 11 and 12.

■ Chapters 10, 11 and 12 (Part 5): these chapters present the resolution of the problems, which correspond to points (3) to (5) of the previous section.

This book is organized in a particular way, from the simplest topics to

the most technical ones. The first part introduces the /r framework and

presents the applicative examples. The second part focuses on the way

to transform a parametrically uncertain plant into an LFT form. The third part focuses on the application of classical p tools. All these three parts are expected to be readable by a large audience, with the excep­tion of chapter 5, which presents methods for computing bounds of the s. s.v.. The reading of this chapter can be nevertheless avoided, since a summary is done at the end of this chapter.

The fourth part is devoted to skewed p problems. To a large extent, this part is here again expected to be readable by a large audience, ex­cept chapter 8 which presents computational methods: a summary is nevertheless done at the end of this chapter.

The last part is the most technical. Unlike what is done before, the theoretical and practical results are presented inside a same chapter. Moreover, when compared to the problems of parts 3 and 4, the prob­lems of part 5 are more sophisticated. Nevertheless, part 5 presents new solutions to difficult engineering problems.

1. GLOSSARY

LFT: Linear Fractional Transformation

LMI: Linear Matrix Inequality

LP: Linear Programming

LTI: Linear Time Invariant

MIMO: Multi Inputs Multi Outputs

m. s.m.: multiloop stability margin

SIDF: Sinusoidal Input Describing Function

SISO: Single Input Single Output

s. s.v.: structured singular value ц