Books and Reports

• Robust Modal Control.
• Robust Flight Control. A Design Challenge.
• Linear Fractional Representation Toolbox (report).

Robust Modal Control with a Toolbox for Use with Matlab®.

Published by Kluwer Academic/Plenum Publishers.
ISBN 0-306-46773-9, February 2002, 334pp.

Links: Kluwer/Springer, Amazon, The MathWorks Book Program.

Key words: pole assignment, eigenstructure assignment, multimodel control, phase control, observer-based control, feedforward design, modal analysis.

Contents.

• The first chapter presents a selection of modal control design techniques. Most results are stated without proof, but discussions are often proposed. I have eliminated all modal techniques that are not reliable for practical control design (for example in my own field of interest: exact pole assignment by output feedback, the Geometric Approach). I have only selected the techniques that were successfully applied.
• The second chapter illustrates the survey given in chapter 1 and the use of the Matlab tools of chapter 3. It is written so as it can be a possible entry point for readers who are not keen on reading before designing. The introduction of this chapter is in fact a summary of the recommendations scattered in Chapter 1. All illustrative examples are relative to aeronautical problems but can be easily adapted to alternative kinds of problems.
• The third chapter is the reference manual for about 50 Matlab functions. Note that the on-line help message of each function contains a describing part "help1" and an illustrative example "help2". A function "rmctdemo" permits the user to run all the illustrative examples proposed in the manual.
• The last chapter gives the proofs of the results stated in the first chapter.

Some points addressed in the book.

• In addition to standard eigenstructure assignment based on decoupling ideas, assigned eigenvectors can be chosen as follows:
  • eigenvectors selected from projection of the open-loop ones,
  • eigenvectors selected from projection of those assigned using an alternative design technique (e.g. mu-synthesis),
  • use of a minimum energy concept.
  These ideas enlarges the field of applications of modal control (that was mostly limited to aeronautics).
• Observer-based feedback design is presented in an original way. First, we suggest to connect the designed observer to the considered system, and then, to design an output feedback law. The measurements considered for that purpose are the original measurements plus the observed signals. This strategy permits us to reduce observer order to its strict minimum, usually less than the "minimum order" considered in more standard work.
• An original technique for dynamic feedforward design is proposed. It can be viewed as a MIMO generalization of SISO pole/zero cancellation. But our MIMO treatment takes also eigenvectors into account.
• Algebraic approach to multimodel design. One of the main interests of this toolbox/book is that it treats in an original and quite simple way the problem of robust control design against real parameter variations. Using a multimodel approach, almost no conservatism is introduced, the feedback gains are simple and, in addition, can be structured. Most alternative robust control design techniques dealing with real parameter robustness ((D,G,K)-iterations, Quadratic stability,...) introduce much conservatism and often lead to very high dimensional controllers that are not realistic.
• Modal analysis is proposed for identifying dominant modes: modal simulation, evaluation of the degree of controllability. This section is essential as our work is based on "dominant mode" treatment.
• Dominant mode analysis combined with dynamic feedback design (i.e. the aforementioned multimodel technique applied to a single model) permits us to reproduce a low dimensional modal version of a controller designed using an alternative approach (e.g. mu-synthesis). Then, modal multimodel design can be applied in order to treat efficiently robustness against real parameter variations. This is an efficient way to combine frequency and time domain techniques.
• Iterative approach to multimodel design. It is a technique which is closer to "continuation techniques" than to non-convex optimization. Convergence properties are quite good.
• A section is devoted to recommendations that are expected to prevent classical errors which resulted in the erroneous conclusion that modal control is a bad approach for robust control design.

Robust Flight Control: A Design Challenge.

Edited by J.F. Magni, S. Bennani and J. Terlouw. Springer Verlag, Lecture Notes in Control and Information Sciences, number 224, 1997.

Download the book (out of print, we are authorized to offer it for download).

Contents. This book summarizes the activities of the GARTEUR FM(AG08) group.

• Introductory part

  • Introduction
    Jan Terlouw and Chris Fielding

• Tutorial part

  • Multi-Objective Parameter Synthesis (MOPS).
    Georg Gruebel and Hans-Dieter Joos
  • Eigenstructure Assignment.
    Lester Faleiro, Jean-François Magni, Jesus M. de la Cruz and Stefano Scala
  • Linear Quadratic Optimal Control.
    Francesco Amato, Massimiliano Mattei and Stefano Scala
  • Robust Quadratic Stabilization.
    Germain Garcia, Jacques Bernussou, Jamal Daafouz and Denis Arzelier
  • H-infinity Mixed Sensitivity.
    Mark R. Tucker and Daniel J. Walker
  • H-infinity Loop Shaping.
    George Papageorgiou, Keith Glover, Alex Smerlas and Ian Postlethwaite
  • mu-Synthesis.
    Samir Bennani, Gertjan Looye and Carsten Scherer
  • Nonlinear Dynamic Inversion.
    Binh Dang Vu
  • Robust Inverse Dynamics Estimation.
    Ewan Muir
  • A Model Following Control Approach.
    Holger Duda, Gerhard Bouwer, J.-Michael Bauschat and Klaus-Uwe Hahn
  • Predictive Control.
    Jan Maciejowski and Mihai Huzmezan
  • Fuzzy Logic Control.
    Gerard Schram, Uzay Kaymak and Henk B. Verbruggen

• RCAM part

  • The RCAM Design Challenge Problem Description.
    Paul Lambrechts, Samir Bennani, Gertjan Looye and Dieter Moormann
  • The Classical Control Approach.
    Jim E. Gautrey
  • Multi-Objective Parameter Synthesis (MOPS).
    Hans-Dieter Joos
  • An Eigenstructure Assignment Approach (1).
    Lester Faleiro and Roger Pratt
  • An Eigenstructure Assignment Approach (2).
    Jesus M. de la Cruz, Pablo Ruipérez and Joaquin Aranda
  • A Modal Multi-Model Approach.
    Carsten Doell, Jean-François Magni and Yann Le Gorrec
  • The Lyapunov Approach.
    Jamal Daafouz, Denis Arzelier, Germain Garcia and Jacques Bernussou
  • An H-infinity Approach.
    Mark R. Tucker and Daniel J. Walker
  • A mu-Synthesis Approach (1).
    Samir Bennani and Gertjan Looye
  • A mu-Synthesis Approach (2).
    Jan Schuring and Rob M.P. Goverde
  • Autopilot Design based on the Model Following Control Approach.
    Holger Duda, Gerhard Bouwer, J.-Michael Bauschat and Klaus-Uwe Hahn
  • Flight Management using Predictive Control.
    Mihai Huzmezan and Jan M. Maciejowski
  • A Fuzzy Control Approach.
    Gerard Schram and Henk B. Verbruggen

• HIRM part

  • The HIRM Design Challenge Problem Description.
    Ewan Muir
  • Design via LQ Methods.
    Francesco Amato, Massimiliano Mattei, Stefano Scala and Leopoldo Verde
  • The H-infinity Loop Shaping Approach.
    George Papageorgiou, Keith Glover and Rick A. Hyde
  • Design of Stability Augmentation System using mu-Synthesis.
    Karin Stahl Gunnarsson
  • Design of a Robust, Scheduled Controller using mu-Synthesis.
    Johan Anthonie Markerink
  • Nonlinear Dynamic Inversion and LQ Techniques.
    Beatrice Escande
  • The Robust Inverse Dynamics Estimation Approach.
    Ewan Muir

• Concluding part

  • The Industrial View.
    Chris Fielding and Robert Luckner
  • An Other View of the Design Challenge Achievements.
    Georg Gruebel
  • Concluding Remarks.
    Samir Bennani, Jean-François Magni and Jan Terlouw
  • Used Nomenclature.
    Anders Helmersson and Karin Stahl Gunnarsson

Linear Fractional Representation Toolbox (Matlab®/Scilab)

• Version 1.3 : Modelling, Order Reduction, Gain Scheduling.
• Version 2.0 : Modelling, Order Reduction.

Key words: LFT realization, order reduction, approximation, interpolation in LFT form, nonlinear system modelling, LFT distance, mu-analysis.

Contents

• Chapter 1 gives some hints for getting started with the toolbox.
• Chapter 2 gives some basic definitions such as "LFT realization" and presents basic manipulations such as the "star product" and so on. The objective of this chapter is to give general guidelines for low order LFT generation. For this purpose, a "step by step" or "object-oriented" approach to LFT generation is proposed. From the discussions given in this chapter, it turns out that two main steps are to be considered. First, realization must be done carefully, especially taking advantage of parameter commutativity (details in Chapter 3). Then, some techniques reminiscent of "minimal realization" for standard dynamic systems can be applied for further order reduction (details in Chapter 4).
• Chapter 3. This chapter is devoted to LFT realization. First, are considered the conversions from coprime factorizations and from state-space realizations to input/output LFTs. After this discussion, only the state-space form is considered. The realization techniques that are treated are: Morton's technique, Horner factorization and the structured tree decomposition. It is recommended to use the tree decomposition that is the most efficient technique, in the free version of the toolbox (a "professional version" proposes more advanced symbolic preprocessing tools).
• Chapter 4. This chapter treats a generalization to LFTs of "minimal realization" of standard dynamic systems. Considering LFTs, the size of what is improperly called a "minimal realization" depends on the initial realization considered before order reduction. The reason for that is that parameter commutativity is not taken into account, and in fact, minimality is truly attained only if parameters do not commute. First, it is shown how standard system "minimal realization" techniques can be applied to LFTs (1-D technique). Then the generalized Kalman decomposition is briefly presented (n-D technique). The second technique lead to the so-called "minimal realization". We conclude this chapter by describing a technique that permits us to evaluate precisely LFT approximation errors, and by the way, to model approximation errors.
• Chapter 5. This chapter considers complex uncertainties.
• Chapter 6. This chapter describes the use of LFTs for modelling the continuum of linearized models of a nonlinear system. The dependency of parameters on the equilibrium surface is also taken into account. Finally is considered the problem of transforming tables of numerical data to LFTs (e.g., aerodynamic coefficients).
• Chapter 7 (not addressed in version 2.0). This chapter takes advantage of the toolbox (manipulation of LFTs like matrices) for feedback design. The feedback gains designed as proposed are in LFT form (which can be viewed as a kind of scheduled gains). Two techniques are proposed. The problem of well-posedness and the practical implementation of such gains are considered.

Some features of the toolbox.

• LFR-objects. Step by step LFR-modelling requires various combinations of elementary LFR-objects. By combinations we mean addition (objects in parallel), multiplication (objects in series), concatenation, inversion and so on. Such operations look like matrix manipulations but the underlying computation is much more complex. The first objective of the toolbox is to perform automatically the above manipulations so that the user only needs to enter simple commands, for example commands of the form:

  lfrs x y z t
  sys = [x+y z*(t+y)]*[1 ; x]*(1+x^2)^(-1)

  Conventional objects like constant matrices and dynamic linear systems are automatically converted to LFR-objects when combined with these new objects.
• Interfacing with symbolic tools (only Matlab version). For LFR-modelling it might be necessary to perform some complex computation such as for example, linearizing a nonlinear system. Performing manually such computation is highly risky. To prevent computational mistakes it is recommended to consider symbolic expressions and to convert them to LFR-objects after all risky treatments are performed by automatic symbolic computation. The toolbox provides an interface between Maple (via the Matlab Symbolic Toolbox) and LFR-objects.
• Interfacing with other tools (only Matlab version). Interfaces with other toolboxes for µ-analysis (mu-Analysis and Synthesis Toolbox, LMI Control Toolbox) are available. Version 2.0 of the LFRT is compatible to some extent with version 3 of the Robust Control Toolbox. In the Scilab version, the objects introduced in version 3 of the Robust Control Toolbox are considered as special cases of our LFR-object.
• Interpolation tools. In some cases, especially in aeronautics, it is necessary to transform numerical data tables to LFR-objects. It is also often necessary to approximate some components of a system, for example the equilibrium surface of a nonlinear system (that is often an implicit equation). In this case, by gridding and interpolation it becomes possible to identify parameter dependency at equilibrium. The toolbox proposes the corresponding tools.
• Order reduction. The toolbox makes available most of the known order reduction techniques evoked in the above description of Chapters 3 and 4.
• Approximate modelling. The toolbox proposes a µ-analysis-based technique for an accurate computation of approximation errors. By the way, approximation errors can be replaced by artificial uncertainties for which variation bounds are precisely known.
• Gain scheduling design (not addressed in version 2.0). A feedback gain given as an LFR-object is a kind of scheduled feedback. The toolbox proposes two approaches for designing feedback gains in LFR-object form.
• Scheduled gains analysis (not addressed in version 2.0). When a feedback gain is in LFR-form, it is straightforward to use µ-analysis for robust stability analysis (using the available interfaces with LMI and µ-toolboxes). In addition, we propose parameter gridding utilities so that classical analysis like bode, nyquist, nichols, step responses, pole map can be performed with minimum efforts. Well-posedness analysis of scheduled gains is also addressed.