A course in robust control theory.

*(English)*Zbl 0939.93001
Texts in Applied Mathematics. 36. New York, NY: Springer. xx, 417 p. (2000).

This textbook is an introduction to robust control theory of linear time-invariant systems.

The introduction motivates the reader concerning the role of uncertainties in control systems described by ordinary differential equation and shows how to represent them.

Chapter one presents some basics of linear algebra, convex sets, linear matrix inequalities (LMI).

Chapter two is a standard introduction to linear control systems from the point of view of the state space representation. Controllability, observability, pole placement, transfer functions are defined and characterized.

Chapter three gives some rudiments of a functional analytic background which provides a framework for the handling of signals. The Fourier transform, Hardy spaces, and the relation between causality and \(H_\infty\) are explained.

Chapter four introduces into elementary realization theory, controllability and observability gramians, balanced realizations, Hankel operators, model reduction and truncation.

Chapter five introduces into (dynamic) feedback stabilization of perturbed linear control systems. Input-output stability and internal stability (i.e. the states of the controller should also tend to zero) are defined, characterized and related. (Note that on page 177 the authors state lemmas where “Figure 5.2 (resp. 5.1) is internally stable”.) Finally, the authors describe stabilization in term of LMI and coprime factorization.

Chapter six deals with \(H_2\) optimal control synthesis. One minimizes the \(H_2\) norm of a linear fractional transformation of a feedback interconnection subject to internal stability constraints. The optimal controller is parameterized by the Riccati equation and the authors use the Hamiltonian matrix formulation to derive their results. A section is devoted to linear matrix inequalities for the synthesis.

Chapter seven characterizes the solution to \(H_\infty\) problems with linear matrix inequalities.

Chapter eight presents first the basic uncertainty model, which is a usual feedback interconnection but the feedback is seen as uncertainty. Moreover a disturbance acts on the system. The notion of robust well-connectedness is introduced and equivalent characterizations in the form of a small gain theorem are provided. These concepts characterize also the structured singular value which is defined. It is difficult to compute it in general.

In chapter nine, one combines the configurations of chapter seven and eight and the goal is to achieve robust stabilization. Characterizations are provided in terms of the structured singular value. Synthesis characterizations via LMI in the context of \(H_\infty\) control are given.

Chapter ten and eleven concentrate on more advanced topics. One handles uncertainty via integral quadratic constraints or one studies a robust \(H_2\) performance problem. The last chapter is devoted to discrete linear parameter-varying systems.

Two appendices describe technicalities used in the text. A welcome index of notations is very detailed and references and an index end this book.

Let us mention that more than half of the book is really standard and does not deal with uncertainty (except chap. 5). This topic is treated from chapter eight on, and at the beginning of chapter ten the authors state: “We have achieved the major goals of our course”, so that around 70 pages can be seen as the hard core of this 417 pages text.

This volume can be recommended to students (mainly in automatic control) because it makes research publications in robust control (with their jargon) accessible with a minimal amount of prerequisites. It is clear and elementary. It might be also used by professionals not directly involved in robust control so that they will have quick access to the recent developments in this field.

The introduction motivates the reader concerning the role of uncertainties in control systems described by ordinary differential equation and shows how to represent them.

Chapter one presents some basics of linear algebra, convex sets, linear matrix inequalities (LMI).

Chapter two is a standard introduction to linear control systems from the point of view of the state space representation. Controllability, observability, pole placement, transfer functions are defined and characterized.

Chapter three gives some rudiments of a functional analytic background which provides a framework for the handling of signals. The Fourier transform, Hardy spaces, and the relation between causality and \(H_\infty\) are explained.

Chapter four introduces into elementary realization theory, controllability and observability gramians, balanced realizations, Hankel operators, model reduction and truncation.

Chapter five introduces into (dynamic) feedback stabilization of perturbed linear control systems. Input-output stability and internal stability (i.e. the states of the controller should also tend to zero) are defined, characterized and related. (Note that on page 177 the authors state lemmas where “Figure 5.2 (resp. 5.1) is internally stable”.) Finally, the authors describe stabilization in term of LMI and coprime factorization.

Chapter six deals with \(H_2\) optimal control synthesis. One minimizes the \(H_2\) norm of a linear fractional transformation of a feedback interconnection subject to internal stability constraints. The optimal controller is parameterized by the Riccati equation and the authors use the Hamiltonian matrix formulation to derive their results. A section is devoted to linear matrix inequalities for the synthesis.

Chapter seven characterizes the solution to \(H_\infty\) problems with linear matrix inequalities.

Chapter eight presents first the basic uncertainty model, which is a usual feedback interconnection but the feedback is seen as uncertainty. Moreover a disturbance acts on the system. The notion of robust well-connectedness is introduced and equivalent characterizations in the form of a small gain theorem are provided. These concepts characterize also the structured singular value which is defined. It is difficult to compute it in general.

In chapter nine, one combines the configurations of chapter seven and eight and the goal is to achieve robust stabilization. Characterizations are provided in terms of the structured singular value. Synthesis characterizations via LMI in the context of \(H_\infty\) control are given.

Chapter ten and eleven concentrate on more advanced topics. One handles uncertainty via integral quadratic constraints or one studies a robust \(H_2\) performance problem. The last chapter is devoted to discrete linear parameter-varying systems.

Two appendices describe technicalities used in the text. A welcome index of notations is very detailed and references and an index end this book.

Let us mention that more than half of the book is really standard and does not deal with uncertainty (except chap. 5). This topic is treated from chapter eight on, and at the beginning of chapter ten the authors state: “We have achieved the major goals of our course”, so that around 70 pages can be seen as the hard core of this 417 pages text.

This volume can be recommended to students (mainly in automatic control) because it makes research publications in robust control (with their jargon) accessible with a minimal amount of prerequisites. It is clear and elementary. It might be also used by professionals not directly involved in robust control so that they will have quick access to the recent developments in this field.

Reviewer: A.Akutowicz (Berlin)

##### MSC:

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93B50 | Synthesis problems |

93B36 | \(H^\infty\)-control |

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93C73 | Perturbations in control/observation systems |

93B35 | Sensitivity (robustness) |

93D21 | Adaptive or robust stabilization |

93D09 | Robust stability |

15A39 | Linear inequalities of matrices |