##
**Linear matrix inequalities in system and control theory.**
*(English)*
Zbl 0816.93004

SIAM Studies in Applied Mathematics. 15. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xii, 193 p. (1994).

This is an excellent, concisely written research monograph which allows to consider a large variety of results on linear matrix inequalities in system and control theory (starting with the famous Lyapunov matrix inequality more than 100 years ago and including results connected with the names of Lur’e and Postnikov in the 1940’s, Kalman, Yakubovich, Popov in the early 1960’s, Zames, Sandberg, J. C. Willems, etc.) in the framework of a unified computational approach. As the authors state in the preface, “the basic topic of this book is solving problems from system and control theory using convex optimization. We show that a wide variety of problems arising in system and control theory can be reduced to a handful of standard convex and quasiconvex optimization problems that involve matrix inequalities. For a few special cases there are ‘analytical solutions’ to these problems, but our main point is that they can be solved numerically in all cases. These standard problems can be solved in polynomial-time (by, e.g., the ellipsoid algorithm of Shor, Nemitovskij, and Yudin), and so are tractable, at least in a theoretical sense. Recently developed interior-point methods for these standard problems have been found to be extremely efficient in practice. Therefore, we consider the original problems from system and control theory as solved”.

The book consists of ten chapters. Chapter 1 is a short introduction, including a list of problems to be handled, and a brief history. In Chapter 2 the above mentioned standard optimization problems are listed and the means to treat them (ellipsoid algorithm, interior point methods) are indicated. Chapter 3 deals with some matrix problems (minimizing the condition number or the norm by scaling; quadratic approximation of a polytopic norm; ellipsoidal approximation).

Chapter 4 to 7 are devoted to linear differential inclusions (LDIs), i.e. systems of the type \(\dot x= A(t) x+ B_ w(t) w+ B_ u(t)u\), \(z= C_ z(t) x+ D_{zw}(t) w+ D_{zu}(t)u\) with \[ \left[\begin{matrix} A(t) & B_ w(t) & B_ u(t)\\ C_ z(t) & D_{zw}(t) & D_{zu}(t)\end{matrix}\right]\in \Omega; \] here \(x\) is the state, \(w\) is an exogenous input signal, \(u\) is the control input and \(z\) is the output, and \(\Omega\) has one of the following three special forms: a single point, a polytope, or the image of a unit ball under a matrix linear-fractional mapping. In Chapter 5 the state properties, in Chapter 6 the input/output properties of LDIs are analyzed, in Chapter 7 the synthesis of state-feedbacks for LDIs is discussed.

Chapter 8 is devoted to Lur’e and multiplier methods, Chapter 9 to systems with multiplicative noise. Chapter 10 deals with ‘miscellaneous problems’: optimization over an affine family of linear systems; analysis of systems with linear time-invariant perturbations; positive orthant stability; linear systems with delays; interpolation problems; the inverse problem of optimal control; system realization problems; nonconvex multi-criterion quadratic problems.

Each chapter is complemented by a detailed ‘notes and references’ section which includes a lot of additional material: proofs, precise statements, elaborations, historical notes, and hints to the extensive bibliography (which, according to the authors, in spite of its size of more than 500 titles does not claim for completeness). The book is primarily intended for the researcher in system and control theory, but it can also serve as a source of application problems for researchers in convex optimization. It can be strongly recommended to everybody interested in the subject.

The book consists of ten chapters. Chapter 1 is a short introduction, including a list of problems to be handled, and a brief history. In Chapter 2 the above mentioned standard optimization problems are listed and the means to treat them (ellipsoid algorithm, interior point methods) are indicated. Chapter 3 deals with some matrix problems (minimizing the condition number or the norm by scaling; quadratic approximation of a polytopic norm; ellipsoidal approximation).

Chapter 4 to 7 are devoted to linear differential inclusions (LDIs), i.e. systems of the type \(\dot x= A(t) x+ B_ w(t) w+ B_ u(t)u\), \(z= C_ z(t) x+ D_{zw}(t) w+ D_{zu}(t)u\) with \[ \left[\begin{matrix} A(t) & B_ w(t) & B_ u(t)\\ C_ z(t) & D_{zw}(t) & D_{zu}(t)\end{matrix}\right]\in \Omega; \] here \(x\) is the state, \(w\) is an exogenous input signal, \(u\) is the control input and \(z\) is the output, and \(\Omega\) has one of the following three special forms: a single point, a polytope, or the image of a unit ball under a matrix linear-fractional mapping. In Chapter 5 the state properties, in Chapter 6 the input/output properties of LDIs are analyzed, in Chapter 7 the synthesis of state-feedbacks for LDIs is discussed.

Chapter 8 is devoted to Lur’e and multiplier methods, Chapter 9 to systems with multiplicative noise. Chapter 10 deals with ‘miscellaneous problems’: optimization over an affine family of linear systems; analysis of systems with linear time-invariant perturbations; positive orthant stability; linear systems with delays; interpolation problems; the inverse problem of optimal control; system realization problems; nonconvex multi-criterion quadratic problems.

Each chapter is complemented by a detailed ‘notes and references’ section which includes a lot of additional material: proofs, precise statements, elaborations, historical notes, and hints to the extensive bibliography (which, according to the authors, in spite of its size of more than 500 titles does not claim for completeness). The book is primarily intended for the researcher in system and control theory, but it can also serve as a source of application problems for researchers in convex optimization. It can be strongly recommended to everybody interested in the subject.

Reviewer: W.Müller (Berlin)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

65K05 | Numerical mathematical programming methods |

90C25 | Convex programming |

34D99 | Stability theory for ordinary differential equations |

93D10 | Popov-type stability of feedback systems |

93D15 | Stabilization of systems by feedback |

93E15 | Stochastic stability in control theory |

15A45 | Miscellaneous inequalities involving matrices |