The methods the authors present are designed to advance the solution of linear controller design problems in light of technological developments that have made sophisticated control algorithms feasible, and in light of developments in computing that make the simulated investigation of a large range of algorithms possible. Their goals are to clarify the formulation of the design problem in terms of fundamental specifications (that is, strictly in input/output terms), rather than in terms of a particular solution design method; to demonstrate that a large class of linear controller design problems are actually convex optimization problems; to use the methods of convex optimization to demonstrate that feasible methods exist to solve the design problems; and to emphasize the determination of limits of performance, or feasibility sets, associated with design problems.

They divide the book into four parts, composed of two, three, five, and four chapters respectively, and preceded by a general introductory chapter. The first part, “A Framework for Controller Design,” lays down the formal framework for the concepts of the system, the configuration, the controller, and design goals and objectives. Chapter 2 does this for the system, the control law, and the control configuration, and chapter 3 presents a uniform framework for describing design goals in terms of Boolean design specifications.

Part 2, “Analytical Tools,” describes norms of signals and systems, used to specify design goals, and investigates geometric properties of design specifications. Chapter 4 covers signal norms as a generalization of Euclidean length. Chapter 5 discusses norms of systems in terms of the size of a transfer function or linear-time-invariant system. Chapter6 covers the geometrical properties that design specifications have, and in particular the closed-loop convex design specification.

Part 3, “Design Specifications,” looks at various types of closed-loop convex design specifications. Chapter 7 deals with the specifications of realizability and internal stability, and demonstrates that the resulting set of controllers has an affine geometric structure. Chapter 8 considers performance specifications and shows that many of these are closed-loop convex. Chapter 9 considers specifications limiting differential sensitivity of a closed-loop system and demonstrates that some are closed-loop convex but the general specification is not. Chapter 10 considers robustness specifications and describes a method for formulating these as norm-bound specifications that are closed-loop convex. Chapter 11 goes through a standard example with an attached two-dimensional affine specification that allows for direct graphical depiction of the design specifications as closed-loop convex.

Part 4, “Numerical Methods,” describes numerical methods for solving the controller design problem. Chapter 12 looks at several families of design problems that can be solved exactly with standard analytical methods, such as the linear quadratic regulator. Chapter 13 describes the basic tools of convex nondifferentiable analysis, such as subgradients, directional derivatives, and supporting hyperplanes, and illustrates how to calculate them for design specifications. Chapter 14 describes algorithms for convex optimization, including cutting-plane algorithms and ellipsoid algorithms. Chapter 15 describes how to form the inner and outer finite-dimensional approximations to the controller design problem and how these can be solved, which in turn provides a method for solving the controller design problem to any desired degree of accuracy. The concluding chapter summarizes the main points, discusses applications, and surveys some history of the main ideas.

The book is a clear and careful statement of some important new ideas in control theory. The authors begin each chapter with a few sentences of summary and end each chapter with a helpful section of notes and references. They conclude the book with a 5-page list of symbols, a list and index of acronyms, a 14-page bibliography, and a complete index. They include a large number of diagrams and graphs, which help make their presentation clear. The design and visual impact of the book are excellent. The authors are careful to make most of their presentation self-contained, although they do presume a mathematically sophisticated reader. The lack of exercises makes the book unsuitable as a text. For the control engineer or applied mathematician interested in control problems, however, the volume seems ideal as a self-study introduction to the application of the methods of convexity to linear controller design.