In control theory, the mathematics of observability are well understood for linear systems. Martinelli addresses the far more complex problem of nonlinear systems, in particular those that include some unknown inputs. His approach is based on the geometric approach to control theory, which models the evolution of both system state (including observable outputs) and the control input as trajectories on a manifold. Generalizing from the Euclidean space that hosts linear systems to a manifold allows representation of nonlinear dynamics.

The two main problems in control theory are controllability (can the available inputs move the system between any two allowable states in finite time?) and observability (does knowledge of the inputs and the outputs determine the state of the system?). The author’s focus is on the latter, which has posed some knotty problems, notably obtaining a complete analytic characterization of the observability of nonlinear systems with unknown inputs. His contribution, which allows him to resolve this problem, lies in the application of intuitions and formalisms from theoretical physics, notably tensor methods from general relativity (naturally suited to reasoning over nonlinear manifolds) and a focus on solving a problem by characterizing the set of transformations under which it is invariant.

An important component of his solution is recognizing that time plays two different roles in control theory. On the one hand, it is an index that synchronizes inputs and outputs, described as “chronological time,” that is associated with a special “sensor,” the system clock. On the other, in systems with drift, it functions as an input to the system, described as “ordinary time.” Including the ordinary time in the system inputs and functions of the chronological time in the state space and system outputs is central to characterizing the group of invariance for control systems.

The introductory chapter describes the observability problem, particularly for nonlinear systems and unknown inputs, and points out the two different roles of time. The second chapter reviews the mathematical tools used in the book: manifolds, tensors, distributions and codistributions, and Lie groups. This section is appropriately labeled “Reminders on Tensors and Lie Groups.” It assumes that the reader is familiar with these concepts, and tersely summarizes them to fix terminology for the rest of the exposition.

Chapters 3 through 5 deal with observability in nonlinear systems. Chapter 3 characterizes the group of invariance appropriate to the problem, drawing on the distinction between the two roles of time to define the chronostate (including a zeroth component dependent on chronological time as well as more conventional state variables) and the chronospace (the corresponding analog of the state space). It characterizes first the required invariances under transformation of the system inputs and outputs without unknown inputs, and then deals with systems with such inputs, merging the input and output transformations into a single group called the “simultaneous unknown input-output (SUIO) transformations group.” Chapter 4 uses this group to construct a theory of nonlinear observability without unknown inputs that extends the basic observability rank condition, and chapter 5 illustrates the theory with a number of applications.

Chapters 6 through 8 extend the theory to deal with unknown inputs. Three appendices cover the proof of a critical theorem in chapter 4, a summary of formalisms on quaternions and rotations used elsewhere in the volume, and defining a canonical form with respect to unknown inputs. The book includes a bibliography through 2020, and a short index.

The major contribution of this volume, beyond the previously unresolved problems in nonlinear control theory that it addresses, is a demonstration of the conceptual power of bringing to bear a new perspective (in this case, theoretical physics). However, this very strength makes the volume difficult to read. Geometric control theory is already a complex discipline. The field has spawned much literature since its introduction in the 1960s, and this volume is clearly addressed to those who are familiar with the discipline and its conventional methods. However, it also assumes familiarity with the constructs from theoretical physics with which it enriches that discipline, and it would be more accessible if those contributions were introduced in a more tutorial fashion. Researchers in geometric control theory will find the volume an important contribution to the literature and a concise summary of many individual publications by the author, but control neophytes who are tantalized by its promise will need to develop their toolbox both in classical geometric control theory and in physics to benefit fully from its insights.