Linear Time-Invariant Systems, Behaviors and Modules

Many engineering students take a required course in systems engineering. Such a course includes topics such as linear time-invariant systems of differential equations, signals, networks, the Laplace transform, transfer functions and a bit of control theory.

 

This book addresses many of these subjects, but it approaches them from a very abstract point of view. Already by the third page of the first chapter homomorphisms from a quotient space into a module of solutions (what the authors call behaviors) has been introduced. Some category theory, covariant and contravariant functors, complexes and exact sequences also appear fairly early. Even Gröbner bases make an appearance. Clearly this is not a text for undergraduate engineering students.

 

The authors’ goals are to derive all the systems-theoretic and related electrical engineering results that are found in standard texts using new mathematical methods (including the module-behavior duality), to provide complete proofs of all results, and to include algorithms that can be implemented in computer algebra systems such as MAPLE.

 

The language of behaviors (as solution sets of a linear system and submodules of a function space) is carried throughout the book. Behavior-module duality replaces the time-frequency duality of more conventional linear systems theory. After the language of behaviors is set up, the authors provide an algebraic characterization of the properties of observability, autonomy, and controllability. These are fundamental aspects of control theory, and the intent here is to analyze the trajectories of controllable behaviors. The authors also treat input-output behaviors and their interconnections at some length and then analyze the transfer matrix operator as an input-output map. 

 

No previous knowledge of systems and control theory is expected. Indeed the authors choose to reconstruct the theory in their own terms and develop all the standard results with their own methods. The authors claim that their approach has several advantages. For example, they say that the behavior-module duality retains information that is lost in the traditional time-frequency domain approach.

 

Those who teach linear systems theory now might have some difficulty recognizing their own subject in this book. It is not clear who the intended readers might be. The mathematical literature in this area has been moving in the direction of greater abstraction, and the authors have taken that considerably further.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.