Hybrid Feedback Control

Classical control theory has been very successful for analysis of closed-loop systems and the creation of feedback controllers. It has provided a collection of very powerful tools, useful primarily when systems operate in either continuous-time or discrete-time. Developments in technology, however, have changed the environment quite a bit. The classical feedback picture of system and controller has evolved into hybrid systems with complex interconnections involving the underlying physics, digital devices, computing systems and the interfaces between them.

 

The dynamics of these hybrid systems requires the associated hybrid control systems to manage the interface between the physics and the digital components to exchange information between continuous-time and discrete-time processes. As they do this hybrid control can reset variables when certain events occur, reconfigure to accommodate sporadic availability of information, and incorporate variables that change instantaneously, logical variables, timers, and memory states as part of its state vector. Hybrid control can also stabilize some continuous-time systems that are not stabilizable by classical continuous-state feedback.

 

The author’s goal is to provide an introduction to hybrid feedback control that is self-contained and provides the reader with enough design tools to get started. As the book progresses he considers the question of how one determines if a system is truly hybrid so that one could avoid using the more complex methods of hybrid feedback when it would be overkill, and more conventional control theory would be applicable.

 

Some of the applications of hybrid control deal with robotic systems such as control of a walking robot. Others - like control of thermostats, automatic transmissions, gasoline engines, and embedded control by microchip of a physical device - are more conventional. Somewhat more exotic applications include air traffic control and control of data flow in a large communication network. The common elements are sensors, actuators, digital devices, and the interplay of discrete and continuous variables. 

 

The book is divided roughly into three parts. The early chapters describe the modeling framework for hybrid controllers, provide examples, and introduce formal notions and tools. The next part introduces hybrid control strategies that use two or more feedback controllers, event-triggered control, and supervisory algorithms used to coordinate a family of hybrid controllers. The last part generalizes to present strategies for general hybrid control with control via passivity and Lyapunov functions.

 

The author suggests that his book is primarily directed to graduate students with backgrounds in classical and modern control theory, nonlinear differential equations, and real analysis. Readers without a command of these subjects would find the text very difficult. The approach throughout is largely theoretical with many theorems and their proofs. Numerical simulation is discussed briefly early in the book; the author notes that most of the models of systems he discusses can be simulated using a hybrid equations toolbox available through the book’s website. Many exercises are provided and there is an extensive bibliography.

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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.