David H. Jacobson discusses the basics of Optimal Control Theory
– the subject of a new book he has written with Jason L. Speyer
Every engineer’s objective is to design the best product or system and to get the highest operating performance out of it, subject of course to practical cost and engineering constraints. The contemporary design of electrical and electronic and other engineering devices and the operation of many complex processes can be enhanced significantly using mathematical and computational techniques of optimization and optimal control.
Systems, such as power-generating and distribution networks, broadband communication systems, aircraft, chemical processes, economic systems, have at the disposal of their designers and operators certain controls which can be modulated so as to enhance some desired property of the system. For example, in commercial aviation, best fuel usage at cruise is an important consideration in profitability of the airlines in these times of harsh business conditions. In economic systems, full employment or the growth of the gross domestic product are measures of economic system performance. These may be enhanced by proper choice of controls such as interest rates determined by the Reserve Bank or altered tax codes devised by the Ministry of Finance.
The essential features of the above-mentioned systems are the dynamic model of a system, the available controls, the measure of system performance, and the constraints under which the system must operate. Models of complex dynamic system often are described by a set of first-order coupled nonlinear differential equations representing the propagation of the state variables as a function of the independent variable, usually time. The vector which describes the state of a physical system at a particular time, the “state vector”, may be composed of position, velocity and acceleration. In the case of an electrical system, the control vector which affects the way in which the solution of the nonlinear differential equation model evolves in time may, for example, be a generator setting or a load-balancing profile. The performance criterion which establishes the effectiveness of the control process on the dynamical system can take many forms. For a power-electrical system, desired performance may be a measure of system efficiency such as primary fuel consumed in generating electricity to meet a specific power demand. The engineering problem to be solved is to optimize the value of the performance criterion subject to the constraints imposed by the system dynamics and other constraints of an engineering nature. Important classes of constraints are those imposed at the termination of an operating shift and path constraints throughout the time of operation of the system. When controlling an aircraft, maximum force constraints or maximum altitude constraints may have to be satisfied.
While optimality conditions which a control strategy must satisfy if it is to be an optimal one are relatively easy to apply, their derivation, in the case of the most general nonlinear models, is highly detailed and abstract and not accessible to most engineers. This is the domain of the mathematician rather than the engineer. This is unfortunate because it is both highly desirable, and satisfying, for engineers to understand the derivations and underlying assumptions of the optimizing techniques which they use. This is in fact essential if they are to apply the right optimization tools for the particular problem at hand.
Primer on Optimal Control Theory (SIAM, April 2010) is a new book by myself and Jason L. Speyer. We set out to get the reader to think about the subject of optimal control at a suitable level of abstraction for engineers. The text is designed to be of value to practitioners and advanced undergraduates and post-graduate students who are studying optimal control theory in Electrical, Mechanical and Chemical Engineering, as well as those students in Computer Science who learn control theory while studying robotics.
The objective of the book is to make optimal control theory accessible to a large class of engineers and scientists who are not mathematicians although they do have a basic mathematical background, and who need to understand and want to appreciate the sophisticated material associated with optimal control theory. Therefore, the material in this new book is presented using elementary mathematics, which is sufficient to treat and understand in a rigorous way the issues underlying the limited class of control problems treated in this text. Although more advanced topics that build on this foundation are covered only briefly, such as inequality constraints, singular control problems and advanced numerical methods, the foundation laid in this accessible text should be adequate to enable the reader to move on to the rich literature on these subjects if he or she has a higher mathematical interest.
The book begins with an example to illustrate simply some of the concepts found in later chapters. These, as well as more advanced topics of optimization in the following chapters are handled using mathematics at a level consistent with that taught in undergraduate and post-graduate engineering courses. Therefore, the treatment in this book is not the most general, but it does cover in an accurate way a large class of optimization problems of practical concern.
Details about the book are available at:
Primer on Optimal Control Theory
by David H. Jacobson and Jason L. Speyer
(Society for Industrial and Applied Mathematics (SIAM), April 2010)
© David H. Jacobson, 2010
SAIEE Past President, SA Math Soc. Past Chairman, Life Fellow IEEE
18 May 2010
David Jacobson (FIEEE) is Director – Emerging Technologies in the Advisory Services practice of PricewaterhouseCoopers LLP, working in the Toronto office. He is the Canadian member of the PwC international technology network which includes the PwC Global Technology Centre resources in California, USA, the UK and Europe. He is a contributing editor of PwC Technology Forecasts and an advisor to the Global Technology Centre. He is a Fellow of the US-based Institute of Electrical and Electronics Engineers (IEEE) and has been a Fellow of the British Institution of Engineering and Technology (IET) and a Chartered Engineer (UK). He has been a member of the advisory board of the emerging technologies journal IEEE Spectrum and is currently on the advisory board of the Canadian journal Telemanagement. He has served as a board member of the Canadian Photonics Consortium. He holds PhD and BSc (With Distinction) degrees in electrical engineering, has been a professor at Harvard University and is an Honorary Professor of the University of the Witwatersrand.