The objective of the research is to understand how a machine or an organism can adapt to a time-varying environment, particularly when it is rapidly time-varying. Under such conditions, conventional adaptation (based on incremental adjustments) may be too slow to cope with the changes. Obviously a new approach is needed, and the multiple-model method is an attractive candidate.
Rapidly varying environments arise in medicine, neuroscience, vision, economics, and other disciplines. The world we live in is an excellent example, where sudden and discontinuous changes are possible. The multiple model based approach described in the proposal can be thought of as a method which attempts to classify (or identify) the different situations that might arise, and proposes the appropriate actions that should be taken.
Intellectual Merit
From a mathematical standpoint, the research proposed opens a new area of adaptive control. It attempts to answer an important theoretical question which will have significant impact on problems that arise in many other disciplines.
To the non-expert it will provide some understanding of the limits to adaptive control. More specifically, it attempts to answer questions related to how fast the environment can change, if we are to adapt successfully to it.
Broader Impacts
The research, when completed, will provide both graduate and undergraduate students, much needed training in new areas of mathematics related to advanced control methods.
Undergraduates at Yale, who are women and underrepresented minority students will be actively encouraged by the PI to participate in the projects during the summers as well as to work on them under this direction for their senior projects.
The PI organizes seminars regularly at Yale, He also delivers both scientific and popular lectures (e.g. Peabody Museum, Whitney Senior Citizen Center) both in the United States and around the world, and organizes the bi-annual Yale Workshop on Adaptive and Learning Systems. The results of the research carried out will be widely disseminated at these different settings. Finally, the mathematical results will be published in leading control journals.
Introduction: By "control of a process" we mean qualitatively the ability to direct, alter, or improve its behavior. A control system is one in which some quantities of interest are maintained around prescribed values, e.g temperature, population, Gross National Product. As such millions of control systems are used around the world every day. Adaptive Control: When a system is completely known, it can be controlled efficiently using the principles developed in control theory. However, if there are large uncertainties in the system, the control has to be gradually altered based on the response of the system. Mathematical models of the system are built and adjusted so that their behavior approximates that of the system to be controlled. Using the model, a suitable control input is generated; the more accurate the model, the better is the control. A doctor diagnosing a patient and prescribing appropriate medication is an example of an adaptive system. As more information concerning the patient becomes available, the doctor is better equipped to prescribe more precise medication. Adaptive Control using Multiple Models: How would one pose the same problem described above if there are several doctors examining the patient? This is what is referred to as the multiple model problem. What control action should the patient take if the different doctors propose different solutions? This is a question that arises every day in a large number of contexts. Stockbrokers suggesting different stocks, real estate agents recommending different properties, different advisors suggesting different courses of action are typical examples. These problems are also similar to the way one makes up one’s mind while listening to a panel discussion of experts on any subject. Multiple- Models and Switching and Tuning: In the past ten years the approach in adaptive control has been to evaluate all the models over a short period of time and choose the control action suggested by that model which is deemed "the best". In the diagnosis analogy discussed earlier, this corresponds to the patient choosing that doctor’s suggestion whose predictions were closest to what transcribed over a preceding interval of time. (We note that future action can be based only on past performance). The New Approach: The research investigated under the new grant differs markedly from earlier methods in two ways. The first is in the number of models required, and the second is in the manner in which the information provided by all the models is used to make a decision. In the analogy described earlier, the patient does not merely follow the recommendations of one doctor but considers all the suggestions made by the different doctors and arrives at a decision regarding his/her future course of action. The method is found to be significantly better, both with respect to speed and accuracy, as compared to earlier methods and is found to be much better suited to situations in which the uncertainty is constantly changing with time. Applications: The need for new tools for reacting effectively to large and/or rapidly changing uncertainties is arising increasingly in widely differing fields including biology, medicine, economics, finance and various engineering problems such as energy management, renewable power generation, aircraft and automotive control, and national security. These problems can be cast as adaptive control problems described earlier and the methods developed are found to be particularly effective while dealing with them. Results: 1. The results have been published widely in several technical reports, refereed journals, and presented at national and international conferences. 2. The PI has given Plenary lectures on the subject at an IFAC (International Federation of Automatic Control) Conference in Antalya, Turkey, and at an international conference in Santiago, Chile, a Centenary Lecture in India and a distinguished lecture at Indiana University. 3. Undergraduates have worked with the PI during the summers, and the PI is also training three doctoral students to continue the work. 4. The basic ideas contained in the new approach have also been disseminated by the PI in several popular lectures.
