- Email: [email protected]
Stochastic Adaptive Control-Integrating Research and Teaching*
11th 11th IFAC IFAC Symposium Symposium on on Advances Advances in in Control Control Education Education 11th IFAC 2016. Symposium on Advances in Control Education June June 1-3, 1-3, 2016. Bratislava, Bratislava, Slovakia Slovakia Available online at www.sciencedirect.com June 1-3, 2016. Bratislava, Slovakia
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Stochastic Stochastic Adaptive Adaptive Control Control -- Integrating Integrating Research and Teaching Research and Teaching ∗∗ ∗∗ Bozenna Pasik-Duncan ∗∗∗ Tyrone E. Duncan Bozenna Bozenna Pasik-Duncan Pasik-Duncan Tyrone Tyrone E. E. Duncan Duncan ∗∗ ∗ ∗ ∗
Department Department Department
of Mathematics, University of Kansas, Lawrence, KS of Mathematics, University of of Mathematics, University of Kansas, Kansas, Lawrence, Lawrence, KS KS 66045 USA, (e-mail: [email protected]) 66045 USA, (e-mail: [email protected]) USA, (e-mail: [email protected]) ∗∗ ∗∗ Department 66045 Mathematics, University of Kansas, Lawrence, KS Mathematics, University of ∗∗ Department of Department of of Mathematics, University of Kansas, Kansas, Lawrence, Lawrence, KS KS 66045 USA, (e-mail: [email protected]) 66045 USA, (e-mail: [email protected]) 66045 USA, (e-mail: [email protected]) Abstract: This paper focuses on innovative methods of teaching stochastic adaptive control Abstract: This paper focuses on innovative methods of adaptive control Abstract: Thiswho paper focusesall on science, innovative methods engineering of teaching teaching stochastic stochastic adaptive(STEM) control with students represent technology, and mathematics with students who represent all science, technology, engineering and mathematics (STEM) with students who represent all science, technology, engineering and mathematics (STEM) disciplines. The Stochastic Adaptive Control course has been developed developed based based on the authors’ disciplines. The Stochastic Adaptive Control course has been on authors’ disciplines. The Stochastic Adaptivethe Control course hasand beenexcitement developed of based on the the adaptive authors’ research area and it demonstrates power, beauty stochastic research area and it demonstrates the power, beauty and excitement of stochastic adaptive research area and it demonstrates the power, beauty and excitement of stochastic adaptive control as aa field that spans STEM. Teaching Teaching is is shown as as a stochastic stochastic process process that that changes changes control as that spans STEM. control asThe a field field thathas spans STEM. Teaching is shown shown as aadepartment stochastic for process that 20 changes in time. course been taught at the mathematics the last years in time. The course has been taught at the mathematics department for the last 20 years in time. The of course has been and taught at the mathematics department for the last 20 years by the team mathematics engineering faculty. The course is very popular, attracts by the team of mathematics and engineering faculty. The course is very popular, attracts by the and team of mathematics andand engineering faculty. The course is very honors popular,theses attracts junior senior undergraduate graduate students, and leads towards for junior undergraduate and graduate students, and towards honors theses for junior and and senior seniorand undergraduate and graduate students, and leads leads towards honors theses for undergraduates, masters and doctoral theses for graduate students. Best practices leading undergraduates, and masters and doctoral theses for graduate students. Best practices leading undergraduates, and masters and doctoral theses for graduate students. Best practices leading to the success are presented. to to the the success success are are presented. presented.
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Keywords: stochastic systems, systems, adaptive control, control, STEM education. education. Keywords: Keywords: stochastic stochastic systems, adaptive adaptive control, STEM STEM education. 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The authors of the paper have an established record The authors of an record The authorscollaboration of the the paper paperin have have an established established record of research the area of stochastic sysof research collaboration in the area of stochastic sysof research collaboration in the area of stochastic systems and control, as well as estimation and stochastic tems and control, as well as estimation and stochastic tems and control, as well as estimation and stochastic adaptive control. Together with some other international adaptive control. Together with some other adaptive control. Together with been some working other international international collaborators the authors have on solving collaborators the authors have been working on solving collaborators the authors have been working on solving stochastic control problems in which a system is described stochastic control problems in which a system is described stochastic control problems in which a system is described by either stochastic differential equations or stochastic by either stochastic differential equations or by either stochastic differential equations or stochastic stochastic partial differential equations with a noise modeled by an partial differential equations with a noise modeled by partial differential equations with a noise modeled by an an ordinary Brownian motion or by a fractional Brownian ordinary Brownian motion or by a fractional Brownian ordinary Brownian motion or by a fractional Brownian motion. The theory of stochastic adaptive control generates motion. The of stochastic adaptive control generates motion. The theory theory ofattractive stochasticproblems adaptive for control generates tremendously many mathematics, tremendously many attractive problems for mathematics, tremendously many attractive problems for mathematics, engineering and computer science as well as economics engineering and computer science as well economics engineering and computer science as open well as as economics and business students. There are many mathematical and business students. There are many open mathematical and business students. There are many open mathematical problems such as the existence of solutions of stochastic problems such as the of solutions of problems such as the existence existence of equation solutions and of stochastic stochastic differential and partial differential there are differential and partial differential equation and there differential and partial differential equation and there are are many challenging numerical and computational problems many challenging numerical and computational problems many challenging numerical and computational problems such as approximate solutions or convergence and rate such as solutions or convergence and such as approximate approximate solutions or convergence and rate rate of convergence of estimators of unknown parameters as of convergence of estimators of unknown parameters as of convergence of estimators of unknown parameters as well as estimates of the so called a Hurst parameter that well as estimates of the so called a Hurst parameter that well as estimates of the so called a Hurst parameter that characterizes aa fractional Brownian motion. The Math 750 characterizes Brownian motion. The 750 characterizes a fractional fractional Brownian motion. The Math Math and 750 course on stochastic adaptive control was developed course on stochastic adaptive control was developed and course on stochastic adaptive control was developed and designed for motivated students from all science, technology, designed for students from science, designed for motivated motivated students(STEM) from all all disciplines science, technology, technology, engineering, and mathematics with the engineering, and mathematics (STEM) disciplines the engineering, and mathematics (STEM) disciplines with with the purpose to increase in a friendly and welcoming way the purpose to increase in a friendly and welcoming way the purpose to increase in a friendly and welcoming way the general awareness of the importance of control and control general awareness of the and control general awareness of cross-disciplinary the importance importance of of control control and control technology and its nature. Stochastic technology and its cross-disciplinary nature. Stochastic technology and its cross-disciplinary nature. Stochastic adaptive control has been recognized naturally by students adaptive control has recognized naturally by adaptive control has been been recognized naturally by students students as a field that spans science, technology, engineering and as a field that spans science, technology, engineering and as a field that spans science, technology, engineering and mathematics (STEM), therefore it has attracted strongly mathematics (STEM), therefore it has attracted strongly mathematics (STEM), therefore it has attracted strongly Research supported part by supported in in part by NSF NSF grant grant DMS DMS 1411412, 1411412, ARO ARO Research Research supported in ,part bySimons NSF grant DMS 1411412, ARO grant grant W911NF-14-10390 W911NF-14-10390 , and and a a Simons Fellowship Fellowship for for the the second second grant W911NF-14-10390 , and a Simons Fellowship for the second author. author. author.
STEM students. The course became quickly popular as a STEM students. The course became quickly as STEM students. The course becametechniques, quickly popular popular as a a powerful resource for ideas, methods, and topics powerful resource for ideas, methods, techniques, and topics powerful resource for ideas, methods, techniques, and topics which when carried on beyond the course lead initially to which when carried on beyond the course lead to which when carried onthen beyond the course lead initially initiallyand to research projects, and with further developments research projects, and then with further developments and research projects, and then with further developments and advances to honors undergraduate, masters and doctoral advances to honors masters and doctoral advances todissertations. honors undergraduate, undergraduate, masters and doctoral theses and The course has been offered every theses and dissertations. The course has been offered every theses and dissertations. The course has been offered every year and has been adapted to current advances in stochastic year and has been adapted to current advances in stochastic year and has been adapted to current advances in stochastic adaptive control following closely the authors research adaptive control following closely adaptive control following closely the the authors authors research research developments in the area. developments in the area. developments in the area. 2. DESCRIPTION OF THE MATH 750–STOCHASTIC 2. OF THE 750–STOCHASTIC 2. DESCRIPTION DESCRIPTION OF CONTROL THE MATH MATHCOURSE 750–STOCHASTIC ADAPTIVE ADAPTIVE CONTROL ADAPTIVE CONTROL COURSE COURSE Stochastic adaptive control theory is concerned with Stochastic adaptive theory is with Stochastic adaptiveof control control theory is concerned concerned with recursive estimation unknown parameters and control for recursive estimation of unknown parameters and control for recursive estimation of unknown parameters and control for systems with uncertainties modeled as random variables or systems with uncertainties modeled as random variables or systems with uncertainties modeled as random variables or random processes. The stochastic adaptive control problem random processes. The stochastic adaptive control problem random processes. The stochastic adaptive control problem is understood as identification and control of unknown is understood as and control of is understood as identification identification and control of unknown unknown stochastic systems. The solution to the stochastic adaptive stochastic systems. The solution to the stochastic adaptive stochastic systems. The solution to the stochastic adaptive control problem consists of the strong consistency of a control problem consists of the strong consistency of a control problem consists of the strong consistency of of a family of estimators for an identification problem and family of estimators for an identification problem and of family of estimators for an identification problem and of self optimality of an adaptive control that the family of self optimality of an control that uses uses the of self optimality oftrue an adaptive adaptive control usesproblem. the family family of estimates as the parameters for aathat control The estimates as the true parameters for control problem. The estimates as the true parameters for a control problem. The theory is motivated by applications in such diverse areas theory is by applications in diverse theory is motivated motivated byand applications in such such diverse areas areas as aerospace guidance control, signal processing and as aerospace guidance and control, signal processing and as aerospace guidance and control, signal processing and communications, manufacturing processes, and financial communications, manufacturing processes, and financial communications, manufacturing processes, and financial economics. The mathematical theory of identification, economics. The mathematical theory of economics. The mathematical theory for of identification, identification, control and stochastic adaptive control models based control and stochastic adaptive control for models control and stochastic adaptive control for models based based on stochastic difference equations such as autoregressive on stochastic difference equations such as autoregressive on stochastic difference equations such as autoregressive processes and stochastic differential equations as Markov processes and differential equations as Markov processes and stochastic stochastic differential equations as Markov diffusion processes have been developed and are presented. diffusion processes have been developed and are presented. diffusion processes have been developed and are presented. The course course main topics: topics: The The course main main topics: (1) Conditional Expectation and their Properties. (1) (1) Conditional Conditional Expectation Expectation and and their their Properties. Properties.
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(2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
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Introduction to Autoregressive and ARMAX models. Linear Systems and their Properties. Linear Filtering Theory, Kalman Filter. Introduction to the Optimal Control Theory. Feedback Control. Martingales and Limit Theorems of Probability Estimation Methods: Least Squares and Maximum Likelihood Method. Introduction to Stochastic Processes: Markov Chains and Brownian Motion as well as fractional Brownian motions. Introduction to System Identification. Identification of Markov Chains and Identification of Linear Systems. Adaptive Control of Markov Chains and Adaptive Control of Linear Systems. Applications of Stochastic Adaptive Control to finance, economics models, telecommunication networks, actuarial sciences, biomedical sciences, and all engineering areas.
Recommended textbooks include: (1) A First Course in Stochastic Processes, S. Karlin, H.M. Taylor (2) Introduction to Stochastic Processes, S. Ross (3) Discrete Time Stochastic Systems, T. Sodestrom (4) Mathematical Theory of Statistics, Hogg/Tanis or Hogg/Craig or any other book (5) Identification and Stochastic Adaptive Control, H.F. Chen, L. Guo (6) Stochastic Systems, Estimation, Identification and Adaptive Control, P.R. Kumar, Pravin Varaiya, Prentice-Hall, 1986; new edition: SIAM, 2016 (7) Stochastic Modeling and Control, M.H.A. Davis, R.B. Vinter (8) On Adaptive Control, (O Sterowaniu Adaptacyjnym) Research Monograph, Habilitation Doctorate Dissertation, (in Polish) B. Pasik, 1986 (9) Linear Systems, P. Antsaklis, A. Michel, McGraw Hill, 1997 (10) Introduction to Mathematical Systems Theory, Linear Systems, Identification and Control, Christian Heij, Andre Ran, Freek van Schagen, Birkhauser Verlag, 2007 (11) Feedback Systems, An Introduction for Scientists and Engineers, K. J. Astrom and R. M. Murray, 2008 The course has been developed by Bozenna Pasik-Duncan based on the author’s habilitation doctorate dissertation written in Polish, and translated into English for the purpose of being used as the lecture notes for the course. The lecture notes have been available to students and have been adapted each year to the current advances in stochastic adaptive control and technology. The course has been taught for the last 25 years by Bozenna Pasik-Duncan with collaboration of Tyrone Duncan. The course requires a very good mathematical understanding in real analysis, probability and statistics as well as good computational skills. The students diversity differs from year to year therefore the way how the course is taught, its content, and illustrative examples of applications differ too.
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3. CLASSROOM TEACHING AS A STOCHASTIC ADAPTIVE CONTROL PROBLEM The classroom with students and their instructor is considered as a controlled system. It is stochastic because there is a lot of randomness in the classroom. Students talk, fall asleep or bring a baby to the classroom, or there is a fire alarm suddenly on during the exam and we must leave the building. In systems theory, we analyze every system carefully. We analyze the existence of a solution and computational aspects of it, we simulate stochastic equations, we collect information, we compose results, etc. We do the same in our teaching. A classroom becomes a scientific laboratory. We collect information. As we introduce ourselves as the course instructors, we ask students to introduce themselves too. The students in the class are unknown to the instructor at the beginning of a semester. They come from different departments with different academic background. The only prerequisites for this course are: good knowledge of real analysis, probability and statistics, very good academic standing, and permission of the instructor. The challenge for the instructor is to learn about students by collecting relevant course information about them. Too many unknowns in the system make the system unknown so learning or identifying the system is a critical issue in the stochastic adaptive control. 4. SHORT BIO AS THE FIRST ASSIGNMENT One of the best practices for learning about the students from the beginning of a semester is to ask them to prepare a well done short bio with information that will help us as instructors to find optimal adaptive strategies. The information should contain students family and academic background, math and science courses taken, any significant recognitions, motivation for taking this particular course, short and long term goals for studying and career, research and real world problem interests, hobbies and favorite things to do during free time. These short bios should be revisited by the instructor as a semester progresses, and students should have an opportunity to update them twice: during the middle and at the end of a semester. We as teachers need to know the students and their interests. This information is important when we design projects for them. There are many unknowns in this system, so we need to estimate (learn) them as in the theory, and at each instant a controller/teacher uses this new estimate in control strategies and adapts the system. In theory we called it adaptation. It is the same in teaching. We collect information, we build a portfolio, we analyze our reports and data after every class, and we also want to do better each time so that in the long run we will do as well as if we knew the system perfectly. In the theory of stochastic adaptive control, this property of adaptive control is called self-tuning. We call this method of teaching scholarship in teaching. We treat teaching as a stochastic process that changes over time, a process with several components such as vision, design, data collection, and data analysis. We integrate teaching and learning. As in the theory, the controller has to learn, so a teacher, as controller, has to learn too, and the system has to learn, meaning the students have to learn.
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5. A CLASS SYLLABUS A class syllabus is prepared as a joint project which is understood as a ”contract” (agreement) between the students and the instructor. Students receive a draft of the class syllabus on the first day of classes. They are given a time to discuss it in groups and they have the opportunity to make their individual modifications. All students have access to all individual modifications. The final version of the syllabus is the result of a collaborative effort. It becomes a ”contract” between students and the instructor. Students play an major role in preparing this important document for which they become co-owners. 6. C 5 : COLLABORATION, COMMUNICATION, CONNECTIONS, CREATIVITY AND CURIOSITY-A KEY FOR A SUCCESS Inviting Tyrone Duncan, an important contributor to stochastic control to teach the introduction to control demonstrates the power of collaboration in the partnership with Bozenna Pasik-Duncan and her passion for stochastic processes and statistics. Thirty years ago an anonymous reviewer of their joint research proposal wrote: ”a marriage made in heaven” meaning that stochastic adaptive control, the topic of their research project = stochastic processes and control. Both instructors participate in a scientific discussion that takes a place in the classroom. Students witness how they both respond to their questions. Students experience and observe two research collaborators in the classroom. They have the first hand learning that collaborative efforts in research and collaborative efforts in teaching are important and rewarding. Communication and writing are equally important. There is a lot of discussion in the classroom to develop good communication skills, and there is a lot of writing in mathematics to develop good skills in writing. Students are expected to read independently, outside of the classroom and to submit written reports from their reading. Stochastic adaptive control taught in this mathematical approach makes it exciting for students with connections in the use of different areas of mathematics and its fundamental concepts such as calculus and limits, linear algebra and vector spaces, difference and differential equations, numerical analysis with convergence and rates of convergence of algorithms, probability with limits theorems of probability, conditional expectations and martingales, stochastic processes, in particular Markov chains and Brownian motion, mathematical statistics with consistency and strong consistency as well as estimation methods, and control theory. This is the only mathematics course at the University of Kansas that brings so many areas of mathematics together. Students are fascinated by building bridges between different fields, the phenomenon which is not observed in a course that focuses on a single area. Control is viewed naturally as a field that spans science, technology, engineering and mathematics (STEM). Curiosity is the most important part of learning in this course as it covers many different topics. Students are engaged in research discussion through curiosity and creativity. They are alert in this course of what they are doing, they ask why things are the way they are, they try different ways to explain to each other what they observe. They are taught to consider how even the most abstract 107
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mathematics they do can be applied to real life problems. The cross-boundaries nature of stochastic adaptive control is exciting for students. It motivates them for being creative in particular while searching for the correct models in the modern real complex world. Reviewing papers is a powerful method to prepare students for writing a scientific project. The recently introduced practice of reviewing a conference paper has turned to be the most effective way of teaching students how to write a scientific project. Students were provided with a sample of very well professionally done reviews. They received a thirty page paper for reviewing with a two week deadline. Students have done outstanding work. Their reviews were very constructive, detailed, providing missing references, providing missing steps in mathematical proofs, indicating weaknesses in interpretations of results and in particular graphs, and amazing remarks on editing. Their reviews have led to a significant improvement of the quality of the paper presentation.
7. GUEST SPEAKERS Guest speakers energize, inspire and motivate students. Former students who took the Stochastic Adaptive Control Course or research collaborators are invited as guest speakers during the middle of a semester to have a conversation with students and share their industrial career experiences. Real world problems generate new mathematical problems. Mathematics of finance and mathematics of networks, as well as understanding of the brain as the most complex system, generate new stochastic control problems, such as the identification, detection and control of stochastic system with a noise modeled by a fractional Brownian motion. Neurologists, mathematicians and engineers from the medical research centers talk about epilepsy and seizures, and they show brain waves and explain how you can use mathematics to detect or predict seizures, and how one can introduce control theory via brain stimulation: openloop versus closed-loop. However, because most people have never experienced an epileptic seizure, they have no feel for these graphs. The musicians can modify a Mozart sonata by adding a noise in the orchestra, duplicating the frequency behavior in powerful ways. Here students discovered amazing creativity and connections of science and art. Bringing research collaborators, especially those from other countries, to the classroom with good preparation for their visits changes students perspectives. It opens their eyes and it awakes their imagination and creativity. They see how math and control can be found everywhere and how often it is hidden. They see the role of the broad and inspiring vision of stochastic adaptive control. They see the power, beauty, and excitement of the cross-boundary nature of control. Students love guest speakers. They find them to be most inspirational and most motivating for continuing studying mathematics and engineering to do research in stochastic adaptive control. In a summary of the reflections from a visitor’s talk a student wrote: ”Today’s visitor is a fine example of how an engineer should proceed with his career and deal with his success. I hope that the inspiration I have drawn from the speaker can help fuel some creativity for me in my field and motivate me to do my work for the betterment of the world.”
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8. VERTICAL AND HORIZONTAL INTEGRATION AND PEER TUTORING In this course with a very diverse group of undergraduate and graduate students, vertical and horizontal integrations as well as peer tutoring play crucial roles by building a community of learners. Vertical integration incorporates students at different levels in their studying and learning approach. It involves and engages undergraduate and graduate students. Horizontal integration incorporates students from various disciplines in learning and in teaching. Through peer tutoring students become teachers for their peers. They enthusiastically volunteer to teach the entire class a topic which they learned well or which they studied independently. A peer tutoring program was created as we identified that at that moment in time, there were some students who were in need of help, and others who could offer that help. This aligns with the important ideas of promoting engagement, communication and an exchange of ideas. Mathematics is often a quiet pursuit, but in this class the students talk with one another, building not only a community of learners but more importantly they are building life-time friendships. One of the most gorgeous, and most rewarding outputs of this C 5 style is collaborative learning and teaching of stochastic adaptive control. 9. FEEDBACK FROM STUDENTS In teaching, we love feedback from students as we love it in stochastic control. Here is a sample of things the instructor did well by recent and former students: ”hands on learning experience with current field research,” ”provide motivation and create interest in the topic,” ”provide many resources in the form of notes, old class projects, suggested reading,” ” letting us participate in real reviewing,” ”guest speaker was a wonderful part of class,” ” the class taught us lots of useful material including the cutting edge of discipline,” ” the class forced us to be independent , it elevated us to the level of independent researchers,” ”by engaging us in research topics this professor makes us feel very independent and very confident,” ”taking this class really opened my eyes to mathematics. Instead of dreading and avoiding proofs, I love them. This professor teaches until we know what is going on and not to follow a schedule she teaches us stochastic adaptive control with a true passion for the subject. She shows us how practical it is in life and how relevant it is for any field of study or career we choose I learned from material, learned from my classmates, learned from myself, and for it all, my life is better. A course with that kind of an impact and outcome is definitely life changing,” ”This professor has used her passion for mathematics and research as a vehicle to inspire a generation of students in the stochastic and control theory. I know this to be true because I am one of these inspired students who is making a career in this area, and there are many others besides me. From her I also learned the value of conducting quality research, and the importance of disseminating it to others through teaching. Additionally, her students come from her course with a firm understanding of maintaining ethics and professional standards in all that we say and do. I have taken these principles to heart, and together with her conscious effort 108
to make each student feel valued and inspired to realize their full potential, apply them in my classrooms and to my graduate students,” ”that appetite for research which she helped nurture and I shared with all others in the stochastic adaptive control course led me to M.I.T. where I completed a Ph.D. in electrical engineering.” 10. A SAMPLE OF THESES MOTIVATED BY THE COURSE A sample of the topics of masters theses and doctoral dissertations as the outcome of research projects initiated in the Math 750-Stochastic Adaptive Control is given now. A stochastic system model for PageRank: parameter estimation and adaptive control. Identification and adaptive control methods for some stochastic systems. From reality to abstract in stochastic processes applied to telecommunications. New approaches in continuous time stochastic adaptive control. Some applications of adaptive control: diffusion approximation and hierarchial approximation. Stochastic adaptive control and its applications to the theory of finance. Adaptive control in queueing systems. On adaptive control of Markov chains. Optimal policy advertising. Ocular artifact removal through ARMAX model system identification using the extended least squares algorithm. Distribution of aggregate claims. Estimation of parameters of branching processes with immigration by adaptive control. Current efforts in computerized epilepsy prediction. Is the stock market efficient? The stochastic effects on a mathematical model for cardiac arrythmia. Bonus-Malus systems in insurance. Biostatistics in Genetics. Markov decision processes and early retirement. A new scheme for traffic estimation and resource allocation for bandwidth brokers. The modeling of risk reserves with skew distributions and control. An optimal trading strategy of a financial portfolio. Numerical methods for parameter estimation in stochastic systems Computational methods for stochastic differential and partial differential equations. Evolutionary dynamics of protean evasion. System identification and parameter estimation in breast cancer models. 11. CONCLUSIONS Students who are broadly prepared, have a higher chance to find attractive jobs. The Math 750-Stochastic Adaptive Control Course prepares students broadly for studying and research. It prepares them for new challenges and new opportunities of the 21st century. These new challenges include: processing information, ability to work with complex systems and diverse groups, ability to handle quick changes, ability to manage diverse groups of people, ability to handle computationally large data set and analyze it using advanced mathematical and statistical methods, ability to understand well all kinds of randomness., ability to balance personal and professional life, ability to be willing to take risks, ability to take advantage of tremendous opportunities. Students and professor have satisfaction from this integrated learning and teaching as the optimization criterion. In this class considered as a controlled system, the optimization criterion is to maximize students’ and professors’ satisfaction from the integrated learning and
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teaching process. The optimal value of the cost criterion is 5 on the scale from 1 to 5 in student evaluations, and all students receive the highest grade in the class based on students’ perfect attendance, group projects, paper reviewing, independent reading and reports, the final project and its presentation. From the syllabus: ”Final Paper is expected to be written in the form and the size of a short research paper, with the title, name, affiliation, abstract, key words, introduction, main results, conclusions, future work, references and acknowledgements if applicable. Ten minute presentations with class discussions will be scheduled.” Integrating research, learning and teaching using the adaptive control approach described above has demonstrated to be most effective. Acknowledgement: Bozenna Pasik-Duncan would like to thank the C21 community of remarkable scholars and educators associated with the University of Kansas Center for Teaching Excellence for inspirations and motivations
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for innovative teaching. The authors wish to thank the reviewers for inspiring and motivating comments. REFERENCES B. Pasik On Adaptive Control Research Monograph, Habilitation Doctorate Dissertation, (in Polish), SGPiS Publications 1986. Mathematics Education of Tomorrow AWM Newsletter Vol. 24. T. E. Duncan and B. Pasik-Duncan Stochastic adaptive control Encyclopedia of Systems and Control Springer 2014. T. E. Duncan and B. Pasik-Duncan Stochastic adaptive control The Control Handbook Taylor and Francis, CRC 2010. B. Pasik-Duncan and M. A. Verleger Education and Qualification for Control and Automation Springer Handbook of Automation Ch. 44, 2009.
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Stochastic Stochastic Adaptive Adaptive Control Control -- Integrating Integrating Research and Teaching Research and Teaching ∗∗ ∗∗ Bozenna Pasik-Duncan ∗∗∗ Tyrone E. Duncan Bozenna Bozenna Pasik-Duncan Pasik-Duncan Tyrone Tyrone E. E. Duncan Duncan ∗∗ ∗ ∗ ∗
Department Department Department
of Mathematics, University of Kansas, Lawrence, KS of Mathematics, University of of Mathematics, University of Kansas, Kansas, Lawrence, Lawrence, KS KS 66045 USA, (e-mail: [email protected]) 66045 USA, (e-mail: [email protected]) USA, (e-mail: [email protected]) ∗∗ ∗∗ Department 66045 Mathematics, University of Kansas, Lawrence, KS Mathematics, University of ∗∗ Department of Department of of Mathematics, University of Kansas, Kansas, Lawrence, Lawrence, KS KS 66045 USA, (e-mail: [email protected]) 66045 USA, (e-mail: [email protected]) 66045 USA, (e-mail: [email protected]) Abstract: This paper focuses on innovative methods of teaching stochastic adaptive control Abstract: This paper focuses on innovative methods of adaptive control Abstract: Thiswho paper focusesall on science, innovative methods engineering of teaching teaching stochastic stochastic adaptive(STEM) control with students represent technology, and mathematics with students who represent all science, technology, engineering and mathematics (STEM) with students who represent all science, technology, engineering and mathematics (STEM) disciplines. The Stochastic Adaptive Control course has been developed developed based based on the authors’ disciplines. The Stochastic Adaptive Control course has been on authors’ disciplines. The Stochastic Adaptivethe Control course hasand beenexcitement developed of based on the the adaptive authors’ research area and it demonstrates power, beauty stochastic research area and it demonstrates the power, beauty and excitement of stochastic adaptive research area and it demonstrates the power, beauty and excitement of stochastic adaptive control as aa field that spans STEM. Teaching Teaching is is shown as as a stochastic stochastic process process that that changes changes control as that spans STEM. control asThe a field field thathas spans STEM. Teaching is shown shown as aadepartment stochastic for process that 20 changes in time. course been taught at the mathematics the last years in time. The course has been taught at the mathematics department for the last 20 years in time. The of course has been and taught at the mathematics department for the last 20 years by the team mathematics engineering faculty. The course is very popular, attracts by the team of mathematics and engineering faculty. The course is very popular, attracts by the and team of mathematics andand engineering faculty. The course is very honors popular,theses attracts junior senior undergraduate graduate students, and leads towards for junior undergraduate and graduate students, and towards honors theses for junior and and senior seniorand undergraduate and graduate students, and leads leads towards honors theses for undergraduates, masters and doctoral theses for graduate students. Best practices leading undergraduates, and masters and doctoral theses for graduate students. Best practices leading undergraduates, and masters and doctoral theses for graduate students. Best practices leading to the success are presented. to to the the success success are are presented. presented.
© 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Keywords: stochastic systems, systems, adaptive control, control, STEM education. education. Keywords: Keywords: stochastic stochastic systems, adaptive adaptive control, STEM STEM education. 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The authors of the paper have an established record The authors of an record The authorscollaboration of the the paper paperin have have an established established record of research the area of stochastic sysof research collaboration in the area of stochastic sysof research collaboration in the area of stochastic systems and control, as well as estimation and stochastic tems and control, as well as estimation and stochastic tems and control, as well as estimation and stochastic adaptive control. Together with some other international adaptive control. Together with some other adaptive control. Together with been some working other international international collaborators the authors have on solving collaborators the authors have been working on solving collaborators the authors have been working on solving stochastic control problems in which a system is described stochastic control problems in which a system is described stochastic control problems in which a system is described by either stochastic differential equations or stochastic by either stochastic differential equations or by either stochastic differential equations or stochastic stochastic partial differential equations with a noise modeled by an partial differential equations with a noise modeled by partial differential equations with a noise modeled by an an ordinary Brownian motion or by a fractional Brownian ordinary Brownian motion or by a fractional Brownian ordinary Brownian motion or by a fractional Brownian motion. The theory of stochastic adaptive control generates motion. The of stochastic adaptive control generates motion. The theory theory ofattractive stochasticproblems adaptive for control generates tremendously many mathematics, tremendously many attractive problems for mathematics, tremendously many attractive problems for mathematics, engineering and computer science as well as economics engineering and computer science as well economics engineering and computer science as open well as as economics and business students. There are many mathematical and business students. There are many open mathematical and business students. There are many open mathematical problems such as the existence of solutions of stochastic problems such as the of solutions of problems such as the existence existence of equation solutions and of stochastic stochastic differential and partial differential there are differential and partial differential equation and there differential and partial differential equation and there are are many challenging numerical and computational problems many challenging numerical and computational problems many challenging numerical and computational problems such as approximate solutions or convergence and rate such as solutions or convergence and such as approximate approximate solutions or convergence and rate rate of convergence of estimators of unknown parameters as of convergence of estimators of unknown parameters as of convergence of estimators of unknown parameters as well as estimates of the so called a Hurst parameter that well as estimates of the so called a Hurst parameter that well as estimates of the so called a Hurst parameter that characterizes aa fractional Brownian motion. The Math 750 characterizes Brownian motion. The 750 characterizes a fractional fractional Brownian motion. The Math Math and 750 course on stochastic adaptive control was developed course on stochastic adaptive control was developed and course on stochastic adaptive control was developed and designed for motivated students from all science, technology, designed for students from science, designed for motivated motivated students(STEM) from all all disciplines science, technology, technology, engineering, and mathematics with the engineering, and mathematics (STEM) disciplines the engineering, and mathematics (STEM) disciplines with with the purpose to increase in a friendly and welcoming way the purpose to increase in a friendly and welcoming way the purpose to increase in a friendly and welcoming way the general awareness of the importance of control and control general awareness of the and control general awareness of cross-disciplinary the importance importance of of control control and control technology and its nature. Stochastic technology and its cross-disciplinary nature. Stochastic technology and its cross-disciplinary nature. Stochastic adaptive control has been recognized naturally by students adaptive control has recognized naturally by adaptive control has been been recognized naturally by students students as a field that spans science, technology, engineering and as a field that spans science, technology, engineering and as a field that spans science, technology, engineering and mathematics (STEM), therefore it has attracted strongly mathematics (STEM), therefore it has attracted strongly mathematics (STEM), therefore it has attracted strongly Research supported part by supported in in part by NSF NSF grant grant DMS DMS 1411412, 1411412, ARO ARO Research Research supported in ,part bySimons NSF grant DMS 1411412, ARO grant grant W911NF-14-10390 W911NF-14-10390 , and and a a Simons Fellowship Fellowship for for the the second second grant W911NF-14-10390 , and a Simons Fellowship for the second author. author. author.
STEM students. The course became quickly popular as a STEM students. The course became quickly as STEM students. The course becametechniques, quickly popular popular as a a powerful resource for ideas, methods, and topics powerful resource for ideas, methods, techniques, and topics powerful resource for ideas, methods, techniques, and topics which when carried on beyond the course lead initially to which when carried on beyond the course lead to which when carried onthen beyond the course lead initially initiallyand to research projects, and with further developments research projects, and then with further developments and research projects, and then with further developments and advances to honors undergraduate, masters and doctoral advances to honors masters and doctoral advances todissertations. honors undergraduate, undergraduate, masters and doctoral theses and The course has been offered every theses and dissertations. The course has been offered every theses and dissertations. The course has been offered every year and has been adapted to current advances in stochastic year and has been adapted to current advances in stochastic year and has been adapted to current advances in stochastic adaptive control following closely the authors research adaptive control following closely adaptive control following closely the the authors authors research research developments in the area. developments in the area. developments in the area. 2. DESCRIPTION OF THE MATH 750–STOCHASTIC 2. OF THE 750–STOCHASTIC 2. DESCRIPTION DESCRIPTION OF CONTROL THE MATH MATHCOURSE 750–STOCHASTIC ADAPTIVE ADAPTIVE CONTROL ADAPTIVE CONTROL COURSE COURSE Stochastic adaptive control theory is concerned with Stochastic adaptive theory is with Stochastic adaptiveof control control theory is concerned concerned with recursive estimation unknown parameters and control for recursive estimation of unknown parameters and control for recursive estimation of unknown parameters and control for systems with uncertainties modeled as random variables or systems with uncertainties modeled as random variables or systems with uncertainties modeled as random variables or random processes. The stochastic adaptive control problem random processes. The stochastic adaptive control problem random processes. The stochastic adaptive control problem is understood as identification and control of unknown is understood as and control of is understood as identification identification and control of unknown unknown stochastic systems. The solution to the stochastic adaptive stochastic systems. The solution to the stochastic adaptive stochastic systems. The solution to the stochastic adaptive control problem consists of the strong consistency of a control problem consists of the strong consistency of a control problem consists of the strong consistency of of a family of estimators for an identification problem and family of estimators for an identification problem and of family of estimators for an identification problem and of self optimality of an adaptive control that the family of self optimality of an control that uses uses the of self optimality oftrue an adaptive adaptive control usesproblem. the family family of estimates as the parameters for aathat control The estimates as the true parameters for control problem. The estimates as the true parameters for a control problem. The theory is motivated by applications in such diverse areas theory is by applications in diverse theory is motivated motivated byand applications in such such diverse areas areas as aerospace guidance control, signal processing and as aerospace guidance and control, signal processing and as aerospace guidance and control, signal processing and communications, manufacturing processes, and financial communications, manufacturing processes, and financial communications, manufacturing processes, and financial economics. The mathematical theory of identification, economics. The mathematical theory of economics. The mathematical theory for of identification, identification, control and stochastic adaptive control models based control and stochastic adaptive control for models control and stochastic adaptive control for models based based on stochastic difference equations such as autoregressive on stochastic difference equations such as autoregressive on stochastic difference equations such as autoregressive processes and stochastic differential equations as Markov processes and differential equations as Markov processes and stochastic stochastic differential equations as Markov diffusion processes have been developed and are presented. diffusion processes have been developed and are presented. diffusion processes have been developed and are presented. The course course main topics: topics: The The course main main topics: (1) Conditional Expectation and their Properties. (1) (1) Conditional Conditional Expectation Expectation and and their their Properties. Properties.
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Introduction to Autoregressive and ARMAX models. Linear Systems and their Properties. Linear Filtering Theory, Kalman Filter. Introduction to the Optimal Control Theory. Feedback Control. Martingales and Limit Theorems of Probability Estimation Methods: Least Squares and Maximum Likelihood Method. Introduction to Stochastic Processes: Markov Chains and Brownian Motion as well as fractional Brownian motions. Introduction to System Identification. Identification of Markov Chains and Identification of Linear Systems. Adaptive Control of Markov Chains and Adaptive Control of Linear Systems. Applications of Stochastic Adaptive Control to finance, economics models, telecommunication networks, actuarial sciences, biomedical sciences, and all engineering areas.
Recommended textbooks include: (1) A First Course in Stochastic Processes, S. Karlin, H.M. Taylor (2) Introduction to Stochastic Processes, S. Ross (3) Discrete Time Stochastic Systems, T. Sodestrom (4) Mathematical Theory of Statistics, Hogg/Tanis or Hogg/Craig or any other book (5) Identification and Stochastic Adaptive Control, H.F. Chen, L. Guo (6) Stochastic Systems, Estimation, Identification and Adaptive Control, P.R. Kumar, Pravin Varaiya, Prentice-Hall, 1986; new edition: SIAM, 2016 (7) Stochastic Modeling and Control, M.H.A. Davis, R.B. Vinter (8) On Adaptive Control, (O Sterowaniu Adaptacyjnym) Research Monograph, Habilitation Doctorate Dissertation, (in Polish) B. Pasik, 1986 (9) Linear Systems, P. Antsaklis, A. Michel, McGraw Hill, 1997 (10) Introduction to Mathematical Systems Theory, Linear Systems, Identification and Control, Christian Heij, Andre Ran, Freek van Schagen, Birkhauser Verlag, 2007 (11) Feedback Systems, An Introduction for Scientists and Engineers, K. J. Astrom and R. M. Murray, 2008 The course has been developed by Bozenna Pasik-Duncan based on the author’s habilitation doctorate dissertation written in Polish, and translated into English for the purpose of being used as the lecture notes for the course. The lecture notes have been available to students and have been adapted each year to the current advances in stochastic adaptive control and technology. The course has been taught for the last 25 years by Bozenna Pasik-Duncan with collaboration of Tyrone Duncan. The course requires a very good mathematical understanding in real analysis, probability and statistics as well as good computational skills. The students diversity differs from year to year therefore the way how the course is taught, its content, and illustrative examples of applications differ too.
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3. CLASSROOM TEACHING AS A STOCHASTIC ADAPTIVE CONTROL PROBLEM The classroom with students and their instructor is considered as a controlled system. It is stochastic because there is a lot of randomness in the classroom. Students talk, fall asleep or bring a baby to the classroom, or there is a fire alarm suddenly on during the exam and we must leave the building. In systems theory, we analyze every system carefully. We analyze the existence of a solution and computational aspects of it, we simulate stochastic equations, we collect information, we compose results, etc. We do the same in our teaching. A classroom becomes a scientific laboratory. We collect information. As we introduce ourselves as the course instructors, we ask students to introduce themselves too. The students in the class are unknown to the instructor at the beginning of a semester. They come from different departments with different academic background. The only prerequisites for this course are: good knowledge of real analysis, probability and statistics, very good academic standing, and permission of the instructor. The challenge for the instructor is to learn about students by collecting relevant course information about them. Too many unknowns in the system make the system unknown so learning or identifying the system is a critical issue in the stochastic adaptive control. 4. SHORT BIO AS THE FIRST ASSIGNMENT One of the best practices for learning about the students from the beginning of a semester is to ask them to prepare a well done short bio with information that will help us as instructors to find optimal adaptive strategies. The information should contain students family and academic background, math and science courses taken, any significant recognitions, motivation for taking this particular course, short and long term goals for studying and career, research and real world problem interests, hobbies and favorite things to do during free time. These short bios should be revisited by the instructor as a semester progresses, and students should have an opportunity to update them twice: during the middle and at the end of a semester. We as teachers need to know the students and their interests. This information is important when we design projects for them. There are many unknowns in this system, so we need to estimate (learn) them as in the theory, and at each instant a controller/teacher uses this new estimate in control strategies and adapts the system. In theory we called it adaptation. It is the same in teaching. We collect information, we build a portfolio, we analyze our reports and data after every class, and we also want to do better each time so that in the long run we will do as well as if we knew the system perfectly. In the theory of stochastic adaptive control, this property of adaptive control is called self-tuning. We call this method of teaching scholarship in teaching. We treat teaching as a stochastic process that changes over time, a process with several components such as vision, design, data collection, and data analysis. We integrate teaching and learning. As in the theory, the controller has to learn, so a teacher, as controller, has to learn too, and the system has to learn, meaning the students have to learn.
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5. A CLASS SYLLABUS A class syllabus is prepared as a joint project which is understood as a ”contract” (agreement) between the students and the instructor. Students receive a draft of the class syllabus on the first day of classes. They are given a time to discuss it in groups and they have the opportunity to make their individual modifications. All students have access to all individual modifications. The final version of the syllabus is the result of a collaborative effort. It becomes a ”contract” between students and the instructor. Students play an major role in preparing this important document for which they become co-owners. 6. C 5 : COLLABORATION, COMMUNICATION, CONNECTIONS, CREATIVITY AND CURIOSITY-A KEY FOR A SUCCESS Inviting Tyrone Duncan, an important contributor to stochastic control to teach the introduction to control demonstrates the power of collaboration in the partnership with Bozenna Pasik-Duncan and her passion for stochastic processes and statistics. Thirty years ago an anonymous reviewer of their joint research proposal wrote: ”a marriage made in heaven” meaning that stochastic adaptive control, the topic of their research project = stochastic processes and control. Both instructors participate in a scientific discussion that takes a place in the classroom. Students witness how they both respond to their questions. Students experience and observe two research collaborators in the classroom. They have the first hand learning that collaborative efforts in research and collaborative efforts in teaching are important and rewarding. Communication and writing are equally important. There is a lot of discussion in the classroom to develop good communication skills, and there is a lot of writing in mathematics to develop good skills in writing. Students are expected to read independently, outside of the classroom and to submit written reports from their reading. Stochastic adaptive control taught in this mathematical approach makes it exciting for students with connections in the use of different areas of mathematics and its fundamental concepts such as calculus and limits, linear algebra and vector spaces, difference and differential equations, numerical analysis with convergence and rates of convergence of algorithms, probability with limits theorems of probability, conditional expectations and martingales, stochastic processes, in particular Markov chains and Brownian motion, mathematical statistics with consistency and strong consistency as well as estimation methods, and control theory. This is the only mathematics course at the University of Kansas that brings so many areas of mathematics together. Students are fascinated by building bridges between different fields, the phenomenon which is not observed in a course that focuses on a single area. Control is viewed naturally as a field that spans science, technology, engineering and mathematics (STEM). Curiosity is the most important part of learning in this course as it covers many different topics. Students are engaged in research discussion through curiosity and creativity. They are alert in this course of what they are doing, they ask why things are the way they are, they try different ways to explain to each other what they observe. They are taught to consider how even the most abstract 107
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mathematics they do can be applied to real life problems. The cross-boundaries nature of stochastic adaptive control is exciting for students. It motivates them for being creative in particular while searching for the correct models in the modern real complex world. Reviewing papers is a powerful method to prepare students for writing a scientific project. The recently introduced practice of reviewing a conference paper has turned to be the most effective way of teaching students how to write a scientific project. Students were provided with a sample of very well professionally done reviews. They received a thirty page paper for reviewing with a two week deadline. Students have done outstanding work. Their reviews were very constructive, detailed, providing missing references, providing missing steps in mathematical proofs, indicating weaknesses in interpretations of results and in particular graphs, and amazing remarks on editing. Their reviews have led to a significant improvement of the quality of the paper presentation.
7. GUEST SPEAKERS Guest speakers energize, inspire and motivate students. Former students who took the Stochastic Adaptive Control Course or research collaborators are invited as guest speakers during the middle of a semester to have a conversation with students and share their industrial career experiences. Real world problems generate new mathematical problems. Mathematics of finance and mathematics of networks, as well as understanding of the brain as the most complex system, generate new stochastic control problems, such as the identification, detection and control of stochastic system with a noise modeled by a fractional Brownian motion. Neurologists, mathematicians and engineers from the medical research centers talk about epilepsy and seizures, and they show brain waves and explain how you can use mathematics to detect or predict seizures, and how one can introduce control theory via brain stimulation: openloop versus closed-loop. However, because most people have never experienced an epileptic seizure, they have no feel for these graphs. The musicians can modify a Mozart sonata by adding a noise in the orchestra, duplicating the frequency behavior in powerful ways. Here students discovered amazing creativity and connections of science and art. Bringing research collaborators, especially those from other countries, to the classroom with good preparation for their visits changes students perspectives. It opens their eyes and it awakes their imagination and creativity. They see how math and control can be found everywhere and how often it is hidden. They see the role of the broad and inspiring vision of stochastic adaptive control. They see the power, beauty, and excitement of the cross-boundary nature of control. Students love guest speakers. They find them to be most inspirational and most motivating for continuing studying mathematics and engineering to do research in stochastic adaptive control. In a summary of the reflections from a visitor’s talk a student wrote: ”Today’s visitor is a fine example of how an engineer should proceed with his career and deal with his success. I hope that the inspiration I have drawn from the speaker can help fuel some creativity for me in my field and motivate me to do my work for the betterment of the world.”
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8. VERTICAL AND HORIZONTAL INTEGRATION AND PEER TUTORING In this course with a very diverse group of undergraduate and graduate students, vertical and horizontal integrations as well as peer tutoring play crucial roles by building a community of learners. Vertical integration incorporates students at different levels in their studying and learning approach. It involves and engages undergraduate and graduate students. Horizontal integration incorporates students from various disciplines in learning and in teaching. Through peer tutoring students become teachers for their peers. They enthusiastically volunteer to teach the entire class a topic which they learned well or which they studied independently. A peer tutoring program was created as we identified that at that moment in time, there were some students who were in need of help, and others who could offer that help. This aligns with the important ideas of promoting engagement, communication and an exchange of ideas. Mathematics is often a quiet pursuit, but in this class the students talk with one another, building not only a community of learners but more importantly they are building life-time friendships. One of the most gorgeous, and most rewarding outputs of this C 5 style is collaborative learning and teaching of stochastic adaptive control. 9. FEEDBACK FROM STUDENTS In teaching, we love feedback from students as we love it in stochastic control. Here is a sample of things the instructor did well by recent and former students: ”hands on learning experience with current field research,” ”provide motivation and create interest in the topic,” ”provide many resources in the form of notes, old class projects, suggested reading,” ” letting us participate in real reviewing,” ”guest speaker was a wonderful part of class,” ” the class taught us lots of useful material including the cutting edge of discipline,” ” the class forced us to be independent , it elevated us to the level of independent researchers,” ”by engaging us in research topics this professor makes us feel very independent and very confident,” ”taking this class really opened my eyes to mathematics. Instead of dreading and avoiding proofs, I love them. This professor teaches until we know what is going on and not to follow a schedule she teaches us stochastic adaptive control with a true passion for the subject. She shows us how practical it is in life and how relevant it is for any field of study or career we choose I learned from material, learned from my classmates, learned from myself, and for it all, my life is better. A course with that kind of an impact and outcome is definitely life changing,” ”This professor has used her passion for mathematics and research as a vehicle to inspire a generation of students in the stochastic and control theory. I know this to be true because I am one of these inspired students who is making a career in this area, and there are many others besides me. From her I also learned the value of conducting quality research, and the importance of disseminating it to others through teaching. Additionally, her students come from her course with a firm understanding of maintaining ethics and professional standards in all that we say and do. I have taken these principles to heart, and together with her conscious effort 108
to make each student feel valued and inspired to realize their full potential, apply them in my classrooms and to my graduate students,” ”that appetite for research which she helped nurture and I shared with all others in the stochastic adaptive control course led me to M.I.T. where I completed a Ph.D. in electrical engineering.” 10. A SAMPLE OF THESES MOTIVATED BY THE COURSE A sample of the topics of masters theses and doctoral dissertations as the outcome of research projects initiated in the Math 750-Stochastic Adaptive Control is given now. A stochastic system model for PageRank: parameter estimation and adaptive control. Identification and adaptive control methods for some stochastic systems. From reality to abstract in stochastic processes applied to telecommunications. New approaches in continuous time stochastic adaptive control. Some applications of adaptive control: diffusion approximation and hierarchial approximation. Stochastic adaptive control and its applications to the theory of finance. Adaptive control in queueing systems. On adaptive control of Markov chains. Optimal policy advertising. Ocular artifact removal through ARMAX model system identification using the extended least squares algorithm. Distribution of aggregate claims. Estimation of parameters of branching processes with immigration by adaptive control. Current efforts in computerized epilepsy prediction. Is the stock market efficient? The stochastic effects on a mathematical model for cardiac arrythmia. Bonus-Malus systems in insurance. Biostatistics in Genetics. Markov decision processes and early retirement. A new scheme for traffic estimation and resource allocation for bandwidth brokers. The modeling of risk reserves with skew distributions and control. An optimal trading strategy of a financial portfolio. Numerical methods for parameter estimation in stochastic systems Computational methods for stochastic differential and partial differential equations. Evolutionary dynamics of protean evasion. System identification and parameter estimation in breast cancer models. 11. CONCLUSIONS Students who are broadly prepared, have a higher chance to find attractive jobs. The Math 750-Stochastic Adaptive Control Course prepares students broadly for studying and research. It prepares them for new challenges and new opportunities of the 21st century. These new challenges include: processing information, ability to work with complex systems and diverse groups, ability to handle quick changes, ability to manage diverse groups of people, ability to handle computationally large data set and analyze it using advanced mathematical and statistical methods, ability to understand well all kinds of randomness., ability to balance personal and professional life, ability to be willing to take risks, ability to take advantage of tremendous opportunities. Students and professor have satisfaction from this integrated learning and teaching as the optimization criterion. In this class considered as a controlled system, the optimization criterion is to maximize students’ and professors’ satisfaction from the integrated learning and
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teaching process. The optimal value of the cost criterion is 5 on the scale from 1 to 5 in student evaluations, and all students receive the highest grade in the class based on students’ perfect attendance, group projects, paper reviewing, independent reading and reports, the final project and its presentation. From the syllabus: ”Final Paper is expected to be written in the form and the size of a short research paper, with the title, name, affiliation, abstract, key words, introduction, main results, conclusions, future work, references and acknowledgements if applicable. Ten minute presentations with class discussions will be scheduled.” Integrating research, learning and teaching using the adaptive control approach described above has demonstrated to be most effective. Acknowledgement: Bozenna Pasik-Duncan would like to thank the C21 community of remarkable scholars and educators associated with the University of Kansas Center for Teaching Excellence for inspirations and motivations
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for innovative teaching. The authors wish to thank the reviewers for inspiring and motivating comments. REFERENCES B. Pasik On Adaptive Control Research Monograph, Habilitation Doctorate Dissertation, (in Polish), SGPiS Publications 1986. Mathematics Education of Tomorrow AWM Newsletter Vol. 24. T. E. Duncan and B. Pasik-Duncan Stochastic adaptive control Encyclopedia of Systems and Control Springer 2014. T. E. Duncan and B. Pasik-Duncan Stochastic adaptive control The Control Handbook Taylor and Francis, CRC 2010. B. Pasik-Duncan and M. A. Verleger Education and Qualification for Control and Automation Springer Handbook of Automation Ch. 44, 2009.