# Dynamical System and Control

By Prof. N. Sukavanam, Prof. D. N. Pandey   |   IIT Roorkee
Learners enrolled: 64
This course ‘Dynamical systems and control’ is a basic course offered to PG students and final year UG students of Engineering/Science background. The objective of this course is to enhance the understanding of the theory, properties and applications of various dynamical and control systems. After completing the course one may be able to understand some of the important aspects of dynamical systems such as mathematical modeling, well posedness (existence, uniqueness and stability) of the considered problem. The participants will also be conversant with the controllability, stabilizability and optimal control aspects of a dynamical system.

Most Dynamical systems-physical, social, biological, engineering are often conveniently expressed (modeled) in the form of differential equations with or without control. Such mathematical models can provide an insight into the behavior of real life system if appropriate mathematical theory and techniques are applied. In this context this course has tremendous applications in diverse fields of engineering and technology.

INTENDED AUDIENCE :  UG/PG students of technical institutions/ universities/colleges.

PREREQUISITES : Basic concepts from Linear Algebra and Ordinary Differential Equations
Summary
 Course Status : Upcoming Course Type : Elective Duration : 12 weeks Start Date : 23 Jan 2023 End Date : 14 Apr 2023 Exam Date : 30 Apr 2023 IST Enrollment Ends : 30 Jan 2023 Category : Mathematics Credit Points : 3 Level : Undergraduate/Postgraduate

### Course layout

Week 1  :  Formulation of physical systems-I, Formulation of physical systems-II, Existence and uniqueness theorems-I, Existence and uniqueness theorems-II, Linear systems-I
Week 2  :  Linear Systems-II, Solution of linear systems-I, Solution of linear systems–II, Solution of linear systems-III, Fundamental Matrix-I
Week 3  :  Fundamental Matrix-II, Fundamental matrices for non- autonomous systems, Solution of non-homogeneous systems , Stability of systems: Equilibrium points, Stability of linear autonomous systems-I
Week 4  :  Stability of linear autonomous systems-II, Stability of linear autonomous systems-III, Stability of weakly non-  linear systems-I, Stability of weakly non-  linear systems-II, Stability of non-  linear systems using linearization
Week 5  :  Properties  of phase portrait, Properties  of orbits, Phase portrait : Types of critical points, Phase portrait of linear differential equations-I, Phase portrait of linear differential equations-II
Week 6  :   Phase portrait of linear differential equations-III, Poincare Bendixson Theorem, Limit cycle , Lyapunov stability-I, Lyapunov stability–II
Week 7  :  Introduction to Control Systems-I, Introduction to Control Systems-II, Controllability of Autonomous Systems, Controllability of Non-autonomous Systems, Observability-I
Week 8  :  Observability-II,  Results on Controllability and Observability,  Companion Form,  Feedback Control-I, Feedback Control-II
Week 9  :   Feedback Control-III, Feedback Control-IV,  State Observer, Stabilizability, Introduction to Discrete Systems-I
Week 10  :  Introduction to Discrete Systems-II,  Lyapunov Stability Theory-I,  Lyapunov Stability Theory-II,  Lyapunov Stability Theory-III, Optimal Control- I
Week 11  :   Optimal Control-II, Optimal Control-III, Optimal Control- IV, Optimal Control for Discrete Systems-I,  Optimal Control for Discrete Systems-II
Week 12  :  Controllability of Discrete Systems, Observability of Discrete Systems, Stability for Discrete Systems, Relation between Continuous and Discrete Systems-I,  Relation between Continuous and Discrete Systems-II

### Books and references

Braun, M. “Differential Equations and Their Applications”, 4th Ed., Springer 2011.2. Stephen Barnett, Introduction to Mathematical Control Theory, Oxford University Press, 1990 3. D. Subbaram Naidu, Optimal Control Systems, CRC Press, 20034. Deo, S.G., Lakshmikantham, V., and Raghvendra, V.,"Text Book of Ordinary Differential Equations”, 2nd Ed., Tata McGraw Hill 2010.5. M. Gopal, Modern Control System Theory, John Wiley & Sons Ltd., 1994. 6. . Simmons G.F., “Ordinary Differential Equations with Applications”, Tata McGraw Hill 2003.

### Prof. N. Sukavanam

IIT Roorkee
Prof. N. Sukavanam received his Ph. D from the Indian Institute of Science, Bangalore in 1985. He served as a Scientist-B at Naval Science and Technological Laboratory, DRDO for two years (1984-86). Then joined as a Research Scientist in the Department of Mathematics, IIT Bombay (1987-90). Worked as a Lecturer at BITS Pilani from 1990 to 1996. In May 1996 he Joined the Department of Mathematics at IIT Roorkee (University of Roorkee at that time) as an Assistant Professor. Currently he is a Professor in the Department of Mathematics IIT Roorkee and Head of the Mathematics from Feb. 2018. His areas of research includes Nonlinear Analysis, Control Theory and Robotics. Professor Sukavanam has published about 80 papers in refereed journals, 30 papers in International Conference Proceedings. He has guided 19 Ph. Ds, 60 M. Sc./M. Phil/MCA Dissertations. Organized International Workshop on Industrial Problems. Developed Pedagogy online course on Mathematics I, offered NPTEL online video course on Dynamical Systems and Control and conducted more than six QIP/Continuing Education courses on Robotics and Control

### Prof. D. N. Pandey

Dr. D. N. Pandey is an Associate Professor in the Department of Mathematics, IIT Roorkee. Before joining IIT Roorkee, he worked as a faculty member in BITS-Pilani Goa campus and LNMIIT Jaipur. His area of expertise includes semigroup theory and functional differential equations of fractional and integral orders. He has already prepared e-notes for the course titled “Ordinary Differential Equations and Special Functions” under e-Pathshala funded by UGC. Also, he has published a book titled “Nonlocal Functional Evolution Equations: Integral and fractional orders, LAP LAMBERT Academic Publishing AG Germany”. He has delivered several invited talks at reputed institutions in India and abroad. He has guided three PhD theses and has published more than 60 papers in various international journals of repute. Currently, he is supervising five research students.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams:
30 April 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Roorkee.It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team