# SC639 - Mathematical Structures for Control

#### Instructor

R. N. Banavar

#### Sections

Only one, this is a core course for Syscon M. Tech and PhD students. (Previously called SC 201)

#### Semester

Autumn’19

#### Course Difficulty

Content is easy - but evaluation is brutally strict.

#### Time Commitment Required

- Lectures - 3 hours per week
- Tutorials - 1.5 hours per week
- Self-study - 2 hours per week

#### Grading

Not great. 4 AAs and 8 ABs out of 73. Most of the grades were BBs or BCs. No fail grades were awarded.

#### Attendance Policy

None. Although the instructor had an interesting policy. He would announce surprise quizzes at the start of a week. The quiz would be conducted anytime that week, in class or in the tutorial. That made for a pretty anxious week and the classes were always packed :).

#### Pre-requisites

None, although a thorough grasp of the contents of MA 105, MA 106 and MA 108 will be a bonus and can help you score much better than your peers. There were points for class participation as well, although it was unclear as to how this was used to grade students, given that attendance wasn’t mandatory.

#### Evaluation Scheme and Weightage

- Four surprise quizzes (55%)
- Four Assignments (20%)
- Class participation (25%)
- No midsem/endsem.

#### Topics Covered in the Course

- Vector Spaces: Linear independence, basis, dimension, subspaces, linear functionals, dual spaces, linear transformations, inner products, normed vector spaces, eigen values and eigen vectors (Builds on MA 106)
- Sequences and series: Sequences, series and convergence, power series, differentiation and integration of a series (Builds on MA 105)
- Multivariable calculus: Taylor’s formula, differentiation of functions of several variables, Gateaux and Frechet derivatives, differentiability of vector-valued functions, the chain rule, Taylor’s formula for functions of several variables. (Advanced content)
- Convexity and Optimization: Unconstrained optimization, first-order and second-order optimality conditions, quadratic forms, affine geometry, convex sets, convex cones, convex functions, optimization on convex sets. (This wasn’t completely covered due to time constraints)

#### Mechanism of Instruction and Teaching Style

The professor used to write stuff on the whiteboard and students would copy it down. No supplementary material was shared. There’s no real way to catch up if you miss a class, because the material is picked up from a wide range of sources. Since the course is intended as a mathematical primer for M. Techs and PhDs, you may find the initial part of the course slow and repetitive. The course picks up pace in the latter half, though. The prof is very enthusiastic and gives personal attention to each student in class, regularly asking people to come up to the board and answer questions. He is also very receptive to doubts and may often spend entire lectures devoted to one doubt (which may be frustrating at times).

#### Assignments and projects in the Course

Tutorials were often detailed and lengthy and sessions often went well overtime (started at 9 PM and went beyond 11 PM). Attendance wasn’t compulsory in tutorials. The prof often supplied additional problems along with self-study and these were very useful while preparing for quizzes. Four assignments were given, each a week to ten days in advance. Fairly descriptive and involved and graded very strictly. Notations might be difficult to understand if not regular in class.

#### Exams

All the exams were open-book, open-notes 100 mark papers. Each paper had 4 questions only. The duration was one hour. The questions are long and descriptive, making it extremely difficult to complete the paper in the allotted time. Notes aren’t of particular use since almost all the questions are proofs. Grading was very tough and strict. Most of the time marks were only awarded in multiples of 5, with a decent proportion of the marks reserved for those who completely and correctly solved the question, so this meant that there was a great difference between people who actually completed a question and those who attempted it. There were quizzes where the highest was 90/100, with the mean in the 20s. Quiz questions were often picked up from Lang or Halmos, and these books don’t have easily available solutions, so it might be a good idea to actually practice problems by writing each step down in order to avoid scoring dismally in the exams.

#### References

- Finite Dimensional Vector Spaces - P. R. Halmos, Springer 1984. (Very important!!)
- Undergraduate Analysis - S. Lang, Springer, 1983. (Very important!!)
- Fundamentals of Optimization - O. Guler, Springer 2010.
- Optimization by Vector Space Methods - D. G. Luenberger, Wiley, 1967.

#### Importance of Course

Very useful, presents multivariable calculus and linear algebra from a methodical and rigorous viewpoint. Prerequisite for several courses with the department, such as SC 607, SC 633, etc.

#### Motivation to take the course

The most fundamental prerequisite before understanding any controls system is a solid background in mathematics. That is what this course aims to provide, as is evident from the name. Since I had a keen interest in abstract math as well as controls systems, I decided to take this course.

#### When did you take the course?

I took it in my 3th semester. I believe that 2nd year is a good time to take up this course but, beware, it requires considerable mathematical rigour (which you may like or lament, depending on you :P)

#### How strongly would you recommend someone for taking this course?

Take the course only if you are interested in mathematics, because rigor is the norm in this course (pun intended). It serves as a solid base for any other courses you may take in SC, as most of them do involve abstract math.

#### When did you take this course? What will be the ideal semester to take this course? Any other course which can be done before this?

Second year first semester is ideal, that’s when I took it.