Course Title and Code

Allied Topics in Control (EE5603)

Course Credit

3-0-0-3 (Lecture-Tutorial-Practical-Total Credits)

Course Category

Elective

Target Programme

MTech, MS, PhD

Target Discipline

All

Prerequisite Course

Linear Algebra and Multivariable Calculus (or equivalent courses)

Course Content

Sr. No. Topics Hours
1 Metrics (distance functions), complete spaces, open set, closed set, interior points, exterior points, corner points, compact set, convex set, connected set, operators, identity, inverse, commutativity, associativity, groups, Abelian groups, Subsets of Euclidean space, Functions and continuity, Sequences and convergence. 8
2 Systems of linear equations, Subspaces and bases, Orthogonal bases and orthogonal projections, Gram-Schmidt process, Linear models and least-squares problems, Eigenvalues and Eigenvectors, Symmetric and positive definite matrices, Singular value decomposition, Norms and inner product. 6
3 Dynamical Systems: ordinary differential equations (ODEs), state space formulation, linear systems, nonlinear systems, equilibrium points, periodic orbits, limit cycles, stability (local and global), center manifold theorem, phase portraits, existence and uniqueness of solutions of ODEs. Brief intro to chaos and bifurcation. 14
4 Intro to Manifolds: transformations, isomorphism, diffeomorphism, inverse transformations, derivatives, partial derivatives, Inverse and Implicit function theorem. Charts, functions, representations, transformations and ODE on manifolds. Vector fields on manifolds, example of control systems on manifolds. 14

Learning Outcomes

At the end of the course, students should be able to study, understand, and analyze nonlinear dynamical systems evolving in different spaces.

Text/Reference Books

Text Books:

  • Mathematical Analysis by Tom M. Apostol, Narosa Publishing House, 1993. ISBN: 0201002884
  • Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes, Applied Mathematical Sciences, 1983. ISBN: 0387908196
  • Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus by Michael Spivak, Taylor and Francis, 1971. ISBN: 1138329398
  • Introduction to Linear Algebra by Gilbert Strang, MA: Wellesley-Cambridge Press, 1993. ISBN: 0980232775

References:

  • Real and Convex Analysis by Erhan Cinlar and Robert J. Vanderbei, Springer Science & Business Media, 2013. ISBN: 1461452562
  • Introductory Functional Analysis with Applications by Erwin Kreyszig, NY: Wiley, 1978. ISBN: 0471504599
  • Finite-Dimensional Vector Spaces by Paul R. Halmos, D Van Nostrand Company, Inc., 1942. ISBN: 0486814866
  • Optimization Algorithms on Matrix Manifolds by Absil, P-A., Robert Mahony, and Rodolphe Sepulchre. Princeton University Press, 2009. ISBN: 0691132984