# IE 614: Linear Systems

**Prerequisite:** Exposure to relevant concepts at undergraduate level and instructor consent** **

**Aim of course:**

To familiarize students with models and theory of linear dynamical systems.

**Major Contents:**

Review of linear algebra: Systems of linear equations, Gaussian elimination, inverse of matrix and transpose. Review of Fields, Vector Spaces and Linear Transformations. Review of concepts of linear independence, basis, and fundamental subspaces associated with matrices and linear transformations, rank-nullity theorem, duality. Eigenvalues and eigenvectors of matrices and linear transformations, characteristic polynomials, Cayley-Hamilton Theorem, diagonalization. of dynamic systems and linear time invariant systems.

Representation of linear dynamical systems using ordinary differential equations (ODEs) and linearization of non-linear dynamical systems. Solution methods of first order, second order and higher order ODEs and systems of ODEs. Solution methods of difference equations, existence and uniqueness theorems, Laplace and z-transforms.

Concepts of Transfer functions, concepts of stability, controllability and observability, canonical forms-Diagonal and Jordan forms.

Applications, examples and case-studies involving simple linear systems.

**Text and References:**

● Chi-Tsong Chen (1998) Linear System Theory and Design, 3rd edition, Oxford University Press.

● Morris W. Hirsch, Robert L. Devaney and Stephen Smale (2004) Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2nd edition, Academic Press.

● Jack Leonard Goldberg, Matthew Boelkins, and Merle Potter . Differential Equations with Linear Algebra, 1 st Edition (2009), Oxford University Press.

● Ferenc Szidarovszky and Terry Bahill (1997) Linear Systems Theory, 2nd edition, CRC Press.

● Robert L. Williams III and Douglas A. Lawrence (2007) Linear State-Space Control Systems,

John Wiley & Sons