E1 241 (AUG) 3:0 Dynamics of Linear Systems



Dynamics of Linear Systems
August-December, 2013

Announcements:
 		Nov 26, 2013: Final examination will be held on December 02, 2013 from 2:00PM to 5:00PM in PE302.
Nov 21, 2013: HW#4 is due on November 26, 2013 at 1:45PM.
Nov 12, 2013: A doodle link for scheduling the final exam has been sent to all students; if you have not received it, send an email to prasantg AT ee.iisc.ernet.in
Oct 29, 2013: Midterm#2 will be held on November 7, 2013 at PE302 from 2:00PM to 3:30PM.
Oct 29, 2013: HW#3 is due on November 5, 2013 at 1:45PM.
Oct 15, 2013: Today's lecture is cancelled.
Oct 8, 2013: HW#2 is due on October 15, 2013 at 1:45PM.
Sep 17, 2013: Midterm#1 will be held on September 19, 2013 at PE302 from 2:00PM to 3:30PM.
Sep 17, 2013: HW#1 solution is sent via email. If you have not received it, send an email to prasantg AT ee.iisc.ernet.in
Sep 12, 2013: If you are taking this course for credit, please send an email (if you have not yet) to prasantg AT ee.iisc.ernet.in before Sep 17, 2013.
Sep 9, 2013: HW#1 is due on September 17, 2013 at 1:45PM.
Sep 3, 2013: Due to administrative reasons, all lectures will be held in PE301 from now (till any further notice).
Aug 6, 2013: Lecture of Aug 13, 2013 is preponed to 1:00pm-2:30pm in EE 301 on Aug 9, 2013 (Friday).
Aug 1, 2013: First lecture will be held at 10:30am in MMCR (EE 217) on Aug 2, 2013 (Friday).
Instructor:
 		Prasanta Kumar Ghosh
Office: EE 200D
Phone: +91 (80) 2293 2694
prasantg AT ee.iisc.ernet.in
Teaching Assistant(s):
  • Abhijit K
    Office: Room 116, Control Systems Lab
    Phone: 9535144467
    abhijitk AT ee.iisc.ernet.in
  • Debasish Nath
    Office: Room No. 207, High Voltage Lab
    Phone: 9482224759
    debasishnath AT ee.iisc.ernet.in
Class meetings:
 		2:00pm to 3:30pm every Tuesday and Thursday (Venue: MMCR, EE 217)
Course Content:
  • Linear algebra, differential equations, linear operator, systems
  • Representation of dynamical system, equilibrium points
  • Solution of state equations - LTI and LTV, canonical realization
  • Stability of systems, Lyapunov matrix equation
  • Observability and controllability, minimal realization, canonical decomposition
  • Linear state variable feedback, stabilization, pole-placement
  • Asymptotic observers, compensator design, and separation principle


Prerequisites:
 		Exposure to linear algebra, Laplace transform, differential equations
Textbooks:
  • Primary Text(s)
    • Chi-Tsong Chen, Linear System Theory and Design, Oxford University Press, 1984
  • Other text(s) of possible interest
    • Joao P. Hespanha, Linear System Theory, Princeton University Press, 2009 (Errata)
    • K. Ogata, Modern Control Engineering, Prentice Hall, 2002
    • Thomas Kailath, Linear Systems, Prentice-Hall, Inc., 1980


Web Links:
 		Predator-Prey
Theorems about power series from Physics 116A, UCSC
Determinants of Block Matrices by John R. Silvester, Dept of Mathematics, Kings's College
Notes on Delta function by Ernesto Estevez Rams, Universidad de la Habana.
On sampling without loss of observability/controllability by Gerhard Kreisselmeier, University of Kassel, Germany.
Operationsl Amplifiers by Aydin Ilker Karsilayan, Texas A&M University.
The operational amplifier and applications by Jose Silva-Martinez, Texas A&M University.
A New Approach to Linear Filtering and Prediction Problems R. E. KALMAN, Transactions of the ASME - Journal of Basic Engineering, 1960.
Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation Ramsey Faragher, IEEE SIGNAL PROCESSING MAGAZINE, September 2012.
Grading:
  • Surprise exam. (30 points) - 15 surprise exams. Closed book. 10 minutes per exam. Each surprise exam is worth 3 points. Missed exams earn 0 points. No make-up exams. The total surprise exam score sums your 10 best surprise exam scores (we ignore five worst scores). Class attendance is mandatory. Unexcused absences get an automatic exam score of zero for that session's exam grade.
  • Assignments - Assignments do not count any grade point. However they must be turned in (before due date). Assignments are meant for learning and preparation for exams. Students may discuss homework problems among themselves but each student must do his or her own work. Cheating or violating academic integrity (see below) will result in failing in the course. Turning in identical homework sets counts as cheating.
  • Midterm exam. (20 points) - 2 midterm exams. Closed book. Missed exams earn 0 points. No make-up exams. The midterm score will be your best score among the two.
  • Final exam. (50 points) - Closed book
Topics covered:
Date
Topics
 Remarks
Aug 2
Course logistics, definition of state-space linear system, illustrative examples of linear systems
 Introductory lecture
Aug 6
Field, vector space, linear dependence, basis, dimension, representation, operators, injection, surjection, bijection.
 -
Aug 8
Linear operators, similarity transformation, companion matrix, rank, nullity
 Surprise Test #1
Aug 9
Sylvester's Theorem, Special matrices, Nilpotency, Eigenvalue, Eigenvectors, distinct eigenvalues.
 -
Aug 20
Gram-Schmidt orthogonalization, Diagonalization, the role of diagonalization in solving linear system equations, multiplicity of eigenvalues
 Surprise Test #2
Aug 22
Matrix polynomial, polynomial matrix, addition and multiplication of polynomial matrix, left, right quotient, remainder, remainder theorem, Cayley-Hamilton theorem
 Surprise Test #3
Aug 27
Laplace Transform, partial differential equation, illustrative problems on diagonalization of matrix, matrix solution of differential equations and Gram-Schmidt orthogonalization
 Lecture by Debasish Nath (TA)
Aug 29
Illustrative problems on linear differential equations, eigenvalues, eigenvectors, Nilpotency, state-space modeling
 Lecture by Abhijit K (TA)
Sep 3
Irreducible, annihilating, minimum polynomial, generalized eigenvectors, chain, Jordan canonical form.
 Surprise Test #4
Sep 5
Jordan canonical form [JCF] (contd..), computational procedure of finding JCF.
 -
Sep 10
Function of matrices, minimum polynomial computation, exponential of a matrix.
 HW#1 assigned
Sep 12
Power series of a function of matrix, Properties of exponential of a matrix, vector norm, Cauchy Swartz Inequality, matrix norm, induced matrix norm, singular value decomposition.
 Surprise Test #5
Sep 17
Continuous and discrete time systems, memoryless, causal system, states of a system, linear, lumped, distributed system, zero-state, zero-input response, convolution, state-space representation, time-invariance.
 Surprise Test #6
Sep 19
-
 Midterm #1
Sep 24
LTI system, rational, irrational transfer function, zeros and poles, system decomposition, linearization
 -
Sep 26
Example of linearization, system composition, discrete-time system, Solution of LTI state-space equation, examples (phase potraits) for different types of A matrices and corresponding steady-state responses in relation to eigenvalues of A.
Surprise Test #7
Oct 1
Discretization, Solution of LTI discrete-time state-space equation, Equivalent state-space equations, zero-state equivalence, zero-input equivalence, Solution to LTV state-space equation, fundamental matrix, state-transition matrix.
Surprise Test #8
Oct 3
Impulse response of LTV state-space equation, LTV discrete-time state-space equation, Equivalent LTV state-space equations.
Surprise Test #9
Oct 8
Computing exponential of matrix with distinct complex eigenvalues and complex eigenvalues with multiplicities.
Surprise Test #10
HW#2 assigned
Oct 10
Stability of continuous and discrete-time LTI SISO system, BIBO stability, stability for systems with proper rational transfer function, stability of MIMO system.
-
Oct 15
Lecture cancelled
-
Oct 17
Marginal and asymptotic stability of LTI continuous and discrete-time system, effect of equivalence transformation on stability properties, Lyapunov theorem for LTI continuous and discrete-time system, stability of LTV system.
-
Oct 22
Controllability of a continuous-time LTI system, controllability matrix, conditions for controllability, controllability Gramian
Surprise Test #11
Oct 24
Controllability of a discrete-time LTI system, Observability of a continuous and discrete-time system, observability matrix, Gramian, Theorem of duality, Effect of equivalence transformation on controllability and observability, controllability and observability of special systems.
Surprise Test #12
Oct 29
Controllability and observability of linear time-varying systems, duality theorem for linear time-varying systems.
HW#3 assigned
Oct 31
Controllability and observability of system with A matrix in Jordan canonical form, Controllability and observability indices.
Surprise Test #13
Nov 5
Canonical decomposition, irreducible system.
Surprise Test #14
Nov 7
-
 Midterm #2
Nov 12
Realizations, controllable and observable canonical form, coprimeness, degree of transfer function.
-
Nov 14
minimal realization and equivalence between minimal realizations, controllability after sampling.
-
Nov 19
state feedback, controllability, observability, stability with state feedback, stabilization.
Surprise Test #15
Nov 21
state estimator, feedback with estimated state, separation principle, compensator design, pole-placement.
HW#4 assigned
Nov 26
Kalman Filter, hidden Markov model (HMM).
-


Academic Honesty:
 		As students of IISc, we expect you to adhere to the highest standards of academic honesty and integrity.
Please read the IISc academic integrity.

Picture:
Thanks to Mr. Pradeep Bansal for the initiative to take the picture