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



Dynamics of Linear Systems
August-December, 2015

Announcements:
 		November 24, 2014: Final examination will be held on December 1, 2015 from 2:00PM to 5:00PM in PE303.
November 17, 2015: HW#4 is due on November 24, 2015 at 2:00PM.
October 29, 2015: Midterm#2 will be held on November 5, 2015 from 2:00pm to 3:30pm in EE302.
October 27, 2015: HW#3 is due on November 3, 2015 at 2:00PM.
September 29, 2015: HW#2 is due on October 6, 2015 at 2:00PM.
September 4, 2015: The lecture of September 10, 2015 is rescheduled at 5:30pm on September 11, 2015 (Friday) in EE 301.
September 3, 2015: There will be lectures by TAs on 5th September and 8th September 2015 from 2:00pm to 3:30pm in EE301.
September 3, 2015: Midterm#1 will be held on September 17, 2015 from 2:00pm to 3:30pm in EE301.
August 28, 2015: HW#1 is due on September 4, 2015 at 5:00PM.
August 21, 2015: There will be a lecture by TA on August 22, 2015 in EE 301 from 10:00am.
August 5, 2015: If you have not sent an email to prasantg AT ee.iisc.ernet.in with subject as E1_241_2015, please do so on or before August 9, 2015.
August 5, 2015: The lecture of August 6, 2015 is rescheduled at 5:30pm on August 7, 2015 (Friday) in EE 301.
July 28, 2015: First lecture will be held in EE 301 on August 3, 2015 (Monday) at 5:30pm.
Instructor:
 		Prasanta Kumar Ghosh
Office: EE 320
Phone: +91 (80) 2293 2694
prasantg AT ee.iisc.ernet.in
Teaching Assistant(s):
  • Sivaram Prasad M
    Office: EE 312
    Phone: 9611379692
    sivaram.443 AT gmail.com
  • Santosh J
    Office: Room number 207, High Voltage lab
    Phone: 9611189949
    santoshj AT ee.iisc.ernet.in
Class meetings:
 		2:00pm to 3:30pm every Tuesday and Thursday (Venue: EE 301)
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
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.
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 3
Course logistics, definition of state-space linear system, illustrative examples of linear systems
 Introductory lecture
Aug 4
Field, vector space, linear dependence, basis, dimension, representation, operators, injection, surjection, bijection, linear operator.
 -
Aug 7
Similarity transformation, Companion form, System of linear equations, rank, nullity, Sylvester's Inequality.
 -
Aug 11
Special matrices, Nilpotency, Eigenvalue, Eigenvectors, distinct eigenvalues, diagonalization, the role of diagonalization in solving linear system of equations.
 Surprise Test #1
Summary
Scores #Students
0.0 11
0.5 15
1.0 17
1.5 3
2.0 1
3.0 1

Aug 13
Multiplicity of eigenvalues, Matrix polynomial, polynomial matrix, addition and multiplication of polynomial matrix, left, right quotient, remainder, remainder theorem.
 -
Aug 18
Cayley-Hamilton theorem, irreducible, annihilating, minimum polynomial.
 Surprise Test #2
Summary
Scores #Students
0.0 6
0.5 10
1.0 15
1.5 5
2.0 6
2.5 2
3.0 3

Aug 20
Generalized eigenvectors, chain, Jordan canonical form.
 Surprise Test #3
Summary
Scores #Students
0.0 2
0.5 9
1.0 5
1.5 2
2.0 6
2.5 3
3.0 18

Aug 22
Subspace, The four Subspaces associated with a matrix A of size m x n, The relationships among the subspaces of A, Method to generate orthogonal vectors from a given LI vectors, Least Squares Solution.
 Lecture by Sivaram Prasad Mudunuri
Aug 25
Computational procedure of finding JCF, Function of matrices, minimum polynomial computation.
 -
Aug 27
Computing polynomial of a matrix, arbitrary function of a matrix using finite degree polynomial, exponential of a matrix.
 Surprise Test #4
Summary
Scores #Students
0.0 5
0.5 27
1.0 4
2.0 4
2.5 4
3.0 2

Sep 1
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.
-
Sep 3
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, LTI system.
 Surprise Test #5
Summary
Scores #Students
0.0 3
0.5 4
1.0 10
1.5 13
2.0 3
2.5 2
3.0 9

Sep 5
Partial differential equations, properties, order, degree, linearity, homogenity, Solutions of typical partial differential equations - Physical significance of Laplace equations, Numerical solution, Analytical solution of Laplace equations in 2 variables.
Lecture by Santosh Janakiraman
Sep 8
Laplace Transforms - definition and existence, Properties of Laplace transforms - linearity, Translation, differentiation, Integration, Convolution, Initial and Final value theorems, Solution of ODEs using Laplace transforms, Circuit examples solution using LT, State space solutions using LT, Solution of 1st order PDEs using method of characteristics.
Lecture by Santosh Janakiraman
Sep 11
LTI system, rational, irrational transfer function, zeros and poles, system decomposition, system composition, linearization.
-
Sep 15
Example of linearization, 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, modes of a dynamical system.
-
Sep 17
Midterm# 1
 Summary
Scores #Students
6-7 2
7-8 3
8-9 2
9-10 5
10-11 3
11-12 3
12-13 4
13-14 4
14-15 2
15-16 3
16-17 4
17-18 6
18-19 3
19-20 2

Sep 22
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.
 Surprise Test #6
Summary
Scores #Students
0.0 17
0.5 26
2 1
3 2

Sep 24
Lecture Cancelled
-
Sep 29
Solution to LTV state-space equation, fundamental matrix, state-transition matrix, Impulse response of LTV state-space equation, LTV discrete-time state-space equation, Equivalent LTV state-space equations.
-
Oct 1
Stability of continuous and discrete-time LTI SISO system, BIBO stability, stability for systems with proper rational transfer function.
 Surprise Test #7
Summary
Scores #Students
0.0 14
0.5 9
1 6
1.5 6
2 1
2.5 1
3 3

Oct 6
stability of MIMO system, 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.
 Surprise Test #8
Summary
Scores #Students
0.0 8
1 2
1.5 4
2 4
2.5 2
3 23

Oct 8
Stability of LTV system, Controllability of a continuous-time LTI system, controllability matrix, conditions for controllability, controllability Gramian.
 Surprise Test #9
Summary
Scores #Students
0.0 9
0.5 8
1 6
1.5 3
2 3
2.5 2
3 12

Oct 13
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, Controllability and observability of linear time-varying systems, duality theorem for linear time-varying systems.
-
Oct 15
Controllability and observability of system with A matrix in Jordan canonical form, introduction to controllability and observability indices.
 Surprise Test #10
Summary
Scores #Students
0.0 11
1 23
1.5 6
3 2

Oct 20
Controllability and observability indices, Introduction to canonical decomposition.
-
Oct 22
Canonical decomposition, irreducible system.
-
Oct 27
Introduction to realizations, controllable and observable canonical form, coprimeness.
 Surprise Test #11
Summary
Scores #Students
0.0 1
0.5 6
1 12
1.5 1
2 6
2.5 5
3 11

Oct 29
Degree of transfer function, minimal realization.
 Surprise Test #12
Summary
Scores #Students
0.0 2
0.5 7
1 3
1.5 7
2 7
3 15

Nov 3
Equivalence between minimal realizations, Controllability after sampling.
-
Nov 5
Midterm# 2
 Summary
Scores #Students
0 14
3-4 1
4-5 3
5-6 1
6-7 3
7-8 2
8-9 3
9-10 3
10-11 7
11-12 5
12-13 2

Nov 10
state feedback, controllability, observability.
-
Nov 12
stability with state feedback, stabilization.
-
Nov 17
state estimator, feedback with estimated state, separation principle.
Surprise Test #13
Summary
Scores #Students
0.0 18
0.5 3
1.0 8
1.5 1
3 10

Nov 19
compensator design, pole-placement.
Surprise Test #14
Summary
Scores #Students
0.0 1
0.5 1
1.0 15
1.5 11
2 3
3 10

Nov 24
Kalman Filter, hidden Markov model (HMM).
Surprise Test #15
Summary
Scores #Students
2 2
2.5 4
3 35











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