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



Dynamics of Linear Systems
August-December, 2014

Announcements:
 		Nov 19, 2014: Final examination will be held on December 05, 2014 from 2:00PM to 5:00PM in PE303.
Nov 19, 2014: HW#4 is due on November 24, 2014 at 3:45PM.
Oct 31, 2014: Midterm#2 will be held on November 03, 2014 at PE301 from 4:00PM to 5:30PM.
Oct 20, 2014: HW#3 is due on October 27, 2014 at 3:45PM.
Oct 1, 2014: HW#2 is due on October 8, 2014 at 3:45PM.
Sep 22, 2014: Lecture of September 24, 2014 will be held at PE301 from 2:00PM to 3:30PM.
Sep 6, 2014: Midterm#1 will be held on September 10, 2014 at PE301 from 4:00PM to 5:30PM.
Sep 3, 2014: HW#1 is due on September 8, 2014 at 3:45PM.
Aug 18, 2014: 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 Aug 20, 2014 with email subject as 'E1_241_2014'.
Aug 12, 2014: Lectures will be held in PE 301 until any further notice.
Aug 11, 2014: Lecture of August 11, 2014 (Monday) will be held in PE 301.
Aug 7, 2014: Lecture timings are changed from 3:30pm-5:00pm to 4:00pm-5:30pm.
Aug 4, 2014: Lecture of August 6, 2014 (Wednesday) will be held in PE 301.
Aug 1, 2014: First lecture will be held in PE 301 on August 4, 2014 (Monday) at 3:15pm.
Instructor:
 		Prasanta Kumar Ghosh
Office: EE 200D
Phone: +91 (80) 2293 2694
prasantg AT ee.iisc.ernet.in
Teaching Assistant(s):
 		Shantanu Chakrabarty
Office: Room 214B, Power System Computer Control Lab
Phone: 9482516504
shantanu AT ee.iisc.ernet.in

Class meetings:
 		4:00pm to 5:30pm every Monday and Wednesday (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
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 4
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, linear operator.
 -
Aug 11
Similarity transformation, Companion form, System of linear equations, rank, nullity, Sylvester's Inequality.
 -
Aug 13
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 5
0.5 5
1.0 3
1.5 3
2.0 7
3.0 7

Aug 18
Multiplicity of eigenvalues, Matrix polynomial, polynomial matrix, addition and multiplication of polynomial matrix, left, right quotient, remainder, remainder theorem.
 Surprise Test #2
Summary
Scores #Students
0.0 5
0.5 9
1.0 6
1.5 6
2.0 1
2.5 4

Aug 20
Cayley-Hamilton theorem, irreducible, annihilating, minimum polynomial.
 Surprise Test #3
Summary
Scores #Students
1.0 5
1.5 5
2.0 7
2.5 12
3.0 3

Aug 23
Inner Product, orthogonality, projection, least squares solution, orthogonality of the subspaces, Orthonormal sets, Gram-Schmidt orthonormalization.
 Lecture by Shantanu Chakrabarty
Aug 25
Examples problems on Nil-potent matrix, Laplace Transform, ON sets, Orthonormal Complements, basis extention.
 Lecture by Shantanu Chakrabarty
Aug 27
Generalized eigenvectors, chain, Jordan canonical form.
 Surprise Test #4
Summary
Scores #Students
0.0 2
1.0 8
1.5 3
2.0 10
2.5 2
3.0 9

Sep 1
Computational procedure of finding JCF, Function of matrices, minimum polynomial computation.
-
Sep 3
Computing polynomial of a matrix, arbitrary function of a matrix using finite degree polynomial, exponential of a matrix.
 Surprise Test #5
Summary
Scores #Students
0.0 5
0.5 6
1.0 2
1.5 1
2.0 2
2.5 5
3.0 11

Sep 8
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 10
Midterm# 1
 Summary
Scores #Students
5-6 1
6-7 1
7-8 2
8-9 2
9-10 4
10-11 5
11-12 1
12-13 3
13-14 1
14-15 1
15-16 3
16-17 5

Sep 13
PDE, Geometric Interpretation of Linear PDEs including conceps of Level sets, Fundamental Solutions of wave equation and laplace equation.
 Lecture by Shantanu Chakrabarty
Sep 15
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.
-
Sep 17
LTI system, rational, irrational transfer function, zeros and poles, system decomposition, system composition, linearization.
 Surprise Test #6
Summary
Scores #Students
0.0 3
0.5 10
1.0 17
1.5 2
2.5 1

Sep 22
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.
 Surprise Test #7
Summary
Scores #Students
0.0 4
0.5 5
1.0 7
1.5 2
2 1
3 13

Sep 24
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.
-
Sep 29
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, stability of MIMO system.
Surprise Test #8
Summary
Scores #Students
0.0 1
0.5 1
1.0 4
1.5 3
2 6
2.5 3
3 9

Oct 8
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 13
Controllability of a continuous-time LTI system, controllability matrix, conditions for controllability, controllability Gramian
Surprise Test #9
Summary
Scores #Students
0.0 2
0.5 1
1.0 4
1.5 3
2 7
2.5 7
3 7

Oct 15
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.
Surprise Test #10
Summary
Scores #Students
1.0 4
1.5 13
2 6
2.5 1
3 6

Oct 20
Controllability and observability of system with A matrix in Jordan canonical form, introduction to controllability and observability indices.
Surprise Test #11
Summary
Scores #Students
0.0 1
1.0 5
1.5 8
2 2
2.5 1
3 12

Oct 22
Controllability and observability indices, Introduction to canonical decomposition.
-
Oct 27
Canonical decomposition.
-
Oct 29
Canonical decomposition, irreducible system, introduction to realizations.
Surprise Test #12
Summary
Scores #Students
1.0 2
1.5 1
2 7
3 19

Nov 03
Midterm# 2
 Summary
Scores #Students
6-7 2
7-8 2
8-9 0
9-10 1
10-11 0
11-12 6
12-13 2
13-14 1
14-15 3
15-16 1
16-17 3
17-18 5
18-19 1
19-20 1

Nov 5
Realizations, controllable and observable canonical form, coprimeness, degree of transfer function.
-
Nov 10
minimal realization and equivalence between minimal realizations.
Surprise Test #13
Summary
Scores #Students
0.0 3
0.5 1
1.0 3
1.5 4
2 9
2.5 8
3 2

Nov 12
Controllability after sampling, introduction to state feedback.
Surprise Test #14
Summary
Scores #Students
0.0 3
0.5 1
1.0 3
1.5 6
2 5
2.5 10
3 1

Nov 17
state feedback, controllability, observability, stability with state feedback, stabilization.
-
Nov 19
state estimator, feedback with estimated state, separation principle, compensator design, pole-placement.
Surprise Test #15
Summary
Scores #Students
0.0 2
0.5 1
1.0 4
1.5 8
2 6
2.5 5
3 3

Nov 24
Kalman Filter, hidden Markov model (HMM).
-










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