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



Dynamics of Linear Systems
August-December, 2016

Announcements:
 		November 15, 2016: Final examination will be held on December 5, 2016 from 10:00AM to 1:00PM in EE B 308.
November 15, 2016: HW#4 is due on November 22, 2016 at 2:00PM.
October 23, 2016: HW#3 is due on October 27, 2016 at 2:00PM.
October 21, 2016: Midterm#2 will be held on November 4, 2016 from 6:00pm to 7:30pm in EE B 308.
October 6, 2016: HW#2 is due on October 13, 2016 at 2:00PM.
September 15, 2016: Midterm#1 will be held on September 21, 2016 from 6:30pm to 8:00pm in EE B 308.
September 1, 2016: HW#1 is due on September 6, 2016 at 2:00PM.
Aug 25, 2016: Class on Aug 30, 2016 (Tuesday) will start at 1:45pm.
July 26, 2016: First lecture will be held in EE B 304 on August 2, 2016 (Tuesday) at 2:00pm.
Instructor:
 		Prasanta Kumar Ghosh
Office: EE C 330
Phone: +91 (80) 2293 2694
prasantg AT ee.iisc.ernet.in
Teaching Assistant(s):
  • Sunil R
    Office: EE C 238
    Phone: +91 9972560535
  • Sanjay Ghosh
    Office: EE C 324
    Phone: +91 7899625171
Class meetings:
 		2:00pm to 3:30pm every Tuesday and Thursday (Venue: EE B 304)
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.
When is a Periodic Discrete-Time System Equivalent to a Time-Invariant One? by Dooren et al.
Newton's Identities and their proofs by Dan Kalman
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 2
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 9
Similarity transformation, Companion form, System of linear equations, rank, nullity, Sylvester's Inequality.
 -
Aug 11
Special matrices, Nilpotency, Eigenvalue, Eigenvectors, distinct eigenvalues, diagonalization.
 -
Aug 16
The role of diagonalization in solving linear system of equations, Multiplicity of eigenvalues, Matrix polynomial, polynomial matrix, addition and multiplication of polynomial matrix, left, right quotient, remainder.
 -
Aug 18
Remainder Theorem, Cayley-Hamilton theorem, irreducible, annihilating, minimum polynomial.
 Surprise Test #1
Summary
Scores #Students
0.0 6
0.125 1
0.25 5
0.5 7
0.625 2
0.75 1
1.0 3
1.25 5
1.375 2
1.625 1
1.75 4
2.0 4
2.125 1
2.25 1
2.5 2
2.75 2
3.0 2

Aug 23
Generalized eigenvectors, chain,.
 Surprise Test #2
Summary
Scores #Students
0.0 3
0.25 1
0.5 7
0.75 7
1.0 14
1.25 1
1.5 4
1.75 3
2.0 3
2.25 1
2.25 1

Aug 25
Jordan canonical form, Computational procedure of finding JCF, Function of matrices, minimum polynomial computation.
 Surprise Test #3
Summary
Scores #Students
0.0 2
0.5 4
1.0 2
1.5 15
2.0 1
2.5 2
2.75 1
3.0 9

Aug 30
Computing polynomial of a matrix, arbitrary function of a matrix using finite degree polynomial, exponential of a matrix.
 -
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.
Surprise Test #4
Summary
Scores #Students
0.0 2
0.5 7
1.0 19
1.25 2
1.5 4

Sep 4
Discussion session on basics of vector spaces, four fundamental subspaces of a matrix, rank of a matrix, eigenvalue and eigenvector (concepts and problems).
 Sanjay Ghosh,
Sunil R (TAs)
Sep 6
Singular value decomposition, continuous and discrete time systems, memoryless, causal system, states of a system, linear, lumped, distributed system, zero-state, zero-input response.
Surprise Test #5
Summary
Scores #Students
0.5 3
1.0 1
1.5 1
2.25 4
2.5 2
3 20

Sep 8
convolution, state-space representation, time-invariance, LTI system, LTI system, rational, irrational transfer function, zeros and poles, system decomposition, system composition.
Sep 15
Linearization, example of linearization, discrete-time system.
Surprise Test #6
Summary
Scores #Students
0.0 10
0.25 1
0.5 11
1.0 8
1.5 1
2.0 1
3.0 1

Sep 20
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, Discretization.

Sep 21
Midterm# 1.
 Average=8.64,
Std Dev=3.23,
Min=0.5, Max=15.5
Summary
Scores #Students
0-2 2
3-4 1
4-5 2
5-6 1
6-7 3
7-8 6
8-9 4
9-10 5
10-11 3
11-12 3
12-13 3
15-16 1

Sep 22
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, Impulse response of LTV state-space equation, solution of LTV discrete-time state-space equation.
Sep 27
Equivalent LTV state-space equations, Stability of continuous and discrete-time LTI SISO system, BIBO stability, stability for systems with proper rational transfer function, stability of MIMO system.

Sep 29
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 systems.
 Surprise Test #7
Summary
Scores #Students
0.0 15
0.25 1
0.5 7
0.75 1
1 4
1.5 2
1.75 1
2 1
2.5 2

Oct 4
Controllability of a continuous-time LTI system, controllability matrix, conditions for controllability, controllability Gramian.
 Surprise Test #8
Summary
Scores #Students
1 3
1.5 2
2 4
2.5 3
2.75 1
3 19

Oct 6
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 #9
Summary
Scores #Students
0 3
0.5 1
0.75 1
1 2
1.25 1
1.5 8
1.75 2
2 13
2.25 2
2.5 1

Oct 13
Controllability and observability of linear time-varying systems, duality theorem for linear time-varying systems, Controllability and observability of system with A matrix in Jordan canonical form, introduction to controllability and observability indices.
-
Oct 18
Controllability and observability indices, Introduction to canonical decomposition.
-
Oct 20
Canonical decomposition, irreducible system.
 Surprise Test #10
Summary
Scores #Students
0.5 6
1 5
1.25 1
1.5 8
1.75 1
2 5
2.5 5
2.75 2

Oct 25
Introduction to realizations, controllable canonical form.
 Surprise Test #11
Summary
Scores #Students
0 6
0.5 6
1 11
1.5 6
2 2
2.5 1
3 2

Oct 27
controllable and observable canonical form, coprimeness, degree of transfer function, minimal realization.
 Surprise Test #12
Summary
Scores #Students
0.5 4
0.75 1
1 9
1.25 3
1.5 3
1.75 1
2 8
2.25 1
2.75 2
3 2

Oct 29
Discussion session on State-Space Solutions and Realizations, Stability, Controllability and Observability.
Sanjay Ghosh,
Sunil R (TAs)
Nov 1
Holiday
-
Nov 3
Minimal realization, Equivalence between minimal realizations, Controllability after sampling.
-
Nov 4
Midterm# 2
-
Nov 8
state feedback, controllability, observability, stability with state feedback, stabilization.
-
Nov 10
state estimator, feedback with estimated state, separation principle.
Surprise Test #13
Summary
Scores #Students
0.5 2
1.0 1
1.5 4
2 2
2.25 6
2.75 1
3 15

Nov 15
state feedback with eigenstructure specification.
Surprise Test #14
Summary
Scores #Students
1.0 8
1.5 1
2 6
2.25 1
2.75 1
3 15

Nov 17
compensator design, pole-placement, Kalman Filter, hidden Markov model (HMM).
Surprise Test #15
Summary
Scores #Students
0.5 1
1.0 5
1.5 22
3 4

Dec 5
Final Exam
 Average=24.38,
Std Dev=5.19,
Min=16.5, Max=36.25
Summary
Scores #Students
16.0-17.5 4
18.5-20.0 2
21.0-22.5 9
23.0-24.5 5
25.5-27.0 8
29.0-31.0 2
33.0-35.0 2
36.0-50.0 2











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