Syllabus
 

 

The purpose of this course is to provide the students with the basic tools of modern linear systems theory: stability, controllability, observability, realization theory, state feedback, state estimation, separation theorem, etc. For time-invariant systems both state-space and polynomial methods are studied. The students will also be introduced to the computational tools for linear systems theory available in MATLAB. The intended audience for this course includes, but is not restricted to, students in circuits, communications, control, signal processing, physics, and mechanical and chemical engineering.

Students are expected to have taken at least one class in control systems and/or dynamical systems and be very familiar with linear algebra and ordinary differential equations. Familiarity with Laplace transforms and classical control methods are strongly recommended.

Co-requisite

ECE 210A Matrix Analysis and Computation

Graduate level-matrix theory with introduction to matrix computations. SVD's, pseudo-inverses, variational characterization of eigenvalues, perturbation theory, direct and iterative methods for matrix computations.

Course's web page
  1. The syllabus and general information relevant to the course is available at

    http://www.ece.ucsb.edu/~hespanha/ece230a-me243a/

  2. Canvas will be used to post and collect homework assignments and solutions.

    PLEASE add you picture to Canvas so that I can associate names to faces.

 

quick links
SYLLABUS
STUDY GUIDE
HOMEWORK ASSIGNMENTS

Academics

 

Instructor

João P. Hespanha

email: hespanha@ece.ucsb.edu
phone: (805) 893-7042
office: Harold Frank Hall, 5157

Office hours: Please email instructor for appointment

Assessment format

Homework – 30%

Mid-term exam – 30% (tentatively on Nov 4; "in class")

Final exam – 40% (Monday, December 9, 2024 8:00 AM - 11:00 AM; Phelps 1437)

Textbook

The course will follow closely:
[1] J. Hespanha. Linear Systems Theory, Second Edition, 2018. (ISBN-13: 978-0691179575). Details available here.

Other recommended textbooks are:

[2]   P. Antsaklis, A. Michel. Linear Systems. McGraw Hill, 1997.
[3]   C.-T. Chen. Linear Systems Theory and Design. Oxford Univ. Press, 3rd ed., 1999. (ISBN 0-19-511777-8)

All students are strongly encouraged to review linear algebra. Chapter 3 of [3] provides a brief summary, but a review of a Linear Algebra textbook (such as [4] below) is preferable, especially if one goes through a few exercises.

[4]   Gilbert Strang Linear Algebra and Its Applications, 1988.

 

Study Guide

 

The following is a tentative schedule for the course. If revisions are needed they will be posted on the course's web page. Students are strongly encouraged to read the corresponding chapter of the textbook prior to each class.

Class Contents Remarks/Supplemental material

Lect #1

9/30

Introduction and course overview

Systems Representation

System representation: input-output, block diagrams

  • Continuous vs. discrete-time
  • Examples

Lect #2

10/2

Where do state-space linear systems come from?

  • Local Linearization
  • Feedback Linearization
  

Lect #3

10/7

Basic system properties: causality, linearity, time-invariance

  • Forced responses
  • Impulse response
  • Transfer function
 

Lect #4

10/9

Impulse response and transfer function for state-space systems

  • Definitions
  • Elementary realization theory for LTI systems
  • Equivalent state-space representations
 

Lect #5

10/14

Solution for state-space linear time-varying (LTV) systems

  • Solution to homogeneous linear systems—Peano-Baker series
  • State-transition matrix
  • Properties of the state transition matrix
  • Solution to nonhomogeneous linear systems—variations of constants formula
  

Lect #6

10/16

Solution for state-space linear time-invariant (LTI) systems

  • Matrix exponential (definition and properties)
  • Computation of matrix-exponentials using the Laplace transform
  • The importance of the determinant of A
 

Lect #7

10/21

Solution to state-space linear time-invariant (LTI) systems (cont.)

  • Jordan normal form
  • Computation of matrix-exponentials using the Jordan normal form
  • Poles with multiplicity larger than one (block diagram interpretations)
 

Lect #8

 

10/23

Stability

Internal stability of continuous-time LTI systems

  • Definitions
  • Eigenvalues condition (block diagram interpretation of multiplicity)
  • Lyapunov Theorem (LMI)
  • Stability of nonlinear systems from local linearization
  

Lect #9

10/28

Input-output stability of LTI systems

  • Definition
  • Time-domain condition
  • Frequency-domain condition
 

Lect #10

10/30

Preview of optimal control

  • Linear quadratic regulator problem
  • Algebraic Riccati equation
  • Optimal state-feedback control
  • Stability
 

11/4

In class midterm exam on the material covered up to (and including)
lecture #9 of the textbook.

  

Lect #11

11/6

Controllability and State Feedback

Reachability and controllability subspaces for LTI systems

  • Controllability matrix
  • Open-loop minimum energy control
  • Controllability matrix
  • Open-loop minimum energy control
 

11/11

Veteran's day (no class)

  

Lect #12

11/13

Controllable systems

  • Definition
  • Controllability matrix test
  • Popov-Belevitch-Hautus (PBH) test
  • Eigenvector/eigenvalue test
  • Lyapunov test (LME)
  • Feedback stabilization based on the Lyapunov test
 

Lect #13

11/18

Canonical decompositions

  • Invariance with respect to equivalence transformations
  • Controllable canonical form for single-input systems
  • Controllable decomposition
 

Lect #14

11/20

Stabilizability

  • Definition
  • Popov-Belevitch-Hautus (PBH) test
  • Eigenvector/eigenvalues test
  • Lyapunov test (LMI)
  • Lyapunov test-based control

Eigenvalue assignment

  • Controllable case
  • Stabilizable case
 

Lect #15

11/25

 Observability and Output Feedback

Observability

  • Observability and constructibility
  • Physical examples & block diagrams
  • Observability/constructibility Gramians
  • Gramian-based reconstruction
  • Duality
  • Observability tests
  

11/27

Thanksgiving holiday (no class)

  

Lect #16

12/2

Output feedback

  • Detectability
  • Observable decomposition
  • Detectability tests
  • State estimation
  • Eigenvalue assignment by output injection
  • Stabilization through output feedback—separation theorem
 

Lect #17

12/4

Minimal realizations

  • Markov Parameters
  • Kalman decomposition Theorem
  • Connection with controllability/observability
  • Equivalence of minimal realizations
  
12/9

Final Exam

The final exam will take place on Monday, December 9, 2024 8:00 AM - 11:00 AM.The exam is closed book, but during the exam you are allowed to consult one letter-size piece of paper with handwritten notes.

 

Homework Assignments

 

Number Posted on Due date Exercises Relevant lectures

#1

 See Canvas  

 

#1, #2

#2

#3, #4

#3

#5, #6, #7

#4

 #8, #9

#5

 #11, #12

#6

 #13, #14
#7 #15, #16, #17