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.


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

The syllabus, homework, solutions to homework, and all other information relevant to the course will be continuously posted at the course's web page. The URL is

quick links




João P. Hespanha

phone: (805) 893-7042
office: Harold Frank Hall, 5157

Office hours: Please email or phone me in advance to schedule for an appointment.

Assessment format

Homework – 30%

Mid-term exam – 30% (tentatively on Nov 12; in class)

Final exam – 40% (Monday Dec 15, 2014; Phelps 1437 from 8-11am)


The course will follow closely:
[1] J. Hespanha. Linear Systems Theory, 2009. (ISBN-13: 978-0-691-14021-6). 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

Lect #1


Introduction and course overview

System representation: input-output, block diagrams

  • Continuous vs. discrete-time
  • Examples

Lect #2


Where do state-space linear systems come from?

  • Local Linearization
  • Feedback Linearization

Lect #3


Basic system properties: causality, linearity, time-invariance

  • Forced responses
  • Impulse response
  • Transfer function

Lect #4


Impulse response and transfer function for state-space systems

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

Lect #5


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


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


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


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 #8



Internal stability of continuous-time LTI systems (cont.)


Lect #9


Input-output stability of LTI systems

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

Lect #10


Preview of optimal control

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




In class midterm exam on the material covered up to (and including) lecture #9 of the textbook (9. Input-output stability of LTI systems).

Practice exam 1 (and solutions)

Practice exam 2 (and solutions)

Solutions to the mid-term exam

Lect #11


Reachability and controllability subspaces for LTI systems

  • Definition, Physical example, and block diagrams
  • Reachability and controllability Gramians

Lect #11 (cont.)


Reachability and controllability subspaces for LTI systems (cont.)

  • Controllability matrix
  • Open-loop minimum energy control

Lect #12


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


Canonical decompositions

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

Lect #14



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

Eigenvalue assignment

  • Controllable case
  • Stabilizable case

Lect #15



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

Lect #16


Output feedback

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

Lect #17


Minimal realizations

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

Final Exam

The final exam will take place on Monday Dec 15, 2014 in Phelps 1437 from 8-11am.

The final exam is closed book, but during the exam you are allowed to consult one letter-size piece of paper with handwritten notes.

Practice final exam 1 (and solutions)

Practice final exam 2

Practice final exam 3


Homework Assignments


Number Posted on Due date Exercises Relevant lectures


10/6/2014 10/20/2014

Download from here

Solutions available here

#1, #2



(revised on 10/20/2014)

Download from here

Solutions available here

#3, #4


10/19/2014 11/3/2014

Download from here

Solutions available here

#5, #6, #7


10/19/2014 11/10/2014

Download from here

Solutions available here

#8, #9




(revised on 11/17/2014)

Download from here

Solutions available here




(revised on 11/17/2014)

Download from here

Solutions available here

#13, #14
#7 10/19/2014

12/12/2014 @ 12noon (email instructor)

(revised on 11/17/2014)

Download from here

Solutions available here

#15, #16, #17