EEC 510 Linear Systems
Chapter 5
Vectors and Linear Vector Spaces
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 155
Introduction
A vector is defined as a quantity that has both a
magnitude and a direction.
An example of such a quantity is veloci
EEC 510 Linear Systems
Chapter 3
State Variables and the State Space
Description of Dynamic Systems
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 25
Introduction
We are now interested in describing the
behavior of a system in terms of its interna
EEC 510 Linear Systems
Chapter 10
Stability
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 349
Introduction
In this chapter, we will be introducing stability
concepts for multivariable systems described
by state variable models.
While we will con
EEC 510 Linear Systems
Chapter 4
Fundamentals of Matrix Algebra
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 111
Introduction
We have already used the matrix notation
extensively in Chapter 3.
In this chapter, we will introduce a more
formal de
EEC 510 Linear Systems
Chapter 13
Design of Linear Feedback Control Systems
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 410
Introduction
In this chapter, we will be investigating the use
of feedback compensation for linear, constant
coefficient
EEC 510 Linear Systems
Chapter 7
Eigenvalues and Eigenvectors
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 266
Introduction
Eigenvalues and eigenvectors (also known as
the proper or characteristic values and
vectors) play an important role in ma
EEC 510 Linear Systems
Chapter 6
Simultaneous Linear Equations
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 233
Introduction
Many problems in engineering require the
solution of a system of simultaneous linear
equations.
E.g., control, system i
EEC 510 Linear Systems
Chapter 1
Background and Preview
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 1
Introduction
Control Theory is a branch of so-called Systems
Theory.
Usually, we are talking about self-regulating, or
feedback systems hence
EEC 510 Linear Systems
Chapter 8
Functions of Square Matrices and the
Cayley-Hamilton Theorem
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 304
Introduction
We have already seen scalar-valued functions of
square matrices such as , Tr(), etc.
In
EEC 510 Linear Systems
Chapter 11
Controllability and Observability for
Linear Systems
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 390
Introduction
In this chapter, we will be investigating
controllability and observability criteria for
linear
EEC 510 Linear Systems
Chapter 12
The Relationship Between State Variable and
Transfer Function Description of Systems
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 95
Introduction
Classical control theory mostly deals with
input-output transfer
EEC 510 Linear Systems
Chapter 9
Analysis of Continuous- and Discrete-Time
Linear State Equations
Dr. Murad Hizlan
Cleveland State University
EEC 510 - 331
Introduction
We already know that we can represent a physical
system in terms of coupled different
1
Homework 4: EEC 510
Problem 1: Using Laplace transform, solve the differential equation x = 2x(t) + b(t)u(t) with the
initial condition x(0).
Problem 2: A system has two inputs u = [u1 , u2 ]T and two outputs y = [y1 , y2 ]T . The input-output
equations
1
Homework 2- Part 1: EEC 510
Problem 1: Let e = [e1 , e2 , , en ]T be a real-valued vector variable. Show that eT e = tr(eeT ) =
e21 + e22 + + e2n .
Problem 2: For the e defined in Problem 1, show that |eT e| = 0. Find |In + eT e|, where In is n n
identi
1
Homework 2- Part 2: EEC 510
Problem 1: Consider that x1 = [1 2 3]T , x2 = [1 2 3]T , and x3 = [0 1 1]T .
a) Show that the set cfw_x1 , x2 , x3 is linearly independent.
b) Generate an orthonormal set using Gram-Schmidt procedure.
c) Express the vecto
1
Homework 3- Part 1: EEC 510
Problem 1:Find the eigenvalues
and eigenvectors and then use similarity transformation to diago
0
2
nalize A =
.
3 5
Problem 2: Find the eigenvectors of
A=
2
3
3 4
4 2 1
, A = 0 6 1 ,
0 4 2
using the bottom-up method (Method
1
Homework 3- Part 2: EEC 510
1 2 0
1 0 .
Problem 1: Using Cayley-Hamilton Theorem, find the inverse of A = 1
2 1 2
Problem 2: Compute eAt for
A=
3 1
2 2
, A=
3 2
0 3
,
Problem 3: Express eAt as MeJt M1 where J is a Jordan form matrix, M is a modal matrix