Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
November 10, 2015
Chapter 5 - Computing Transition Matrices
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department

Laplace Transform Methods
Michael A. Carchidi
October 6, 2014
1. Definitions
Given a function f (t) that is defined for all values of < t < , and another
function K(s, t) that is defined for all < t < , and all < s < , the integral
transform of f (t), wit

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
September 24, 2015
Chapter 4 - Transition Matrix Properties
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department

ESE500 - Linear Systems Theory (Master Final Exam)
Fall Semester, 2015
M. Carchidi
Problem #1 (15 points)
Determine a minimal time-invariant linear system which has the weighting
pattern
G(t, ) = cos(t ).
Hint: Try to be smart when choosing t0 and tf and

ESE500 - Linear Systems Theory (Exam #1)
Fall Semester, 2015
M. Carchidi
Instructions
1.) You must do any four (4) of the following six (6) problems while in class.
2.) You may take the other two (2) problems home and hand them in at the
BEGINNING of clas

Solutions Manual
LINEAR SYSTEM THEORY, 2/E
Wilson J. Rugh
Department of Electrical and Computer Engineering
Johns Hopkins University
PREFACE
With some lingering ambivalence about the merits of the undertaking, but with a bit more dedication than
the first

University Of Pennsylvania
Department of Electrical and Systems Engineering
ESE500 Linear Systems (Course Outline)
Instructor: Dr. Michael A. Carchidi
-Textbook:
1.)
Linear System Theory by Wilson J. Rugh
(Required)
(Prentice Hall, 2th Edition @1996,
ISBN

ESE500 - Linear Systems Theory (Exam #1)
Fall Semester, 2012
M. Carchidi
Problem #1 (25 points) - Solving a Linear System Completely
Solve completely for x1 (t) and x2 (t) given that
d
dt
"
x1 (t)
x2 (t)
#
=
"
4t 1 1 2t
4t 2 2 2t
#"
x1 (t)
x2 (t)
#
and
"

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
August 5, 2012
Chapter 13 - Controller and Observer Forms
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
December 17, 2012
Chapter 9 - Controllability and Observability
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Departm

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
September 24, 2015
Chapter 3 - State Equation Solutions
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of E

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
October 18, 2015
Chapter 2 - State Equation Representation
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department o

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
November 4, 2015
Chapter 6 - Internal Stability Analysis
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
October 18, 2015
Chapter 2 - State Equation Representation
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department o

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
September 24, 2015
Chapter 3 - State Equation Solutions
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of E

ESE500 - Linear Systems Theory (Homework #6)
Fall Semester, 2016
M. Carchidi
Problem #1 (20 points)
Show (without solving the system) that
d
dt
"
x1 (t)
x2 (t)
#
=
"
2t2 t 1
1 t2
2
5t 2t + 1 2t2 t 1
#"
x1 (t)
x2 (t)
#
is uniformly exponentially stable for

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
December 2, 2015
Chapter 10 - Realizability
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of Electrical
an

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
November 10, 2015
Chapter 5 - Computing Transition Matrices
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
November 18, 2015
Chapter 9 - Controllability and Observability
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Departm

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
November 9, 2015
Chapter 20 - Discrete-Time State Equations
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department

Functions of a Square Matrix
Michael A. Carchidi
October 6, 2014
1. Positive Powers of a Square Matrix
In this note, we consider the problem of computing an analytic function f
of a square n n matrix A. A function f (z) is called analytic at a point z0
if

ESE500 - Linear Systems Theory (Ph.D. Final Exam)
Fall Semester, 2012
M. Carchidi
Problem #1 (20 points)
Consider the time-invariant linear system described by the state equation
dx(t)
= FAx(t)
dt
where F is symmetric and positive definite and A is such t

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
October 6, 2012
Chapter 4 - Transition Matrix Properties
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
August 5, 2012
Chapter 12 - Input-Output Stability
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department of Electr

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
October 8, 2012
Chapter 21 - Computing Transition Matrices
The following notes are written by Dr. Michael A. Carchidi for the Linear
System Theory Course (ESE 500) taught through the Department o

ESE500 - Linear Systems Theory (Extra Credit Problems)
Fall Semester, 2012
M. Carchidi
Extra Credit Problem #1 (10 points)
Suppose that
A(t) = f (t)B + g(t)C
where B and C are constant matrices satisfying BC = CB and f and g
are continuous functions of th

ESE500 - Linear Systems Theory (Final Exam)
Fall Semester, 2012
M. Carchidi
Problem #1 (20 points)
Determine all conditions on the real parameter a so that the 33 symmetric
matrix
1 a 1
Q(a) =
a 3a 3
1 3 2
is positive semi-definite.
Problem #2 (20 poin

ESE500 - Linear Systems Theory (Homework #4)
Fall Semester, 2013
M. Carchidi
Problem #1 (20 points)
Compute the transition matrix for the discrete system
x1 (k + 1)
1
k1 1k
x1 (k)
x2 (k + 1) = 2 2k 3k 2 2 2k x2 (k)
2 2k 2k 2 2 k
x3 (k + 1)
x3 (k)
for k

ESE500 - Linear Systems Theory (Homework #1)
Fall Semester, 2013
M. Carchidi
Problem #1 (20 points)
Consider the non-linear equations
dx(t)
= u(t) x(t)xT (t)u(t)
dt
where is a constant. If this system is linearized by setting
(t)
x (t) = x(t) x
and
u (t)

ESE500 - Linear Systems Theory (Homework #5)
Fall Semester, 2013
M. Carchidi
Problem #1 (20 points)
Consider the T -periodic system
d
dt
"
x1 (t)
x2 (t)
#
=
"
3 4
2 3
#"
x1 (t)
x2 (t)
#
+
"
1 0
0 1
#"
sin(t)
cos(t)
#
.
a.) (10 points) Determine T and all