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

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

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

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

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 (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
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 #2)
Fall Semester, 2015
M. Carchidi
Problem #1 (20 points) - Computing The Norm of a Matrix
Given the 2 2 matrix function
1
16t3 et + 9 sin(t) 12t3 et 12 sin(t)
,
A(t) =
25 12t3 et 12 sin(t) 9t3 et + 16 sin(t)
dete

ESE500 - Linear Systems Theory (Homework #1)
Fall Semester, 2015
M. Carchidi
Problem #1 (20 points)
a.) (5 points) Place the dierential equation
3
y (t) + 4y 3 (t) = u(t)
is the state-space form
d
dt
"
x1 (t)
x2 (t)
#
=
"
f1 (x1 , x2 , u, t)
f2 (x1 , x2 ,

ESE500 - Linear Systems Theory (Homework #5)
Fall Semester, 2015
M. Carchidi
Problem #1 (20 points)
a.) (5 points) Show that if
x(t) =
m1
X k
k=0
t
vk et ,
k!
for constant n 1 columns vk (k = 0, 1, 2, ., m 1), is required to be a
solution to the ODE
dx(t)

ESE500 - Linear Systems Theory (Homework #3)
Fall Semester, 2015
M. Carchidi
Problem #1 (15 points) - The Inverse of A (t, )
Setting
X(t, ) = A (t, )A (, t)
and using the fact that
A (t, )
= A(t)A (t, )
t
and
A (t, )
= A (t, )A( )
show that
X(t, )
X(t, )

ESE500 - Linear Systems Theory (Homework #7)
Fall Semester, 2015
M. Carchidi
Problem #1 (20 points)
Show that the two systems
d
dt
"
x1 (t)
x2 (t)
with
and
with
#
=
"
1 2
3 2
#"
x1 (t)
x2 (t)
#
+
"
t
1
#
d
dt
"
#
y1 (t)
1 2 "
x1 (t)
y2 (t) = 3 4
x2 (t)

ESE500 - Linear Systems Theory (Homework #6)
Fall Semester, 2015
M. Carchidi
Problem #1 (25 points)
The linear state equation
dx(t)
= A(t)x(t) + B(t)u(t)
with
x(t0 ) = x0
dt
is called reachable on [t0 , tf ] if for any x0 = 0 and given any n 1 vector
xf ,

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
September 17, 2014
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
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 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

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
October 8, 2012
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 o

Notes on Linear System Theory
(ESE 500)
Michael A. Carchidi
June 7, 2012
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 Elec

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