Kiselyuk 1
Constantine was the son of Helena and Flavius Constantinius. Ever since his youth, signs were evident of his future greatness. He was responsible for reuniting a fractured empire. The original story of Constantine's conversion to Christia

10 LINEAR SYSTEMS THEORY
where 6 is the angular position of the motor, J is the moment of inertia. F is the
coefﬁcient of viscous friction, and T is the torque applied to the motor.
a) Generate a two—state linear system equation for this motor in the form

8 LINEAR SYSTEMS THEORY
0 0 1/5
1 t t2/2
= 0 1 t
0 0 1
From the third expression in Equation (1.72) we obtain
eAt : Qez‘itQ—l
The eigendata of A are found to be
A : {0,0,0}
1 0 0
v = 0 ‘ 1 , 0
0 0 1
Actually we can note that A is already in Jordan form, w

CHAPTER 1
#
Linear systems theory
_—._._._—-—-———-—
Problems
Written exercises
1.1 Find the rank of the matrix [ U 0
Solution
The rank of a matrix A can be deﬁned as the dimension of the largest submatrix
consisting of rows and columns of A whose determ

18 PROBABiLlTY THEORY
Using integration by parts we obtain
13(372) = 3326_”E30+f Zace‘mdac
0
2 GO
: —f (Lace—"'de
a 0
2-
= em
a
2
: F
d).
02 = E(a:2)—a':2
: 3—2
e).
2/0!
P(:E*USISCE+O') = / {re—‘1de:
0
2 lie—2
m 0.85
2.6 Derive an expression for the skew

6 UNEAR SYSTEMS THEORY
H
2
A[1+At+(g:)+.]
: AeAt
This proves the ﬁrst equality. After writing the third expression of the above
sequence of equations, we can bring the common factor A out to the right to obtain
‘5 At (Atlz
_ : I At A
rite [ + + 2!

2 LINEAR SYSTEMS THEORY
Solution
Suppose A and B are given as
Then we see that
AB : [albl-i-agbg a1b2+a2b3]
(1251 + (1352 (12172 + G353
albl ‘l' 9-2b2 ($251+ (1ng
@152 Jr a2b3 Clsz + 0,3453
BA=[
'We see that AB : BA if (11le +0.2 b3 = (2.2131 + (13 b2. Th

4 LINEAR SYSTEMS THEORY
1.5 Consider the matrix
A _ a b
_ b (3
Recall that the eigenvalues of A are found by ﬁnd the roots of the polynomial
PH) : |AI — A]. Show that PM) = 0. (This is an illustration of the Cayley—
Hamilton theorem [Bay99, Che99, KaiOO].

12 LINEAR SYSTEMS THEORY
Simulate from f : 0 to t 2 5 using step sizes of 0.1 and 0.2. Tabulate the
RMS value of the error between the numerical and analytical solutions for
the capacitor voltage for each of your six simulations.
Solution
3.).
a = m2R+LﬁT

16 PROBABILITY THEORY
2.2 Suppose a deck of 52 cards is randomly divided into four piles of 13 cards
each. Find the probability that. each pile contains exactly one ace [GreDl].
Solution:
Consider the ﬁrst pile. There are a. total of 52-choose-13 possible