ME 232
Lecture # 4
1 Input-output Differential Equation Models
As we have seen, many systems are modeled by differential equations that relate the inputs u to
the outputs y. So far, we have mainly looked at single-input single-output systems modeled by
li

1
2
Summary
ME 232 Advanced Control Systems I
Lecture 15
Controllable and Unobservable Subspaces
Kalman Canonical Staircase Decompositions
(ME232 Class Notes pp. 97-102)
and
(Additional Notes section 5)
Subspaces, range and null spaces (review)
The contro

1
2
Summary
ME 232 Advanced Control Systems I
Lecture 16
Singular value decomposition
meaning
algorithm
Balanced realizations
meaning
algorithm
proof
Singular Value Decomposition &
Balanced Realizations
(ME232 Class Notes pp. 92-96)
3
Matrix induce

ME 232
Fall 2016
Midterm 1
NAME:
ID # :
#1
#2
#3
#4
#5
#6
TOTAL
12
8
12
24
6
12
74
Instructions:
1 Write your name and student ID number.
2 Read the questions carefully.
3 Write your solution clearly. Please, please please .
4 This exam has 6 questions wo

ME 232
Solutions # 3
1 State-space Realizations
Write the given system in H(s) in matrix form:
x1
0
x1
0
1
x1
=
u, y = 2 1
+
x2
1
x2
4 5
x2
or more compactly as
0
1 0
H(s) 4 5 1
2
1 0
Taking Laplace Transforms of the state equations we get
sX(s) x(0

1
2
Summary
ME 232 Advanced Control Systems I
Lecture 13
Definition of controllability
Cayley-Hamilton theorem
The controllability theorem
examples
proof
remarks
Controllability and Observability
of Discrete Time Systems
The finding the controllabi

ME 232
Homework # 2
Issued: September 01, 2016
Due: September 09, 2016
1 Laplace Transforms
Assume that f (t) = 0 for t < 0. Find the Laplace Transforms of
(a) f (t) = 2et + 3e2t
(b) f (t) = et sin(5t)
Z t
(c) f (t) =
e5 sin(3 )d
0
(d) f (t) = t2 e2t
at

ME 232
Solutions # 4
1 DC Gain
(a) From the block diagram,
Y = P (s) [F R + K(s) (R Y )]
so the transfer function from r to y is
Gry (s) = (I + P (s)K(s)1 (P (s)F + P (s)K(s)
Notice that this is a matrix, and the ordering of the terms matter (since typica

ME 232
Lecture # 3
1 Geometric Sums
Some basic results:
N
X
k
a =
k=0
if |a| < 1,
(1 aN +1 )/(1 a)
N +1
X
ak =
k=0
if a 6= 1
if a = 1
1
1a
2 Sequences
Let us use k as the time-index.
We consider one-sided sequences of real numbers:
f = (f0 , f1 , )
Two sp

ME 232
Lecture # 2
1 Some Important Functions
dn f
For a function f (t), write f [n] = n .
dt
Unit step , unit impulse .
The unit impulse can be regarded as a limit of pulses with area 1.
Given functions f, g, the convolution h = f g is defined by
Z
h(t)

ME 232
Lecture # 5
1 From State-space to Transfer Functions
It is easy to compute the transfer function from the realization
x(t)
= Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
Using the s-notation, we can rewrite these equations as
sx = Ax + Bu
=
(sI A)x = Bu
=
x

ME 232
Lecture # 11
1 Computing Matrix exponentials.
We may compute matrix exponentials via the method outlined in the previous lecture.
Example 1 Let
2 3
A=
1 2
e
At
Jt
= Te T
1
=T
et 0
0 et
=
1.5et 0.5et 1.5et + 1.5et
0.5et 0.5et 0.5et + 1.5et
Example 2

ME 232
Lecture # 1
1 Logistics
Go over Homework, Midterms, Grading, Office Hours, Discussion, Pre-requisites, etc.
2 Classical Control Review
Go over ingredients - plant, model, controller, noises, disturbances, reference/commands.
Stability, robustness,

1
ME 232 Advanced Control Systems I
Lecture 14
Summary
2
Definition of controllability
The controllability theorem
Finding the controllability grammian via the Lyapunov
Controllability and Observability
of Continuous Time Systems
equation when A is Hur