ECE 451
LECTURE 1
Preface
This lecture reviews several topics from ECE 350, i.e. state variables, controllability and observability, as
well as state variable feedback. It is assumed that students attending this class have already been
exposed to this mat
ECE 451
LECTURE 3
Preface
This lecture introduces the method of dynamic programming which can be used to solve optimal control
problems. This method was developed by R. Bellman during the early 1960s and was meant to leverage
the new digital computer tech
ECE 451
LECTURE 5
Preface
This lecture continues our discussion of the method of dynamic programming. We start by solving an
optimal control problem, which was given as an exercise at the end of Lecture 4. Next, we proceed to a
practical optimal control p
ECE 451
LECTURE 7
Preface
This lecture continues our discussion of the method of dynamic programming and its application to
continuoustime systems resulting in the classic HamiltonJacobiBellman (HJB) partial differential
equation. We first consider an
Solution example of an ordinary differential equation (ODE) by:
1)
Classical ODE method.
2)
Laplace transformation method.
Given an ODE:
+ 4 + 3 =
Initial condition: = 0; = 0;
(1)
=0
Solution 1:
The expression at the right side of the ODE suggests th
The following figure shows a translational mechanical system. Derive a set of differential equations that
describes all variables in this system. Draw the block diagram assuming that:
1. Force F is the input and the displacement X1 is the output.
2. Force
Control is concerned with dynamics in a wider sense:
 seeking understanding of the dynamic behaviour of systems (mechanical, electrical, chemical,
biological, economic, )
 with a view to operate the system in some desired manner variables (position, vel
Topic 5 (Mathematical Modelling of Various Systems)
Electrical System:
1) Resistance, R
2) Inductance, L
V (t ) = Ri (t )
V (t ) = L
i (t ) = GV (t ), where
i (t ) =
1
G=
R
3) Capacitance
1 t
i (t )dt + V (0 )
C 0
d
i (t ) = C V (t )
dt
d
i (t )
dt
V (t )
Topic 3 (Mathematical Modelling of SpringMassDamper System)
The dynamic performance of physical systems is obtained by utilizing the physical laws of
mechanical, electrical, fluid and thermodynamic systems. In order to obtain the transfer functions
for
Topic 2 (Mathematical Modelling of System Dynamics Part I)
 set of equations that represents the dynamics of a physical system accurately or, at least it is able
to describe the response or behaviour of the system approximately
 generally mathematical m
ECE 451
LECTURE 6
Preface
This lecture continues our discussion of the method of dynamic programming and our example of its
application to the solution of the discrete regulator problem. We consider a numerical example of the
linear regulator. We next con
ECE 451
LECTURE 4
Preface
This lecture continues our discussion of the method of dynamic programming. We start by solving an
optimal control problem, which was introduced at the end of Lecture 3. Next, we proceed to generalize
the particular solution obta
ECE 451
Homework 1
Due Wednesday, September 7, 2011
1. Use the Smith chart to find the following quantities for the transmission line circuit below:
(a)
The SWR on the line
(b)
The reflection coefficient at the load
(c)
The load admittance
(d)
The input i
Homework 2  Solutions
Problem 1 (Steer Exercise 4.1)
(a) o c / f 3 108 / (18 109 ) 16.67 mm
(b) g g / r (16.67mm) / 64 2.08 mm
(c) l g / 4 (2.08 mm) / 4 0.52 mm
Problem 2 (Steer Exercise 4.5)
R=100 /m; L=85 nH/m, C=150 pF/m; G =1 S/m
R j L G jC
(a) j
ECE 451
Homework 3
Solutions
Problem 1
(a) For Pi Network
V1
I1
Z11
Z 21
Y11
V2
I1
I2 0
I 2 0
Y21
2Z A Z B
V1
ZA ZA ZB
ZA
I1Z11
Z A ZB
I1
I1
V1 V 0
2
V1
I1
ZZ
I1 A B
Z A ZB
Z A Z A ZB
Z 22 (symmetry)
2Z A Z B
2
ZA
2Z A Z B
Z12 (reciprocit
ECE 451
Homework 3
Due Wednesday October 5, 2011
1. Derive the Z and Y matrices for the following twoport networks:
(a)
(b)
2. A fourport network has the scattering matrix shown below
(a)
Is this network lossless?
(b)
Is this network reciprocal?
(c) Wha
Homework 4  Solutions
Problem 1 (Steer 61)
(a) No, S12 S21
(b)
Maximum available power condition:
12
V / 50 1 mW V 0.316V E 2V 0.632V
2
(c)
Since S12 = 0, in = S11 = 0.25
2
PR IN PIN (0.25) 2 1 mW 62.5W
(d)
Zo = 50
s 1 S11 1 S22 S12 S21
Z11 1 S11 1 S
ECE 451
Homework 5
Problem 1
Scope/DVM
detector
a3
b3
Zs
Solutions
Directional Coupler
+
Vs
a1
b1

ZL
b2
a2
Zo
(a)
d
b3
a3
S33
S31
1
S23
S13
a1
S32
b2
bs
S21
S11
S22
L
S
S12
a2
b1
(b)
Assume L = 0
First order loops: sS11, sS31dS13, dS33.
Second order loo
ECE 451
Homework 5
Due Wednesday November 2, 2011
Scope/DVM
detector
b3
Zs
Directional Coupler
+
Vs

a3
a1
b1
ZL
Zo
b2
a2
Figure 1
1. Consider the measurement set up shown in Figure 1. Choose Zo as the reference
impedance of the system.
a)
b)
Draw the co
ECE 451
Homework 6
Solutions
Problem 1
The voltage and current on a lossless transmission line are
V ( z ) V e z V e z
I ( z ) Yo V e z V e z
Let port 1 be located at z=l with voltage and current
V1 V e l V e l
and
I1 Yo V e l V e l
Let port 2 be locat
ECE 451
Homework 6
Due Wednesday, November 16, 2011
1. Find the admittance matrix Y of a lossy transmission line. Express your answers
in terms of the propagation constant , the length of the line l and the
characteristic admittance, Yo.
2. A 2cm section
NAME _
MIDTERM EXAM SOLUTIONS
ECE 451
October 17, 2011
12:00 12:50 p.m.
Instructions: Write your name and section where indicated. Show all work. Indicate the units of
your answers.
Useful constants :
o = 8.85 1012 F/m =
o = 4 107 H/m
1 GHz = 109 Hz
Pro
ECE 451
LECTURE 2
Preface
This lecture introduces the optimal control problem. The concepts of an admissible control, an admissible
trajectory, and the general form of the performance measure are presented. We also review the basic
forms of performance me
ECE 350: Principles of Automatic Control
UIC: FALL 2010
LECTURE 1
Summary
In this first lecture, we introduce the subject of control engineering and the role that the control engineer
serves in overall system creation. Key to the subject of control engine
Topic 6 (Block Diagram Analysis)
A block diagram of a system is a pictorial representation of the function performed by each
component and of the flow of signals within a system. In brief, a block diagram is a sequence
of causes and effects. Differing fro