Solution, HW #1
ECE 450, Fall 2012
6.1.1. Find the matrices A, B, C and D for the state space representation of the circuit
shown. Use v1 and v2 for the state variables. VA and VB are input voltages. Remember
dv ( t )
that the current across a capacitor i
Solution, HW #2
ECE 450, Fall 2012
6.1.5 A system is described by the transfer function
s+3
H ( s) = 2
.
s + 9 s + 20
Find the step response using state space methods.
Solution
A corresponding differential equation is
y '+ 9 y '+ 20 y = f '+ 3 f .
We begi
Solution, HW #4
Fall 2012
7.1.1 Find the sensitivity function for the unity feedback system defined in Fig. 7.1.4
assuming:
1000
H ( s) =
.
s + 10
What is the frequency range of disturbances that can be harmful to this system?
Solution
The sensitivity fun
7.3 Feedback System Steady State Errors
We are also interested in achieving desired steady-state values for the system
output. In this section, we identify the three types of system output steady state errors.
Figure 7.3.1. A unity feedback loop
For the u
Solution, HW #3
ECE 450, Fall 2012
6.4.1 Explain how you would estimate a transfer function for the system of problem
6.3.1.
Solution
From the statement of the problem, we can see that
B = [ 1 0] , C = [ 0 1] , D = [ 0] .
As a first order approximation, I
Exam I, ECE 450 Fall 2011 This is a take home exam. It must be turned in to room BEL 323 by 4:30 pm, Friday, September 23, 2011. During this time, you are not to discuss it with anyone or ask assistance of anyone on any aspect of the exam. Any questions s
6. State Space Analysis
State space analysis uses matrix algebra for systems analysis rather than just
differential equations. It is particularly useful for higher order systems and systems with
multiple inputs or multiple outputs.
6.1 Introduction
For th
6.2 Diagonalization
Associated with any rectangular matrix A are eigenvalues i and eigenvectors u i
such that
Au i = i u i .
It can be very desirable to have an A matrix that is diagonalized. For instance, if
0
A= 1
,
0 2
then
s 1
sI A =
0
0
,
s 2
an
6.3 The State Transition Matrix
The discussion in this section is taken primarily from section 8.2.1 of Gajic.
Let us look at a 1st order DE:
dx ( t )
(6.3.1)
= ax ( t ) + bf ( t ) .
dt
cfw_In the following discussion, a will be a negative number. The Lap
6.4 State Space and the Transfer Function
The basic state equations are
&
x = Ax + Bf (t ), x ( 0 ) = x 0 ,
y = Cx ( t ) + Df ( t ) .
We have since established the solution:
1
1
X ( s ) = ( sI A ) x ( 0 ) + ( sI A ) BF ( s ) .
(6.4.1 a)
(6.4.1 b)
(6.4.2)
6.5 Time-Domain Simulation of State Space
Start with our basic state space equations:
&
x = Ax + Bf (t ), x ( 0 ) = x 0 , y = Cx .
(6.5.1)
st
We are all familiar with the 1 order approximation of a time-domain derivative
dx ( t ) x ( t + t ) x ( t )
.
dt
6.6 Discrete-Time Models
In discrete time the state equations become
x [ k + 1] = Ad x [ k ] + Bd f [ k ]
y [ k ] = C d x [ k ] + Dd f [ k ]
For instance, for a 2nd order difference equation
y [ k + 2] + a1 y [ k + 1] + a0 y [ k ] = b1 f [ k + 1] + b0 f [
7. Feedback Control
7.1 Introduction to Feedback
The following system is referred to as open loop:
Figure 7.1.1 An open loop system
By closing the loop, i.e., adding a feedback which gives me a closed loop,
Figure 7.1.2. A closed loop, or feedback system
7.2 Transient Response
7.2.1. Transient Response of Second-order Systems
Look at Fig. 7.2.1. In the open loop transfer function:
a. K is the system static gain
b. T is the system time constant
Figure 7.2.1.
The closed loop transfer function is
Y ( s)
K /T
7.4 Feedback System Frequency Characteristics The open control loop frequency transfer function is defined by G( ) H ( ) The closed-loop transfer function is M ( ) = H ( ) 1+ G ( ) H ( )
(7.4.1) (7.4.2)
Look at the denominator of Eq. (7.4.2). If the quant
Memorandum
Date: Sept. 20, 2012
To: Professor Dennis Sullivan
From: John Smith
Subject: Grading of problem #2, Exam 1.
Dear Professor Sullivan:
I would like to request a review of the grading of problem number two on our
last exam. I believe I had the cor
6.5.1: plot of y(t) using analytical and state space simulation
Analytical
2
1
0
-1
0
2
4
6
8
10
6
8
10
8
1
0
sec
Simulated
2
1
0
-1
0
2
4
sec
6.5.2: plot of H(s) response to u(t) u(t-2)
SH
i
m
yt
()
2
1
0
1
0
2
4
6
sc
e
s mfz a d t p ep neo x a d 2
u o i
Additional Information for Outreach Students ECE 450, Spring 2012
Despite what is said in the lectures, Outreach students are not required to submit the homework for grading. We have found the logistics of this to be too great a problem. But understand th