Automatic Control Systems
Homework 1 Solutions
(a)
2-16)
G (s )
Spring 2012
(b)
5
s 5
(c)
G (s )
2
4s
s
2
4
1
G( s)
s2
4
2
s 4s 8
(d)
1
G( s)
2
s 4
2-18 (a)
g (t ) u s ( t ) 2u s (t 1) 2 u s ( t 2) 2u s (t 3)
G(s)
1
s
gT (t ) u s (t ) 2 u s ( t 1)
Stability and Routh Criterion
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the mode
Block Diagrams
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
Performance
Final Exam on
Monday, 7 Dec from 8:00-10:00
1
Course Description (FIT catalog)
Stresses both classical and modern
control methodologies. Includes
frequency and time-domain
representation of linear systems,
stability analysis and design
techniques.
2
Spec
Chris Pagan Control Systems HW # 3 Page 1
3.
> help rlocus
rlocus Evans root locus.
rlocus(SYS) computes and plots the root locus of the single-input,
single-output LTI model SYS. The root locus plot is used to analyze
the negative feedback loop
+-+
->O->
1. Which Bode plot represents a Lead controller?
a.
G
b.
G
0 dB
0 dB
G
G
0
c.
0
d.
G
G
0 dB
0 dB
G
0
2. What System Type is
( ) =
a. Type 0
10(2+4)
3 +42 +6
G
0
?
b. Type 1
c. Type 2
d. Type 3
3. A Nyquist plot with any clockwise (CW) encirclements of th
Root Locus
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
Performance and
Lead Controller Design
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
Perf
Nichols Chart
1
Nichols Chart
2
3
Nichols Chart Example
Read Mr by noting value of
highest CL magnitude contour
that is tangent to the curve
Read BW where curve crosses -3 dB
3 dB lower than
mag at =0. In
this case, mag
was 0 dB at
=0.
Read GM where curve
MAE 4014
Control Systems
Dr Brian Kish
1
Course Description (FIT catalog)
Stresses both classical and modern
control methodologies. Includes
frequency and time-domain
representation of linear systems,
stability analysis and design
techniques.
2
Textbooks
Differential Equations
1
Importance of Differential Equations
Applying Newtons 2nd Law to dynamic systems often results in
differential equations
e(t)
(t)
B
k
2
Methods for Solving Differential Equations
ODE solver in Matlab (this section)
Laplace Tran
Laplace Transforms
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
Performa
Building Transfer Functions
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
Partial Fraction Expansion
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
Lead-Lag Controller Design
1
Why is this important?
1. Derive a model for the system
2. Simplify the model to a linear system with constant
coefficients (needed for this class)
3. Laplace transform the model into a transfer function
4. Analyze the model
1. Which Bode plot represents a Lead controller?
@ IGI b- |G|
MB ' f
OdB
OdB a,
Va 0 0
103(Zs+4) 9
2. WhatSystemTVPeisG(5) = ss+452+6 '
d. Type 3
3. A Nyquist plot with any clockwise (CW) encirclements of the -1 point (i.e. N>O) means:
a. The system is s
I. (2 pts) The response of the pole on the gure below would be?
. load
w J4
2. (2 pts) The response of the pole on the gure below would be?
Imag
Real
Imag
><
Real
0 J4 1
3_ (2 pts) Given the plant G (S) = $6)
Calculate the angle deciency (9%) to get the
Exam 2 Review
1
Exam ROE
Closed everything
No electronics
Bags on floor, hats off
Only a pen or pencil
2
Exam Format
20 Multiple Choice (3 pts each)
5 True / False (2 pts each)
10 Matching (3 pts each)
3
Big Picture
From the syllabus: Analyze its char
1. Which Bode plot represents a Lead controller?
a.
G
b.
G
0 dB
0 dB
G
G
0
c.
0
d.
G
G
0 dB
0 dB
G
0
2. What System Type is ()
a. Type 0
=
10(2+4)
3 +42 +6
G
0
?
b. Type 1
c. Type 2
d. Type 3
3. A Nyquist plot with any clockwise (CW) encirclements of the
MAE 4014 Homework #6 (due Mar 24)
Given:
G ( s )=
16 ( s+ 0.1 )
s ( s+0.01 ) ( s2 +3 s +9 )
1. Put G(s) in Bode Form:
G ( j )=
K B ( 1+ j T z 1 )
[ ( ) ( )]
j ( 1+ j T p 1 ) 1+2
j
j
+
n
n
2
a.
b.
c.
d.
e.
(1 pt) What is the value of n?
(1 pt) What is the