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)
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
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
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
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
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
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
Time Domain Specifications
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 Controller Design
Using Bode Plots
1
Why is this important?
1. Acquire (or steal) a system
2. Excite the system with sine waves of various frequencies
(one-at-a-time)
3. Record the magnitude and phase at each frequency
4. Analyze the system (using da
State-Space Methods
1
Systems with multiple inputs and multiple outputs
Thus far, weve examined single-input, single-output (SISO) systems
input
output
G
Next, well examine multiple-input, multiple-output (MIMO) systems
We could still use transfer functio
Lag Controller Design
Using Bode Plots
1
Why is this important?
1. Acquire (or steal) a system
2. Excite the system with sine waves of various frequencies
(one-at-a-time)
3. Record the magnitude and phase at each frequency
4. Analyze the system (using dat
Lead-Lag Controller Design
Using Bode Plots
1
Why is this important?
1. Acquire (or steal) a system
2. Excite the system with sine waves of various frequencies
(one-at-a-time)
3. Record the magnitude and phase at each frequency
4. Analyze the system (usin
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
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
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
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
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->
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
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
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