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ECE 345 / ME 380Introduction to Control Systems
Lecture Notes 2
August 27, 2013
Learning Objectives
• Find the Laplace transform and inverse Laplace transform
• Find the transfer function of a system from a di↵erential equation
• Solve a di↵erential equation by using its transfer function
• Find the transfer function for
– LTI electrical networks– LTI mechanical systems (translational and mechanical)– LTI electromechanical systems– gear systems with and without loss
• Produce analogous electrical and mechanical circuits
• Linearize a nonlinear system to find its transfer function
References:
• Nise Chapter 2
1
Outline
• The Laplace transform
• Transfer functions
– Electrical systems– Mechanical systems– Geared systems
• Mechanical-electrical system analogs
• Linearization
2
The Laplace Transform
Modeling
• Three elements: input, output, and the system (process)
• Di↵erential equations often model systems of interest, but can becumbersome mathematically
• Laplace transforms facilitate block-diagram modeling of systems andsubsystems
3
The Laplace Transform
• The method of Laplace transforms converts a calculus problem (thelinear di↵erential equation) into an algebra problem.
• Pierre-Simon Laplace (1749-1827)• Integral transformation (similar to the Fourier transform)
L [f(t)] = F (s) =
Z 1
0�f(t)e
�st
dt
• Multiplication by the Laplace variable s corresponds to di↵erentiation inthe time domain
• Inverse Laplace transformation
L�1[F (s)] =
1
2⇡j
Z�+j1
��j1F (s)e
st
ds = f(t)u(t)
4
The Laplace Transform
Properties of the Laplace transform
• Linearity
• Di↵erentiation
• Final value theorem
• Initial value theorem
• Region of convergence
Important Laplace transform pairs
• Impulse function
• Step function
• Exponential decay
• Sine and cosine
• Damped oscillations
5
The Laplace Transform
Clicker question
Find L�1h
3s4+2s2
i.
A. L�1h
3s4+s
2
i=
32 cos(2t)
B. L�1h
3s4+s
2
i= 3 cos(2t)
C. L�1h
3s4+s
2
i=
32 cos
�p2t
�
D. L�1h
3s4+s
2
i=
32 sin(2t)
E. L�1h
3s4+s
2
i= 4 cos
⇣3p2t
⌘
6
The Laplace Transform
Clicker question
Use the method of Laplace transforms to solve the di↵erential equation
dy
dt
= �y + 3e
�2t, y(0) = 1
with non-zero initial condition. What is the solution y(t) for t � 0?
A. y(t) = 4e
�t � 3e
�2t
B. y(t) = e
�t � e
�2t
C. y(t) = �3e
�t
+ 4e
�2t
D. y(t) = e
�4t+ 3e
�2t
Hint: Start by taking the Laplace transform of both sides of the equation.
7
Transfer functions
Goal: Algebraically relate input and output• Consider an n
th order di↵erential equation with initial conditions equalto zero, that relates the input u(t) and output y(t)
d
n
y
dt
n
+ a1d
n�1y
dt
n�1+ · · ·+ a
n
y = b0d
m
u
dt
m
+ b1d
m�1u
dt
m�1+ · · ·+ b
m
u,
• Take the Laplace transform of both sides
s
n
Y (s)+a1sn�1
Y (s)+· · ·+a
n
Y (s) = b0sm
U(s)+b1sm�1
U(s)+· · ·+b
m
U(s)
• And rearrange to obtain
Y (s)
R(s)
=
�b0s
m
+ b1sm�1
+ · · ·+ b
m�1s+ b
m
�
(s
n
+ a1sn�1
+ · · ·+ a
n�1s+ a
n
)
8
Transfer functions
Goal: Algebraically relate input and output
• The transfer function is the ratio G(s) =
Y (s)R(s).
• Note that this means that Y (s) = G(s)R(s), so the output signal canbe calculated in a straightforward manner (multiplication!).
9
Transfer functions
Clicker question
Consider the system represented by
dy
dt
+ 2y(t) = r(t), y(0) = 0
What is the system response y(t) to a step input r(t) = u(t)?
A. y(t) = 2e
�t for t � 0, y(t) = 0 for t < 0
B. y(t) =
12 �
12e
�2t for t � 0, y(t) = 0 for t < 0
C. y(t) =
12 +
12e
�4t for t � 0, y(t) = 0 for t < 0
D. y(t) =
�12 �
12e
�t/2�u(t)
Hint: First find the transfer function.
10
Transfer functions
Electrical systems• Mesh analysis = Kirchho↵’s voltage law• Nodal analysis = Kirchho↵’s current law• Identify integro-di↵erential equations directly, or use• Laplace transforms
– Impedances often more convenient for mesh analysis (Nise, p. 51)– Admittances often more convenient for nodal analysis (Nise, p. 55)
11
Transfer functions
Electrical systems• Ideal operational amplifier
– May be inverting or non-inverting– No current at the input terminals that flows into the op-amp
(Input impedance is infinite)– Voltage at the two input terminals is tied
• Convenient way to build, implement, and realize transfer functions• Analog controllers
12
Transfer functions
13
Transfer functions
Mechanical systems – translational elements• Spring-mass-damper systems• Dynamic simulator, courtesy of Professor Bonnie Heck, Georgia Tech:
http://users.ece.gatech.edu/⇠bonnie/book1/applets/suspension/MSDdemo.htm
• Choose “positive” direction, then apply Newton’s second law.
14
Transfer functions
Mechanical systems – rotational elements
• Spring-inertial load-damper systems
• Choose “positive” direction, then apply Newton’s second law.
15
Transfer functions
Geared systems• For lossless gears, T2
T1=
✓1✓2
=
N2N1
Electromechanical systems• DC motor
16
Mechanical / Electrical System Analogs
Clicker question
Which of the following pairs could be analogs?
A. RLC parallel circuit and spring-mass system
B. RC series circuit and a spring-mass system
C. LC series circuit and spring-mass system
D. LC series circuit and damper-mass system
E. Not very confident about any of these answers
17
Linearization
Linearity of f(x)
• Scaling: f(↵x) = ↵x
• Superposition: f(x1 + x2) = f(x1) + f(x2)
For example: f(x) = 3x
Examples of nonlinearities
18
Linearization
Solution:
• Linearize y = f(x) for x near x0
• Taylor’s series approximation• Works for (x� x0) “small enough”
f(x)� f(x0) ⇡@f
@x
����x=x0
· (x� x0)
19
Linearization
Frictionless rigid pendulum
• Linearize the dynamics ⌧(✓) = MgL sin ✓ around ✓ = 0.
⌧(✓)� 0 ⇡ MgL
@ sin ✓
@✓
����✓=0
(✓ � 0)
�⌧ ⇡ MgL · �✓
• with �⌧ = ⌧ , �✓ = ✓
20
Linearization
Clicker question
Consider again the pendulum dynamics
⌧(✓) = MgL sin ✓
What is linearization of this system when the pendulum is upright, e.g.,around ✓ = ⇡?
A. �⌧ ⇡ �MgL · �✓, �⌧ = ⌧ , �✓ = ✓ � ⇡
B. �⌧ ⇡ MgL · �✓, �⌧ = ⌧ �MgL, �✓ = ✓ � ⇡
C. �⌧ ⇡ �MgL · �✓, �⌧ = ⌧ � ⇡, �✓ = ✓
D. ⌧ ⇡ �MgL · �✓, �✓ = ✓ � ⇡/2
21
