Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Transfer function
Lecture 3
Solution to ODEs via Laplace transform
Models for systems
• electrical
• mechanical
• electromechanical
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Time response
• Transient
• Steady state
Frequency response
• Bode plot
Stability
• RouthRouth-Hurwitz
• Nyquist
Block diagrams
Linearization
Design
Design specs
Root locus
Frequency domain
PID & LeadLead-lag
Design examples
(Matlab simulations &) laboratories
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Laplace transform (review)
An advantage of Laplace transform
ƒ One of most important math tools in the course!
ƒ Definition: For a function f(t) (f(t)=0 for t<0),
ƒ We can transform an ordinary differential
equation (ODE) into an algebraic equation (AE).
t-domain
(s: complex variable)
s-domain
AE
ODE
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f(t)
2
0
t
F(s)
Solution to ODE
ƒ We denote Laplace transform of f(t) by F(s).
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Partial fraction
expansion
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Properties of Laplace transform
Differentiation (review)
Example 1
ODE with initial conditions (ICs)
t-domain
1. Laplace transform
s-domain
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Example 1 (cont’d)
2. Partial fraction expansion
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Example 1 (cont’d)
unknowns
3. Inverse Laplace transform
Multiply both sides by s & let s go to zero:
Similarly,
If we are interested in only the final value of y(t),
y(t), apply
Final Value Theorem:
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Example 2
ƒ S1
In this way, we can find a rather
complicated solution to ODEs easily by
using Laplace transform table!
ƒ S2
ƒ S3
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Example: Newton’s law
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EX. Air bag and accelerometer
ƒ Tiny MEMS accelerometer
M
ƒ Microelectromechanical systems (MEMS)
We want to know the trajectory of x(t).
x(t). By Laplace transform,
(Total response) = (Forced response) + (Initial condition response)
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(Pictures from various websites)
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Ex: Mechanical accelerometer
Ex: Mechanical accelerometer (cont’d)
ƒ We would like to know how y(t) moves when unit
step f(t) is applied with zero ICs.
ƒ By Newton’s law
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Ex: Mechanical accelerometer (cont’d)
ƒ Suppose that b/M=3, k/M=2 and Ms=1.
ƒ Partial fraction expansion
Summary & Exercises
ƒ
Solution procedure to ODEs
1. Laplace transform
2. Partial fraction expansion
3. Inverse Laplace transform
ƒ Inverse Laplace transform
ƒ
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Amplitude
-0.1
ƒ
-0.2
Next, modeling of physical systems using
Laplace transform
Exercises
ƒ
-0.3
ƒ
Derive the solution to the accelerometer problem.
E2.4 of the textbook in page 135.
-0.4
-0.5
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Time [sec]
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