Course roadmap
ME451: Control Systems
Modeling
Analysis
Laplace transform
Transfer function
Lecture 4
Modeling of electrical systems
Models for systems
• electrical
• mechanical
• electromechanical
Block diagrams
Dr. Jongeun Choi
Department of Mechanical Engineering
Michigan State University
Linearization
Time response
• Transient
• Steady state
Frequency response
• Bode plot
Stability
• RouthRouth-Hurwitz
• Nyquist
Design
Design specs
Root locus
Frequency domain
PID & LeadLead-lag
Design examples
(Matlab simulations &) laboratories
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Controller design procedure (review)
Disturbance
Ref.
Controller
Actuator Input
plant
Output
Sensor
4. Implemenation
Mathematical model
ƒ Representation of the input-output (signal)
relation of a physical system
Input
1. Modeling
Controller
2
Physical
system
Output
Modeling
Mathematical model
3. Design
Model
2. Analysis
ƒ A model is used for the analysis and design of
control systems.
ƒ What is the “mathematical model”
model”?
ƒ Transfer function
ƒ Modeling of electrical circuits
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Important remarks on models
A Brief Control History
ƒ 1788: James Watt’
Watt’s flyfly-ball governor
ƒ Modeling is the most important and difficult task
in control system design.
ƒ No mathematical model exactly represents a
physical system.
ƒ Mechanical feedback control of steam supply to an engine
ƒ Do not confuse models with physical systems!
ƒ In this course, we may use the term “system” to
mean a mathematical model.
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The Block Diagram
The Block Diagram
ƒ Communication tool for Engineering Systems
ƒ Blocks Connect to form systems
ƒ Composed of Blocks with
with inputs and outputs
ƒ Outputs of one block becomes input to another
Load “Disturbance”
“Input”
Steam
Flow
“Block”
Steam
Engine
“Output”
Engine
Speed
Valve
Angle
“Input”
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Steam
Valve
Steam
Flow
“System”
Steam
Engine
Engine
Speed
“Output”
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The Block Diagram
Transfer function
ƒ Blocks Connect to form systems
ƒ A transfer function is defined by
ƒ Outputs of one block becomes input to another
Laplace transform of system output
“System”
Steam
Valve
Valve
Angle
“Output”
Steam
Flow
Steam
Engine
Laplace transform of system input
Engine
Speed
Flyball
Sensor
ƒ A system is assumed to be at rest. (Zero initial
condition)
“Feedback”
Pivot
Setpoint
“Input”
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Models of electrical elements:
(constitutive equations)
Impulse response
ƒ Suppose that u(t) is the unit impulse function
and system is at rest.
Resistance
Inductance
v(t)
v(t)
ƒ The output g(t) for the unit impulse input is called
impulse response.
ƒ Since U(s)=1, the transfer function can also be
defined as the Laplace transform of impulse
response:
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i(t)
i(t)
i(t)
i(t)
i(t)
i(t)
System
Capacitance
R
v(t)
v(t)
L
v(t)
v(t)
C
Laplace
transform
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Impedance
Kirchhoff’s Voltage Law (KVL)
ƒ Generalized resistance to a sinusoidal
alternating current (AC) I(s)
I(s)
I(s)
ƒ Z(s): V(s)=Z(s)I(s)
Z(s)
Z(s)
Element
Time domain
ƒ The algebraic sum of voltage drops around any
loop is =0.
V(s)
V(s)
Impedance Z(s)
Resistance
Inductance
Capacitance
Memorize!
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Kirchhoff’s Current Law (KCL)
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Impedance computation
V1(s)
ƒ Series connection
ƒ The algebraic sum of currents into any junction
is zero.
I(s)
I(s)
V2(s)
Z1(s)
Z2(s)
V(s)
V(s)
ƒ Proof (Ohm’
(Ohm’s law)
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Impedance computation
Modeling example
R1
i(t)
i(t)
ƒ Parallel connection
I1(s)
Input v1(t)
Z1(s)
I(s)
I(s)
R2
v2(t) Output
C
Z2(s)
I2(s)
ƒ Proof (Ohm’
(Ohm’s law)
ƒ Kirchhoff voltage law (with zero initial conditions)
V(s)
V(s)
ƒ KCL
ƒ By Laplace transform,
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Modeling example (cont’d)
i(t)
i(t)
Input v1(t)
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Example: Modeling of op amp
R1
Zf(s)
(s)
R2
I(s)
I(s)
v2(t) Output
-
Zi(s)
(s)
C
Input Vi(s)
(s)
i
vd
+
If(s)
(s)
-
Rule1: i =0
Rule2: vd=0
Vo(s)
(s) Output
ƒ Transfer function
ƒ Impedance Z(s): V(s)=Z(s)I(s)
ƒ Transfer function of the above op amp:
(first(first-order system)
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Modeling example: op amp
R2
i(t)
i(t) R1
Input vi(t)
(t)
C
-
i =0
-
i
vd
+
Modeling exercise: op amp
C2
C1
Vd=0
vo(t)
(t) Output
R1
ƒ By the formula in previous two pages,
i =0
+
Vd=0
-
-
i
vd
Input vi(t)
(t)
R2
vo(t)
(t) Output
Find the transfer function!
(first(first-order system)
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Summary & Exercises
More exercises in the textbook
ƒ Find a transfer function from v1 to v2.
ƒ Modeling
ƒ Modeling is an important task!
ƒ Mathematical model
ƒ Transfer function
ƒ Modeling of electrical systems
ƒ Find a transfer function from vi to vo.
ƒ Next, modeling of mechanical systems
ƒ Exercises
ƒ Do the problems in page 23 of this lecture note.
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