ECE317 : Feedback and Control
Lecture:Modeling of electrical & mechanical systems
Dr. Richard Tymerski
Dept. of Electrical and Computer Engineering
Portland State University
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Course roadmap
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Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
Controller design process (review)
• What is the “mathematical model”?
• Transfer function
• Modeling of electrical & mechanical systems
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plantInput OutputRef.
Sensor
ActuatorController
Disturbance
1. Modeling
Mathematical model
2. Analysis
Controller
3. Design
4. Implementation
Mathematical model
• Representation of the input-output (signal) relation of a physical system
• A model is used for the analysis and design of control systems.
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Physical
system
Model
Modeling
Input Output
Important remarks on models
• Modeling is the most important and difficult task in control system design.
• No mathematical model exactly represents a physical system.
• Do not confuse models with physical systems!
• In this course, we may use the term “system” to mean a mathematical model.
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Course roadmap
6
Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
Transfer function
• A transfer function is defined by
• A system is assumed to be at rest. (zero initial condition)
• Transfer function is a generalization of “gain” concept.
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Laplace transform of system output
Laplace transform of system input
input output
Impulse response
• Suppose that r(t) is the unit impulse function and system is at rest.
• The output g(t) for the unit impulse input is called unit impulse response.
• Since R(s)=1, the transfer function can also be defined as the Laplace transform of impulse response:
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System
Course roadmap
9
Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
Models of electrical elements
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v(t)
i(t)
R
Resistance CapacitanceInductance
v(t)
i(t)
L v(t)
i(t)
C
Laplace
transform
Impedance
Modeling example
• Kirchhoff voltage law (with zero initial conditions)
• By Laplace transform,
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v1(t)
i(t)
R2Input
R1
v2(t) Output
C
Modeling example (cont’d)
• Transfer function
• If output is i(t), then ….
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v1(t)
i(t)
R2Input
R1
v2(t) Output
C
(first-order system)
Modeling example (cont’d)
• Impedance method• Replace electrical elements with impedances.
• Deal with impedances as if they were resistances.
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V1(s) R2Input
R1
V2(s) Output
1/(sC)
Impedance computation
• Series connection
• Parallel connection
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V(s)
I(s)
Z1(s) Z2(s)
V(s)
I(s) Z1(s)
Z2(s)
Operational amplifier (op-amp)
• Electronic voltage amplifier
• Basic building block of analog circuits, such as• Voltage summation (math operation)
• Voltage integration
• Various transfer functions (Signal conditioning, filtering)
• Ideal op-amp (does not exist, but is a good approximation of reality)
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-
+
i-
vd voutVd=0
i-=0
Modeling of op-amp
• Impedance Z(s): V(s)=Z(s)I(s)
• Transfer function of the above op amp:
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vdVi(s)
I(s)
Input
Zi(s)
Vo(s) Output
-
+
i-
Vd=0
i-=0Zf(s)
Modeling example: op-amp
• By the formula in previous two slides,
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vi(t)
i(t)
R2
Input
R1
vo(t) Output
C
-
+
i-
vd
vd=0
i-=0
(first-order system)
Course roadmap
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Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
Translational mechanical elements
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Mass DamperSpring
M
f(t)x(t) f(t)
x1(t)
Kx2(t)
f(t)
f(t)x1(t)
B x2(t)
f(t)
Summary
• Modeling• Modeling is an important task!
• Mathematical model
• Transfer function
• Modeling of electrical & mechanical systems
• Next lecture, block diagram reduction
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