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

2

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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