control systems - lehigh universityinconsy/lab/css/me389/lectures/lecture01_modeling.pdf2 classical...

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Classical Control – Prof. Eugenio Schuster – Lehigh University 1

Control Systems

Lecture 01 Modeling


What is Control for?

Feedback control can help: •  reference following (tracking) •  equilibrium regulation •  disturbance rejection •  stability •  performance

First step in this design process: DYNAMIC MODEL

CHANGING DYNAMIC BEHAVIOR

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Modeling Approaches

•  First Principles Modeling

o  Uses fundamental physics law

•  System Identification

o  Uses forced input-output


First-Principles Dynamic Models

MECHANICAL SYSTEMS:

xvaxv

xbuxm

==

=

−=velocity

acceleration

maF = Newton�s law

mbsm

UV

mu

vmb

vo

o

+=!!!!! →!=+

1Transform Laplace

Transfer Function

sdtd→

Real world is more complex than static maps → We need Dynamic Models

3


First-Principles Dynamic Models MECHANICAL SYSTEMS: αIF = Newton�s law

2

2 sin

mlI

Tlmgml c

=

==

=

+−=

θωα

θω

θθ

angular velocity

angular acceleration

moment of inertia

2sin2sinmlT

lg

mlT

lg cc =+!! →!=+

≈θθθθ θθ

Linearization


First-Principles Dynamic Models

2mlT

lg c=+ θθ

θ

θ=

=

2

1

x

x

212

21

mlTx

lgx

xx

c+−=

=

Reduce to first order equations:

uxlgx

mlTu

xx

x c!"

#$%

&+

!!"

#

$$%

&

−=⇒≡!"

#$%

&≡

10

010

, 22

1 State Variable Representation

General case:

JuHxyGuFxx+=

+=

4


First-Principles Dynamic Models ELECTRICAL SYSTEMS:

Kirchoff�s Current Law (KCL): The algebraic sum of currents entering a node is zero at every instant

Kirchoff�s Voltage Law (KVL)

The algebraic sum of voltages around a loop is zero at every instant

iR

vR vC

iC

vL

iL + + +

)()()()( tGvtitRitv RRRR =⇔=

Resistors:

)0()(1)()()(0

C

t

CCC

C vdiC

tvdttdvCti +=⇔= ∫ ττ

∫ +=⇔=t

LLLL

L idvL

tidttdiLtv

0

)0()(1)()()( ττ

Capacitors:

Inductors:



+ -

RI RO

A(vp-vn)

+

-

ip

in

iO vO

vn

vp

+

+

+ 0==

=

np

np

iivv

∞→−= AvvAv npO ),(

OP AMP:

To work in the linear mode we need FEEDBACK!!!

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Inverting integrator

K v1 vO

∫

( ) ( ) ∫−=⇒∞=tIOO dv

CRvtvR

01

2 )(10)( µµOC

RCK 1

−=

+ -

R1 C v1 vO

R2

IOO v

CRv

CRdtdv

12

11−=

KCL:


First-Principles Dynamic Models ELECTRO-MECHANICAL SYSTEMS: DC Motor

me

at

KeiKTθ=

=

armature current

shaft velocity

torque

emf

0=+++−

+−=

edtdiLiRv

TbJ

aaaa

mmm θθ

Obtain the State Variable Representation

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First-Principles Dynamic Models HEAT-FLOW:

( )

qC

T

TTR

q

1

121

=

−=

Temperature Difference Heat Flow

Thermal resistance Thermal capacitance

( )IoI

I TTRRC

T −""#

$%%&

'+=

21

111


First-Principles Dynamic Models FLUID-FLOW:

outin wwm −=

Mass rate

Outlet mass flow Inlet mass flow

( )outin wwA

hhAm −=⇒=ρ

ρ1

A: area of the tank ρ: density of fluid h: height of water

Mass Conservation law

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Identified Dynamic Models First-order Dynamic Systems – Step Response:

H (s) = 1s+σ

⇒ h(t) = e−σ t

στ

1= Time Constant

Impulse Response

00

<

>

σ

σ Stable

Unstable


Identified Dynamic Models First-order Dynamic Systems – Step Response:

Y (s) = Ks+σ

1s⇒ y(t) = K

σ1− e−σ t( ) Step Response

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Time [s]

Out

put

B = Kσ⇒ K =σB

0 0.5 1 1.5 2−20

−15

−10

−5

0

Time [s]

ln((f

latto

p−ou

tput

)/fla

ttop) σ = −m

m: slope

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)cos()( φω += tAtu )(sG ))(cos()( ωφωω jGtAjGyss ∠++=

Stable Transfer Function

•  After a transient, the output settles to a sinusoid with an amplitude magnified by and phase shifted by . •  Since all signals can be represented by sinusoids (Fourier series and transform), the quantities and are extremely important. •  Bode developed methods for quickly finding and for a given and for using them in control design.

)( ωjG )( ωjG∠

)( ωjG )( ωjG∠

)( ωjG )( ωjG∠)(sG

Identified Dynamic Models General-order Dynamic Systems – Frequency Response:


Identified Dynamic Models General-order Dynamic Systems – Frequency Response:

•  Find and from the experiment

•  Infer the transfer function

)( ωjG )( ωjG∠

)(sG

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Classical Control – Prof. Eugenio Schuster – Lehigh University

Model Classification

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Spatial Dependence

Lumped parameter system Ordinary Diff. Eq. (ODE)

Distributed parameter system Partial Diff. Eq. (PDE)

Linearity

Linear

Nonlinear

Temporal Representation

Continuous-time

Discrete-time

Domain Representation

Time

Frequency

)(tff = ),( xtff =

Model Representation Control Technique


Spatial Dependence

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Distributed Parameter Systems PDE

Lumped Parameter Systems ODE

nnee

ne SVn

rnDr

rrtn

+!"#

$%& −

∂∂

∂∂

−=∂∂ 1 ( )rtSn ,

00=

∂∂

=r

e

rn a

eare nn ==

aeare nn =

=

nee

e Sndtdn

+−=τ1 )(tSn

( ) 00 ee nn =

1 Reduction

2 Keep the PDE representation (problem specific)

Control: Interior Boundary

Linearity: Nonlinear/Linear

( )rtnn ee ,= ( )tnn ee =

Linear/Nonlinear Distributed Parameter Control

Linear/Nonlinear Lumped Parameter Control

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Linearity

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Nonlinear (ODE) Systems Linear (ODE) Systems

( )( )uxthy

uxtfx,,,,

=

=

Nonlinear Control

utDxtCyutBxtAx)()()()(

+=

+=

( ) ( ) ( )( )uuyytgy mnn ,,,,,, 11 −−=

1 Linearization

2

DuCxyBuAxx

+=

+=LTI

LTV

( )( )uxhyuxfx,,

=

=Autonomous

Non-Autonomous

( ) ( ) ( ) ububyayay mm

nn

n0

110

11 ++=+++ −

−−

−

[ ]

yu

xxx Tn1= states

input

output

Linear Control

Output/State Feedback

Estimation: How to estimate states from input/output?

Keep the nonlinearities


Linearity

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Particular type of nonlinearities: Constraints

Anti-windup Techniques

1 A priori

2 A posteriori

DuCxyBuAxx

+=

+=LTI

Constraint is considered for control design

Constraint is NOT considered for control design

)(),(, ysatusatyu →

iii xxx <<

input/output constraints state constraints

11



21

Continuous-time Systems Discrete-time Systems

System Identification

( ) ( ) ( ) ububyayay mm

nn

n0

110

11 ++=+++ −

−−

−

[ ] ( )[ ] ( ) [ ] ( )[ ]TmkubkTubTnkyaTkyakTy monk −++=−++−+ − ][11

)()()()()()(tDutCxtytButAxtx

+=

+=LTI

[ ] [ ] [ ][ ] [ ] [ ]kDukCxky

kBukAxkx+=

+=+1LTI

Sampled-Data Systems

Sampling Time

System Identification: How to create models from data? Fault Detection and Isolation: How to detect faults from data?



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Continuous-time Systems Discrete-time Systems ( ) ( ) ( ) ububyayay m

mn

nn

01

101

1 ++=+++ −−

−− [ ] [ ] [ ] [ ]mkubkubnkyakyaky monk −++=−++−+ − ][11

( ) ( ) )()( 011

1011

1 sUbsbsbsYasasas mm

nn

n +++=++++ −−

−−

011

1

011

1

)()()(

asasasabsbsbsb

sUsYsT n

nn

n

mm

mm

++++++++

== −−

−−

)())(()())((

21

21

n

m

pspspszszszsK

−−−−−−

=

Laplace Transform Z Transform

KzpT

i

i

TF poles zeros gain

( ) ( ) )()(1 111 sUzbbsYzazaza m

mon

nn

n−−−

− ++=++++

nn

nn

mmo

zazazazbb

zUzYzT −−

−

−

++++++

== 1111)(

)()(

)())(()())((

21

21

n

m

pzpzpzzzzzzzK

−−−−−−

=

ωjs =

)()()( ωωω jTejTjT ∠=

Tjez ω=

)()()(TjeTTjTj eeTeT

ωωω ∠=Frequency Response

Modern Control

Classical Control

12


Optimality

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Continuous-time Systems Discrete-time Systems ( ) ( ) ( ) ububyayay m

mn

nn

01

101

1 ++=+++ −−

−− [ ] [ ] [ ] [ ]mkubkubnkyakyaky monk −++=−++−+ − ][11

Optimal Control

utDxtCyutBxtAx)()()()(

+=

+=

kkkkk

kkkkk

uDxCyuBxAx

+=

+=+1

( )∑−

=

++1

021

21min

N

kkk

Tkkk

TkNN

TNu

uRuxQxxSxk

( )∫ ++T

TTT

T

tudttutRtutxtQtxTxSTx

0)(

)()()()()()(21)()(

21min


Model Uncertainties

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How to deal with uncertainties in the model?

1

2 Adaptive Control

Design for a family of plants

Update model (controller) in real time

Robust Control

B Model-based control

A Non-model-based control

PID Extremum Seeking

Robust & Adaptive Control

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Model Classification

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Spatial Dependence

Lumped parameter system Ordinary Diff. Eq. (ODE)

Distributed parameter system Partial Diff. Eq. (PDE)

Linearity

Linear

Nonlinear


Continuous-time

Discrete-time


Time

Frequency

)(tff = ),( xtff =

Model Representation Control Technique


Dynamic Model

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MECHANICAL SYSTEM: αIT = Newton�s law

22 sinmlT

lg

mlb c+−−= θθθ

Which are the equilibrium points when Tc=0?

At equilibrium: πθθθθ ,0sin00 =⇒−=⇒==lg

cTblmgI +−−= ωθα sin

damping coefficient

angular velocity

angular acceleration

moment of inertia 2mlI ===

=

θωα

θω

Open loop simulations: pend_par.m, pendol01.mdl

Stable

Unstable

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Linearization

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( )uxxxxfx nnn ,,,,, )1()2()1()( …−−=Dynamic System:

( )oo uxf ,,0,,0,00 …= Equilibrium

Denote oo uuuxxx −=−= δδ ,

( )uuxxxxxfx oonnn δδδδδδ ++= −− ,,,,, )1()2()1()( …

Taylor Expansion

( )

uuf

xxf

xxf

xxf

xxf

uxfx

oooooo

oooo

uxuxux

n

uxn

n

uxnoo

n

δδδ

δδδ

,,,0,,0,0

)1(

,,0,,0,0)1(

)2(

,,0,,0,0)2(

)1(

,,0,,0,0)1(

)( ,,0,,0,0

∂

∂+

∂

∂+

∂

∂+

+∂

∂+

∂

∂+≈ −

−−

−

……

……

…


Linearization

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( )uxfx ,=Dynamic System:

( )oo uxf ,0 = Equilibrium

Denote oo uuuxxx −=−= δδ ,

( )uuxxfx oo δδδ ++= ,Taylor Expansion

( ) uufx

xfuxfx

oooo uxuxoo δδδ

,,,

∂∂

+∂∂

+≈

⇒∂∂

≡∂∂

≡oooo uxux u

fGxfF

,,, uGxFx δδδ +≈

mn RuRx ∈∈ ,

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Linearization

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oo

oo

oo

oo

uxm

nn

m

ux

uxn

nn

n

ux

uf

uf

uf

uf

ufG

xf

xf

xf

xf

xfF

,1

1

1

1

,

,1

1

1

1

,,

!!!!!

"

#

$$$$$

%

&

∂∂

∂∂

∂∂

∂∂

=∂∂

≡

!!!!!

"

#

$$$$$

%

&

∂∂

∂∂

∂∂

∂∂

=∂∂

≡

uGxFx δδδ +≈


Linearization

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What happens around θ=0?

By Taylor Expansion:

( ) ( ) yytohyy ≈⇒+= sin...sin

⇒= yθ ( ) 22 sinmlTy

lgy

mlby c+−−=

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

sin(x)

22 mlTy

lgy

mlby c+−−=

Linearized Equation:

y

sin(

y)

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Linearization

31

What happens around θ=π?

By Taylor Expansion:

( ) ( ) xxtohxx ≈⇒+= sin...sin

⇒+= xπθ ( ) 22 sinmlTx

lgx

mlbx c++−−= π

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

sin(x)

22 mlTx

lgx

mlbx c++−=

( ) 22 sinmlTx

lgx

mlbx c++−=

Linearized Equation:

x

sin(

x)

control systems - lehigh universityinconsy/lab/css/me389/lectures/lecture01_modeling.pdf2 classical...

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