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Course Control theoryDirk SöffkerLU-0: Preliminary remarks

Chair ofDynamics andControl

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Course Control TheoryWiSe 2014/15

Room: SG 135Time: Fr 3.00 – 6.30 pm (lecture and exercise)

Practical exercise: 2nd part of semester

Assistants: Xi Nowak, M.Sc.;WEB: http://www.uni-due.de/srs

Manuscript

Note 1: The collected material is prepared for the use only in connection with the lecture.It is not allowed to use this material outside the lecture of Prof. Söffker.

Note 2: The reprinted figures are coming – if nothing different is mentioned - from the textbook of Prof. Lunze and are free for use in connection with this textbook-based course.

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Content of the course- Requirements- Contents- Relations to other lectures of the chair Dynamics and Control (Bachelor, Master)

Idea behind the the course- Manusscript presented with tablet pc (#former semesters) - Text book >>Ogata/Lunze (Library)

Theoretical and practical exercises- theoretical exercise: included within the course- practical exercise: 3 practical each of 4 hours in June- add. Exercise about control technique contents May 2nd-6th

Consulting hours- Th 10.00 - 11.30am

Exercises tasks- Lunze / Ogata /others

Exam>> as usual (2 hours written exam in february/march 2015)Cop

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0.2 Add. remarks I

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0.2 Add. remarks Ib

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0.2 Add. remarks Ic

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0.2 Add. remarks Id

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0.2 Add. remarks II

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0.2 Add. remarks IIb

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0.2 Add. remarks IIc

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Course Control theoryDirk SöffkerLU-1-2: Introduction, description, behavior

Chair of Dynamics and Control

I

LU-1 Introduction to MIMO systems

What is the content of lecture units 1 and 2?

>> You should learn about the prerequisites of the course, especially about some terms and methodsto use the time to exam to learn from books if neccessary.

>> You should get sensible for the MIMO topic.> This is illustrated with static and dynamic examples.

>> The differences between SISI and MIMO are declared.> Representation, Methods

>> The main differences between SISO and MIMO are developed by changing theviewpoint from SISO to MIMO > State space representation, Frequency domain, Couplings of elements, Time behavior and integrationCop

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1.1 Characterisctics of MIMO systems (MIMO: Multi Input - Multi Output)Inputs and Outputs:

Complex dynamics:

Coupled systems:

Result:

Practical examples I:

- Heat exchanger

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Practical examples II:

- Elastic beam (robot beam)

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1.2 Representation of values- m-dim. input vector u(t) - r1-dim. output vector y(t)

- r2-dim. vector of reference w(t) - p-dim. disturbance vector d(t)

> written form

> spoken form

> A special case: the SISO system

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1.3 Analysis and control goals in the context of MIMO systems

- SISO control design goals : stability, reference control, disturbance compensation, dynamic requirementsplus- Analysis of the dynamics Goal: Use of couplings > decoupling

- Analysis of the inner couplings What acts how?

(- Guaranty of the functionality of the control loop) Robustness against failures of actuators and sensors)

1.4 Transformation/Use of the SISO methods to analysis and synthesis of MIMO-systems possible?

- Description (state space , transfer function , weighting function )

- Description tools to describe the dynamic behavior (poles , zeros , Hurwitz )

- Signal models (internal model principle ) ( works for single transfer pathes)

---

- Root locus

- Frequency domain-based control approaches ( but: more complex relations)

1.5 New ideas and methods:

- Analysis of internal couplings

- Design of control structures using internal couplings

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LU-2 Description and behavior of linear MIMO systems

2.1 Description techniques for SISO systems

- Differential equations

- Transfer functions

- State space description

2.2 Description techniques for MIMO systems- Set of linear differential equations> Vector differential equation

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- Canonical regular form (diagonal canonical form)> (effect of inputs, effect of outputs, modal measurements)

> Calculating modes (eigenvalue equation: eigenvalues, eigenvector equation: eigenvectors)

> Decomposing the system (calculating the modes)

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2.3 Description of the MIMO-system in frequency domain

- Transfer function matrix (be careful > MIMO-case)

- Frequency domain description of a MIMO systems> elementwise analog to the SISO case

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- Connections between the frequency domain and the state space

> Transfer behavior in state space: Rosenbrock system matrix

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2.4 Parallel and series network of MIMO transfer elements

2.5 (Time-)behavior of MIMO systems(of the state-space description)

- Equations of motion

- Output equations of motion

- Instationary (transient) and stationary (# constant) behavior

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S

What you should learm from LU 1-2:

>> The fact that MIMO systems link several inputs and outputs with one another leads to complex dynamics, i.e. based on the dynamic couplings. From this it results that the known fast controller design strategy(SISO: PID-approaches etc.), which are used in SISO approaches and which realizes a simple connectionbetween the controller input and system output, can not be used with MIMO systems.

>> A system analysis (stability, modal characteristics, observability, controllability) alwaysprecedes the controller design (synthesis) in the case of MIMO systems.

>> The state space representation permits a systematic representation of all linear MIMO systems. The analysis methods are independent from the number of the states as well as from the dynamics.

>> The transfer function matrix shows the dynamic effects between the several inputs and outputs.

>> The structural properties are shown in the time domain using the state space representation, in the frequency domain using the Rosenbrock matrix.

>> The state space representation is the linear and explicit special case of the general representationfor the time integration of dynamic systems (in explicit form).

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Uni

vers

ity o

f D

uisb

urg-

Esse

n Cha

ir o

f D

ynam

ics

and

Con

trol

U

niv.

-Pro

f. D

r.-I

ng.

Dirk

Söf

fker

C

ours

e C

ontr

ol T

heo

ry/

Con

trol

Tec

hn

iqu

e

S

prin

g/W

inte

r 20

14/1

5

Term

s an

d D

efin

itio

ns

Sys

tem

:

Purp

ose-

orie

nted

par

t of

the

rea

l w

orld

. Th

e sy

stem

is

in int

erac

tion

with

the

rea

l w

orld

, w

here

by f

rom

the

rea

l wor

ld in

puts

are

act

ing

to t

he s

yste

m a

nd o

utpu

ts a

re a

ctin

g fr

om

the

syst

em t

o th

e en

viro

nmen

t.

Ph

ysic

al v

aria

ble

s /

ph

ysic

al v

alu

es:

In

tera

ctio

n qu

aliti

es (

qual

ities

/ s

tate

s) w

ithin

a s

yste

m o

r fr

om t

he s

yste

m t

o th

e en

viro

nmen

t, in

tech

nica

l sy

stem

s us

ually

of

phys

ical

nat

ure.

Phy

sica

l va

lues

are

usu

ally

de

fined

in c

onne

ctio

n w

ith t

he p

urpo

se o

f re

late

d sy

stem

s to

be

cons

ider

ed.

The

variab

les

to b

e co

nsid

ered

in c

ontr

ol a

re u

sual

ly s

cala

rs o

r ve

ctor

s. T

he m

ain

prop

erty

of

variab

les

cons

ider

ed w

ithin

sys

tem

dyn

amic

s or

con

trol

is t

he t

ime

depe

ndin

g be

havi

or.

Con

trol

var

iab

le /

Ou

tpu

t (s

ing

le in

pu

t –

sin

gle

ou

tpu

t sy

stem

):

Var

iabl

e of

the

sys

tem

to

be c

ontr

olle

d. T

he v

aria

ble

shou

ld b

e us

ually

fix

or

variab

le.

In

Sin

gle

Inpu

t -

Sin

gle

Out

put

(SIS

O)

syst

ems

the

outp

ut (

here

onl

y on

e ex

ists

) is

the

co

ntro

l var

iabl

e.

Op

en lo

op s

yste

m:

Ope

n lo

op s

yste

ms

are

wor

king

so

that

the

inp

uts

of t

he s

yste

ms

are

affe

ctin

g th

e ou

tput

s de

pend

ing

on t

he d

ynam

ic p

rope

rtie

s of

the

sys

tem

and

no

t vi

ce v

ersa

. Th

e ch

arac

terist

ic p

rope

rty

of a

n op

en l

oop

syst

em i

s th

e no

n cl

osed

beh

avio

r of

the

sig

nal

flow

. C

lose

d lo

op s

yste

m:

With

in

a cl

osed

lo

op

syst

em

the

cont

rol

variab

le

is

usua

lly

onlin

e m

easu

red

and

com

pare

d w

ith a

giv

en d

esired

/ r

efer

ence

var

iabl

e (w

hich

als

o ca

n be

zer

o).

The

resu

lt is

gi

ven

to t

he i

nput

in

the

way

tha

t it

is a

ctin

g so

tha

t th

e co

ntro

l va

riab

le t

ends

to

the

desi

red

valu

e. T

his

real

izes

the

fee

dbac

k an

d cl

oses

the

loop

. Th

e ch

arac

terist

ic p

rope

rty

of a

clo

sed

loop

sys

tem

is

the

clos

ed b

ehav

ior

of t

he s

igna

l flo

w.

This

cha

nges

the

dyn

amic

beh

avio

r of

the

ove

rall

syst

em a

nd a

llow

s th

e au

tom

atic

co

mpe

nsat

ion

of d

istu

rban

ces

actin

g to

the

sys

tem

and

affec

ting

the

cont

rol v

aria

ble.

D

istu

rban

ce (

vari

able

/ v

alu

e):

From

th

e en

viro

nmen

t to

th

e sy

stem

ac

ting

variab

le,

whi

ch

influ

ence

s th

e co

ntro

l va

riab

le in

an

unw

ante

d w

ay.

Inp

ut

(var

iab

le /

val

ue)

: Var

iabl

e ac

ting

dire

ct t

o th

e sy

stem

and

cha

nges

the

out

put

of t

he s

yste

m.

In o

pen

loop

sy

stem

the

inp

ut v

aria

ble

acts

fro

m t

he e

nviron

men

t to

the

sys

tem

, in

clo

sed

loop

sy

stem

s th

e in

put

is t

he o

utpu

t of

the

con

trol

ler

and

the

inpu

t of

the

sys

tem

to

be

cont

rolle

d.

Des

ired

var

iab

le /

valu

e /

ref

eren

ce v

aria

ble

/va

lue:

Var

iabl

e fr

om t

he e

nviron

men

t to

the

con

trol

ler,

whi

ch g

ives

the

ref

eren

ce f

or t

he o

utpu

t as

the

val

ue t

o be

con

trol

led.

C

on

tro

l dev

iati

on

/ c

ontr

ol d

iffe

ren

ce:

Var

iabl

e de

notin

g th

e di

ffer

ence

be

twee

n th

e de

sire

d an

d th

e co

ntro

l va

riab

le.

The

cont

rol d

evia

tion

is a

n in

tern

al v

aria

ble

insi

de t

he c

ontr

olle

r.

Pla

nt

(sys

tem

to

be

con

trol

led

):

The

plan

t is

the

sys

tem

to

be c

ontr

olle

d (c

lose

d lo

op)

or t

he s

yste

m t

o be

affec

ted

by t

he

inpu

t (o

pen

loop

).

Co

ntr

olle

r:

The

cont

rolle

r is

the

par

t of

the

sys

tem

, w

hich

aff

ects

the

pla

nt u

sing

an

actu

ator

. Fr

om a

pr

actic

al p

oint

of

view

con

trol

ler

and

actu

ator

are

dis

tingu

ishe

d, f

rom

a t

heor

etic

al p

oint

th

e co

ntro

l un

it co

mbi

nes

the

cont

rol

and

the

actu

ator

and

is

calle

d co

ntro

ller.

Fro

m a

pr

actic

al p

oint

of

view

the

tec

hnic

al c

ontr

ol u

nit

ofte

n al

so c

onsi

sts

of t

he u

nit

for

com

pariso

n of

des

ired

and

con

trol

var

iabl

e.

Tran

sfer

ele

men

t:

Abs

trac

t un

ders

tand

ing

of a

gen

eral

sys

tem

with

dyn

amic

pro

pert

ies,

whi

ch c

an b

e th

e sy

stem

to

be c

ontr

olle

d or

the

con

trol

ler

or a

noth

er e

lem

ent

of t

he t

rans

fer

chai

n. T

he

impo

rtan

t as

pect

is

th

at

the

tran

sfer

el

emen

t is

a

syst

em,

whe

reby

th

e dy

nam

ic

prop

ertie

s of

the

tra

nsm

ittan

ce f

rom

inp

ut t

o ou

tput

are

of

inte

rest

. Th

e re

leva

nt a

spec

t is

the

dyn

amic

cha

ract

eriz

atio

n of

the

tra

nsfe

r be

havi

or.

Plan

t an

d co

ntro

llers

are

in

this

w

ay n

othi

ng m

ore

than

tra

nsfe

r el

emen

ts,

the

deno

tatio

n pl

ant

or c

ontr

olle

r re

sults

fro

m

the

func

tion

of t

he t

rans

fer

elem

ent

with

in t

he lo

op.

Tran

sfer

sys

tem

: Abs

trac

t un

ders

tand

ing

of a

set

of

elem

ents

bui

ldin

g up

a n

ew e

lem

ent

> s

yste

m.

Act

uat

or:

Act

uato

r is

the

phy

sica

l de

vice

in

fron

t of

the

pla

nt u

sed

for

phys

ical

exc

itatio

n of

the

pl

ant

to a

ffec

t th

e pl

ant.

In

the

theo

retic

al /

abs

trac

t co

nsid

erat

ion

the

actu

ator

is p

art

of

the

tran

sfer

ele

men

t co

ntro

ller.

M

easu

rem

ent

dev

ice:

Th

e ph

ysic

al d

evic

e us

ed f

or m

easu

ring

of

the

outp

ut /

the

con

trol

var

iabl

e is

cal

led

mea

sure

men

t de

vice

/ u

nit.

In

the

theo

retic

al /

abs

trac

t co

nsid

erat

ion

the

dyna

mic

al

prop

ertie

s of

usu

al n

egle

cted

, so

the

mea

sure

men

t de

vice

is

unde

rsta

nd i

s an

ide

al

tran

sfer

ele

men

t w

ithou

t an

y ki

nd o

f dy

nam

ic p

rope

rtie

s. F

or p

ract

ical

con

side

ratio

ns t

he

dyna

mic

s pr

oper

ties

of t

he m

easu

rem

ent

unit

as w

ell

as o

f th

e ac

tuat

or h

as t

o be

co

nsid

ered

. R

efer

ence

val

ue:

Pa

ram

eter

of

the

refe

renc

e va

riab

le

Co

ntr

ol v

alu

e:

Para

met

er o

f th

e co

ntro

l var

iabl

e --

----

- R

emar

k:

The

give

n te

rms

and

defin

ition

s ar

e us

ed f

rom

diff

eren

t au

thor

s et

c. in

a d

iffer

ent

way

. Th

e im

port

ant

aspe

ct i

s th

e ve

rbal

/ter

m-b

ased

sep

arat

ion

from

sig

nals

and

phy

sica

l va

riab

les,

fro

m s

yste

ms

and

beha

vior

s, f

rom

the

phy

sica

l/te

chni

cal

real

izat

ion

and

from

th

e m

athe

mat

ical

abs

trac

t m

appi

ng.

For

exam

ple

equ

atio

ns a

re u

sed

to d

escr

ibed

the

be

havi

or o

f sy

stem

s, t

he o

utpu

t va

riab

les

of t

echn

ical

sys

tem

s sh

ow a

lso

this

beh

avio

r.

In o

lder

tex

t bo

oks

and

prac

tical

orien

ted

liter

atur

e th

is d

istin

ctio

n is

oft

en n

ot r

ealiz

ed.

It is

a go

od ide

a to

und

erst

and

the

clea

r se

para

tion

betw

een

real

ity,

the

map

ping

-bas

ed

mat

hem

atic

al r

epre

sent

atio

n to

be

open

for

fut

ure

adva

nced

info

rmat

ion

scie

nce

orie

nted

co

ntro

l and

inte

ract

ion

appr

oach

es.

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