Linear Optimal control - uniroma1.it B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989 C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993

Optimal Control

Lecture

Prof. Daniela Iacoviello

Department of Computer, Control, and Management Engineering Antonio Ruberti

Sapienza University of Rome

26/10/2015 Controllo nei sistemi biologici

Lecture 1

Pagina 2

Prof. Daniela Iacoviello

Department of computer, control and management

Engineering Antonio Ruberti

Office: A219 Via Ariosto 25

http://www.dis.uniroma1.it/~iacoviel

Prof.Daniela Iacoviello- Optimal Control

Grading

Project + oral exam

The exam must be concluded before the second

part of Identification that will be held by Prof. Battilotti

Grading

Project + oral exam

Example of project:

- Read a paper on an optimal control problem

- Study: background, motivations, model, optimal control,

solution, results

- Simulations

You must give me, before the date of the exam:

- A .doc document

- A power point presentation

- Matlab simulation files

The exam must be concluded before the second

part of Identification that will be held by Prof. Battilotti

THESE SLIDES ARE NOT SUFFICIENT

FOR THE EXAM: YOU MUST STUDY ON THE BOOKS


Part of the slides has been taken from the References indicated below

References B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989 C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993 L. Evans, An introduction to mathematical optimal control theory, 1983 H.Kwakernaak , R.Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972 D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 2004 D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton University Press, 2011 How, Jonathan, Principles of optimal control, Spring 2008. (MIT OpenCourseWare: Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA.


Lecture outline

• Calculus of variations

Lagrange problem

Goal: solve the problem of founding of Carthage.

According to a legend the locals said to Dido and her followers,

that wanted to stop in Africa, that they could have the area that a

plow would circumscribe in a day


Lagrange (Torino 1736, Paris 1813)


http://it.wikipedia.org/wiki/File:Joseph_Louis_Lagrange.jpg

The Lagrange problem Problem 1

Let us consider the linear space and define the admissible set:

Introduce the norm:

and consider the cost index:

with L function of C2 class.

RRRC )(1

111 ),(,),(:)(,, vf

viiii RDTTzRDttzRRRCTtzD

T

t

i

i

dtttztzLTtzJ ),(),(),,(

TttztzTtz itt

i )(sup)(sup,,


Find the global minimum (optimum)

for J over D:

An extremum is NON-singular if

ooi

o Ttz ,,

DTtzTtzJTtzJ iioo

io ),,(,,,,


],[insingularnonis)(

**

*

2

2

Tttz

Li

Theorem 1 (Lagrange). If is a local minimum then

In any discontinuity point of

the following conditions are verified:

DTtz i *** ,,

Tttz

L

dt

d

z

Li

T ,0

**

Euler equation

t *z

****

tttt

zz

LLz

z

LL

z

L

z

L

Weierstrass- Erdmann conditions


Moreover, transversality conditions are satisfied:

• If are open subset we have:

• If are closed subsets defined respectively by

such that

fi DD

0000**

**

**

**

Tt

T

T

T

t

LLz

L

z

L

i

i

fi DD

0),(0),( ffii ttzttz

fi

ii TTzrg

ttzrg

**

),(),(


for fi RR

**

****

**

**

,

)(,

)(

Tz

z

LL

tz

z

LL

Tzz

L

tzz

L

T

Ti

T

t

T

Ti

T

t

i

i


• If the sets are defined by the function of σ components of C1 class such that


fi DandD

0)),(,),(( TTzttzw ii

*

),(,),( TTzttz

wrg

ii

R

Tz

w

z

L

tz

w

z

L T

Ti

T

ti

****

)(,

)( **

**

**

,T

wz

z

LL

t

wz

z

LL T

Ti

T

ti


Consider Problem 1 with

fixed

If are closed sets in

defined by the C1 functions

Tandti


fi DandD1vR

1dimension of,0),(

1dimension of,0),(

f

i

vTTz

vttz ii

With affine functions and

If the sets are defined by the function with σ components of C1

class affine with respect to such that


f

o

i

o

i Tzrg

tzrg

)()(

and

fi DandD

))(),(( Tztzw i

)(),( Tztz i

o

i Tztz

wrg

)(),(

The function L must be convex with respect to

Find the global minimum (optimum)

for J over D:

oz

DzzJzJ o


)(),( tztz

Theorem 2. is the optimum if and only if

In any discontinuity point of

the following conditions are verified:

Dzo

Tttz

L

dt

d

z

Li

Too

,0

Euler equation

t *z

o

t

o

t

o

t

o

t

zz

LLz

z

LL

z

L

z

L

Weierstrass- Erdmann conditions


Moreover, transversality conditions are satisfied:


• If are closed subsets defined respectively by

Such that

fi DD

0000 ****

o

T

o

t

To

T

To

t

LLz

L

z

L

ii

fi DD

0)(0)( Tztz i

f

o

i

o

ii TTzrg

ttzrg

),(),(


for fi RR

T

T

T

t

oT

o

T

o

i

T

t

zz

LLz

z

LL

Tzz

L

tzz

L

i

i

0,0

)(,

)(

*

*


If the sets are defined by the function affine with respect to such that


fi DandD

))(),(( Tztzw i

)(),( Tztz i

*

)(),( Tztz

wrg

i

R

Tzz

L

tzz

L T

ti

T

t fi

****

)(,

)( **


Let us consider the linear space

and define the admissible set

of dimension

RRRC )(1

kdtttztzhttztzgRDTTz

RDttzRRRCTtzD

T

t

v

f

v

iiii

i

),(),(0),(),(,),(

,),(:)(,,

1

11

g


v

The Lagrange problem consider the cost index:

T

t

i

i



Define the augmented lagrangian:

ttztzhttztzgt

ttztzLtttztz

TT ),(),(),(),()(

),(),(),(,,),(),( 00


Theorem 3(Lagrange). Let such that

If is a local minimum for J over D,

then there exist

not simultaneously null in such that:

•

DTtz i *** ,,

**

*

,)(

Ttttz

grank i

**

**

,0 Tttzdt

d

zi

T

*** ,, Ttz i


RTtCR i **0**0 ],,[,

],[ *Tti

•

where are cuspid points for

• Moreover, transversality conditions are satisfied:

****

kkkk tttt

zz

zzzz

kt

*z



• If are closed subset defined respectively by

such that

fi DD

0000**

**

**

**

Tt

T

T

T

ti

izz

fi DD

0),(0),( TTzttz ii

fi

ii TTzrg

ttzrg

**

),(),(


for fi RR

**

*

*

***

**

**

,

)(,

)(

Tz

ztz

z

Tzztzz

T

Ti

T

t

T

Ti

T

t

i

i


If the sets are defined by the function affine with respect to

such that


fi DandD

))(),(( Tztzw i

)(),( Tztz i

*

),(,),( TTzttz

wrg

ii

R

T

wz

zt

wz

z

Tz

w

ztz

w

z

T

Ti

T

t

T

Ti

T

t

i

i

****

****

**

**

,

)(,

)(


Let us consider the linear space

and define the admissible set

of dimension with

RRRC )(1

],[,),(),(0),(),(

,,)(,)(,,1

Tttkdtttztzhttztzg

openDandDDtzDtzttCzD

i

T

t

fifiiifi

i

g


v

The Lagrange problem

ti T fixed

g and h linear functions in

L C2 function convex with respect to

Consider the cost index:

T

t

i

i



],[),(),( fi ttttztz

],[),(),( fi ttttztz

Define the augmented lagrangian:

ttztzhttztzgt

ttztzLtttztz

TT ),(),(),(),()(

),(),(),(,,),(),( 0


Theorem 4 (Lagrange). Let such that

is an optimal normal (λ0 =1)

solution if and only if

Dzo

Ttttz

grank i

o

,)(


Dzo

Tttzdt

d

zi

T ,0

**

in the instants for which

are open we have:

fi tt and/or


fi DD and/or

To

z0

Documents

Linear Optimal control - uniroma1.it B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989 C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993