linear optimal control - uniroma1.itiacoviel/materiale/... · d. liberzon, "calculus of...

Optimal Control

Lecture

Prof. Daniela Iacoviello

Department of Computer, Control, and Management Engineering Antonio Ruberti

Sapienza University of Rome

12/10/2016 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 2

Grading

Project + oral exam

3

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

4

THESE SLIDES ARE NOT SUFFICIENT

FOR THE EXAM: YOU MUST STUDY ON THE BOOKS


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

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)

Lagrange education was completed at the university of Torino, Berlin and Paris. He was one of the major Mathematician of the 17th, important for his studies in calculus of variations and in astronomy.


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 ),(,),(:)(,, vfff

viiifi RDttzRDttzRRRCttzD

f

i

t

t

fi dtttztzLttzJ ),(),(),,(

fitt

fi tttztzttz )(sup)(sup,,


Find the global minimum (optimum)

for J over D:

An extremum is NON-singular if

of

oi

o ttz ,,

DttzttzJttzJ fifiof

oi

o ),,(,,,,

Prof.Daniela Iacoviello- Optimal Control

],[insingularnonis)(

**

*

2

2

fi tttz

L

10

Theorem (Lagrange). If is a local

minimum then

1)

2) In any discontinuity point of

Weierstrass-Erdmann condition

3) Transversality conditions different cases

depending on the nature

of the boundary conditions

Dttz fi *** ,,

fiT ttt

z

L

dt

d

z

L,0

**

Euler equation

t*z

11

SCHEME of the theorems

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

In any discontinuity point of

the following conditions are verified:

Dttz fi *** ,,

fiT ttt

z

L

dt

d

z

L,0

**

Euler equation

t *z

****

tttt

zz

LLz

z

LL

z

L

z

L

Weierstrass- Erdmann condition


1

2

Moreover, transversality conditions are satisfied:

• If are open subset we have:

• If are closed subsets defined respectively by

such that

fi DD

0000**

**

**

**

fi

fi

tt

T

t

T

t

LLz

L

z

L

fi DD

0),(0),( ffii ttzttz

fff

iii ttz

rgttz

rg

**

),(),(


3

for fi RR

**

****

**

**

,

)(,

)(

f

T

ti

T

t

f

T

ti

T

t

tz

z

LL

tz

z

LL

tzz

L

tzz

L

fi

fi


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


fi DandD

0)),(,),(( ffii ttzttzw

*

),(,),( ffii ttzttz

wrg

R

tz

w

z

L

tz

w

z

L

f

T

ti

T

t fi

****

)(,

)( **

**

**

,f

T

ti

T

t t

wz

z

LL

t

wz

z

LL

fi

15


Consider Problem 1 with

fixed

If are closed sets in

defined by the C1 functions

fi tandt


fi DandD1vR

1dimension of,0),(

1dimension of,0),(

f

i

vTTz

vttz ii

16

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

fi

o

i tzrg

tzrg

)()(

and

fi DandD

))(),(( fi tztzw

)(),( fi tztz

o

fi tztz

wrg

)(),(

17

The function L must be convex with respect to

Find the global minimum (optimum)

for J over D:

oz

DzzJzJ o


)(),( tztz

18

Theorem 2. is the optimum if and only if

In any discontinuity point of

the following conditions are verified:

Dzo

fiT

oo

tttz

L

dt

d

z

L,0

Euler equation

t*z

o

t

o

t

o

t

o

t

zz

LLz

z

LL

z

L

z

L

Weierstrass- Erdmann condition


Moreover, transversality conditions are satisfied:


• If are closed subsets defined respectively by

Such that

fi DD

0000 **

**

o

t

o

t

To

t

To

tfi

fi

LLz

L

z

L

fi DD

0)(0)( fi tztz

f

o

ffi

o

ii ttzrg

ttzrg

),(),(


Then for

fi RR

T

t

T

t

o

f

To

t

o

i

T

t

fi

fi

zz

LLz

z

LL

tzz

L

tzz

L

0,0

)(,

)(

*

*


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


fi DandD

))(),(( fi tztzw

)(),( fi tztz

o

fi tztz

wrg

)(),(

R

tzz

L

tzz

Lo

f

T

o

t

o

i

T

o

t of

oi

)(,

)(

22


Let us consider the linear space

and define the admissible set

of dimension

of dimension

RRRC )(1

kdtttztzhttztzgRDttz

RDttzRRRCttzD

f

i

t

t

vfff

viiifi

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

,),(:)(,,

1

11

g


v

23

h

The Lagrange problem consider the cost index:

with L scalar function of C2 class.

f

i

t

t



Define the augmented lagrangian:

ttztzhttztzgt

ttztzLtttztz

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

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


Theorem 3(Lagrange). Let be such that

If is a local minimum for J over D,

then there exist

not simultaneously null in such that:

•

Dttz fi *** ,,

**

*

,)(

fi ttttz

grank

****

,0 fiT ttt

zdt

d

z

*** ,, fi ttz


RttCR fi **0**0 ],,[,

],[ *fi tt

26

•

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**

**

**

**

fi

fi

tt

T

t

T

t zz

fi DD

0),(0),( ffii ttzttz

f

ff

i

ii ttzrg

ttzrg

**

),(),(


Size of g Size of χ

for fi RR

**

*

*

***

**

**

,

)(,

)(

f

T

ti

T

t

f

T

ti

T

t

tz

ztz

z

tzztzz

fi

fi


If the sets are defined by the function

affine with respect to

such that


fi DandD

))(),(( fi tztzw)(),( fi tztz

*

),(,),( ffii ttzttz

wrg

R

t

wz

zt

wz

z

tz

w

ztz

w

z

f

T

ti

T

t

f

T

ti

T

t

fi

fi

****

****

**

**

,

)(,

)(

30


Let us consider the linear space

and define the admissible set

of dimension with

RRRC )(1

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

,)(,)(,,1

fi

t

t

fiiifi

tttkdtttztzhttztzg

DtzDtzttCzD

f

i

g


v

31

ti tf fixed

g and h affine functions in

L C2 function convex with respect to

Consider the cost index:

f

i

t

t



],[),(),( fi ttttztz

],[),(),( fi ttttztz

32

Define the augmented lagrangian:

ttztzhttztzgt

ttztzLtttztz

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

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


Theorem 4 (Lagrange). Let such that

is an optimal normal solution

if and only if

Dzo

fi

o

ttttz

grank ,

)(


Dzo

fiT ttt

zdt

d

z,0

**

34

in the instants for which

are open we have:

fitt and/or


fiDD and/or

To

z0

35

linear optimal control - uniroma1.itiacoviel/materiale/... · d. liberzon, "calculus of...

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