Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture3.pdf · Performance...

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

Some projects studied in 2014-15

Application of Optimal Control to malaria: strategies and simulations

Performance compare between LQR and PID control of DC Motor

Optimal Low-Thrust LEO (low-Earth orbit) to GEO (geosynchronous-Earth orbit)

Circular Orbit Transfer

Controllo ottimo di una turbina eolica a velocità variabile attraverso il metodo

dell'inseguimento ottimo a regime permanente

OptimalControl in Dielectrophoresis

On the Design of P.I.D. Controllers Using Optimal Linear Regulator Theory

Rocket Railroad Car

………

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.


Calculus of variation and optimal control

Problem 1

Let us consider the dynamical system described by:

f function of C2 class

),,( tuxfx


x(t) state vector in Rn

u(t) control vector in Rp

ii xtx )( known

1 nf


0),( ff ttx

Vectorial function of C1 class

of dimension

0),,( tuxq


of dimension

Define the cost index

With L function of C2 class

f

i

t

t

f dttuxLtuxJ ,,),,(

Assume the norm:

ft

t

ttttf ttudutxtxtux

i

)(sup)(sup)(sup)(sup,,

AIM: Find

the instant

the control

the state

that satisfy the previous equations and

minimize the cost index

oft


)(0 RCuo

)(1 RCxo

DEFINE the scalar function

the Hamiltonian function


),,()(),,(),,,,( 00 tuxfttuxLtuxH T

Theorem 1

Let be an admissible solution such

that


)*,*,( *ftux

**

*

,),(

)),((

fiaactive

fff

ttttu

qrk

ttxrk

IF is a local minimum

there exist

not simultaneously null in such that:


)*,*,( *ftux

],[],,[, *0**1**0 fifi ttCttCR

],[ *fi tt

***

***

*

0

TT

TT

u

q

u

H

x

q

x

H


T

tft

T

tf

f

j

jj

f

f

f

f

tH

Rtx

t

t

jtuxqt

*

*

*

*

*

*

*

*

*

,))((

)(*

0)(

,...,2,1,0)*,*,()(


The discontinuity of may occurr only in the points where has a discontinuity and

** and

t *u

** tt

HH


Problem 2: Consider Problem 1 assuming L, f, q not depending on t Theorem 2 Let be an admissible solution

)*,*,( *ftux

**

**

**

*

,,

problemsstationaryforMoreover

uofpointcontinuityanyfor

fi

T

tttcH

t

q

t

H

dt

dH

Problem 3

Let us consider the dynamical system described by:

f function of C2 class

),,( tuxfx


x(t) state vector in Rn

u(t) control vector in Rp

ii xtx )( known

Calculus of variation and optimal control

1 nf


0),( ff ttx


of dimension

f

i

t

t

Kdtttutxh ),(),(


of dimension


With L function of C2 class

f

i

t

t

f dttuxLtuxJ ,,),,(

Assume the norm:

ft

t

ttttf ttudutxtxtux

i

)(sup)(sup)(sup)(sup,,

AIM: Find

the instant

the control

the state

that satisfy the previous equations and

minimize the cost index

oft


)(0 RCuo

)(1 RCxo

DEFINE the scalar function

the Hamiltonian function


ttutxh

tuxfttuxLtuxH

T

T

),(),(

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

Theorem 3

Let be an admissible solution such

that


)*,*,( *ftux

fff ttx

rk

*

)),((

IF is a local minimum

there exist

not simultaneously null in such that:


)*,*,( *ftux

T

T

u

H

x

H

*

**

0

RttCR fi **1**0 ],,[,

],[ *fi tt

T

tf

t

T

tf

f

f

f

f

f

tH

Rtx

t

*

*

*

*

*

*

*

,))((

)(*


The discontinuity of may occurr only in the points where has a discontinuity and

*

kt *u

**

kk ttHH


Problem 4: Consider the linea r system Assume: fixed

utBxtAx )()(

Functions of C2 class

fi tti

i xtx )(

nRor pointfixedabeing)( fff DDtx

0),,( tuxqVectorial function of C2 class

of dimension CONVEX



))((),,(),( f

t

t

txGdttuxLuxJ

f

i

Functions of C2 class

CONVEX

Functions of C3 class- CONVEX


Theorem 4 Let be an admissible solution such that is a normal optimal solution

),( oo ux

fia

oactive ttttu

qrk ,),(

),( oo ux

0)(

,...,2,1

,0),,()(

0

t

j

tuxqt

u

q

u

H

x

q

x

H

o

ooj

oj

ooToT

ooToT

o


AND IF

oT

ff

o

tdx

dGt

)()(

nf RD

Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture3.pdf · Performance...

Documents

Transcript of Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture3.pdf · Performance...