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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
Prof.Daniela Iacoviello- Optimal Control
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.
Prof.Daniela Iacoviello- Optimal Control
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
Prof.Daniela Iacoviello- Optimal Control
Lagrange (Torino 1736, Paris 1813)
Prof.Daniela Iacoviello- Optimal Control
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,,
Prof.Daniela Iacoviello- Optimal Control
Find the global minimum (optimum)
for J over D:
An extremum is NON-singular if
ooi
o Ttz ,,
DTtzTtzJTtzJ iioo
io ),,(,,,,
Prof.Daniela Iacoviello- Optimal Control
],[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
Prof.Daniela Iacoviello- Optimal Control
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
**
),(),(
Prof.Daniela Iacoviello- Optimal Control
for fi RR
**
****
**
**
,
)(,
)(
Tz
z
LL
tz
z
LL
Tzz
L
tzz
L
T
Ti
T
t
T
Ti
T
t
i
i
Prof.Daniela Iacoviello- Optimal Control
• If the sets are defined by the function of σ components of C1 class such that
Prof.Daniela Iacoviello- Optimal Control
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
The Lagrange problem Problem 2
Consider Problem 1 with
fixed
If are closed sets in
defined by the C1 functions
Tandti
Prof.Daniela Iacoviello- Optimal Control
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
Prof.Daniela Iacoviello- Optimal Control
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
Prof.Daniela Iacoviello- Optimal Control
)(),( 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
Prof.Daniela Iacoviello- Optimal Control
Moreover, transversality conditions are satisfied:
• If are open subset we have:
• 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
),(),(
Prof.Daniela Iacoviello- Optimal Control
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
)(,
)(
*
*
Prof.Daniela Iacoviello- Optimal Control
If the sets are defined by the function affine with respect to such that
Prof.Daniela Iacoviello- Optimal Control
fi DandD
))(),(( Tztzw i
)(),( Tztz i
*
)(),( Tztz
wrg
i
R
Tzz
L
tzz
L T
ti
T
t fi
****
)(,
)( **
The Lagrange problem Problem 3
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
Prof.Daniela Iacoviello- Optimal Control
v
The Lagrange problem consider the cost index:
T
t
i
i
dtttztzLTtzJ ),(),(),,(
Prof.Daniela Iacoviello- Optimal Control
Define the augmented lagrangian:
ttztzhttztzgt
ttztzLtttztz
TT ),(),(),(),()(
),(),(),(,,),(),( 00
Prof.Daniela Iacoviello- Optimal Control
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
Prof.Daniela Iacoviello- Optimal Control
RTtCR i **0**0 ],,[,
],[ *Tti
•
where are cuspid points for
• Moreover, transversality conditions are satisfied:
****
kkkk tttt
zz
zzzz
kt
*z
Prof.Daniela Iacoviello- Optimal Control
• If are open subset we have:
• 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
**
),(),(
Prof.Daniela Iacoviello- Optimal Control
for fi RR
**
*
*
***
**
**
,
)(,
)(
Tz
ztz
z
Tzztzz
T
Ti
T
t
T
Ti
T
t
i
i
Prof.Daniela Iacoviello- Optimal Control
If the sets are defined by the function affine with respect to
such that
Prof.Daniela Iacoviello- Optimal Control
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
****
****
**
**
,
)(,
)(
The Lagrange problem Problem 4
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
Prof.Daniela Iacoviello- Optimal Control
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
dtttztzLTtzJ ),(),(),,(
Prof.Daniela Iacoviello- Optimal Control
],[),(),( fi ttttztz
],[),(),( fi ttttztz
Define the augmented lagrangian:
ttztzhttztzgt
ttztzLtttztz
TT ),(),(),(),()(
),(),(),(,,),(),( 0
Prof.Daniela Iacoviello- Optimal Control
Theorem 4 (Lagrange). Let such that
is an optimal normal (λ0 =1)
solution if and only if
Dzo
Ttttz
grank i
o
,)(
Prof.Daniela Iacoviello- Optimal Control
Dzo
Tttzdt
d
zi
T ,0
**
in the instants for which
are open we have:
fi tt and/or
Prof.Daniela Iacoviello- Optimal Control
fi DD and/or
To
z0
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