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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
Prof.Daniela Iacoviello- Optimal Control 5
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 6
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
Prof.Daniela Iacoviello- Optimal Control 7
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.
Prof.Daniela Iacoviello- Optimal Control 8
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,,
Prof.Daniela Iacoviello- Optimal Control 9
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
Prof.Daniela Iacoviello- Optimal Control 12
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
**
),(),(
Prof.Daniela Iacoviello- Optimal Control 13
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
Prof.Daniela Iacoviello- Optimal Control 14
• If the sets are defined by the function of σ components of C1 class such that
Prof.Daniela Iacoviello- Optimal Control
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
The Lagrange problem Problem 2
Consider Problem 1 with
fixed
If are closed sets in
defined by the C1 functions
fi tandt
Prof.Daniela Iacoviello- Optimal Control
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
Prof.Daniela Iacoviello- Optimal Control
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
Prof.Daniela Iacoviello- Optimal Control
)(),( 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
Prof.Daniela Iacoviello- Optimal Control 19
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
tfi
fi
LLz
L
z
L
fi DD
0)(0)( fi tztz
f
o
ffi
o
ii ttzrg
ttzrg
),(),(
Prof.Daniela Iacoviello- Optimal Control 20
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
)(,
)(
*
*
Prof.Daniela Iacoviello- Optimal Control 21
If the sets are defined by the function affine with respect to such that
Prof.Daniela Iacoviello- Optimal Control
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
The Lagrange problem Problem 3
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
Prof.Daniela Iacoviello- Optimal Control
v
23
h
The Lagrange problem consider the cost index:
with L scalar function of C2 class.
f
i
t
t
fi dtttztzLttzJ ),(),(),,(
Prof.Daniela Iacoviello- Optimal Control 24
Define the augmented lagrangian:
ttztzhttztzgt
ttztzLtttztz
TT ),(),(),(),()(
),(),(),(,,),(),( 00
Prof.Daniela Iacoviello- Optimal Control 25
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
Prof.Daniela Iacoviello- Optimal Control
RttCR fi **0**0 ],,[,
],[ *fi tt
26
•
where are cuspid points for
• Moreover, transversality conditions are satisfied:
****
kkkk tttt
zz
zzzz
kt
*z
Prof.Daniela Iacoviello- Optimal Control 27
• If are open subset we have:
• 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
**
),(),(
Prof.Daniela Iacoviello- Optimal Control 28
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
Prof.Daniela Iacoviello- Optimal Control 29
If the sets are defined by the function
affine with respect to
such that
Prof.Daniela Iacoviello- Optimal Control
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
The Lagrange problem Problem 4
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
Prof.Daniela Iacoviello- Optimal Control
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 dtttztzLttzJ ),(),(),,(
Prof.Daniela Iacoviello- Optimal Control
],[),(),( fi ttttztz
],[),(),( fi ttttztz
32
Define the augmented lagrangian:
ttztzhttztzgt
ttztzLtttztz
TT ),(),(),(),()(
),(),(),(,,),(),( 0
Prof.Daniela Iacoviello- Optimal Control 33
Theorem 4 (Lagrange). Let such that
is an optimal normal solution
if and only if
Dzo
fi
o
ttttz
grank ,
)(
Prof.Daniela Iacoviello- Optimal Control
Dzo
fiT ttt
zdt
d
z,0
**
34
in the instants for which
are open we have:
fitt and/or
Prof.Daniela Iacoviello- Optimal Control
fiDD and/or
To
z0
35