Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture4.pdf · Lev...
Transcript of Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture4.pdf · Lev...
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Linear Optimal Control
Lecture 4 Daniela Iacoviello
Department of Computer and System Sciences “A.Ruberti”
Sapienza University of Rome
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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
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Grading
Project + oral exam
The exam must be concluded before the second
part of Identification that will be held by Prof. Battilotti
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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
![Page 5: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture4.pdf · Lev Semenovich Pontryagin (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician.](https://reader033.fdocuments.us/reader033/viewer/2022051913/6003e4df5ce4eb1704635512/html5/thumbnails/5.jpg)
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
………
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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
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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
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R.F.Hartl, S.P.Sethi. R.G.Vickson, A Survey of the Maximum Principles for Optimal Control Problems with State Constraints, SIAM Review, Vol.37, No.2, pp.181-218, 1995
Prof.Daniela Iacoviello- Optimal Control
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Pontryagin
Lev Semenovich Pontryagin (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician.
Prof.Daniela Iacoviello- Optimal Control
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The Pontryagin principle
Problem 1: Consider the dynamical system:
with:
tuxfx ,,
niURCx
ff
RUtuRtx
n
i
pn
,...,2,1,,
)(,)(
0
Prof.Daniela Iacoviello- Optimal Control
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Assume fixed the initial control instant and the initial and final values :
Define the performance index :
with
fi
i xTxxtx )()(
))((),(),(,, TxGduxLTuxJ
T
ti
niURCx
LL n
i
,...,2,1,, 0
Prof.Daniela Iacoviello- Optimal Control
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Determine the value
the control and the state
that satisfy the dynamical
system, the constrain t on the control, the
initial and final conditions and minimize the
cost index
,itT
)(0 RCuo
)(1 RCxo
Prof.Daniela Iacoviello- Optimal Control
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Hamiltonian function
uxftuxLuxH T ,)(,,,, 00
Prof.Daniela Iacoviello- Optimal Control
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The Pontryagin principle
Theorem 1 (necessary condition):
Assume the admissible solution is a minimum
there exist a constant
and a n-dimensional vector
not simultaneously null such that :
*** ,, Tux
00
*1* ,TtC i
0*
*
*
H
x
HT
U
ttutxHttxH
,)(,),(),()(,,),( **
0
****
0
*
Prof.Daniela Iacoviello- Optimal Control
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The Pontryagin principle
Problem 2: Consider the dynamical system:
with:
tuxfx ,,
niURCx
ff
RUtuRtx
n
i
pn
,...,2,1,,
)(,)(
0
Prof.Daniela Iacoviello- Optimal Control
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Assume fixed the initial control instant and the
initial state
while for final values assume:
where is a function of dimension
of C1 class.
Define the performance index :
with
i
i xtx )(
))((),(),(,, TxGduxLTuxJ
T
ti
niURCx
LL n
i
,...,2,1,, 0
0)( Tx nf
Prof.Daniela Iacoviello- Optimal Control
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Determine the value
the control and the state
that satisfy the dynamical
system, the constrain on the control, the initial
and final conditions and minimize the cost
index .
,itT
)(0 RCuo
)(1 RCxo
Prof.Daniela Iacoviello- Optimal Control
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Theorem 2 (necessary condition):
Consider an admissible solution such that
If it is a minimum there exist a constant
and an n-dimensional vector
not simultaneously null such that :
*** ,, Tux
00
*1* ,TtC i
fTdx
drank
*
)(
Prof.Daniela Iacoviello- Optimal Control
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0*
*
*
H
x
HT
U
ttutxHttxH
,)(,),(),()(,,),( **
0
****
0
*
Moreover there exists a vector such that:
T
Tdx
dT
*
)()(
fR
Prof.Daniela Iacoviello- Optimal Control
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The Pontryagin principle
Problem 3: Consider the dynamical system:
with:
tuxfx ,,
niURCx
ff
RUtuRtx
n
i
pn
,...,2,1,,
)(,)(
0
Prof.Daniela Iacoviello- Optimal Control
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Assume fixed the initial control instant and the
initial state
while for final values assume:
where is a function of dimension
of C1 class.
Define the performance index :
with
i
i xtx )(
T
ti
duxLTuxJ ),(),(,,
niRURCt
L
x
LL n
i
,...,2,1,,, 0
0),( TTx 1 nf
Prof.Daniela Iacoviello- Optimal Control
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Determine the value
the control and the state
that satisfy the dynamical
system, the constrain t on the control, the
initial and final conditions and minimize the
cost index .
,itT
)(0 RCuo
)(1 RCxo
Prof.Daniela Iacoviello- Optimal Control
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Theorem 3:
Consider an admissible solution
such that
IF it is a minimum there exist a constant
and an n-dimensional vector
not simultaneously null such that :
*** ,, Tux
00
*1* ,TtC i
fTTx
rank
*
),(
Prof.Daniela Iacoviello- Optimal Control
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RkkdH
Hx
HT
t
T
,,
**
**
*
U
ttutxHttxH
,)(,),(),()(,,),( **
0
****
0
*
Moreover there exists a vector such that:
T
T
T
TH
TxT
**
*
**
)()(
fR
Prof.Daniela Iacoviello- Optimal Control
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The Pontryagin principle
Problem 4: Consider the dynamical system:
with:
fixed
tuxfx ,,
RURCt
f
x
ff
RUtuRtx
n
pn
0,,
)(,)(
Prof.Daniela Iacoviello- Optimal Control
i
i xtx )(
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For the final values assume: where is a function of dimension of C1 class. Assume the constraint with Define the performance index : with
T
ti
duxLTuxJ ),(),(,,
RURCt
L
x
LL n
0,,
0),( TTx 1 nf
Prof.Daniela Iacoviello- Optimal Control
kduxh
T
ti
),(),(
niRURCt
h
tx
hh n ,...,2,1,,
)(, 0
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Determine the value
the control and the state
that satisfy the dynamical
system, the constrain t on the control, the
initial and final conditions and minimize the
cost index .
,itT
)(0 RCuo
)(1 RCxo
Prof.Daniela Iacoviello- Optimal Control
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Hamiltonian function
)),(),((,)(,,,, 00 ttutxhuxftuxLuxH TT
Prof.Daniela Iacoviello- Optimal Control
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Theorem 4 (necessary condition):
Consider an admissible solution
such that
IF it is a local minimum
there exist a constant
and an n-dimensional vector
not simultaneously null such that :
*** ,, Tux
00
*1* ,TtC i
fTTx
rank
*
),(
Prof.Daniela Iacoviello- Optimal Control
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,
*
*
T
x
H
U
ttutxHttxH
,)(,),(),()(,,),( **
0
****
0
*
Moreover there exists a vector such that: The discontinuities of may occur only in the instants in which u has a discontinuity and in these instant the Hamiltonian is conitnuous
T
T
T
TH
TxT
**
*
**
)()(
fR
Prof.Daniela Iacoviello- Optimal Control
*
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The Pontryagin principle - convex case
Problem 5: Consider the dynamical linear
system:
with A and B of function of C1 class; assume fixed the initial and final instants and the initial state,
Assume
where U is a convex set.
utBxtAx )()(
n
T RTxorfixedxTx )()(
TttRUtu i
p ,)(
Prof.Daniela Iacoviello- Optimal Control
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Define the performance index :
with
L convex function with respect to x(t), u(t) in
per ogni
G is a scalar function of C2 class and convex
with respect to x(T)
)(),(),(, TxGduxLuxJ
T
ti
niTtURCt
L
x
LL i
n
i
,...,2,1,,,, 0
URn Ttt i ,
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Determine the control
and the state
that satisfy the dynamical system, the
constraint on the control, the initial
and final conditions and minimize the cost
index .
TtCu io ,0
TtCx io ,1
Prof.Daniela Iacoviello- Optimal Control
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Theorem 5 (necessary and sufficient condition):
Consider an admissible solution
such that
It is a minimum normal (i.e.λ0 =1) if and only if
there exists an n-dimensional vector
such that :
oo ux ,
TtC i
o ,1
f
o
TTxrank
),(
Prof.Daniela Iacoviello- Optimal Control
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U
ttutxHttxH ooooo
,)(),(),()(,),(
Moreover if
oT
o
Tdx
dGT
)()(
nRTx )(
oT
o
x
tuxH
),,,(
Prof.Daniela Iacoviello- Optimal Control
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Remark
If the set U coincides with Rp the minimum
condition reduces to :
0
u
H
Prof.Daniela Iacoviello- Optimal Control
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Example 3 (from L.C.Evans) Control of production and consumption Consider a factory whose output can be controlled. Let’s set: x(t) the amount of output produced at time t , 0 ≤ t. Assume we consume some fraction of the output at each time and likewise reinvest the remaining fraction u(t) . It is our control, subject to the constraints 0 ≤ u(t) ≤ 1
Prof.Daniela Iacoviello- Optimal Control
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The corresponding dynamics are: The positive constant k represents the growth rate of our reinvestment. We will chose K=1. Assume as cost index the function: The aim is to maximize the total consumption of the output
0)0(
0),()()(
xx
ktxtkutx
T
dttxtuuJ
0
)()(1))((
Prof.Daniela Iacoviello- Optimal Control
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We apply the Pontryagin Principle; the Hamiltonian is: The necessary conditions are:
xuxuuxH 1,,
1)()()()(max)(),(),(
)()()(
0)(1)()(1)(
10
ttxtutxttutxH
txtutx
Tttut
u
Prof.Daniela Iacoviello- Optimal Control
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From the last equation we have, since x(t)>0: From the equation of the costate, since , by continuity we deduce for t<T, t close to T, that : thus for such values of t. Therefore and consequently: More precisely so long as and this holds for:
1)(0
1)(1)(
tif
tiftu
0)( T
0)( tu1)( t
1)( t
tTt )(tTt )( 1)( t
TtT 1Prof.Daniela Iacoviello- Optimal Control
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For times with t near T we have Therefore the costate equation yields: Since we have for all And over this time interval there are no switchings
1Tt
1)( tu
)(1)(1)( ttt
1)1( T 1Tt
1)( 1 tTet
Prof.Daniela Iacoviello- Optimal Control
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Therefore: For the switching time Homework: find the switching time
Tttif
ttiftu
*
*
*
0
01)(
1* Tt
Prof.Daniela Iacoviello- Optimal Control
*t T
Optimal solution: we should reinvest all
the output
(and therefore consume anything)
up to time t* and afterwards we should
consume everything
(and therefore reinvest nothing) Bang-bang control