Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture4.pdf · Lev...
Transcript of Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture4.pdf · Lev...
Linear Optimal Control
Lecture 4 Daniela Iacoviello
Department of Computer and System Sciences “A.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
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
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
Pontryagin
Lev Semenovich Pontryagin (3 September 1908 – 3 May 1988) was a Soviet Russian mathematician.
Prof.Daniela Iacoviello- Optimal Control
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
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
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
Hamiltonian function
uxftuxLuxH T ,)(,,,, 00
Prof.Daniela Iacoviello- Optimal Control
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
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
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
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
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
0*
*
*
H
x
HT
U
ttutxHttxH
,)(,),(),()(,,),( **
0
****
0
*
Moreover there exists a vector such that:
T
Tdx
dT
*
)()(
fR
Prof.Daniela Iacoviello- Optimal Control
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
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
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
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
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
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 )(
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
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
Hamiltonian function
)),(),((,)(,,,, 00 ttutxhuxftuxLuxH TT
Prof.Daniela Iacoviello- Optimal Control
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
,
*
*
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
*
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
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 ,
Prof.Daniela Iacoviello- Optimal Control
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
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
U
ttutxHttxH ooooo
,)(),(),()(,),(
Moreover if
oT
o
Tdx
dGT
)()(
nRTx )(
oT
o
x
tuxH
),,,(
Prof.Daniela Iacoviello- Optimal Control
Remark
If the set U coincides with Rp the minimum
condition reduces to :
0
u
H
Prof.Daniela Iacoviello- Optimal Control
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
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
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
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
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
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