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Optimal Control
Prof. Daniela Iacoviello
Examples
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Example
Consider a simple model of phamacodynamic:
x1= quantity of drug in the gastrintestinal section
x2= quantity of drug in the blood
r=reference value of the quantity of drug in the blood
u1 flow rate of the drug by oral administration
u2 flow rate of the drug by administered intravenously
Assume
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Minimize:
* * *
Solution of the first state equation:
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Obviously:
x2=x’2 +x’’2
intravenous
Transferred from
the gastrointestinal compartment
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Since:
Define:
It can be defined the following tracking problem:
0,0,
0,00,
''2
''2
QTTtQtx
QTtQtx
It could be verified that:
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The problem is convex and the Lagrangian is strictly convex
necessary and sufficient conditions and,
if the solution exists, it is unique
By using the theory of optimal tracking:
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where:
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where
If the admissibility condition is satisfied then the couple
Is the unique optimal solution of the subproblem
0)(2 tuo
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To determine Q>0 it is possible to solve another
quadratic problem:
where are suitable affine functions depending
on the Problem; the cost index is strictly convex:
If a solution exists
it is unique and is given by
the condition:
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Exercise
Consider the system:
The aim is to have:
with a bounded control
minimizing
)()( tutx 0)0( x
1,1)( ff ttx 1,0)( tu
ft
f dttxtutuJ0
)()(3,
The final instant is not fixed the problem is not convex
The Hamiltonian is: )()()()(3)(,1),(),( tuttxtuttutxH
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Ctt )(*
The costate equation yields
The minimum condition is:
The problem may be analyzed for four different cases:
nd the transverality conditions will be particularized:
1,0)(1)()(1)( **** ttttu
Ctsigntu 112
1)(*
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1)(* t
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1) . Three alternatives are possible:
1Ca) . From Ctsigntu 11
2
1)(*
01 Ct
0)(* tu 0)(* tx Not possible
b) . There exists a discontinuity instant
for the control with
1,2 C 1,0t
Ct 1
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2C
Not possible
c) . 2C 1,0,01 tforCt
1)(* tu ttx )(*acceptable
In case 1) we have obtained one normal extremum:
1)(* tu ttx )(*1ft
2,)( CCtt
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2) . The t The transversality condition
yields:
1C
01)1(* C
0)(* tu 0)(* tx Not possible
3) . The transversality condition yields:
0
**
*
T
ft t
Hf
1)(,1 ff txt
From the stationarity of the problem:
4C 312
1)(* tsigntu
01)1(
)1(
**
C
x
T
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AS far as the value of is concerned two alternatives are possible:
i) . From
*ft
3,1* ft 312
1)(* tsigntu
ttx )(*
Not possible
1)(* tu
1)( *** ff ttx
ii) 3* ft
with 3)( * ftx Not possible
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4) . . The transversality condition yields:
0)( *** Ctt ff
1)(,1 ff txt
0)(3 **** ft
txHf
From 312
1)(* tsigntu
There exists an instant of discontinuity
for the control such that:
** ,01 ff ttt
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From the transversality condition: 13)( ** ff ttx
With: 1)( tt
In case 4) we have obtained one normal extremum:
The value of the cost index evaluated at point of case 4) is bigger than the
value of the cost index evaluated at the solution of point found in case 1).
1)(* tu ttx )(*1ft
2,)( CCttThe normal extremum is:
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Prof.Daniela Iacoviello- Optimal Control
As far as the abnormal solution is concerned, for the case
all the solutions with must be rejected .
For
0)( ** ft
1)( * ftx 0)(* Ct
0)( * ftxNot acceptable:
1ftNot possible with the
condition 1ftIf one obtains
with:
1)(* tu ttx )(*1ft
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1)( * ftx
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1)(* tu ttx )(*1ft
The unique possible solution is the extremum
That is both a normal and abnormal solution.
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