parametric optimal control problems
DESCRIPTION
UNIVERSIDADE de AVEIRO Departamento de Matematica, 2005. PARAMETRIC OPTIMAL CONTROL PROBLEMS. Olga Kostyukova. Institute of Mathematics National Academy of Sciences of Belarus Surganov Str.11, Minsk, 220072 e’mail [email protected]. OUTLINE. Problem statement; - PowerPoint PPT PresentationTRANSCRIPT
• Problem statement;• Solution structure and defining elements;• Solution properties in a neighborhood of regular point;• Solution properties in a neighborhood of irregular point: • construction of new Lagrange vector;
• construction of new structure and defining elements;
• Generalizations.
OUTLINE
Family of parametric optimal control problems:
0
0
0
( ( ) ) ( ( ) ( ) ( ) ( ) ( ) ( )) min
( ) ( ) ( ) ( ) ( )OC( )
( ) ( ) ( ) ( ) ( ) [0 ]
(0) ( ) ( ( )
(1)
(2)
(3)
(4)
(
) 0, {1 }
( ) 1 5)
t
i
f x t x t D x t u t R u t dt
x t A x t B u t
d x t g u t t t T t
x x f x t
h h h
h h
i M m
u t t T
hh h h
h h
0, , ( ) ( ) 0, ( ) ( ) 0, ( ) ( ) ( )n r h h h h h h hx R u R D D R R A B x
( ) 0,..., ; ( ) [ ]ni h hf x i m t x R t hT h h
are given functions,
[ ]h hh is a parameter.
Problem statement
Optimal control and trajectory for problem ( )OC h
( ) ( ( ) ) ( ) ( ( ) )u u t t T xh h h x t t Th
The aims of the talk are
• to investigate dependence of the performance index and
( ) ( )h hu x on the parameter h;
• to describe rules for constructing solutions to ( )OC h
for all [ ]h h h
Terminal control problem OC(h)
( ), ( )hu xh
0 ( ( )) min
( ) ( ) ( ) (0) ( )OC( )
( ( )) 0 {1 }
( ) 1 [0
1
]
( )i
f x t
x t Ax t bu t x z
f x t i M … m
u t t T t
hh
[ ],n hx R u R h h is solution to the problem OC(h),
functions ( ) 0 , are convex.if x i M
Maximum Principle
my R
0 ( ) ( )() 3)(
f x f xA t y
x x
( )u h to be optimal in ОС (h) In order for admissible control
it is necessary and sufficient that a vector
exists such that the following conditions are fulfilled
0 ' ( ( )) (20 )y y f x ht
1( ( )) ( ) max ( ( ))
ut y x t bu th h t y x t bu th T
Here ( ), , , ,m nt y x t T y R x R is a solution to system
Denote by ( ) mY h R the set of all vectors y, satisfying (2), (3)
(4( ) [ ] ).Y hh h h h
• The set ( )Y h is not empty and is bounded for [ ]hh h
and consider the mapping
• The mapping (4) is upper semi-continuous.
Let ( ) ( ( ) ) ( ).ih hy i M hy Y Denote by
( ( ) ) ( ( ) ( ))h h h ht y t y x t b t T
the corresponding switching function.
{ ( ) 1,..., ( )} { ( ( ) ) 0}j h ht j p ht T t hy
1( ) ( ) 1,..., ( ) 1j jt t jh h hp
( ( ) ( ) )( ) { {1 ( )} 0}j h h hy
L p ht
jt
h
1 1( ) 1 if ( ) 0 ( ) 0 if ( ) 0l t lh h h t h
( ) ( )( ) 1 if ( ) ( ) 0 if ( )
+0 ±1,
hp p hh hl t t l t t
k( )= u(
h h
=h | )h
0 ( ) { ( ) ( ) 0}.a ih hM i M hy
( ) { ( ( | )) 0},a ihM i M f t hx
Zeroes of the switching function:
Active index sets:
Double zeroes:
Solution structure:
0( ) { ( ) ( ) ( ) ( ) ( ) ( ) ( )}aS p k M l lh h h h h h h hM L
Defining elements:
( ) ( ( ) 1,..., ( ) ( ))jh h hj p hQ t y
Regularity conditions for solution ( )u h (for parameter h)
0( ) ( ) ( ) ( ) ( ) 0h h h hl L hM l
Lemma 1. Property of regularity (or irregularity) for control ( )u h
does not depend on a choice of a vector ( ) ( )y Yh h
Suppose for a given 0 [ [h hh we know
• solution 0( )u h to problem 0( ),OC h
• a vector 0 0( ) ( )y Yh h
• corresponding structure 0( )S h and defining elements 0( ).Q h
The question is how to find
0( ) ( ) ( ) for ( )?h h hu Q hhS E
is a sufficiently small right-side neighborhood of 0( )E h
the point 0.h
Here
Solution Properties in a Neighborhood of
Regular Point
0( ) : { ( ) ( ) ( ) ( ) ( ) ( ) ( )}aS p k M lh h h h l Mh Lh h h
Solution structure does not change:
0 0 0 0 0 0 0 00{ ( ) ( ) ( ) ( ), ( ) ( ) ( )} : ( )ap k M l lh h M L Sh h h h h h
Defining elements
with initial conditions
0 0 0 0( 0) ( ) 1,..., ( 0) ( )j jt t j p yh h h hy
0 0 0ap k( ) ( ), M ( )ah hk hp M
are uniquely found from defining equations
( ) ( ( ), 1,... ( ))jQ t jh p hyh
ap( ( ), | , k M, ) 0Q h h
a( , | ) ,
( , 1,
p,k
..., ; ) ;
,M p m
p mj
Q R
Q t j
h
p y R
where
Optimal control ( )u h for ОС(h):
1( ) ( 1) [ ( ) ( )[
0,... ,
jj jh hu t k t t ht
j p
0 1( ) 0 ( )ph ht t t
Construction of solutions in neighborhood of irregular point
The set consists of more than one vector.0( )Y h
0 0 0 0 0 0( 0) ( ) ( 0) ( ) ( 0) ( )y y S S Qh h h Qh h h
The first Problem: How to find 0( 0) ?y h
The second Problem: How to find 0 0( 0), ( 0)?S Qh h
0h
0Costruction of vector ( 0)hy
Theorem 1. The vector 0( 0)y h is a solution to the problem
0 0 0min(0 ( )) ( 0) (SI)( )y x t Yh hz yh
The problem (SI) is linear semi-infinite programming problem.
The set 0( )Y h is not empty and is bounded
the problem (SI) has a solution.
Suppose that the problem (SI) has a unique solution y
0( 0)hy y
0 0New Lagrange vector ( 0) ( ) is foundh hy y y
00
0( ) ( ( ) )t t y h h t T
New switching function 0( )( ) t y ht t T Old switching function
0 0Construction of new ( 0) ).and ( 0S h hQ
A) What indices i M are in the new set of active
0( 0)?a hM
B) How many switching points 0( 0)p h will new
0( 0)hu optimal control
indices
have?
0( 0)?a hM Form the index sets
0{ ( ( )) 0},a iM i M f x ht
0 { 0} { 0}.a a i a a iM i M y M i M y
It is true that 0 0( ) ( 0) ( 0)a a a aM M M M Mh h ‚
0
0
0
( 0)
( 0)
a
a
a
M
i M
M M
h
h
‚
?
A): How to determine
\
\
0( 0)?p h
0
Let 1,..., be zeroes of new unperturbed switching function
) ( )(
jt j p
t y tt h T
0 0{1,2,..., }, { : ( 0 | ) ( 0 | )}.R j jhJ p J j J u t u t h
7, {2,4,5,7}Rp J
B): How to determine
0
1,..., ( ) are zeroes of perturbed switching functi( )
( ( ) )
on
, , 0.
jt
t
j p
t T hhh h
h
hy
h
h
0 0*
1,... are zeroes of unperturbed switching function
( ) , with ( 0)) ,(
jt
t h h
j p
t y t T y y
7, ( ) 8p p h
*For each , , a) or b) ?*j Rt j J \ J
Using known vector 0( 0) ,z h
and sets 0, , ,a a RM M J J
form quadratic programming problem (QP):
min( ) ( ) ( ) 2S
I s g s Dg s s Ds
00 0( ( )
( ) 00
aR
a
ij
MJ
if x tg Js s
h
Mj
x i
‚
0where ( ) ( ) ( 0) .js s j gJ s hF sz B
Theorem 2. Suppose that there exist finite derivatives
0 0( 0) ( 0)
1,... (, )jdt
h h
dyj
d
h hp
d
Then the problem (QP) has a solution which can be uniquely found using derivatives
0 0( )js s j J 0 0( ).i ai M
Then derivatives are uniquely calculated by 0 0, .s
( .)
( )
Suppose the problem (QP) has a unique optimal solution:
primal and dual
Let (QP) have unique optimal plans 0 0 0 0( ), ( ).j i as s j J i M
0 ?A): ai M , \ ?B): j Rt j J J
We had problems:
Solution of problem A):
0 00Index belongs to ( 0) if 0.a a ii M M h
0 00Index does not belong to ( 0) if 0.a a ihi M M
0
0
situation ) if 0,
situation ) if 0.
j
j
a s
b s
Solution of problem B):
0 a0 0
Consequently
( 0) { 0}: Ma a a iM M ih M
( ) )0 (0( p0) :hp J J
( ) (0)
where
{ 0} { 0 0 }R R j R j jJ J j J J s J j J J s t t ‚ ‚
( ) (0)Put { 1 } { },j j jt j … p t j J t j J
1, 1 1,j jt t j … p
1 10 0( 0 ) if 0 ( 0 ) ifk 0k hu u tht
0 0 0 0 a( 0) { ( 0)
New structur
p k( 0 M) ( 0) }
e
ah h h hS p k M
0(
New defining element
0) ( 1,..., )p
s
jhQ Q t j y
Theorem 3. Let h0 be an irregular point and the problem (QP) have a unique solution. 0 0( )Then for E h hh \
problems ОС(h) have regular solutions with constant structure
0 a( ) ( 0) p, ;M }: ,{ khShS defining elements Q(h) are uniquely found from
0( 0) ,with initial condition Q Qhs
p pa( , | ) , ( , 1,..., ; )p, k, p ;M m m
jwhere Q R Q t j Rh y
optimal control ( )u h is constructed by the rules
1( ) ( 1) [ ( ) ( )[ 0,. , p.k . ,jj ju t th hth t j
p0 1( ) 0 ( )t t th h
ap, k,( ( ), | ) 0Mdefining equations h hQ
On the base of these results the following problems are investigated and solved
differentiability of performance index and solutions to problems
( ), [ ] with respect to parameter ;h hhO h hC
path-following (continuation) methods for constructing solutions to a family of optimal control problems;
fast algorithms for corrections of solutions to perturbed problems
0 0
0
( ), [ ] with respect to small variations of a
parameter ;
OC h hhh h h
h
construction of feedback control.
• Kostyukova O.I. Properties of solutions to a parametric linear-quadratic optimal control problem in neighborhood of an irregular point. // Comp. Math. and Math. Physics, Vol. 43, No 9, 1310-1319 (2003).• Kostyukova O.I. Parametric optimal control problems with a variable index. Comp. Math. and Math. Physics, Vol. 43, No 1, 24-39 (2003).• Kostyukova, Olga; Kostina, Ekaterina. Analysis of properties of the solutions to parametric time-optimal problems. // Comput. Optim. Appl. 26, No.3, 285-326 (2003).• Kostyukova, O.I. A parametric convex optimal control problem for a linear system. // J. Appl. Math. Mech. 66, No.2, 187-199 (2002).• Kostyukova, O.I. An algorithm for solving optimal control problems. // Comput. Math. and Math. Phys. 39, No.4, 545-559 (1999).• Kostyukova, O.I. Investigation of solutions of a family of linear optimal control problems depending on a parameter. // Differ. Equations 34, No.2, 200-207 (1998).
Results of these investigations are presented in the papers: