Presentazione di PowerPointiacoviel/Materiale/... · Theorem: Necessary conditions for to be an optimal solution are that there exist a constant and an n-dimensional function not

Optimal Control

Prof. Daniela Iacoviello

Lecture 7

19/11/2019 Pagina 2

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Pagina 219/11/2019Prof. D.Iacoviello - Optimal Control

SINGULAR SOLUTIONS

Definition:

Let be an optimal solution of the above problem,

and the corresponding multipliers.

The solution is singular if there exists a subinterval

in which the Hamiltonian

is independent from at least one component of in

( )of

oo tux ,,

oo 0

( ) ''','',' tttt ( )tttxH ooo ),(,,),( 0

( )'',' tt

19/11/2019Prof. D.Iacoviello - Optimal Control Pagina 3

SINGULAR SOLUTIONS

Theorem:

Assume the Hamiltonian of the form

Let be an extremum and the corresponding

multipliers such that is dependent on any

component of in any subinterval of

A necessary condition for to be a singular extremum is that

there exists a subinterval

such that:

( )*** ,, ftux**

0

),,,,(),,,(),,,(),,,,( 002010 tuxNtxHtxHtuxH +=

),,,,( *0

** txN ],[ fi tt

'''],,[]'','[ tttttt fi

]'','[,0),,,( 02 ttttxH =

( )*** ,, ftux


The linear minimum time optimal control

Let us consider the problem of optimal control of a linear system

with fixed initial and final state,

constraints on the control variables and

with cost index equal to the length of the time interval


Problem: Consider the linear dynamical system:

With and the constraint

Matrices A and B have entries with elements of class

Cn-2 and Cn-1 respectively, at least of C1 class .

The initial instant ti is fixed and also:

)()()()()( tutBtxtAtx +=

pn RtuRtx )(,)(

Rtpjtu j = ,,...,2,1,1)(

0)(,)( == fi

i txxtx


The aim is to determine the final instant

the control

and the state

that minimize the cost index:

)(RCu oo

Rtof

)(1 RCxo

if

t

t

f ttdttJ

f

i

−== )(


Theorem: Necessary conditions for to be an optimal

solution are that there exist a constant

and an n-dimensional function

not simultaneously null and such that:

Possible discontinuities in can appear only in

the points in which has a discontinuity.

Moreover the associated Hamiltonian is a continuous function

with respect to t and it results:

( )*** ,, ftux

0*

0

fi ttC ,1*

pjRBuB

A

j

pTT

T

,...,2,1,1:***

**

=

−=

**u


0*

* =ft

H

Proof: The Hamiltonian associated to the problem is:

Applying the minimum principle the theorem is proved.

( ) BuAxtuxH TT ++= 00 ,,,,


STRONG CONTROLLABILITY

Strong controllability corresponds to the

controllability in

any instant ti,

in any time interval and

by any component of the control vector


STRONG CONTROLLABILITY

Let us indicate with the j-th column of the matrix B .

The strong controllability is guaranteed by the condition:

where :

)(tb j

( )

i

n

jjjj

ttpj

tbtbtbtG

=

=

,,...,2,1

0)()()(det)(det )()2()1(

nktbtAtbtb

tbtb

k

j

k

j

k

j

jj

,...,3,2)()()()(

)()(

)1()1()(

)1(

=−=

=

−−

Generic column of matrix B(t)


Remark:

In the NON-steady state case the above condition is sufficient for

strong controllability.

In the steady case it is necessary and sufficient and may be written

in the usual way:

( ) pjbAAbb jn

jj ,...,2,10det 1 =−


Characterization of the optimal solution


Characterization of the optimal solution

Theorem: Let us consider the minimal time optimal problem .

If the strong controllability condition is satisfied and if an

optimal solution exists

it is non singular.

Every component of the optimal control is piecewise constant assuming

only the extreme values

and the number of discontinuity instants is limited.

1


Proof:

the non singularity of the solution is proved by contradiction

arguments.

If the solution were singular, from the necessary condition of the previous

theorem:

there would exist a value

and a subinterval

such that:

pjR

BuB

j

p

TT

,...,2,1,1:

***

=

pj ,...,2,1

],[]'','[ ofi tttt

]'','[,0)()( ttttbt j

oT =


Pagina 1619/11/2019Prof. D.Iacoviello - Optimal Conrol

Deriving successively this expression and using

we obtain:

On the other hand must be different from zero in

every instant of the interval , otherwise, from the

necessary condition

it should be null in the whole interval and in particular

]'','[,0)()( ttttGt j

oT =

)(to

** TA−=

0)( =fo t

** TA−=

( )1,...,2,1],'','[,0)()(

)()( )1(−===

+nittttbt

dt

tbtd ij

oT

i

joTi

Pagina 1719/11/2019Prof. D.Iacoviello - Optimal Conrol

From the necessary condition:

it would result that also which is absurd.

Therefore, from the condition:

it results that

which contradicts the hypothesis of strong controllability.

the solution is non singular.

0** =ft

H

00 =o

]'','[,0)()( ttttGt j

oT =

]'','[0)(det ttttG j =

From the non singularity of the optimal solution it results that the

quantity

can be null only on isolated points .

Therefore the necessary condition

implies

)()( tbt j

oT

pjRBuB j

pTT ,...,2,1,1:*** =

],[,...,2,1

,)()()(

ofi

joTo

j

tttpj

tbtsigntu

=

−=


To demonstrate that the number of discontinuity instants is

finite, we can proceed by contradiction argument that

would easily imply

which contradicts the hypothesis of strong controllability.

In fact, let us assume that at least for one component of the

control it is not true.

Then an accumulation point of instants exists and,

for continuity argument



)(tuoj

ofi tt ,

0)()( = joT b

)( jk

t

For the continuity of between and

there exists an instant, say , in which

Therefore, for continuity, also

Analogously:

That is in contraddiction with the strong controllability hypothesis.



0)()( =tbt joT

)( jk

t)(1

jk

t+

)( jk

t ( )0

)()(=

dt

tbtd joT

( )0

)()(=

dt

tbtd joT

( ))1(,...,1,0,0

)()(−== ni

dt

tbtd

i

joTi

A control function that assumes only the limit values

is called bang-bang control

and

the instants of discontinuity are called

commutation instants


UNIQUENESS RESULT


Theorem (UNIQUENESS):

If the hypothesis of strong controllability is satisfied, if an

optimal solution exists it is unique.

Proof: the theorem is proved by contradiction arguments.


Definition

Let be the space of piecewise constant function defined as

follows:

Let be a function such that there exists a sequence

such that , with the exception of isolated points, one has:

is a measurable function.

The space of measurable function is linear.


( , )i fS t t

1

, 1,2,...,

( )0

j j

m

j

j

c t I j m

s tt I

=

=

=

, ,j i f j kI t t I I =

( )z t ( ) ( , )ki fs S t t

,i ft t t

( )lim ( ) ( )k

ks t z t

→=

( )z t

( , )i fM t t

Remark

These results (characterization of the solution and uniqueness)

hold also to the case in which the control is in

the space of measurable function defined on the space of

real number R.


Existence of the optimal solution

Theorem: if the condition of strong controllability is satisfied

and an admissible solution exists, then there exists a

unique optimal solution nonsingular and the control is

bang-bang.

Proof: the existence theorem is proved on the basis of the

following results.


The following result holds:

Theorem: Let assume that the control functions belong to the space of

measurable functions.

If an admissible solution exists then the optimal solution exists.

Proof: let us consider the non trivial case in which the number

of admissible solution is infinite and let us indicate with the

minimum extremum of the instants corresponding to the

admissible solutions

The problem is: is there a minimum?

oft

ft


infof ft t=

It is possible to built a sequence of admissible solutions

such that

Let us indicate with the transition matrix of the linear

system. It results:

( ) of

kf

kf

kk tttux )()()()( ,,,

of

kf

ktt =

→

)(lim

( ) ,t

( ) ( )

( ) ( ) ( ) 0)()(,lim)()(,,lim

,,lim)()(lim

)()()()(

)()()()(

)(

=+−+

−=−

→→

→→

dttutBttdttutBtttt

xtttttxtx

k

t

t

kf

k

k

t

t

of

kf

k

ii

ofi

kf

k

of

kkf

k

kk

f

o

f

o

f

i


( ) ( )

( ) ( ) dttutBttxtttx

dttutBttxtttx

k

t

t

of

ii

of

of

k

k

t

t

kf

ii

kf

kf

k

of

i

kf

i

)()(,,)(

)()(,,)(

)()(

)()()()()(

)(

+=

+=

0

It is possible to built a sequence of admissible solutions

such that

Let us indicate with the transition matrix of the linear

system. It results:

( ) of

kf

kf

kk tttux )()()()( ,,,

of

kf

ktt =

→

)(lim

( ) ,t

( ) ( )

( ) ( ) ( ) 0)()(,lim)()(,,lim

,,lim)()(lim

)()()()(

)()()()(

)(

=+−+

−=−

→→

→→

dttutBttdttutBtttt

xtttttxtx

k

t

t

kf

k

k

t

t

of

kf

k

ii

ofi

kf

k

of

kkf

k

kk

f

o

f

o

f

i


( ) ( )

( ) ( ) dttutBttxtttx

dttutBttxtttx

k

t

t

of

ii

of

of

k

k

t

t

kf

ii

kf

kf

k

of

i

kf

i

)()(,,)(

)()(,,)(

)()(

)()()()()(

)(

+=

+=

0

Since it results:

Let us indicate by the function truncated on the interval

This function is limited and L2 , since

The sequence of functions

belongs to the space M of measurable function, L2 and such that

Therefore this sequence admits a subsequence, indicated still with ,

weakly converging to a function

,....2,1,0)()()( == ktx

kf

k0)(lim )( =

→

of

k

ktx

)(ku ofi tt ,

)(ku

ofij tttpjtu ,,,...,2,1,1)( =


)(ku

Muo

.,...,2,1,1)()(

pjtukj =

)(ku

This implies that for any measurable function in L2 it results:

Weak convergence

If we indicate by the evolution of the state corresponding

to the input from the initial condition we have:

Weak convergence

h

=→

o

f

i

o

f

i

t

t

oT

t

t

kT

kdttuthdttuth )()()()(lim )(

oxou


( ) ( )

( ) ( ) )()()(,,

)()(,lim,)(lim )()(

of

o

t

t

oof

ii

of

t

t

kof

k

ii

of

of

k

k

txdttutBttxtt

dttutBttxtttx

o

f

i

o

f

i

=+=

+=

→→

Since the elements of

are measurable and L2

( ) )(, tBttof

Therefore, taking into account the expression

we deduce :

An admissible measurable control L2 exists able to transfer

the initial state to the origin.

The optimal solution is

0)(lim )( =→

of

k

ktx

0)( =of

o tx

( )of

oo tux ,,


The results about the characterization of the control may be extended

to the case in which the control belong to the space of measurable

functions:

Therefore it results in:

If the strong controllability condition is satisfied and

if an optimal control in the space of measurable function exists

then

this solution is unique,

non singular and

the control is bang bang type

Remark

The existence of the optimal solution is guaranteed only for the

couples for which the admissible solution exists( )ii xt ,

19/11/2019 Pagina 33

Existence of the optimal solution

Theorem: if the condition of strong controllability is satisfied and

an admissible solution exists

with the control in the space of measurable functions,

then

there exists a unique optimal solution

nonsingular and

the control is bang-bang.


Minimum time problem for steady state system

Let us consider the minimum time problem in case of

steady state system.

In this case it is possible to deduce results about the

number of commutation points


Problem: Consider the linear dynamical system:

With and the constraint

The initial instant ti is fixed and also:

)()()( tButAxtx +=

pn RtuRtx )(,)(

Rtpjtu j = ,,...,2,1,1)(

0)(,)( == fi

i txxtx


The aim is to determine the final instant

the control

and the state

that minimize the cost index:

)(RCu oo

Rtof

)(1 RCxo

if

t

t

f ttdttJ

f

i

−== )(


Theorem:

Let us consider the above time minimum problem and

assume that the

control function belong to the space of measurable function;

if the system is controllable

there exists a neighbor Ω of the origin such that for any

there exists an optimal solution.ix


Proof:

We will show that it is possible to determine a neighbor

Ω of the origin such that for any there exists an admissible

solution.

On the basis of the previous results this implies the existence of the optimal

solution.

Let us fix any tf>0. The existence of a control able to transfer the initial

state to the origin at the instant tf corresponds to the possibility

of solving :

ix

ix


with respect to the control function u.

Let assume for the control the expression:

and substitute it in the preceeding equation:

( ) ( )0)( =+

−−f

i

fif

t

t

tAittAdBuexe

=−

−−−

f

i

Ti

t

t

ATAiAtdeBBexe Gramian

matrix

pfi

AT RtteBuT

= − ,,)(


From the controllability hypothesis, the Gramian matrix is not singular,

therefore:

a measurable admissible control exists if the initial state is sufficiently

near the origin.

fi

iAt

t

t

ATAAT

tt

xedeBBeeBu i

f

i

TT

−=

−

−

−−−

,)(

1


=−

−−−

f

i

Ti

t

t

ATAiAtdeBBexe

pfi

AT RtteBuT

= − ,,)(

Remark

From results obtained in the non steady state case it results that

an optimal non singular solution exists

and the control is bang-bang type.

Prof.Daniela Iacoviello- Optimal Control


Theorem: Let us consider the above time minimum problem in the

steady state case and assume that the control function belongs to

the space of measurable function M(R).

If the system is controllable and

the eigenvalues of matrix A have negative real part

there exists an optimal solution whatever the initial state is.


Proof:

Whatever the initial state is chosen, an admissible solution can be defined

as follows:

• Let Ω be a neighbor of the origin constituted by the original state

for which an admissible solution exists;

• Let Ω’ be a closed subset in Ω

• From the asymptotic stability of the system it is possible to reach Ω’ in a

finite time with null input, from any initial state


Ω

Proof:


as follows:

• Let Ω be a neighbor of the origin constituted by the original state for

which an admissible solution exists;


• From the asymptotic stability of the system it is possible to reach Ω’ in a

finite time with null input, from any initial state


Ω

Ω’

Proof:


as follows:

• Let Ω be a neighbor of the origin constituted by the original state for

which an admissible solution exists;


• From the asymptotic stability of the system it is possible to reach Ω’ in

a finite time with null input, from any initial state


Ω

Ω’x’

Remark

From the results obtained in the non steady state case

it results that an optimal non singular solution exists and the control is

bang-bang type.

Question: how many commutation instants are possible in

the optimal time control problem?


Theorem: If the controllability condition is satisfied and

if all the eigenvalues of the matrix A are real non positive,

then

the number of commutation instants for any components

of control is less or equal than n-1, whatever the inititial state is.

Proof: from the previous results, we have:

where the costate in the steady state case is:

ofij

oToj tttpjbtsigntu ,,,...,2,1,)()( =−=


0

i

ttA iT

et )(0 )(−−

=

Denoting by and the eigenvalues of A and their molteplicity,

the r-th component of can be expressed as follows:

where prs is a polynomial function of degree less than ms.

Substituting this expression in the bang bang control we obtain:

k km0

nretpt

k

s

tt

rsris ,...,2,1,)()(

1

)(0 ===

−−


ofi

k

s

ttjs

k

s

ttn

r

rsjroj

tttpj

etPsign

etpbsigntu

is

is

,;,...,2,1

,)(

)()(

1

)(

1

)(

1

=

−=

−=

=

−−

=

−−

=

Polynomial function

of degree less than ms

The argument of the sign function has at most

m1 + m2+ …. + mk-1 =n-1

real solutions.

Therefore the control has at most n-1 roots.o

ju


Remarks

1. If the eigenvalues of A are not all real the number of

switching instants is bounded but not uniformly.

1. In general it is not possible to establish an esplicit relation

between the control and the state


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Presentazione di PowerPointiacoviel/Materiale/... · Theorem: Necessary conditions for to be an optimal solution are that there exist a constant and an n-dimensional function not