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Optimal Control
Prof. Daniela Iacoviello
Lecture 7
20/11/2017 Pagina 2
THESE SLIDES ARE NOT SUFFICIENT
FOR THE EXAM: YOU MUST STUDY ON THE BOOKS
Part of the slides has been taken from the References indicated below
Pagina 2 20/11/2017 Prof. 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
20/11/2017 Prof. 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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 4
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 5
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 6
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
)(
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 7
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.
*** ,, ftux
0*
0 fi ttC ,1*
pjRBuB
A
j
pTT
T
,...,2,1,1:***
**
**u
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 8
Moreover the associated Hamiltonian is a continuous function
with respect to t and it results:
Proof: The Hamiltonian associated to the problem is:
Applying the minimum principle the theorem is proved.
0*
* ft
H
BuAxtuxH TT 00 ,,,,
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 9
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)
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 10
Remark:
Strong controllability corresponds to the
controllability in
any instant ti,
in any time interval and
by any component of the control vector
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 11
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 12
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.
Moreover every component of the optimal control is
piecewise constant assuming only the extreme values
and the number of discontinuity instants is limited.
1
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 13
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 14
Pagina 15 20/11/2017 Prof. D.Iacoviello - Optimal Conrol
Deriving successively this expression and using
we obtain:
or:
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 16 20/11/2017 Prof. 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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 17
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
]'','[0)(det ttttG j
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 18
)(tuoj
ofi tt ,
0)()( joT b
)( jk
t
For the continuity of between and
there exists an instant, say , in which
Therefore also
Analogously:
That is in contraddiction with the strong controllability
hypothesis.
]'','[0)(det ttttG j
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 19
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 20
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.
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 21
Remark
This result holds also to the case in which the control is in
the space of measurable function defined on the space of
real number R, M(R).
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 22
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.
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 23
The following result holds:
Theorem: Let assume that the control functions belong to the
space M(R).
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
oft
ft
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 24
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 25
dttutBttxtttx
dttutBttxtttx
k
t
t
of
ii
of
of
k
k
t
t
kf
ii
kf
kf
k
o
f
i
k
f
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 of measurable function M , L2 and such
that
Therefore this sequence admits a subsequence, indicated still
with , weakly converging to a function
,....2,1,0)()()( ktx
kf
k 0)(lim )(
of
k
ktx
)(ku
ofi tt ,
)(ku
ofij tttpjtu ,,,...,2,1,1)(
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 26
)(ku Muo
.,...,2,1,1)()(
pjtukj
)(ku
This implies that for any measurable L2 function it results:
If we indicate by the evolution of the state corresponding
to the input from the initial condition we have:
h
o
f
i
o
f
i
t
t
oT
t
t
kT
kdttuthdttuth )()()()(lim )(
oxou
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 27
)()()(,,
)()(,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 ,,
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 28
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 ,
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 29
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.
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 30
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 31
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 32
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
)(
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 33
Theorem:
Let us consider the above time minimum problem and
assume that the control function belong to the space of
measurable function M(R);
if the system is controllable
there exists a neighbor Ω of the origin such that for any
there exists an optimal solution.
ix
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 34
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 35
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
,,)(
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 36
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 37
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 38
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.
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 39
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
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 40
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.
Question: how many commutation instants are possible in
the optimal time control problem?
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 41
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,)()(
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 42
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 km
0
nretpt
k
s
tt
rsris ,...,2,1,)()(
1
)(0
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 43
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 + m12+ …. + mk-1 =n-1
real solutions.
Therefore the control has at most n-1 roots.
o
ju
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 44
Remarks
1. If the eigenvalues of A are not all real the number
of switching instants is bounded but not uniformly.
2. In general it is not possible to establish an esplicit
relation between the control and the state
20/11/2017 Prof. D.Iacoviello - Optimal Control Pagina 45
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