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Optimal Terminal State Control with Disturbance Rejection for Systems with Time Delay
Copyright ~ IFAC New Trends in De&igD of Control Systems. Smolenice. Slovak Republic. 1994
OPTIMAL TERMINAL STATE CONTROL WITH DISTURBANCE REJECTION FOR SYSTEMS WITH TIME DELAY F. AMATO and A. pmONTI Uaiveraiti degli Studi di Napoli Federico 11, Dipartimellto di IlIiormatica e S;'&eutica, via Claudio 21, 80125 Napoli, Italy
Abstract. In this paper we consider the optimal termiraal .tate control problem for a linear time-varying system with time delay and subject to unknown, square integrable disturbances. Using a differential games approach, we provide a necessary and sufficient condition for the existence of an optimal control law which minimizes the weighted terminal state attenuating the effects of the admissible disturbances under a prespecified level. Key Words. Linear systems; time-varying systems; terminal state control; delays; differential games; disturbance rejection.
1
Introduction
Our main result (Section 3) is a necessary and sufficient condition for the existence of the op.timal control law; this condition requires that a certain Riccati differential equation admits a pO&itive aemidefinite solution. In particular we show that our original problem has a solution if and only if a certain differential game with time delay has; then, using a procedure suggested in Hanalay (1968), we transform the original game in a game without delay, which can be solved via standard techniques (see Basar, 1989). The resulting control law, if existing, is given by the sum of two contributes: a traditional memory less plus a distributed delay state feedback.
In the last years there has been an increasing interest in the study of linear time-varying sy&tems. The effort has been devoted to extend the main techniques developed for linear timeinvariant systems to the time-varying setting; see, for instance, Tadmor (1990) and Ravi et al (1991) in the 'Hoc context, Hinrichsen et al (1989) and Amato et al (1994) for what concerna the theory related to the quadratic stability and stabilization issues. While the above techniques have been presented only recently, the solution of the standard linear optimal control problem for time-varying systems dates back to the Sixties (see Athans and Falb, 1966). In this framework Alekal et al (1971) dealt with the particular case in which there is a de/a, term in the state equation. In this note, starting from the theory developed in the differential game setting in Hanalay (1968), we consider the more general situation in which the system to be controlled contains a delay element and is also subject to an unknoWD, square integrable disturbance; we focus on the problem of finding a linear state feedback control law which minimizes the weighted terminal state attenuating, at the same time, the effects of the admissible disturbances under a prespecified level.
Notation Let 0:=[0, T]; we denote by £2(0) the set of the real vector-valued functions which are square integrable on [0, T] and by 1I'1I2,R the usual norm in £2(0) weighted by the symmetric positive definite time-varying matrix R(t), i.e.
writing simply lIull2 when R is the identity matrix. Given a vector z, IIzllQ denotes the usual
29
Euclidean norm weighted by the symmetric itive definite matrix Q.
2
3
p0s-
The next lemma connects Problem 2 with the differential game theory. Lemma 1 IIGII < 'Y if and only if for .ome 6 > ara. for .11 w e £2(0) - to}
°
Problem Statement
Consider the linear time-varying system with time delay in the form %(t) = A(t)z(t) + F(t)z(t - r) +B(t)u(t) + G(t)w(t) z(.) = 0, t e [O,Tj, • e [-r, 0]
Main Result
lI(z(T), U)II~I>R - 'Y2I1wll~ Proof. If IIGII that
(2a) (2b)
sup [ .. ec2-{0}
where z(t) e JRn is the state, u(t) e JRm is the control input and wet) E JR' is the disturbance, which is assumed to be a member of £2(0). For w = it makes sense to consider the following terminal.tate linear optimal control problem (see Alekal et ai, 1971 and the bibliography therein). Problem 1 Find the linear state feedback control K : z - u, £2(0) _ £2(0), which minimizes the cost functional
(8)
°
< 'Y then there exists c > such
lI(z(T), U)IIQJ'~l2 IIwlb
From this follows that for all w
°
J(u):=lIz(T)II~J + lIull~.R
< -62I1wll~
2
2
= 'Y - c
e £2(0) - to}
lI(z(T), U)II~J.R - 'Y2I1wll~ $ -c2I1wll~
°
(3)
Vw
e £2(0) - to} (11)
From this immediately follows that IIGII
< 'Y.
min
max J(u, w)
ueC2(o) .. eC2(O)
QJ and R(t) being positive definite matrices.
X
£2(0)(5)
lI(z(T), u)IIQI>R IIwlb
J(u, w):= (lI(z(T), U)II~I>R - 'Y2l1wlln
Lemma 2 Coruider the differential game con-
d~
.;'ting of the .tate e9"ation %(t)
(6)
L(zo, u, w) = lI(z(T), U)II~J.R - 'Y2I1wll~ . Thera a) there em" a .add/e poirat fudback .ol"tion if .nd oral, if the Riccati differential e9"ation
Problem 2 Given a positive number 'Y find, if existing, a state feedback linear control K : z u, £2(0) _ £2(0) such that
< 'Y
= B(t)u(t) + G(t)w(t) z(O) = zo
and co.t "'ndioraal
The problem we consider in this paper is stated precisely in Problem 2.
IIGII
(13)
Before stating the main result we need the following lemma.
Consequently the norm of the operator G is fined as sup
(12)
where
Observe that, given an, linear state feedback control law K : z - u, £2(0) _ £2(0), system (2) uniquely defines the linear operator
.. eC2(O)-{0}
•
By virtue of Lemma 1 we focus on the study of the differential game with time de/a,
(4)
IIGII:=
(10)
Equation (10) implies (8) with 6 E (0, c). Conversely, suppose that, for some 6 > and for all w e £2(0) - to}, (8) holds. Dividing by IIwll~ (w # 0) we have
In this note we shall consider the more general situation in which the extra term depending on the disturbance w is present. In order to state precisely the problem we deal with, let us consider the Cartesian product set JRn X £2(0) equipped with the norm
G: w E £2(0) - (z(T), u) = (z(T), K(z» E JRn
(9)
- Pet)
(7)
= pet)
(:2
G(t)GT (t)
-B(t)R- 1 (t)B T (t») pet)
In the next section we shall solve Problem 2 using a differential game approach.
P(T)
30
= QJ
(14)
admit. a po.itive .emidefinite .ol."ion. In tAil cue tlae optimal .tndegie. are given 6,
wlere
UJ(t) = _R-l(t)BT (t)P(t)z(t) (ISa) I
-T
wJ(t) = 2G (t)P(t)z(t);
R-1(t)BT (t)XT (T, t)P(t)X(T, t) z(t) - R-1(t)BT (t)XT (T, t)P(t)
Ke(t) = .cSF
'+7'
1
x,
(ISb)
"'(
X(T,a)F(a)z(a-r)da
and X(T, t) .atilfie. 6) defined
(16a) (16b)
X(T, t) = -X(T, t)A(t) - X(T, t + r)F(t + r) X(T,T) = I, X(T,t) = 0 . t > T (23)
z'[ P(O)zo + lIuoll~ R -"'(2I1woll~
Proof. We use a procedure suggested in Hanalay (1968) to transform the original differential game with delay in a differential game without delay. Let X(T, t) the solution of (23) and
= u-uJ Wo = w-wJ, Uo
we Aave
L(zo, u, w) =
.
(17)
Proof. a) See Basar (1989). b) First note that
(18) Completing the squares in (18) one obtains
Moreover
d
z(T) = z(T)
dt zT (t)P(t)z(t) = U'[ (t)R(t)uo(t) - "'(2w'[ (t)wo(t) +r2wT (t)w(t) - uT (t)R(t)u(t) (19)
=0 (27) J(u, w) = lI(z(T), u)II~/.R - "'(2I1wll~ (28) Note that, in the new variable z, (12) is a differential game without delay. From statement a) in Lemma 2 follows that the game (12) has a feedback saddle point solution if and only if the Riccati differential equation (14) admits a positive aemidefinite solution; substituting in (14) the expressions for B and G given in (2S), equation (21) is readily obtained. Now the optimal strategy (ISa), in terms of z, can be written in the form (22); we will show that, under this control law, IIGII < "'( . Indeed from statement b) in Lemma 2, with zo = 0 and Uo = 0, we have that
"'('wT (t)w(t)
= u'[ (t)R(t)uo(t) - "'(2w'[ (t)wo(t) ~20) Integrating both sides, part b) of the lemma fol~s. • Now we are ready to state the main result of the paper. Theorem 1 Pro6/em f iI .olvab/e if and only if tlae following Riccati differential equation
J(u, w) = -"'(2I1woll~ where
- P(t) = P(t)( ~X(T, t)G(t)GT(t)X T (T, t)
I
(29) (30)
"'(
-X(T, t)B(t)R-l(t)BT (t)X T (T, t)}P(t)
Under the control law (22), w and Wo are the input and the output of the system i(t) = -B(t)R- 1(t)BT (t)P(t)z(t)
(21)
admit6 a positive .emidefinite .olution P. In tlail cue a control law .atil/ying (7) lau tlae .tMlet.re
u(t) = -Ke(t)z(t) - .cSF[Z](t)
-T
.
Wo = w - 2G (t)P(t)z(t)
"'(
P(T) = QJ
(26)
z(O)
From (19) follows that
+ uT (t)R(t)u(t) -
(24)
i(t) = X(T, t)z(t).+ X(T, t)i(t) +X(T, t + r)F(t + r)z(t) -X(T, t)F(t)z(t - r) = X(T, t)B(t)u(t) + X(T, t)G(t)w(t) := B(t)u(t) + G(t)w(t) (2S)
d
d
X(T,a)F(a)z(a- r)da.
Differentiating (24) we obtain
dt zT (t)P(t)z(t)=uj (t)R(t)uJ(t) - "'(2wj (t)wJ(t) -2uT (t)R(t)uJ(t) + 2"'(2wT (t)wJ(t)
dt zT (t)P(t)z(t)
'+7'
1
z(t) = X(T,t)z(t)+,
+G(t)w(t) Wo
(22)
31
=
(31a)
_~GT(t)P(t)z(t) + w(t) . (3Ib) r
Athans M. and P. L. Falb (1966). OptimGI COAtrol, AA IAtrodaction to the T"eorr GAd It. AppliutioA&, McGraw-Hill, New York.
This system is invertible and the inverse system is
i(t)
= (-B(t)R-1(t)B T (t)P(t) +
B~
T . and G. J. Olsder (1989). D,nGmic NOAcoopt!ndive GGme Theorr, Academic Press, New York.
~G(t)GT(t)P(t»Z(t) + G(t)WO(t)
"'I
(32a) (32b)
Hanalay A. (1968). Differential Games with Delay. SIAM J. COAtrol, 6, 579-593.
Since system (32) cannot have finite escape time we can conclude that there exists IJ > 0 such that for all w E £2(0)
Hinrichsen D., A. Hilchmann and A. J. Pritchard (1989). Robustness of stability of linear systems. J. Differ. Eqra., 6, 219-250.
IIwll~ $ 1J211woll~ .
Ravi R., K. M. Nagpal and P. P. Khargonekar (1991). 1{,oo control of linear time-varying systems: a state-space approach. SIAM J. COAtr. Opt., 29, 1394-1413.
w =
1
-T
,G (t)P(t)z(t) + wo(t) .
"'I
(33)
Hence
J(u,w)
$
-
(;r IIwll~
< -6I1wll~,
Tadmor G. (1990). Worst-case design in time domain: the maximum principle and the standard 1{,oo problem. MCSS, 3, 301-324.
(34)
:I
for all 6 E (0,;;') and for all w E £2(0). - The proof follows from Lemma 1. • It is important to note that the optimal control law is given by the sum of two contributes: a traditional memoryless plus a distributed delay state feedback.
4
Conclusions
In this paper, given a linear time-varying system with delay and subject to square integrable disturbances, we have considered the problem of finding a linear state feedback control law which minimizes the weighted terminal state attenuating the effects of the disturbances under a prespecified level. The problem is solvable if and only if a certain Riccati differential equation has a positive semidefinite solution. Future research will be devoted to solve the more general optimal control problem over the whole interval [0, TJ.
5
References
Alekal Y., P. Brunovsky and D. H. Chyung, E. B. Lee (1971). The quadratic problem for systems with time delays. IEEE Trcuu. Aat. Cordr., AC-16, 673-687. Amato F., A. Pironti and S. Scala (1994) . Quadratic stabilization and disturbance attenuation for linear time-varying systems. Procuding3 0/ the IEEE Worhhop OA Ro6..t COAtrol viG VGria6/e Stnacture GAd LYGpunov Technique., Benevento.
32
OPTIMAL TERMINAL STATE CONTROL WITH DISTURBANCE REJECTION FOR SYSTEMS WITH TIME DELAY F. AMATO and A. pmONTI Uaiveraiti degli Studi di Napoli Federico 11, Dipartimellto di IlIiormatica e S;'&eutica, via Claudio 21, 80125 Napoli, Italy
Abstract. In this paper we consider the optimal termiraal .tate control problem for a linear time-varying system with time delay and subject to unknown, square integrable disturbances. Using a differential games approach, we provide a necessary and sufficient condition for the existence of an optimal control law which minimizes the weighted terminal state attenuating the effects of the admissible disturbances under a prespecified level. Key Words. Linear systems; time-varying systems; terminal state control; delays; differential games; disturbance rejection.
1
Introduction
Our main result (Section 3) is a necessary and sufficient condition for the existence of the op.timal control law; this condition requires that a certain Riccati differential equation admits a pO&itive aemidefinite solution. In particular we show that our original problem has a solution if and only if a certain differential game with time delay has; then, using a procedure suggested in Hanalay (1968), we transform the original game in a game without delay, which can be solved via standard techniques (see Basar, 1989). The resulting control law, if existing, is given by the sum of two contributes: a traditional memory less plus a distributed delay state feedback.
In the last years there has been an increasing interest in the study of linear time-varying sy&tems. The effort has been devoted to extend the main techniques developed for linear timeinvariant systems to the time-varying setting; see, for instance, Tadmor (1990) and Ravi et al (1991) in the 'Hoc context, Hinrichsen et al (1989) and Amato et al (1994) for what concerna the theory related to the quadratic stability and stabilization issues. While the above techniques have been presented only recently, the solution of the standard linear optimal control problem for time-varying systems dates back to the Sixties (see Athans and Falb, 1966). In this framework Alekal et al (1971) dealt with the particular case in which there is a de/a, term in the state equation. In this note, starting from the theory developed in the differential game setting in Hanalay (1968), we consider the more general situation in which the system to be controlled contains a delay element and is also subject to an unknoWD, square integrable disturbance; we focus on the problem of finding a linear state feedback control law which minimizes the weighted terminal state attenuating, at the same time, the effects of the admissible disturbances under a prespecified level.
Notation Let 0:=[0, T]; we denote by £2(0) the set of the real vector-valued functions which are square integrable on [0, T] and by 1I'1I2,R the usual norm in £2(0) weighted by the symmetric positive definite time-varying matrix R(t), i.e.
writing simply lIull2 when R is the identity matrix. Given a vector z, IIzllQ denotes the usual
29
Euclidean norm weighted by the symmetric itive definite matrix Q.
2
3
p0s-
The next lemma connects Problem 2 with the differential game theory. Lemma 1 IIGII < 'Y if and only if for .ome 6 > ara. for .11 w e £2(0) - to}
°
Problem Statement
Consider the linear time-varying system with time delay in the form %(t) = A(t)z(t) + F(t)z(t - r) +B(t)u(t) + G(t)w(t) z(.) = 0, t e [O,Tj, • e [-r, 0]
Main Result
lI(z(T), U)II~I>R - 'Y2I1wll~ Proof. If IIGII that
(2a) (2b)
sup [ .. ec2-{0}
where z(t) e JRn is the state, u(t) e JRm is the control input and wet) E JR' is the disturbance, which is assumed to be a member of £2(0). For w = it makes sense to consider the following terminal.tate linear optimal control problem (see Alekal et ai, 1971 and the bibliography therein). Problem 1 Find the linear state feedback control K : z - u, £2(0) _ £2(0), which minimizes the cost functional
(8)
°
< 'Y then there exists c > such
lI(z(T), U)IIQJ'~l2 IIwlb
From this follows that for all w
°
J(u):=lIz(T)II~J + lIull~.R
< -62I1wll~
2
2
= 'Y - c
e £2(0) - to}
lI(z(T), U)II~J.R - 'Y2I1wll~ $ -c2I1wll~
°
(3)
Vw
e £2(0) - to} (11)
From this immediately follows that IIGII
< 'Y.
min
max J(u, w)
ueC2(o) .. eC2(O)
QJ and R(t) being positive definite matrices.
X
£2(0)(5)
lI(z(T), u)IIQI>R IIwlb
J(u, w):= (lI(z(T), U)II~I>R - 'Y2l1wlln
Lemma 2 Coruider the differential game con-
d~
.;'ting of the .tate e9"ation %(t)
(6)
L(zo, u, w) = lI(z(T), U)II~J.R - 'Y2I1wll~ . Thera a) there em" a .add/e poirat fudback .ol"tion if .nd oral, if the Riccati differential e9"ation
Problem 2 Given a positive number 'Y find, if existing, a state feedback linear control K : z u, £2(0) _ £2(0) such that
< 'Y
= B(t)u(t) + G(t)w(t) z(O) = zo
and co.t "'ndioraal
The problem we consider in this paper is stated precisely in Problem 2.
IIGII
(13)
Before stating the main result we need the following lemma.
Consequently the norm of the operator G is fined as sup
(12)
where
Observe that, given an, linear state feedback control law K : z - u, £2(0) _ £2(0), system (2) uniquely defines the linear operator
.. eC2(O)-{0}
•
By virtue of Lemma 1 we focus on the study of the differential game with time de/a,
(4)
IIGII:=
(10)
Equation (10) implies (8) with 6 E (0, c). Conversely, suppose that, for some 6 > and for all w e £2(0) - to}, (8) holds. Dividing by IIwll~ (w # 0) we have
In this note we shall consider the more general situation in which the extra term depending on the disturbance w is present. In order to state precisely the problem we deal with, let us consider the Cartesian product set JRn X £2(0) equipped with the norm
G: w E £2(0) - (z(T), u) = (z(T), K(z» E JRn
(9)
- Pet)
(7)
= pet)
(:2
G(t)GT (t)
-B(t)R- 1 (t)B T (t») pet)
In the next section we shall solve Problem 2 using a differential game approach.
P(T)
30
= QJ
(14)
admit. a po.itive .emidefinite .ol."ion. In tAil cue tlae optimal .tndegie. are given 6,
wlere
UJ(t) = _R-l(t)BT (t)P(t)z(t) (ISa) I
-T
wJ(t) = 2G (t)P(t)z(t);
R-1(t)BT (t)XT (T, t)P(t)X(T, t) z(t) - R-1(t)BT (t)XT (T, t)P(t)
Ke(t) = .cSF
'+7'
1
x,
(ISb)
"'(
X(T,a)F(a)z(a-r)da
and X(T, t) .atilfie. 6) defined
(16a) (16b)
X(T, t) = -X(T, t)A(t) - X(T, t + r)F(t + r) X(T,T) = I, X(T,t) = 0 . t > T (23)
z'[ P(O)zo + lIuoll~ R -"'(2I1woll~
Proof. We use a procedure suggested in Hanalay (1968) to transform the original differential game with delay in a differential game without delay. Let X(T, t) the solution of (23) and
= u-uJ Wo = w-wJ, Uo
we Aave
L(zo, u, w) =
.
(17)
Proof. a) See Basar (1989). b) First note that
(18) Completing the squares in (18) one obtains
Moreover
d
z(T) = z(T)
dt zT (t)P(t)z(t) = U'[ (t)R(t)uo(t) - "'(2w'[ (t)wo(t) +r2wT (t)w(t) - uT (t)R(t)u(t) (19)
=0 (27) J(u, w) = lI(z(T), u)II~/.R - "'(2I1wll~ (28) Note that, in the new variable z, (12) is a differential game without delay. From statement a) in Lemma 2 follows that the game (12) has a feedback saddle point solution if and only if the Riccati differential equation (14) admits a positive aemidefinite solution; substituting in (14) the expressions for B and G given in (2S), equation (21) is readily obtained. Now the optimal strategy (ISa), in terms of z, can be written in the form (22); we will show that, under this control law, IIGII < "'( . Indeed from statement b) in Lemma 2, with zo = 0 and Uo = 0, we have that
"'('wT (t)w(t)
= u'[ (t)R(t)uo(t) - "'(2w'[ (t)wo(t) ~20) Integrating both sides, part b) of the lemma fol~s. • Now we are ready to state the main result of the paper. Theorem 1 Pro6/em f iI .olvab/e if and only if tlae following Riccati differential equation
J(u, w) = -"'(2I1woll~ where
- P(t) = P(t)( ~X(T, t)G(t)GT(t)X T (T, t)
I
(29) (30)
"'(
-X(T, t)B(t)R-l(t)BT (t)X T (T, t)}P(t)
Under the control law (22), w and Wo are the input and the output of the system i(t) = -B(t)R- 1(t)BT (t)P(t)z(t)
(21)
admit6 a positive .emidefinite .olution P. In tlail cue a control law .atil/ying (7) lau tlae .tMlet.re
u(t) = -Ke(t)z(t) - .cSF[Z](t)
-T
.
Wo = w - 2G (t)P(t)z(t)
"'(
P(T) = QJ
(26)
z(O)
From (19) follows that
+ uT (t)R(t)u(t) -
(24)
i(t) = X(T, t)z(t).+ X(T, t)i(t) +X(T, t + r)F(t + r)z(t) -X(T, t)F(t)z(t - r) = X(T, t)B(t)u(t) + X(T, t)G(t)w(t) := B(t)u(t) + G(t)w(t) (2S)
d
d
X(T,a)F(a)z(a- r)da.
Differentiating (24) we obtain
dt zT (t)P(t)z(t)=uj (t)R(t)uJ(t) - "'(2wj (t)wJ(t) -2uT (t)R(t)uJ(t) + 2"'(2wT (t)wJ(t)
dt zT (t)P(t)z(t)
'+7'
1
z(t) = X(T,t)z(t)+,
+G(t)w(t) Wo
(22)
31
=
(31a)
_~GT(t)P(t)z(t) + w(t) . (3Ib) r
Athans M. and P. L. Falb (1966). OptimGI COAtrol, AA IAtrodaction to the T"eorr GAd It. AppliutioA&, McGraw-Hill, New York.
This system is invertible and the inverse system is
i(t)
= (-B(t)R-1(t)B T (t)P(t) +
B~
T . and G. J. Olsder (1989). D,nGmic NOAcoopt!ndive GGme Theorr, Academic Press, New York.
~G(t)GT(t)P(t»Z(t) + G(t)WO(t)
"'I
(32a) (32b)
Hanalay A. (1968). Differential Games with Delay. SIAM J. COAtrol, 6, 579-593.
Since system (32) cannot have finite escape time we can conclude that there exists IJ > 0 such that for all w E £2(0)
Hinrichsen D., A. Hilchmann and A. J. Pritchard (1989). Robustness of stability of linear systems. J. Differ. Eqra., 6, 219-250.
IIwll~ $ 1J211woll~ .
Ravi R., K. M. Nagpal and P. P. Khargonekar (1991). 1{,oo control of linear time-varying systems: a state-space approach. SIAM J. COAtr. Opt., 29, 1394-1413.
w =
1
-T
,G (t)P(t)z(t) + wo(t) .
"'I
(33)
Hence
J(u,w)
$
-
(;r IIwll~
< -6I1wll~,
Tadmor G. (1990). Worst-case design in time domain: the maximum principle and the standard 1{,oo problem. MCSS, 3, 301-324.
(34)
:I
for all 6 E (0,;;') and for all w E £2(0). - The proof follows from Lemma 1. • It is important to note that the optimal control law is given by the sum of two contributes: a traditional memoryless plus a distributed delay state feedback.
4
Conclusions
In this paper, given a linear time-varying system with delay and subject to square integrable disturbances, we have considered the problem of finding a linear state feedback control law which minimizes the weighted terminal state attenuating the effects of the disturbances under a prespecified level. The problem is solvable if and only if a certain Riccati differential equation has a positive semidefinite solution. Future research will be devoted to solve the more general optimal control problem over the whole interval [0, TJ.
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References
Alekal Y., P. Brunovsky and D. H. Chyung, E. B. Lee (1971). The quadratic problem for systems with time delays. IEEE Trcuu. Aat. Cordr., AC-16, 673-687. Amato F., A. Pironti and S. Scala (1994) . Quadratic stabilization and disturbance attenuation for linear time-varying systems. Procuding3 0/ the IEEE Worhhop OA Ro6..t COAtrol viG VGria6/e Stnacture GAd LYGpunov Technique., Benevento.
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