Systems & Control Letters 18 (1992) 371-381 371 North-Holland

The/2-control problem for time-varying discrete systems

Vlad Ionescu Department of Automatic Control, Polytechnic Institute of Bucharest, Romania

Martin Weiss Research Institute for Informatics, Bucharest, Romania

Received 15 March 1991 Revised 5 September and 22 December 1991

Abstract: This paper deals with the /Z-problem for time-varying discrete systems. The problem considered is a deterministic counterpart of the well-known LQG problem. An adequate lZ-seminorm for linear bounded input-output operators is introduced in order to express the optimality criterion. The solution is obtained by successively reducing the original problem to simpler ones putting in evidence the Separation Principle of the optimal controller.

Keywords: Discrete time-varying system;/Z-optimization; Riceati equation; lZ-seminorm; dual system.

I. Introduction

The Linear Quadratic Gaussian (LQG) problem has been deeply investigated both for continuous and discrete-time systems, either in time-invariant or in time-varying cases (see for example [4,6]). Starting with [8,9], this problem received a frequency domain significance for the time-invariant case within the H 2 approach. This approach provided also a deterministic interpretation for the optimal properties of the solution to the LQG problem. More exactly, the LQG compensator, obtained through stochastic considerations, provides the minimum H Z-norm of the resulting closed loop transfer matrix with respect to the class of all stabilizing compensators.

The aim of this paper is to extend this deterministic H Z-approach to the case of time-varying discrete systems. For this purpose an adequate seminorm which is a counterpart of the classical H Z-norm is introduced. The proposed solution follows the two-Riccati procedure exposed in [2] and the technical machinery is essentially based on the Kalman-Szeg6-Popov-Yakubovi tch system of equations (see [7]). The paper is organized as follows. Section 2 is devoted to some basic questions including the /Z-seminorm of a linear bounded input-output operator associated to a linear exponentially stable time-varying discrete system as well as elements of the discrete-time varying Riccati equation (DTVRE) theory. Section 3 contains the problem formulation. Section 4 presents the solutions for three particular problems. The reduction of the original problem to these, in order to obtain the solution, is presented in Section 5. Concluding remarks are exposed in Section 6.

2. Preliminaries

Let M = ( M k ) k ~ Z be a sequence of bounded m Xp matrices. We shall say that M is: (a) uniformly monic (epic) if there exists u > 0 such that M T M k > v I ( M k M T > ~ I ) for all k ~ Z.

Correspondence to: Vlad Ionescu, Department of Automatic Control, Polytechnic Institute of Bucharest, 313 Splaiul Independentei, 77206 Bucharest, Romania.

0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved 1

372 V. lonescu, M. Weiss / 12-control for time-varying discrete systems

(b) uniformly invertible if it is uniformly monic and uniformly epic. In this case M k is invertible with bounded inverse with respect to k.

Any sequence (u(k) ) k ~z , u ( k ) ~ R m, will be deno ted by u. Let

k k ~ Z I

Consider the doubly infinite block matr ix T = [Tij]i, j ~ z, Tij ~ Rp×m, ~ j = 0 for i < j and ][ Tij [] _< pqi - j for i > j where 0 < q < 1, and define the sequence y via

i

y ( i ) = ~ Ti ju( j ) (1) j = --oo

for each u ~ 12(Z, Rm). It can be easily shown that I[ y [I 2 < /x I[ u II 2 where # >/0 is independen t of u. Consequent ly a l inear bounded ope ra to r T : 12(Z, R m) --~ 12(Z, R p) associated to the matr ix T, y = Tu is well defined. The adjoint ope ra to r of T deno ted T * acts as

( T * y ) ( i ) = Y'~T,. fy(j) , i ~ Z . j=i

It follows that T * is associated to the doubly infinite matr ix T T, the t ranspose of T. Let tij := trace(T, fT i j ) = trace(T/jTi T) (i.e. the squared Frobenius norm of T/j) and define

1 k k 1 ~ k

11Tl122=limsup - Y'. ~ t i j = l i m s u p - - Y~. ~-'.tij. (2) k - ~ 2k + 1 j=_k i=-k k - ~ 2k -]- i j= -k i=j

It can be immedia te ly checked that II T II 2 is a s eminorm and will be t e rmed the 12-seminorm of the ope ra to r T. Since II Tij II -< pq,-l, i >j, it follows that

ti j < p q i - j , i >j , (3)

for an adequa te v. Hence

k k p

E tij < E pqi- j < u E qi j _ i=j i=j i=j 1 - q

and consequent ly II T II 2 _< v / ( 1 - q) which shows that [I T II 2 is finite. Let e i := col(0 . . . . . 1 . . . . 0) E R m, 1 < i < m, and let u ij ~ 12(Z, R m) be def ined by

( k uiJ(k) := ~e~, = i ,

~0, k4=i,

and denote by

yij = TuiJ ~ 12( Z , R p)

the cor responding output . Hence

k

YiJ(k) = ( T u i J ) ( k ) = E TksuiJ(s) = Tkiej S ~ o o

and

]]yiJ(k)[]2= ~ e T T k T T k i e j = t r a c e ( T T T a i ) = t k i . j= l j= l

A useful result is given by:

(4)

(5)

(6)

V. Ionescu, M, Weiss / 12-control for time-varying discrete systems 373

Lemma 1 . (a ) l iT* 112 = IITIIz;

1 L ~ [l yij 112, ( 7 ) (b) II T II ~ --- lims_,=sup 2s + 1 i= -s i= 1

We omit the proof as it is based on usual limsup properties and straightforward manipulations involving inequality (3).

Consider the discrete-time linear system (A, B, C, D) with time-varying coefficients:

x ( k + 1) = A k x ( k ) + Bku(k ), (8)

y(k) = Ckx(k) +Dku(k),

where AI, ~ R "x', Bk E R n x m , Ck ~ R p x n , Dk ~ R p x " and A = (Ak)k~z, B = (Bk)l,~z, C = (Ci,)k~z, D = (Dk) k ~ z are bounded sequences.

Let

I, k = i , qbk'i:= A k - l " " Ai, k > i, (9)

be the causal evolution operator associated to the equation x(k + 1 ) = A k x ( k ) . For an input sequence u = (u( i ) ) i >_ s, the forced evolution of (8) (x(s) = 0) is given by

k y ( k ) = ( T ~ u ) ( k ) = ~_,Tkju(j ), k > s , (10)

j=s

where

D j, k = j , Tkj:= Cj6k,j+IBj, k >j, (11)

and T s is called the causal input-output operator of (8) at moment s and is defined for all s in Z. Introduce the notation

G I j (12)

If A defines an exponentially stable evolution, i.e. there exist p > 1 and 0 < q < 1 such that

]1 (~k,i 1] -~< pqk-i for k > i,

(8) defines through (1) with (11) a linear bounded input-output operator T from 12(Z, R") into RP). For such an operator we use the notation

then 12(Z,

T= Ck Dk k~Z

and it is, as usual, called the node associated to the system (8) following a terminology of [1]. Note that for a given s, we have from (10) and (11) that

T'=Ps+TIIe([s , ~), R m) (14)

where 12(Z, R r) = 12((-o% s - 1], R r) ~12([s, oo), R r) for any integer r > 0 and P+ is the orthogonal projection of I2(Z, R p) onto 12([s, oo), Rp). Hence, if A defines an exponentially stable evolution, T s is the causal Toeplitz operator associated to T at moment s as defined in [3].

374 V. Ionescu, M. Weiss / 12-control for time-varying discrete systems

Assume that A defines an exponential ly stable evolution. Then, according to the above considera- tions, t h e / 2 - s e m i n o r m of the l inear bounded i n p u t - o u t p u t ope ra to r T in t roduced through (8) and (11) is well defined. Consider now that D k = O, Vk ~ Z. Then using L e m m a 1 we get f rom (6) and (11),

II T I[ 2 = lim sup - -

s--+~ It yii 112

2 s + l i = _ s j = 1

= lim sup - - 1 ~ I ly iJ (k) l l2=l imsup - ~ t k i 2 s + l i = s j = l k= ~ s---,~ 2 s + l i = _ s k= i

= lim sup - - , _ , ~ 2s q- 1 i= - s k = i + 1

T T i B i ) trace( B~( dk.i+l) C k CkCbk,i+

= lim sup - - trace(BTQi+,B~), (15) s - ~ 2 s + l i = s

where Q = (Qi ) i • z is the unique global and bounded solution to the Lyapunov equat ion

Qi =ATQi+IAi + ciTci, i ~ Z ,

and cor responds to the (causal) observabili ty Gramian . Using now point (b) of L e m m a 1 appl ied to T* , we have

II T * II 2 = lira sup - - ~ tik s--,~ 2 s + 1 i = - s k = - ~

1 ~ i - I

= lim sup - - s---,~¢ 2S -J- i i = - s k = - ~

1 ~ trace(CiPiCT)= IITII 2 = lim~,+~sup 2s + 1 i= - s

, B T T T trace(C~faik+,Bk k (6i,k+l) Ci )

(16)

where P = (Pi)i • z is the unique global and bounded solution to the Lyapunov equat ion

Pi+ I = AiPiA~ + BiBi v, i ~ Z,

and cor responds to the (causal) controllabil i ty Gramian . The above developed theory can be dualized by considering the dual of system (8),

w(k) =A~w(k + 1) + C[v (k + 1),

z ( k ) =B~w(k + 1) + D ~ v ( k + 1), (17)

which is in reversed time. If A = (A k)k • Z defines an exponential ly stable evolution, then A T = x (Ak)k~Z defines an exponent ial ly antistable evolution (for w(k) = ATw(k + 1)) and the l inear bounded i n p u t - o u t - put (anticausal) ope ra to r of (17) is the adjoint of (13) and has the same /2-seminorm as follows f rom L e m m a 1.

Two systems (A t, B l, C 1, D 1) and (A 2, B 2, C 2, D 2) are said to be Lyapunov equivalent if there exists a uniformly invertible matr ix sequence S = (Sk) k ~ z such that

- l , 2 2 , = Bk=Sk+lBk, Ck=Ck(Sk) , D2=D~.

Two Lyapunov-equiva lent systems have the same causal i n p u t - o u t p u t ope ra to r associated to each m o m e n t s.

V. Ionescu, M. Weiss / 12-control for time-varying discrete systems 375

Our approach is essentially based on the theory of Riccati equations. We introduce the discrete time-varying Riccati equation (DTVRE) for control

(AkXk+IB k + + T -1 T T - -AkXk+lAk- - L~)(Rk BkXk+lBk) (Lk +BkXk+lAk) +Qk ( 1 8 ) Xk-- T T

and the DTVRE for estimation

_ T _ ( 1 9 ) Yk+l --AkYkAk ( AkYk CT + LT)(Rk + CkYkCT)-I( Lk + CkYk AT) + Qk

where R k > vI (u > 0), QT = Qk and L k are uniformly bounded. Let

Fk= - ( R k + B T X k + I B k ) - ' ( L T + BTXk+IAk)

and

= - + c~Y~G ) • H~, ( A ~ Y k C ~ + L T ) ( R k T -1

We say that X = (Xk.> O) k ~ z and Y = (Yk > O)k ~ Z are stabilizing (positive semidefinite) solutions for the DTVRE's if they are global and bounded solutions and ( A k + BkFk)k~ z and (AI, + H k C k ) k ~ z define exponentially stable evolutions. Concerning the existence conditions for the stabilizing solutions to the DTVRE's for arbitrary Q, L and R sequences, we mention that they can be expressed through the so called 'positivity condition' (see [3,5]).

3 . P r o b l e m f o r m u l a t i o n

Let

~ Ck I Dk ]

A k

= C~ k>s 2

- Ck

B~ B 2 ]

0 D~ 2 ,

D 21 0 k >_s

(20a)

be the linear time-varying discrete model for the generalized plant. We consider the problem of finding a linear time-varying controller

T~= ¢ , s ~ Z , (21a) tc l 0

u 2 = T~Y2, (21b)

such that: (S) The resulting system is internally stable, i.e. its A-matrix sequence defines an exponentially stable

evolution. (O) The 12-seminorm of the resulting linear bounded input-output operator from /-/1 to Yl denoted

by Tylu 1 is minimum. Note that we preserved the system and compensator structures: D 1 1 = 0, 2 2 c D k - 0 , D k=O, as

encountered in the classical LQG setting (see [4]). In the sequel we shall examine three particular problems which will lead us to the solution of the

general case called the Output Feedback (OF) problem.

376 K lonescu, M. Weiss / 12-control for time-varying discrete systems

4. Three particular problems

4.1. The Disturbance Estimation (DE) problem

Consider for (20) the following assumptions: (DEl) ( Di21k E =, ( Dzl), ~ z are uniformly invertible. (DE21 (Ak -B:(D,21)-1C;)k,Z, (Ak -B,2(D:2)-1C,&Z define exponentially stable evolutions.

We have:

Theorem 1. If T satisfies (DEl) and (DE21 then

T” I A, -Rj(D,21)-1C;-R;(D;2)-1C; B;(D;‘)-’ copt =

- (Ok’)-‘C; 0 krs

(24

is the optimal controller and it makes

II Ty,,,,OPt II 2 = 0

(perfect rejection of exogenous input u, 1.

Proof. Connecting to system (20) the controller (22) we get the resulting system

Tysl,,,, = B;(D~~~‘c;

\

A, - B;(~~‘~~~~L&*)-‘C; 1 3 k~s

I

and after the Lyapunov transformation

kEZ,

it becomes the node

Ty,,,,= -(:D:;,-‘c~ *;:F;;;& 1 TkEZ \ I

since the resulting system is internally stable because of (DE2). After suppressing the uncontrollable and the unobservable stable parts, the resulting input-output operator is obviously null. q

4.2. The Disturbance Feedforward (DF) problem

Let us consider now the assumptions: (DFl) (D;‘)k t z is uniformly invertible, (DL21k t z is uniformly manic. (DF2) (Ak -B,$D;l)-LC,$kEZ defines an exponentially stable evolution.

We have:

Theorem 2. Zf (20) satisfies (DFl) and (DF2) and the DTVRE for control

xk =A;xk+lAk - (A;X,+,B; + (qTDL2)

X((D;2)TD:2+ (B:)Txk+,B:)-1((D:2)TC: + (B;)TXk+lAk) + (C;)Tc; (23)

V. lonescu, M. Weiss / 12-control for time-varying discrete systems 377

has a stabilizing positive semidefinite solution and

F 2 12 T 12 ( O k ) ~ ~x'ck ] t . .k..l_(nk) X k + l A k ) ) + X k + l B 2 ) - l [ [ lll2"~T i,-,1 2 T

is the stabilizing feedback matrix sequence, then the optimal controller is

l 2+BkF£ lq,1/, /121 ] - 1 - Bk ( D k ) Ck "-'k t " k 1 T~Sp t = Ak 1 2 1 - 2 2

F 2 0 k>_s

which provides the optimal performance

1 k [[Ty,u,optll 2= l i m s u p - - 2 trace((B~i)TXi+lB]) •

k ~ 2 k + l i= -k

(24)

P r o o f . If X = (Xk)g~ z is the stabilizing positive semidefinite solution to the D T V R E (23) then there exist uniformly invertible V=(Vk)k~ z and bounded W = ( W k ) k ~ z such that the following system of equations is satisfied:

( / ~ 1 2 ~ T / ) 12 .4- 2 T 2 • -'k ! "-'k -- (Bk) Xk+lBk = VkVVk, (25a)

(c~)TD~2 T 2 W~Vk (258) + A k X k + I B k =

(c~)Tc~ + ATkXk+IAk - X k = W ~ W k, (25c) and F ~ = -- V k l W k ~

System (25) is termed the Kalman-SzegS-Popov-Yakubovi tch system (KSPY) (see [7]). We introduce now the fictitious output

~ ( k ) = Wkx( k ) + Vku2( k ) (26)

and denote the generalized plant with inputs u~ and u 2 and outputs 331 and Y2 by

= 0 V k . (27) ?s

C k 0 k>__s

It is easy to notice that system (27) satisfies the conditions (DE1) and (DE2). Furthermore, every stabilizing controller for (27) is a stabilizing controller for (20). According to Theorem 1, system (24) is the optimal stabilizing controller for (27) and it makes Tyl,; = 0. Thus T[opt is a stabilizing controller for T s in (20). Let us prove now the optimality of (24). To this end, consider an arbitrary stabilizing controller for (20). Then

[[ Y l ( k ) l [ 2 = IIc2x(k) + D~au2(k)II 2

=xT( k ) ( c l ) T C l x ( k ) + uT( k )( O~2)T C~x( k )

+xT(k)(C T ) Dj, u2(k ) +uT( ,2 T ,2 . k ) ( D k ) Dk u2(k)

Using KSPY (25) we have

[1Yl(k) II 2 = II Vku2(k ) + W k x ( k ) II 2 + x T ( k ) X k x ( k ) _ x T ( k + 1)Xk+lx( k + 1)

+ u T ( k ) ( B ~ ) T x k + , x ( k + 1) + x T ( k + 1)Xk+,BgUl(k ) - u T ( k ) ( B ~ ) T x k + , B ~ u , ( k ) .

378 V. lonescu, M. Weiss / 12-control for time-varying discrete systems

Compute now II Ty~u, II 2 using (b) of Lemma 1. For, consider the inputs

u~J(k) := (ej, k =i, O, k ~ i ,

for each moment i ~ Z and each input channel j = 1 . . . . . m r Denote by y{J(k), u~(k) and ~{J(k) the corresponding outputs of (20), (21) and (27), respectively, and by xiJ(k) the corresponding state evolution.

From (26) it follows

II y~(~)II = = II ;~,(k)II 2 + ( x , ~ ( ~ ) ) ~ x ~ ( k ) - ( x ~ ( ~ + 1))Txk+lxiJ(k + 1)

+ ( u ~ J ( k ) ) T ( 1 T ij 1) + B~) Xk+,x (k + (xiJ(k + l))TXk+lBlu~J(k) 1 T - (u~(~) )~(B~) x~ +,~-l~"~' k ) ,

Summing with respect to k and taking into account that xiJ(k) = O, u~J(k) = 0 for k < i and x~J(i + 1) = B]u~J(i) = B]ej and u~ffk) = 0 for k ~ i w e get

11 Y~' 1122 = II ;~' I1~ + eT(B/ l ) T Xi+ ,B)ej. Hence

ml ml

E 11 ' 'ij[12 = y l 112 E [1)~J1122 + trace((B])TXi+lB]) (28) j = l j=l

and taking limsup of the mean for i = - s to i = s on both sides of (28), we obtain the inequality

1 L trace((B1)wXi+tB]) (29) II Tylu , 112 >- lim sup 2s +~-1- i= S~°° --S

and equality in (29) is attained for (24) which nullates E~'21 II 91 j I1~ for all i ~ Z and hence it is the desired solution. []

4.3. The Output Estimation (OE) problem

This is the dual of the previous DF problem. The assumptions considered are: (OE1) (D12) k e z is uniformly invertible, (D~l)k ~ z uniformly epic. (OE2) (A k _Bk(D k 2 12)-lCk)k~zl defines an exponentially stable evolution.

The following result is directly obtained by dualizing Theorem 2 and using (a) of Lemma 1.

Theorem 3. If (20) satisfies (OE1) and (OE2) and the D T V R E for estimation

1 21 T Yk+I=AkYkAT--(AkYk(c2)T+Bk(Dk ) )

/ FI21/ /~21"~ T CkYg(Ck)2 2 T X . . X T ×t~'k ~=k I + t ~ +Bk(Bk)

has a stabilizing positive semidefinite solution and

H2k = _(AkYk(c~)T+ 1~1[/~21]T]t.21/" t-~21] T 2 2 T -1 ~ , ~ , j ~ . ~ ~ . .~ ~ +c~Y~(C~) ) is the stabilizing injection matrix sequence, then the optimal controller is

( I 2 12 2 2 AI , -Bk(Dk ) C k+H/,C k H~

TcSpt = ( 3 1 2 ) - 1Ckl 0 ]k >s

(3o)

(31)

V. lonescu, M. Weiss / 12-control for time-varying discrete systems 379

which provides the optimal performance

1 k

][Tyluloptll 2= l i m s u p - - E trace(C/~Y~(c~)T) • k--,~ 2 k + l i = _ k

5. The Output Feedback (OF) problem

Consider now the general problem. The following result obtains.

Theorem 4. I f the DTVRE's (23) and (30) have stabilizing positive semidefinite solutions X = ( X k ) k ~ z , Y = (Yk)k ~ Z and (F~) k ~ z , (H~)k ~ z are the corresponding stabilizing feedback and injection matrix sequences, respectively, then the optimal controller is

2 2 2 2 _H 2] A k + B k F[ + Hg C k

ZcSpt = F 2 0 ]k>s (32)

and it makes

1 ,)] I ITy iu ,op t l l2=l imsup - ~ [ t r a c e ( C ) Y i ( c l ) T ) + t r a c e ( ( B i ) T X i + l B .

k - ,~ 2 k + 1 i= - k

Proof. We follow the main lines in the proof for Theorem 2. After introducing the fictitious output Y l given by (26), the system 7 ~s given by (27) satisfies the conditions (OE1) and (OE2) and its associated DTVRE for estimation coincides with (30). According to Theorem 3, T~opt in (31) receives the form given in (32) and consequently the latter is the optimal stabilizing controller for (28). Hence it is stabilizing for (20). Since relation (28) still remains valid, it follows from Theorem 3 that

II TylUl II 2 ~ lim sup - - k--+~

1 k 2k + 1 i= -k trace C C + trace B X i + I B

with equality attained for controller (32) which is thus optimal. 13

6. Concluding remarks

A straightforward solution for the discrete/2-control problem in the time-varying case was presented. Our approach underlines the separation property of the /2-optimal controller. Some final remarks are now in order:

1. Let T be a node and T i its associated Toeplitz operator at moment i. Using (4), (5) and (6) let us define the 12-seminorm of T i as

o o

liT i 112 2= ~ Ilyij112= E ~ IlyiJ(k)H 2 j=l k=-oo j=l

= ~ t k i = t r a c e ( B F Q i + l B i ) , i ~ Z . k=i

(33)

380 v. lonescu, M. Weiss / 12-control for time-carying discrete systems

We have also

i I1(zi) II 2 = E t i ~= t rac e (C i P i C~) , i ~ Z . (34)

k _ - o o

Note that in general

lIT i 1124: I I (T i ) * 112. (35)

Note also that in the time-invariant case, when t k i = t k_~ for k > i and tki = 0 for k < i, (33) is precisely the classical H2-norm and (35) becomes an equality.

According to Lemma 1, we have

1 ~L, 1 s I I Z l l 2 = l i m s u p - z., I lTi l122=l imsup - ~] I I (T i )* l l z z= l lT* l l~ . (36)

i---,~ 2i + 1 i-~= 2i + 1 i = - s i = - - s

Consider now the D F problem and an arbitrary stabilizing controller (21) for (20). Let Ty,, 1 and T~1,1 be the resulting node and its associated Toepli tz opera tor at momen t i, respectively. From Theorem 2, it follows (see (28)) that (24) provides the minimum of i II Tfl,u, II 2 for all i ~ Z. Nevertheless, the /2-semi-

norm of the Toeplitz opera tor Tjlu, used for the H~-problem in the t ime-invariant case is not appropr ia te for our purposes because of (35) which does not allow us to join together the D F and O E problems in order to solve the O F problem. This is the reason for introducing the seminorm in (2).

2. As we ment ioned in the case of the D F problem, (24) provides the minimum of II Tj,u I II 2 for all i. The existence of such a minimum can be shown to be strongly related to the 'positivity condi t ion '

>~6((u 2 , u 2 ) i + ( x , x ) i ) , 6 > 0 , i ~ Z , (37) u 2 ' U R uz i

for all u 2 ~ lZ([i, ~), R m) and x ~ 12([i, ~), R n) linked by

x ( k + l ) = A k x ( k ) + B ~ u 2 ( k ), x ( i ) = O ,

where (a , b)i ~]k T 1 T 1 I T 12 = = = ((Ck) Dk ) k ~ z , ~z" ((C k) Ck) k L R = ((D~2)TD~2) k But (37) is a >~iakbk, O ~z , necessary condit ion for existence of the stabilizing solution to D T V R E (23) (see [3,5]). Note also that according to (36), if II T i II 2 is minimized for all i then II T II 2 is minimized, but the converse is not t rue in general. Hence the necessity of the existence condit ion for the stabilizing solution of D T V R E (23) as well as of D T V R E (30) remains an open problem.

Acknowledgement

The authors are indebted to Professor A. Halanay for many useful suggestions.

References

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V. lonescu, M. Weiss / 12-control for time-varying discrete systems 381

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