Robust Stabilizing Output Feedback Nonlinear Model Predictive Control byUsing Passivity and Dissipativity

Han Yu, Feng Zhu, Meng Xia and Panos J. Antsaklis

Abstract— Motivated by the passivity-based nonlinear modelpredictive control (NMPC) scheme reported in [1], in this paper,we propose a robust stabilizing output feedback NMPC schemeby using passivity and disspativity. Model discrepancy betweenthe nominal model and the real system is characterized bycomparing the outputs for the same excitation function, andwith this kind of characterization, we are able to compare thesupply rate between the nominal model and the real systembased on their passivity indices. Then, by introducing specificstabilizing constraint based on the passivity indices of thenominal model into the MPC, we show that our proposedNMPC scheme can stabilize the real system to be controlled.

I. INTRODUCTION

Model predictive control (MPC), as an effective controltechnique to deal with multi-variable constrained controlproblems, has been widely adopted in a variety of industrialapplications. The success of MPC can be attributed to its ef-fective computational control algorithm and its ability to im-pose various constrains when optimizing the plant behavior.Unlike the conventional feedback control, MPC allows oneto first compute an open-loop optimal control trajectory byusing an explicit model over a specified prediction horizon,then only the first part of the calculated control trajectory isactually implemented and the entire process is repeated forthe next prediction intervals. For extensive surveys on MPC,one can refer to [2], [3], [4], [5] and the references therein.

Although MPC has many advantages, several issues, suchas feasibility, closed-loop stability, nonlinearity and robust-ness still need to be studied. If either models of the plant orconstraints are nonlinear, nonlinear MPC (NMPC) schemesare required to be used. However, it was pointed out in [6]that NMPC does not always guarantee closed-loop stability.Moreover, robustness of MPC is also an issue when modeluncertainty or noise appears [5]. Model uncertainty usuallyexists in MPC because the model used for prediction cannotperfectly match the real dynamics of the plant to be con-trolled.

On the other hand, passivity theory is a powerful tool inanalysis and control of nonlinear systems [7], [8], [9], [10].Recently, a passivity-based NMPC scheme is proposed in[1], motivated by the relationship between optimal controland passivity as well as by the relationship between optimalcontrol and NMPC. It is shown that close-loop stabilityand feasibility can be guaranteed by introducing specificpassivity-based constraints.

The authors are with the Department of Electrical Engi-neering, University of Notre Dame, Notre Dame, IN, 46556,USA, hyu@nd.edu, fzhu1@nd.edu, mxia@nd.edu andantsaklis.1@nd.edu.

Motivated by [1], in this paper, we propose a robuststabilizing output feedback NMPC scheme by using passivityand disspativity. Instead of assuming both the nominal modeland the plant are passive systems with the same dynamicsas reported in the previous work, we assume that the modelfor prediction and the actual plant dynamics are dissipative(which are more general than passive systems since theycould be non-passive), and they do not have to possess thesame dynamics. Model discrepancy between the nominalmodel and the real system is characterized by comparing theoutputs for the same excitation function. With this charac-terization of model discrepancy, we are able to compare thesupply rate between the nominal model and the real systembased on their passivity indices [11]. Then, by introducingspecific stabilizing constraint into the MPC based on thepassivity indices of the nominal model, we can show thatthe control input calculated using the nominal model canguarantee stability of the plant to be controlled.

The rest of this paper is organized as follows: in SectionII, background material on passivity and dissipativity isprovided; in Section III, we give a brief review on theresults of passivity-based NMPC; in Section IV, modeldiscrepancy between the nominal model and the real systemis characterized, and conditions under which the supply rateof the real system is bounded above by the supply rateof the nominal model is provided; our proposed stabilizingoutput feedback NMPC scheme is presented in Section V;simulations are provided to validate our results in SectionVI; finally, conclusions are made in Section VII.

II. BACKGROUND ON DISSIPATIVE AND PASSIVESYSTEMS

We first introduce some basic concepts on passive anddissipative systems. Consider the following control system,which could be linear or nonlinear:

H :

!x(t) = f(x(t)) + g(x(t))u(t)

y(t) = h(x(t), u(t))(1)

where x(t) ! X " Rn, u(t) ! U " Rm and y(t) ! Y "Rm are the state, input and output variables, respectively,and X , U and Y are the state, input and output spaces,respectively. The representation !(t, t0, x0, u(t)) is used todenote the state at time t reached from the initial state x0 atthe time t0 under the control u(t).

For an easy understanding on the concepts of dissipativityand passivity it is convenient to imagine that system H is aphysical system with the property that its energy can only beincreased through the supply from an external source. The

definitions below give the generalizations of such physicalproperties.

Definition 2.1: (Supply Rate [12]) The supply rate"(t) = "(u(t), y(t)) is a real valued function defined onU # Y , such that for any u(t) ! U and x0 ! X andy(t) = h(!(t, t0, x0, u(t)), u(t)), "(t) satisfies

" t

t0

|"(#)|d# < $. (2)

Definition 2.2: (Dissipative System[12]) System H withsupply rate "(t) is said to be dissipative if there exists anonnegative real function V : X % R+, called the storagefunction, such that, for all t & t0 & 0, x0 ! X and u ! U ,

V (xt)' V (x0) (" t

t0

"(#)d#, (3)

where xt = !(t, t0, x0, u(t)) and R+ is a set of nonnegativereal numbers. If V is C1, then we have V ( "(t), )t & 0.

Definition 2.3: (Passive System [12]) System H is saidto be passive if there exists a storage function V such that

V (xt)' V (x0) (" t

t0

uT (#)y(#)d#. (4)

If V is C1, then

V ( uT (t)y(t), )t & 0. (5)Definition 2.4: (IF-OFP systems [13]) System H is said

to be Input Feed-forward Output Feedback Passive(IF-OFP)if there exists a storage function V such that

V (xt)' V (x0)

(" t

t0

#uT (#)y(#)' $yT (#)y(#)' %uT (#)u(#)

$d#.

(6)

for some $, % ! R.For the rest of this paper, we will denote an m'inputsm'outputs dissipative system with supply rate (6) by IF-OFP(%, $)m and we will call (%, $) the passivity indices ofthe system.

Remark 2.5: A positive % and a positive $ indicate thatthe system has an excess of passivity; otherwise, the systemis lack of passivity. In the case when % > 0 or $ > 0,the system is said to be input strictly passive (ISP) or outputstrictly passive(OSP) respectively; if either % or $ is negative,then the system is non-passive. Clearly, if a system is IFP(%)or OFP($), then it is also IFP(%'&), or OFP($' &), )& > 0.

Lemma 2.6: [14] The domain of $, % in IF-OFP system(6) is ! = !1 * !2 with !1 = {$, % ! R|$% < 1

4}, !2 ={$, % ! R|$% = 1

4 ; $ > 0}.Proof: The proof shown in this paper is a little bit

different from the one shown in [14], and it is provided inthe Appendix I.

III. PASSIVITY-BASED NMPCDifferent to many other NMPC schemes, which achieve

stability by enforcing a decrease of the control Lyapunovfunction (CLF) along the solution trajectory, stability isachieved for the passivity-based NMPC scheme by usinga nonlinear input-output constraint, which is implemented

as an additional condition within the NMPC set-up. Thepassivity-based NMPC scheme in [1] is given by

minu(·)

" tk+Tp

tk

#q(x(#)) + u(#)Tu(#)

$d#

s.t.

%&'

&(

x(t) = f(x(t)) + g(x(t))u(t)

y(t) = h(x(t))

uT (t)y(t) + yT (t)y(t) ( 0,

(7)

where tk denotes the time instant at which state measurementof the controlled system is available to the MPC, q(x(t)) isa positive semi-definite function and Tp denotes the finitetime horizon for prediction. The passivity-based constraintuT (t)y(t) + yT (t)y(t) ( 0 guarantees stability of the plantto be controlled, where the dynamics of the plant is assumedto be passive and zero state-detectable. Feasibility is alsoguaranteed due to the already known stabilizing outputfeedback control law u = 'y. It is also shown in [1] thatas Tp % 0, the passivity-based NMPC recovers the knownstabilizing output feedback u = 'y.

Motivated by the work reported in [1], we propose arobust stabilizing output feedback NMPC scheme in thispaper. The nominal model and the real plant are assumedto be IF-OFP, and they do not have to possess the samedynamics. Compared to the previous work, our results canalso be applied to a class of non-passive systems, and modeldiscrepancy between the real system and the nominal modelcan be accommodated as well. We will discuss our proposedscheme in the following sections in details.

IV. CHARACTERIZATION OF MODEL DISCREPANCY

Consider two systems denoted by " and )" as shownin Fig.1. One can view )" as an approximation of ", and)" describes some behavior of interests of ". A commonused measure for judging how well )" approximates " is tocompare the outputs for the same excitation function u. Wedenote the difference in the output by #y. The error may bedue to the modeling, linearization or model reduction, etc.For a “good” approximation, we require that the “worst”case #y over all control inputs u be small. Thus )" is agood approximation of " if there exists a positive constant' > 0 such that

+#y+t ( '+u+t, )u and )t & 0, (8)

where + ·+t denotes the truncated L2-norm update to time t.One can conclude that the value of ' characterizes the modeldiscrepancy between " and )".

Remark 4.1: One can verify that for linear systems, ' isan upper bound on the H! norm of the difference in thetransfer functions of systems " and )".

Assume that system " is IF-OFP(%, $)m, and system )" isan approximation of system " which is IF-OFP()%, )$)m. Itcan be shown that under some conditions, the supply rate ofsystem " is always bounded above by the supply rate of )".Those conditions are summarized in Lemma 4.2.

Fig. 1. Model Approximation

Lemma 4.2: Consider system " and system )" as shownin Fig.1, where " is IF-OFP(%, $)m and system )" is IF-OFP()%, )$)m. If (8) holds and there exists a ( > 0 such that

% ' )% & '2

(+ ' + b

$ ' )$ & ()$2(9)

where b = 2max{0, )$'2}, then

uT y ' $yT y ' %uTu ( uT )y ' )$)yT )y ' )%uTu. (10)Proof: Let " = uT y ' $yT y ' %uTu, )" = uT )y '

$)yT )y ' )%uTu, then

)" ' " = uT )y ' $)yT )y ' )%uTu' (uT y ' $yT y ' %uTu)

= 'uT#y ' )$yT y + 2)$yT#y ' )$#yT#y

' )%uTu+ $yT y + %uTu,(11)

with 'uT#y & ''+u+22 and '2)$yT#y & '!2

" +u+22 '()$2+y+22, based on (11), we can further get

)"'" & (%')%' '2

('')+u+22+($')$'()$2)+y+22')$+#y+22.

(12)So for the case )$ ( 0, if % ' )% & !2

" + ' and $ ' )$ & ()$2,then )" & "; for the case )$ > 0, if % ' )% & !2

" + ' + '2)$and $ ' )$ & ()$2, then )" & ". This completes the proof.

Remark 4.3: Lemma 4.2 provides conditions under whichthe supply rate of system " is upper bounded by the supplyrate of )". Those conditions are related to the bound on themodel discrepancy between " and )" (which, in our case,is characterized by '), and is also related to their passivityindices (as indicated by % & )% + !2

" + ' + b and $ & )$ +()$2). The conditions on the bound of passivity indices canbe relaxed if the bound on the model discrepancy is small(i.e., ' is small, thus )" approximates the behavior of " wellfor the same excitation function).

V. ROBUST STABILIZING OUTPUT FEEDBACK NMPCFOR IF-OFP SYSTEMS BY USING PASSIVITY AND

DISSIPATIVITY

In this section, we first study the problem of stabilizationof IF-OFP systems by using static output feedback gains.This result is important for us to derive the stabilizingcondition in our proposed NMPC scheme. We then proposea NMPC scheme by using passivity and dissipativity whenthere is no model discrepancy between the nominal modeland the real system. Finally, we extend this result to the case

when model discrepancy between the nominal model and thereal system can be characterized in the way as discussed inSection IV.

A. Stabilization of IF-OFP Systems by Using Static OutputFeedback Gain

Lemma 5.1: If system H is IF-OFP($, %), then therealways exists an output feedback stabilizing control u(t) =r(t) 'Ky(t), where K ! R, r(t), y(t) ! L2, such that theclosed-loop system is L2 stable from r(t) to y(t). Moreover,if the system is also zero-state detectable, then with r(t) = 0,the closed-loop system is asymptotically stable.

Proof: Since u(t) = r(t)'Ky(t), we can get

V (xt)' V (x0) (" t

0

*+r(#)'Ky(#)

,Ty(#)

' $yT (#)y(#)' %+r(#)'Ky(#)

,T +r(#)'Ky(#)

,-d#

=

" t

0

#(1 + 2%K)rT (#)y(#)' (K + $ +K2%)yT (#)y(#)

' %rT (#)r(#)$d#,

(13)thus

V (xt)' V (x0) (" t

0

.|1 + 2%K|+r(#)+2+y(#)+2

+ |%|+r(#)+22 ' (K + $ +K2%)+y(#)+22/d#.

(14)

If K + $ +K2% > 0, then we can obtain

V (xt)' V (x0) (" t

0

#. |1 + 2K%|2

2(K + $ + %K2)+ |%|

/+r(#)+22

' K + $ + %K2

2+y(#)+22

$d#,

(15)which further yields

" t

0

K + $ + %K2

2+y(#)+22d#

(" t

0

. |1 + 2K%|2

2(K + $ + %K2)+ |%|

/+r(#)+22d# + V (x0),

(16)which shows that the closed-loop system is L2 sta-ble from r(t) to y(t). With r(t) = 0, we have0 t0

K+#+$K2

2 +y(#)+22d# ( V (x0). With V (x0) beingbounded, we can further conclude that limt"! y(t) = 0.Asymptotic stability follows from that the system H is zero-state detectable. Now it remains to show that there alwaysexists K such that K + $ + %K2 > 0. Assume that theredoes not exist a K such that K + $ + %K2 > 0. This canonly happen when % < 0. Let p(K) = K + $ + %K2, thenone can find that with % < 0, p(K) has a global maximumat K = ' 1

2$ , and maxK{p(K)} = 4#$#14$ . In view of

Lemma 2.6, with % < 0, we have $ ! !1, which yieldsmaxK{p(K)} = 4#$#1

4$ > 0. This implies that there existsK such that p(K) > 0 when % < 0, which completes theproof.

Remark 5.2: It can be shown that:

• if % = 0, then we can choose K > '$;• if % > 0, then we can choose K > #1+

$1#4#$

2$ orK < #1#

$1#4#$

2$ ;• if % < 0, we can choose #1+

$1#4#$

2$ < K <#1#

$1#4#$

2$ .So based on the passivity indices (%, $), we can find therange of stabilizing output feedback gains for the system.

B. Stabilizing Output Feedback NMPC for IF-OFP Systemswith no Model Discrepancy

Motivated by the passivity-based NMPC scheme reportedin [1], we extend this scheme to the more general cases,where the systems to be controlled are IF-OFP. In view ofLemma 5.1, we can conclude that it is always possible tofind a range of stabilizing output feedback gains for an IF-OFP system based its passivity indices. We first considerthe case when there is no model discrepancy between thesystem to be controlled and the nominal model being usedfor prediction. The scheme of stabilizing output feedbackNMPC for IF-OFP systems with no model discrepancy isgiven by:

minu(·)

" tk+Tp

tk

#q(x(#)) + u(#)TRu(#)

$d#

s.t.

%&&&'

&&&(

x(t) = f(x(t)) + g(x(t))u(t)

y(t) = h(x(t), u(t))

uT (t)y(t)' $yT (t)y(t)' %uT (t)u(t)

( 'K+#+$K2

2 yT (t)y(t),

(17)

where R is a positive definite matrix, the stabilizing outputfeedback gain K should be chosen based on the indices (%, $)of the system such that K + $ + %K2 > 0, which has beendiscussed in Remark 5.2.

Theorem 5.3: The output feedback NMPC scheme pro-posed in (17) can asymptotically stabilize system (1) if itis IF-OFP(%, $)m with a continuously differentiable storagefunction and is zero-state detectable.

Proof: The proof is very similar to the proof provided in[1]. First, we need to show that the NMPC scheme proposedin (17) for the system (1) which is IF-OFP(%, $)m is alwaysfeasible; second, we need to show that the NMPC schemewill stabilize the system asymptotically. Actually, feasibilityis guaranteed due to the known output feedback stabilizingcontrol law u(t) = 'Ky(t). Let V be the storage function ofsystem (1). With the differentiable storage function V and thestability constraint uT (t)y(t) ' $yT (t)y(t) ' %uT (t)u(t) ('K+#+$K2

2 yT (t)y(t), one can obtain V ( uT (t)y(t) '$yT (t)y(t) ' %uT (t)u(t) ( 'K+#+$K2

2 yT (t)y(t). Usingthe fact that system (1) is zero-sate detectable, asymptoticstability follows from the result shown in Lemma 5.1.

C. Stabilizing Output Feedback NMPC for IF-OFP Systemswith Model Discrepancy

Consider system " and system )" as shown in Fig.1, where)" is an approximation of ". In NMPC, " represents the realsystem to be controlled, and )" represents the nominal model

used in MPC for prediction. System " is IF-OFP(%, $)m andsystem )" is IF-OFP()%, )$)m. Since )" is an approximation of", ()%, )$) is not necessarily equal to (%, $). In this case, weneed to rectify the stabilizing output feedback NMPC schemeproposed in Section V-B as:

minu(·)

" tk+Tp

tk

#q()x(#)) + u(#)TRu(#)

$d#

s.t.

%&&&'

&&&(

)x(t) = )f()x(t)) + )g()x(t))u(t))y(t) = )h()x(t), u(t))uT (t))y(t)' )$)yT (t))y(t)' )%uT (t)u(t)

( 'K+!#+!$K2

2 )yT (t))y(t),

(18)

where !)x(t) = )f()x(t)) + )g()x(t))u(t))y(t) = )h()x(t), u(t))

(19)

is the state-space model of )", and K is chosen based on()%, )$) such that K + )$+ )%K2 > 0 (see Remark 5.2 on howto choose the range of K).

Due to possible model mismatch between " and )", thecontrol action generated through the NMPC (18) may not beable to stabilize the real system ". Intuitively, stabilizationresults may still hold if )" is a good approximation of ".Theorem 5.4 provides sufficient conditions under which theNMPC scheme provided in (18) can still stabilize the realsystem ".

Theorem 5.4: Consider system " and system )" as shownin Fig.1, where " is IF-OFP(%, $)m with a continuouslydifferentiable storage function V and is zero-state detectable;system )" is IF-OFP()%, )$)m with a continuously differen-tiable storage function )V and is also zero-state detectable. If(8) holds and there exists a ( > 0 such that

% ' )% & '2

(+ ' + b

$ ' )$ & ()$2(20)

where b = 2max{0, )$'2}, then the NMPC scheme providedin (18) is also a stabilizing MPC for the system ".

Proof: If (8) and (20) are satisfied, then in view ofLemma 4.2, we can conclude that uT (t)y(t)'$yT (t)y(t)'%uT (t)u(t) ( uT (t))y(t)')$)yT (t))y(t)')%uT (t)u(t). With thestabilization condition uT (t))y(t)' )$)yT (t))y(t)' )%uT (t)u(t)( 'K+!#+!$K2

2 )yT (t))y(t) in the NMPC, we can conclude that

)V ( uT (t))y(t)' )$)yT (t))y(t)' )%uT (t)u(t)

( 'K + )$ + )%K2

2)yT (t))y(t) ( 0,

(21)

andV ( uT (t)y(t)' $yT (t)y(t)' %uT (t)u(t)

( uT (t))y(t)' )$)yT (t))y(t)' )%uT (t)u(t)

( 'K + )$ + )%K2

2)yT (t))y(t) ( 0,

(22)

since )V & 0 and V & 0, this implies that limt"! )V =

0 and limt"! V = 0 (otherwise, if limt"! )V < 0

and limt"! V < 0, then V and )V will eventuallybecome negative). Observing from (21), we can directlyconclude that limt"! )y(t) = 0, and asymptotic stabil-ity of the model )" follows from the assumption that)" is zero-state detectable. Since 0 = limt"! )V (limt"!{uT (t))y(t) ' )$)yT (t))y(t) ' )%uT (t)u(t)} ( 0, withlimt"! )y(t) = 0, we can get limt"! u(t) = 0. With0 = limt"! V ( limt"!{uT (t)y(t) ' $yT (t)y(t) '%uT (t)u(t)} ( 0 and limt"! u(t) = 0, we can furtherconclude that limt"! y(t) = 0. Asymptotic stability of thesystem " follows from the assumption that " is zero-statedetectable.

Remark 5.5: One can see that the basic idea behind thisrobust stabilizing NMPC scheme is that if the supply rate ofthe real system is upper bounded by the supply rate of thenominal model, then by introducing the stabilizing conditionprovided in Theorem 5.4, we are able to stabilize the realsystem as well. And in view of the discussions providedin Section IV, this bound on the supply rate is related tobound on the model discrepancy between the real systemsand the nominal models, and their own passivity indices.This approach may appear to be conservative at the firstlook, but one should be aware that the nominal model usedfor prediction can always be adapted in order to meet thoseconditions.

VI. EXAMPLE

Example 6.1: In this example, the dynamics of the realsystem is given by

1x1

x2

2=

1'2 0.10.2 1

2 1x1

x2

2+

10.11

2u

y = 0.1x1 + x2,

(23)

while the nominal model is given by

1)x1

)x2

2=

1'1 0.20.3 0.95

2 1)x1

)x2

2+

10.120.96

2u

)y = 0.12)x1 + 0.96)x2.

(24)

In this case, one can verify that $ = '1.05, % = '0.04, )$ ='1.2, )% = '0.2, ' = 0.0451, which satisfy the conditionsprovided in Lemma 4.2, and we choose K = 2.5 based onRemark 5.2. By using the proposed NMPC scheme providedin Theorem 5.4, we get the simulation results by usingJModelica[15], where the state measurements of the plant aresent to the NMPC at every 0.5s, while the prediction periodof NMPC is 2s. In the cost function, q()x) = 0.5)x2

1 + 0.3)x21,

and R = 0.5. The simulation results are shown in Fig.2-Fig.4.

Fig. 2. State of the Plant and State of the Nominal Model

Fig. 3. Output and Control Input

Fig. 4. Supply Rate and Cost Comparison

Fig.2 compares the state of the plant and the state of thenominal model; Fig.3 shows the output and the control inputof the plant and the nominal model; Fig.4 compares their

supply rate and the cost, where " denotes the supply rate ofthe real system, and )" denotes the supply rate of the nominalmodel; one can see that " is always upper bounded by )".

VII. CONCLUSION

In this paper, we propose a robust nonlinear model pre-dictive control scheme by using passivity and dissipativity.Compared with the previous results on passivity-based MPCreported in [1], our proposed scheme is more general becauseit can also be applied to a class of non-passive systems, andmodel discrepancy can be accommodated as well. By intro-ducing specific stabilizing constraint based on the passivityindices of the nominal model into the MPC, we show thatour proposed NMPC scheme can guarantee the stability ofthe real system to be controlled.

APPENDIX IPROOF OF LEMMA 2.6

Proof: If $, % ! ! = !3 * !4 with !3 = {$, % !R|$% & 1

4 ; $ < 0} and !4 = {$, % ! R|$% > 14 ; $ > 0},

degenerate cases occur. In case $, % ! !3, multiplying (6)with $ < 0 and taking the square complement it follows

$#V (xt)' V (x0)

$+

" t

0

#$2+y(#)+22 ' $uT (#)y(#)

+1

4+u(#)+22 + ($% ' 1

4)+u(#)+22

$d# & 0.

(25)

Let )1 = '$min{V (x0)}, then $#V (xt) ' V (x0)

$(

'$V (x0) ( )1, in view of (25), we have

)1 +

" t

0

#33$y(#)' 1

2u(#)

3322+ ($% ' 1

4)33u(#)

3322

$d# & 0,

(26)which is satisfied for any pair of (u, y), since $% ' 1

4 & 0imposing no restriction to the system’s input-output behavior.In case $, % ! !4, multiplying (6) with $ > 0 and taking thesquare complement it follows

V (xt)' V (x0) ( '1

$

" t

0

#33$y(#)' 1

2u(#)

3322

+ ($% ' 1

4)33u(#)

3322

$d# ( 0,

(27)

which indicates that V (xt) ( V (x0), )t & 0. Thus

0 = max4V (xt)' V (x0)

5

( '1

$

" t

0

#33$y(#)' 1

2u(#)

3322+ ($% ' 1

4)33u(#)

3322

$d#,

(28)which can be only satisfied for u(t) = 0 since $% ' 1

4 > 0.The proof is completed.

ACKNOWLEDGEMENT

The support of the National Science Foundation under theCPS Large Grant No. CNS-1035655 is gratefully acknowl-edged.

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