fuzz-ieee 2007 2007 ieee international conference on fuzzy systems tutorial 2: model based fuzzy...

FUZZ-IEEE 2007

IEEE International Conference on

Fuzzy Systems

Tutorial 2: Model based Fuzzy Logic Control: Overview and Perspectives

1

Dept of Manufacturing Engineeringand Engineering Management

City Universityof Hong Kong

Model Based Fuzzy Control: Overview and Perspectives

Gang FENG and Xiaojun ZENG

City University of Hong KongManchester University

[email protected]@manchester.ac.uk



Acknowledgements

Financial support: Australian Research CouncilResearch Grants Council of Hong KongCity University of Hong KongUniversity of New South Wales Collaborators:

Profs. N. Rees, S.G. CaoDrs. C.K. Chak, Z.X. Han, M. Cheng, L. Wang, C.L. ChenMs./Mr. M. Chen, T.J. Zhang, B. Yang, et al.



Outline• Brief review of fuzzy logic control• T–S fuzzy model • Representation theory of T-S fuzzy models• Stability analysis based on common, piecewise, and fuzzy

quadratic Lyapunov functions• Stabilization based on common quadratic Lyapunov functions• Stabilization based on piecewise quadratic Lyapunov

functions• Stabilization based on fuzzy quadratic Lyapunov functions• Future perspectives



Fuzzy control systems: Structure

Fuzzy inputs

Knowledge Base

Defuzzification Interface

Inference Engine

Fuzzification Interface

Controller inputs

Fuzzy outputs

Controller outputs



Fuzzy control techniques: review

Conventional fuzzy logic control Fuzzy PID Neuro-fuzzy control or fuzzy-neuro control Fuzzy sliding mode control Adaptive fuzzy control T-S model based control



Conventional fuzzy logic control

Mamdani and Assilian’s [199], [200] Heuristic and model free Applications:

warm water plant [137]heat exchanger [229]robot [10], [289], [314], [319], stirred tank reactor [146], traffic junction [237]steel furnace [153] aircraft [58], [161], network traffic management and congestion control [131], [169]bioprocesses [111]

2



Conventional fuzzy logic control

The key idea for stability analysis and control design:to regard a fuzzy controller as a nonlinear controller and to embed the problem of fuzzy control systems into conventional nonlinear system stability theory.

Typical approaches:describing function approach [136]cell-state transition [132]Lure’s system approach [59][208]Popov’s theorem [91] circle criterion [226][244]conicity criterion [69]

Limitation: Consistency in design and performance



Fuzzy PID

Direct-action type: [201]within the conventional PID conceptMamdani fuzzy controller as a two input PI controller

Indirect-action type: [107][344]“gain-scheduling” fuzzy PID

controller gains change as operating condition varies Fuzzy + PID



Fuzzy PID

Advantages:Nonlinear PIDBetter performance

Other topics:Self-tuning fuzzy PID controllers [202][215]Optimal fuzzy PID control [113][270]

LimitationConsistency in design and performance



Neuro-Fuzzy Control

Neural Networks: learning capabilities and high computation efficiency

Fuzzy Systems:expert knowledge representation

Both:Store knowledge or data, robust with uncertainties

Fuzzy control augmented by neural networksflexibility, data processing capability, and adaptability



Neuro-Fuzzy Control

Fuzzy control augmented by neural networksprocess of fuzzy reasoning realized by neural networks

Advantage: No information of the plants is required Limitations:

(1) systematic analysis of stability(2) Learning convergence

Other topics:tuning parameters in neuro-fuzzy controller via genetic algorithm [72], [209]self-organizing or adaptive neuro-fuzzy control [63], [177]



Fuzzy Sliding Mode Control

Fuzzy control similar to a modified sliding mode controller [234]

Fuzzy control + Sliding mode controltaking advantages of both techniques

Fuzzy sliding surface: elimination of chattering [94] Fuzzy control + supervisory sliding mode control

[80][206] Advantages:

(1) Easier stability analysis and control design (2) Robustness

3



Adaptive Fuzzy Control

Similar to neural networks [246] Fuzzy basis functions [301] Fuzzy systems [326] [336] Universal function approximators Key idea:

replacing the unknown nonlinear functions of the controlled systems with fuzzy systemslinear regression with unknown parameters

Robust adaptive control Semi-global stability



Motivation for model-based FLC

Conventional fuzzy control lacks systematic guidelines Many nonlinear systems are linear in local

regions Dynamic fuzzy models represent a large

class of nonlinear systems There are many mature design techniques in

conventional control field



Fuzzy model based control

Basic idea: integration of conventional control theory and fuzzy logic concept



Fuzzy dynamic models or T-S model

ml

txCty

atuBtxAtSxThen

FiszANDFiszIfR

l

lll

lqq

ll

,...,2,1

)()(

)()()(

...: 11

==

++=



Definition

The l-th fuzzy local dynamic model is a binary set defined as FLDMl : = (µl(x), (Al, Bl, Cl,al)), where (Al, Bl, Cl,al) represents the

crisp input-output relationship or dynamic properties of the system at a crisp point.



Using singleton fuzzifier, product fuzzy inference, centre-average defuzzifier

Dynamic global fuzzy model

==

==

==

==

=++=

m

lll

m

lll

m

lll

m

lll

aaCC

BBAA

txCty

atuBtxAtSx

11

11

)(,)(

,)(,)(

)()()()()()()()()(

µµµµ

µµµµ

µµµµ

4



Remarks

The fuzzy dynamic model includes two kinds of knowledge The model has the structure of a two level

control system Similar to the concept of ordinary piecewise

linear approximation methods in nonlinear control



Stability analysis

• Common quadratic Lyapunov functions (CQLF)• Piecewise quadratic Lyapunov functions (PQLF)• Fuzzy quadratic Lyapunov functions (FQLF)

)()()1(1

txAztx lm

ll

==+ µ



Common Quadratic Lyapunov Functions

PxxxV T=)(

Lyapunov functions:

Stability condition:

0)()()1()1()( <−++=∆ tPxtxtPxtxxV TT



0<− PPAA lTl Ll ∈

0<

−−

XXA

XAX

l

Tl Ll ∈

Stability conditions (LMIs):

Or

Remark: General conservative especially for highly nonlinear systems

Common Quadratic Lyapunov Functions



Example I- stability analysis

−=

−=

−=

−=

8.01.04.01

,8.0125.0

4125.01

,8.0275.0

4875.01,

8.03.05.01

76

21

AA

AA

No CQLF can be found to confirm its stability!



Membership functions

-4 -3 -2 -1 0 1 2 3 4 0

0.2

0.4

0.6

0.8

1

1.2

x1

1µ2µ 3µ 4µ

5µ 6µ7µ

5



Piecewise Quadratic Lyapunov Functions -I

limizzzS ill ≠=>= ,,...,2,1),()(| µµSpace partitions:

System model in local region:

))x(t)(A(A1)x(t ll µ∆+=+ lSz(t)∈

Ll ∈lAlAll EEAA ΤΤ ≤∆∆ )]([)]([ µµ




xPxxV lT=)( lSz ∈

Lyapunov functions:


0)()()1()1()( <−++=∆ txPtxtxPtxxV jT

lT




0)(

<

−−+−

lll

lTllA

TlAlll

Tl

PIAP

PAEEPAPALl ∈

0)(

<

−−+−

jlj

jTllA

TlAllj

Tl

PIAP

PAEEPAPAΩ∈jl,


,,,)1(,)(|),(: jlLjlStzStzjl jl ≠∈∀∈+∈=Ω




Ll ∈

Ω∈jl,

Stability condition (alternative):

0

0

0)( <

−−−

−

IXE

IXXAEXAXX

llA

lll

TlAl

Tlll

00

0)( <

−−−

−

IXE

IXXA

EXAXX

llA

jll

TlAl

Tlll



Piecewise Quadratic Lyapunov Functions -II

∈

=+)(

)()()1(lKk

kk txAztx µ

Space partitions:

System model in local region:

lSz(t)∈

crisp (operating) and fuzzy (interpolation) regions

Ll ∈




xPxxV lT=)( lSz ∈

Lyapunov functions:


0)()()1()1()( <−++=∆ txPtxtxPtxxV jT

lT

6



0<− lklTk PAPA Ll ∈ )(lKk ∈

0<− lkjTk PAPA Ω∈),( jl )(lKk ∈





Ll ∈ )(lKk ∈

Ω∈),( jl )(lKk ∈


Stability condition (alternative):

0<

−−

llk

Tkll

XXA

AXX

0<

−−

jlk

Tkll

XXA

AXX



Example I- Stability analysis

−=

−=

−=

−=

8.01.04.01

,8.0125.0

4125.01

,8.0275.0

4875.01,

8.03.05.01

76

21

AA

AA

PQLF can be found to confirm its stability!



Example I- Stability analysis

=

0330.00063.00063.00189.0

1P

=

0343.00070.00070.00183.0

3P

=

0376.00086.0

0086.00176.05P

=

0427.00094.0

0094.00168.07P

=

0333.00066.00066.00185.0

2P

=

0356.00078.00078.00179.0

4P

=

0403.00090.0

0090.00170.06P



Remarks

PQLF –I based partition introduces uncertainty terms and thus leads to conservatism

PQLF-II reduces to CQLF based approach if all the Lyapunov matrices are chosen to be the same

PQLF based approaches are less conservative than those based on CQLFs, but computation is more demanding



Fuzzy Quadratic Lyapunov Functions

=

=m

ll

Tl xPxzxV

1)()( µ

Lyapunov functions:


0)()()1()1()( <−++=∆ txPtxtxPtxxV jT

lT

7



Fuzzy Quadratic Lyapunov Functions

0<− lljTl PAPA Llj ∈,


0<

−−

jll

Tlll

XXAAXX

Or

Llj ∈,



Remarks

• FQLF based approach reduces to CQLF based approach if all the Lyapunov matrices are chosen to be the same

• FQLF based approaches are less conservative than those based on CQLFs, but computation is more demanding

• Continuous time case leads to difficulty• Stability analysis is much more involved with affine

terms, [140], [141], [77], [128], and [295]



Stabilization via CQLFs

)()(

...: 11

txKtuTHEN

FiszandFiszIFC

ll

lqq

ll

−=

)()()()(11

tKxtxKtutu lm

lll

m

ll −=−==

==µµ

Smooth controller:

+=+==

m

ljlllj

m

jtxKBAtx

11)()()1( µµ

Closed loop system:



Stabilization condition

0<

−++−

XQBXA

BQXAX

jll

Tl

Tj

Tl Ljl ∈,

1−= XQK ll

Controller gains:

LMIs:



Remarks

• Many improved approaches• Reducing the number of LMIs• Reducing the conservatism• Or both• [142], [192], [198], [267], [275], [286], [304]



Stabilization via CQLFs

lStz ∈)(

Switching controller:

)()( txKtu l= lStz ∈)( Ll ∈

)()))(()(()1( txKBBAAtx lllll µµ ∆++∆+=+

Closed loop system:

Ll ∈

8



Stabilization condition switching controller

Ll ∈

1−= XQK ll

Controller gains:

LMIs:

0

21

00

021

0

00)(<

−

−

−−++−

IQE

IXE

IXQBXA

EQXEBQXAX

llB

lA

lll

TlB

Tl

TlA

Tl

Tl

Tl



Remarks

• Switching controller approach has less number of LMIs

• Introduction of uncertainty terms might lead to conservatism

• Extension to H-infinity, H2 and other controller designs

• Extension to observer design, filter design• Extension to time-delay systems and uncertain fuzzy

systems



Examples

Inverted pendulum Ball-beam system Bench-mark robust nonlinear control

problem Backing-up control of truck-trailer MIMO coal pulverizer mill



Problems

Tend to be conservative Common quadratic Lyapunov functions

might not exist for highly nonlinear systems



Example II Stabilization

=

=

−=

−=

01

,10

,14.05.01

,15.04.01

21

21

BB

AA

No stabilization solution can be found by CQLF based approaches!



Region Partitions

S1

S2

S3

S4

X2

X1

9



Stabilization via PQLFs

limizzzS ill ≠=>= ,,...,2,1),()(| µµSpace partitions:

System model with switching controller in local region:

Ll ∈)()))(()(()1( txKBBAAtx lllll µµ ∆++∆+=+

xXxxV lT 1)( −= lSz ∈

Lyapunov functions:



Stabilization Conditions

Ll ∈

0

21

00

021

0

00)(<

−

−

−−++−

IQE

IXE

IXQBXA

EQEXBQAXX

llB

llA

lllll

TlB

Tl

TlAl

Tl

Tl

Tlll



Stabilization Conditions-cont.

0

21

00

021

0

00)(

<

−

−

−−++−

IQE

IXE

IXQBXA

EQEXBQAXX

llB

llA

jllll

TlB

Tl

TlAl

Tl

Tl

Tlll

Ω∈jl,

LlXQK lll ∈= − ,1

Controller gains:



Stabilization via PQLFs

Space partitions:

System model with switching controller in local region:

lSz(t)∈ Ll ∈

crisp (operating) and fuzzy (interpolation) regions

+=+∈ )(

)()))((()1(lKk

lkkk txKBAtztx µ



Stabilization Conditions

)(lKk ∈

0<

−++−

llklk

Tk

Tl

Tkll

XQBXA

BQAXXLl ∈ )(lKk ∈

0<

−++−

jlklk

Tk

Tl

Tkll

XQBXA

BQAXXΩ∈),( jl

LlXQK lll ∈= − ,1

Controller gains:



Remarks

• Improved approaches to reducing number of LMIs or conservatism

• Extension to H-infinity, H2 and other controller designs

• Extension to observer design, filter design• Extension to time-delay systems and uncertain

fuzzy systems

10



New Piecewise Lyapunov FunctionsContinuous time Case

1

0

1

0

,

,

,,

)(

LlFTFP

LlTFFP

LlSxxPx

LlSxxPxxV

lT

ll

lT

ll

llT

llT

∈=

∈=

∈∈∈∈=

jljl SSxxFxF ∩∈= ,

Lyapunov function:

Boundary condition:



New Piecewise Lyapunov FunctionsContinuous time Case

• It is difficult to develop controller design approaches via LMI techniques due to the special structure of the Lyapunov matrices

• BMI based approaches have been developed



Example II Cont.

=

=

−=

−=

01

,10

,14.05.01

,15.04.01

21

21

BB

AA



Example II: Solution with PQLFs

[ ] [ ]4038.00444.1,0444.14038.0

9607.18576.08576.02186.3

,2186.38576.08576.09607.1

21

21

−−=−=

=

−−

=

KK

PP



Example III - Henon map

( ) ( ) ( ) ( ) ( )( ) ( )

=+++++−=+

txtx

ttutxtxtx

12

2211

1

2sin01.04.13.01 π

( ) ( ) ( ) ( ) ( )( ) ( )

=++++−=+

tete

ttutetete

12

2211

1

2sin01.03.01 π

( ) ( )txte 11 = ( ) ( ) 3/1422 += txte



Fuzzy model

111 : FiseIfR

212 : FiseIfR

)()()()1( 222 tvDtuBteAteThen ++=+

)()()()1( 111 tvDtuBteAteThen ++=+

( )( ) ( )

−=d

teteF 1

11 121 ( )( ) ( )

+=dte

teF 112 1

21

30=d

,

11



Henon map - Parameters

=

013.0

1

dA

−=

013.0

2d

A

==

01

21 BB

==

0

01.021 DD

( )ttv π2sin)( =



Open Loop Response



Solutions

−−

=9243.10144.00144.00317.0

1P

=

4147.80567.0

0567.00315.02P

[ ]3.00246.301 −−=K [ ]3.09762.292 −=K



Closed loop responses



Stabilization via FQLFs

=

=m

lll txKtu

1)()( µ

Smooth controller:

Closed loop system:

+=+==

m

ljlllj

m

jtxKBAtx

11)()()1( µµ



Stabilization via FQLFs

=

−=m

ll

Tl xXxzxV

1

1)()( µ

Lyapunov functions:

12



Stabilization conditions

0<

−++−−

ijljl

Tl

Tj

Tl

Tjj

Tjl

XQBGA

BQAGGGXLlji ∈,,

LjGQK jjj ∈= − ,1

Controller gains:

LMIs:



Remarks

• Improved approaches to reducing number of LMIs or conservatism

• Extension to H-infinity and H2 limited• Extension to observer design limited



Stabilization via FQLFs :Continuous time case

difficultyx

xPxxxVm

ll

Tl

→→

= =

µ

µ1

)()(



Universal fuzzy controllers

))(),(()( tutxftx =

))(),((ˆ)( ttxgtu µ=

=

=m

lll txgtttxg

1))(()())(),((ˆ µµ

)))((ˆ),(()( txgtxftx =



Preliminary Results

Fuzzy dynamic models with affine terms are universal function approximator Fuzzy controllers are universal controllers

for a class of nonlinear systems



Challenges

What kind of nonlinear systems can be represented by FDM? How to determine FDM? Fuzzy controller design: piecewise Lyapunov

function or fuzzy Lyapunov function, other techniques How to use as much information of FDM as

possible? Universal fuzzy controllers Adaptive control based on FDM

13



Challenges

• Whether can fuzzy controllers, which are designed to stabilize T-S fuzzy models,stabilize the original nonlinear systems? How to achieve it if possible? Time-delay nonlinear systems Hybrid control systems• Stochastic systems Applications



Conclusions

Conventional control theory can be combined with fuzzy logic control Constructive designs are possible with

guaranteed performance for FLC





TS FUZZY MODEL AND UNIVERSAL FUNCTION APPROXIMATION

• General Result: Able to approximate any continuous nonlinear function on any compact set of interest.

• Let be continuous on compact set , for any ,– there exists a standard (i.e., Mamdani) fuzzy system

– there exists an affine T-S fuzzy system

such that

)(xf X

=

=m

lll axxF

1

)()( µ

ε<−∈ |)()(|max xFxfXx

0>ε

))(()(1

xAaxxF l

m

lll +=

=

µ




• Particular Result for control: Able to approximate any continuous nonlinear function with zero as its equilibrium point.

• Let be continuous differentiable on compact set with , there exists a linear T-S system

such that

)(xf X

ε<−∈ |)()(|max xFxfXx

xAxxF l

m

ll

=

=1

)()( µ

)0.,.( ≡laei0)0( =f




• Why? – The reason is that, when continuous differentiable

on compact set with , there exists a continuous matrix function as

such that . Using a Mamdani fuzzy system to approximate leads to the above result.

X

xxGxf )()( =)(xG

=∂

∂

≠−

=+

++

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

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

),...,(1

11

1

jj

nji

jj

njinjji

nij

xifx

xxf

xifx

xxfxxxf

xxg

)(xG

)(xf0)0( =f

14



Affine nonlinear control systems and T-S fuzzy control systems

Recall dynamic (affine) T-S fuzzy model

here assume , other cases will consider later. Question 1: What nonlinear systems can be

represented by above T-S fuzzy control systems?

xz =

==

==

==

==

=++=

m

lll

m

lll

m

lll

m

lll

axaCxC

BxBAxA

txCty

atuBtxAtSx

11

11

)()(,)()(

,)()(,)()(

)()()()()()()()()(

µµµµ

µµµµ

µµµµ




Answer to Question 1: Let

be a nonlinear system with (compact set) and zero as its equilibrium point. Then it can be approximated to any degree of accuracy by an affine T-S fuzzy model if and only if it is an affine nonlinear system. That is,

)](),([)( tutxNtSx =

)()]([)]([)](),([)( tutxgtxftutxNtSx +==

Xtx ∈)(




Why? can be approximated by an affine T-S fuzzy system

can be approximated by a standard fuzzy system

Therefore

can be approximated by

)(xf

)()()()()()( tuBtxAatSx µµµ ++=

xAaxAaxxf l

m

lllA )()())(()(

1

µµµ +=+==)(xg

l

m

ll BxB

=

=1

)()( µµ

)()]([)]([)( tutxgtxftSx +=




Questions needed to be further addressed Question 2. Most existing stability analysis and

stabilization methods assume that fuzzy control systems are

That is, Then the question is what nonlinear systems can be represented by such fuzzy control systems (called linear T-S fuzzy control systems from now on)?

0)( =xa

==

==

+=m

lll

m

lll BxBAxA

tuBtxAtSx

11

)()(,)()(

)()()()()(

µµµµ

µµ




Problems needed to be further addressed Question 3. Given a nonlinear observation function

, whether it can be approximated by fuzzy observation function

to any degree of accuracy?

=

==m

lll CxCtxCty

1

)()()()()( µµµ

)]([)( txHty =




Answer to Question 2: Let

be an affine nonlinear system with (compact set) and zero as its equilibrium point. If continuous differentiable, then it can be approximated to any degree of accuracy by a linear T-S fuzzy model as

Xtx ∈)()()]([)]([)](),([)( tutxgtxftutxNtSx +==

)(xf

==

==

+=m

lll

m

lll BxBAxA

tuBtxAtSx

11

)()(,)()(

)()()()()(

µµµµ

µµ

15




Answer to Question 3: Let nonlinear observation function be continuous differentiable with zero as its equilibrium point. then it can be approximated to any degree of accuracy by a linear T-S fuzzy observation model as

)]([)( txHty =

=

==m

lll CxCtxCty

1

)()()()()( µµµ




Why? can be approximated by a linear T-S fuzzy system

can be approximated by a standard fuzzy system

can be approximated by a linear T-S fuzzy system

)(xfxAxAx l

m

ll )()(

1

µµ ==)(xg

l

m

ll BxB

=

=1

)()( µµ

)(xH

xCxCxm

lll )()(

1

µµ ==




Summarize Answers to Questions 2 and 3: Let

be a continuous differentiable nonlinear system. Then it can be approximated to any accuracy by a linear T-S fuzzy control system

if and only if it is an affine nonlinear system. That is,

)]([)()](),([)( txHtytutxNtSx ==

)()]([)]([)](),([ tutxgtxftutxN +=

===

===

=+=m

lll

m

lll

m

lll CxCBxBAxA

txCtytuBtxAtSx

111

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

)()()()()()()()(

µµµµµµ

µµµ




Summarize Answers to Questions 2 and 3: Another representation:

That is, the original nonlinear system can be stabilizable if

is robust stabilizable under uncertainty

CBA ECEBEA

txCCtxHty

tuBBtxAA

tutxgtxftSx

εµεµεµµµ

µµµµ

≤∆≤∆≤∆∆+==

∆++∆+=+=

|)(|,|)(|,|)(|)()]()([)]([)(

)()]()([)()]()([)()]([)]([)(

)()()()()()()()( txCtytuBtxAtSx µµµ =+=

)(),(),( µµµ CBA ∆∆∆




Remarks: If a given nonlinear system is continuous

differentiable, then the representation capabilities of simpler linear T-S fuzzy systems are the same as affine T-S systems This means that more commonly existing methods for

stabilization based on linear T-S fuzzy systems are enough in most applications when a considered nonlinear system is an affine one.




Remarks: On the other hand, affine T-S fuzzy systems are useful If a given nonlinear system is only continuous, as some

continuous only nonlinear affine systems can not be approximated by linear T-S fuzzy systems. For example:

Affine T-S models require less local systems as the approximation accuracy of affine T-S systems are 3rd order whereas linear T-S systems are 2nd order (related to the width of fuzzy partitions)

)(|)(|)( tutxtx +=

16



General nonlinear control systems and T-S fuzzy control systems

As shown before that linear or affine T-S fuzzy systems are enough and only enough to represent affine nonlinear control systems. For a general (non-affine) nonlinear system, more

general T-S model are needed. That is, the following question needs to be solved Question 4. If a given nonlinear control system is

not affine one, how to represent it by fuzzy control system model?




Answers to Questions 4. Let

be a continuous differentiable nonlinear system on (compact set) with zero as its

equilibrium point. Then it can be approximated to any degree of accuracy by a general T-S fuzzy control system

where

)](),([)( tutxNtSx =

==

==

+=m

lll

m

lll BzBAzA

tuBtxAtSx

11

)()(,)()(

)()()()()(

µµµµ

µµ

UXtutx ×∈)](),([

),( uxz =




Why? can be approximated by a linear T-S fuzzy system

),( uxN

uBxAuBxAux ll

m

ll )()(][),(

1

µµµ +=+=

The rule base of such general T-S fuzzy control system can be written as

where

mltuBtxAtSxThen

UisuANDUisuFisxANDFisxIfR

ll

lmm

llqq

ll

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

,...,...: 1111

=+=

∏∏==

=m

jj

lj

q

ii

lil uUxFux

11

)()(),(µ




Remarks So far it is shown that it is not difficult to represent a

general nonlinear control system by a T-S fuzzy control system However, the control design becomes much

complicated and this is generally speaking still an open problem



fuzz-ieee 2007 2007 ieee international conference on fuzzy systems tutorial 2: model based fuzzy...

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