TUTORIAL on
Networked Control Systems with Delay
Cicsyn2010
2nd International Conference on Computational Intelligence,
Communication Systems and Networks
Liverpool, UK, July 29th, 2010
Vasilis Tsoulkas: Center for Security Studies, Athens, Greece
&
Dept. of Mathematics, University of Athens
Research Group:
- Pantelous Athanasios., University of Liverpool,
- Dritsas Leonidas.,
- Halikias George,
Hellenic Airforce Academy
City University, London, UK.
1
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Introduction – General Features
NCS Modeling - the issue of network induced delays
Discretization of NCS dynamics
Decomposing the Uncertain Delay (nominal and
uncertain parts) and the NCS dynamics
Robust Stability Analysis based on the augmented
closed-loop vector (“ξ”)
Design of a Simple Output Tracking Controller
Investigation of Robust Tracking Performance via
Simulation - Numerical Examples for Networked
Stable and Unstable systems
Conclusions & Topics for further study
2
Schematics of Networked Control Systems
Networked control systems (NCSs) are spatially distributed
systems for which the communication between sensors,
actuators, and controllers is supported by a shared
communication network. Hespanha et al.: “Survey of Recent
Results in Networked Control Systems” (Proceedings of the IEEE,
Vol. 95, No. 1, January 2007)
3
Motivation & Some Benefits

Easy and low cost installation, wiring, maintenance,
configuration
 Distributed Controllers and Plant with low cost
distributed sensors and actuators are all coupled over
the same Real Time communications network
 The distributed nature of elements offers great
flexibility of architectures.
 Applicable in a wide variety of fields such as: Remote
surgery, mobile sensor networks, UAV’s, Space teleoperations and Robotics.
4
Distributed Networked System
5
Control networks are indicated by solid lines,
and diagnostics networks are indicated by dashed lines.
6
1. Introduction
 Feedback control systems wherein the control loops are closed
through a real-time network are called Networked Control
Systems (NCSs)
 Defining feature of NCS: Information (reference input, plant
output, control input, etc.) is exchanged using a network among
control system components (sensors, controllers, actuators,
7
7
etc.).
1. Introduction

Νetwork Induced Delays
Information flow in the control loop is delayed due to
– buffering,
– access contention (the time a node waits until it gets access to the
network),
– computation delay (assume absorbed into {τca (k)} )
– propagation (“transmission”) delays.
–
Network-induced delays in NCS appear in the information flow
between (“k” denotes the dependence on the kth sampling period).
 A).
The sensor and the controller {τsc (k)}, (controller
receives “outdated” information about process behavior)
 B).
The controller and the actuator {τca (k)}, (control action
cannot be applied “on time” and the controller does not know the
exact instance the calculated control signal will be received by the
actuator)
8
1.
Introduction
Νetwork Induced Delays
When a static linear time
invariant controller is
employed, can lump the
delays τsc (k), τca (k),
into
τk= τsc (k)+τca (k).
Network-induced delays in NCS
between the sensor and the
controller {τsc (k)}, and between
the controller and the actuator
{τca (k)}, (“k” denotes
the dependence on the kth
sampling period).
9
1.Introduction
Tracking Control Design for NCS
The Usual Approach for NCS Analysis & Design:
– design a controller ignoring the network, then
– analyze stability, performance and robustness with respect to the
effects of network-delays and scheduling policy…(usually via the
selection of an appropriate scheduling protocol).
The issue of Tracking Control over Networks has not been
adequately met
– very limited published work on NCS Tracking !!!
– the majority of NCS publications concerns regulation ,
(“design a controller which brings the output/state to “0” )
– many results on tracking for Time Delayed Systems (TDS) but
cannot be applied “as is” to NCS due to the “Networkcentric” nature of NCS e.g.
 special nature of delays in NCS
 the fundamental issue of “Packet Loss/Drops”
 Scheduling, Quality of Service, Middleware
10
1.Introduction
Tracking Control Design for NCS
Concerning NCS Robust Tracking Performance…
– only preliminary results - no strict mathematical proofs
– yet…useful “lessons learned” through extensive simulations on
S.I.S.O systems
– we investigate both constant unknown or time-varying uncertain
delays with known bounds
– we do not take into account the network delays in the tracking
controller design process…
– “a posteriori” analysis of stability, performance and
conservatism of results
– we do not take into account “packet drops”
– Analysis & Synthesis in the continuous time domain
– No need to assume knowledge of the P.D.Fs (not a stochastic
approach)
11
2. NCS Modeling
NCS with network-induced delays in the actuation and sensor path
Assumptions made …
• the dynamics of the NCS under investigation is a combination of a
continuous–time LTI plant with a discrete–time controller.
• Time Invariant controller  can lump τsc (k), τca (k), into τk= τsc (k)+τca
(k).
• Single source of uncertainty and performance degradation  the
lumped transmission delay τk.
12
• No plant uncertainties or nonlinearities - No packet drops
2. NCS Modeling - Assumptions
In Practice…
 the dynamics of the NCS under investigation is a
combination of a continuous–time uncertain/nonlinear plant
with a discrete–time (“sampled-data”) controller.

The sampler is time-driven, whereas both controller and
actuator are event-driven, (=they update their outputs as
soon as they receive a new sample).

Some packets are lost or intentionally dropped (contain
obsolete/useless info)
13
The delays τksc , τkca , τk < h

τksc = τsc (k) is the delay experienced by a state or output
sample x(kh), y(kh), sampled at time instance “kh” and
presented –after a delay τksc to the event–driven remote
controller for control computation purposes.
 τkca =τ ca (k) is the delay experienced by the control–action,
computed immediately after its reception at time instance
kh+ τksc until it is transmitted via the network to the Z.O.H
(and finally presented to the event–driven actuator).
 The computation delay is absorbed into τ kca
14
The delays τksc , τkca , τk < h

τ k : Total delay within the kth sampling period,
i.e. the time from the instant when the sampling node
samples sensor data from the plant to the instant when
actuators exert a control action –whose computation
was based on this sample– to the plant.
τk= τksc + τkca
(since a static time invariant control law is employed)
Known Bounds:
0 ≤ τ min < τk < τ max ≤ h

15
NCS Timing Diagram (τk < h) for “short” (τk < h)
+ bounded delay 0 ≤ τ min < τk < τ max ≤h
16
2. NCS Modeling
Difficulties in case of Discrete “Sampled Data” Controller
û(t) is the “most recent” control action presented to the event–driven
actuator at the time instance t within a sampling period [kh, kh + h)
& can take two values ûk or ûk-1
 û(t) experiences a “jump” at the uncertain or unknown time
instance kh+ τ k , changing from ûk-1 into ûk (uncertain actuation
instance)
 Very Complicated Dynamics  Impulse Delayed Systems,
Asynchronous Dynamical Systems, Hybrid Systems, etc even for
the regulation case (r=0)

17
2. NCS Modeling - the issue of network-induced delays
NCS Timing Diagram form Zhang &
Branicky paper (IEEE Control
Systems Magazine, Febr.2001).
Possible misconceptions if symbols
are not adequately clarified…
Authors clarify that the confusing
symbol “u(kh)” denotes the actuation
that takes place at kh+ τk and its value
is u(kh) = -Kx(kh)
Hence (unless τk is constant) it is not possible to treat the ensuing NCS in
a standard “sampled data” or “time-delayed” setting. Instead a “hybrid”
setup should rather be used, as for example the one presented in
P. Naghshtabrizi and J. P. Hespanha, “Stability of network control systems with
variable sampling and delays” in Proc. of the 44th Annual Allerton Conf. on
Communication, Control, and Computing, 2006.
18
CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
Introduction – General Features
NCS Modeling - the issue of network-induced delays
Discretization of NCS dynamics
Decomposing the Uncertain Delay (nominal and
uncertain parts)
Robust Stability Analysis based on the closed-loop
augmented vector (“ξ”)
Design of A Simple Output Tracking Controller
Investigation of Robust Tracking Performance via
Simulation - Numerical Examples for Networked
Stable and Unstable systems
Conclusions & Topics for further study
19
3. Descretization of NCS state equation with “small delay”
x(t )  Ac x(t )  Bcuˆ(t ) and y(t )  Cc x(t )
τk < h  xk = x(kh)
xk+1 = Φ xk + Γ0(τk) ûk + Γ1 (τk) ûk-1
(Σ1)

notation xk, xk-1,… denotes the values x(kh), x(kh-h), … of the
periodically sampled discrete–time signal coming out of the
sampler. The same notation for yk, yk-1,…

We keep the “hat” notation for ûk , ûk-1 as a reminder of the
asynchronous, (“jump”) nature of these signals.

Õn … is an n-column zero vector, In is the n x n identity matrix, 0n
is the n x n zero matrix.

MT is the transpose of a matrix. M > 0 (< 0) means that M is
positive (negative) definite.
20
3. Discretization of state equation dynamics of NCS
(Comments)
xk+1 = Φ xk + Γ0(τk) ûk + Γ1 (τk) ûk-1
(Σ1)
 Exact Discretization between equidistant sampling
instances  finite dimensional dynamics…
The uncertain time varying delay τk can still take any
(out of infinite) values within the allowable interval
 the uncertainty of τk  generates an uncertainty in the
actuation instance 
System matrices (Γ0(τk), Γ1(τk)) are uncertain
Presence of a delayed input term ûk-1
21
Exact Discretization despite the “jump’’ nature of û(t)
xk = x(kh), Φ = exp(Ach)
State Equation: x(t )  Ac x(t )  Bcuˆ (t ), y (t )  Cc x(t ) (1)
leads to
kh  k
x(kh  h)  exp( Ac h) x(kh) 

kh  h
exp( Ac (kh  h - s )) Bcuˆk -1 
Define the three matrices ,0 , 1
 exp( Ac h), 1 ( )
k

λ

exp( Ac (kh  h - s )) Bc ds,  0 ( )
λ
kh  h

exp( Ac (kh  h - s )) Bc ds
kh  
kh
x(kh  h)  x(kh)  0 (t )uˆk  1 (t )uˆk -1 , where: 0 ( ) 
k
exp( Ac (kh  h - s )) Bcuˆk ds
kh  k
kh
kh  

k
k
h  k

exp( Ac  ) Bc d  , 1 ( ) 
k
h
 exp( A  ) d 
c
c
h - k
0
we have used the following:   kh  h - s ( so d   -ds since h const.) and changing the variable of integration into
d   -ds, the new limits of integration are (h -  k ) and 0 so we get the simplified expression for Γ0 :
0 ( k ) 
kh  h

kh 

exp( Ac (kh  h - s)) Bc ds 
0
h- k
 exp( Ac )c d (- )  0 exp( Ac )cd 
h- k
22
Exact Discretization despite the “jump’’ nature of û(t)
xk = x(kh), Φ = exp(Ach)
• Similarly from the definition of Γ1, using the same change of variables as
previously
k
(  kh  h - s  d   -ds ) the new limits of integration are h and (h -  ) so we get :
1 ( ) 
k
kh  

h - k
exp( Ac (kh  h - s)) Bc ds 
kh

exp( Ac  )c d (- ) 
h

exp( Ac  )c d 
h - k
h
Moreover assuming there is no uncertainty on output matrix we have : y (kh)  Cc x(kh) or yk  Cc xk
A usefull identity for calculating integrals of matrix exponential functions
( such as  0 ( k )).
Xn x n
exp( 
 0m x n
To Compute  0 ( k ) 
h - 

 (X t)
Yn x n 
e
t
)


0m x m 
0m x n
t
e

Ydr 


( Xr )
0
Im
e( Ac  ) c d  we can use above identity as:
0
 0T 
 0 ( )  [ I n 0 ] e
  Identity I.
1 
where 0 01 x n the zero row vector with n columns
k
T
Ac Bc 
( 
(h -  k ) )
0
0 

23
Exact Discretization despite the “jump’’ nature of û(t)
xk = x(kh), Φ = exp(Ach)
• Notice that :
(3.A).
(3.B).
From 2nd equation : 1 ( k ) 
h

h-
e( Ac )c d  
0

h- k
k
h
e( Ac )c d    e( Ac )c d 
0
h
= - 0 ( k )   e( Ac )c d 
0
24
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Introduction – General Features
NCS Modeling - the issue of network-induced delays
Descretization of NCS dynamics equation
Decomposing the Uncertain Delay (nominal and
uncertain parts) and the NCS dynamics
Robust Stability Analysis based on the closed-loop
augmented vector (“ξ”)
Design of A Simple Output Tracking Controller
Investigation of Robust Tracking Performance via
Simulation - Numerical Examples for Networked
Stable and Unstable systems
Conclusions & Topics for further study
25
4.
Decomposing the uncertain delay of the system
(into nominal & uncertain part)
Examples:
τo = τ min
τo = τ max
τo = τ avg



τo is chosen as constant and known («semi-arbitrary»)
Use of “Min Max” techniques for selection of τo
The nominally delayed system, Stability Analysis and
Controller Synthesis depend on the (user’s) choice of τo
26
4.
Decomposing the uncertain delay of the system
k
(into nominal & uncertain part)0 ( )
(4.C).
so the nominal part of  0 (  ) is
and the uncertain part of is
0 (  )
ΔΓ 0 (  )
h  
0
e( Ac  ) c d  and
h  k

h 
e( Ac ) c d 

27
4.
Decomposing the uncertain delay of the system (into
nominal & uncertain part)  ( k )
1
4.D
h  0

The Uncertain part is: 1 ( k )
e( Ac  ) c d  and
h  k
the Nominal Part is : Γ1 ( 0 )
h

e( Ac  ) c d 
h  0
28
4. Decomposing the uncertain delay of the system
(into nominal & uncertain part)
h
The nominal part can be calculated from: 1 ( 0 )  0 ( 0 )   e( Ac )c d 
0
Moreover from previous ΔΓ0 ( k ) and 1 ( k ) we observe the following relation
between the two uncertain matrices:
1 ( k )   0 ( k ),  k [ min , max ] 4.E
29
Contents
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
NCS Modeling - the issue of network-induced delays
Descretization of NCS dynamics equation
Decomposing the Uncertain Delay (nominal and
uncertain parts)
Robust Stability Analysis based on the augmented
closed-loop vector (“ξ”)
Design of Simple Output Tracking Controller
Investigate Robust Tracking Performance via
Simulation
Conclusions & Topics for further study
30
5. Robust Stability Analysis based on the closed-loop
vector augmented (“ξ”)
- Closing the loop



THE AUGMENTED CLOSED–LOOP STATE VECTOR
(“ACLSV”) “ξ”
xk+1 = Φ xk + Γ0(τk) ûk + Γ1 (τk) ûk-1
ûk = -Ksf xk
Static State Feedback (SSF)
ûk-1 = -Ksfxk-1
Closed Loop Dynamics
 xk+1= [Φ- Γ0(τk) Ksf] xk + [- Γ1 (τk) Ksf ] xk-1
only periodically sampled state vector values {xk+1, xk, xk-1 }
are present
31
5. Robust Stability Analysis based on the closed-loop vector
(“ξ”)
Define the augmented
“sampled data”
closed-loop state vector
Assuming there is no uncertainty on output matrix it holds:
yk  Ck xk  Cc [ I n 0n ] k  [Cc 0] k  C o k
0n  is the zero square matrix (n x n).
32
5. Robust Stability Analysis based on the closed-loop vector (“ξ”)
The Closed loop Matrix A sf  Asf ( k , K sf ,  )  R 2 n x 2 n is a function of :
the uncertain delay τ k
the preselected gain K sf and
the predetrmined nominal delay το
The above matrix relation is
manageable and Robust Control
Methods now can be used.
Using 4C, 4D, 4E : Asf can brake down to a nominal time invariant part A osf
and an uncertain part  sf ( k ) as :
  0 (  ) sf -1 (  ) sf  -ΔΓ0 (  ) sf -ΔΓ1 (  ) sf 
Asf  
+

In
0n
0n
0n

 

Asfo  sf
33
Contents
1. Introduction
2. NCS Modeling - the issue of network-induced
delays
3. Discretization of NCS dynamics
4. Decomposing the Uncertain Delay (nominal and
uncertain parts) and NCS dynamics
5. Robust Stability Analysis based on the augmented
closed-loop vector (“ξ”)
6. Design of Simple Output Tracking Controller
7. Investigate Robust Tracking Performance via
Simulation
8. Conclusions & Topics for further study
34
6. Design of Simple (Set Point) Tracking Controllers
 SPCT = Set Point Tracking Controller(s)
The reference signal to be tracked by the output is (piecewise)
constant (a “set point”)
–Assumption:both the plant and the controller under
investigation are continuous–time LTI systems
–Since the controller is time invariant, can lump the delays τsc (k), τca
(k), into τk= τsc (k)+τca (k).
– A “naïve” tracking controller consists of two parts: Feedback &
Feedforward u(t) = -Kx(t)+Fr
 The feedback part (-Kx(t)) assures closed-loop stability
 The feedforward part (Fr) assures that the static gain is “1”
(Stable Transfer Function from r to y)
35
6. Design of Simple (Set Point) Tracking Controller
•Suffers from three drawbacks (“naïve”):
• the plant must not contain integrators (system matrix A is
nonsingular)
• cannot handle disturbances and/or model uncertainties (it
is NOT Robust)
• Number of inputs ≥ Number of outputs (“overactuation”) 36
Contents
1. Introduction
2. NCS Modeling - the issue of network-induced
delays
3. Descretization of NCS dynamics equation
4. Decomposing the Uncertain Delay (nominal and
uncertain parts)
5. Robust Stability Analysis based on the closed-loop
vector (“ξ”)
6. Design of Simple Output Tracking Controller
7. Investigate Robust Tracking Performance via
Simulation
8. Conclusions & Topics for further study
37
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
• A “benevolent” stable & minimum phase (=zeros in LHP) system
with infinite Gain Margin and…
• “Lightly Damped” = stable poles close to the Imaginary axis 
“damping ratio” is small  damped oscillative open-loop behaviour
(typical in aerospace and “flexible space structure” applications)
•SPTC was designed via LQR with R=1, Q=1000*I2
u(t) = -30.63 x1(t) - 30.63 x2(t) + 31.63*r
gives “perfect tracking” in the absence of delays
38
7. Robustness of Tracking Performance
Nmerical Example 1: a networked stable & minimum phase system
with constant delay
The Networked Version with constant delay τk
τsc =τca = 0.0131 s  τk= τsc +τca=0.0262s
•Assuming that τk ≤ h this delay corresponds (for the discrete time
control case) to a sampling frequency of 38Hz a relatively “slow
sampling”…
• “slow sampling” is typical for NCS (fast sampling  increases # of
packets  increases network traffic  increases chances for
collisions  packet loss/drops)
39
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with constant delay
The Networked Version with constant delay τk
τsc =τca = 0.0131 s  τk= τsc +τca=0.0262s
• 7th order Pade Approximation used in simulations for the
constant time-delay
•Reference Signal(s) r are (piecewise) constant:
• combination of step functions or
• square pulse with period slower than the system’s time
constants
• Simulation needs time for Instability to occur (see next Figs)
40
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with constant delay
41
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with constant delay
42
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with uncertain (time-varying) delay
• The Networked Version with uncertain time-varying delay τk
varying between
τmin = 0 and τmax = 0.0312s < h
corresponding to a sampling frequency of 32 Hz
• Implementation used in simulations:
τk = τo + δ τunc , |δ| < 1
with τo = τavg = (τmax + τ min )/2 = 0.0156 s being the “mean value” (a
constant “nominal” delay) and |δ|<1 being a random variable of
uniform distribution.
43
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with uncertain (time-varying) delay 0=τmin ≤ τk ≤ τmax = 0.0312s
An instance of the actual
uncertainly varying delay
used in simulations
τk = 0.0156 + 0.0156* δ
|δ| < 1
τk = τo + δ τunc , |δ| < 1
τo = τavg = (τmax + τ min )/2
44
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with uncertain (time-varying) delay
τk = 0.0156 + 0.0156 *δ |δ| < 1
45
7. Robustness of Tracking Performance
Numerical Example 1: a networked stable & minimum phase system
with uncertain delay
46
7. Robustness of Tracking Performance
Numerical Example 2: a networked unstable system
•SPTC was designed via LQR with R=1, Q=100*I2
u(t) = -9.05 x1(t) -10.78 x2(t) + 10.05 *r
gives “perfect tracking” in the absence of delays
•The “Q” matrix was selected “small” in order to avoid high
feedback gains…and yet
47
7. Robustness of Tracking Performance
Numerical Example 2: a networked unstable system with constant delay
τk= τsc +τca=0.0155s
48
7. Robustness of tracking performance…….some
comments

Many more simulation results with different 2nd order
“benchmark” S.I.S.O systems from the literature (not shown)

But….we can deduce useful conclusions (despite the lack of
a mathematically rigorous approach)

Clearly a more sophisticated approach is needed for the
design of tracking controllers for NCS

We cannot “pretend” that the “delays are not there” - must
take them into account in the design phase.

We can not compromise stability (avoid large gains) - rule
of thump for Time-Delayed -Systems (mid ’50s result !!!)
49
CONCLUSIONS AND FUTURE WORK
 1. The constant delay case (contrary to intuition) is as
detrimental to tracking performance as the varying delay case.
2. The feedback gain must be kept “small”.
–If an LQR design is employed : extensive trial-and-error
simulations with various “Q” matrices must be carried out for the
entire delay range to ensure (at least) stability.
–Tracking for the case of unstable plants and/or lightly damped
plants is not trivial.
3. For Unstable plants it is always difficult to enforce tracking
(with or without delays).
4.
When implementing the tracking controllers in discretetime special attention is needed due to (1) the interplay between
sampling period and delay and (2) the “asynchronous / jump”
nature of the control signal
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Last Minute Thoughts – Dynamical Systems with Time Delays

Consider the time delay systems:
x(t )  Ax(t )  Ad x(t  d ) (S1 ) x(t )  Ax(t )  Ad x(t  d (t )) (S2 )
x(t )   (t ), t [-d , 0]
x(t )   (t ), t  [-d M ,0]
state x(t)  R n and  (t) is the continuous I.C.
Asumptions :1. d (t ) is a Cont. function satisfying 0  d(t)  d M
with d M a constant positive scalar. It is the upper bound of d(t)
2. d(t) is a differ. function satisfying 0  d(t)  d M and d(t)    1.
d M is given previously and  is the u.b. of d(t).
3. d(t) is a differ. function satisfying 0  d m  d (t )  d M and d(t)   .
d m , d M are given pos. scalars representing l. u. bounds of interval time varying
delay d(t) and ρ as given previously.
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Theorem 1.( S1 ) is asymptotically stable if
there exist matrices P > 0 and Q > 0
 A T P  PA  Q PAd 
such that : 
0
T
Ad P
-Q 

where the Lyapunov - Krasovskii functional was used:
t
V(t) = xT (t ) Px(t ) 

xT ( s )Qx( s )ds
t d
Theorem 2. Under assumption 2, (S2 ) is As. Stable if there exist matrices P > 0
 A T P  PA  Q
PAd 
and Q > 0 such that : 
0
T
Ad P
-(1- )Q 

the Lyapunov - Krasovskii functional of the form
t
V(t) = xT (t ) Px(t ) 

t d (t )
xT ( s)Qx( s)ds is used.
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CONCLUSIONS AND FUTURE WORK





Generalize achieved results for
– MIMO NCS plants with multiple delays, Parametric
Uncertainties & Actuator constraints
The use of Robust Control Methodologies (H∞ or
“Guaranteed Cost”) for the design of Feedback Gain
The employment of Integral Action (apart from feedback
and feedforward terms) in the tracking control
Algorithm(s).
Investigate Specific Applications: Aerospace &
Robotics (Teleoperation)
NCS’s indeed constitute a very interesting and rich field
of control systems both in theoretical results as well as in
future applications.
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THANK YOU
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