Time-Delay Systems for - Inriaresearchers.lille.inria.fr › ~jrichard › pdfs ›...

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Time-Delay Systems forNetworked Control problems

Networks such as Internet or Wireless 802.11 present great advantages forflexible and low-cost networking. However, they are not as reliable as CANs,and integrating them in control applications, while preserving someperformance, constitutes an interesting challenge.

It will be explained how using delay models allows for catching many of theeffects introduced by the presence of such unreliable networks in the controlloops (such as variable transmission times, lost packets, non-uniformscheduling and sampling).

Various theoretical techniques allow for analyzing the resulting models. Some ofthem need the delays to be constant, which can be obtained by using waitingstrategies involving buffers and leading to time-driven solutions. Some othertolerate fast varying delays and allow for event-driven, faster solutions.Two experimental cases, remote control with state prediction and bilateralteleoperation over Internet, will illustrate the talk.

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Time-Delay Systems forNetworked Control problems

Jean-Pierre Richard

http://researchers.lille.inria.fr/~jrichard

[email protected]

CEIT’2015 – Tlemcen – May 2015

EMETTEUR 00 MOIS 2011

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The team(permanent staff)

LOTFI DENIS THIERRY WILFRID ANDREY J.-PIERRE ROSANE GANG ALEX NICOLAI CHRISTOPHE LAURENTIU ANNEMARIE

BELKOURA EFIMOV FLOQUET PERRUQUETTI POLYAKOV RICHARD USHIROBIRA ZHENG KRUSZEWSKI CHRISTOV FITER HETEL KOKOSY

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• Motivation and examples

• Modelling

• Sampling and delay

• A crude example

• Control : to buff, or not to buff? A selection of results

• Conclusions


• Modelling


• A crude example


• Conclusions

OverviewOverview

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Tele surgery: the Lindbergh operation, 07/09/2001

Constant RTT< 200 msecDistance:17 000 km

Com. cost ≈ 160k$/month

« The only restriction to the development of long-distance tele-surgery has to do, still today, with itscost. For tele-surgery, you must use a transcontinental ATM line, that you have to book during 6monthes, at the price of about 1 million dollars. » Prof. J. Marescaux, Le Monde, January 6, 2010

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otherexamples…

Bilateral Teleoperation

Avoiding electrical cabling

Remote Monitoring

Robot collaboration

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and an example from everyday life…

Natural loop of audio-phonatory control

TO BE OR NOT TO BE, THAT IS THE QUESTION: WHETHER

‘TIS NOBLER IN THE MIND TO SUFFER THE SLINGS AND ARROWS OF OUTRAGEOUS FORTUNE. OR TO

TAKE ARMS AGAINST A SEA OF TROUBLES, AND BY OPPOSING,

END THEM? TO DIE: TO SLEEP; NO MORE; AND BY A SLEEP

TO SAY WE END

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and an example from everyday life…

Networked loop of audio-phonatory ctrl

G…

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Some surveys…• Control methodologies in Networked Control Systems

Y. Tipsuwan, M.Y. Chow, Control Eng. Practice 11, 1099-111, 2003

• Networked Control System: a brief survey T.C. Yang, IEE Proc. Control Theory Appl., 153 (4), 2006

• Control over Wireless Networks K.H. Johansson, 25th Benelux Meeting on S&C, The Netherland, 2006

• A survey of recent results in Networked Control SystemsJ.P. Hespanha, P. Naghshtabrizi, Y. Xu, Proc. of the IEEE, 95 (1), 2007

• Trends in Networked Control SystemsS. Zampieri, 17th IFAC World Congress, Seoul, 2008

• A switched system approach to exponential stabilization through com. network A.Kruszewski, WJ.Jiang, E.Fridman, J.P.Richard, A.Toguyeni, IEEE CST, 20 (4) 2012

nice (wrt 2007)

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not forgetting…

C. Canudas de Wit, T. Divoux, P. Fraisse, D. Georges, J.P. Georges, G. Juanole, A. Lelevé, F. Lepage, F. Michaut, G. Mouney, W. Perruquetti,

J.P. Richard, E. Rondeau, O. Sename, A. Seuret, E. Wittrant.

2007

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• Modelling


• A very basic example


• Conclusions

Overview©

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Single-loop NCS

Tipsuwan 2003

Tipsuwan 2003

What about multi-loop?

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Yang 2006

Multiple-loop NCS according to [Yang 2006]:

Yang 2006

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Multiple-loop NCS according to [Hespanha 2007]:

Hespanha 2007

Hespanha 2007

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Single-loop NCS.

Measurement

channel

1 - Controller 2 - PlantActuating

channel

Network

… the talk will be limited to this case, too.

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Single-loop NCS

Measurement

channel


channel

Network

Types of networks:

�dedicated (ControlNet, DeviceNet) : frequent transmission of small packets � guaranteed time but €�Ethernet, wifi : rare transmission of bigger packets � no guarantee for delay but €

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Effects of the network on the closed-loop control

IssueTranslation in control terms

ConcernedNot

concerned

limited bandwidth

quantification, quantité d’info par seconde limitée (Shannon, maximum bit rate)

systèmes à ressources énergie limitée (UAVs, réseaux de µ-capteurs ou de µ-actionneurs…)

envoi par paquets type Inter/Ethernet, Bluetooth… 1 bit ou 300 � idem ATM=384, Ethernet>368, Bluetooth>499

sampling, coding, scheduling, transmission, asynchronism

retards variables,estimés si modèle ou time-stamps

systèmes à transmission par paquets

réseau spécialisé (ControlNet, DeviceNet)

non partagé

packet losseséchantillonnage asynchrone, retard variable

sans fils, UDP

protocoles type TCP (mais peu utile : perte de temps pour infos périmées)

out-of-sync clocks

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ConcernedNot

concerned

limited bandwidth

quantification, limited quantity of info per second (Shannon, maximum bit rate)

limited energyresource systems(UAVs, WSN, µ-sensorsor µ-actuators, aerosp…)

packet-transmiss. type Inter/Ethernet, Bluetooth… 1 bit or 300 � idemATM=384, Ethernet>368, Bluetooth>499


retards variables,estimés si modèle ou time-stamps

systèmes à transmission par paquets

réseau spécialisé (ControlNet, DeviceNet)

non partagé


sans fils, UDP


out-of-sync clocks

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ConcernedNot

concerned

limited bandwidth





variable delays,estimated if there is a model, or time-stamps

packettransmission systems

dedicated and unshared netw. (ControlNet, DeviceNet)


sans fils, UDP


out-of-sync clocks

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ConcernedNot

concerned

limited bandwidth








packet lossesasynchronoussampling, variable delay

wireless, UDP-like protocols

TCP-like protocols(but generally useless: time wasting for outdatedinfo)

out-of-sync clocks

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ConcernedNot

concerned

limited bandwidth








packet lossesasynchronoussampling, variable delay

wireless, UDP-like protocols

TCP-like protocols(but generally useless: time wasting for outdatedinfo)

out-of-sync clocks delays (at least) internetcontrol-dedicatednetw. (CAN bus…)

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In such a framework, let’s show we have the equation:

transmission + sampling + packet loss = 1 delay .transmission + sampling + packet loss = 1 delay .transmission + sampling + packet loss = 1 delay .transmission + sampling + packet loss = 1 delay .

Measurement

channel


channel

Network

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Transmission delays

RTT (40km)Mean = 82 msMaxi = 857 msMini = 1 ms

Lille-Lens (France, 40 km) – Source: PhD WJ. Jiang 2008

To give an idea on the delay variation…

France - France

one week of RTT…

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One week of RTT…

Tunis-Lille (1640 km) – Source: S. Belhaj, ENSIT 2009

Transmission delays


RTT (1640km)Maxi = 415 msMini = 70 ms

RTT (1640km)Maxi = 415 msMini = 70 ms

France – North-Africa

one week of RTT…

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Transmission delays

unshared CAN 2m: 200 µsec

bluetooth 2m: 40 msec

Internet: 100-400 msec

orbital stations: 0.4-7 sec

underwater 1.7km: >2 sec


Other RTT approximated values

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• Motivations and examples

• Modelling




• Conclusions

Overview

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ContinuousProcess

Computer

A/DConverter

D/A Converter+ ZoH

x(t) x(nT)

u(t)=u

Aperiodic sampling

u*(nT)n

ContinuousProcess

Remote(or non prioritary)

Computing

A/DConverter

D/A Converter+ZoH

x(t)

u(nT(t))communication network

(or task scheduling,or packet loss...)

Sampling via Delay / 1

A

Integration, z-transform, everything’s ok, etc.

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Sampling via Delay / 2Time-varying sampling:� any consequence?

[Zhang, Branicky, Phillips. - IEEE Ctrl.Syst.Mag. 2001]

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Sampling via Delay / 3Time-varying sampling:� any consequence?

[Gu, Kharitonov, Chen - Birkhauser 2003]

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Models for aperiodic sampling

Hybrid / Impulsive syst.Input-Output approach

Input delay systems

"Tutorial on Arbitrary and State-Dependent Sampling "C. FITER, H. OMRAN, L. HETEL, J.P. RICHARD

European Control Conf., Strasbourg, France, 2014

"A robust stability framework for time-varying sampling"C. FITER, L. HETEL, W. PERRUQUETTI, J.P. RICHARD

Automatica, 45: 56-64, 2015.

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Input delay systems

Mirkin TAC 07, Fujioka Aut 09, Kao ACC 14

Omran-Hetel-Richard-Lamnabhi ECC 13 & Aut 14…

Input-Output approach Hybrid / Impulsive syst.



e(t) = y(t) - y(sk )

+ robust control techniques

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Input delay systems



Dullerud -Lall S&CL 99, Hu et al. TSMC 03,

Nešić -Teel TAC 04, Naghshtabrizi-Hespanha-Teel S&CL 08 …

+ stability techniques for HyS

discontinuous Lyapunov functions

Molchanov-Bauer TAC 99, Hetel-Daafouz-Iung TAC 06,

Fujioka TAC 09, Cloosterman et al. Aut 10,

Hetel-Kruszewski-Perruquetti-Richard TAC 11

Hybrid switched systems + convex embedding

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Input delay systems

Mikheev-Sobolev-Fridman AutRem.Ctrl 1988, Nešić -Teel-Kokotović CDC1998,Fridman-Seuret-Richard Aut 04, Fridman Aut 10, Seuret Aut 12, Karafyllis-Krstić TAC 12,Mazenc-Malisoff-Dinh Aut 13

+ stability techniques for TDS

x(tk) = x(t −[t − tk] ) = x(t − h(t) )

Lyapunov-Krasovskii functionals

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Thus… another statement of the packet loss problem

• maximum nb of successively lost packets � hmax

• piecewise-continuous delay with

delay


1 lost packet ↓

2 lost packets ↓ ↓

0 ≤ h(t) ≤ hmax

h(t) ≤ 1dtd

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How fast should we sample?

Another question in networked systems: How to reduce the number of the sent packets?


versus

Reduce the packets nb so to:- better schedule other tasks- reduce the communication cost- save wireless sensor batteries

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"A robust stability framework for time-varying sampling“ C. FITER, L. HETEL, W. PERRUQUETTI, J.P. RICHARDAutomatica , 45: 56-64, 2015.



Various ways of controlling the sampling instants:

� Event-triggering: dedicated sensor hardware (send yk when…)(Donkers-Heemels 2012; Lunze-Lehmann 2010; Postoyan-Anta-Nesic-Tabuada, 2011; Tabuada, 2007)

� Self-triggering: the previous without dedicated hardware (online estimation)(Anta-Tabuada, 2010, 2012; Mazo-Anta-Tabuada, 2010; Wang-Lemmon, 2009, 2010)

� Periodic event-triggering: the previous but estimation at periodical instants(Arzén 1999; Heemels-Donkers-Teel, 2013; Peng-Yang, 2013; Postoyan-Anta-Heemels-Tabuada-Nesic, 2013; Yue-Tian-Han, 2013)

� State-dependent sampling: partition of the state space into conic regions (off-line) (Fiter-Hetel-Perruquetti-Richard, 2012, 2015)

see cross-links and tools in

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State-dependent sampling (state x k )

"A state-dependent sampling for linear state feedback"C. FITER, L. HETEL, W. PERRUQUETTI, J.P. RICHARD

Automatica 48 (8), p.1860-1867, 2012


� Self-triggering

� Complexity is off-line

� LRF technique

Max next sampling computed off-line ( β-stab.) � often less than with periodic sampling

state-spaceangle θ

x2

x1

angle θ (rad) �Max

sam

plin

g τ

(sec

)� slow↑

↓

fast

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Hybrid systems modeling (Teel et al.)"Robust stab. of bilinear syst. under aper. sampled-data ctrl "H. OMRAN, L. HETEL, J.P. RICHARD, F. LAMNABHI-LAGARRIGUE

Automatica 50 (4), p.1288-1295, 2014


� constructive

� LMI

or Dissipativity–based (Fujioka)

"On stability of input-affine nonlinear syst. with sampled-data ctrl"H. OMRAN, L. HETEL, J.P. RICHARD, F. LAMNABHI-LAGARRIGUE

ECC2013 Zurich, 2013

� constructive

� SoS

+ the case of Bilinear/Nonlinear Systems

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transmission time + access time + packet loss + sampling…

Measurement

channel


channel

Network

... let’s sum up until now:

variable delay h1

variable delay h2

= 2 variable delays

can be estimated by Plant(time stamps + packet nb)

can be estimated by Controller (time stamps + packet nb)

known / unknown ?

Hyp: Clock Synchro NTP, GPS

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• Modelling


• A crude example


• Conclusions

Overview

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controller poles = -1 and -15observer poles = -2 and -20

A crude example

driveu position y

target

y ref

-

State Feedb. ( )with

Observer ( )

K

L

Net

internetwifi…

Theory (when neglecting the network effect)©

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A crude example

LTI model with order 2state feedback gain ( )

and observer gain ( )computed by pole assignement

without paying attention to the net (here, Internet over 40km)

KL

yref

yremote

.yobserver^

yobserver^

Controller poles = -1 and -15Observer poles = -2 and -20

Experimental results (with network)

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A crude example

drivevoltage u measured angle x

+-

gap

ε = 0 – x

Interpretation on a simplified model

speed

target anglexc = 0

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measured angle xutarget anglexc = 0

+-

ctrl.channel.� delay~h/2

meas.channel� delay~h/2transmitted angle x(t-h/2)

ε

transmitted controlε(t-h/2)

A crude example

remote feedback:

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t

x

x0

0

?

?

t0

t1

t2 t3 t4 t6t5 t7 t8 t9 t10

Exercise…

A crude example

Exercise… for my students, don’t worry ;-)

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A crude example

w.r.t.

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… and mind the variable delays!

1 π/2

constant : ∀h ∈ [0,1] iff ∈ grey zone

variable : asymptot. stable iff ∈ yellow zone :

a

b

(T=1)

2

2

0

-2

-2

2

1

2

stable h(t)<1 - unstable h=cte<1

unstable h(t)<1 - stable h=cte<1

=1

A crude example

for

if

if

1 2 3 4 5 6 7 8 91

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Few words about Lyapunov method for TDS

� need of specific adaptations:1) functions of Lyapunov-Razumikhin (not here)

2) functionals of Lyapunov-Krasovskii

ODE :

(« usual » quadratic)

FDE :

cross terms

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Illustration of the LKF approach (Lyapunov-Krasovskii functionals)

(quad + integral)

Lyapunov for TDS

Remarks on the conservatism: 1) here, a « pessimistic » condition since it is i.o.d. - independent of the delay h…

2) linked with the majoration of the crossed terms, here :

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More general…Lyapunov for TDS

x vector + variable delay

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Still more general…ongoing extensions for nonlinear TDS systems

Lyapunov for TDS

"Development of homogeneity concept for time-delay systems"D. EFIMOV, W. PERRUQUETTI, J.P. RICHARDSIAM J. Control & Opt., 2014 http://hal.inria.fr/hal-00956878

“Weighted homogeneity for TDS: Finite-time and independent of delay stability"D. EFIMOV, A. POLYAKOV, W. PERRUQUETTI, J.P. RICHARDIEEE TAC 2015, 60:1-6 https://hal.inria.fr/hal-01145321

"Implicit Lyapunov-Krasovskii Functionals for stability analysis and control design of TDS"A. POLYAKOV, D. EFIMOV, W. PERRUQUETTI, J.P. RICHARDIEEE TAC, to appear, 2015

HOMOGENEITY IMPLICIT LYAPUNOV FUNCTIONS[Korobov 1979, Polyakov 2012][Rothschild&Stein 1976]

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A link with finite time

Few words on convergence types

;-)

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• Modelling




• Conclusions

Overview©

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to buff, or not buff?

1rst solution = act « as if » constant� [Niemeyer & Slotine 98][Huang & Lewis 03] � approximation

[Azorin et al. 03][Fattouh & Sename 03] etc.

control process

buffer

buffer

2nd solution = force thedelay tobe constant� therefore, maximize it : � thanks to a buffer � time-driven

� then, apply classical techniques:�predictor (Smith) [Lelevé & Fraisse 01]

�error eqn. obeying a retarded model [Estrada-Marquez-Moog 07]…

�

Network delay: variable, asymmetric �

4th solution = keep the variable delay…

� model & predict the delay [Witrant et al. 07] � event-driven

� complexify the Lyapunov technique[Seuret-Richard 08][Kruszewski et al.11,12] [Demirel et al.13][Zhang PhD12][Zhang et al.14]

3rd solution (intermediate)just one buffer

[Seuret PhD 06] [Jiang et.al 08]

1

2

3

4

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4. Buffer-free control (event-driven)A. Seuret, J.P. Richard - IFAC Seoul 2008 (theory only)

A. Kruszewski, W.J. Jiang, E. Fridman, J.P. Richard, A. Toguyeni - IEEE T. CST 2011 (with experiments)

Set value

Observer

ControllerImplementation

card

Sensors

GPS/NTP

h1

h2

Master Slave

Zero-Holder

Zero-Holder

τ1

Net

wor

k

τ2

GPS/NTP

GainK

Gain L

buffer

4

buffer

knownthx to

time-stamps

UDP

• defines the target• receives Slave’s output• observes Slave’s state• computes & sends control

• receives & applies control• sends measured output

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Experimental set-up

MasterSlaveMiabot

SlavePC Bluetooth

(2m)

Internet

(40km)

Night-timevs

Day-time

Internet: Bluetooth:120 ms

60 ms

A switched system approach to exponential stabilization through communication network.A. KRUSZEWSKI, W.J. JIANG, E. FRIDMAN, J.P. RICHARD, A.TOGUYENI. IEEE TCST, 20 (4) 2012

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One week of RTT…

Link max delay ↔ provable performance

0.5sec

0.05sec

The achievable exponential rate depends on hm ,which motivates some adaptation w.r.t. QoS...

e−αt

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globaldelay

states

chatter+ obs.IC

α = 1 α = 3α = 0

� buffer and gains K, L adapted to the QoS (delay ranges)

Note the predictor effect(compensates the network delay)

α = 3

Experimental results with one buffer (time-driven)x 10ms

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control

(without buffer)

(with buffer)

states

delays

switches

Experimental results without buffer (event-driven)

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force & position tracking (haptic rendering)PhD Bo Zhang (2012) + Zhang, Kruszewski, Richard, Int.J.Control 2014

Proxy

Network

MasterOperator EnvironmtSlave

Controller

( )mF t

( )sF t ( )sx t( )hF t ( )eF t

( )eF t

( )mx t

( )( )2eF t tτ−

( )hF t

( )1( )m tx t τ−

( )1ˆ ( )h tF t τ−

( )px t

• Proxy synchronized on Master• Slave synchronized on Proxy• Combines LKF with H

∞

• Reduction of the delay effect• Accurate haptic rendering

(environnement’s force)

5) Natural continuation: teleoperation (event driven)5©

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« combines LKF with H∞

» means…

( ) ( )( )

e

h

F tw t

F t

=

( )( )( )

( ) ( )

s

m

s m

t

x t t

t t

θθ

θ θ

= −

&

&

= z(t)

5) Natural continuation: teleoperation (event driven)

perturb.forces

w(t)

positionerror

z(t)

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• Modelling




• Conclusions

Overview

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Conclusion-Summary

� Control with a network model [Witrant 05] predictor with variable delay

� General case (without network model) :

� with 2 buffers [Estrada 08] retarded error equation

� with 1 buffer [Seuret 06] state feedback + observ./predict.

� + adapt. to QoS [Jiang-Kruszewski-R-T 08] ‘ ‘

� with 0 buffer [S-R 08, Jiang-K-R-T 09] ‘ ‘

� Bilateral teleoperation with performance (alternative to passivity)

� with 0 buffer [Zhang-K-R 14] state feedback + obs + H∞

� Quite general problems formulated thx to delay/switched models

⇒ design in the presence of variable delays

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Other healthy readings:"Deteministic and stochastic approaches to supervisory control design for NS with time-varying com. delays "B. DEMIREL, C. BRIAT, M. JOHANSSON - Nonlinear Anaysis: Hybrid Systems , 10, 94-110, 2013

"Master-Slave synchronization for two inverted pendulums with communication time-delay“H.J. ESTRADA-GARCIA, L.A. MARQUEZ-MARTINEZ, C.H. MOOG - Springer LNCIS vol.388, Chap. Part V, 403-414. 2009

"A switched system approach to exponential stabilization through communication network"A.KRUSZEWSKI, W.JIANG, E.FRIDMAN, J.P.RICHARD, A.TOGUYENI - IEEE TCST,20 (4) 887-900, 2012 http://hal.inria.fr/inria-00602327/fr/

"Control design for teleoperation over unreliable networks: A predictor-based approach"A. KRUSZEWSKI, B. ZHANG, J.P. RICHARD Springer , in Delay Systems: From Theory to Numerics and Appl., 2013

"Control of a remote system over network including delays and packet dropout"A. SEURET, J.P. RICHARD, IFAC'08 , 17th IFAC World Congress, Seoul, South Korea, July 2008

"A refined input delay approach to sampled-data control”E. FRIDMAN, Automatica , 48 (2), 2010

"SOS for sampled-data systems“A. SEURET, M. PEETS, IFAC’11, 18th IFAC World Congress, Milano, Italy, 2011

"A novel stability analysis of linear systems under asynchronous samplings”. A. SEURET, Automatica 48 (1), 177-182, 2011

"Discrete and intersample analysis of systems with aperiodic sampling"L. HETEL, A. KRUSZEWSKI, W. PERRUQUETTI, J.P. RICHARD, IEEE TAC 56 (7), p.1696 - 1701, 2011

"Remote Stabilization via Communication Networks with a Distributed Control Law" E. WITRANT, C. CANUDAS DE WIT, D. GEORGES, M. ALAMIR, IEEE TAC, 52, 1480-1485, 2007

"A novel control design for delayed teleoperation based on delay-scheduled Lyapunov-Krasovskii functionals“ B. ZHANG, A. KRUSZEWSKI, J.P. RICHARD - Int. J. Control , 87 (8), 1694-1706, 2014. http://hal.inria.fr/hal-00939507

“A Historical View of Network Traffic Models”M. WILSON, online course http://www.cse.wustl.edu/~jain/cse567-06/traffic_models2.htm

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