Event-based Control:Theory and Application

JAN LUNZE

Ruhr-Universität Bochumemail: Lunze@atp.rub.de

Overview

1 Introduction to event-based control

2 Event-based state feedback

3 Analysis of the event-based control loop

4 Experimental evaluation

5 Event-based PI control

6 Conclusion and outlook

Introduction to event-based control

Event-based control

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Aim: Reduction of the network traffic.

Event-based control

Why is the theory of sampled-data control not applicable?

No zero-order hold input.

No periodic sampling! no discrete-time model available.

Sampling theorem possibly violated.

It is necessary to develop a new theory of event-based control.

Literature survey• Deadband control(OTANEZ, MOYNE, TILBURY, American Control Conf., 2002)Do not send new data as long as ||x(t) " x(tk)|| < e

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• Quantised state feedback• Self-triggered control(TABUADA, IEEE Trans. Autom. Control, 2007)Estimate the next event time: tk+1 = h(x(tk), tk)

Literature survey• Deadband control• Quantised state feedback(GRÜNE, JUNGE, Syst. Control Lett., 2005)(DE PERSIS, IFAC World Congress, 2008)

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• Self-triggered control(TABUADA, IEEE Trans. Autom. Control, 2007)

Literature survey• Deadband control• Quantised state feedback• Self-triggered control(TABUADA, IEEE Trans. Autom. Control, 2007)Estimate the next event time: tk+1 = h(x(tk), tk)

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Literature survey

Evaluation of existing approaches:Many approaches result from an extension of sampled-datacontrol and do not answer the three basic questions

Many approaches do not consider any kind of disturbanceand model uncertainties.

Almost all approaches use a zero-order hold as controlinput generator.

ExperimentWhat is a typical behaviour of an event-based control loop?

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Experiment

Discrete-time control:

2

Experiment

Event-based control:

-2

2

Control aim„Ultimate boundedness“ („practical stability“):Hold the state x(t) inside a set !d:

x(t) # !d, $t % T(x0), x0 # !1, d(t) # [dmin, dmax]

d

1

Then the set !d is said to be robust positively invariant.

Event-based state feedback

Event-based state feedbackWhy do we need information feedback?

... to stabilise an unstable plant,

... to compensate model uncertainties,

... to attenuate disturbances.

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Event-based state feedbackPlant:

x(t) = Ax(t) + Bu(t) + Ed(t), x(0) = x0y(t) = Cx(t)

Assumptions:

Linear, asymptotically stable plant,

no model uncertainties,

synchronised clocks, no transmission delays,

no computational restrictions.

Event-based state feedback

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Main idea: The event-based controller should mimic thebehaviour of a continuous state-feedback controller withadjustable precision.

||x(t) " xSF(t)||!& emax

Control input generator

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xSF(t) = (A" BK)! "# $

¯A

xSF(t) + Ed(t)

u(t) = "KxSF(t)

Behaviour of the continuous feedback loop after the statexSF(tk) at time tk is known:

u(t) = "Ke¯A(t " tk)xSF(tk)"

% t

tkKe

¯A(t " !)Ed(!) d!, t % tk

Control input generator

u(t) = "Ke¯A(t " tk)xSF(tk)"

% t

tkKe

¯A(t " !)Ed(!) d!, t % tk

'

Control input generator mimics the continuous state feedback:

u(t) = "Ke¯A(t " tk)x(tk) "

% t

tkKe

¯A(t " !)Edk d!, t % tk

Control input generatorThis input is generated by the following system:

xs(t) = Axs(t) + Edk, xs(t+k ) = x(tk), t % tku(t) = "Kxs(t)

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Event generator% . ! /

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Event generator initiates an information exchange whenever

|| x(tk) " xs(tk!)! "# $

x!(tk)

|| = e

Disturbance estimator

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For constant disturbancesd(t) = d, t # [tk!1, tk)

the state difference is

x(tk) " xs(tk) =

% tk

tk!1eA(t " !)E (d" dk!1) d!

= A!1&

eA(tk+1 " tk) " I'

E (d " dk!1).

Disturbance estimator

The „mean“ magnitude d of the disturbance in the time intervalt # [tk!1, tk] is used as disturbance estimate dk in the timeinterval t % tk:

Disturbance estimator

d0 = 0

dk = dk!1 +&

A!1&

eA(tk " tk!1) " I'

E'+ (

x(tk) " xs(t!k ))

Event-based control algorithm

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Summary of the event-based control algorithmCheck the difference ||x(t) " xs(t)|| until an event is detected.Then:

1 Send the information x(tk) from the event generator to thecontrol input generator.

2 Determine the disturbance estimate dk.3 Reinitialise the control input generator: xs(t+k ) = x(tk)

Event-based control algorithm

Behaviour of the event-based control loop:

x ts( ), ( )x t

{e

t0 t1 t2t

Summary

Three novelties of this methods with respect to literature:The control input generator is not a zero-order hold, butdetermines exponential inputs u(t).The event generator compares the behaviour of theevent-based control loop with some reference system(model of the continuous feedback loop).A disturbance estimate is used to adapt the event-basedloop to the unknown disturbance d(t).

Analysis of the event-based control loop

Analysis of the event-based control loopClosed-loop system between two consecutive eventst # [tk, tk+1):

*

x(t)xs(t)

+

=

*

A "BKO A

+ *

x(t)xs(t)

+

+

*

EO

+

d(t) +

*

OE

+

dk*

x(tk)xs(t+k )

+

=

*

x(tk)x(tk)

+

y(t) = (C O)

*

x(t)xs(t)

+

State transformation*

x!(t)xs(t)

+

=

*

I "IO I

+ *

x(t)xs(t)

+

Analysis of the event-based control loop

Transformed state-space model*

x!(t)xs(t)

+

=

*

A OO A

+ *

x!(t)xs(t)

+

+

*

EO

+

(d(t) " dk)! "# $

d!(t)

+

*

OE

+

dk

*

x!(tk)xs(t+k )

+

=

*

0x(tk)

+

Analysis of the event-based control loop

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d!(t) = d(t) " dk affects the (uncontrolled) plant.For a good approximation

d(t) " dk ( 0, for t % tk

no communication is necessary.

Analysis of the event-based control loopEvent-based control loop for small disturbance dd(t)There is no event for t > 0 if

||x(t) " xs(t)|| = ||

% t

0eA(t " !)E (dd(!) " d0) d! || < e.

Reformulation of the event condition:

maxt"0

||

% t

0eA(t " !)E (dd(!) " d0) d! ||

& maxt"0

% t

0||eA(t " !)E|| d! ·max

t"0||dd(t)||

&

% #

0||eA!E|| d! · |d|.

Analysis of the event-based control loop

Theorem (Insensitivity to small disturbances)

d(t) = dd(t)||d(t)|| & 1

For sufficiently small disturbance magnitude |d| the eventgenerator does not generate any event for t > 0.

|d| <e

, #

0 ||eA!E|| d!

Analysis of the event-based control loopCommunication frequency described by

Tmin = mink

(tk+1 " tk)

Assumption:

||d!(t)|| & d!max for t % 0

Theorem (Bounded sampling rate)Minimum time between two communication events: Tmin % Twith

% T

0||eA!E|| d! =

ed!max

Analysis of the event-based control loopEvent-based control vs. continuous state-feedback controlFor e(t) = x(t) " xSF(t) we get

e(t) = x(t) " xSF(t)= Ax(t) " BKxs(t) + Ed(t) " AxSF(t) " Ed(t)= (A" BK)e(t) + BK (x(t) " xs(t))

! "# $

x!(t)

.

ande(t) = Ae(t) + BKx!(t), e(0) = 0

||e(t)|| & e% #

0||e A!BK|| d!

Analysis of the event-based control loop

Theorem (Bounded approximation error)

||x(t) " xSF(t)|| & emaxwith

emax = e ·% #

0||e A!BK|| d!

The event-based control loop mimics the continuous loop withadjustable accuracy emax.

x(t) # !s(xSF(t)) = {x | ||x" xSF(t)|| & emax}

Analysis of the event-based control loop

0 20 40 60 80 100-2

-1

0

1

2

t

xk() x ks( )

x k( )

x kSF( )

Analysis of the event-based control loop

Summary:The event-based control loop mimics the continuousstate-feedback loop with adjustable precision

Increase the precision emax by decreasing the event threshold eConsequence: Increase of communication frequency 1

Tmin

Experimental evaluation

Experimental evaluation

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Experimental evaluation

Design steps:

1 Design a state-feedback controller K with gooddisturbance attenuation properties.

2 Implement the event-based controller (control inputgenerator, event generator, disturbance estimator)

3 Choose the event threshold e and evaluate the performanceof the event-based control loop.

Experimental evaluationSimulation results

t0=0 t0

,

,

,

4

Experimental evaluationSimulation results

emax = e ·, #0 ||e A!BK|| d! = 2.26

dmax = e,"

0 ||eA!E|| d!= 0.0114

Experimental evaluationExperimental results

Experimental evaluation

Evaluation:

Considerable reduction of the communication.

Robustness against model uncertainties.

Necessary extensions:... for set-point tracking... for quantitative robustness evaluation... for nonlinear plants.

Event-based PI control

Event-based PI control

Constant external signals:

d(t) = d, w(t) = w

Set-point tracking:

limt$#

||y(t) " w|| = 0.

Event-based PI controlPlant:

x(t) = Ax(t) + Bu(t) + Ed(t), x(0) = x0y(t) = Cx(t)

Continuous PI controller:

xr(t) = y(t) " w(t), xr(0) = xr0u(t) = "KIxr(t) " KPx(t)

Known result: If the closed-loop system is asymptoticallystable, set-point tracking occurs.

Event-based PI controlPI-control input generator:

Model

-K=- K K( )P I

u( )t

dk

xs( )t

w( )t

u( )tx( )t

k

dk

Disturbance

estimator

xs( )t

ys( )t

xsr( )t

Event-based PI control

*

xs(t)xsr(t)

+

=

*

A" BKP "BKIC O

+ *

xs(t)xsr(t)

+

! "# $

xsI(t)

+

*

O"I

+

w(t) +

*

EO

+

dk

xs(t+k ) = x(tk)u(t) = "(KP KI)xsI(t)

Only the state xs(t) is updated at event time tk.

Event generator and disturbance estimator remain the same.

Set-point tracking propertiesFor large disturbance d, an event is generated at time t1 and thedisturbance is estimated correctly:

d1 = d.

Hence, set-point tracking occurs: limt$# ||y(t) " w|| = 0.

For small disturbance d, no event occurs:

||x!(t)|| = ||A!1&

eAt " I'

Ed|| < e, t % 0.

Hence, the output remains in the set

limt$#

y(t) # {||y" w|| & ||C||e}

Set-point tracking properties

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0

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t0=0 t1 t01000 4000even

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Further extensions

(Lehmann, PhD Thesis 2011):

Event-based output feedback using a state observer

Robustness against communication delay and packet loss

Data-rate constraints! communication of quantisedinformation

Discrete-time implementation

Conclusion and outlook

Conclusions and outlook

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# . ! /

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Main idea: Mimic the continuous feedback system by theevent-based control loop with adjustable precision

Conclusions and outlook

% . ! /0 & - # $

! . ! /

# . ! /

% % ! " /

In which situation should information be transmitted?... if ||x(t) " xs(t)|| = e

Which information should be transmitted?x(tk)

How should the control input be generated?... use the model of the continuous loop

Conclusions and outlookInteresting open problems:

What happens if the network introduces time delay andpackage loss?

How should event-based control be implemented withoutsynchronous clocks?

How can event-based control be extended to multiloopsystems?

Further extensions to unstable plants, nonlinear systems, outputfeedback etc.

Conclusions and outlookTo read:

J. Lunze, D. Lehmann: A state-feedback approach toevent-based control, Automatica (2010)D. Lehmann, J. Lunze: Extension and experimentalevaluation of an event-based state-feedback approach,Control Engineering Practice 19 (2011)J. Lunze: Event-based control: A tutorial introduction,SICE Journal 11 (2010).L. Grüne, F. Müller, S. Jerg, O. Junge, M. Post, D.Lehmann, J. Lunze: Two complementary approaches toevent-based control, Automatisierungstechnik 58 (2010).D. Lehmann: Event-Based State-Feedback Control, PhDThesis, Ruhr-Universität Bochum, 2011.