CDC 2005 1 Control over Wireless Communication Channel for Continuous-Time Systems C. D. Charalambous…

CDC 2005 1

Control over Wireless Communication Channel for Continuous-Time Systems

C. D. CharalambousECE Department University of Cyprus, Nicosia, Cyprus.Also, School of Information Technology and Engineering,University of Ottawa, OttawaCanada

Stojan Denic and Alireza FarhadiSchool of Information Technology and Engineering,University of Ottawa, OttawaCanada

2CDC 2005

Overview

Problem Formulation

Necessary Condition for Stabilizability

Optimal Encoding/Decoding Scheme for Observability

Optimal Controller, Sufficient Condition for Stabilizability

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Problem Formulation

4CDC 2005

Problem Formulation

Block diagram of control/communication system

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Problem Formulation

Plant.

where and are Borel measurable and bounded, and .

Throughout, we assume that there exists a unique solution, such that , where

),0(),()()()()()()( xtdwtGdttutBdttxtAtdx

],0[:,],0[: TBTA ],0[: TG),(~ 000 VxNx

x}|)(|}{);];,0([{

0

20

T

tt dttxEandadaptedisxTCx

6CDC 2005

Problem Formulation

Channel. The communication Channel is an AWGN, flat fading, wireless channel given by

We assume for a fixed sample path

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

TtytdvdtxxtFttztdy

T

T

dtxxtFttzE

dtxxtFttz

0

22

0

22

.|),~,,(|))(,(

,1}|),~,,(|))(,(Pr{

7CDC 2005

Problem Formulation

Bounded Asymptotic and Asymptotic Observability in the Mean Square Sense. Let .Then, the system is bounded asymptotically (resp. asymptotically) observable, in the mean square sense, if there exists encoder and decoder such that

Bounded Asymptotic and Asymptotic Stabilizability in the Mean Square Sense. The system is bounded asymptotically (resp. asymptotically) stabilizable, in the mean square sense, if there exists a controller, encoder and decoder, such that

]))(~)([()( ,,0

2 yttxtxEt

.).,0)(lim.(..,)(lim saPtrespsaPttt

)..,0]|)([|lim.(.,.,]|)([|lim ,02

,02 saPtxErespsaPtxE t

tt

t

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Necessary Condition for Existence of Stabilizing Controller

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Necessary Condition for Bounded Asymptotic Stabilizability

Control/communication system

10CDC 2005 1- C. D. Charalambous and Alireza Farhadi

Necessary Condition for Bounded Asymptotic Stabilizability

Theorem. A necessary condition for the existence of a bounded asymptotic stabilizing controller is given by

For the case of AWGN channel (e.g., ), the necessary condition is reduced to the following condition

000

][,][AifAifA

AACah

][ACa

)()( tth

1

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Optimal Encoding/Decoding Scheme for Observability

12CDC 2005 2-C. D. Charalambous and Stojan Denic

Optimal Encoding/Decoding Scheme

Theorem. Suppose the transmitter and receiver are subject to the instantaneous power constraint,Then the encoder that achieves the channel capacity, the optimal decoder, and the corresponding error covariance, are respectively given by

2

PxxtFE ]|),~,,([| 2

.)0(,)0(~

,}))(,()(2exp{)(

}))(,()(2exp{)0(),,(

),(),,())(,()()(),,(~)(),,(~

)),,(~)((),,(

),~,,(

0*

0*

0

22

0

2

0

**

***

**

*

VVxx

dsPduuuzduuAsG

PdssszdssAVytV

tdyytPVttzdttutBdtytxtAytxd

ytxtxytVPxxtF

t t

s

t

s

tt

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Necessary and Sufficient Condition for Observability

Theorem. i) When , a sufficient condition for bounded asymptotic observability in the mean square sense is given by

(1)

while, a necessary condition for bounded asymptotic observability is given by

(2)

ii) When , (1) is a sufficient condition for asymptotic observability in the mean square sense, while, when , condition (2) is a necessary condition for asymptotic observability in the mean square sense.

0)( tG

..,0..,)]([),(2

2 saPteatAtzP

..,0..,)]([),(2

2 saPteatAtzP

0)( tG00 V

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Necessary and Sufficient Condition for Observability

Remark. In the special case of AWGN ( ), for which the channel capacity is , the conditions (1) and (2) are reduced to the following conditions, respectively.

1z2PCa

.0..,)]([

,0..,)]([

teatAC

teatAC

a

a

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Optimal Controller, Sufficient Condition for Stabilizability

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Optimal Controller

Problem. For a fixed sample path, the output feedback controller is chosen to minimizes the quadratic pay-off

Assumption. The noiseless analog of the plant is completely controllable or exponentially stable.

}0));(,({ Tsssz

.},)](|)(|)(|)([|{1

0

22 TdttRtutQtxET

JT

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Optimal Controller

Solution. According to the classical separation theorem of estimation and control, the optimal controller that minimizes the pay-off subject to a flat fading AWGN channel and linear encoder is separated into a state estimator and a certainly equivalent controller given by

.0)(lim),()(2)()()()()(

),()()()(),,,(~)()(

122

1*

TPtPtAtRtBtPtQtP

tPtBtRtKytxtKtu

T

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Optimal Controller

Corollary. For a fixed sample path of the channel, it follows that if the observer and regulator Ricatti equations have steady state solution and , respectively, the average criterion

can be expressed in the alternative form

where for the time-invariant case, it reduced to

)(tV )(tP

})](|)(|)(|)([|{1lim0

2*2

T

TdttRtutQtxE

TJ

})]()()()()([{1lim0

22

T

TdttRtKtVtGtP

TJ

.. 22 RKVGPJ

19CDC 2005 1- C. D. Charalambous and Alireza Farhadi

Conditions for Stabilizability

Proposition. Consider the time-invariant analog of plant and assume it is controllable or exponentially stable. Then, for a fixed sample path of the channel, we have the followingsi) Assuming and as , by using the certainly equivalent controller, and as .

ii) Assuming and as , by using the certainly equivalent controller, and as .

0G VtV )( t2|)(| txE

2|)(| tuE t

0G 0)( tV t

t0|)(| 2tuE0|)(| 2txE

1

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Sufficient Condition for Stabilizability

Theorem. Consider the time-invariant analog of plant and assume it is controllable or exponentially stable. Then, a sufficient condition for bounded asymptotic stabilizability and asymptotic stabilizability, in the mean square sense is given by

Remark. For the special case of AWGN channel, this condition is reduced to

..,0..,][))(,(2

2 eaPteaAttzP

.][ ACa

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Conclusion

For the class of scalar diffusion process controlled over AWGN flat fading channel, we built optimal encoder/decoder which achieves channel capacity and minimizes the mean square error.

Since the separation principle holds, the optimal encoder/decoder scheme and the certainly equivalent controller leads to the optimal strategy.

For the future work, it is interesting to build encoder which is independent of the decoder output. Also, it would be interesting to extend the results to the case when there is also AWGN flat fading communication link between the controller and the plant.

22CDC 2005

References

[1] C. D. Charalambous and Alireza Farhadi, Control of Continuous-Time Systems over Continuous-Time Wireless Channels, 2005 (preprint).

[2] C. D. Charalambous and Stojan Denic, “On the Channel Capacity of Wireless Fading Channels”, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, December 2002.