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Control over Wireless Communication Channel for Continuous-Time Systems

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### Control over Wireless Communication Channel for Continuous-Time Systems

### Optimal Controller, Sufficient Condition for Stabilizability

C. D. Charalambous

ECE Department

University of Cyprus, Nicosia, Cyprus.

Also,

School of Information Technology and Engineering,

University of Ottawa, Ottawa

Canada

Stojan Denic and Alireza Farhadi

School of Information Technology and Engineering,

University of Ottawa, Ottawa

Canada

Overview

- Problem Formulation
- Necessary Condition for Stabilizability
- Optimal Encoding/Decoding Scheme for Observability
- Optimal Controller, Sufficient Condition for Stabilizability

Problem Formulation

- Block diagram of control/communication system

Problem Formulation

- Plant.

where and are Borel measurable and bounded, and .

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

Problem Formulation

- Channel. The communication Channel is an AWGN, flat fading, wireless channel given by
- We assume for a fixed sample path

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

Necessary Condition for Bounded Asymptotic Stabilizability

- Control/communication system

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

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

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.

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.

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.

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

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

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 followings

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

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

as .

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

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.

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.

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