Control over Wireless Communication Channel for Continuous-Time Systems

1. Control over Wireless Communication Channel for Continuous-Time Systems 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

2. Overview • Problem Formulation • Necessary Condition for Stabilizability • Optimal Encoding/Decoding Scheme for Observability • Optimal Controller, Sufficient Condition for Stabilizability

3. Problem Formulation

4. Problem Formulation • Block diagram of control/communication system

5. Problem Formulation • Plant. where and are Borel measurable and bounded, and . Throughout, we assume that there exists a unique solution, such that , where

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

7. 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

8. Necessary Condition for Existence of Stabilizing Controller

9. Necessary Condition for Bounded Asymptotic Stabilizability • Control/communication system

10. 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

11. Optimal Encoding/Decoding Scheme for Observability

12. 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

13. 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.

14. 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.

15. Optimal Controller, Sufficient Condition for Stabilizability

16. 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.

17. 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

18. 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

19. 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 .

20. 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

21. 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.

22. 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.