1. CDC 2005 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. 2 CDC 2005 Overview Problem Formulation Necessary Condition for Stabilizability Optimal Encoding/Decoding Scheme for Observability Optimal Controller, Sufficient Condition for Stabilizability

3. CDC 2005 3 Problem Formulation

4. 4 CDC 2005 Problem Formulation Block diagram of control/communication system

5. 5 CDC 2005 Problem Formulation Plant. where and are Borel measurable and bounded, and. Throughout, we assume that there exists a unique solution, such that, where

6. 6 CDC 2005 Problem Formulation Channel. The communication Channel is an AWGN, flat fading, wireless channel given by We assume for a fixed sample path

7. 7 CDC 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

8. CDC 2005 8 Necessary Condition for Existence of Stabilizing Controller

9. 9 CDC 2005 Necessary Condition for Bounded Asymptotic Stabilizability Control/communication system

10. 10 CDC 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

11. CDC 2005 11 Optimal Encoding/Decoding Scheme for Observability

12. 12 CDC 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

13. 13 CDC 2005 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. 14 CDC 2005 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. CDC 2005 15 Optimal Controller, Sufficient Condition for Stabilizability

16. 16 CDC 2005 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. 17 CDC 2005 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. 18 CDC 2005 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. 19 CDC 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 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. 20 CDC 2005 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. 21 CDC 2005 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. 22 CDC 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 41 st IEEE Conference on Decision and Control, Las Vegas, December 2002.