C. D. Charalambous Dep. of ECE University of Cyprus Nicosia, Cyprus

1. Control of Jump Linear Systems Over Jump Communication Channels – Source-Channel Matching Approach S. Z. Denic Dep. of ECE University of Arizona Tucson C. D. Charalambous Dep. of ECE University of Cyprus Nicosia, Cyprus

2. Control Over Communication Channel • Partially observed uncontrolled source is modulated by FSM chain; channel does not use feedback • Partially observed controlled source is modulated by FSM chain; channel uses feedback Communication Channel Sink Decoder Dynamical System Encoder Sensor Collection and Transmission of Information (Node 1) Capacity Limited Link Reconstruction with Distortion Error (Node 2)

3. Objectives • Design encoders, decoders, controllers to achieve control and communication objectives • Establish a separation principle between communication and control system design

4. References • Tatikonda, Sahai, and Mitter, “Stochastic linear control over a communication channel,” 2004. (and Ph.D. Theses) • Nair, Dey, and Evans,“Communication limited stabilisability of jump Markov linear systems,” 2002. • Nair, Dey, and Evans, “Infimum data rates for stabilising Markov jump linear systems,” 2003.

5. Overview • Problem formulation • Necessary conditions for observability and stabilizability over causal communication channels • Source-channel matching • Conclusions

6. Problem Formulation • Problem formulation • Information Measures • Necessary conditions for observability and stabilizability over • causal communication channels • Source-channel matching • Conclusions

7. Problem Formulation • Block diagram of control/communication system Independent Crucial

8. Problem Formulation • Encoder, decoder, controller are causal • Communication channel with feedback with feedback

9. Problem Formulation Communication System Performance Measure Definition: (Reconstruction in probability). Consider a control-communication system of Fig. 1. For a given δ ≥ 0 there exist an encoder and decoder (and control sequence) such that Definition: (Reconstruction in r-th mean). Consider a control-communication system of Fig. 1. There exist an encoder and decoder (and control sequence) such that where D ≥ 0 is finite.

10. Problem Formulation Control System Performance Measures Definition: (Stabilizability in probability). Consider a control-communication system of Fig. 1. For a given δ ≥ 0 there exist a controller, encoder and decoder such that Definition: (Stabilizability in r-th mean). Consider a control-communication system of Fig. 1. There exist a controller, encoder and decoder such that where D ≥ 0 is finite.

11. Information Measures Restricted Self-Information • Causality of Stochastic Kernels • Restricted Self-Mutual Information (RND)

12. Information Measures Restricted Mutual-Information (Directed Information) • Expectation of Restricted self-Mutual Information • This is Directed Information [Massey]

13. Information Measures Information Capacity and Rate Distortion • Information Channel Capacity • Information Rate distortion

14. Information Measures Assumption: FSM Chain is irreducible, Aperiodic, Homogeneous (Ergodic) Standard Detectability and Stabilizability Conditions of Linear Quadratic Gaussian Theory Hold Uniformly over the States of the FSM Chain.

15. Necessary conditions for reconstruction and stabilizability over causal communication channels • Problem formulation • Information Measures • Necessary conditions for reconstruction and stabilizability over • causal communication channels • Source-channel matching • Conclusions

16. Necessary conditions for reconstruction and stabilizability over causal communication channels • [Linder-Zamir 1994] Consider the following form of distortion measure , where Then, a lower bound for is given by where It follows and under some conditions, this lower bound is exact for

17. Necessary conditions for reconstruction and stabilizability over feedback communication channels Application to the Jump System:

18. Necessary conditions for reconstruction and stabilizability over causal communication channels • A necessary condition for reconstruction and stabilizability in probability is given by

19. Necessary conditions for reconstruction and stabilizability over causal communication channels is the covariance matrix of the Gaussian distribution which satisfies A necessary condition for reconstruction and stabilizability in r-th mean is given by

20. Source-Channel Matching • Problem formulation • Necessary conditions for observability and stabilizability over • causal communication channels • Source-channel matching • Conclusions

21. Source-Channel Matching • Consideruncontrolled system (U = 0). • Source signal processing. Define innovations process • Conditioned on the state S=s, K is orthogonal Gaussian process with • Compress the innovations process K and send it through the communication channel

22. Source-Channel Matching • Decoding. • Reconstruction with distortion D can be achieved by setting

23. Source-Channel Matching • Equivalent channel is given by • The transmission rate is given by where R(D) is the rate distortion between

24. Source-Channel Matching • Mean square error • The channel capacity • Since a sufficient condition for the reliable transmission is C>R(D) then is indeed compression

25. Source-Channel Matching • Controlled system Y Partially observed dynamical system (source) Innovations generator K AGN Separation Can be Shown Between Control and Communication System Design; optimality of LGQ pay-off. Mean square estimator Controller

26. Conclusions • Different Information Patters • Separation principle holds for Gaussian control and communication channels • Uncertain control systems and channels