CDC 2006, San Diego 1 Control of Discrete-Time Partially- Observed Jump Linear Systems Over Causal Communication Systems C. D. Charalambous Depart. of ECE University of Cyprus Nicosia, Cyprus S. Z. Denic Depart. of ECE University of Arizona Tucson

2 CDC 2006, San Diego Control Over Communication Channel Block diagram of a control-communication problem The source is partially observed jump system and communication channel is causal Sensor Communication Channel Decoder Capacity Limited Link Collection and Transmission of Information (Node 1) Reconstruction with Distortion Error (Node 2) Dynamical System Encoder Sink

3 CDC 2006, San Diego Critical Features Critical features from the communication and control point of view  Amount of data produced by a particular source (sensor) – source entropy  Capacity of the communication channel  Controllability and observability of the controlled system

4 CDC 2006, San Diego Nair, Dey, and Evans,“Communication limited stabilisability of jump Markov linear systems,” In Proc. 15th Ini. Symp.Math. The. New. Sys., U. Notre Dame, USA, Aug Nair, Dey, and Evans, “Infimum data rates for stabilising Markov jump linear systems,” in Proc. 42th IEEE Conf Dec. Contr., pp , C. D. Charalambous, “Information theory for control systems: causality and feedback,” in Workshop on Communication Networks and Complexity, Athens, Greece, August 30-September 1, References [Plenty More]

5 CDC 2006, San Diego Overview Problem formulation Causal communication channels and systems  Mutual information for causal channels  Data processing inequalities for causal communication channels  Capacity for causal communication channels  Rate distortion for causal communication channels  Information transmission theorem Necessary conditions for observability and stabilizability over causal communication channels Conclusions

CDC 2006, San Diego 6 Problem Formulation Problem formulation Causal communication channels and systems Necessary conditions for observability and stabilizability over causal communication channels Conclusions

7 CDC 2006, San Diego Problem Formulation Block diagram of control/communication system independent

8 CDC 2006, San Diego Problem Formulation Encoder, Decoder, Controller are causal Communication channel causality with feedback

9 CDC 2006, San Diego Problem Formulation System performance measures Definition 2.1: (Observabilit in Probability). The system is observable in probability if for any D, δ ≥ 0 there exist an encoder and decoder such that Definition 2.2: (Observability in r-th mean). The system is observable in r-th mean if there exist an encoder and decoder such that where D ≥ 0 is finite.

10 CDC 2006, San Diego Problem Formulation System performance measures Definition 2.3: (Stabilizability in probability). The system is stabilizable in probability if for any D, δ ≥ 0 there exist a controller, encoder and decoder such that Definition 2.4: (Stabilizability in r-th mean). The system is asymptotically stabilizable in r-th mean if there exist a controller, encoder and decoder such that where D ≥ 0 is finite.

CDC 2006, San Diego 11 Causal Communication Channels and Systems Problem formulation Causal communication channels and systems  Mutual information for causal channels  Data processing inequalities for causal communication channels  Capacity for causal communication channels  Rate distortion for causal communication channels  Information transmission theorem Necessary conditions for observability and stabilizability over causal communication channels Conclusions

12 CDC 2006, San Diego Causal Communication Channels and Systems Lemma 3.2: Let denote the self-mutual information when the RND is restricted to a non-anticipative or causal feedback channel with memory. Then, the restricted mutual information is given by Directed Information

13 CDC 2006, San Diego Causal Communication Channels and Systems Remark: In general, causal mutual information is not symmetric Data processing inequality for causal channels

14 CDC 2006, San Diego Causal Communication Channels and Systems Channel capacity based on the causal mutual information Rate distortion based on the causal mutual information

15 CDC 2006, San Diego Causal Communication Channels and Systems Theorem 4.1: (Information Transmission Theorem) Suppose the different communication blocks in Fig. 1 form a Markov chain. Consider a control-communication system where the communication channel is restricted to being causal. A necessary condition for reconstructing a source signal up to a distortion level D from is given by

CDC 2006, San Diego 16 Necessary conditions for observability and stabilizability over causal communication channels Problem formulation Causal communication channels and systems  Mutual information for causal channels  Data processing inequalities for causal communication channels  Capacity for causal communication channels  Rate distortion for causal communication channels  Information transmission theorem Necessary conditions for observability and stabilizability over causal communication channels Conclusions

17 CDC 2006, San Diego Necessary conditions for observability and stabilizability over causal communication channels Lemma 4.2. Consider the following single letter distortion measure, where Then, a lower bound for is given by where It follows and under some conditions, this lower bound is exact for

18 CDC 2006, San Diego Necessary conditions for observability and stabilizability over causal communication channels Theorem 4.3. Consider a jump control-communication system where is the observed process at time t. Let be a steady state distribution of the underlying Markov chain. Introduce the following notation A necessary condition for asymptotic observability and stabilizability in probability is given by

19 CDC 2006, San Diego Necessary conditions for observability and stabilizability over causal communication channels is the covariance matrix of the Gaussian distribution which satisfies A necessary condition for asymptotic observability and stabilizability in r-th mean is given by

20 CDC 2006, San Diego Conclusion General necessary conditions for observability and stabilizability for jump linear systems controlled over a causal communication channel are derived. Causal Information Theory is Essential for Channels with Feedback and Memory Different criteria for observability and stabilizability corresponds to different necessary condition. Future work  Sufficient conditions (design encoders and decoders)  Channel-source matching