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

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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

- 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)

- Design encoders, decoders, controllers to achieve control and communication objectives
- Establish a separation principle between communication and control system design

- 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.

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

Problem Formulation

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

- Block diagram of control/communication system

Independent Crucial

- Encoder, decoder, controller are causal
- Communication channel with feedback

with feedback

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.

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.

Restricted Self-Information

- Causality of Stochastic Kernels
- Restricted Self-Mutual Information (RND)

Restricted Mutual-Information (Directed Information)

- Expectation of Restricted self-Mutual Information
- This is Directed Information [Massey]

Information Capacity and Rate Distortion

- Information Channel Capacity
- Information Rate distortion

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.

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

- [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

Application to the Jump System:

- A necessary condition for reconstruction and stabilizability in probability is given by

is the covariance matrix of the Gaussian distribution which satisfies

A necessary condition for reconstruction and stabilizability in r-th mean is given by

Source-Channel Matching

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

- 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

- Decoding.
- Reconstruction with distortion D can be achieved by setting

- Equivalent channel is given by
- The transmission rate is given by
where R(D) is the rate distortion between

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

- 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

- Different Information Patters
- Separation principle holds for Gaussian control and communication channels
- Uncertain control systems and channels