Control of jump linear systems over jump communication channels source channel matching approach
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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. Control Over Communication Channel.

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Control of jump linear systems over jump communication channels source channel matching approach

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


Control over communication channel
Control Over Communication Channel Channels – Source-Channel Matching Approach

  • 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)


Objectives
Objectives Channels – Source-Channel Matching Approach

  • Design encoders, decoders, controllers to achieve control and communication objectives

  • Establish a separation principle between communication and control system design


References
References Channels – Source-Channel Matching Approach

  • 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.


Overview
Overview Channels – Source-Channel Matching Approach

  • Problem formulation

  • Necessary conditions for observability and stabilizability over causal communication channels

  • Source-channel matching

  • Conclusions


Problem formulation

Problem Formulation Channels – Source-Channel Matching Approach

  • Problem formulation

  • Information Measures

  • Necessary conditions for observability and stabilizability over

  • causal communication channels

  • Source-channel matching

  • Conclusions


Problem formulation1
Problem Formulation Channels – Source-Channel Matching Approach

  • Block diagram of control/communication system

Independent Crucial


Problem formulation2
Problem Formulation Channels – Source-Channel Matching Approach

  • Encoder, decoder, controller are causal

  • Communication channel with feedback

with feedback


Problem formulation3
Problem Formulation Channels – Source-Channel Matching Approach

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.


Problem formulation4
Problem Formulation Channels – Source-Channel Matching Approach

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.


Information measures
Information Measures Channels – Source-Channel Matching Approach

Restricted Self-Information

  • Causality of Stochastic Kernels

  • Restricted Self-Mutual Information (RND)


Information measures1
Information Measures Channels – Source-Channel Matching Approach

Restricted Mutual-Information (Directed Information)

  • Expectation of Restricted self-Mutual Information

  • This is Directed Information [Massey]


Information measures2
Information Measures Channels – Source-Channel Matching Approach

Information Capacity and Rate Distortion

  • Information Channel Capacity

  • Information Rate distortion


Information measures3
Information Measures Channels – Source-Channel Matching Approach

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

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


Necessary conditions for reconstruction and stabilizability over causal communication channels1
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


Necessary conditions for reconstruction and stabilizability over feedback communication channels
Necessary conditions for reconstruction and stabilizability over feedback communication channels

Application to the Jump System:


Necessary conditions for reconstruction and stabilizability over causal communication channels2
Necessary conditions for reconstruction and stabilizability over causal communication channels

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


Necessary conditions for reconstruction and stabilizability over causal communication channels3
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


Source channel matching

Source-Channel Matching over causal communication channels

  • Problem formulation

  • Necessary conditions for observability and stabilizability over

  • causal communication channels

  • Source-channel matching

  • Conclusions


Source channel matching1
Source-Channel Matching over causal communication channels

  • 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


Source channel matching2
Source-Channel Matching over causal communication channels

  • Decoding.

  • Reconstruction with distortion D can be achieved by setting


Source channel matching3
Source-Channel Matching over causal communication channels

  • Equivalent channel is given by

  • The transmission rate is given by

    where R(D) is the rate distortion between


Source channel matching4
Source-Channel Matching over causal communication channels

  • Mean square error

  • The channel capacity

  • Since a sufficient condition for the reliable transmission is C>R(D) then

    is indeed compression


Source channel matching5
Source-Channel Matching over causal communication channels

  • 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


Conclusions
Conclusions over causal communication channels

  • Different Information Patters

  • Separation principle holds for Gaussian control and communication channels

  • Uncertain control systems and channels


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