CDC 2006, San Diego 1 Control of Discrete-Time Partially- Observed Jump Linear Systems Over Causal Communication Systems C. D. Charalambous Depart. of

CDC 2006, San Diego

1

Control of Discrete-Time Partially- Observed Jump Linear Systems Over Causal Communication Systems

C. D. CharalambousDepart. of ECE University of Cyprus Nicosia, Cyprus

S. Z. Denic Depart. of ECE University of ArizonaTucson

2CDC 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 CommunicationChannel Decoder

Capacity Limited Link

Collection and Transmission of Information (Node 1)

Reconstruction with Distortion Error (Node 2)

DynamicalSystem Encoder Sink


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


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 2002. Nair, Dey, and Evans, “Infimum data rates for stabilising Markov jump linear systems,” in Proc. 42th IEEE Conf Dec. Contr., pp. 1176-1181, 2003. C. D. Charalambous, “Information theory for control systems: causality and feedback,” in Workshop on Communication Networks and Complexity, Athens, Greece, August 30-September 1, 2006.

References [Plenty More]


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

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Problem Formulation

Problem formulationCausal communication channels and systemsNecessary conditions for observability and stabilizability over causal communication channelsConclusions


Problem Formulation

Block diagram of control/communication system

1 0

10

1

0 0 0

0 0

,

, ,...,

Pr |

0,

0,

,

, , , :

t t t t t t t

t t t t t

t t Mt

t j t i ij

t k

t l

t t t

X A S X B S W N S U X X

Y C S X D S V

S S

S S p

W N I

V N I

X N x Q

W V S X t N

independent


Problem Formulation

Encoder, Decoder, Controller are causal

Communication channel causality

10 0 0

1 10 0 0 0

10 0 0

1 10 0 0 0

, ,

, , ,

, ,

| , | , ,

t t tt

t t t tt

t t tt

n t t tt t

U Y U S

Z c Y Z Z S

Y d Z Y S

P dZ z z P dZ z z n t

with feedback


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.

1

0

1lim Pr ,t

k ktk

Y Y Dt

1

0

1lim , 0t r

k ktk

E Y Y D rt


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.

1

0

1lim Pr 0 ,t

ktk

X Dt

1

0

1lim 0 , 0t

rkt

k

E X D rt

CDC 2006, San Diego

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Causal Communication Channels and Systems

Problem formulationCausal communication channels and systems

Mutual information for causal channels Data processing inequalities for causal communication

channels Capacity for causal communication channelsRate distortion for causal communication channels Information transmission theorem

Necessary conditions for observability and stabilizability over causal communication channelsConclusions



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

1 10 01 1

0 0 10

|; log

T TT T

R T

p z zi Z Z

p z

1 10 0

10

|T T

T

p z z

p z

1 1 1 1 10 0 0 0 0 0

0; ; ; |

TT T T T i iC R i

iI Z Z E i Z Z I Z Z Z

Directed Information



Remark: In general, causal mutual information is not symmetric

Data processing inequality for causal channels

1 1 1 10 0 0 0; ;T T T T

C CI Z Z I Z Z

0 0 0 0 0 0 0 0

0 0 0

; ; ; ;

; , ,

n n n n t n t tC

t tC

I Z Z I Z Z I Y Z I Y Y

I Y Y n t N



Channel capacity based on the causal mutual information

Rate distortion based on the causal mutual information

0 0

0

1 1lim lim sup ;T TC T C

T T p TZ

C C I Z ZT T

0 0|0 0

1 1lim lim inf ;T TC T C

T T p MT TY Y

R D R D I Y YT T



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

C CR D C

tY tZ

CDC 2006, San Diego

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Problem formulationCausal communication channels and systems

Mutual information for causal channels Data processing inequalities for causal communication

channels Capacity for causal communication channelsRate distortion for causal communication channels Information transmission theorem

Necessary conditions for observability and stabilizability over causal communication channelsConclusions



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

0 00

1( , ) ( )TT T

T i ii

Y Y Y YT

( ) : [0, )pi iY Y R

1 ( )TR DT

01 1( ) ( ) max ( ),T

T S Sh GD

R D H Y H hT T

{ : [0, ); ( ) 1, ( ) ( ) }.pD

p pR RG h R h y dy y h y dy D

*0( ) ( ) ( )C SR D H h

0.D



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

ptY R

,

1log 2 log[(2 ) det ]2 2

logdet

pC g

tr tri Si

MpC e e

C i Q C i D i D i p i

S

p s

1 1 10 0 0| , ,t t t

t t tX X E X Y U S 1 1 1

0 0 0| , ,t t ttrt t tQ E X X Y U S

lim tt

Q Q



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

,

log 2 log log( ( ) ).2 ( )

logdet

p pr rC

d

tr tri Si

Mp r pC e e p rDdVr

C i Q C i D i D i p i

g*( ) ~ (0, ),( )pgh y N y R

*

|| ||( ) .

yh y dy D


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

Documents

CDC 2006, San Diego 1 Control of Discrete-Time Partially- Observed Jump Linear Systems Over Causal Communication Systems C. D. Charalambous Depart. of