information theory for control systems: causality and … information theory for causal systems...

IntroductionInformation Theory for Causal Systems

Control of Dynamic Systems over Finite Capacity ChannelsExample: Necessary and Sufficient Conditions

ConclusionsReferences

Information Theory for Control Systems: Causalityand Feedback

Charalambos D. Charalambous

Department of Electrical and Computer EngineeringUniversity of Cyprus

E-mail: chadcha@ucy.ac.cy

Workshop on Communication Networks and ComplexityAthens, Greece, 2006

Outline

1 Introduction

2 Information Theory for Causal SystemsDefinition of Control/Communication SubsystemsMutual Information for Causal SystemsCausal Rate DistortionData Processing InequalitiesInformation Transmission Theorem

3 Control of Dynamic Systems over Finite Capacity ChannelsAsymptotic Stability and Observability

4 Example: Necessary and Sufficient ConditionsAsymptotic Stability and Observability

5 Conclusions

6 References

Outline

1 Introduction

2 Information Theory for Causal Systems

3 Control of Dynamic Systems over Finite Capacity Channels

4 Example: Necessary and Sufficient Conditions

5 Conclusions

6 References

Introduction I

Control/CommunicationSystem

Information Theory forCausal Control Systems

Channel Capacity withFeedback (Causal)Rate Distortion (Causal)

Feedback Control

ObservabilityStability

Plant(Information

Source)Encoder

Discrete Time Channel

Decoder

tY~Controller

Introduction II

Objective

Design Encoders, Decoders, Controllers to Achieve Control andCommunication Objectives

Trade Offs

What are the Trade Offs Between Control and CommunicationDesign Objectives?

Separation Principle

Does Separation Hold Between Communication and ControlSystem Design?

Introduction III

Structure of Presentation

Information Theory for Causal Systems

Control of Stochastic Systems over Limited Capacity Channels

Partially Observed Linear Stochastic Control Systems

Definition of Control/Communication SubsystemsMutual Information for Causal SystemsCausal Rate DistortionData Processing InequalitiesInformation Transmission Theorem

Outline

1 Introduction

5 Conclusions

6 References

Definition of Subsystems I

Information Source

The information source is identified by the joint probability

distribution P(dY T ), Y T 4= (Y0, . . . ,Y

T−1)

Communication Channel

The communication channel is modeled by a feedback channelwith memory via the family of stochastic kernels{P(dZt ; z

t , z t−1)}Tt=0, t,T ∈ N+, where Z t = z t is the specificrealization of the channel input, and Z t−1 = z t−1 is a specificrealization of the previous channel outputs.

Definition of Subsystems II

Causality Y T C→ ZN .

Given any two sequences Y T and ZN ,T ,N ∈ N+ we shall say thatthe stochastic kernel connecting Y T to ZN is causal if and only ifP(dZt ; y

n, z t−1) = P(dZt ; yt , z t−1),∀n > t, n, t ∈ N+.

Definition of Subsystems III

Causal Channel, Feedback, Compression

ZT C→ ZT is equivalent toP(dZt ; z

n, z t−1) = P(dZt ; zt , z t−1), ∀n > t, where

t, n ∈ N+, which means the communication channel isnon-anticipative or causal.

ZT C← ZT is equivalent to P(dZt ; zn, z t−1) = P(Zt ; z

t , z t−1),∀n > t, where t, n ∈ N+, which means that thecommunication channel is used with non-anticipative or causalfeedback.

Y T C→ Y T is equivalent toP(dYt ; y

t−1, yn) = P(dYt ; yt−1, y t), ∀n > t, where

t, n ∈ N+, which means that the source reproductionstochastic kernel is non-anticipative or causal.

Restricted Mutual Information for Causal Systems I

Shannon’s Self-Mutual Information

Given a communication channel with input ZN−1 and outputZN−1, the self-mutual information is defined by

i(ZN−1; ZN−1)4= log

P(dZN−1;ZN−1)

P(dZN−1)

= logP(dZN−1, dZN−1)

P(dZN−1)× P(dZN−1),

P(dZN−1;ZN−1)

P(dZN−1)is the RND between the stochastic kernel

P(dZN−1;ZN−1) and distribution P(dZn−1).

Symmetry: i(ZN−1; ZN−1) = i(ZN−1;ZN−1)

Restricted Mutual Information for Causal Systems II

Mutual Information

I (ZN−1; ZN−1) = EP(dZN−1,dZN−1)i(ZN−1; ZN−1)

= I (ZN−1;ZN−1).

The Shannon information capacity for the time horizon N is

CN4= sup

P(dZN−1)∈DN−1CI

I (ZN−1; ZN−1)

Restricted Mutual Information for Causal Systems III

Restricted Self-Mutual Information

Given a channel with input ZN−1 and output ZN−1, the restrictedself-mutual information is defined by

iR(ZN−1; ZN−1)4= log

P(dZN−1;ZN−1)

P(dZN−1)|R

= logP(dZN−1, dZN−1)

P(dZN−1)× P(dZN−1)|R

which denotes the logarithm of the restricted RND associated with

ZN C→ ZN .

Restricted Mutual Information for Causal Systems IV

The Restricted Mutual Information

I (ZN−1; ZN−1)|R4= EP(dZN−1,dZN−1)iR(ZN−1; ZN−1)

=N−1∑i=0

I (Z i ; Zi |Z i−1) =N−1∑i=0

logP(dZi ;Z

i , Z i−1)

P(dZi ; Z i−1)× P(dZ i , dZ i )

Symmetry Fails: I (ZN−1; ZN−1)|R 6= I (ZN−1;ZN−1)|RTighter Bound: I (ZN−1; ZN−1)|R ≤ I (ZN−1; ZN−1)

Restricted Mutual Information for Causal Systems V

Relation to Previous Work: Directed Information

H. Marko, 1973 (IEEE Communication Systems)

J. Massey, 1990 (ISIT)

G. Kramer, 1998 (Ph.D. Thesis)

S. Tatikonda, 2000 (Ph.D. Thesis)

Causal Rate Distortion I

Shannon’s Rate Distortion

Let Y T−1 and Y T−1 denote the source and reproduction outputs,respectively, and denote the distortion measure

{P(dY T−1; yT−1);EρT (Y T−1, Y T−1) ≤ Dv

}The information rate distortion function for the time horizon T isdefined by

RDT (Dv )

4= inf

P(dY T−1;yT−1)∈DTDC

I (Y T−1; Y T−1)

Causal Rate Distortion II

The Minimizing Kernel is Non-Causal

The infimizing reproduction stochastic Kernel can be factoredas P(dY T−1; yT−1) =

∏T−1i=0 P(dYt ; y

t−1, yT−1), which is anon-causal operation on the source data.

Causal Rate Distortion III

Rate Distortion for Causal Systems

Let Y T−1 and Y T−1 denote the source output and thereproduction of the source output, respectively, and

{P(dY T−1; yT−1);

T−1∑t=0

Eρt(Yt , Y t) ≤ Dv

}The restricted information rate distortion function is defined by

RDT (Dv )|R

4= inf

P(dY T−1;yT−1)∈DTDC

I (Y T−1; Y T−1)|R

Causal Rate Distortion IV

The Minimizing Kernel is Causal

The infimizing reproduction stochastic Kernel is given by

P∗(dYt ; yt , y t−1) =

esρt(y t ,y t−1,Yt)P∗(dYt ; yt−1)∫

Ytesρt(y t ,y t−1,Yt)P∗(dYt ; y t−1)

Data Processing Inequalities I

Data Processing Inequalities for Causal Channels

Suppose Y T−1 → ZN−1 → ZN−1 → Y T−1 → UT−1 form aMarkov chain.For Zn C→ Zn,Y t C→ Y t , t, n ∈ N+ then

I (Zn; Zn) ≥ I (Zn; Zn)|R ≥ I (Y t ; Zn) ≥ I (Y t ; Y t) ≥ I (Y t ; Y t)|R

Information Transmission Theorem I

Information Transmission Theorem

Suppose Y T−1 → ZN−1 → ZN−1 → Y T−1 → UT−1 form aMarkov chain.For Zn C→ Zn,Y t C→ Y t , t, n ∈ N+

CN |R ≥ RDT (Dv )|R

Asymptotic Stability and Observability

Outline

1 Introduction

5 Conclusions

6 References

Asymptotic Stability and Observability I

Asymptotic Observability

Consider the general control/communication system.The system is asymptotically observable in r -mean if there exist acontrol sequence and an encoder and a decoder such thatlimT→∞

∑T−1t=0 Eρ(Yt , Yt) ≤ Dv , where ρ(Y , Y ) = ||Y − Y ||,

r > 0 and Dv ≥ 0 is finite.

Asymptotic Stability and Observability II

Asymptotic Stabilizability

Consider the general control/communication system in which,Yt = Ht + Υt , where Υt , t ∈ N+ is a function of the measurementnoise.This system is asymptotically stabilizable in r -mean if there exist acontroller, encoder, and decoder such thatlimT→∞

∑T−1t=0 Eρ(Ht , 0) ≤ Dv , where ρ(H, 0) = ||H − 0||r ,

r > 0 and Dv ≥ 0 is finite.

Asymptotic Stability and Observability III

Necessary Conditions for Asymptotic Observability andStabilizability

Suppose Y T−1 → ZN−1 → ZN−1 → Y T−1 → UT−1 form aMarkov chain.

For Zn C→ Zn,Y t C→ Y t , t, n ∈ N+ theNecessary Conditions are

C|R ≥ RD(Dv )|R ≥ HS(Y)− log edr + log(

dVdΓ(dr )

where HS(Y) is Shannon Entropy Rate of Output Process Y T .

Outline

1 Introduction

5 Conclusions

6 References

Asymptotic Stability and Observability I

Stochastic Control System

Xt+1 = AXt + NUt + BWt , X0 = X ,

Yt = Ht + DVt , Ht = CXt

AGN Communication Channel

Zt = Zt + Wt , 0 ≤ t ≤ T

where with power constraint 1T

∑T−1i=0 E ||Zt ||2 ≤ P.

Asymptotic Stability and Observability II

Channel Does Not Assume Feedback

The encoder is an innovations encoder without channel feedback

Zt = Yt − E [Yt |Y t−1,Ut−1]

Necessary and Sufficient Condition for Asymptotic Observability

C|R ≥ RD(Dv )|R =1

2log Λ∞ −

2log Dv

where Λ∞ is the steady state value of the innovations errorcovariance

Asymptotic Stability and Observability III

The Sufficiency is obtained by matching the source to thechannel

HS(Y) ≥ 12 log(2πeD2) + max{0, log |A|}

This encoder only works for stable control systems (thedecoder cannot stabilize the control system

Next, we present one that works!

Asymptotic Stability and Observability IV

AGN Communication Channel With Feedback

Encoder (uses channel feedback)

Zt = A0(Zt−1,Ut−1) + A1(Z

t−1,Ut−1)Yt

Decoder (Least-Squares Decoder)

Yt|t−1 = E(Yt |Z t−1,Ut−1

(Xt |Z t−1,Ut−1

)Controller (LQG)

Ut = KtYt|t−1

Outline

1 Introduction

5 Conclusions

6 References

Conclusions I

Causality of Rate Distortion

Separation Principle Holds for Gaussian Control andCommunication Channels

Uncertain Control Systems and Channels (done some work)

Nonlinear Stochastic Control Systems (future work)

Outline

1 Introduction

5 Conclusions

6 References

References I

C. E. Shannon, The Mathematical Theory of Communication,Bell Systems Technical Journal, vol. 27, pp. 379–423,pp. 623–656, 1948.

R. L. Dubrushin, Information Transmission in Channel withFeedback, Theory of Probability and its Applications, vol. III,No. 4, pp. 367–383, 1958.

H. Marko, The Bidirectional Communication Theory-AGeneralization of Information Theory, IEEE Transactions onCommunication Systems, vol. 21, pp. 1345–1351, 1973.

References II

J. Massey, “Causality, Feedback and Directed Information”, inthe in Proceedings of the 1990 IEEE International Symposiumon Information Theory and its Applications, pp. 303–305,Nov.27–30, Hawaii, U.S.A.1990.

S. Tatikonda, Control Under Communication Constraint,Ph.D. Thesis, Department of Electrical Enginnering andComputer Science, MIT., September 2000.

G. Kramer, Directed Information for Channels with Feedback,Ph.D. Thesis, Swiss Federal Institute of Technology, Diss. ETHNo. 12656, 1998.

References III

S. Tatikonda and S. Mitter, Control Over Noisy Channels,IEEE Transactions on Automatic Control, vol. 49, No. 7,pp. 1196–1201, 2004.

S. Tatikonda and S. Mitter, Control Under CommunicationConstraints, IEEE Transactions on Automatic Control, vol. 49,No. 7, pp. 1056–1068, 2004.

S. Yang and A. Kavcic and S. Tatikonda, Feedback Capacityof Finite-State Machine Channels, IEEE Transactions onInformation Theory, vol. 51, No. 3, pp. 799–810, 2005.

information theory for control systems: causality and … information theory for causal systems...

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