CDC 2007 1

Robust Control of a Class of Feedback Systems Subject to Limited Capacity Constraints

Alireza FarhadiSchool of Information Technology and Engineering,University of Ottawa, Ottawa, Canada

C. D. CharalambousElectrical and Computer Engineering Department University of Cyprus, Nicosia, Cyprus

2CDC 2007

Overview

Uncertain Control System

Reliable Communication for Uncertain Sources Described via a Relative Entropy Constraint

Uncertain Fully Observed Controlled System

CDC 2007 3

Problem Formulation

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Problem Formulation

Control/communication system subject to the uncertainty in the source

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Problem Formulation

Information Source: The information source is described by the probability measure which depends on the control sequence as shown in the Figure. It is assumed that the density function belongs to the class

TY

T dYfdYP T)(

TYf

1 1 1 1 1

1'

0

1 1( ) ; ( || ) [ ]

2T T T T T

T

SU c t t tY Y Y Y Yt

f D g f H f g R E Y M YT T

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Problem Formulation

Example: Uncertain class of fully observed Gauss Markov systems

where

is i.i.d. is the perturbed noise random process and is the signal to be controlled.

[Pra-Meneghini-Runggaldier 96, Ugrinovskii-Petersen 99]

,,

, 01

tttt

ttttt

XHHY

XXWBBWNUAXX

td

tm

tm

to

td

t WHVxNXWWUX ,),,(~,,,, 000

tWW WN ,0),,0(~ tH

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Problem Formulation

Nominal system:

It can be shown that for the sequence

tttt

tttt

XHHY

XXBWNUAXX

,

,, 01

1 1

2' 1

0

1( || ) [ ].

2T T

T

t W tY Yt

H f g E W W

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Problem Formulation

The objective is to design encoding/decoder and stabilizing controller which guarantee uniform mean square reconstruction and stability.

Uniform mean square reconstruction:

For :

.0,2

,||~

||1

suplim1

0)( 11

v

v

T

t

rtt

gDfT

Dr

DYYET

TYSUTY

ttt HY

.0,2

,||||1

suplim1

0)( 11

v

v

T

t

rt

gDfT

Dr

DHET

TYSUTY

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Problem Formulation

Shannon: Consider a communication system without feedback. A necessary condition for mean uniform reconstruction of is given by

Above result can be extended to channels and sources which use feedback from the output of the decoder to the input of the encoder ( the control/communication system), provided 1) The capacity with and without feedback are the same. 2) The rate distortion of a source without using feedback from the decoder to the encoder is the same as the one that uses such feedback, and the reconstruction kernel is causal. (generalizations [Charalambous 2006]

r TY

).())()(

log()( , vrSr

d

vd

r DRrD

d

rd

dV

r

r

dC

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Uncertain Source Described Via Relative Entropy Constraint

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Rate Distortion for a Class of Sources

Information Source: Consider a class of sources which produce orthogonal zero mean output process

described via the following relative entropy constraint

where the density function is Gaussian. That is,

The rate distortion for this class is defined by the following minimax problem

where

tK dt ;

]2

1[)||(

1;)(

1

0

'1

1111

T

ttttfcKKKKSU KMK

TERgfH

TfgD

TK

TTTT

1TKg

).,...,,,0(~ 1101 TK diagNg T

),~

;(supinf)( 11

)();~

(,

11

11

TT

gDfkKdPvrT KKIDR

TKSUTKDC

TT

.||~

||1

);;~

(1

0

211

v

T

ttt

TTDC DKKE

TkKdP

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Rate Distortion for a Class of Sources

It can be shown that above minimax problem is equivalent to the following maximin problem

Consider the case of (the vector case was treated in the paper). Under assumption of , we have

where is unique solution to the following equation

).~

;(infsup)( 11

);~

()(

sup11

11

TT

kKdPgDfvT KKIDR

DCTT

TKSUTK

tK 1,...,2,1,0,1)( TtM tt

,

1

1,min,log

2

1)(

*

***

1,...,1,0

1

0

*sup

tt

ttt

Ttv

T

t v

tvT Ms

sD

DDR

0* s

.)1log(2

111log

2

1 1

0**

*

c

T

ttt RM

Tss

s

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Computation of Robust Shannon Lower Bound

It can be shown that when and , the robust Shannon lower bound is given by

Thus, for , the robust Shannon lower bound is an exact approximation of . That is,

TTlim MMT

T

lim

vvrS D

Ms

s

DR

1

1

log2

1)(

*

*

,

tt

t

tv Ms

sD

1

1min

*

*

)(1

lim)( supsupvT

Tv DR

TDR

).2log(2

1)()()( ,

supvrvrSv eDDRDR

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Realization of a Communication Link Matched to the Uncertain Source

Next, consider the following AWGN channel.

Under assumption of and , it can be shown that if the encoder multiplies by

( ) and transmits it under transmission power constraint , where is the unique solution of the following equation , we have

On the other hand, if the decoder multiplies the channel outputs by

, we have an end to end transmission with distortion

.)(,

),,0(~~

,~~

2ttt

ctttt

PZEZ

WNorthogonalWWZZ

TTlim MMT

T

lim

tK *1,

t

vt

v

ctt

D

D

W

tt

t

tv Ms

sD

1

1min

*

*

ttt

t

t

t

v

ctttt P

MD

WKEZE

1

1)()(

*

*222

0* t

ctttt

t RM

)1log(2

111log

2

1**

*

).()( ,sup

vrSv DRDRC

c

vt W

Dt

.)~

(1

supsuplim1

0

2

)( 11

v

T

ttt

gDfTDKKE

TTKSUTK

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Uncertain Fully Observed Controlled Gauss Markov System

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Uncertain and Nominal System

Consider the following uncertain system

The corresponding nominal system is

The uncertain system is subject to the following sum quadratic uncertainty constraint.

tttt

ttttttt XHHY

XXWBBWNUAXXP

,

,,:);),(,( 01

0

tttt

tttttt XHHY

XXBWNUAXX

,

,,:);),(,( 01

0

].)(2

1[)(;0)(;

1

0

'2

0

1'2220 c

T

ttt

T

ttWtP

TTTtt RMXXWW

TEWgWgW

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Computing the Robust Entropy

For and , consider the following robust entropy problem

Suppose , then above problem is equivalent to the following robust entropy problem

111)( T

YT dYfdYP T

111)( T

YT dYgdY T

).(1

sup)(1

1

11

1

)(

*

T

TYSUTY

T YSgDf

Yr fHT

fHT

11)( WW BBBB

.][

2][log

1][

2

1supmin)(

1 1

0

'2

0

1'

0

*,1

2

0

*

1

T

tttPcYP

T

ttWtP

Ws

sYr MYYE

T

ssRgE

TWWE

T

sfH

TT

T

tt

T

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Computing the Robust Entropy

Theorem: Consider the robust entropy problem.

Let for some . Then,

i)

where is a real symmetric solution of

and is the minimizing solution of the following equation

11 )1()( WW sBBBB 0s

tttWWt AXBBBsBBBBW 11

111* ])1()([

t

.

])1()([

1

11

111

11

sM

SMABBBsBBBBBAAA

T

ttWWttt

0s

1

100

0

* )(2

1)(

2

1min)(

T

tWtc

sBBtrac

TVtrac

TsRsz

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Computing the Robust Entropy

Theorem (Continued): The robust entropy rate is given by

where is the solution of the following Algebraic Riccati equation appearing in the estimation and control problems

)(

2

1min)det(log

2

1)2log(

2)(

0Wc

sWcr BBtracsRBBe

dsR

H

.])1()([ 111 sMABBBsBBBBBAAA WW

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Optimal Controller Design

Let

The objective is to design an encoder, decoder and controller for mean square stability subject to the following cost functional.

where

.,...,~

,...,~

;;: 0,01)(

ttUt

Utt

odottt UUKKGGUUU

),,(

1supinflim 1,01,0

0)(; 22

0

1

TTWgWUT

UXJTTT

tt

T

).0(,)||||||(||2

1),(

1

0

221,01,0

HUXEUXJ

T

tHttPTT

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Optimal Controller Design

Below figure illustrates encoding and stabilizing schemes for uniform observability and robust stability. The encoder, decoder and controller will be an extension of the results of previous section and [1].

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Conclusion

Our motivation for considering the relative entropy uncertainty description is that it gives as a special case a constraint on the energy of the uncertainty. Such uncertainty description has been considered in [2].

Under certain conditions the robust Shannon lower bound is a tight bound for uniform observability.

For future work Separation Theorem Uncertain Channels and Sources

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References

[1] I. R. Petersen, M. R. James and P. Dupuis, Minimax Optimal Control of Stochastic Uncertain Systems with Relative Entropy Constraints, IEEE Transactions on Automatic Control, vol. 45, No. 3, pp. 398-412, March 2000.

[2] C. D. Charalambous and Alireza Farhadi, Stochastic Control of General Discrete Time Partially Observed Systems over Finite Capacity Communication Channels: LQG Optimality and Separation Principle, submitted to Automatica, July 2007, under review.

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References for Control/Communication

Tatikonda, Mitter, Sahai, Elia, Nuno, Dahleh,

Yuksel, Basar, Girish, Dey, Evans, Liberzon, …