INFORMATION-THEORETIC VIEW OF CONTROL

Prateep Roy, Arben C¸ ela

Universit

e Paris-Est, ESIEE, D

epartement Syst

emes Embarqu

es, Paris, France

Yskandar Hamam

F’SATIE-TUT, Pretoria, South Africa

Keywords:

Information Theory, Shannon Entropy, Mutual-Information, Control, Bode Sensitivity Integral.

Abstract:

In this paper we are presenting the information-theoretic explanation of Bod

e Sensitivity Integral, a funda-

mental limitation of control theory, controllability grammian and the issues of control under communication

constraints. As resource-economic use of information is of prime concern in communication-constrained

control problems, we need to emphasize more on informational aspect which has got direct relation with un-

certainties in terms of Shannon Entropy and Mutual Information. Bode Integral which is directly related to the

disturbances can be correlated with the difference of entropies between the disturbance-input and measurable

output of the system. These disturbances due to communication channel-induced noises and limited bandwidth

are causing the information packet-loss and delays resulting in degradation of control performances.

1 INTRODUCTION

In recent years, there has been an increased interest

for the fundamental limitations in feedback control.

Bode’s sensitivity integral ( Bode Integral, in short ) is

a well-known formula that quantiﬁes some of the lim-

itations in feedback control for linear time-invariant

systems. In (Sandberg and Bernhardsson, 2005), it is

shown that there is a similar formula for linear time-

periodic systems.

In this paper, we focus on Bode integral of control

theory and Shannon Entropy of information theory

because the latter is a stronger metric for uncertainty

which hinders control of a system.

It has been known that control theory and informa-

tion theory share a common background as both the-

ories study signals and dynamical systems in general.

One way to describe their difference is that the fo-

cal point of information theory is the signals involved

in systems while control theory focuses more on sys-

tems which represent the relation between the input

and output signals. Thus, in a certain sense, we may

expect that they have a complementary relation. For

this reason, many researchers have consecrated stud-

ies on the interactions of the two theories : Control

Theory and Information Theory.

In networked control systems, there are issues related

to both control and communication since communica-

tion channels with data losses, time delays, and quan-

tization errors are employed between the plants and

controllers (Antsaklis and Baillieul, 2007). To guar-

antee the overall control performance in such systems,

it is important to evaluate the quantity of information

that the channels can transfer. Thus, for the analysis

of networked control systems, information theoretic

approaches are especially useful, and notions and re-

sults from this theory can be applied. The results

in (Nair and Evans, 2004) and (Tatikonda and Mit-

ter, 2004) show the limitation in the communication

rate for the existence of controllers, encoders, and de-

coders to stabilize discrete-time linear feedback sys-

tems.

The focus of information theory is more on the signals

and not on their input-output relation. Thus, based on

information theoretic approaches, we may expect to

extend prior results in control theory. One such result

can be found in (Martins et al., 2007), where a sen-

sitivity property is analyzed and Bode’s integral for-

mula (Bode H., 1945) is extended to a more general

class of systems. A fundamental limitation of sensi-

tivity functions is presented in relation to the unstable

poles of the plants.

Roy P., Ã

Gela A. and Hamam Y.

INFORMATION-THEORETIC VIEW OF CONTROL.

DOI: 10.5220/0002166600050012

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page

ISBN: 978-989-674-001-6

2 PROBLEM FORMULATION

Networked control systems suffer from the drawbacks

of packet losses, delays and quantization in particular.

These cause degradation of control performances and

under some conditions instability. Uncertainties due

to packet losses, delays, quantization, communication

channel induced noises etc. have a great inﬂuence on

the system systems performance. If we consider only

the uncertainties induced by channel noise and quan-

tization we may write:

˙x(t) = Ax(t) + Bu(t) + w(t); (1)

y(t) = Cx(t) + Du(t) + v(t);

where A ∈ R

n×n

is the system or plant matrix and

B ∈ R

n×q

is the control or input matrix. Also, x(t)

is the state, u(t) is the control input, y(t) is the out-

put, C is the output or measurement matrix, D is the

Direct Feed matrix, w(t) and v(t) are the external dis-

turbances and noises of Gaussian nature respectively.

Our aim is to achieve better control perfor-

mance of system by tackling these uncertainties

using Shannon’s Mutual-Information, Information-

Theoretic Entropy and Bode Sensitivity. We present

the information-theoretic model of such uncertainties

and their possible reduction using information mea-

sures.

3 PRELIMINARIES

By means of the connection between Bode integral

and the entropy cost function, paper (Iglesias, 2001)

provided a time-domain characterization of Bode in-

tegral. The traditional frequency domain interpreta-

tion is that, if the sensitivity of a closed-loop system is

decreased over a particular frequency range typically

the low frequencies the designer ”pays” for this in in-

creased sensitivity outside this frequency range. This

interpretation is also valid for the time-domain char-

acterization presented in (Iglesias, 2001) provided

one deals with time horizons rather than frequency

ranges. Time-domain characterization of Bode’s in-

tegral shows how the frequency domain trade-offs

translate into the time-domain. Under the usual con-

nection between the time and frequency domains:

low (high) frequency signals are associated with long

(short) time horizons. In Bode’s result, it is important

to differentiate between the stable poles, which do not

contribute to the Bode sensitivity integral and the un-

stable poles, which do. Time-varying systems which

can be decomposed into stable and unstable compo-

nents are said to possess an exponential dichotomy.

What the exponential dichotomy says is that the norm

of the projection onto the stable subspace of any orbit

in the system decays exponentially as t → ∞ and the

norm of the projection onto the unstable subspace of

any orbit decays exponentially as t → −∞, and fur-

thermore that the stable and unstable subspaces are

conjugate. The existence of an exponential dichotomy

allows us to deﬁne a stability preserving state space

transformation (a Lyapunov transformation) that sep-

arates the stable and unstable parts of the system.

3.1 Mutual Information

Shannon’s Mutual information is just the information

carried by one random variable about the other. It is

a quantity in the time domain. Mutual Information

I(X;Y ), between X as the input variable and Y as the

output variable, has the lower and upper bounds given

by the following:

R(D) = RateDistortion = MinI(X;Y) (2)

C = CommunicationChannelCapacity = MaxI(X;Y )

(3)

where D is the distortion which happens when infor-

mation is compressed (i.e. fewer bits are used to rep-

resent or code more frequent or redundant informa-

tions) and entropy is the limit to this compression i.e.

if one compresses the information beyond the entropy

limit there is a high probability that the information

will be distorted or erroneous. This is as per Shan-

non’s Source Coding Theorem. We code more fre-

quently used symbols with fewer number of bits and

vice-versa.

Mutual information is also the difference of entropies,

where entropy is nothing but the measure of uncer-

tainty. Just as entropy (Middleton, 1960) in physical

systems tends to increase in the course of time, the

reverse is true for information about an information

source : as information about the source is processed,

it tends to decrease with time, becoming more cor-

rupt or noisy until it is evidently destroyed unless ad-

ditional information is made available. Here, infor-

mation refers to the case of desired messages.

3.2 Shannon Entropy

Shannon proposed a measure of uncertainty in a dis-

crete distribution based on the Boltzmann entropy of

classical statistical mechanics. He called it the en-

tropy and deﬁned as follows.

We have to take into account the statistics of the al-

ternatives by replacing our original measure of the

number of alternatives by the more general expression

deﬁning the entropy as follows:

H = −

∑

log

) (4)

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

where p

is the probability of the alternative i. The

above quantity is known as the binary entropy in bits

as we use logarithmic base of 2 (with logarithmic base

e the entropy is in nats), and was shown by Shannon

to correspond to the minimum average number of bits

needed to encode a probabilistic source of N states

distributed with probability p

. Intuitively, H can also

be considered as a measure of uncertainty : it is min-

imum, and is equal to zero, when one of the alterna-

tives appears with probability one, whereas it is max-

imum and equals to log

N when all the alternatives

are equiprobable so that p

for all i.

The term entropy is associated with the uncertainty

or randomness whereas information is used to reduce

this uncertainty. Uncertainty is the main hindrance

to control and if we can reduce the uncertainty by

getting the relevant information and utilizing the in-

formation properly so as to achieve the desired con-

trol performance of the system. Many researchers

have posed the same question: How much informa-

tion is required for controlling the system based on

observed informations in the case where these infor-

mations are passed through communication channels

in a networked based system?

Mutual Information I(X;Y ) and Entropies H(X),

H(Y ) and joint entropy H(X ,Y ) are related as :

I(X;Y ) = H(X) + H(Y ) −H(X,Y)

where H(X) is the uncertainty that X has about Y ,

H(Y ) is the uncertainty that Y has about X , and

H(X,Y ) is the uncertainty that X and Y hold in com-

mon. Information value degrades over time and en-

tropy value increases over time in general. The con-

ditional version of the chain rule (Cover and Thomas,

2006) :

I(X;Y ) = H(X) −H(X|Y ) = H(Y ) −H(Y |X) ; valid

for any random variables X and Y .

Mutual information I(X;Y ) is the amount of uncer-

tainty in X, minus the amount of uncertainty in X

which remains after Y is known”, which is equivalent

to ”the amount of uncertainty in X which is removed

by knowing Y ”. This corroborates the intuitive mean-

ing of mutual information as the amount of informa-

tion (that is, reduction in uncertainty) that each vari-

able is having about the other.

The conditional entropy H(X|Y ) or read as condi-

tional entropy of X knowing Y or conditioned on Y ,

is often interpreted in communication theory as rep-

resenting an information-loss (the so-called equivo-

cation of Shannon (Shannon, 1948)), which results

from subtracting the maximum noiseless capacity

I(X; X) = H(X) of a communication channel with in-

put X and output Y from the actual capacity of that

channel as measured by I(X;Y ).

3.3 Bode Integral

Physically an intrinsically stable system needs no in-

formation on its internal state or the environment to

assure its stability. So, if we consider a well de-

signed stable feedback control system with distur-

bances or/and noises as inputs and performance sig-

nals as outputs then it not needed to have extra feed-

back loop to assure its stability. We may say the same

thing for systems which are intrinsically open-loop

stable. For example, a pendulum with non-zero fric-

tion coefﬁcient subject to a perturbation will return

back to the equilibrium position after a transient pe-

riod without any need of extra information. For unsta-

ble systems the mutual information between the initial

state and the output of the system is related to its un-

stable poles.

The simplest (and perhaps the best known) result is

that, for an open loop stable plant, the integral of the

logarithm of the closed loop sensitivity is zero; i.e.

∞

( jω)

dω = 0

Where, S

and ω being the sensitivity function and

frequency respectively.

Now, we know that the logarithm function has the

property that it is negative if

< 1 and it is positive

> 1. The above result implies that set of fre-

quencies over which sensitivity reduction occurs (i.e.

where

< 1) must be matched by a set of frequen-

cies over which sensitivity magniﬁcation occurs (i.e.

where

> 1). For a stable rational transfer function

L ( jω), sensitivity is deﬁned as S ( jω) =

1+L( jω)

. This

has been given a nice interpretation as thinking of sen-

sitivity as a pile of dirt. If we remove dirt from one

set of frequencies, then it piles up at other frequencies.

Hence, if one designs a controller to have low sensi-

tivity in a particular frequency range, then the sensi-

tivity will necessarily increase at other frequencies -

a consequence of the weighted integral always being

a constant; this phenomenon has also been called the

Water-Bed Effect (pushing down on the water bed in

one area, raises it somewhere else).

For linear systems Bode Integral is the difference in

the entropy rates between the input and output of the

systems which is an information-theoretic interpreta-

tion. For nonlinear system (if the open loop system

is globally exponentially stable and has fading mem-

ory) this difference is zero. Fading Memory Require-

ment is used to limit the contributions of the past val-

ues of the input on the output. Entropy of the signals

in the feedback loop help provide another interpreta-

tion of the Bode integral formula (Zang and Iglesias,

2003)(Mehta et al., 2006) as follows. Shannon En-

INFORMATION-THEORETIC VIEW OF CONTROL

Figure 1: (a) Feedback loop and (b) Sensitivity function.

tropy - Bode Integral Relation can be rewritten as :

(x)−H

(d) =

2π

−π





jω





dω =

∑

log(p

)

(5)

Where S



jω



is the transfer function of the feedback

loop from the disturbance d to output x and p

’s are

unstable poles (|p

| > 1) of the open-loop plant; S is

referred to as the sensitivity function for an open-loop

plant gain P and a stabilizing feedback controller

gain C, S is given by S =

1+PC

. Sensitivity shows

how much sensitive is the observable output state to

input disturbance. Here, H

(x) and H

(d) denote the

conditional entropy of the random processes associ-

ated with the output x and disturbance d respectively

as per Figure1(Mehta et al., 2006).

Consider a random variable x ∈ ℜ

of continuous

type with entropy associated with this is given by

H(x) := −

ℜ

p(x)ln p(x)dx;

where p(x) being the probability density func-

tion of x and the conditional entropy of order n is

deﬁned as

H(x

k−1

,. .., x

k−n

) := −

ℜ

p(.)ln p(.)dx

where p(.) = (x

k−1

,. .., x

k−n

This conditional entropy is a measure of the un-

certainty about the value of x at time k under the

assumption that its n most recent values have been

observed. By letting n going to inﬁnity, the condi-

tional entropy of x

is deﬁned as

) := lim

n→∞

H(x

k−1

,. .., x

k−n

) assuming

the limit exists. Thus the conditional entropy is

a measure of the uncertainty about the value of x

at time k under the assumption that its entire past

is observed. Difference of conditional entropies

between the output and input is nothing but the Bode

sensitivity integral which equals the summation of

logarithms of unstable poles.

For a stationary Markov process, conditional entropy

(Cover and Thomas, 2006) is given by

H(x

k−1

,. .., x

k−n

) = H(x

k−1

4 RELATED WORK

It is well known that the sensitivity and complemen-

tary sensitivity functions represent basic properties

of feedback systems such as disturbance attenuation,

sensor-noise reduction, and robustness against uncer-

tainties in the plant model. Researchers have worked

earlier on the issues of relating the entropy and the

Bode Integral and complementary sensitivity. Refer

to work in (Sandberg and Bernhardsson, 2005), (Mar-

tins et al., 2007), (Bode H., 1945), (Freudenberg and

Looze, 1988), (Zang, 2004), (Iglesias, 2002), (Igle-

sias, 2001), (Zang and Iglesias, 2003), (Mehta et

al., 2006), (Sung and Hara, 1989), (Sung and Hara,

1988), (Okano et al., 2008), (Jialing Liu, 2006). In

(Iglesias, 2002) the sensitivity integral is interpreted

as an entropy integral in the time domain, i.e., no

frequency-domain representation is used.

One has to gather relevant information, transmit the

information to the relevant agent, process the infor-

mation, if needed, and then use the information to

control the system. The fundamental limitation in

information transmission is the achievable informa-

tion rate (i.e. a fundamental parameter of Information

Theory), the fundamental limitation in information

processing is the Cramer-Rao Bound (CRB) which

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

deals with Fisher Information Matrix (FIM) in Esti-

mation Theory, and the fundamental limitation in in-

formation utilization is the Bode Integral (i.e. a funda-

mental parameter of Control Theory), seemingly dif-

ferent and usually separately treated, are in fact three

sides of the same entity as per the paper (Liu and Elia,

2006). Even Kalman et al. in their paper (Kalman et

al., 1963) have stated that Controllability Grammian

Matrix W-matrix is analogous to FIM and the deter-

minant det W is analogous to Shannon Information.

These research work motivated us to investigate some

important correlations amongst mutual information,

entropy and design control parameters of practical im-

portance rather than just concentrating on stability is-

sues.

5 INFORMATION INDUCED BY

CONTROLLABILITY

GRAMMIAN

In general, from the viewpoint of the open-loop sys-

tem, when the system is unstable, the system ampli-

ﬁes the initial state at a level depending on the size

of the unstable poles (Okano et al., 2008). Hence, we

can say that in systems having more unstable dynam-

ics, the signals contain more information about the

initial state. Using this extra information (in terms of

mutual information between the control input and the

initial state) we can reduce the entropy (uncertainty)

and thus rendering more easy the observation of ini-

tial state.

Suppose that we have a feedback control system in

which control signal is sent through a network with

limited bandwidth. We will consider the case where

the state of the system is measurable and the con-

troller can send the state of the system without error.

Under these conditions we may write:

˙x(t) = Ax(t) + Bu

∗

(t); (6)

∗

(t) = −K

x(t) + u

(t);

where K

, u

∗

(t) and u

(t), represent, respectively, the

feedback controller gain, the applied control input

and control error due to quantization noise of limited

bandwidth network. In the sequel we are supposing

that the control signal errors are caused by Gaussian

White Noise which may be given by u

(t) =

√

δ(t).

So we may write :

˙x(t) = (A −BK

)x(t) + Bu

(t); (7)

∗

(t) = −K

x(t) + u

(t);

or more compactly:

˙x(t) = A

x(t) + Bu

(t); (8)

where,A −BK

= A

∗

(t) = −K

x(t) + u

(t);

The feedback system (8) is a stable one which is

perturbed by quantization errors or noises due to the

bandwidth limitation.

Lemma : The controllability grammian matrix

W of system (8) is related with the information-

theoretic entropy H as follows (Mitra, 1969):

H(x,t) =

{

detW(D,t)

}

(1 + ln 2π) (9)

(Where D being the Diagonal Matrix (positive deﬁ-

nite symmetric matrix) with D

being the ith diagonal

element. Here unit impulse inputs are considered.)

= Average apriori uncertainty of the state x at

time t for an order n of the system.

where

W(D,τ) =

BDB

for a system modeled as (8).

Proof of Eqn.(9): Referring to (Cover and Thomas,

2006) we are providing the proof. The input of

(8) being Gaussian White Noise, the state of the

system is with probability density having mean-value

¯x(t) = e

x(0) and Covariance Matrix Σ at time t is

given by

Σ = E



(x − ¯x)(x − ¯x)



= W(D,t).

In a more detailed form :

x(t) = e

x(0) +

Bu(t)dt

{

x(t)

}

= E



x(0) +

Bu(t)dt



T here f ore, ¯x(t) = e

x(0)

where ¯x(t) denotes the mean value of x(t) and Covari-

ance Matrix

Σ = E



(x − ¯x)(x − ¯x)



⇒ Σ = E[



x(0) +

Bu(t)dt −e

x(0)





x(0) +

Bu(t)dt −e

x(0)



]

INFORMATION-THEORETIC VIEW OF CONTROL

Therefore, Σ =

BDB

dt = W(D,τ).

where, u

(t) =

√

δ(t) i.e. weighted impulses

and D being the Diagonal Matrix (positive deﬁnite

symmetric matrix) with D

being the ith diagonal

element. Here unit impulse inputs are considered.

p(x,t) =

(2π)

n/2

{

detW(D,t)

}

1/2

[−1/2

(x−¯x)

−1

(D,t)(x−¯x)

]

(10)

Now, for multidimensional continuous case, entropy

(precisely differential entropy) of a continuous ran-

dom variable X with probability density function f (x)

( if

∞

−∞

f (x)dx = 1 ) is deﬁned (Cover and Thomas,

2006) as

Differential Entropy h(X) = −

f (x) ln f (x)dx;

where the set S for which f (x) > 0 is called the sup-

port set of X.

As in discrete case, the differential entropy depends

only on the probability density of the random variable

and therefore the differential entropy is sometimes

written as h( f ) rather than h(X). Here, we call dif-

ferential entropy as H(x,t) and f (x) as p(x,t) which

are correlated as

H(x,t) = −

p(x,t)ln p(x,t)dx (11)

Using equation (10) in equation (11) we get

H(x,t) = −

p(x,t)[−

(x − ¯x)

−1

(D,t)(x − ¯x)

−ln

(2π)

n/2

{

detW(D,t)

}

1/2

]dx

H(x,t) =



∑

i, j



−

)(W

−1

(D,t))

i j

−

)





ln[

{

(2π)

{

detW(D,t)

}

]



∑

i, j



−

)(X

−

)(W

−1

(D,t))

i j





ln[

{

(2π)

{

detW(D,t)

}

]

∑

i, j





−

)(X

−

)



−1

(D,t))

i j



ln[

{

(2π)

{

detW(D,t)

}

]

∑

(W(D,t))

−1

(D,t))

i j

ln[

{

(2π)

{

detW(D,t)

}

]

∑



(W(D,t))(W

−1

(D,t))



j j

ln[

{

(2π)

{

detW(D,t)

}

]

∑

j j

ln[

{

(2π)

}{

detW(D,t)

}

]

(Where I

j j

is the Identity Matrix )

ln[

{

(2π)

}{

detW(D,t)

}

]

{

(2π)

}

{

detW(D,t)

}

{

(2π)

}

{

detW(D,t)

}

H(x,t) =

{

detW(D,t)

}

(1 + ln 2π)

Since Controllability Grammian is independent

of co-ordinate system and so is the Mutual Informa-

tion, we try to draw the analogy between the two.

Based on the equation (9) we can write the entropy

reduction as

∆H(x,t) =

∆[ln

{

detW(D,t)

}

]

This shows that the entropy reduction which is

same as uncertainty reduction is dependent on

Controllability Grammian only. Other term being

constant for constant n, gets canceled.

Therefore, ∆H(x,t) = H(x(t

),t

) −H(x(t

),t

)

{

detW

)

}

−

{

detW

)

}

⇒ ∆H(x,t) =



detW

)

detW

)



(12)

For simplicity we denote ∆H(x,t) by ∆H, W

)

by W

and W

) by W

Therefore, ∆H =

det(

)

⇒ det(

) = e

2(∆H)

Using the above expression along with the con-

cept of mutual information being the difference of

the entropy and the residual conditional entropy i.e.

I(X;U) = H(X ) − H(X|U) (gain in information is

reduction in entropy), we can conclude that Mutual

Information I(X ;U) between the state X and control

input U denoted simply by Shannon Information I

is given by this ∆H which can be expressed further as

Finally,

det(

) = e

2(∆H)

= e

(13)

We may conclude that the uncertainty reduction

which is directly related to the ∆H(x,t) will reduce

the variance of the state with respect to the steady-

state if ∆H(x,t) converges to zero. The only inﬂuence

we have on the control signal is related to that of feed-

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

back gain, to be chosen such that the norm of grammi-

ans, represented by det(W (D

,t)) converge rapidly to

their norm to inﬁnity det(W (D

∞

,∞)). We will detail

the related approach in a future paper.

6 CONCLUSIONS

This paper has addressed some new ideas concerning

the relation between control design and information

theory. Since the networked control system has com-

munication constraints due to limited bandwidth or

noises, we must have to adopt a policy of resource al-

location which enhances the information transmitted.

This may be done possible if we know the character-

istics of the networks, the bandwidth constraints and

that of the dynamical system under study.

As demonstrated the grammian of controllability con-

stitute a metric of information theoretic entropy with

respect to the noises induced by quantization. Reduc-

tion of these noises is equivalent to the design meth-

ods proposing a reduction of the controllability gram-

mian norm. In the case of bandwidth constraints it

takes its full interest which will be demonstrated in

a future paper. Future work in this direction would

be also to propose an information-theoretic analysis

for enhancing the zooming algorithm proposed (Ben

Gaid and C¸ ela, 2006) and optimal allocation of com-

munication bandwidth which maximizes the systems’

performances based on Controllability Grammians.

Illustration of these results by simulation and / or ex-

perimental veriﬁcation of the theoretical approaches

is the objective of our work.

REFERENCES

Sandberg H. and Bernhardsson B.(2005), A Bode Sensitiv-

ity Integral for Linear Time-Periodic Systems, IEEE

Trans. Autom. Control, Vol. 50, No. 12, December.

Antsaklis P. and Baillieul J. (2007), Special Issue on the

Technology of Networked Control Systems, Proc. of

the IEEE, Vol. 95, No. 1.

Nair G. and Evans R.(2004), Stabilizability of stochastic

linear systems with ﬁnite feedback data rates, SIAM

J. Control Optim., Vol. 43, No. 2, p. 413-436.

Tatikonda S. and Mitter S.(2004), Control under communi-

cation constraints, IEEE Trans. Autom. Control, Vol.

49, No. 7, p. 1056-1068.

Martins N., Dahleh M., and Doyle J.(2007), Fundamental

limitations of disturbance attenuation in the presence

of side information, IEEE Trans. Autom. Control, Vol.

52, No. 1, p. 56-66.

Bode H.(1945), Network Analysis and Feedback Ampliﬁer

Design, D. Van Nostrand, 1945.

Iglesias P.(2001), Trade-offs in linear time-varying systems

: An analogue of Bode’s sensitivity integral, Auto-

matica, Vol. 37, No. 10, p. 1541-1550.

Middleton D.(1960), An Introduction to Statistical Commu-

nication Theory, McGraw-Hill Pub., p. 315.

Cover and Thomas(2006), Elements of Information Theory,

2nd Ed., John Wiley and Sons.

Shannon C.(1948), The Mathematical Theory of Commu-

nication, Bell Systems Tech. J. Vol. 27, p. 379-423, p.

623-656, July, October.

Zang G. and Iglesias P.(2003), Nonlinear extension of

Bode’s integral based on an information theoretic in-

terpretation, Systems and Control Letters, Vol. 50, p.

1129.

Mehta P., Vaidya U. and Banaszuk A.(2006), Markov

Chains, Entropy, and Fundamental Limitations in

Nonlinear Stabilization, Proc. of the 45th IEEE Con-

ference on Decision & Control, San Diego, USA, De-

cember 13-15.

Freudenberg J. and Looze D.(1988), Frequency Domain

Properties of Scalar and Multivariable Feedback Sys-

tems, Lecture Notes in Control and Information Sci-

ences, Vol. 104, Springer-Verlag, New York.

Zang G.(2004), Bode’s integral extensions in linear time-

varying and nonlinear systems, Ph.D. Dissertation,

Dept. Elect. Comput. Eng., Johns Hopkins Univ.,

USA.

Iglesias P.(2002), Logarithmic integrals and system dy-

namics : An analogue of Bode’s sensitivity integral

for continuous-time time-varying systems, Linear Alg.

Appl., Vol. 343-344, p. 451-471.

Sung H. and Hara S.(1989), Properties of complementary

sensitivity function in SISO digital control systems,

Int. J. Control, Vol.50, No. 4, p. 1283-1295.

Sung H. and Hara S.(1988), Properties of sensitivity and

complementary sensitivity functions in single-input

single-output digital control systems, Int. J. Control,

Vol. 48, No. 6, p. 2429-2439.

Okano K, Hara S. and Ishii H.(2008), Characterization of a

complementary sensitivity property in feedback con-

trol: An information theoretic approach, Proc. of the

17th IFAC-World Congress, Seoul, Korea, July 6-11,

p. 5185-5190.

Jialing L.(2006), Fundamental Limits in Gaussian Channels

with Feedback: Conﬂuence of Communication, Esti-

mation and Control, Ph.D. Thesis, Iowa State Univer-

sity.

Liu J. and Elia N. (2006), Convergence of Fundamental

Limitations in Information, Estimation, and Control,

Proceedings of the 45th IEEE Conference on Decision

& Control, San Diego, USA, December.

Kalman R., Ho Y. and Narendra K.(1963), Controllability

of linear dynamical systems, Contributions to Differ-

ential Equations, Vol. 1, No. 2, p. 189-213.

Mitra D.(1969), W-matrix and the Geometry of Model

Equivalence and Reduction, Proc. of the IEE, Vol.

116, No.6, June, p. 1101-1106.

INFORMATION-THEORETIC VIEW OF CONTROL

Ben Gaid M. M., and C¸ ela A.(2006), Trading Quantiza-

tion Precision for Sampling Rates in Networked Sys-

tems with Limited Communication, Proceedings of

the 45th IEEE Conference on Decision & Control, San

Diego, CA, USA, December 13-15.

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics