back gain, to be chosen such that the norm of grammi-
ans, represented by det(W (D
i
,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
11