rojas2009

Systems & Control Letters 58 (2009) 353–358
Contents lists available at ScienceDirect
Systems & Control Letters
journal homepage: www.elsevier.com/locate/sysconle
Signal-to-noise ratio performance limitations for input disturbance rejection in
output feedback control
Alejandro J. Rojas∗
ARC Centre of Excellence for Complex Dynamic Systems and Control, The University of Newcastle, Callaghan NSW 2308, Australia
article info
Article history:
Received 22 July 2008
Received in revised form
10 December 2008
Accepted 5 January 2009
Available online 1 February 2009
Keywords:
Signal to noise ratio
Control over communication networks
Performance limitations
Input disturbance rejection
Stability of linear systems
abstract
Communication channels impose a number of obstacles to feedback control. One recent line of research
considers the problem of feedback stabilization subject to a constraint on the channel signal-to-noise
ratio (SNR). We use the spectral factorization induced by the optimal solution and quantify in closed-
form the infimal SNR required for both stabilization and input disturbance rejection for a minimum
phase plant with relative degree one and memoryless additive white Gaussian noise (AWGN) channel.
Finally we conclude by presenting a closed-form expression of the difference between the infimal AWGN
channel capacity for input disturbance rejection and the infimal AWGN channel capacity required only
for stabilizability.
©2009 Elsevier B.V. All rights reserved.
1. Introduction
Fundamental limitations in control design have been an
important area of research for many years, [1,2]. Recently, the
study of fundamental limitations has been extended to problems
of control over communication networks, see for example [3,
Theorem 4.6], [4], as well as the special issue [5] and the recent
survey by Nair et al. [6].
Communication channels impose additional limitations to
feedback, such as constraints in transmission data rate and
bandwidth, and effects of noise and time-delay. One line of recent
research introduced a framework to study stabilizability of a
feedback loop over channels with a signal to noise ratio (SNR)
constraint [7,8]. These papers obtained the infimal SNR required
to stabilize an unstable linear time invariant (LTI) plant over
an additive white Gaussian noise (AWGN) channel. A distinctive
characteristic of the SNR approach is that it is a linear formulation,
suited for the analysis of robustness using well-developed tools [9].
For the case of LTI controllers and minimum phase plant models
with no time delay, these conditions match those derived in [10]
by application of Shannon’s theorem [11, Section 10.3].
Different techniques are used in [7,8] depending on whether
stabilization is achieved by state feedback or by output feedback.
A common framework for both state and output feedback cases
is proposed in [12], where it is shown that both problems can
∗Tel.: [phone].
E-mail address: [email protected].
be solved as a linear quadratic Gaussian (LQG) optimization.
Specifically, the optimal control problem arising from the infimal
SNR constrained control problem may be posed as an LQG
optimization with weights chosen as in the loop transfer recovery
(LTR) technique (see [13–17]). Doing so not only allows a unified
treatment of the state and output feedback cases, but also suggests
how performance considerations may be analyzed in addition to
stabilization.
The first contribution of the present paper is to obtain the
infimal SNR constrained solution in closed-form for stabilizability
and input disturbance rejection in the case of memoryless AWGN
channels. The second contribution is that by means of the infimal
SNR constrained solution we quantify the difference between
the infimal channel capacity ˆ Cfor performance and the infimal
channel capacity required for stabilizability only, see [10]. This
difference between channel capacities has been identified in [4],
based on information theoretic arguments, as a key element
representing a fundamental limitation in control over network
performance. To the best knowledge of the author the channel
capacity difference imposed by the input disturbance rejection has
not been quantified in closed-form before.
Of the two possible configurations for the location of the
idealized communication channel, we consider the case of an
AWGN communication channel over the measurement link. Such
a setting is common in practice and arises, for example, when
sensors are far from the controller and have to communicate
through a communication network.
We neglect all pre- and post- signal processing involved in
the communication link, which is then reduced only to the
0167-6911/$ – see front matter ©2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2009.01.001

354 A.J. Rojas / Systems & Control Letters 58 (2009) 353–358
Fig. 1. Feedback control stabilization of a discrete-time unstable plant subject to
input disturbance over a discrete-time AWGN channel.
communication channel itself (see Fig. 1). The reason for this is
motivated by the goal of quantifying the fundamental limitations
imposed by the communication channel. The results in [8], which
also do not consider explicitly encoder and decoder, give strength
to the argument by agreeing with the results in [10], which
considers explicitly an encoder and decoder.
The paper is organized as follows: Section 2introduces the
paper assumptions and the problem definition. In Section 3we
proceed to quantify the infimal SNR and the channel capacity
difference for stabilizability and input disturbance rejection in the
case of memoryless AWGN channels, as in Fig. 1. In Section 4we
conclude with final remarks on the present work.
A preliminary version of the present results has been commu-
nicated in [18].
Terminology: Let D−,¯ D−,D+and ¯ D+denote respectively the
open unit-disk, closed unit-disk, open and closed unit-disk
complements in the complex plane C, with ∂Dthe unit-disk itself.
Let Rdenote the set of real numbers, R+the set of positive real
numbers, R+
othe set of non-negative real numbers and R−the set
of real negative numbers. Let Z+denote the set of positive integers.
Define j=√−1.
A discrete-time signal is denoted by x(k),k=0,1,2, . . .,
and its Z-transform by X(z),z∈C. The expectation operator is
denoted by E. A rational transfer function of a discrete-time system
is minimum phase if all its zeros lie in ¯ D−, and is non minimum
phase if it has zeros in D+. We define the H∞space as a (closed)
subspace of L∞with functions that are analytic and bounded in
D+. The RH∞space consists of all proper and real rational stable
transfer functions. The norm of a system P(z)in H∞is given
by kPk∞=supθ∈[−π,π) P(ejθ). In the discrete-time setting we
define the L2space as the space of functions f:ej[−π,π) →Csuch
that kfk2
2=1
2πRπ
−π|f(ejθ)|2dθ < ∞. Define the H2space as a
(closed) subspace of L2with functions f(z)analytic in D+. Finally,
we define the H⊥
2space as the orthogonal complement of H2in L2,
that is the (closed) subspace of functions in L2that are analytic in
D−. If ais in C,¯ arepresents its complex conjugate and aH=¯ aTthe
Hermitian (i.e., the transposed complex conjugate of a). By general
convention we have 0! = 1.
2. Assumptions and problem formulation
We consider the discrete-time feedback system depicted in
Fig. 1. The AWGN channel is characterized by two parameters: the
admissible input power level of the channel, P, and the channel
additive noise process n(k).
2.1. Assumptions
General assumptions involved in the present discussion, which
will be in place unless stated otherwise, are
Plant model assumptions: throughout the present work, if not
stated otherwise, it is assumed that the plant model G(z)is a
strictly proper rational function with the following properties:
– relative degree ng=1,
– all its zeros have moduli less than 1,
–munstable poles, |ρi|>1, each with multiplicity ni,∀i=
1,...,m.
Channel additive noise process: the channel additive noise
process is labeled n(k)and it is a zero-mean i.i.d. Gaussian white
noise process with variance σ2.
Input disturbance process: the input disturbance process is
labeled d(k)and it is a zero-mean i.i.d. Gaussian white noise
process with variance σ2
d.
Notice that if we lift the assumption of G(z)minimum phase,
then we would be required to invoke an inner factorization
argument similar to the one presented in [16,19].
2.2. Problem formulation
We assume that C(z)is such that the closed loop system is
stable in the sense that, for any distribution of initial conditions, the
distribution of all signals in the loop will converge exponentially
rapidly to a stationary distribution. The channel input power, see
Fig. 1, is required to satisfy an arbitrary imposed power constraint
P>Ey2(1)
for some predetermined power level P, where Ey2stands for
limk→∞ Ey2(k)and it is introduced to ease the notation. Under
the assumption of stationarity, as in [20, Section 4.4], the power in
the channel input may be computed as
Ey2=1
2πZπ
−π|Tyn(ejω)|2σ2dω+1
2πZπ
−π|Tyd(ejω)|2σ2
ddω,
where
Tyn(z)= − C(z)G(z)
1+C(z)G(z),Tyd(z)=G(z) 1+C(z)G(z),(2)
are the transfer functions that relate y(k)with n(k)and y(k)with
d(k). Since the feedback control system is stable and recalling the
definition of the L2norm, we have
Ey2=
Tyn
2
2σ2+
Tyd
2
2σ2
d.
Thus, the power constraint (1) at the input of the channel translates
into a channel SNR bound given by the H2norm of Tyn(z)and Tyd(z)
P
σ2>
Tyn
2
2+
Tyd
2
2
σ2
d
σ2.(3)
An expression similar to Eq. (3) can be found in [21, Section VI] for
output disturbance rejection.
From (3) we observe that a fundamental limitation on the
channel SNR will be given by the infimum of 
Tyn
2
2and
Tyd
2
2. This observation is the foundation of the infimal SNR for
stabilizability with input disturbance rejection LTI problem.
Problem 1 (Infimal SNR for Stabilizability with Input Disturbance
Rejection LTI Problem). Find a proper rational stabilizing controller
C(z)such that the feedback control loop is stable and the transfer
functions in (2) achieve the infimum possible constraint (3)
imposed on the admissible channel SNR.
The search for the optimal stabilizing controller ˆ C(z)that
achieves the infimal H2norm of Tyn(z)and Tyd(z)can be performed
via the machinery of LQG estimation with LTR at the output.
The procedure of LQG optimization with LTR at the output
involves the solution of two Riccati equations, one associated

A.J. Rojas / Systems & Control Letters 58 (2009) 353–358 355
with the design of the observer and another with the design
of the regulator. If we were to perform the full design for the
output feedback loop we would have to design two pairs of
weighting matrices, one pair for the observer’s Riccati equation and
a second pair for the regulator Riccati equation. The LTR procedure
simplifies the LQG design by pre-assigning the weights for the
regulator Riccati equation as a cheap control design.
It is well known that as a result of the LQG/LTR approach, for a
minimum phase plant model with relative degree one, the output
feedback optimal sensitivity function ˆ S(z)=1/(1+G(z)ˆ C(z))
recovers the observer’s design. That is, for a minimum phase plant
model with relative degree one, we are able to recover the design
for the observer at the output. This remark is very useful since a
simple spectral factorization analysis can now be applied to obtain
the closed loop characteristic polynomial whenever the infimal
controller ˆ C(z)is in place, see for example [22, Section 6.4.3] and
also [18, Section V].
3. Infimal channel signal to noise ratio for input disturbance
rejection
Consider that ˆ C(z), the controller that solves Problem 1, is
in place. It is possible to verify that such controller will induce a
spectral factorization given by
ˆ
S−1(z) m
Y
i=1ρi
zini!ˆ
S−T(z−1)=1+G(z)σ2
d
σ2G(z−1). (4)
From Eq. (4) we have that the plant model G(z), together with σ2
and σ2
d, will determine ˆ
S(z). Notice though that the stable poles
of G(z)will also play a role in (4), as different from the case of
stabilizability with no input disturbance rejection where only the
unstable poles of G(z)played a role (see [22]).
We now quantify the infimal SNR required for stabilizability
and input disturbance rejection for the case of memoryless AWGN
channels. To do so we use the spectral factorization result in (4)
and specify the plant model to be
G(z)=q(z)
p(z)=q(z)
m
Q
i=1
(z−ρi)ni
.(5)
The polynomial q(z)is assumed known, with degree n1+n2+···+
nm−1 and all its solutions are in D−. As we stated before, we are
ultimately attempting to characterize the particular ˆ S(z)that takes
part into the infimal SNR for stabilizability with input disturbance
rejection LTI solution. Notice that ˆ S(z)must contain the munsta-
ble plant poles ρi, including multiplicity, as non minimum phase
(NMP) zeros to guarantee the internal stability of the closed loop.
Thus, we only have to obtain from (4) the location of the poles of
ˆ
S(z)
ˆ
S−1(z) m
Y
i=1ρi
zini!ˆ
S−T(z−1)
=p(z)p(z−1)+q(z)σ2
d
σ2q(z−1)
p(z)p(z−1).(6)
From (6) we recognize that the poles of ˆ S(z), labeled zi,zi∈D−i=
1,···m, with multiplicities n1,n2,···,nm, are the Pm
i=1niSchur’s
solutions of
p(z)p(z−1)+q(z)σ2
d
σ2q(z−1)=0,(7)
a polynomial of degree 2 Pm
i=1ni. It can be shown that the
other Pm
i=1nisolutions of (7) are the reflections of each zi, that is
1/zi,1/zi∈D+i=1,···m. We have then that a characterization
of the optimal ˆ S(z)is given by
ˆ
S(z)=
m
Y
i=1z−ρi
z−zini
.(8)
We stress that, although we do not have a closed-form for each zi,
they can be computed by any of the many currently available al-
gorithms for the purpose of finding the solutions of a polynomial,
thus for all purposes we consider them as known quantities. Finally
notice, also from (7), that as σ2
d→0, each ziwill tend to one of the
unstable plant poles mirrored images 1/ρi. By means of (8) we are
able to quantify the infimal SNR for stabilizability with input dis-
turbance rejection.
Theorem 2 (Infimal SNR for Stabilizability with Input Disturbance
Rejection). Assume the plant to be as in (5) and the channel model to be
a memoryless AWGN channel as in Fig. 1. Then, for the feedback loop
to be stabilizable and ensure input disturbance rejection, the channel
SNR must satisfy
P
σ2>
m
Y
i=1
(ρi)2ni−1+
m
Y
i=1
(ρi)2ni
×
m
X
i=1
ni
X
l=1
mi,l
(l−1)!
m
X
j=1
nj
X
p=1
dl−1
dzl−1mj,pzp−1 (1−z¯ zj)pz=zi
+σ2
d
σ2
m
X
i=1
ni
X
l=1
gi,l
(l−1)!
m
X
j=1
nj
X
p=1
dl−1
dzl−1gj,pzp−1 (1−z¯ zj)pz=zi
,
(9)
where
mi,l=1
(ni−l)!
dni−l
dzni−l
m
Q
j=1
(z−1/ρj)nj
m
Q
j=1
j6=i
(z−zj)nj
z=zi
,
gi,l=1
(ni−l)!
dni−l
dzni−l(q(z))|z=zi.
(10)
Proof. We proceed by considering the function spaces L2,H2,H⊥
2,
and RH∞, with the stability region given by the open unit disk
in the complex plane. Introduce a coprime factorization such that
G(z)=N(z)/M(z), where
N(z)=q(z)
m
Q
i=1
(1−zρi)ni
,M(z)=
m
Y
i=1z−ρi
1−zρini
,
and the parameterization of all stabilizing controllers (see [23, pp.
64–65])
C(z)=(X(z)+M(z)Q(z))/(Y(z)−N(z)Q(z)),
where X(z)and Y(z)satisfy the Bezout identity, N(z)X(z)
+M(z)Y(z)=1 and Q(z)is the Youla parameter, see for example
[24]. From (2) and (3) we have
P
σ2>kTynk2
2+σ2
d
σ2kTydk2
2.
Notice that for the infimal solution with input disturbance
rejection S(z)is equal to ˆ S(z), thus
P
σ2>kNX +NM ˆ
Qk2
2+σ2
d
σ2
q(z)
m
Q
i=1
(z−zi)ni
2
2
.(11)

356 A.J. Rojas / Systems & Control Letters 58 (2009) 353–358
We start by analyzing the term NX +NM ˆ Qwhich can be factorized
as
kNX +NM ˆ Qk2
2= kM−1NX +Nˆ
Qk2
2,(12)
since M(z)is all-pass. Introduce now the following decomposition
M−1(z)N(z)X(z)=Γ⊥(z)+Γ(z),
where Γ⊥is in H⊥
2and Γis in H2and replace in (12)
kNX +NM ˆ Qk2
2= kΓ⊥k2 2+ kΓ+Nˆ
Qk2
2.(13)
The term kΓ⊥k2
2is the infimal SNR for stabilizability with no input
disturbance rejection which, from applying Theorem III.1 in [8],
we know is equal to Qm
i=1(ρi)2ni−1. Also directly from (13) we
can observe that the Youla parameter ˜ Q(z)achieving the infimal
SNR for stabilizability with no input disturbance rejection would be
given by −Γ(z)N−1(z), since this choice would ensure the squared
H2norm of Γ+N˜ Qto be zero. Therefore we can observe that
kNX +NM ˆ Qk2
2=
m
Y
i=1
(ρi)2ni−1+k−N˜ Q+Nˆ Qk2
2.
Reintroduce now M(z), since it is all-pass, into the squared H2
norm term and add and subtract the term N(z)X(z)
kNX +NM ˆ Qk2
2=
m
Y
i=1
(ρi)2ni−1
+k − NM ˜ Q+NM ˆ Q+NX −NXk2
2.
Rearrange terms and recognize ˜ T(z)=NX +NM ˜ Qand ˆ T(z)=
NX +NM ˆ Q
kNX +NM ˆ Qk2
2=
m
Y
i=1
(ρi)2ni−1+ kˆ T−˜ Tk2
2.
Since ˆ T(z)=1−ˆ S(z)and ˜ T(z)=1−˜ S(z)and ˜ S(z)=ˆ S(z)|σ2
d=0=
Qm
i=1z−ρi
z−1/ρiniwe have
kNX +NM ˆ Qk2
2=
m
Y
i=1
(ρi)2ni−1
+
m
Y
i=1z−ρi
z−zini
−
m
Y
i=1z−ρi
z−1/ρini
2
2
.
By recognizing and extracting M(z)we can claim that
kNX +NM ˆ Qk2
2=
m
Y
i=1
(ρi)2ni−1+
m
Y
i=1
(ρi)2ni
×
m
Q
i=1
(z−1/ρi)ni−
m
Q
i=1
(z−zi)ni
m
Q
i=1
(z−zi)ni
2
2
.(14)
Applying partial fraction expansion on the term inside the RHS
squared H2norm and then invoking the Residue Theorem (see for
example [25, pp. 169–172]) we obtain
kNX +NM ˆ Qk2
2=
m
Y
i=1
(ρi)2ni−1+
m
Y
i=1
(ρi)2ni
×
m
X
i=1
ni
X
l=1
mi,l
(l−1)!
m
X
j=1
nj
X
p=1
dl−1
dzl−1mj,pzp−1 (1−z¯ zj)pz=zi
,(15)
with mi,las in (10). Notice that the term Qm
i=1(z−zi)niin the
numerator of the squared H2norm in (14) does not contribute to
the residue since it and its derivatives in zare zero at each z=zi.
We now focus on the second term in (11), for which applying
partial fraction expansion and then invoking the Residue Theorem
gives
σ2
d
σ2
m
X
i=1
ni
X
l=1
gi,l
(l−1)!
m
X
j=1
nj
X
p=1
dl−1
dzl−1gj,pzp−1 (1−z¯ zj)pz=zi
,(16)
with gi,las in (10). Finally we observe that the result in (15), plus
the result in (16), give the expression in (9) which concludes the
proof. 
Theorem 2 quantifies the infimal SNR for stabilizability with input
disturbance rejection. Notice how if σ2
d=0, then the last term in
the RHS of (9) vanishes, whilst the second term vanishes as well
since zi=1/ρiin (14), and we are left with the infimal LTI SNR
for stabilizability as in [8, Theorem III.1]. Let us introduce a simple
example in which we can explicitly account for the pole of ˆ S(z).
Example 3. Consider the plant to be G(z)=K/(z−ρ), with
ρ∈R+,ρ > 1 and K∈R+. From (4) we have that
1+G(z)σ2
d
σ2G(z−1)=
z2+−ρ−1
ρ−K2σ2
d
ρσ 2z+1
(z−ρ) z−1
ρ,
therefore z1, the solution for the numerator and pole of ˆ S(z), is
given by
z1=
ρ+1
ρ+K2σ2
d
ρσ 2−r−ρ−1
ρ−K2σ2
d
ρσ 22
−4
2,(17)
the only solution that satisfies z1∈D−. This can be seen to be true
from the fact that
ρ+1
ρ+K2σ2
d
ρσ 2>2,
as long as ρ∈R+, which in turn allows us to claim that z1<1.
With z1known, applying Theorem 2 gives
P
σ2> ρ2−1+ρ2(z1−1/ρ)2
1−z2
1+σ2
d
σ2
K2
1−z2
1
,(18)
from which we observe that as σ2
d→0, then z1→1/ρ and
P
σ2> ρ2−1,
the infimal LTI SNR for stabilizability with no input disturbance
rejection result, see for example [8, Theorem III.2].
As a comparison with the LQG/LTR approach, consider now the
input disturbance variance σdto be in [0,10], the plant unstable
pole ρto be 2, the plant gain Kto be 10 and the channel noise
variance σ2to be 1. In Fig. 2 we can observe the difference between
z1obtained as in Eq. (17) and z1obtained from equation an LQG/LTR
approach. Since the expression in (17) for z1in this example is
exact, any difference shown in Fig. 2 can be in principal adjudicated
to possible numerical issues involved in the LQG/LTR approach. As
σdincreases so does the numerical error value. Also, since there
is a numerical error in z1, there will be a numerical error in the
infimal SNR expression as in (18), although in the present case we
do not show it since it falls below the eps precision number of
2.2204 ·10−016 for Matlab.
Remark 4. Notice from the previous example that, by extending
the stabilizability problem to input disturbance rejection, the gain
of the plant model Kalso plays a role in the performance limitation.

A.J. Rojas / Systems & Control Letters 58 (2009) 353–358 357
Fig. 2. Difference between z1obtained from Eq. (17) and z1obtained from equation
an LQG/LTR approach.
The capacity of a communication channel C, defined as the
maximum of the mutual information between the channel input
and output (see [11, p. 241]), is also used to characterize a
communication channel. For the case of a memoryless AWGN
channel, the channel capacity is given by C=1
2log21+P
σ2bits
per transmission, and is thus completely determined by its SNR.
Notice that, as stated in [26], the presence of feedback does not
increase the capacity of a memoryless AWGN channel.
Corollary 5 (Channel Capacity Difference). Consider a plant model as
in (5) and a memoryless AWGN channel as in Fig. 1. Then the infimal
channel capacity ˆ Cfor stabilizability with input disturbance rejection
must satisfy
ˆ
C−
m
X
i=1
nilog2ρi=1
2log2
1+γ1+σ2
d
σ2
γ2
m
Q
i=1
(ρi)2ni
,(19)
where
γ1=
m
X
i=1
ni
X
l=1
mi,l
(l−1)!
dl−1
dzl−1 m
X
j=1
nj
X
p=1
mj,pzp−1
(1−z¯ zj)p!z=zi
,
and
γ2=
m
X
i=1
ni
X
l=1
gi,l
(l−1)!
dl−1
dzl−1 m
X
j=1
nj
X
p=1
gj,pzp−1
(1−z¯ zj)p!z=zi
,
with mi,land gi,las in (10).
Proof. Directly from Theorem 2, the definition of the capacity
for an AWGN channel and the fact that it does not increase with
feedback. 
The result in Corollary 5 quantifies the difference between
the infimal channel capacity ˆ Cfor stabilizability with input
disturbance rejection and the infimal channel capacity for
stabilizability Pm
i=1nilog2ρi. This difference has been shown in [4,
Corollary 4.4] to represent a fundamental limitation in control
over networks performance. Our present contribution is therefore
to explicitly quantify in closed-form such performance limitation.
Finally, we conclude by reprising Example 3 to compute with
Corollary 5 the channel capacity difference.
Example 6. In Example 3 we quantified the infimal SNR for
stabilizability with input disturbance rejection. In the present
example we apply the result from Corollary 5 to obtain the channel
capacity difference, which is then given by
ˆ
C−log2ρ=1
2log21+(z1−1/ρ)2
1−z2
1+σ2
d
σ2
K2
(1−z2
1)ρ2.
Observe that, as expected, when σ2
d=0 (that is no input
disturbance process is present), the RHS of the above expression
is zero and the channel capacity matches the infimal channel
capacity for stabilizability of log2ρ.
4. Conclusion and remarks
In the present paper we have addressed the infimal SNR for
stabilizability and input disturbance rejection LTI problem for the
case of AWGN channels.
By studying the spectral factorization induced by the optimal
solution to the SNR for stabilizability with input disturbance
rejection LTI problem, we have quantified in closed-form the
infimal LTI SNR for a class of minimum phase unstable plant models
with relative degree one, a memoryless AWGN channel and direct
feedthrough of the input disturbance process. We have shown
how the obtained SNR approaches the stabilizability result of [8,
Theorem III.1] as the variance of the input disturbance process
vanishes. Finally, we have used the infimal SNR result to quantify
the memoryless AWGN channel capacity in the context of recent
information theoretic results from [4].
Future work will include extending the closed-form quantifica-
tion of the infimal SNR for stabilizability with input disturbance re-
jection to more general plant models, channel models and filtered
input disturbance processes, as well as different injection points
for such disturbance processes.
References
[1] H.W. Bode, Network Analysis and Feedback Amplifier Design, Von Nostrand,
Princeton, NJ, 1945.
[2] I. Horowitz, Synthesis of Feedback Systems, Academic Press, 1963.
[3] N. Elia, When Bode meets Shannon: Control-oriented feedback commu-
nication schemes, IEEE Transactions on Automatic Control 49 (9) (2004)
1477–1488.
[4] N.C. Martins, M.A. Dahleh, Fundamental limitations of performance in the
presence of finite capacity feedback, in: Proceedings of the 2005 American
Control Conference, Portland, USA, 2005, pp. 79–86.
[5] Special issue on networked control systems, IEEE Transactions on Automatic
Control 49 (9) (2004).
[6] G.N. Nair, F. Fagnani, S. Zampieri, R.J. Evans, Feedback control under data rate
constraints: An overview, in: Proceedings of the IEEE (special issue on The
Emerging Technology of Networked Control Systems), January 2007.
[7] R.H. Middleton, J.H. Braslavsky, J.S. Freudenberg, Stabilization of non-
minimum phase plants over signal-to-noise ratio constrained channels, in:
Proceedings of the 5th Asian Control Conference, Melbourne, Australia, 2004.
[8] J.H. Braslavsky, R.H. Middleton, J.S. Freudenberg, Feedback stabilisation over
signal-to-noise ratio constrained channels, IEEE Transactions on Automatic
Control 52 (8) (2007) 1391–1403.
[9] K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, 1996.
[10] G.N. Nair, R.J. Evans, Stabilizability of stochastic linear systems with finite
feedback data rates, SIAM Journal on Control and Optimization 43 (2) (2004)
413–436.
[11] T.M. Cover, J.A. Thomas, Elements of Information Theory, John Wiley & Sons,
1991.
[12] J.S. Freudenberg, J.H. Braslavsky, R.H. Middleton, Control over signal-to-noise
ratio constrained channels: Stabilization and performance, in: Proceedings
of the 44th IEEE Conference on Decision and Control and European Control
Conference, Seville, Spain, December 2005.
[13] J.M. Maciejowski, Asymptotic recovery for discrete-time systems, IEEE
Transactions on Automatic Control 30 (6) (1985) 602–605.
[14] G. Stein, M. Athans, The LQG/LTR procedure for multivariable feedback control
design, IEEE Transactions on Automatic Control 32 (1987) 105–114.
[15] M. Kinnaert, Y. Peng, Discrete-time LQG/LTR techniques for systems with time
delays, Systems and Control Letters 15 (1990) 303–311.
[16] Z. Zhang, J.S. Freudenberg, Discrete-time loop transfer recovery for systems
with nonminimum phase zeros and time delays, Automatica 29 (2) (1993).
[17] A. Saberi, B.M. Chen, P. Sannuti, Loop Transfer Recovery: Analysis and Design,
Springer-Verlag, 1993.
[18] A.J. Rojas, J.S. Freudenberg, R.H. Middleton, J.H. Braslavsky, Input disturbance
rejection in channel signal-to-noise ratio constrained feedback control, in:
Proceedings of the 2007 American Control Conference, Seattle, USA, June 2008.
[19] Z. Zhang, Loop transfer recovery for nonminimum phase plants and ill-
conditioned plants, Ph.D. Thesis, The University of Michigan, 1990.