Anytime Reliable Codes for Stabilizing Plants over Erasure Channels

Abstract

The problem of stabilizing an unstable plant over a noisy communication link
is an increasingly important one that arises in problems of distributed control
and networked control systems. Although the work of Schulman and Sahai over the
past two decades, and their development of the notions of "tree codes" and
"anytime capacity", provides the theoretical framework for studying such
problems, there has been scant practical progress in this area because explicit
constructions of tree codes with efficient encoding and decoding did not exist.
To stabilize an unstable plant driven by bounded noise over a noisy channel one
needs real-time encoding and real-time decoding and a reliability which
increases exponentially with delay, which is what tree codes guarantee. We
prove the existence of linear tree codes with high probability and, for erasure
channels, give an explicit construction with an expected encoding and decoding
complexity that is constant per time instant. We give sufficient conditions on
the rate and reliability required of the tree codes to stabilize vector plants
and argue that they are asymptotically tight. This work takes a major step
towards controlling plants over noisy channels, and we demonstrate the efficacy
of the method through several examples.

Full text

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Anytime Reliable Codes for Stabilizing Plants over
Erasure Channels
Ravi Teja Sukhavasi
Babak Hassibi
Abstract—The problem of stabilizing an unstable plant over a
noisy communication link is an increasingly important one that
arises in problems of distributed control and networked control
systems. Although the work of Schulman and Sahai over the past
two decades, and their development of the notions of “tree codes”
and “anytime capacity”, provides the theoretical framework for
studying such problems, there has been scant practical progress in
this area because explicit constructions of tree codes with efficient
encoding and decoding did not exist. To stabilize an unstable plant
driven by bounded noise over a noisy channel one needs real-time
encoding and real-time decoding and a reliability which increases
exponentially with delay, which is what tree codes guarantee. We
prove the existence of linear tree codes with high probability and,
for erasure channels, give an explicit construction with an expected
encoding and decoding complexity that is constant per time instant.
We give sufficient conditions on the rate and reliability required
of the tree codes to stabilize vector plants and argue that they
are asymptotically tight. This work takes a major step towards
controlling plants over noisy channels, and we demonstrate the
efficacy of the method through several examples.
I. INTRODUCTION
In control theory, the output of a dynamical system is
observed and a controller is designed to regulate its behavior.
The controller needs to react and generate control signals in
real-time. In most traditional control systems, the controller
and the plant are colocated and hence there is no measure-
ment loss. There are increasingly many applications such as
networked control systems [1] and distributed computing [2]
where systems are remotely controlled and where measurement
and control signals are transmitted across noisy channels. This
necessitates a need to reliably communicate the measurement
and control signals by correcting for the errors introduced by the
channels. Although Shannon’s information theory is concerned
with reliable transmission of a message from one point to
another over a noisy channel, the reliability is achieved at
the price of large delays which may lead to instability when
they occur in the feedback loop of a control system. Hence,
we need practical real-time encoding and decoding schemes
with appropriate reliability for controlling systems over lossy
networks.
Consider a control system with a single observer that com-
municates with the controller over a lossy communication
channel and where the feedback link from the controller to
the plant is noiseless. When the channel is rate-limited and
deterministic, significant progress has been made (see eg., [3],
[4]) in understanding the bandwidth requirements for stabilizing
open loop unstable systems. When the communication channel
is stochastic, [5] provides a necessary and sufficient condition
on the communication reliability needed over such a channel
to stabilize an unstable scalar linear process, and proposes the
notion of feedback anytime capacity as the appropriate figure of
merit for such channels. In essence, the encoder is causal and
the probability of error in decoding a source symbol that was
transmitted d time instants ago should decay exponentially in
the decoding delay d.
Although the connection between communication reliability
and control is clear, very little is known about error-correcting
codes that can achieve such reliabilities. Prior to the work
of [5], and in a different context, [2] proved the existence
of codes which under maximum likelihood decoding achieve
such reliabilities and referred to them as tree codes. Note
that any real-time error correcting code is causal and since it
encodes the entire trajectory of a process, it has a natural tree
structure to it. [2] proves the existence of nonlinear tree codes
yet gives no explicit constructions and/or efficient decoding
algorithms. Much more recently [6] proposed efficient error
correcting codes for unstable systems where the state grows only
polynomially large with time. So, for linear unstable systems
that have an exponential growth rate, all that is known in the
way of error correction is the existence of tree codes which
are, in general, non-linear. Moreover, the existence results are
not with a “high probability”. When the state of an unstable
scalar linear process is available at the encoder, [7] and [8]
develop encoding-decoding schemes that can stabilize such a
process over the binary symmetric channel and the binary
erasure channel respectively. But little is known in the way
of stabilizing partially observed vector-valued processes over
a stochastic communication channel.
The subject of error correcting codes for control is in its
relative infancy, much as the subject of block coding was after
Shannon’s seminal work in [9]. So, a first step towards realizing
practical encoder-decoder pairs with anytime reliabilities is to
explore linear encoding schemes. We consider rate R =
causal linear codes which map a sequence of k-dimensional
binary vectors {bτ}∞
vectors {cτ}∞
a code is anytime reliable if there exist constants β > 0,η > 0
and a delay do> 0 such that at all times t, P?ˆbt−d|t?= bt−d
The contributions of this paper are as follows: 1. We show
that linear tree codes exist and further, that they exist with a
high probability. 2. For the binary erasure channel, we propose
a maximum likelihood decoder whose average complexity of
decoding is constant per each time iteration and for which the
probability that the complexity at a given time t exceeds KC3
decays exponentially in C. 3. We also prove asymptotically tight
k
n
τ=0to a sequence of n−dimensional binary
τ=0where ctis only a function of {bτ}t
τ=0. Such
?≤
η2−βnd.
arXiv:1103.4438v1 [cs.SY] 23 Mar 2011

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E
N
C
O
D
E
R
C
H
A
N
N
E
L
D
E
C
O
D
E
R
b1
b2
bt
c1= f1(b1)
c2= f2(b1,b2)
ct= ft(b1,...,bt)
z1
z2
zt
ˆb1|1
ˆb1|2,ˆb2|2
ˆb1|t,...,ˆbt|t
............
Fig. 1.Causal encoding and decoding
sufficient conditions on the rate R and exponent β needed to
stabilize vector-valued processes over a noisy channel. As a
consequence, we can efficiently stabilize a partially observed
unstable linear process over a binary erasure channel without
any channel feedback.
In Section II, we introduce the notation and set up the
problem. In Section III, we introduce the ensemble of time
invariant codes and show that they are anytime reliable with
a high probability. In Section IV, we present a simple decoding
algorithm for the BEC and in Section V, we derive sufficient
conditions for stabilizing unstable linear systems over noisy
channels. We present some simulations in Section VIII to
demonstrate the efficacy of the decoding algorithm.
II. PROBLEM SETUP
We will begin by introducing some notation
1) For any matrix F, F ? abs(F), i.e., Fi,j= |Fi,j|.∀ i,j
2) λ(F) ? largest eigen value of F in magnitude.
3) For a vector x, x(i)denotes the ithcomponent of x.
4) 1m? [1,...,1]T, i.e., a column with m 1’s.
5) For w,v ∈ Rm, w ≷ v denotes component-wise inequality.
Consider the following m−dimensional unstable linear system
with scalar measurements. Assuming that the system is observ-
able, without loss of generality, it can be cast in the following
canonical form.
xt+1= Fxt+ But+ wt,yt= Hxt+ vt
(1)
where
F =



−a1
−a2
...
−am−1
−am
1
0
...
0
1
...
0
...
0
...
...
0
...
...
1
0



,H = [1,0,...,0]
where λ(F) > 1, ut is the control input and, wt and vt
are bounded process and measurement noise variables, i.e.,
?wt?∞ <
polynomial of F is zn+ a1zn−1+ ... + am.
The measurements {yt} are made by an observer while the
control inputs {ut} are applied by a remote controller that is
connected to the observer by a noisy communication channel.
Naturally, the measurements y0:t−1 will need to be encoded
by the observer to provide protection from the noisy channel
while the controller will need to decode the channel outputs
to estimate the state xt and apply a suitable control input ut.
This can be accomplished by employing a channel encoder at
the observer and a decoder at the controller. For simplicity, we
W
2
and ?vt?∞ <
V
2. Note that the characteristic
will assume that the channel input alphabet is binary. Suppose
one time step of system evolution in (1) corresponds to n
channel uses1. Then, at each instant of time t, the operations
performed by the observer, the channel encoder, the channel
decoder and the controller can be described as follows. The
observer generates a k−bit message, bt ∈ {0,1}k, that is a
causal function of the measurements, i.e., it depends only on
y0:t. Then the channel encoder causally encodes b0:t∈ {0,1}kt
to generate the n channel inputs ct ∈ {0,1}n. Note that the
rate of the channel encoder is R = k/n. Denote the n channel
outputs corresponding to ctby zt∈ Zn, where Z denotes the
channel output alphabet. Using the channel outputs received so
far, i.e., z0:t ∈ Znt, the channel decoder generates estimates
{ˆbτ|t}τ≤t of {bτ}τ≤t, which, in turn, the controller uses to
generate the control input ut+1. This is illustrated in Fig. 1.
Note that we do not assume any channel feedback. Now, define
?
Thus, Pe
the past.
Definition 1 (Anytime reliability): We say that an encoder-
decoder pair is (R,β,do)−anytime reliable if
Pe
Pe
t,d= Pmin{τ :ˆbτ|t?= bτ} = t − d + 1
t,dis the probability that the earliest error is d steps in
?
t,d≤ η2−nβd, ∀ t,d ≥ do
(2)
In some cases, we write that a code is (R,β)−anytime reliable.
This means that there exists a fixed do> 0 such that the code
is (R,β,do)−anytime reliable.
WewillshowinSection
(R,β)−anytime reliability is a sufficient condition to stabilize
(1) in the mean squared sense2. In what follows, we will
demonstrate causal linear codes which under maximum
likelihood decoding achieve such exponential reliabilities.
V(Theorem5.1)that
III. LINEAR ANYTIME CODES
As discussed earlier, a first step towards developing practical
encoding and decoding schemes for automatic control is to study
the existence of linear codes with anytime reliability. We will
begin by defining a causal linear code.
Definition 2 (Causal Linear Code): A causal linear code is
a sequence of linear maps fτ: {0,1}kτ?→ {0,1}n, τ ≥ 0 and
hence can be represented as
fτ(b1:τ) = Gτ1b1+ Gτ2b2+ ... + Gττbτ
(3)
where Gij∈ {0,1}n×k
We denote cτ ? fτ(b1:τ). Note that a tree code is a more
general construction where fτ need not be linear. Also note
that the associated code rate is R =
is equivalent to using a semi-infinite dimensional block lower
triangular generator matrix, Gn,R, whose entries are clear from
(3) or equivalently as a semi-infinite dimensional block lower
triangular parity check matrix, Hn,R (the parity check matrix
k
n. The above encoding
1In practice, the system evolution in (1) is obtained by discretizing a
continuous time differential equation. So, the interval of discretization could
be adjusted to correspond to an integer number of channel uses, provided the
channel use instances are close enough.
2can be easily extended to any other norm

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satisfies Hn,RGn,R= 0.)
Hn,R=



H11
H21
...
Hτ1
...
0 ...
0
...
...
...
...
...
...
Hττ
...
...
...
...
0
...
H22
...
Hτ2
...



(4)
where3Hij∈ {0,1}n×nand n = n(1−R). In order to ensure
that the code rate is equal to the design rate R =k
to be full rank for every t, where Ht
principal minor of Hn,R. This will happen if Hii is full rank
for all i. The existence results that follow imply the existence
of anytime reliable Hn,Rwhose code rate is same as the design
rate.
We will present all our results for binary input output symmet-
ric channels4. The Bhattacharya parameter ζ for such channels
is defined as
∞
?
−∞
where z and X denote the channel output and input, respec-
tively. We will begin by proving the existence of such codes
that are (R,β)−anytime reliable over a finite time horizon, T,
i.e., under ML decoding Pe
then prove their existence for all time. Due to space limitations,
proofs for all the results in this section are presented in a
companion paper, [10].
n, Ht
n,Rneeds
n,Ris the nt × nt leading
ζ =
?
p(z|X = 1)p(z|X = 0)dz
d,t≤ η2−βd, ∀ t ≤ T. We will
A. Finite Time Horizon
Over a finite time horizon, T, a causal linear code is
represented by a block lower triangular parity check matrix
Hn,R,T∈ {0,1}nT×nT. The following Theorem guarantees the
existence of a Hn,R,T that is (R,β)−anytime relable.
Theorem 3.1: For each time T > 0, rate R and exponent β
such that
R < 1 − log2(1 + ζ),
β < H−1(1 − R)
existsa
(R,β)−anytime reliable.
H−1(1−R) is the smaller root of the equation H(x) = 1−R,
where H(.) is the binary entropy function. Theorem 3.1 proves
the existence of finite dimensional causal linear codes, Hn,R,T,
that are anytime reliable for decoding instants upto time T.
In the following subsection, we demonstrate the existence of
semi-infinite causal linear codes, Hn,R, that are anytime reliable
for all decoding instants. We also show that such codes drawn
from an appropriate ensemble are anytime reliable with a high
probability. The key is to impose a Toeplitz structure on the
parity check matrix.
and
?1
?
log2
ζ
?
code
+ log2
?21−R− 1??
H(n,k,T)
therecausallinearthatis
3While for a given generator matrix, the parity check matrix is not unique,
when Gn,Ris block lower, it is easy to see that Hn,Rcan also be chosen to
be block lower.
4which can be easily extended to more general memoryless channels
B. Time Invariant Codes
Consider causal linear codes with the following Toeplitz
structure

...
Hτ
Hτ−1
...
HTZ
n,R=


H1
H2
0...
0
...
...
...
...
...
...
H1
...
...
...
...
0
...
H1
...
...



The superscript TZ in HTZ
tained from Hn,R in (4) by setting Hij = Hi−j+1 for i ≥ j.
Due to the Toeplitz structure, we have the following invariance,
Pe
analogous to the convolutional structure used to show the
existence of infinite tree codes in [2]. The code HTZ
be referred to as a time-invariant code. This time invariance
obviates the need to union bound over all time t and hence
allows us to prove that such codes which are anytime reliable
are abundant.
Definition 3 (The ensemble TZp): The ensemble TZp of
time-invariant codes, HTZ
full rank binary matrix and for τ ≥ 2, the entries of Hτ are
chosen i.i.d according to Bernoulli(p), i.e., each entry is 1 with
probability p and 0 otherwise.
Note that H1being full rank implies that Ht
every t. For the ensemble TZp, we have the following result
Theorem 3.2 (Abundance of time-invariant codes): For any
rate R and exponent β such that
n,Rdenotes ‘Toeplitz’. HTZ
n,Ris ob-
t,d= Pe
t?,dfor all t,t?. The notion of time invariance is
n,Rwill
n,R, is obtained as follows, H1 is any
n,Ris full rank for
R < 1 −
log2(1 + ζ)
log2(1/(1 − p)),
and
?
β < H−1(1 − R)
?
log2
?1
ζ
+ log2
?(1 − p)−(1−R)− 1??
if HTZ
P?HTZ
Note that by choosing p small, we can trade off better rates
and exponents with sparser parity check matrices. Note that for
BEC(?), ζ = ? and for BSC(?), ζ = 2??(1 − ?). For the Binary
p =
1−2log2(√?+√1 − ?). It turns out that this can be strengthened
as follows.
Theorem 3.3 (Tighter bounds for BSC(?)): For any rate R
and exponent β such that
R < 1 − H(?), β < KL?H−1(1 − R)?min{?,1 − ?}?
if HTZ
P?HTZ
IV. DECODING OVER THE BEC
Owing to the simplicity of the erasure channel, it is possible to
come up with an efficient way to perform maximum likelihood
decoding at each time step. We will show that the average
n,Ris chosen from TZp, then
n,Ris (R,β,do) − anytime reliable?≥ 1 − 2−Ω(ndo)
Symmetric Channel (BSC) with bit flip probability ? and for
1
2, the threshold for rate in Theorem 3.2 becomes R <
n,Ris chosen from TZ 1
n,Ris (R,β,do) − anytime reliable?≥ 1 − 2−Ω(ndo)
2, then

Page 4

complexity of the decoding operation at any time t is constant
and that it being larger than KC3decays exponentially in C.
Consider an arbitrary decoding instant t, let c = [cT
be the transmitted codeword and let z = [zT
the corresponding channel outputs. Recall that Ht
the nt × nt leading principal minor of Hn,R. Let ze denote
the erasures in z and let Hedenote the columns of Ht
correspond to the positions of the erasures. Also, let ˜ zedenote
the unerased entries of z and let˜He denote the columns of
Ht
condition on ze, Heze =
˜He˜ ze. Since ˜ ze is known at the
decoder, s ?˜He˜ ze is known. Maximum likelihood decoding
boils down to solving the linear equation Heze = s. Due
to the lower triangular nature of He, unlike in the case of
traditional block coding, this equation will typically not have
a unique solution, since He will typically not be full rank.
This is alright as we are not interested in decoding the entire
ze correctly, we only care about decoding the earlier entries
accurately. If ze = [zT
earlier time instants while ze,2 corresponds to the latter time
instants. The desired reliability requires one to recover ze,1with
an exponentially smaller error probability than ze,2. Since He
is lower triangular, we can write Heze= s as
?
Let H⊥
H⊥
diag(I,He,22), we get
?
If [HT
can be recovered exactly. The decoding algorithm now sug-
gests itself, i.e., find the smallest possible He,22 such that
[HT
Algorithm 1.
1,...,cT
t]Tdenote
n,Rdenotes
t]T
1,...,zT
n,Rthat
n,Rexcluding He. So, we have the following parity check
e,1, zT
e,2]T, then ze,1 corresponds to the
He,11
He,21
0
He,22
??
ze,1
ze,2
?
=
?
s1
s2
?
(6)
e,22denote the orthogonal complement of He,22, ie.,
e,22He,22 = 0. Then multiplying both sides of (6) with
He,11
e,22He,21
H⊥
?
ze,1=
?
s1
e,22s2
H⊥
?
(7)
e,11(H⊥
e,22He,21)T]Thas full column rank, then ze,1
e,11(H⊥
e,22He,21)T]Thas full rank and it is outlined in
Algorithm 1 Decoder for the BEC
1) Suppose, at time t, the earliest uncorrected error is at a
delay d. Identify zeand Heas defined above.
2) Starting with d?= 1,2,...,d, partition
ze= [zT
e,1zT
e,2]Tand He=
?
He,11
He,21
0
He,22
?
where ze,2correspond to the erased positions up to delay
d?.
3) Check whether the matrix
rank.
4) If so, solve for ze,1in the system of equations
?
5) Increment t = t + 1 and continue.
?
He,11
e,22He,21
H⊥
?
has full column
He,11
e,22He,21
H⊥
?
ze,1=
?
s1
e,22s2
H⊥
?
A. Complexity
Suppose the earliest uncorrected error is at time t−d+1, then
steps 2), 3) and 4) in Algorithm 1 can be accomplished by just
reducing Heinto the appropriate row echelon form, which has
complexity O?d3?. The earliest entry in zeis at time t− d+ 1
which is Pe
complexity is at most K?
complexity being Kd3is at most η2−nβd. The decoder is easy
to implement and its performance is simulated in Section VIII.
Note that the encoding complexity per time iteration increases
linearly with time. This can also be made constant on average
if the decoder can send periodic acks back to the encoder with
the time index of the last correctly decoded source bit.
implies that it was not corrected at time t−1, the probability of
d−1,t−1≤ η2−nβ(d−1). Hence, the average decoding
d>0d32−nβdwhich is bounded and
is independent of t. In particular, the probability of the decoding
V. SUFFICIENT CONDITIONS FOR STABILIZABILITY
Consider an unstable m−dimensional linear system whose
state space equations in canonical form are given by (1), i.e.,
λ(F) > 1, and recall that the characteristic polynomial of F is
zn+ a1zn−1+ ... + am. Suppose the observer does not have
any feedback from the controller, in particular, it does not have
access to the control inputs. Then we can stabilize such a system
in the mean squared sense over a noisy channel provided that
the rate R and exponent β of the (R,β)−anytime reliable code
used to encode the measurements satisfy the following sufficient
condition.
Theorem 5.1 (No Feedback to the Observer): It is possible
tostabilize(1)inthemean
(R,β)−anytime code provided (F,B) is controllable and
R > Rn=1
nlog2
i=1
If the observer knows the control inputs, it turns out that one
can make do with lower rates. This is stated as the following
Theorem
Theorem 5.2 (Observer Knows the Control Inputs): When
the observer has access to the control inputs, it is possible to
stabilize (1) in the mean squared sense with an (R,β)−anytime
code provided (F,B) is controllable and
squaredsensewithan
m
?
|ai|, β > βn=2
nlog2λ(F)
(8)
R > Rf
n= argmin
r
?λ(FDnr) < 1?
nlog2λ(F)
(9a)
β > βf
n=2
(9b)
where Dnr= diag(2−nr,1,...,1). Moreover
Rf
n≤1
nlog2max
?
|am|2m−1,max
1≤i≤m−1|ai|2i
?
(10)
The superscript f in Rf
fact that the observer has access to the control inputs. Before
proceeding further, we will give a brief outline of the proofs for
Theorems 5.1 and 5.2 (details are in Section VII). At each time
t, using the channel outputs received received till t, we bound
the set of all possible states that are consistent with the estimates
of the quantized measurements using a hypercuboid, i.e., a
ndenotes ‘feedback’ to emphasize the

Page 5

region of the form?xt∈ Rm|xmin,t|t≤ xt≤ xmax,t|t
wise. If ∆t|t = xmax,t|t− xmin,t|t, then from Lemma 7.1,
∆t+1|t= F∆t|t+ W1m. The anytime exponent is determined
by the growth of ∆t in the absence of measurements, hence
the bound βn= βf
determined by how fine the quantization needs to be for ∆tto
be bounded asymptotically.
?, where
xmin,t|t,xmax,t|t ∈ Rmand the inequalities are component-
n= 2log2λ(F). The bound on the rate is
A. The Limiting Case
The sufficient conditions derived above are for the case when
the measurements are encoded every time step. Alternately, one
can encode the measurements every, say ?, time steps, and
consider the asymptotic rate and exponent needed as ? grows.
Note that this amounts to working with the system matrix F?.
So, one can calculate this limiting rate and exponent by writing
the eigen values of F, {λi}m
scale. The following asymptotic result allows us to compare
the sufficient conditions above with those in the literature (eg.,
see [3], [5], [11]).
Theorem 5.3 (The Limiting Case): Write the eigen values of
F, {λi}m
converge to R∗, and βnand βf
?
In addition, the upper bounds on Rf
R∗.
Proof: See Section C of the Appendix.
For stabilizing plants over deterministic rate limited channels,
[3] showed that a rate R > R∗, where R∗is as in (11), is neces-
sary and sufficient. So, asymptotically the sufficient conditions
for the rate R in Theorems 5.1 and 5.2 are tight. Though the
above limiting case allows one to obtain a tight and an intuitively
pleasing characterization of the rate and exponent needed, it
should be noted that this may not be operationally practical.
For, if one encodes the measurements every ? time steps, even
though Theorem 5.3 guarantees stability, the performance of the
closed loop system (the LQR cost, say) may be unacceptably
large because of the delay we incur. This is what motivated us to
present the sufficient conditions in the form that we did above.
i=1, as λi = µn
iand letting n
i=1, in the form λi= µn
i. Letting n scale, Rnand Rf
nconverge to β∗, where
n
R∗=
i:|µi|>1
log2|µi|, β∗= 2log2max
i
|µi|
(11)
nin (10) also converges to
B. A Comment on the Trade-off Between Rate and Exponent
Once a set of rate-exponent pairs (R,β) that can stabilize
a plant is available, one would want to identify the pair that
optimizes a given cost function. Higher rates provide finer
resolution of the measurements while larger exponents ensure
that the controller’s estimate of the plant does not drift away;
however, we cannot have both. One can either coarsely quantize
the measurements and protect the bits heavily or quantize them
moderately finely and not protect the bits as much. One can
easily cook up examples using an LQR cost function with the
balance going either way. Studying this trade-off is integral to
making the results practically applicable.
VI. TIGHTER BOUNDS ON THE ANYTIME EXPONENT
From Theorem 5.1, using the technique outlined in the previ-
ous section, one needs an exponent nβ ≥ 2logλ(F). It turns out
that a smaller exponent of 2log2λ(F) suffices. The idea is to
alternately bound the set of all possible states that are consistent
with the estimates of the quantized measurements using an
ellipsoid E(P,c) ?
can be seen as an extension of the technique proposed in
[12] to filtering using quantized measurements. If m = 1,
λ(F) = λ(F). So, let m ≥ 2.
In view of the duality between estimation and control, we can
focus on the problem of tracking (1) over a noisy communica-
tion channel. For, if (1) can be tracked with an asymptotically
finite mean squared error and if (F,B) is stabilizable, then it is
a simple exercise to see that there exists a control law {ut}
that will stabilize the plant in the mean squared sense, i.e.,
limsuptE?xt?2< ∞. In particular, if the control gain K is
chosen such that
stabilize the plant, where ˆ xt|tis the estimate of xtusing channel
outputs up to time t. Hence, in the rest of the analysis, we will
focus on tracking (1). The control input uttherefore is assumed
to be absent, i.e., ut= 0.
We will first present a recursive state estimation algorithm
using the channel outputs and then state the sufficient conditions
needed for the estimation error to be appropriately bounded
using such a filter. Recall that the channel outputs corresponding
to the coded bits ct∈ GFn
suppose using {zτ}τ≤t−1, we have xt∈ E(Pt|t−1, ˆ xt|t−1). Note
that, since H = [1,0,...,0], the measurement update provides
information of the form x(1)
may call a slab. E(Pt|t, ˆ xt|t) would then be an ellipsoid that con-
tains the intersection of the above slab with E(Pt|t−1, ˆ xt|t−1),
in particular one can set it to be the minimum volume ellipsoid
covering this intersection. Lemma A.1 gives a formula for
the minimum volume ellipsoid covering the intersection of an
ellipsoid and a slab. Note that the width of the slab above tends
to be smaller if the observer has access to the control inputs than
when it does not. For the time update, it is easy to see that for
any ? > 0 and Pt+1= (1+?)FPt|tFT+W2
contains the state xt+1whenever E(Pt|t, ˆ xt|t) contains xt. This
leads to the following Lemma. For convenience, we write Pt
for Pt|t−1.
Lemma 6.1 (The Ellipsoidal Filter): Whenever
contains x0, for each ? > 0, the following filtering equations
give a sequence of ellipsoids?E(Pt|t, ˆ xt|t)?that, at each time
Pt+1= (1 + ?)FPt|tFT+W2
Pt|t= btPt− (bt− at)Pte1eT
eT
1Pte1
?x ∈ Rm|?x − c,P−1(x − c)? ≤ 1?. This
√2F + BK is stable, then ut= Kˆ xt|twill
2are zt∈ Zn. Let x0∈ E(P0,0) and
min,t|t≤ x(1)
t
≤ x(1)
max,t|t, which one
4?1m, E(Pt+1,F ˆ xt|t)
E(P0,0)
t, contain xt.
4?1m, ˆ xt+1= F ˆ xt|t
1Pt
, ˆ xt|t= ξt
(12a)
Pte1
?eT
1Pte1
(12b)
where at,bt and ξt can be calculated in closed form using
Lemma A.1.
Using this approach, we get the following set of sufficient
conditions. The proofs are similar to the proofs of Theorems