Feedback Control Performance over a Noisy
Communication Channel
J. S. Freudenberg
EECS Department
University of Michigan
Ann Arbor MI 48109
jfr@eecs.umich.edu
R. H. Middleton
Hamilton Institute
National University of Ireland Maynooth
Co Kildare, Ireland
richard.middleton@nuim.ie
Abstract— We consider the problem of minimizing the variance
in the output of a plant that is driven by a Gaussian disturbance
using measurements of the plant output obtained from a Gaussian
channel. For the special case in which only the variance at the
terminal time is penalized, we derive an optimal linear time-
varying communication and control strategy, and argue that
nonlinear strategies cannot achieve better performance.
I. INTRODUCTION
Recent years have seen much interest in the limitations
imposed on a feedback system by the presence of a noisy
communication channel in the feedback path, as depicted in
Figure 1. One problem is to determine the minimal channel
capacity required to stabilize an open loop unstable plant.
The solution to this problem is known for noise-free data
rate limited channels [1]–[3] and additive Gaussian noise
channels [4]. A more difficult problem is that of determining
the optimal performance, in terms of disturbance attenuation,
that is achievable given the presence of a noisy channel with
fixed capacity.
encoder
decoder/
controller
B
C
(zI-A)
-1
E
Σ
Σ
d
k
s
k
y
k
x
k
n
k
r
k
u
k
Fig. 1. Feedback control over a noisy communication channel.
In this paper the plant is assumed to be linear with initial
state and white noise disturbance input that are Gaussian
random variables, and the channel is assumed Gaussian. We
characterize performance in terms of the variance of the
plant output, and seek an encoder and decoder/controller to
minimize the output variance at a specified terminal time.
It is important to note that this problem has a nonclassical
information pattern, in that the encoder and decoder/controller
do not have access to the same information [5]. For such
problems, the “separation” property of estimation and control
need not hold [5], and the optimal communication and control
strategies need not be linear [6]. Situations in which linear
strategies may indeed be linear are studied in [7], [8]. We
shall show that for our special measure of performance, i.e.,
output variance at a terminal time, the optimal control and
communication strategies are in fact linear. We also show that,
although there is no conflict between estimation and control
for our problem, nevertheless the control and communication
strategies must be carefully coordinated.
In Section II we define the problem of minimizing plant
output variance at a fixed terminal time, and derive a formula
for the optimal control to be applied at the last possible instant.
This formula shows that the control problem reduces to one of
estimating a single linear combination of states. In Section III
we suppose that the encoder has access to additional informa-
tion, and derive a linear time-varying strategy to estimate that
combination of states used in the optimal control. We argue
in Section IV that a nonlinear strategies cannot achieve better
performance. In Section V we return to the original problem
and show how to achieve the same performance achieved
in Section III but without the additional information at the
encoder. The strategies used are again linear and time-varyng.
Conclusions are presented in Section VI.
Other researchers have studied feedback control perfor-
mance over a communication channel; a partial review follows.
The authors of [9] derive a lower bound on a measure of dis-
turbance attenuation that is stated in terms of channel capacity;
however, the realistic assumption of causality is not invoked in
the proof. The authors of [10] study performance limitations
imposed by a vector Gaussian channel, with one channel per
state of the plant. The author of [11] relates the problem of
feedback stabilization over a communication channel to that
of communication over a channel with feedback. The authors
of [12], [13] consider performance constraints imposed by
noise free, data rate limited channels. The authors of [14]
study the joint optimum design of communication and control
strategies for feedback over noisy channels, and show that
linear strategies are optimal only for first order linear systems
with Gaussian noise and quadratic cost.
978-1-4244-2270-8/08/$25.00 ©2008 IEEE
232
II. PRELIMINARIES
The plant in Figure 1 is described by
x
k+1
= Ax
k
+ Bu
k
+ Ed
k
, x
k
∈ R
n
, u
k
, d
k
∈ R (1)
y
k
= Cx
k
, y
k
∈ R, (2)
where x
0
is a zero mean Gaussian random variable with
covariance Σ
0|−1
and d
k
is a stationary zero mean Gaussian
white noise sequence with variance σ
2
d
. The Gaussian channel
has input power constraint E{s
2
k
} ≤ P and additive white
noise with variance σ
2
n
. The encoder f
k
() may be nonlin-
ear and time-varying, and the channel input at time k is
allowed to depend on the sequence of current and previous
plant outputs: s
k
= f
k
(y
k
), where y
k
= {y
0
, y
1
, . . . , y
k
}.
The decoder/controller g
k
() may also be nonlinear and time-
varying, and the control input at time k is allowed to depend
on the sequence of current and previous channel outputs:
u
k
= g
k
(r
k
), where r
k
= {r
0
, r
1
, . . . , r
k
}.
The design problem is to choose the encoder f
k
() and
decoder/controller and g
k
(), k = 0, . . . , N, to minimize the
variance of the plant output at a terminal time k = N + 1.
The optimal value of this cost function is given by
J
∗
= inf
f
k
,g
k
E{y
2
N+1
}.
Denote the conditional expectation of y
N+1
given the channel
output sequence r
N
by
ˆy
N+1 |N
= E{y
N+1
|r
N
}
and the resulting estimation error by ˜y
N+1 |N
= ˆy
N+1 |N
−
y
N+1
. It follows immediately from properties of the condi-
tional expectation (cf. [15, p. 97]) that the variance of the
conditional estimation error provides a lower bound on the
optimal cost:
J
∗
≥ E{˜y
2
N+1 |N
}. (3)
Under mild hypotheses, the lower bound (3) can be achieved.
Proposition 1 Assume that CB 6= 0. Then the control
u
N
= −(CB)
−1
CAˆx
N| N
, (4)
where ˆx
N| N
= E{x
N
|r
N
}, yields E{y
2
N+1
} = E{˜y
2
N+1 |N
}.
Moreover,
E{y
2
N+1
} = E{(CA˜x
N| N
)
2
} + (CE)
2
σ
2
d
. (5)
Proof: Substituting (4) into the state equations (1)-(2)
shows that
y
N+1
= CAx
N
− CAˆx
N| N
+ CEd
N
,
and thus ˆy
N+1 |N
= 0 and y
N+1
= ˜y
N+1 |N
. The identity
(5) follows because ˜x
N| N
and d
N
are independent random
variables.
We have shown that the optimal control to apply at time N
sets the variance of the plant output to its theoretical minimum,
given by the variance of the conditional estimation error (5).
The term (CE)
2
σ
2
d
in (5) is due to the disturbance at time
N, which is uninfluenced by the control and communication
strategies because it has not yet propagated to the encoder.
Hence the estimation problem reduces to that of minimizing
E{(CA˜x
N| N
)
2
}, and the problem of minimizing the variance
of the plant output reduces to that of minimizing the variance
in the conditional estimate of the single linear combination of
states CAx
N
. Iterating the state equations (1)-(2) shows that
CAx
N
= CA
N+1
x
0
+
N−1
X
j=0
CA
N−j
Ed
j
+
N−1
X
j=0
CA
N−j
Bu
j
.
Since the control signal is known at the decoder, the only
information that needs to be communicated over the channel
is the “message”
m
N
= CA
N+1
x
0
+
N−1
X
j=0
CA
N−j
Ed
j
. (6)
The problem of choosing the encoder f
k
(), k = 0, . . . , N and
decoder/controller g
k
(), k = 0, . . . , N − 1 thus reduces to that
of obtaining the best possible estimate of m
N
given N + 1
uses of the channel. If we denote the estimate and estimation
error by ˆm
N| N
= E{m
N
|r
N
} and ˜m
N| N
, then the variance
in the plant output is given by
E{y
2
N+1
} = E{ ˜m
N| N
)
2
} + (CE)
2
σ
2
d
. (7)
An obvious complication is that the complete message is not
available until k = N, the last time step at which the channel
may be used.
III. AN ENCODER WITH MORE INFORMATION
In the feedback system of Figure 1, the encoder has access
only to the sequence of plant outputs. Suppose instead, as
shown in Figure 2, that the encoder has access to perfect
measurements of the plant state, the plant input, and feedback
from the channel output. We now propose a strategy for
estimating the message m
N
under the assumption that the
encoder has this additional information.
B
(zI-A)
-1
E
Σ
d
k
s
k
x
k
r
k
u
k
encoder
decoder/
controller
C
y
k
Σ
n
k
Fig. 2. Encoder with access to the plant state, plant input, and channel
output.
We first show that access to the plant state and control input
allows us to compute the primitive random variables z
0
,
CA
N+1
x
0
and d
0
, d
1
, . . . , d
N−1
. Clearly access to the state
allows z
0
to be computed at k = 0. Access to the state and
control input allows the encoder to determine the sequence of
disturbance inputs. Choose an arbitrary row vector F such that
233
F E 6= 0. Then the disturbance at time k may be computed at
time k + 1:
d
k
= (F E)
−1
(F x
k+1
−F Ax
k
−F Bu
k
), k = 0, . . . , N −1.
The ability to compute the primitive random variables, in turn,
allows the encoder to form an estimate of the message (6) at
each time step:
m
0
= CA
N+1
x
0
(8)
m
1
= m
0
+ CA
N
Ed
0
(9)
.
.
.
m
N
= m
N−1
+ CAEd
N−1
. (10)
Note that m
k
is the best possible estimate of m
N
given the in-
formation available at time k: m
k
= E{m
N
|z
0
, d
0
, . . . , d
k−1
}.
We next propose to transmit the estimates (8)-(10) over the
communication channel, taking appropriate advantage of the
noiseless feedback link to improve the quality of transmission.
To do so, we adapt a technique for communicating over a
channel with feedback that is described in [16, pp. 479-481]
and [17, pp. 166-168], and depicted in Figure 3. The idea is
Σ
n
k
r
k
γ
k
s
k
λ
k
Σ
Σ
z
-1
-
μ
μ
k
^
μ
k-1
^
μ
k-1
~
Fig. 3. Communication over a channel with noiseless feedback.
to use a Gaussian channel N + 1 times for the purpose of
communicating a single message µ, assumed to be a Gaussian
random variable with zero mean and variance σ
2
µ
. Define
the conditional estimate ˆµ
k
, E{µ|r
k
}, k = 0, . . . , N, the
estimation error ˜µ
k
, µ− ˆµ
k
, and set ˆµ
−1
= 0. Then choosing
λ
k
so that E{s
2
k
} = P , and γ
k
= (1/λ
k
)(1+ σ
2
n
/P )
−1
results
in an estimation error at time N with variance E{˜µ
2
N
} =
σ
2
µ
(1+P/σ
2
n
)
−(N+1)
, the minimum possible according to rate
distortion theory [16], [17].
The scheme depicted in Figure 3 is not directly applicable
to our situation, because the message m
N
is unavailable at
the beginning of channel transmission, and we must instead
transmit the estimates m
k
. We thus modify the scheme in
Figure 3 by noting that the sequence m
k
can be modeled as
the response of a discrete integrator to initial condition z
0
and
input v
k
= CA
N−k
Ed
k
, a white noise sequence with variance
σ
2
k
= (CA
N−k
E)
2
σ
2
d
.
A Kalman filter [18] to estimate the state of the integrator
m
k
given the sequence r
k
= λ
k
m
k
+ n
k
has the form
ˆm
k|k
= ˆm
k|k−1
+ L
k
(r
k
− λ
k
ˆm
k|k−1
),
ˆm
k+1|k
= ˆm
k|k
where
L
k
=
λ
k
M
k|k−1
λ
2
k
M
k|k−1
+ σ
2
n
, (11)
and M
k|k−1
= E{ ˜m
2
k|k−1
} satisfies the Riccati difference
equation
M
k+1|k
= M
k|k−1
−
λ
2
k
M
2
k|k−1
λ
2
k
M
k|k−1
+ σ
2
n
+ σ
2
k
, (12)
with initial condition M
0|−1
= CA
N+1
Σ
0|−1
A
(N+1)T
C
T
.
Suppose we adjust λ
k
at each time step so that λ
2
k
M
k|k−1
=
P . Then (11) and (12) reduce to
L
k
=
1
λ
k
P
P + σ
2
n
,
M
k+1|k
= M
k|k−1
σ
2
n
P + σ
2
n
+ σ
2
k
.
The resulting estimation scheme may be implemented over a
channel with noiseless feedback as shown in Figure 4.
Σ
n
k
r
k
L
k
s
k
λ
k
Σ
Σ
z
-1
-
^
m
k|k-1
^
~
m
k
m
k|k-1
m
k|k
v
k
Σ
z
-1
m
k-1
Fig. 4. Communicating the output of a discrete integrator over a channel
with feedback.
Note the similarity of the communication system in Figure 4
to that in Figure 3. In each figure λ
k
is chosen to satisfy the
channel power constraint with equality. Once λ
k
is chosen, the
formulas for γ
k
in Figure 3 and L
k
in Figure 4 are identical.
The only difference is the presence of the disturbance sequence
v
k
in Figure 4. With this disturbance present, the error in
the estimate ˆm
N| N
may be obtained by iterating the Riccati
equation (12):
E{ ˜m
2
N| N
} = M
0|−1
σ
2
n
P + σ
2
n
N+1
+
N−1
X
j=0
σ
2
j
σ
2
n
P + σ
2
n
N−j
.
(13)
Were this disturbance not present, the estimation error would
equal that achieved with the procedure described in [16], [17].
We have shown how to use the additional information
present at the encoder in Figure 4 to obtain an estimate of
the “message” m
N
defined by (10), and whose estimation
error has variance (13). As noted in the discussion preceding
(10), the control signal is known at the decoder and may be
used to obtain an estimate of CAx
N
from that of m
N
; the
estimation error will be the same for these two quantities.
Hence substituting (13) into (5) and using the definitions of
M
0|−1
and σ
2
k
shows that if we apply the optimal control (4)
at time k = N, then the plant output will satisfy
E{˜y
2
N+1 |N
} = CA
N+1
Σ
0|−1
A
(N+1)T
C
T
σ
2
n
P + σ
2
n
N+1
+ σ
2
d
N
X
j=0
(CA
N−j
E)
2
σ
2
n
P + σ
2
n
N−j
. (14)
Note that the control inputs at times k = 0, . . . , N − 1 are
irrelevant and with no loss of generality may be set to zero.
234
IV. OPTIMALITY OF LINEAR TIME-VARYING
COMMUNICATION AND CONTROL
In Section III we increased the amount of information
available to the encoder and derived linear time-varying com-
munication and control strategies that resulted in the output at
time k = N + 1 having variance (14). It remains to determine
whether nonlinear communication and control strategies may
yield a smaller variance. Were the disturbance not present, the
arguments from rate distortion theory used in [16], [17] would
suffice to prove that there is no advantage to use of nonlinear
strategies; however, our problem is more complicated due to
the presence of the disturbance input. To address this question,
we use arguments based on the concept of entropy power,
introduced by Shannon in [19].
The entropy of a scalar zero mean Gaussian random variable
x with variance σ
2
is completely determined by its variance
[20]: h(x) = (1/2) log 2πeσ
2
. For a non-Gaussian random
variable, the relation between entropy and variance is only
an inequality: h(x) ≤ (1/2) log 2πeσ
2
. The latter fact com-
plicates attempts to use entropy to study the propagation of
variance through potentially nonlinear transformations. The
entropy power of a random variable, defined as N(x) =
(1/2πe)e
2h(x)
, has the interpretation of being the variance of
a Gaussian random variable with the same entropy as that of
x [19]. It follows that the entropy power of x is a lower bound
on the variance of x, with equality holding precisely when x
is Gaussian. The concept of entropy power and the associated
entropy power inequality [20] were used in [2] to derive a
lower bound on the variance of the (possibly vector valued)
state of a plant under control over a noise-free data rate limited
channel using a nonlinear time-varying encoder and decoder.
The analysis of [2] was adapted in [21] to derive a similar
lower bound for control over Gaussian channels. It is noted in
[12], however, that bounds on variance obtained using entropy
power arguments will be tight only in special cases, including
that of a scalar system.
With the preceding for motivation, we have noted that with
additional information at the encoder, the problem of using
control to minimize variance at a terminal time reduces to that
of sending estimates of m
N
over the channel. The encoding
and decoding schemes we used to do so were linear and time-
varying. By adapting the arguments of [21], however, it may
be shown that nonlinear control and communication strategies
offer no improvement. Indeed, denote the random conditional
entropy [21] of the integrator state m
k
conditioned on past
channel outputs by h(m
k
|r
k−1
), and the random conditional
entropy power by
N(m
k
|r
k−1
) = (1/2πe)e
2h(m
k
|r
k−1
)
.
Let n
k
= E{N(m
k
|r
k−1
)}. Then [20] n
k
≤ E{ ˜m
2
k|k−1
}, and
the results of [21] may be adapted to show that n
k
satisfies
the difference inequality
n
k+1
≥
σ
2
n
/(P + σ
2
n
)
n
k
+ σ
2
k
.
Since m
0
is Gaussian, n
0
= M
0|−1
, and since ˜m
k+1|k
=
˜m
k|k
, it follows that
E{ ˜m
2
N| N
} ≥ n
N+1
≥ M
0|−1
σ
2
n
P + σ
2
n
N+1
+
N−1
X
j=0
σ
2
j
σ
2
n
P + σ
2
n
N−j
.
(15)
Comparing (15) with (13), we see that use of nonlinear com-
munication and control strategies is not effective at reducing
the variance in the estimate of m
N
below the level achievable
with linear strategies. By arguments similar to those used to
derive (14), the same conclusion holds for the variance of the
plant output E{˜y
2
N+1 |N
}. Hence use of linear communication
and control strategies is optimal for the system in Figure 2,
whose encoder has access to additional information.
V. THE ORIGINAL PROBLEM
We now return to the original problem depicted in Figure 1.
We shall show that the performance achieved in Section III
with additional information available at the encoder is in fact
also achievable without such information. We saw in Sec-
tion IV that nonlinear strategies offer no improvement when
additional information is available. It follows that since we can
achieve the same level of performance without the additional
information, nonlinear strategies also offer no advantage for
our original problem setup.
Our first result is to show that the encoder need not have
access to the plant input or channel output to achieve the level
of performance that was obtained using this information in
Section III. The following result is proven in [22].
Proposition 2 Assume that the encoder has access to the
states of the plant. Choose the channel input to satisfy
s
k
= λ
k
H
k
x
k
, (16)
where
H
k
= CA
N+1 −k
, k = 0, . . . , N, (17)
and, if H
k
Σ
k|k−1
H
T
k
6= 0,
λ
k
H
k
Σ
k|k−1
H
T
k
= P, k = 0, . . . , N. (18)
Choose the control law
u
k
= −(H
k+1
B)
−1
H
k+1
Aˆx
k|k
, k = 0, . . . , N, (19)
with ˆx
k|k
given by
ˆx
k+1|k
= Aˆx
k|k−1
+ Bu
k
+ AL
k
(r
k
− λ
k
H
k
ˆx
k|k−1
),
ˆx
k|k
= ˆx
k|k−1
+ L
k
(r
k
− λ
k
H
k
ˆx
k|k−1
),
where
L
k
= λ
k
Σ
k|k−1
H
T
k
/(P + σ
2
n
), (20)
and Σ
k|k−1
is the solution to the Riccati difference equation
Σ
k+1|k
= AΣ
k|k−1
A
T
−
AΣ
k|k−1
H
T
k
H
k
Σ
k|k−1
A
T
H
k
Σ
k|k−1
H
T
k
P
P + σ
2
n
+ σ
2
d
EE
T
.
235
Then at time k = N + 1, the mean square value of the plant
output satisfies E{y
2
N+1
} = E{˜y
2
N+1 |N
}, where E{˜y
2
N+1 |N
}
is given by (14).
Our second result, also proven in [22], shows that under
reasonable hypotheses access to the plant state is also unneces-
sary. Instead of measuring the states to form the channel input,
we let the encoder have access to the state of an estimator
whose input is a noise-free measurement of the plant output.
In general estimators must have access to the plant input; as
we shall see, this is not the case in the present situation.
To distinguish these state estimates from those obtained by
processing the channel output, we denote them by ˆx
0
k|k−1
=
E{x
k
|u
k−1
, y
k−1
} and ˆx
0
k|k
= E{x
k
|u
k−1
, y
k
}. The next
result, taken from [22], shows that access to the plant input is
unnecessary for the purpose of computing the estimate ˆx
0
k|k
.
Proposition 3 Assume that E = B, that CB 6= 0, and that
the transfer function G(z) = C(zI − A)
−1
B has no zeros
outside the closed unit circle. Then the state estimate ˆx
0
k|k
may be obtained from the recursion
ˆx
0
k|k
= Aˆx
0
k−1|k−1
+ L
f
(y
k
− CAˆx
0
k−1|k−1
),
where where L
f
= B(CB)
−1
. Furthermore, if F
x
(z) denotes
the transfer function from the plant output y
k
to the state
estimate ˆx
0
k|k
, then F
x
(z)G(z) = (zI − A)
−1
B.
It follows from Proposition 3 that the response of the state
estimate ˆx
0
k|k
to disturbance and control inputs is identical
to the response of the system states to these signals. It
follows that if the estimator is initialized with the plant initial
condition, ˆx
0|−1
= x
0
, then ˆx
0
k|k
= x
k
, ∀k. If the initial plant
state is unknown, then ˆx
0
k|k
→ x
k
as k → ∞.
We have now shown that the level of performance achieved
in Section III using an encoder that has access to the plant
state, plant input, and channel output can be achieved without
this additional information.
VI. CONCLUSION
We have presented a solution to the problem of minimizing
the variance of the plant output at a specified terminal time
using measurements of the plant output that are obtained
from a Gaussian channel in the feedback loop. Our solu-
tion involves linear time-varying communication and control
strategies. There is no separation between the problems of
communication and control, because the two strategies must
be carefully synchronized, as shown in Proposition 2. Yet there
is also no conflict, because the control input is chosen to aid
in the estimation process up until the last time step, when
estimation is no longer an issue and the control (4) may be
used that sets the plant output equal to the estimation error.
As examples presented in [22] show, the transient response
associated with control and communication strategies designed
to minimize variance at a terminal time may be very poor.
This is unsurprising, and strategies that are suboptimal at the
terminal time but possess better transient performance need
to be explored. Of course, penalizing transient as well as
terminal variance will introduce conflict between control and
estimation, and thus nonlinear strategies may prove superior.
ACKNOWLEDGMENT
The authors would like to acknowledge many useful dis-
cussions with Sandeep Pradhan and Achilleas Anastasopoulos
of the University of Michigan, and Nuno Martins of the
University of Maryland.
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