23-08-2014, 10:44 AM
Escape time formulation of state estimation and stabilization with
intermittent communication Project Report
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Abstract
The problems of state estimation and feedback
stabilization of an unstable system including a communications
channel are quantified as escape or survival times, which yield
stochastic processes describing the time of first exit of the state
estimate error or of the state itself from a specific domain.
The system complications introduced by communications –
intermittency, channel noise, etc – are evaluated using a Markov
hitting time formulation. This is compared to and contrasted
with earlier analyses which considered: the behavior of Kalman
filters with intermittent data, the evaluation of the minimal
number of bits required for mean square stabilization, and the
Large Deviations analysis of probability theory. The main result
shows the the escape time is characterized by a Markov chain
which is amenable to explicit analysis in the linear gaussian
case
INTRODUCTION
We study the problems of state estimation and output
feedback stabilization of an unstable linear time-invariant
system including a single communications link.
xk+1 = Axk + Buk + wk, (1)
yk = Cxk + Duk + vk, (2)
Here, as usual, xk, uk, yk, wk, vk are the system state,
input, output, process noise and measurement noise signals
of dimensions n, p, m, n, m respectively and [A, B, C, D]
are the system matrices of conformable dimensions. As in
[1], [2], the intermittency of the communication channel is
modeled by the random 0-1 {!k} sequence.
Definition 1 (Escape time): Given a closed domain D ⇢
Rd and a stochastic process {⇠k : k = 1,... } on Rd with
initial condition ⇠0, the escape time is defined to be
⌧e =
(
arg mink ⇠k 62 D,
1, if ⇠k 2 D 8k.
Sometimes the escape time is called the ‘first exit time’, or
‘hitting time’ We shall be concerned with the escape time
for the state process, xk, or the output process, yk, of (1-2)
when the control input is causally computed.
Clearly the escape time is a random variable provided the
infinite value has zero probability. We have the following
simple result, applicable in the linear gaussian case and
independent of the system matrices [A, B, C, D].
Lemma 1: If the noise process {wk} in (1) is ergodic and
possesses a density function of unbounded support in all
Kalman state estimation
The state estimate used in Assumption 1.3 is derived
from the Kalman filtering equations commencing from initial
values xˆ1|0 and ⌃1|0, with xˆ1|0 independent from {wk} and
{vk}. The Kalman filtering equations for an estimator with
intermittent observations described by {!k} are as follows.
Note the !k appears explicitly solely in the Kalman gain Lk.
measurement update
Lk = !k
⇥
⌃k|k!1CT (C⌃k|k!1CT + R)
!1⇤
,
xˆk|k = ˆxk|k!1 + Lk(yk ' Cxˆk|k!1),
⌃k|k = ⌃k|k!1 ' LkC⌃k|k!1.
time update
xˆk+1|k = Axˆk|k + Buk
⌃k+1|k = A⌃k|kAT + Q,
filtered error
x˜k|k = xk ' xˆk|k,
prediction error
x˜k+1|k = xk+1 ' xˆk+1|k,
x˜k+1|k = (A ' ALkC)˜xk|k!1 + wk ' ALkvk.
(3)
The Kalman predictor error equation (3) will be central to
both the estimator and the output escape time formulations
below. We note that, when !k = 0 then Lk = 0 and
(3) is unstable and driven by wk. We have the following
MARKOV CHAIN ANALYSIS
General Markov Chain model
Following Definition 1, given a closed domain D ⇢ Rd
and a stochastic process {⇠k : k = 1,... } on Rd, the
escape time of ⇠k from D is described by a Markov chain.
This is a general result concerning adapted processes and is
not limited to linear gaussian systems nor to hyperspherical
domains.
Theorem 1: For the stochastic process {⇠k : k = 1,... }
the random variable
Jk =
(
1, if ⇠k 2 D and Jk!1 = 1,
0, otherwise, (4)
is a Markov process and, denoting
⇧k =
Pr(Jk = 1)
Pr(Jk = 0)*
,
is described by the Markov chain.
⇧k+1 =
↵k 0
1 ' ↵k 1
*
⇧k,
↵k =
Pr(Jk+1 = 1, Jk = 1
+
+
+
Jk!1 = 1)
Pr(Jk = 1
+
+
+
Jk!1 = 1)
(5)
The escape time of the stochastic process {⇠k} is the value
of k when Jk = 0 for the first time.
Proof: From the definition (4), Jk satisfies the Markov
property.
Pr(Jk+1
+
+
+
Jk,...J1) = Pr(Jk+1
+
+
+
Jk),
ESTIMATOR ESCAPE TIME ANALYSIS
Kalman predictor escape time with intermittent data
The estimator error dynamics are given by (3) and do not
depend on the feedback control sequence, uk. The complete
behavior is given by the following.
Lk = !k[⌃k|k!1CT (C⌃k|k!1CT + R)
!1]
x˜k+1|k = (A ' ALkC)˜xk|k!1 + wk ' ALkvk
⌃k+1|k = (A ' ALkC)⌃k|k!1AT + Q
Adopting the Markov Chain method of Section III, we
compute Pr(
(
(x˜k!j+1|k!j
(
(
2 G, j = 0 ...k) from the
initial condition x˜1|0 ⇠ N(0, M¯ ), where M¯ is the starting
covariance, which might be taken to be that in Lemma 2 or
some other suitable value. The starting probability vector is
given by
J1 =
Pr(
(
(x˜1|0
(
(
2 G)
Pr(
(
(x˜1|0
(
(
2 > G)
*
.
The joint covariance of x˜k+1|k and x˜k|k!1 is given by
⌃(˜xk+1|k, x˜k|k!1) 4
= ⌃k+1|k (A ' ALkC)⌃k|k!1
⌃k|k!1(A ' ALkC)T ⌃k|k!1
*
.
This yields the following Markov chain parameter.
↵k = Pr(
(
(x˜k+1|k
(
(
2 G, (
(x˜k|k!1
(
(
2 G)
Pr(
(
(x˜k|k!1
(
(
2 G) ,
=
⇥
✓ 0
0
*
, ⌃(˜xk+1|k, x˜k|k!1), D ⇥ D)
◆
⇥(0, ⌃k|k!1, D) ,(17)
where domain D = {x˜ : kx˜k2 G} and ⇥(µ, ⌃, F) is
the multivariate normal distribution function for an N(µ, ⌃)
process on the set F
COMMENTS AND CONCLUSIONS
We have presented an approach to the study of state
estimation with intermittent measurements. This is based on
escape time analysis and computation, which is compared to
earlier works in estimation and stabilization with differing
descriptions of the properties – finite expected covariance
and mean-square stabilization – and of the communication
system – intermittent but exact and quantized but certain.
The central result is that the escape time can be described
by a Markov chain, which in the linear gaussian case is easily
computed. This yields much less conservative evaluation of
behavior than covariance calculations. In the full version of
this paper [26], these issues will be used to examine the
bitrate assignment questio