Appl. Math. Lett. Vol. 11, No. 4, pp. 27-32, 1998
© 1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0893-9659/98 $19.00 + 0.00
Pergamon
PII: S0893-9659(98)00051-2
Linear Unbiased State Estimation under
R a n d o m l y Varying B o u n d e d Sensor Delay
E. YAZ
Electrical Engineering Department, University of Arkansas
Fayetteville, AR 72701, U.S.A.
eyl©engr, uark. edu
A. RAY
Mechanical Engineering Department, The Pennsylvania State University
University Park, PA 16802, U.S.A.
axr2©psu, edu
(Received June 1997; revised and accepted October 1997)
A b s t r a c t - - T h e motivation for the work reported in this paper accrues from the necessity of finding
stabilizing control laws for systems with randomly varying bounded sensor delay. It reports the development of reduced-order linear unbiased estimators for discrete-time stochastic parameter systems
and shows how to parametrize the estimator gains to achieve a certain estimation error covariance.
Both finite-time and steady-state estimators are considered. The results are potentially applicable to
state estimation for stabilizing output feedback control systems. (~) 1998 Elsevier Science Ltd. All
rights reserved.
K e y w o r d s - - C o n t r o l and estimation, Stochastic processes, Random delay.
INTRODUCTION
This paper presents a continuation of our previous work [1] which provided the framework for
a partial solution to the problem of stabilizing randomly delayed control systems (in the mean
square sense) via Grammian assignment by static output feedback to a stochastic-parameter
model. The motivation for the work results from the necessity of finding stabilizing control laws
for output feedback control systems with randomly varying distributed delays. In the present
work, reduced-order estimators are presented for potential application to state-estimate feedback
control schemes. Random delays could be induced in the sensor data by an asynchronous timedivision-multiplexed network which serves as a data communications link between the spatially
dispersed components of the integrated decision and control system such as the vehicle management system of future generation aircraft [2]. In this context, the key issue is that filters and
controllers designed for non-networked systems may not satisfy the performance and stability requirements in the delayed environment of network-based systems. Therefore, a state-estimation
methodology is needed for compensation of the randomly varying sensor delays.
Since, in this work, the state estimation problem for a model involving randomly varying
bounded sensor delays is reformulated as a stochastic parameter estimation, the analysis is based
on the previous work in the latter area. Special cases of such models have been considered
for minimum variance estimation by Nahi [3] in an uncertain measurement context where the
The work reported in this paper was supported in part by the National Science Foundation under Grants ECS9322798 and CMS-9531835.
N,qt | ] - 4 - 8
Typeset by ~4h/~TEX
27
28
E. YAZ AND A. R A Y
measurement may or may not contain the signal with certain a priori probabilities and later by
several investigators [4-7] for more general types of random parameters. Yaz [7-10] considered
such models for stable full-order stochastic observer designs. Robustness issues and guaranteed
convergence rates were discussed in [11-13]. Reduced order observer design was presented in [14],
and [15] was on the application to a nonlinear version of the model in [3]. In the following
development, the sensor delay model is presented first. Then, all linear unbiased finite-time
fixed-order estimators are characterized. It is shown how linear unbiased minimum variance
estimators can be obtained. Sufficient conditions for a convergent estimator are also established
under steady-state conditions.
MODEL
EQUATIONS
The control system under consideration consists of a continuous-time plant (where some of
the states may not be directly measurable) and the data acquisition system of a discrete-time
controller which share a communications network with other subscribers [16]. Furthermore,
the plant is subjected to random disturbances and the sensor data is contaminated with noise.
Typically, both the sensor and controller data are subjected to randomly varying bounded delays
induced by the network before they arrive at their respective destinations. In the present context,
only the sensor delay in the state estimation problem is considered. The plant dynamics and
delayed sensor outputs are modeled as reported in [17]:
Plant Model:
Zk+l = ¢bkXk + F~vk,
Delayed Sensor Model:
(1)
Yk = Ckxk + F~(Vk,
(2)
Yk = (1 - ~k)~Jk + ~k~lk-1,
where Xk E R n is the plant state vector to be estimated; Ck and F~ are varying deterministic
matrices; vk is a zero mean white noise sequence at the plant input having covariance Vk with
arbitrary probability distribution such that: Trace Vk < co, V k >_ 0; the measurement t)k E R p
is contaminated by zero mean white noise wk with covariance Wk having arbitrary probability
distribution such that: Trace Wk < c~, V k >_ 0; ~k is a binary white noise sequence having the
expected values E{¢k} = ak and E { ~ } = ak, where Prob {~k = 1} = ak'. This assumption of
one-step sensor delay is based on the rationale that, following the standard practice of communication network design [16], the induced data latency from the sensor to the controller is restricted
not to exceed the sampling period. The random processes Xo, Vk, (Vk, and Ck are assumed to be
mutually independent. It is also assumed that F ~ W k F 2T > 0, Vk > 0. Based on (1) and (2), a
compact representation of the plant and measurement system is given as:
•~k : =
[xk]
Xk-1
Ak :=
In
°0]
0 2 := [(1 - C k ) F 2
C'k : = [ ( 1 - ~k)C'k
W k :-~
io ].
,l~k_ l
(3)
,
and the augmented plant model becomes
Plant Model:
Sensor Model:
xk+l = Akxk + D~vk,
Yk = CkYck + D2wk,
(4)
(5)
where the nonwhite noise wk is of zero mean and it has the covariance:
[? Wk-101
(6)
Bounded
Sensors
REDUCED-ORDER
Delay
29
ESTIMATION
Consider the compact form of system and measurement equations (4) and (5) with the necessary
definitions (3) and (6). We would like to estimate only a part of the composite state vector
=
Xk-1
:= T
Xk-1
.
(7)
The objective is to estimate xh by using a linear estimator of the following form:
(8)
:~h+l = Ki:~h -[- K~yh+I,
which is an n-dimensional (reduced order) estimator for a 2n-dimensional model. Although the
form of the observer may suggest that the measurement has to be processed immediately because
of the use of Yh+I in generating the estimate xh+l, equation (8) can be rewritten by introducing
the auxiliary variable
vh = :~h K~-lYh
(9)
-
in the following way:
oh+, = g l (oh + KLlyh)
(lO)
Equation (10) uses Yk to find the one-step-ahead prediction 75h+1 so that it can be used in actual
implementation, and the necessary estimate 2h+l can be calculated from (9). However, we will
use (8) in our development due to better mathematical tractability. We use eh to denote the
resulting estimation error:
ek+l
----- X k + l
-- ~:k+l
= T (AhSzh + Dtvh ) - Kli:k - K 2 [Ch+, (Ak~k + Divh) + Dk2+iwk+l]
= g~eh + (TAb - K i T - K~Ch+IAh) xh + ( T - g~Ch+l) Divh _ KhDh+w
lh+12
(II)
.
To have an unbiased estimator, we have to let
E{xo} =/:0
and
TAk - K I T - K~Ch+IAh = 0.
(12)
The above equation, upon substitution from (3) and (7), reduces to
(1)h - -
K 1 -
K 2
(O:h+lCh + (1 - - O~k+l)Ok+le~}k): O,
and letting
g l = ~k - g 2 (ak+lCk + (1 - ak+l)Ck+l~k) ,
(13)
equation (11) yields
ek+l = [~k - K2 ( +(1 _c~k+l~k
÷
+ ( T - K~Ck+I) Divk.
Denoting the estimation error covariance matrix as Pk := E{eke-[}, the recursive relation
follows:
o +1o
+ E { [ ( T - K 2 C k + I ) D~VkD~T] ~ + K~E [ [ C~+IAh"~kA'~C~+'T
x ( T - K~Ch+I) T
" [ I . + D h 2+ l ~ ' . , h~+ l D h +2 1T
])
1, 1 -r
]} K~'r,
(15)
30
E. YAZ AND A. RAY
where we define )~k - E{£'k~T}. Substituting (3) and completing the square, we obtain the
following equation for the filter gain matrix:
P~+I ¢kPkff~-~
_
_
~
z~Ik ,V,k . t~
" 1k T
+(1 - o~k+l)dk+lF~VkF~ T
~T
1
T "T
]
]
J
1
1T ~T
+(1 - oLk+I)F~VkF~
Ck+ 1
(C~k+lOk +(1--O~k+l)Ok+lCk)Pk(c~k+lCk +(1--O~k+l)Ck+lCk) T ]
+E C~+IAk2kATC~+I T + (1 -- olk+l)Ck+lF1VkrlTOTk+ 1
+ak+lF 2kWkF~2T + (1 -ak+l)F~+lWk+lF~+2
T / 2 a Ar r / 2 T
:= - K 2 N T _ NkK 2T + .'~k,vlknk
-
=
-
K
)T
K~ T
J
T
_
(16)
where we define
K S := NkM~ 1.
(17)
Equation (16) is obtained by using definitions in (3) and the existence of the inverse in (17) is
guaranteed because rE'~k2T,T.r
vvkrB-~2
k T > 0, Vk > 0. Substituting (17) into (16) and rearranging yield:
L~I_,-: = P~+~ - CkPk¢-: - F~,~f~ T + N~M;~N-[
---- (K~: - NkMk 1) Mk (K~ - NkME z) T
(18)
with Lk 6 R nxp because the right-hand side in (18) is positive semi-definite and of rank not
exceeding p. The optimal gain in the minimum variance sense is obviously given by (17) from
the above discussion and its use in (16) gives rise to the expression (18) for minimal estimation
error covariance with Lk = 0, V k > 0. Alternatively,
Pk+l = ~kPk~[ + F~VkF 1T - N k M k l N T.
(19)
The covariance matrix in (19) is iteratively found by starting with Po = E{eoe[}. The optimal
value of g ~ , i.e., KS, is computed based on Po, and then g 1 is computed from g 2 by using (13).
For a suboptimal gain, in general, (18) yields
LkL T = (g~ - g k M i 1) Mk (Z~ - NkMk 1) T
(20)
This equation can be decomposed as in [15] into:
LkUk = (Kk -- KS) ~
(21)
for a nonsingular square root of ~k and some orthogonal matrix sequence Uk. Solving for suboptimal gains yields:
K~ = ikUkv/--~k-1 + N k i k 1.
(22)
REMARK. There is a dimensional reduction not only in the estimator in (8) and error covariance
matrix Pk in (18) but also in the second moment )fk of the state that is not necessary to iterate
in full because the term in Mk containing ~Tk reduces to:
fC e
×
A )~ A T ' e
{Ck+l¢kXkOkCk+l _ Ck+1,kXkCk-
1
T
"T
T
1-T
=
_
CkXk,k~1
T'Tc£+I +
Ck+'XkC;+1h],
"
1 ~T
(23)
Bounded Sensors Delay
31
w h e r e X 1 = E{xkx'~} e R nxn is e v a l u a t e d b y using (1) from t h e following recursive relation:
1
1T
X1+1 = ~)kX~q~ + F~VkF~
(24)
T h e r e s u l t s a r e now s u m m a r i z e d as follows.
THEOREM 1. Given t h e models (3)-(6) and the linear unbiased reduced o r d e r estimator (8)-(10),
all assignable estimation covariances a r e given by (18) where Nk a n d Mk a r e defined in (16). The
g a i n s that g u a r a n t e e a n assignable covariance matrix a r e given by (22) where the term dependent
on t h e second m o m e n t o f the s t a t e v e c t o r is given by (23) a n d (24). The m i n i m u m possible e r r o r
covariance is given by (19) a n d the corresponding optimal estimator gain is found from (17).
INFINITE-TIME (STEADY-STATE) ESTIMATION
L e t us a s s u m e t h a t all d e t e r m i n i s t i c a l l y t i m e - v a r y i n g q u a n t i t i e s are c o n s t a n t a n d all r a n d o m l y
v a r y i n g d i s t u r b a n c e s are w e a k l y s t a t i o n a r y (with c o n s t a n t first two m o m e n t s ) . U n d e r t h e s e
a s s u m p t i o n s , it is p o s s i b l e to o b t a i n s t e a d y - s t a t e results for a s y m p t o t i c a l l y s t a b l e s y s t e m s b e c a u s e
Pk a n d K k can converge to r e s p e c t i v e c o n s t a n t m a t r i c e s o n l y if t h e X~ sequence converges. T h i s
c a n b e achieved if t h e s p e c t r a l r a d i u s p((I)) < 1 so t h a t t h e r e exists a unique X 1 = X 1-r >_ 0 t h a t
solves:
c~
X 1 -- (I)Xl(I)T -b F 1 V F 1T = E
(~kF1VF1T~:"
(25)
k=0
T h e p r o o f of convergence of t h e p r o p o s e d s t e a d y - s t a t e e s t i m a t o r is a s u b j e c t of f u t u r e research.
New a l g o r i t h m s a r e b e i n g t e s t e d for realistic a p p l i c a t i o n s [2] a n d t h e results will b e p r e s e n t e d in a
f o r t h c o m i n g p u b l i c a t i o n . E x t e n s i o n of this d e l a y c o m p e n s a t i o n a p p r o a c h is u n d e r c o n s i d e r a t i o n
for solving t h e c o m p l e t e p r o b l e m of d i s t r i b u t e d r a n d o m delays in b o t h sensors a n d a c t u a t o r s [18].
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E. YAZ AND A. RAY
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