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Department of Mechanical and Aerospace Engineering Technical Report No. 3046, October 1998.Princeton University, Princeton, NJ.
State Estimation with ARMarkov Models
Ryoung K. Lim 1
Columbia University, New York, NY 10027
Minh Q. Phan 2
Princeton University, Princeton, NJ 08544
Richard W. Longman 3
Columbia University, New York, NY 10027
Abstract
The ARMarkov models were originally developed for adaptive neural control, and
later for predictive control, and state-space identification. Recently, an interaction matrix
formulation has been developed that explains the internal structure of the ARMarkov
models and their connection to the state-space representation. Using the interaction
matrix formulation, we show in this paper how a state estimator can be identified directly
from input-output data. The conventional approach is to design such a state estimator
from knowledge of the plant, and the difficult-to-obtain process and measurement noise
statistics. A numerical example compares the identified state estimator with an optimal
Kalman filter derived with perfect knowledge of the plant and noise statistics.
1 Graduate Student, Department of Mechanical Engineering.2 Assistant Professor, Department of Mechanical and Aerospace Engineering, Dissertation Advisor.3 Professor, Department of Mechanical Engineering, Dissertation Co-Advisor.
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1. Introduction
State estimation is an important element of modern control theory. Given a
known model of the system under the influence of process and measurement noise with
known statistics specified in terms of their covariances, it is well known that the Kalman
filter is an optimal state estimator in the sense that its state estimation error is minimized.
In practice, it is difficult to design such an optimal estimator because neither the system
nor the noise statistics can be known exactly. From the point of view of system
identification, information about the system and the noise statistics are embedded in a
sufficiently long set of input-output data. Thus it would be advantageous to be able to
obtain such an estimator directly from input-output data without having to identify the
system and the noise statistics separately. This is the problem of observer identification.
Recently, a class of models known as ARMarkov models has been developed in
the context of adaptive neural control, Ref. 1. The term ARMarkov refers to Auto-
Regressive model with explicit Markov parameter coefficients. ARMarkov models form
a bridge between the common ARX model (Auto-Regressive model with eXogenous
inputs) where the Markov parameters are implicit, and the non-auto-regressive pulse
response model where every coefficient is a Markov parameter. Later the ARMarkov
models are used for state-space system identification, Refs. 2-4. In particular, it was
found that when the ARMarkov models are used to identify the system Hankel matrix,
the true or effective order of the system can be detected more effectively than with an
ARX model. This issue has been investigated extensively in Ref. 4. In fact, with an ARX
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model it is also possible to identify a state-space model together with an observer gain as
shown in Refs. 5 and 6. But with this technique it is not always possible to determine
the order of the state space realization by Hankel singular value truncation alone and a
separate post-identification model reduction procedure must be used. With ARMarkov
models, we have the opportunity to identify state estimators with true or effective orders
without having to invoke a separate model reduction step as normally required. This is
one motivation for the present paper. It is clear that for state estimation, efficient
detection of the dimension of the effective state space model is important because it is
computationally a burden to have a state estimator with unnecessarily large dimensions.
The ARMarkov model used in Ref. 4 is based on the development of the
interaction matrix in Ref. 7 that can be explained in terms of a generalization of the well-
known Cayley-Hamilton theorem. In all of these developments the role of the interaction
matrix has been to justify the structures of various input-output models and the
relationship among the coefficients of these models, but there has been no need to identify
the interaction matrix itself from input-output data. The second motivation of this paper
is an investigation of the role of this interaction matrix in the context of state estimation as
opposed to treating it as a convenient mathematical construct. In this work, we establish
that fact that identifying a state estimator amounts to identifying this interaction matrix.
Another interesting aspect of this formulation is that this interaction matrix based state
estimator has a non-standard form, different from the usual form of a Luenberger observer
or a Kalman filter. However, it will be shown that this new form is very convenient both
from the point of view of state estimator design as well as from the point of view of its
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identification from input-output data. This point makes up the third motivation for this
work.
In this paper we will quickly derive such a state estimator using the interaction
matrix. We then show how this state estimator can be identified from input-output data.
This identification will first be derived in the deterministic (noise-free) setting. Then a
stochastic analysis will be carried out that shows why the calculations involved in the
deterministic case are indeed justified in the stochastic case. Following the theoretical
justification, a numerical example illustrates how an identified state estimator using the
technique developed here compares to that of an optimal Kalman filter designed with
perfect knowledge of the system and perfect knowledge of the noise statistics. In
particular, we show that the output residuals obtained with this identified state estimator
indeed match the residuals of the optimal Kalman filter.
2. State Estimation by Interaction Matrix
In the following we briefly derive a state estimator via an interaction matrix.
Consider an n-th order, r-input, m-output discrete-time model of a system in state-space
format
x k Ax k Bu k
y k Cx k Du k
( ) ( ) ( )
( ) ( ) ( )
+ = += +
1 (1)
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By repeated substitution, we have for some p ≥ 0 ,
x k p A x k u k
y k x k u k
pp
p p
( ) ( ) ( )
( ) ( ) ( )
+ = +
= +
C
O T (2)
where u kp( ) and y kp( ) are defined as column vectors of input and output data going p
steps into the future starting with u k( ) and y k( ) , respectively,
u k
u k
u k
u k p
y k
y k
y k
y k p
p p( )
( )
( )
( )
, ( )
( )
( )
( )
=+
+ −
=+
+ −
1
1
1
1
M M (3)
For a sufficiently large p, C in Eq. (2) is an n × pr controllability matrix, O is a pm n×
observability matrix, T is a pm pr× Toeplitz matrix of the system Markov parameters,
C = Ap−1B, K, AB, B[ ] ,
O =
C
CA
M
CAp−1
,
T =
D 0 0 L 0
CB D O O M
CAB CB D O 0
M O O O 0
CAp−2B K CAB CB D
(4)
As long as pm n≥ , it is guaranteed for an observable system that an interaction matrix M
exists such that A Mp + =O 0 . The existence of M ensures that for k ≥ 0 an expression
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for x k p( )+ exists where the state variable is completely eliminated from the right hand
side of Eq. (2),
x(k + p) = Apx(k) + Cup(k)
= Apx(k) + Cup(k) + M Ox(k) + T up(k)[ ] − Myp(k)
= Ap + MO( )x(k) + C + MT( )up(k) − Myp(k)
= C + MT( )up(k) − Myp(k)
(5)
Shifting the time indices back by p time steps, we have for k ≥ p an expression that
relates the current state of the system in terms of p past input and p past output
measurements,
x(k) = α ii=1
p
∑ u(k − i) + βii=1
p
∑ y(k − i) (6)
where α α αp M, , , K 2 1[ ] = +C T , and
β β βp M, , , K 2 1[ ] = − . Note that these
formulas are applicable to both the single-input single-output and multiple-input
multiple-output cases.
Equation (6) is a state estimator, but not in the standard form of a Luenberger
observer or a Kalman filter. In fact, this non-standard form is quite convenient from both
perspectives of design and identification. If an observable state-space model of the
system is known, to design this state estimator, one simply forms the controllability
matrix C , the Toeplitz matrix T , compute an interaction matrix M from A Mp + =O 0 ,
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pm n≥ . When p is chosen such that pm n> then M is no longer unique, in which case
the solution M Ap= − +O where O+ denotes the pseudo-inverse of O, produces a
minimum-norm solution for M, whose elements are precisely the output gains of this
state estimator. We have focused on state estimation, but output estimation can be
similarly derived. It is simply,
y k C M u k p CMy k p Du kp p( ) ( ) ( ) ( )= +( ) − − − +C T (7)
and the minimum norm solution for CM is − +CApO , whose elements are the output
gains for the output estimator is CM CAp= − +O .
On the other hand, if the system is unknown but input-output data is available,
then the parameter combinations C + MT and M can be computed for the state
estimator. This is shown in the next section.
3. Identification of State Estimator with ARMarkov Models
We must first derive an input-output expression that involves C + MT and M
by combining Eq. (5) with the output expression in Eq. (2),
y k x k u k
M u k p My k p u k
p p
p p p
( ) ( ) ( )
( ) ( ) ( )
= +
= +( ) − − − +
O T
O C T O T (8)
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Equation (8) is a family of ARMarkov models. The first member of the family is
an ARX model for y k( ), which is a special case of an ARMarkov model. All remaining
members are expressions for y k( )+1 , …, y k p( )+ −1 which are true ARMarkov models.
Each ARMarkov model is different from another in that it has an increasing number of
Markov parameters appearing explicitly as coefficients. We need not one but such a
family of ARMarkov models to solve for C + MT and M . Furthermore, we assume to
have only output measurements and not the full state, therefore the above input-output
expression does not have C + MT and M appear explicitly as coefficients, but the
combinations O C + MT( ) and −OM (and T ). Since O is not known, the identification
of C + MT and M from input-output data is in fact a non-linear problem. Fortunately,
an exact solution can be found without any kind of iterations as shown below.
For simplicity define A = O C + MT( ), B = −OM , then A , B , and T can be
identified from input-output data as
A B T[ ] = ( )+YV VVT T
(9)
where u kp2 ( ) combines u kp( ) with u p kp( )+ for convenience,
Y y p y p y pp p p= + +[ ]( ) ( ) ( )1 L l (10)
V
u u u
y y yp p p
p p p
=
2 2 20 1
0 1
( ) ( ) ( )
( ) ( ) ( )
L l
L l
(11)
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To find C + MT( ) and M from A and B we must first find O. Taking advantage of the
internal structure of the coefficients A , B as revealed by the interaction matrix, the
combination H = OC can be computed from
H = A + BT (12)
The observability matrix O is obtained by a singular value decomposition of
H = U Vn n nTΣ , where n is the order of the system,
O = Un nΣ1 2/
(13)
Since the state-space representation is uncertain up to a similarity transformation of the
state vector, we generally have O in a different set of coordinates from O but they are
related to each other by a similarity transformation, O O= T . The identified parameter
combinations, however, are invariant with respect to such a transformation,
A
B
= +( ) = +( ) = +( )
= − = − =
− −
−
O C T O C T O C T
O O O
M T T T M M
M TT M M
1 1
1
(14)
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Hence, as long as p is chosen to be sufficiently large such that the observability matrix O
or O has rank n, the needed parameters C T+( )M and M to construct a state estimator
can be found from
C T O O O+( ) −[ ] = [ ]−M M
T T( ) 1 A B (15)
We thus see that it is not necessary to produce a realization of A, B, C in the
above steps of extracting a state estimator from input-output data. However, this step
can be easily done as well, and would be necessary if we desire to put the state estimator
in a different set of coordinates than that chosen by the above realization, such as the
modal coordinates. Any realization algorithm can be used for that purpose. Here we
review the realization provided by the Eigensystem Realization Algorithm (ERA), Ref. 8.
The procedure calls for the extraction of two Hankel matrices H( )0 and H( )1 from H ,
H( )0
1
2 1 2
=
+
CB CA B
CA B CA B
n
n n n
L
M M M
L
,
H( )1
1
2 1 2
1
1 1
=
+
+ + +
CAB CA B
CA B CA B
n
n n n
L
M M M
L
(16)
The matrix H has the Markov parameters of increasing order going from left to right,
whereas H( )0 and H( )1 are typically defined with the higher order Markov parameters
going from right to left. To maintain the standard notation, a trivial rearrangement of the
Markov parameters is needed in forming H( )0 and H( )1 . A s-th order state-space model
is A U Vs sT
s s= − −Σ Σ1 2 1 21/ /( )H , B is the first r columns of Σs sTV1 2/ , C is the first m rows of
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Us sΣ1 2/ , and s r n m n≤ + +( )min ( ), ( )1 21 1 . The matrix Us and Vs are made up of s left and
right singular vectors of H( )0 , respectively. The diagonal matrix Σs is made up of s
corresponding singular values of H( )0 . With perfect data H( )0 has exactly n positive
singular values (all remaining singular values are identically zero), where n is the true
minimum order of the system, s n= . Otherwise, the user can specify the order of the
state-space model by s, the number of Hankel singular values to retain.
The particular set of coordinates of the realization in O has a special property
that it becomes internally balanced when p is large and the system is stable, Ref. 9. A
realization is said to be internally balanced if and only if the controllability and
observability grammians are equal to each other, and both equal to Σn . As mentioned, the
realization can also be put in another user-specified set of coordinates. We simply need
to compute the corresponding observability matrix O in that set of coordinates and use
Eq. (15) to obtain the corresponding state estimator.
Stochastic Analysis
In the previous section the identification of the state estimator is derived in the
deterministic setting. Now we will consider the situation where data is corrupted by
process and measurement noise, and show how the same deterministic calculations are
justified in the stochastic case. Consider the case where process and measurement noise
are present in Eq. (1). The corresponding version of Eq. (2) is
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x k p A x k u k v k
y k x k u k w k
pp
p p p
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
+ = + +
= + +
C
O T (17)
Although Eq. (17) is p-step ahead, i.e., it relates x k p( )+ to x k( ) with sampling interval
∆t , input u k( ) , and output y k( ), it can be thought of as one-step ahead with sampling
interval p t∆ , input u kp( ) , and output y kp( ). Hence it admits an estimator of the form
with some gain K,
√ ( ) √ ( ) √ ( ) ( ) √ ( ) ( )
√ ( ) √ ( ) ( )
√ ( ) √ ( ) ( )
x k x k K y k y k x k Ke k
x k A x k p u k p
y k x k u k
p p p
pp p
p p
+ − −
− +
−
= + −[ ] = +
= − + −
= +
C
O T
(18)
The quality of the estimation certainly depends on the gain K. For a choice of K, e k( ) is
the corresponding output residual defined to be the difference between estimated output
√( )y k and measured output y k( ), e k y k y k( ) √( ) ( )= − . The expression for the state in Eq.
(18) can be written as
√ ( ) √ ( ) √ ( ) ( ) ( )
√ ( ) √ ( ) ( ) ( ) ( )
√ (
x k A x k p K y k p y k p u k p
A x k p A K x k p u k p y k p u k p
A A K x k p
pp p p
p pp p p
p p
− −
−
−
= − + − − −( )( ) + −
= − + − − + − − −( ) + −
= +( ) −
C
O T C
O )) ( ) ( )+ +( ) − − −C TA K u k p A Ky k ppp
pp
(19)
The above equation is somewhat subtle for the following reason. Due to the presence of
the term A A Kp p+( )O , it is a p-step ahead state estimator, i.e., it estimates √ ( )x p− from
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√ ( )x− 0 , then √ ( )x p− 2 from √ ( )x p− , and so on. However, if A A Kp p+( )O is zero (which
will be justified later), then this equation can be used to provide state estimation every
single step beginning with √ ( )x p− , then √ ( )x p− +1 , √ ( )x p− + 2 , and so on. When combined
with the output expression we have
y k Ax k p A K u k p A Ky k p u k e kpp
pp
p p( ) ƒ√ ( ) ( ) ( ) ( ) ( )= − + +( ) − − − + +−O O C T O T (20)
where ƒA A A Kp p= + O . From a given set of input-output data of sufficient length we can
form the data matrices Y and V as defined in Eqs. (10) and (11), then
Y V AX E= [ ] + − +A B T O ƒ√ (21)
where A = +( )O C TA Kp , B = −OA Kp , √ √ ( ) √ ( ) √X x x x− = ( )[ ]− − −0 1 K l , and
E e p e p e p
e p e p e p
e p e p e p
e p e p e pp p p
= + +[ ] =
+ ++ +
+ +
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1
1
1
1
1 1 1
2 2 2K l
K l
K l
M M K M
K l
(22)
The residual matrix E has the following interpretation when ƒA = 0. The first residual
sequence e k( ) ( )1 , k p p p= + +, , .., 1 l is associated with the ARX model derived from
the first m rows of Eq. (20). The second residual sequence e k( ) ( )2 is associated with the
first ARMarkov model derived from the second m rows of Eq. (20). Similarly, the
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remaining residual sequences e kj( ) ( ) , j = 3 through p, are associated with the remaining
ARMarkov models. Because not a single but a family of models are being used in the
input-output map of Eq. (20), we do not have a single but rather a family of residuals.
Pre-multiplying Eq. (21) by V T and re-arranging it yields,
YV VV AX V EVT T T T−[ ] = − +A B T O ƒ√ (23)
Now let us impose conditions on K so that the observer in Eq. (19) possesses desirable
properties. If K is chosen such that ƒA = 0 then by choosing
A B T[ ] = ( )+YV VVT T (24)
the left hand side of Eq. (23) vanishes. Referring back to Eq. (21), with ƒA = 0, the
solution given in Eq. (24) is exactly the one that minimizes the Euclidean norm of E which
is the sum of the squares of the residuals for the entire data record. Furthermore, from
Eq. (23), this solution also results in EV T = 0 , which can be written explicitly as
e p e p e p
e p e p e p
e p e p e p
u u p u p u
p p p
T T T T( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) (1 1 1
2 2 2
1
1
1
0 1 2+ ++ +
+ +
−
K l
K l
M M K M
K l
K K pp
u u p u p u p
u u p u p u p
T T T T
T T T T
−+
− + + − +
=
1
1 1 2
1 2 1
0
)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
K K
M K M M K M
l K l l K l
(25)
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e p e p e p
e p e p e p
e p e p e p
y y p
y y p
p p p
T T
T T
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) (
1 1 1
2 2 2
1
1
1
0 1
1
+ ++ +
+ +
−
K l
K l
M M K M
K l
K
K −−
− +
=
1
1
0)
( ) ( )
M K M
l K ly y pT T
(26)
Let us now examine the implications of Eqs. (25) and (26) for the j-th member of the
residual sequences e kj( ) ( ) . When the length of the data record tends to infinity, the above
equations imply for a stationary random process,
E e k u k i e k u k i i p pj
Tj
k p
pT
( ) ( )( ) ( ) lim ( ) ( ) , ,..., , , , ..,−{ } = − = = − +→∞
=
+
∑l
l
l
10 1 0 1 2 (27)
E e k y k i e k y k i i pj
Tj
k p
pT
( ) ( )( ) ( ) lim ( ) ( ) , , , ..,−{ } = − = =→∞
=
+
∑l
l
l
10 1 2 (28)
where E .{ } denotes the expectation operator. Thus, if the data record is sufficiently long,
the identified family of ARMarkov models has its combined residuals minimized, and in
particular, Eqs. (27) and (28) state that the residual for each member of the family
becomes uncorrelated with input and output data. Recall these results are obtained while
imposing the condition on K such that ƒA A A Kp p= + =O 0. We note here that this
condition can also be satisfied if p is large and the system is stable so that Ap ≈ 0 .
By referring back to the deterministic formulation, it is clear that the interaction
matrix M plays the same role as A Kp at every step of the derivation, including the
condition A Mp + =O 0 . Hence, the deterministic calculations are indeed justified in the
stochastic case.
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Illustration
Consider a chain of three masses connected by springs and dampers with force
input to the first mass and position measurements of the last two masses. The state-
space matrices for this dynamical system are
AI
M K M C=
− −
× ×− −
03 3 3 31 1 , B =
×
×
0
1
0
3 1
2 1
, C = [ ]×0 1 1 01 3 , D = 0
The state vector is made up of positions and velocities of the three masses in the
following order, x x x x x x xT= [ ]1 2 2 1 2 3, , , « , « , « , and the mass, stiffness, and damping matrices
are
M
m
m
m
=
1
2
3
0 0
0 0
0 0
, C
c c c
c c c c
c c
=+ −
− + −−
1 2 2
2 2 3 3
3 3
0
0
, K
k k k
k k k k
k k
=+ −
− + −−
1 2 2
2 2 3 3
3 3
0
0
where m Kg1 0 5= . , m m Kg2 3 1= = , k k k N m1 2 3 10= = = , c c c N m1 2 3 0 35= = = . sec .
The sampling interval is 0.1 sec. The system is excited by random input, and the output
data is corrupted by significant process and measurement noise. In modal coordinates,
each of the modal states is corrupted by about 3-5% process noise (measured in terms of
standard deviation ratios of noise to signal), and the outputs are corrupted by about 15%
measurement noise. Because this is a simulation, we can actually compute the noise
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statistics given in terms of their covariances. From exact knowledge of the system and the
computed process and measurement noise covariances, a Kalman filter is designed. The
Kalman filter represents the “best” or optimal estimation that can be achieved for the
given system with the known noise statistics.
Next we use the above set of noise corrupted input-output data to identify a state
estimator with the procedure described in this paper, and this is done without knowledge
of the system and without knowledge of the embedded process and measurement noise
statistics. As mentioned in the introduction, ARMarkov models are effective in capturing
the true or effective system order. Order determination is achieved by examining the
quality of the identified state space model in reproducing the identification data for
various model order selection by Hankel singular value truncation. This is shown in Fig. 1
which indicates that the system order is six, which is indeed the case.
Figure 2 shows the actual measured (noise-corrupted) outputs together with an
overlay of the results of the optimal Kalman filter estimation and the identified 6-th order
state estimator. Let us examine the first output. The jagged curve is the measured noise-
corrupted output. The smooth curve represents the optimal filtering by the Kalman
filter. Note that result obtained with the identified state estimator closely follows the
Kalman filter result. The same pattern is observed for the second output. Figure 3 shows
a comparison of the residuals itself, for each of the two outputs. Recall that the Kalman
filter results are derived with exact knowledge of the system and noise statistics, whereas
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the identified state estimator is derived from input-output data alone. Figure 4 shows the
auto-correlation of the Kalman filter residual and our identified state estimator residual.
In addition to comparing filtered outputs, we compare the filtered modal states
and this is shown in Fig. 5. Since the Kalman filter minimizes the state estimation error, it
is interesting to see how the identified state estimator compares to this optimal result.
Keeping in mind that in the presence of noise, identification can never perfectly extract
the system model and noise statistics with finite data records but with increasing data
length and p, improvement in the identification should be expected. Recall that increasing
p is beneficial because it helps making the residuals more and more uncorrelated with
identification data as shown in the theoretical section. Indeed, it is shown in Table 1 that
the norm of the state estimation error (and of output prediction error) of the identified
state estimator approaches that of the Kalman filter with increasing data length and p.
These illustrations indicate that the proposed state estimator identified from input-output
data does indeed approach the optimal Kalman filter designed with perfect knowledge of
the system and perfect knowledge of process and measurement noise statistics.
Conclusions
In this paper we have shown how ARMarkov models can be used to identify a
state estimator from input-output data. This work extends previous development of
ARMarkov models for system identification and control applications. Being able to
identify a state estimator from data is significant in view of the standard approach of
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designing such a state estimator from an assumed model of the system and the noise
statistics.
A key ingredient that makes this possible is the recent derivation of ARMarkov
models through the use of an interaction matrix. In our previous work, the role of the
interaction matrix is to justify the existence of various input-output models and to
establish various relationship among the identified coefficients. For control and system
identification problems, there is no need to recover the interaction matrix itself. In this
work, we proceed one step further by actually recovering the interaction matrix, and
explaining it in the context of a state estimator. This state estimator is not in the standard
form of a Luenberger observer or a Kalman filter but in this new form, this state estimator
is easy to design from a known model, and if the model is not known, it can be identified
from input-output data.
References
1. Hyland, D.C., “Adaptive Neural Control (ANC) Architecture - a Tutorial,”
Proceedings of the Industry, Government, and University Forum on Adaptive Neural
Control for Aerospace Structural Systems, Harris Corp., Melbourne, FL, 1993.
2. Akers, J.C., and Bernstein, D.S., “ARMARKOV Least-Squares Identification,”
Proceedings of the American Control Conference, Albuquerque, NM, 1997, pp. 186-
190.
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3. Akers, J.C., and Bernstein, D.S., “Time-Domain Identification Using
ARMARKOV/Toeplitz Models,” Proceedings of the American Control Conference,
Albuquerque, NM, 1997, pp. 191-195.
4. Lim, R.K., Phan, M.Q., and Longman, R.W., “State-Space System Identification with
Identified Hankel Matrix,” Department of Mechanical and Aerospace Engineering
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21
0 20 40 60 80 1000 . 8
1
1.2
1.4
1.6
1.8P
redi
ctio
n er
ror
Selected state space model order
Fig. 1. Order determination
22
-0.05
0
0.05
100 102 104 106 108 110
ARMarkovKalmanMeasured
Out
put
1
-0.05
0
0.05
100 102 104 106 108 110
ARMarkovKalmanMeasured
Out
put
2
Time (sec.)
Fig 2. Measured and estimated outputs by ARMarkov and Kalman filter.
23
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
100 102 104 106 108 110
ARMarkovKalman
Res
idua
l out
put 1
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
100 102 104 106 108 110
Res
idua
l out
put 2
Time (sec.)
Fig. 3. ARMarkov and Kalman filter output residuals.
24
0 10 20 30 40 50
ARMarkovKalman
Aut
o-co
rrel
atio
n (r
esid
ual
1) 1.2
0
10 -4
1.0
0.8
0.6
0.4
0.2
0.2
0 10 20 30 40 50
Aut
o-co
rrel
atio
n (r
esid
ual
2)
1.2
0
10-4
1.0
0.8
0.6
0.4
0.2
0.2
Number of time shifts
Fig. 4. Auto-correlation of output residuals.
25
- 4
-2
0
2
4
100 105 110 115 120
ARMarkov Kalman ActualS
tate
2
- 8
-6
-4
-2
0
2
4
6
8
100 105 110 115 120
Sta
te 6
Time (sec.)
- 5
0
5
100 105 110 115 120
Sta
te 4
Fig. 5. True and estimated modal states by ARMarkov and Kalman filter.
26
p 30 30 100
Data Length 4000 20000 20000
ARMarkov 0.0090 0.0038 0.0025State Residual
Kalman 0.0046 0.0021 0.0021
ARMarkov 2.2864 1.0122 1.0099Output Residual
Kalman 2.2584 1.0210 1.0091
Table 1. Comparison of state and output residuals by ARMarkov and Kalman filter.