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.

2

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)

5

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

6

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)

9

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)

15

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.

16

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

17

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

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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

Technical Report No. 3045, Princeton University, Sept. 1998.

5. Juang, J.-N., Phan, M., Horta, L.G., and Longman, R.W., “Identification of

Observer/Kalman Filter Markov Parameters: Theory and Experiments,” Journal of

Guidance, Control, and Dynamics, Vol. 16, No. 2, 1993, pp. 320-329.

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pp. 175-252.

7. Phan, M.Q., Lim, R.K., and Longman, R.W., “Unifying Input-Output and State-

Space Perspectives of Predictive Control,” Department of Mechanical and Aerospace

Engineering Technical Report No. 3044, Princeton University, Sept. 1998.

8. Juang, J.-N., and Pappa, R.S., “An Eigensystem Realization Algorithm for Model

Parameter Identification and Model Reduction,” Journal of Guidance, Control, and

Dynamics, Vol. 8, No.5, 1985, pp. 620-627.

9. Juang, J.-N., and Lew, J.-S., “Integration of System Identification and Robust

Controller Designs for Flexible Structures in Space,” Proceedings of the AIAA

Guidance, Control, and Navigation Conference, Portland, OR, 1990, pp. 1361-1375.

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.