JOURNAL OF OPTIMIZATION THEORY AND APPLICATION: Vol. 79, No. 1, OCTOBER 1993
Linear System Identification
via an Asymptotically Stable Observer
M. PI-IAN,I L. G. HORTA,2 J. N.
JUANG, 3 AND
R. W. LONGMAN4
Communicated by L. Meirovitch
Abstract. This paper presents a formulation for identification of
linear multivariable systems from single or multiple sets of input-output
data. The system input-output relationship is expressed in terms of
an observer, which is made asymptotically stable by an embedded
eigenvalue assignment procedure. The prescribed eigenvalues for the
observer may be real, complex, mixed real and complex, or zero corresponding to a deadbeat observer. In this formulation, the Markov
parameters of the observer are first identified from input-output data.
The Markov parameters of the actual system are then recovered from
those of the observer and used to realize a state space model of the
system. The basic mathematical formulation is derived, and numerical
examples are presented to illustrate the proposed method.
Key Words. System identification, observer identification, pole
placement, state space realization, Markov parameters, observer
Markov parameters,
1. Introduction
The basic purpose of system identification is to develop a mathematical model of a physical system based on its i n p u t - o u t p u t data. One is often
concerned with linear models, since m a n y real systems can be described by
linear or approximately linear equations. Linear systems can be represented
in the state space format where the relationship between the input and
~Senior Engineer, Lockheed Engineering and Sciences Company, Hampton, Virginia.
2Aerospace Engineer, Spacecraft Dynamics Branch, NASA Langley Research Center,
Hampton, Virginia.
3principal Scientist, Spacecraft Dynamics Branch, NASA Langley Research Center, Hampton,
VirNnia.
4Professor of Mechanical Engineering, Columbia University, New York, New York.
59
0022-3239/93/1000.0059507.00/0 © 1993 Plenum Publishing Corporation
60
JOTA: VOL 79, NO. 1, OCTOBER 1993
output variables are described via an intermediate quantity called the
state variable. As a part of system identification theory, realization methods
are concerned with the problem of finding a minimal order state space
representation of a linear system when its sampled pulse response functions
are known. The pulse response samples are also known as the Markov
parameters. Current methods, such as the eigensystem realization algorithm
(ERA, Refs. 1 and 2) have been successfully applied to the identification of
large flexible structures. Since state space realization methods require that
the system Markov parameters be known as a starting point, the problem
of determining the Markov parameters from input-output data is one of
fundamental importance in system identification. In practice, if the
response of a system to a certain rich input is available, the fast Fourier
transform (FFT) technique is normally used to compute its Markov
parameters. However, the process of transforming the data to the frequency
domain by the FFT technique places stringent requirements on the characteristics of the data record such that the input must be very rich to ensure
computational accuracy. The role of the Markov parameters in system
identification is reviewed and discussed extensively in Ref. 3.
A time-domain system identification method is developed in Ref. 4 by
means of a description of the original system via an observer. An important
distinguishing feature of the proposed approach as opposed to previous
development is that the system is identified indirectly via an observer,
which is made asymptotically stable by an eigenvalue assignment procedure. Instead of identifying the Markov parameters of the system directly,
the method first identifies the Markov parameters of an associated observer
from data. The approach avoids direct identification of the system Markov
parameters, which can exhibit very slow decay for lightly damped flexible
structures. The role of the observer is not to provide estimates of the
system states but rather to provide by design a set of asymptotically stable
equations whose parameters can be easily identified. The discrete-time
observer eigenvalues or poles are prescribed a priori and they are required
to be real and distinct, with magnitudes less than one. The system Markov
parameters are then recovered from those of the identified observer and
used to realize a state space model of the system. It is known that in
practice the assignment of observer poles should not be restricted to
real poles alone. The main objectives of this paper are to show that it is
indeed possible to allow more general placement of observer poles in the
identification problem, and to explain how this can be accomplished.
The basic outline of this paper is as follows: First, a simplified reformulation of the original identification algorithm with placement of real
eigenvalues is presented. Second, extensions to the cases of complex or
mixed real and complex eigenvalue assignment are then formulated. Third,
JOTA: VOL. 79, NO. 1, OCTOBER 1993
61
a special version of the identification algorithm using a deadbeat observer
is presented. This is a case of particular interest because of its simplicity
and effectiveness. The method requires only a minimum amount of inputoutput data, and the number of identified observer Markov parameters is
reduced to a minimum set. Fourth, numerical examples are provided to
illustrate the basic characteristics of the algorithm. The method developed
here is applicable for data from either a single set or multiple sets of
experiments. To study the exact nature of the identification procedure
under ideal circumstances, this paper is confined to purely deterministic
results. In later developments, when process and measurement noises are
present, the relationship between the identification algorithm with a deadbeat observer presented in this paper and the stochastic Kalman filter algorithm of Ref. 5 is established in Ref. 6. A procedure to improve observer
and Kalman filter identification results by whitening the residual sequence
is presented in Ref. 7. Often of interest in practice is the identification of a
model in a prescribed frequency range. Such a development of the algorithm using frequency-weighted observer Markov parameters is formulated
in Ref. 8.
2. System Description
In this section, various input-output descriptions of a linear system
are described. With the state space model, the relationship between the
input and output variables is expressed in terms of the system Markov
parameters, which relate current output to the past and present input
values. When an observer is introduced to the set of state space equations,
the current output is related to the past output values in addition to the
past and present input values. This relationship is expressed in terms of the
observer Markov parameters.
2.1. State Space Model. First, consider a general discrete multivariable linear system expressed in the space space format
x(i + 1 ) = Ax(i) + Bu(i),
(la)
y(i) = Cx(i) + Du(i),
(lb)
where x ( i ) e R " , y ( i ) e R q, u ( i ) e R m. Let x(O) denote the initial state at
i = O. An input-output description of the above system can be obtained
from (1) as
i--1
y(i) = CA~x(O) + ~ CA ~....... iBu(~) + Du(i).
(2)
62
JOTA: VOL. 79, NO. 1, OCTOBER 1993
Note that the first term on the right-hand side of the above
dependent on the initial condition x(0). The products CA i-~by Y;_ ~_ 1, and D are known as the Markov parameters of
From (2), the input-output description of the system with
condition becomes
equation is
IB, denoted
the system.
zero initial
i--1
y(i)= ~
Y ~ u ( i - ' c - 1)+Du(i),
(3)
"c=O
where y(i) is expressed in terms of Y0 up to Yi t and the direct transmission term D. In general, this description requires i + 1 Markov parameters
to describe the output at time step i. If the system is asymptotically stable
such that the Markov parameters Yp, Yp + 1, Yp +2. . . . can be neglected for
some p, then at time steps i>~ p, the input-output description can be
approximated with a finite set of Markov parameters as
p--1
y(i)~ ~
Y~u(i-r-1)+Du(i).
(4)
"t'=O
It is important to note that, for a finite-dimensional system, there is
only a finite number of independent system Markov parameters. Therefore,
the system Markov parameters used in the description of (4) are not
necessarily independent. For stffficiently damped systems, (4) is a valid
description of the input-output relationship provided that p is chosen
sufficiently large such that the approximation holds. However, for lightly
damped systems such as large flexible space structures, such an inputoutput model requires a very large number of system Markov parameters,
which would not be computationally attractive for system identification. In
fact, if the system is marginally stable or unstable, such a description is no
longer possible.
2.2. Observer Model. In the following, a procedure is described to
express the state space model in (1) as an autoregressive model with a finite
number of Markov parameters. The Markov parameters are later shown
to be of an observer system, which is made asymptotically stable by
eigenvalue assignment. This observer model is then used to developed an
identification method for the system described by (1).
To construct an observer model, add and subtract the term My(i) to
the right-hand side of the state equation in (1),
x(i + 1) = Ax(i) + Bu(i) + My(i) -- My(i)
= (A + M C ) x(i) + ( B + MD) u(i) - My(i).
(5)
JOTA: VOL. 79, NO. 1, OCTOBER 1993
63
For notational simplicity, define
B= [B+MD, -M],
~1 = A + M C ,
vii)=?(;)]
Ly(i)J"
(6)
Then, the original system becomes
x(i + 1 ) = Ax(i) + By(i),
(7a)
y(i) = Cx(/) + Du(i).
(7b)
The input-output description of the above system is
i--1
y(i) = CAix(O) + ~
Yi ..... t v(z) + Du(i),
(Sa)
"r=0
where
(8b)
Consider the case where the system (7) is made asymptotically stable by
placing the eigenvatues of A inside the unit circle in the complex plane such
that 1gp, lTp+ 1, Yp+ 2. . . . can be neglected for some p. Then at time steps
i~> p, the input-output description can be approximated with a reduced
set of parameters {Y0, Y1. . . . . Fp_l, D}. The following equality then
approximately holds:
p--1
y(i)= ~
Y~v(i-z-1)+Du(i),
i>~p.
(9)
z=O
If the pair (A, C) is observable, then a matrix M that places the eigenvalues
of .g at any particular (symmetric) configuration always exists. For the case
of lightly damped systems, this procedure transforms the set of an
otherwise large number of Markov parameters {D, Y0, Y,, Y2. . . . } to an
approximately equivalent reduced set {D, Y0, Y1,-.., F;_ 1} by selecting
appropriate eigenvalues for A. Furthermore, for a sufficiently large p, the
influence of a nonzero initial condition on the output at time steps i>7 p
can be neglected. The model (9) is used to develop the identification
method presented herein, and the eigenvalue assignment step is achieved
implicitly through processing of the measured input-output data. To see
that (9) is a speciaI autoregressive moving average (ARMA) model, define
a delay operator q t, applied to a variable z(i), to be
q - *z(i) ~ z(i - 1).
(10)
64
JOTA: VOL. 79, NO. 1, OCTOBER 1993
The system input output relation can be written in the usual deterministic
ARMA model format as
A(q
1) y ( i ) = B ( q - l )
(lla)
u(i),
with the polynomials of the delay operators A(q -1), B ( q - l ) given as
A(q-1)=I+CMq-I+C~Mq-
2+ ... + C A P - IMq -p,
(llb)
... + C A P - I B ' q -p,
(llc)
B(q-1)=D+CB'q-I+CAB'q-2+
where
A=A+MC,
B' = B + M D .
(1 ld)
Furthermore, in the special case where the matrix M is such that
A = A + M C is deadbeat (i.e., . ~ k - 0 , k>~ p), (9) holds exactly, In such a
case, all the eigenvalues of A are at the origin in the complex plane, i.e.,
they are zero. Such a matrix M always exists provided that the pair (A, C)
is observable. This case will be revisited in Section 5 in relation to the
identification problem.
3. Relation of the System to an Observer Model
The role of the matrix M in the above development can be interpreted
in terms of an observer model. Consider the system (1). It has an observer
of the form
2(i + 1) = A2( i) + Bu(i) - M [ y ( i) - ~(i)],
(12a)
f~(i) = C2(i) + Du(i).
(12b)
It can be shown that, from (12) and (1),
)~(i + 1) = A2(i) + Bu(i) -- M [ y ( i ) - C2(i) - Du(i) ]
= (A + M C ) 2(i) + ( B + M D ) u ( i ) - My(i).
(13)
Defining the state estimation error e(i) = x(i) - 2(0, we obtain the equation
that governs e(i),
e(i + 1 ) = A x ( i ) + Bu(i) -- [(A + M C ) 2(i) + (B + M D ) u(i) - M y ( i ) ]
= (A + M C ) e(i).
(14)
JOTA: VOL. 79, NO. 1, OCTOBER 1993
65
From (14), if M is chosen such that A + M C is asymptotically stable, then
limi~ ~ e(i)= 0; i.e., the estimated state 2(i) tends to the true state x(i) as
i tends to infinity. Equation (13) then becomes
x(i + 1) = (A + M C ) x(i) + (B + M D ) u(i) - My(i),
(15)
which is exactly the same as (5).
From this analysis, the matrix M can be interpreted as an observer
gain. The parameters ~ _ ~_, = C.4 i *- ~/7 in (Sb) are the Markov parameters of an observer system, hence they are referred to as observer Markov
parameters. In the identification process, these are the parameters to be
identified. Once they are determined, the actual system Markov parameters
can be recovered. There is an algebraic relationship between the Markov
parameters of the observer system and those of the actual system. This
result is established in the following section.
4. Relationship between the System Markov Parameters and the Observer
Markov Parameters
As before, let the Markov parameters of the observer system be
denoted by I~ and the Markov parameters of the actual system by Y~.
Recall that
Y~ = CA~B
= [C(A + MC)~(B + MD), - C(A + M C ) ~ M ]
- [17} I>, I?}2)].
(16)
From the second equation in (16), the Markov parameter CB of the system
is simply
Yo = CB = C(B + M D ) - ( C l ~ ) D
= YoO) + Yo(2)D.
(17)
To obtain the Markov parameter CAB, first consider the product 17~~),
I~ ~)= C(A + M C ) ( B + M D )
= CAB + C M C B + C(A + M C ) MD.
Hence,
YI = C A B
= ~'+
~o(~)ro + ~#)D.
(18)
66
JOTA: VOL. 79, NO. 1, OCTOBER 1993
Similarly, to obtain the Markov parameter CA2B, consider the product F2(1),
Y (2 1 )- -- C(A + M C ) 2 ( B + M D )
= C(A 2 .-1-M C A + A M C + M C M C ) ( B + M D )
= CAZB + C M C A B + C(A + M C ) M C B
+ C(A + M C ) 2 MD.
Therefore,
Y2 = CA 2B
= Y(21)- C M C A B - C(A + M C ) M C B - C(A + M C ) 2 M D
= :F2(') + Yo(2)Y1+ Y~2)Y0 + Y2(2)D.
(19)
By induction, the general relationship between the actual system Markov
parameters and the observer Markov parameters can be shown to be
"c--I
Y~=Y~I)+ ~, F}2)Y~ i - l + Y--(2)
~ D.
(20)
i=0
For a finite-dimensional system, knowledge of a sufficient number of actual
system Markov parameters is adequate to deduce a state-space realization
of the system of interest. Physical aspects of the model such as natural
frequencies, damping ratios, and mode shapes can then be found.
5. System Identification via an Asymptotically Stable Observer
Consider the multivariable system in (1). The input-output relation in
terms of the Markov parameters of an observer system is given in (9),
which can be rewritten as
p--1
Y(i)= ~ ( C A ~ B ' ) u ( i - ~ - I )
p--1
-
~
(CA~M)y(i-T
- 1)+Du(i),
(21a)
"c=0
where
Y~ = CA~B = [CA~B ' - CALM],
B' = B + M D .
(21b)
An algorithm that computes the coefficients of the ARMA model and
at the same time places the eigenvalues of A at prescribed locations can be
JOTA: VOL. 79, NO. 1, OCTOBER 1993
67
derived. These eigenvalues may be real, complex, or a combination of both.
The eigenvalues of .~ may also be placed at the origin, which corresponds
to a deadbeat observer. Each of these cases is considered in the following
sections.
5.1. Real Eigenvalne Assignment. Let the prescribed eigenvalues of
A = T - I A T be denoted by 2 , i = t, 2 , . . . , n, with
A = diag(21, 22 . . . . . 2,).
(22)
Then the products CTUB', C A l M become
CA'B '= CT 1A~TB',
CALM= CT-IATTM.
If the elements of C* - C T - 1, B* =--TB', M * ==T M are written explicitly as
/m*7
~T
C * = [c~), c~2),...,c~'J,
B*--
b(*~. ,
hi*--
.(2) ,
Lb '. J
(23)
Lm . j
*T
~T
where c~) denotes the ith column of the matrix C* and b(o,
m(i),
i = 1, 2 , . . . , n, denote the ith row of the matrices B*, M*, respectively,
then the products in (21b) may be expressed as
r ) (,) 7
~l,m
I
,](~)
C ~ B ' ---- Lrp*
~ , r ~~.' ( 2 ) u~.,r
L.(1)c,(t),
(2),
• . . ,
A*r-11"-"~2'm
I
J
•
/
/
c*
(n)C'(n)
,
[.- ~ n , m , J
F',I(~c) 7
/'-°l,q
/~(~) l/
t'(l)"~(1),
--L'(2)'"(2)~
" "',
--~'(n)
(n) J
/
/
/~(~)/
k_ "-~n, q ._l
where ~,,~
(~) and 21,~ are m x m and q x q diagonal matrices of the eigenvalue
2~ repeated m and q times, respectively, i.e.,
!~) = diag(2~, 2~,
"--~t,m
_2!')=
diag(2~., 2;,
t, q
•
"
•
•
" "
,
)o~). . . . .
~)qxq"
(24a)
(24b)
68
JOTA: VOL 79, NO. 1, OCTOBER 1993
With the following simplifying definitions:
_~ =
[-~*
jq*T
I_W(l)C'(l),
,.,* / . * T
t~(2)v(2),
[ _ .v,( 1 ) H~,T
/~=
"(1)~'
,1~)
-
=
.., ,.,,r
--~'(2)'r~'(2)~.
rL ~t ~.l ,)r n , ;L~2,
(~)m ~
(25a)
.~.* / ~ * T 1
' • ' , "(n)U(n)A,
- . . :,
--c*> ,.( ) a
(25b)
(25c)
~.(') 1 r,
• • • ~ :.-n, Ftl.a
(25d)
Eq. (21a) becomes
p-1
p--I
y(i)=g ~ _2~)u(i-r-1)+_fl ~ ~(q~)y(i-r--1)+Du(i),
(26a)
y(i) =TF(/- 1),
(26b)
or
where u(i), y(i) are m x 1, q x 1 input and output vectors, respectively. The
above equation is in a linear form with the unknown observer parameters
in the matrices _~, fl, D with
F_~(i-i) l
7 = [g, fl, D],
_F(i- 1)=/q~(i- 1)/,
L- u(i) J
(27)
where
p
1
~(i-- 1)= ~ )_.~)u(i--z-- 1)=~_mU_(i--p),
(28a)
z=O
p--1
~(i-- 1) = ~ ~.~)y(i-- r-- 1) ~q)_,(i- p),
=
(28b)
and
3.=
r~(p-u
(o)
~(x)
/1, m , ),,, ]
_~(mP - 2 )
~ i ~ { I),> "-.'l(p-2)2,
,m)'(P2 ...... _)(1))(1)l,m I,,,
mimxm]
×
~
- 2, m
2(d-i)
k..-- n , m
- 2, m
(29)
"-'2,m
,~(p-2>
- n,m
* " "
;o)
t-~n,m
I....
Similarly,
_-3q
=
[g~"-'>,_q
~(.-2)
.....
.-~q. ) , ~ ) ] ,
(30)
JOTA: VOL. 79, NO. 1, OCTOBER 1993
69
which has the same general structure as ~.~, except that the block matrices
are of dimensions q x q. The mp x 1 input history vector u ( i - p) and the
qp x 1 output history vector _y(i- p) are defined as
_y(i-P)=ly(i_2)[.
p)-- [u(i_ 2)|,
L u ( i - 1)3
(3e)
Ly(i- t)l
The unknown observer parameter matrix y in (26b) can be solved from a
set of input-output data of sufficient length l by a batch-type solution as
y=y_F +,
(32)
where the superscript + denotes the pseudo-inverse, and
y = [y(p), y(p+ 1) . . . . . y(p+I)],
(33a)
12 = [ F ( p - 1), [ ( p ) . . . . . [ ( p + l - 1)],
(33b)
Alternatively, for on-line implementation, the solution to the observer
parameter matrix y may be obtained recursively (Refs. 9 and 10),
~(i) =_~(i- t ) + 6_~(i- 1),
(34a)
where
l+F_(i_l)rgt(i_2)F(i_l )
A_,(i-I)=\,3
j,
(34b)
9t(i - 2) _F(i- 1 ) F ( i - 1 )r ~II(i - 2)
9 t ( i - 1) = 9 t ( i - 2 ) -
l+F_(i_l)r91(i_2)F(i_l)
(34c)
The observer Markov parameters f~, r = 0, 1, 2 , . . . , can be reconstructed
according to
f~ = CA'B = C3"[B ', - M ]
Finally, the actual system Markov parameters can be computed from the
reconstructed observer Markov parameters according to (20) as
z
r,=
1
S
i=O
_
(z)
~: -
1
--~-gm +fl(
),(~'rz - - t - - 1
_ ~ ~g"
a
"-~q
\i=O
,~
+Z~q')D]
-
(36)
70
JOTA: VOL. 79, NO. 1, O C T O B E R
1993
5.2. Complex Eigenvalue Assignment. The complex eigenvalue
assignment for the multiple-input multiple-output case can be derived by
setting A = T-~A~ T, where A~ is given as
A~=diag
([ ---~'c19 ~;] ' [ --02
a2 e)2]
~ a,,12 c°,,12l~.
(37)
cr2 ' ' " ' L - ° 9 . / 2
a.l~31
The prescribed complex-conjugate pairs of eigenvalues are denoted
2i=ai+j~i,
i=1,2 .....
n/2.
Using the same notation for vectors formed by the columns and rows of C*
and B*, respectively, we can express the products in (21b),
CA~B'= C*A~B*
:Ul
~(z)dr,* 1.t*T2_r.* h*T'~..Lr.~(z)[,~*
l~*T
~,* l ~ * T ]
\t~(1)v(1)~(2)~'(2)~'1
~.~(1)~'(2)--t~(2)v(1) !
~ (z)(r.,
/~,T
u 2 t'--(3)u(3)
+
,
,T
+ c(4)b(4) ) +
tr(r)(~,*
h* T
-L
21- t " n l 2 t t ' ( n - - 1 ) ~ ( .
1)"
o9(C)(c~3)b~4"~' -
C~4)b~3~)
r.~(z')[C*
*T--r~*
h *T
c;)bL~) +'~',~l=t
(..... ~)b(.)
"(,,W(.-1))
(,)
=_~_2 ....
Cfi~M= C*A;M*
~(z)(~*
r.a*T~_t.*
v~*T]~_o~('c)it.@
m*T
r.* rta*T~t
~-- ~ 1 t t ~ ( 1 ) ' e ' ( 1 ) t L . ( 2 ) " , ( 2 ) I ~ ~ 1 t , t ~ ( 1 ) ' " ( 2 ) - - ~ ( 2 ) H ' ( 1 ) ?
~(z)[~*
rv~*T~_ ~* ~.Tx,±
~.~(z)/r, , v v ~ * T
"~ t~' 2 ~.t'(3)"~(3) - - t ' ( 4 ) H t ( 4 ) I ~ v J 2 \ t ' ( 3 ) " t (4)
+ ~(z)/.~*
~t~*T
,
*T
/~*T
/,*
.~* v ~ * T ' ~
t'(4)"~(3) J
r,~ (r) 1 . , *
na*T
i)'"(n--1) + C ( n ) m ( n ) ) + ~ , , I z t ~ ( n - l ) ' ' ' ( n )
Unl2t~(n
~*
v~*T
--<'(n)'"(n
1))
where
_~Xc =
*
*T
/~*T
,**
~=1,:
/~:gT
(n-- l ) U ( n )
.....
J~c
=
-
~,
--
(38a)
C~n)b~nT--1)3,
*
*T
~,
~,T
o* ~*T
E c ( 1 ) m ( 1 ) 71- t~(2)~'J"~(2) , t . ( 1 ) H , ( 2 )
...
,
C~gn
(.) - r a " )
*T
1)b(m
,.,.)
~* taa*T
t-(2),,,(1) , • • •
(38b)
~* wa*T
- ~(n)'"(n-l)l,
,~.)
-~ c,m - - L-'~I , m~ ~- l,m~ ~ 2 , m t
2(z) :
c,q
~*T
[ c ( 1 ) b ( 1 ) + t ' ( 2 ) c ' ( 2 ) ~ t~(1)c'(2) - - t.(2)c,(1)~ • • •
a(,)
r,~ ( z )
-I T,
(38c)
" " " ~ ~- n/2,m ~ ""Jn/2,m3
(38d)
r.(~)
r,.(z) . ( z )
.(z)
rn(~r) -'IT
L~-" l,q~ ~- l.q~ ~- 2,q~ " " " ~ ~- n/2.q~ ~- n / 2 , q d
(0)
a-i,m=Imxm,
0)(.0)
=t,m :0
....
--(0) __
°-i,q - - I q x q ,
o9
(°) = 0 q x q "
- i,q
(38e)
JOTA: VOL 79, NO. 1, OCTOBER t993
71
The matrices q~,(~),,,and m!
~=,,., are m x m diagonal matrices formed by the
elements G}~) and col~) repeated m times respectively, i.e.,
~) = diag(al °, ~r(~)
~i
~ " " • ~ cr~)~
--i
Jmxm~
(39a)
~-i,m
m-i,m
~i
.....
°i
(39b)
). . . . . .
where al ~ and col") are the elements associated with the complex eigenvatue
pair
i = 1, 2 . . . . .
Z i = a i ~ j(Di,
n/2,
defined as
ol')J •
(40)
Similar definitions apply for q},~ and _col~) simply by replacing m by q.
Equation (21a) becomes
p--1
p--,1
-~c, m U ( 1 - -
"C - - 1 )
-t- _~ "
~=0
E
,:~_(~>) , q y ( i - ~:-- 1 ) + Du(i), (41)
~=0
or simply,
y ( i ) =7,:F_~(i- 1),
(42)
where
[-~,(~-
1)]
Fc(i-- 1) = / ~ ( i - - 1)
y c = E_~c,~ , D],
,
(43a)
u_(i--p),
(43b)
~ ( i - - t ) = ~ g(~q>y(i--~-- 1)=~_~,qy(i--p).
(43c)
L u(i)
=
2~,mU(t
~--
~:=0
p--1
r=O
The vectors u ( i - p ) , _ y ( i - p ) are defined in (31), and
-- ,
~--c,q
-
= Eg~,p-" 1)
,
~ (p--2)
"--'c,q
• • " ~ "-~c,m~ - e , m J ~
,''
,~(1) ~(0)
• ' "-'c,q' /-~c,q]'
(44a)
(44b)
72
JOTA:
VOL
79, N O .
t, O C T O B E R
1993
Expressed in terms of the real and imaginary parts of the prescribed eigenvalues, the matrix ~--e,mis
- - _o'~Pm 1)
0.(p -- 2)
......
0,,.{
1)
- 1,m
Im
co(P - 1)
1,m
o-(p- D
(D
P - 2)
- (1,m
......
(D(
1)
- 1,m
Omx m
( p - - 2)
-o -2,m
......
,.r(1)
-~
2,rn
Imx m
_o4f
(p--2)
(-D2,m
......
~.,~
- ~ 2(1)
,m
Omxm
q(p-2)
n/2,m
......
,.r(1)
~n/2,m
Imxm
O_.)(p 1)
n/2, m
......
~-n/2,m
r,~(1)
Om×m
-
-%_
=
-
2,m
"
r,~(P-- I)
~- n / 2 , m
1,m
x m
--
(45)
The structure for ~c,q is similar to (45). A solution to (41) can be similarly
obtained by replacing _7 and _F in (32) by y~ and _F~, respectively. The same
solution can be obtained recursively by replacing ~ and _F(i- 1 ) in (34) by
_~ and F ~ ( i - 1 ) , respectively. The observer Markov parameters and the
actual Markov parameters can then be computed as
Y, = CA~B = C 2 " [ B ', - M ]
L~--'C'-YC, m ~ ~'c=.'c,q J
"r
(46)
L--T
I
• ,- 1
*~
Y~ D
i=0
= -~c'-'c,m
~ ,t(~) + _fl~
,a(~) ~gz
"-~c,q
t
i
1
+ Z (~)
(47)
"
0
5.3. Mixed Real and Complex Eigenvalue Assignment. Among n prescribed eigenvalues, let n~ denote the number of prescribed real eigenvalues
2~, i = I, 2 , . . . , nr, and n, the number of prescribed complex eigenvalues
a~ + jo~, i= 1, 2 . . . . . n~/2. Then, write
~= T-lAsT,
where
Am = diag(2~. 22, - - •, 2.~, [
--(D 1
--0
20"2J
k--OOne/2
ffnc/2.J,/
(48)
and define
= D,
=
(49a)
_LL
~ (~ )r-I T
-'re, m--
k ~m
~
~'c,m-I
~
-tn, q
(49b)
JOTA: VOL, 79, NO. 1, OCTOBER 1993
73
where
r~,* i,,7" ct:) b (,2T) ,
~=
kt.(1)t.,(1)~
/7=
_ r ~k t*~ ( 1 ) "~t , r( I )
-
, ~ ,(r2 ) ,
.,,
, t'(2)'"
• . . ,
h*r
.
~ c = k t Fgc*
(nr+l)t..(nr+1)-l-C~nr+2)
tic= _ FL \/ c .( n r + l ) 'm" (~nTr + l )
(50a)
~*
t'(nr) *,*'-1
t/(nr)3,
• . .
c *(nr)~'t(nr)A~
,~*rl
(50b)
b ,( nrr + 2 ) J , , -..,(C*n
1 ) b (, nr ) - - t . (,~,
n ) c ,,(, n, r_ l ) ) ] ,
-t- C*( n r + 2 ) tH (¢~T
n r + 2 ) ) ' ' "' '
-
[,,*
~8¢ r,ta~cT
. . . . t~(.
t) m , T(.)-~(~),,,(,,-,))],
).~)
-
= r J(,)
(*)2(,)
k'--'l,m'
r~(*)
rer(o
}(z) .~ per(r)
"-"c,q
l-~" l , q '
;(*)
,;(=)
,,,(,)
(50e)
(5Of)
l r
,~(~) ,.,(~)
rn(z)
er0:)
r,~O:)
-'-'-'-'~1 , q ' -'x"2, q ' -'~2, q . . . .
q{O)
t,m =Imxm,
(50d)
7"
] ,
- (o
*---~2,,rt' " " " ' -J~. . . .
~(~)
(50C)
m{
°) = 0 . . . .
.~- t,m
er(~)
'
co<~)
qr,
(50g)
er(z)
r,~('r)
]
~-nct2, q' ~-'nc/Z, qa T
qi,(°)--I
q -- qxq~
(50h)
(0) =
~- i,q
0q×q.
(50i)
Equation (21a) may be expressed as
y(i)
=2'mFm(i- 1),
(51)
where
ym=[gm,_~.~,D],
_Fro(i-I)=
[
1) t
g.,(i-1) ,
~_m(i--
u(i)
1)= ~ _ 2 ~ , ) m u ( i - - r - 1)= ~m, mU_(i--p) ,
~m(i--
(52)
(53a)
r=O
p
_q%(i-- 1) =
I
2"m,
(~)q Y ( l"-- Z - - 1)=~,,,,qy(i--p)._
Z
(53b)
The matrix ~ .... includes elements formed from both real and complex
prescribed' eigenvalues,
~,.m
=
m
(54a)
L~<,mJ
where
f
~(p
1)
2(p-l)
-2m
. . .2(p"
. . . 1)
fl(p-2)
. . .
)(1)
]mxm--
2(p-2) . . . -2(
1)
2,m
Im× m
_ ,
)~5,~22)
"'"
Im×~,-
-2,m
"-'n~,m
)(*)
(54b)
JOTA: VOL. 79, NO. 1, OCTOBER 1993
74
(p-l)
- - _0"1, m
69(p- - 1,vet
2)
......
0.)(
p
1,m
2)
......
-
t)
1,m
(p--2)
-O'2,m
if(p-- I)
- 2,m
~(p-1)
2'~--c,rn ~
o.(p
(.0(
p
2, m
- 2,m
t,~(P-~ 1)
~- n c / 2 , m
t.r(i )
Im
~ l , m
.(1)
--
xm
Omxm
-'~" 1 , m
......
t'r (1)
-'Z 2,rn
[rn
2) . . . . . .
(.0
(1)
- 2,m
Omxm
o-(p-2)
nc/2,m
. . . . . .
~ nc/2, m
rr(1)
~(p-1)
nc/2,m
. . . . . .
~- n c / 2 , m
'(~)
xm
Imx
(54c)
m
Om×m
The diagonal matrices ".'-~i,m,~(z)
~-i,m,rr('c)~-i,m(Z)in ~m_,m are of dimensions m x m,
i = 1, 2 . . . . . n~, or n~/2, and ~ = 1, 2 , . . . , p - 1. Similar structures apply for
~m,q which is c o m p o s e d of the matrices =,;,q,~(~)=e,q,
' ~ ) O~,q(~)of dimensions q x q
instead. Again, (51) is in linear form, and the u n k n o w n parameter matrix
?,~ can be solved for from i n p u t - o u t p u t data. The observer M a r k o v
parameters and the actual system Markov parameters are then computed
as
p=r~,
2(~)
A' . ~ ( z ) q _ l - ~ ( 1 )
F(2)l
(55)
l ~ 2(~
(~)
y~=~= ' m z~(n
- m , q Y " : - - t•- 1 +)_,~,qD
~
: m , m 4-tim
-- -
(56)
"
i=0
5.4. Deadbeat Eigenvalue Assignment. In the deadbeat case, all eigenvalues of the observer are placed at the origin. The corresponding, Markov
parameters vanish identically after a finite number of terms, i,e., ]2~ -= 0 for
r = p, p + 1, p + 2 , . . . . The input-output description is directly given in
terms of observer Markov parameters as
p--I
y(i)= ~
p
1
~ l ) u ( i - ~ - l ) - ~ Y~2)y(i-~-l)+Du(i)
~=0
"c=O
= yaF_a(i- 1),
(57)
where
P"-'l
F_a(i-1 ) = lY(i--p)[.
ya = [gJ, _fla, D],
~a= []~oo), ~(I)
11 '
L- u(i)
...
'
V(')
" p - - l Jl'
~d
= [ ~(2)
(58)
.j
1~'7(2)
1 ~
...
yp(2_)1]. (59)
From (57), the observer Markov parameters can be solved for directly
from input-output data by replacing _7 and _F in (32) by yd and _Fa or _~and
_F(i-1) in (34) by _?a and Fa(i-1), respectively. The actual system
Markov parameters are then recovered by (20).
JOTA: VOL. 79, NO. 1, OCTOBER 1993
75
A remarkable feature of the deadbeat solution is that, to recover
additional system Markov parameters from the identified observer Markov
parameters, one simply invokes the deadbeat condition by setting the extra
observer Markov parameters to zero and proceeds with the calculation in
(20) to recover as many system Markov parameters as desired. Another
way to view this result is that the infinite (but not all independent) number
of system Markov parameters has been compressed into a finite number of
corresponding deadbeat observer Markov parameters. Once these observer
Markov parameters are found, one can actually perform the reverse
process to recover all the original system Markov parameters.
6. State Space Realization by ERA
A state space model of the system from the recovered Markov
parameters can be obtained by the eigensystem realization algorithm,
which is outlined in this section. The algorithm begins with an r x s block
data matrix called the Hankel matrix and denoted by H(z),
H(z)=
Y~
Y~+I " ' "
Y.c+s- [
"]
Y~+'
Y~+2 " ""
Y~+,
1.
g~+~ I
Y~+~ "'"
Y~+/+~-~.J
(60)
The order of the system is determined from the singular value decomposition of H(0),
H(0) :
u ~ v ~ = ulsl v~r,
(61)
where the columns of U1 and V1 correspond to the positive singular values
in $1. The matrix $1 is an n x n diagonal matrix of the positive singular
values in Z. Defining a q x rq matrix EqT and an rn x sm matrix Emr made up
of identity and null matrices of the form
Erq=[Iq×q, Oq×<r_~)q],
E mr =
[I~× m Om×,,s-t>m],
(62)
a discrete-time minimal order realization of the system can be shown to be
Ar = S;1/2U~H(1 ) VI S i- 1,,2
(63a)
Br = S ? 1/2VT(Em,
(63b)
C r = r*~q'~,~'l
T l c ' [ K*--l/2 -
(63c)
This is the basic ERA formation. To use ERA in the present identification
procedure, the entries that make up the data matrix given in (60) are
76
JOTA: VOL. 79, NO. 1, OCTOBER 1993
precisely the recovered system Markov parameters Y~, ~=0, 1, 2. . . . .
which are computed from the identified observer Markov parameters
described in previous sections.
An important aspect in connection with system realization from
Markov parameters is the order of the minimal realization. In the noisefree case, given a sufficient number of Markov parameters, the minimal
order n of the system is equal to the rank of the Hankel matrix H(0), which
has n positive singular values in E, and the remaining singular values are
zero. Therefore, the second equality in (61) holds exactly. In practice, if the
data contain noises, the singular-value decomposition step in the above
realization procedure is normally used to determine this order. Certain
smaller singular values are attributed to noises and truncated. The number
of retained singular values then determines the system order. The matrices
U1 and V~ made up of the corresponding retained columns of U and V in
the singular value decomposition of H(0) are used in (63) to obtain a
realization.
7. Computation Steps
This section reviews the basic steps involved to implement the
identification procedure developed in this paper. The related equations are
identified in each step of the process.
Step 1. Assume an order for the system to be identified, denoted by
n. Choose an order for the ARMA model, denoted by p, and select the
prescribed eigenvalues of the observer. For the real and complex eigenvalue
assignment procedures, p is normally several times larger than the assumed
order of the system n. Specifically, the value of p chosen must be consistent
with the prescribed eigenvalues for the observer as described in the
following:
(a)
for real eigenvalues, select n real eigenvalues 2i, i = 1, 2 , . . . , n,
such that ~P ~ 0;
(b)
for complex eigenvalues,
complex-conjugate pairs,
~i ~ (~i ~ j(A) i,
such that
I ffi
(A)i~P~o;
--(.l) i (7iA
the eigenvalues must appear in
JOTA: VOL. 79, NO. l, OCTOBER 1993
77
(c)
for a combination of real and complex eigenvalues, the same
rules apply;
(d) for deadbeat observers, however, all eigenvalues are set to be
zero; no explicit specification of the eigenvalues for this case is
necessary; only a selection of the order p of the ARMA model is
required.
For asymptotic stability, all prescribed real or complex eigenvalues
must have magnitudes less than one.
Step 2. Compute the observer parameters. The corresponding equations used for each case are outlined as follows. For observers with
assigned real eigenvalues, (34) is used. For observers with complexeigenvalues, the recursive equations are obtained by replacing _~ by _~c and
F ( i - 1 ) by F_c(i-1) in (34). For observers with mixed real and complex
eigenvalues, replace ~(i) by _'~m(i) and F ( i - 1 ) by Fro(i-!). For deadbeat
observers, replace ~(i) by 27a(i) and _F(i-1) by Fa(i-1). For off-line
computation, (32) is used for observers with assigned real eigenvalues.
Analogous replacements for the entries in (33b) for other cases are obvious.
Step 3. Reconstruct the observer Markov parameters from the
identified observer parameters. For observers with real eigenvalues, (35) is
used. For observers with complex eigenvalues, (46) is used. Similarly, for
observers with both real and complex eigenvalues, (55) is used. For deadbeat observers, the identified parameters are precisely the observer Markov
parameters, and no reconstruction of the observer Markov parameters is
needed for this case.
Step 4. Recover the system Markov parameters from the observer
Markov parameters. The general equation is given in (20), which is then
specialized to various cases. For observers with real eigenvatues, (36) is
used. For observers with complex eigenvatues, (47) is used. For observers
with both real and complex eigenvalues, (56) is used. For deadbeat
observers, (20) directly applies.
Step 5. Realize a state space model for the identified system from the
recovered system Markov parameters in Step 4 above. The basic equations
for ERA are summarized in (60) (63).
8. Numerical Examples
The theoretical development sections present a formulation that uses
observers and eigenvalue placement to recover the system Markov
78
JOTA: VOL. 79, NO. -1, OCTOBER 1993
parameters which are the pulse response samples of a linear system. The
fundamental idea in the developed identification procedure is to identify
parameters of an observer rather than those of the actual system. From the
observer parameters the true system parameters can be recovered. The
observer eigenvalues or poles determine the observer pulse response (or
observer Markov parameters) decay rate. By making the pulse response of
the observer system decay sufficiently fast through the placement of its
poles, one can truncate the pulse response after a finite number of time
steps. In fact, for the deadbeat observer, its pulse response vanishes identically after a finite number of time steps. This is in contrast with the pulse
response of the system where one has no control over its decay rate. The
following examples are provided to illustrate certain key features of the
identification procedure developed in this paper for the complex and deadbeat cases. For a complete study, the readers are referred to Ref. t 1.
A model obtained by finite-element analysis of the Mini-mast truss
structure is used as an example system (Ref. 12). The 10th order mathematical model consists of the first two bending modes with practically the
same frequencies, the first torsional mode, and the second two bending
modes, again with practically the same frequencies. The model has two
inputs and two outputs. The inputs are two torque wheels for the x and y
axes, and the outputs are two displacement sensors mounted at the top the
structure as shown in Fig. 1. The system frequencies and the associated
damping factors, expressed as the real parts of the eigenvalues, are listed in
Table 1. The input-output data is simulated using random inputs for
6 sec. The system is discretized at a sampling rate of 33.3 Hz, and an
input-output history of 200 points is recorded for system identification.
The analytical model contains five modes, but practically only three of
them are controllable and observable from any given input-output pair.
The mathematical model is given in the Appendix.
Consider the case where all prescribed observer poles are complex and
the order of the system is underspecified in the algorithm. The complex
Table 1. Damping and frequencies of the truss
structures.
Mode
number
Damping
(%)
Frequency
(Hz)
1
2
3
4
5
0.090
0.091
0.329
0.383
0.387
0.800
0.801
4.364
6.104
6.157
JOTA: VOL. 79, NO. 1, OCTOBER 1993
79
poles are evenly distributed within the unit circle with a radius r = 0.5,
the initially assumed order is 4, and the window width p is set to 40. The
recursive versions of the identification equations are used in the following
examples. The initial observer parameter estimates are assumed to be zero.
To start the algorithm, the projection matrix 2 ( - 1) is set to a large value
to reflect the degree of uncertainty of the initial guess. The top row of Fig. 2
shows the parameter convergence histories and the variance distribution
for the recursive least-squares solution, which are the diagonal elements of
the projection matrix 9~(i) at the end of 160 time steps. The parameters
seem to have reached constant values, but some variationsare still observed.
At any time step, the prediction error is defined to be the difference
TIP PLATE
Bay 18
\,\
/i
\\
((
TWAy
X ~
//
\\
Y
20.2 M
/7
_M
/
/g
DISPLACEMENTSENSORS
Bayst0, 14, 18
D10B
D14B
B D 1 8 ~B
C/D10C
\
/D14C
\
/
o1,o
D14A "
D18A
Fig. 1. Mini-mast truss structure showing x and y torque wheel inputs TWAx, TWAy, and
displacement outputs D18A, DI8B on bay 18 tip plate.
80
JOTA: VOL 79, NO. 1, OCTOBER 1993
100
I
0.S
50
>"
~D
0
"~
100
c~
i
0.4
>" 0.2
150
0
50
190
t50
0
0
200
5
Time steps
10
15
20
Parameter number
40
20
0.8 - ~
0.6
-2o
~0.4
-40
0.2
-60
i
i
i
50
100
150
0
0
.........
200
50
Time steps
Singular value number
2x10 "6
C~
~'~ -100
x
o
:
.
,,
,
2
"~ -15o
-1
-20
.............
i
i
2
4
....
6
10-z
10-1
100
l0 t
102
Frequency (Hz)
Time (sec)
4 x10-4 . . . . . .
200
~,T
i~TirUln
i
i ~llul,
•
~ ~l,llql
I"~'N'TI'I
"~ 100
i
0
~'
-2
-40
',
",
~-~
i~, '
0
-100
4
Time (sec)
6
-2°°o-,
,oo
,o,
,o,
Frequency (Hz)
Fig. 2. Identification of a 10th order system with complex eigenvalue assignment: n =4,
p=40, r = 0.5.
JOTA: VOL. 79, NO. 1, OCTOBER I993
81
20OO
40
20
. ~ 1000
~
>
-20
-40
0
50
109
150
50o
0
00
I0
0
30
20
Parameter number
Time steps
40
0.8
20
0.6
-20
-40
-60
0.2
0
50
100
150
O
0
200
2 xlo~
E
50
Singular value number
Time steps
-50
i
~
, , i,.,.,,
., . ' r r - m m - - ' v - v w ' r v ~
,~ -100
o
-150
-2
0
2
4
000-2
6
4 xlO~
N
2
~
0
102
~o I00
-100
t0-2
0
Time (see)
Fig. 3.
101
-8
~
.4
100
Frequency (Hz)
Time (sec)
~-~
lift
1@1
10o
101
102
Frequency (Hz)
Identification of a 10th order system with complex eigenvalue assignment: n = 6,
p = 40, r = 0.5.
82
JOTA: VOL. 79, NO. 1, OCTOBER 1993
between the true output value and the predicted output based on the
estimated model available at that time step. As expected, because of order
underspecification, the lack of freedom in the identified parameters
prevents the prediction error from converging to zero as shown in the
second row of Fig. 2, resulting in the fluctuation of the parameter estimates.
The identified observer parameters along with the prescribed poles are then
used to compute the observer Markov parameters, from which the system
Markov parameters are recovered. Shown in the second row of Fig. 2 are
the singular values obtained from the singular value decomposition step in
ERA. Counting the number of nonzero singular vaues, the identified system
order is found to be 8. In general, for a multiple-output system, the maximum order of the system that can be recovered is equal to the assumed
order times the number of outputs (Refs. 3, 6, or 11). The 8th order state
space model of the system obtained using (63) represents an approximation
of the original system of order I0 by a system of order 8. The bottom four
plots in Fig. 2 show results for the second output comparing the identified
state space model with the true system model. Results for the first output
is similar and not shown here. Included in this group are comparisons of
identified and actual pulse responses; actual displacement history used in
the identification and its reconstruction using the identified model; and
frequency response functions. There are two curves in each of the four plots
in this group; the solid curve corresponds to actual data and the dashed
curve to reconstruction. It can be seen that the identified system pulse
response, the reconstructed resPonse , and the frequency response functions
obtained with the identified reduced-order model only approximate the
actual responses.
Figure 3 shows the results obtained when the assumed order is
increased to 6. Since the system has two outputs, the maximum system order
that can be recovered is 12. This is a case where more than enough freedom
is allowed to identify the original system order of 10. The parameters
converge to constant values as shown in the top left plot in Fig. 3. Note
that, when the identified parameters are not all independent, the large
variances do not imply inaccuracies in the parameter estimates. This merely
means that, for the specified order, the identified set of observer parameters
is not unique. The identified observer parameters and the prescribed
observer poles are then used to reconstruct the observer Markov parameters from which the system Markov parameters are recovered. At the
realization step, it is found that the system is identified correctly in spite of
the redundancy in the set of estimated parameters. The prediction error
converges to zero and the system order is correctly identified to be 10.
Again, using (63), a 10th order state space model of the system is obtained.
The pulse response, the reconstructed response, and the frequency response
JOTA: VOL. 79, NO, 1, O C T O B E R 1993
2000
I0
83
¥
r
I0
20
8
5
©
(.1
1500
100o
0
o
>
-5
t
-I0
0
,,
50
i
~
100
150
500
0
.....
200
0
Time steps
30
Parameter number
20
0,8
o
0.6
0
"~-0 0.4
-10
0.2
-20
0
5
i
i
100
150
0
200
0
n
50
Singular value number
Time steps
2xlO -~
o -100
-150
,2
2
4
-200
10
6
4!xlO'~
i
g
104
10o
I01
t0 2
Frequency (Hz)
Time (sec)
200
I
100
~'~
e~
-100
-2:
-4
o
Fig. 4.
0
i
,~
Time (sec)
2
-200 ~
10-2
lO-t
......................... t
10o
lO t
102
Frequency (Hz)
Identification of a 10th order system with deadbeat eigenvalue assignment: p = 6,
84
JOTA: VOL. 79, NO. 1, OCTOBER 1993
functions computed from this model match those obtained from the actual
model exactly as shown in the bottom four plots of Fig. 3, where the two
curves in each plot representing the actual and reconstructed responses
overlap.
Similar results are obtained with the deadbeat algorithm as shown in
Fig. 4. The deadbeat algorithm is considerably simpler in that the observer
Markov parameters are identified directly from input-output data. These
examples show that overparametrization, if any, does not affect the final
results. The system can still be correctly identified as the algorithm returns
an identified model of minimal order at the realization step.
9. Concluding Remarks
This paper formulates an algorithm for the identification of linear multivariable systems from general input-output data. Data from either single
or multiple sets of experiments can be used. For each data set, the initial
condition may be arbitrary and need not be known. The procedure identifies the Markov parameters of an observer system instead of those of the
actual system. The actual system Markov parameters are recovered from
the observer Markov parameters, and then used to realize a minimal state
space model of the system. The embedded eigenvalue assignment procedure
is used to specify the observer asymptotically stable poles. The prescribed
poles may be real, complex, or mixed real and complex. When all the
prescribed poles are placed at the origin, this results in an identification
algorithm with a deadbeat observer. In each of these cases, the observer
Markov parameters are related to the input-output data by a linear relationship. Therefore, they can be easily solved for in one step for off-line
computation, or recursively for on-line computation. Identification procedures for multiple-input multiple-output systems are formulated, and
numerical examples are presented to illustrate the basic characteristics of
the developed method.
10. Appendix: Mini-Mast Truss Structure
The continuous-time system matrices are listed here. For ease of
presentation, the matrices are subdivided and given below,
A = [A1, A2],
where
J O T A : V O L . 79, N O .
"-
8.918 x 10 -2
- 1.330 x 10 4
1.303 x 10 - 4
-8.912x10
1 . 5 4 0 x 1 0 -4
5.035
- 4 . 1 0 0 x 10 -5
-3.238x
At=
10
-5.032
3
- 2 . 0 9 3 x 10
2.660 x 1 0 - 2
-1.015x10
1.020 x 1 0 - 1
- 2 . 6 2 7 x 10 -2
5 . 4 9 8 x 10 -3
2
-I,412x10
1
-2.110x
10 -~
1 . 0 1 0 x 10 -2
3.884x10
2
2
- 9 . 2 0 5 x 10 . 2
7.388 × t 0 -3
2.748 x 10 - 3
_2.6t7xi0-2
-t.999x
10 - 2
- - 6 . 0 6 2 x 10 - 3
-3.766x
- 1 . 5 4 0 x 10 2
- - 3 . 2 5 1 x 10
10 -2
1.075x10-2
_1.040x10-1
1.546 x 10 -3
- 2 . 7 4 8 x 10 - 2
3.791 x 10 -3
- 3 . 2 8 3 x 10 . 3
- 2 . 5 6 7 x I0 -2
1,491 x 10 -2 , ,
-9.892x
3.740 x 1 0 - 2 ~,,
10 -3
3 . 8 5 9 x 10 - 2
1.079 x l O - Z
--5.495 x 10 -3
2.055 x 10 - 2
_2.179x10-2
- - 6 . 5 1 7 x t 0 -3
- 9 . 4 4 7 x 10 -3
1.774 x 10 - 2
1.114 x 10 . 2
--2.519 x I 0 -2
- 3 . 3 3 0 x 10 -1
- 9 . 8 8 4 x 10 -3
2.238 x 10 -2
1 . 1 1 7 x 10 2
- 2 . 1 2 5 x 10 -2
- 3.763 x 10 - 1
5.972 x 10 - i
-- 1.320 x 10 - 2
- 2 . 8 3 4 x 10
-5.956x
3
2 . 3 4 5 x 10 -3
3
10
t
-38.364
2.657 x 10 3
- 2 . 1 0 1 x 10
-
1
10
--~ 2.364 x 10 - 3
-1.052 x 10-4
- 2 . 4 8 8 x 10 - 4
t.107 x 10 -4
2.455 x 10 - 4
1.667 x 10 3
9.519 x 10 `-4
D = ~0"000
-- 1.010
- - 4 . 6 5 6 x 10 -2
-3.912x10
-1
5 . 9 8 6 × 1 0 -1
x
10 - l
38.660
- 5 . 9 6 9 x 10 -1
-3.943 x lO-lu,
3
- - 2 . 0 1 5 x 10 -3
, , - 8 . 9 1 7 x 10 4
-38.660
38.364
1.996 x 1 0 - 3 , ,
-2.360x
1.999 x 10 -3
1 . 6 3 0 x 10 3
- - 3 . 7 9 0 x 10 - t
4.638x10 -a
1.011 x 10
- 2 . 3 4 9 x 10 . 3
- 9 . 0 9 5 x 10 - 4
1 . 5 5 4 x 10 3
9.180 x 1 0 - 4
1.509 x 10 -3 ,,,
0.000
kO.OOO0.oo0£
C = [-C D C 2 ] ,
where
C1=[
1.119x10
2
- 9 . 1 1 4 x 10 -3
C
1
27.420
_9.974x10-2
1.005 x 10 -1
- 1.403 x 10 -3
--27.420
2.839 x 10 -3
B=
4
1.335 x 10 -4
1.043 x 10 - I
9.106 x 10 -4
1.309 x 10 -2
-I.293x10
3.540 x 10 -3
_2.691x10-z
1.214 x 1 0 - 3
"
-2
-9.535x10
1.527 x 10
"-
-9.212x10
2.468 x 1 0 - 2
2
85
5.032
- 4 . 0 4 8 x 10 -3
- 2 . 5 1 4 x 10 - 2
1993
4.756 x !0 -5
- 7 . 5 9 6 x 10 . 3
m _ 1.549 x 10 -3
As=
3
.- 5.035
_1A74xt0-4
4.008 x 10 -3
--9,585 x 10 - 2
•" -
2
1, O C T O B E R
~-9.177x10
4.016x10 3
1.122x10 -2
7 . 6 2 0 x 10 -3 - 9 . 1 3 6 x
3 _4.321×10-4-2.448x10
2 = L _ 9 . 3 2 6 x 10 -3 - 2 . 4 2 7 x 10 '3
-4.025x10
10 3 - 7 . 6 3 9 x
1.965 x 10
-3
10 -3
3
4.669 x 10 --4
3
2.423 x 1 0 - 3
-9.167x10
-3]
--9.311 x 1 0 - 3 J '
2.393 x 1 0 - 3 1
_ 1.990 x t 0 - 3 j "
86
JOTA: VOL. 79, NO. 1, OCTOBER 1993
In the numerical examples, the above continuous-time model is discretized
at a sampling frequency of 33.3 Hz corresponding to a sampling interval of
0.03 sec.
References
1. HO, B. L., and KALMAN, R. E., Effective Construction of Linear State Variable
Models from Input~Output Data, Proceedings of the 3rd Annual Allerton
Conference on Circuits and Systems Theory, Monticello, Illinois, 1965.
2. JUANG, J. N., and PAPPA, R. S., An Eigensystem Realization Algorithm for
Modal Parameter Identification and Model Reduction, Journal of Guidance,
Control, and Dynamics, Vol. 8, No. 5, pp. 620--627, 1985.
3. PHAN, M., JUANG, J. N., and LONGMAN, R. W., On Markov Parameters in
System Identification, NASA TM-104156, 1991.
4. PHAN, M., JUANG, J. N., and LONGMAN,R. W., Identification of Linear Multi-
variable Systems by Identification of Observers with Assigned Real Eigenvalues,
Journal of the Astronautical Sciences, Vol. 40, No. 2, pp. 261-279, 1992.
5. CHEN, C. W., et al., Integrated System Identification and Modal State Estimation for Control of Large Flexible Space Structures, Journal of Guidance,
Control, and Dynamics, Vol. 15, No. 1, pp. 88-95, 1992.
6. JUANG, J. N., et al., Identification of Observer/Kalman Filter Markov
Parameters: Theory and Experiments, Journal of Guidance, Control, and
Dynamics, Vol. 16, No. 2, pp. 320-329, 1993.
7. PHAN, M., et al., Improvement of Observer/Katman Filter Identification by
Residual Whitening, Proceedings of the 8th VPI&SU Symposium on Dynamics
and Control of Large Structures, Edited by L. Meirovitch, Virginia Polytechnic
Institute and State University, Blacksburg, Virginia, 1991.
8. HORTA, L. G., et al., Frequency Weighted System Identification and LinearQuadratic Controller Design, [Proceedings of the AIAA Guidance, Navigation,
and Control Conference, New Orleans, Louisiana, 199 I]. Journal of Guidance,
Control, and Dynamics, Vol. 16, No. 2, pp. 330-336, 1993.
9. GOODWIN, G. C., and SIN, K. S., Adaptive Filtering, Prediction, and Control,
Prentice-Hall, Englewood Cliffs, New Jersey, 1984.
10. LJUNG, L., and SODERTROM,T., Theory and Practice of Reeursive Identification,
MIT Press, Cambridge, Massachusetts, 1983.
11. PHAN, M., et al., Identification of Linear Systems by an Asymptotically Stable
Observer, NASA TP-3164, 1992.
12. TANNER, S. E., et al., Mini-Mast CSI Testbed User's Guide, NASA TM-102630,
1992.