A NEW METHOD TO OBTAIN A UNIQUE SOLUTION
IN SYSTEM IDENTIFICATION
Atsushi Hirano and Masayoshi Misawa
Department of Mechanical Engineering
Shizuoka University
Johoku 3-5-1
Hamamatsu, Shizuoka 432-8561, Japan
φ̄
φi
ψ
ABSTRACT
This paper deals with the uniqueness of identified mass
and stiffness matrices in system identification. Using the
constrained minimization theory, these matrices are so
identified as to minimize the Euclidean norm of the matrices. It is shown that there are two problems and their
causes in the current identification methods. One problem is that current methods can never give a unique mass
matrix, and the other is that the identified mass matrix cannot provide mass properties of structures. This
paper proposes constraints considered in the mass matrix identification to solve these problems. It is shown
that the identified stiffness matrix is always unique if the
identified mass matrix is unique as well. The proposed
method is applied to a space deployable truss structure
to demonstrate the effectiveness of the method.
NOMENCLATURE
A
E
IX , IY , IZ
IXY , IY Z , IZX
K
KA
M
MA
M0
R
XG , Y G , ZG
f
m
nA
nT
∆M
Λ
Ω
φ
diagonal matrix with arbitrary
constant as its elements
unit matrix
moment of inertia of a structure
product of inertia of a structure
identified stiffness matrix
initial stiffness matrix
identified mass matrix
initial mass matrix
rigid-body matrix defined by Eq.
(24)
rigid-body mode matrix
center of gravity of a structure
natural frequency
weight of a structure
total number of analytical modes
number of measured modes
= M − MA
matrix with Lagrange multiplier
λij as its elements
measured frequency matrix
modal matrix
1
modal matrix defined by Eq. (8)
ith mode
Lagrange function
sum of squares of all elements of matrix
INTRODUCTION
Stiffness is one of the most important design requirements for space structures. Space structures in orbit have to meet specified frequency requirements to
avoid coupling with both their control systems and other
equipment. Structural designs to satisfy these requirements are relatively easy for small structures, however it
becomes difficult to meet the frequency requirements for
large structures. Natural frequencies decrease as the size
of a space structure increases, and they come close to the
specified requirements. Therefore, we need more accurate predictions to confirm the frequency margin during
the design phase. Finite element analysis is a powerful
tool to obtain dynamic characteristics for space structures. Finite-element models of those structures will
need a large number of degrees of freedom as they become larger. In such cases, it is very difficult to find
accurate solutions due to modeling errors of initial finiteelement models.
System identification is a key technology in predicting
the accurate dynamic behavior of structures. Several
types of methods[1∼6] have been proposed to establish
the analytical models mating with vibration test results.
One of these methods assumes an initial, undamped,
finite-element model which is characterized by its mass
and stiffness matrices[3∼6] . The mass and stiffness matrices are corrected by minimizing the Euclidean norm
subject to constraints. These constraints include mode
orthogonality, the symmetry of mass and stiffness matrices, and the connectivity of finite-elements as proposed
by Kabe[7] .
First of all, this paper describes some problems in the
current mass matrix identification. One problem is that
the identified mass matrix is not unique and the other
1445
is that we cannot obtain mass properties of real structures from the identified mass matrix. This means that
the mass distribution of the analytical model is different from that of the real structures. Next, the cause of
these problems is clarified. The cause is the use of mode
shapes in the constraints and because the mode shape
is not uniquely defined, any mode shape can be multiplied by a nonzero constant without changing its physical meaning. As a result, there exist an infinite number
of identified mass matrices that satisfy the constraints.
Next, a new method is proposed to solve these problems. Finally, a numerical example is given to demonstrate that the proposed method is effective for system
identification.
2
CURRENT METHOD
(1)
From Eq. (1), one may obtain
φT ∆M φ = E − P
(2)
where
P = φT MA φ
(3)
We can not obtain a unique identified mass matrix from
Eq. (2) because φ is not a square matrix. Therefore, the
mass matrix is identified by minimizing the Euclidean
norm M − MA subject to the mode orthogonality
conditions. Using the constrained minimization theory,
the Lagrange function ψ is written by
−1
−1
ψ = MA 2 ∆MMA 2 nA
nA κij M − M T ij
+
+
i=1 j=1
nT
nT i=1 j=1
λij φT ∆M φ − E + P ij
(4)
where κij and λij are Lagrange multipliers. Equation
(4) is differentiated with respect to each element of ∆M
and set to zero. We obtain
1
∆M = − MA φΛT φT MA
(5)
2
We can find Lagrange multipliers by substituting Eq. (5)
into Eq. (2) as
Λ = −2P −1 (E − P ) P −1
(6)
Substitution of Eq. (6) into Eq. (5) yields ∆M , and the
mass matrix is identified as
M = MA + MA φP −1 (E − P ) P −1 φT MA
PROBLEMS AND THEIR CAUSES IN
THE CURRENT METHOD
One problem is that the identified mass matrix is not
unique. The cause of this problem is the use of mode
shapes in the constraints. Because the mode shape is
not uniquely defined, any one of the mode shapes can be
multiplied by a nonzero constant without changing its
physical meaning:
φ̄ = φA
(8)
If we assume that there exists another mass matrix M̄
satisfying Eq. (2), the following equation is obtained.
φ̄T ∆M̄ φ̄ = E − P̄
(9)
where
Berman’s method[4] is a typical one in system identification. This method has the potential to update finiteelements model with large degrees of freedom. This
method considers constraints of the mode orthogonality:
φT (MA + ∆M ) φ = E
3
(7)
P̄ = φ̄T MA φ̄
(10)
In the same way as the Berman’s method, the identified
mass matrix M̄ is expressed as
M̄ = MA + MA φP −1 A−2 − P P −1 φT MA
(11)
Equation (11) indicates that there exist an infinite number of identified mass matrices because all the elements
of the diagonal matrix A are arbitrary. Therefore, we
cannot obtain a unique mass matrix. Buruch[8] also
points out the same problem when all modes are considered (the mode matrix is square and its dimension is
equal to the number of degrees of freedom of the analytical mass matrix).
The cause of the problem is that the mode shape is the
relative ratio of amplitude at each point on the structures. Therefore, the mode shape can be normalized
in several different expressions. Equation (1) is just a
typical one of these expressions. In other wards, it is
unreasonable to introduce Eq. (1) into the constraints
that the mass matrix must satisfy because it is the constraints imposed on the mode shape. This is the reason
why the identified mass matrix is not unique.
The other problem is that the identified mass matrix
cannot provide mass properties of structures mating with
the measured values. The cause of this problem results in
using only several measured modes to identify the mass
matrix. It is difficult to determine mass distribution of
the structures by using several modes. This means it
cannot be assured that the identified mass matrix approaches the true mass matrix of real structures. As a
result, the identified mass matrix gives physically unreasonable results, which states that the weight of the
structure has different values in different directions as
described later in the numerical example.
On the other hand, the identified stiffness matrix is
unique even though any normalized mode is used in its
identification. It can be understood that the uniqueness
1446
of the mass matrix depends on the constraints as shown
in Eq. (11). Therefore, we confirm the uniqueness of the
stiffness matrix by checking the constraints. The symmetry requirement, the mode orthogonality conditions
and the dynamic equation are selected as constraints.
The modal matrix satisfies the mode orthogonality conditions:
φT Kφ = φT M φΩ 2
(12)
It should be noted that modal masses are not normalized
to be one. If another stiffness matrix K̄ exists which
satisfies Eq. (12), we obtain
T
T
φ̄ K̄ φ̄ = φ̄ M φ̄Ω
2
(13)
Substitution of Eq. (8) into Eq. (13) yields
AT φT K̄φA = AT φT M φAΩ 2
(14)
Because A and Ω 2 are diagonal matrices, Eq. (14) can
be rewritten as
AT φT K̄φA = AT φT M φΩ 2 A
(15)
Pre-multiplying and post-multiplying Eq. (15) by A−1
give
T
T
φ K̄φ = φ M φΩ
2
4
4.1
PROPOSED METHOD
Constraints
It can be seen from Eq. (11) that the mode orthogonality
constraints, the off-diagonal terms of the mode orthogonality matrix φT M φ, do not affect the uniqueness of
the identified mass matrix. Therefore, we consider the
mode orthogonality as the constraints in mass matrix
identification:
φTi M φj = 0
(i = j)
(21)
Symmetry requirements are also considered as
M = MT
(22)
Mass matrix is assembled by using geometry data and
material data of finite elements. So the mass matrix
has the potential to express the mass property of real
structures if the modeling error is small. It is reasonable
to introduce the mass property into constraints from a
physical point of view. Rigid-body mass matrix, calculated from the mass matrix and rigid-body modes, gives
the mass property of the structure. Since each element
of the rigid-body matrix is unique, a unique mass matrix can be identified by considering the mass property
constraints
R T M R = M0
(23)
(16)
where M0 is a matrix with a measured value of mass
property as its element, and is expressed as
From Eqs. (12) and (16), we obtain


m
K̄ = K
(17)

 0
m
sym.



 0
0
m
(24)

M
=
Therefore, mode normalization gives no effect on the
0

 0
−ZG m
YG m
IX


mode orthogonality conditions.

 ZG m
0
−XG m IXY
IY
−YG m XG m
0
IXZ IY Z IZ
We can reach the same result for the constraint of the
dynamic equation. In a similar way, another solution K̄
As described earlier, the identified stiffness matrix is
satisfies the following equation.
unique if the identified mass matrix is unique. That
is, even though constraints include mode shapes, we can
K̄ φ̄ = M φ̄Ω 2
(18)
identify a unique stiffness matrix. Therefore, the modal
stiffness, mode orthogonality, symmetry, and the dySubstitution of Eq. (8) into Eq. (18) gives
namic equation are used the same as in the Berman’s
method:
K̄φA = M φAΩ 2
φT Kφ = φT M φΩ 2
(25)
= M φΩ 2 A
(19)
Post-multiplying Eq. (19) by A−1 , we obtain
K̄φ = M φΩ 2
(20)
The constraint of the dynamic equation is not affected by
mode normalization either. Baruch[8] says that we cannot obtain a unique stiffness matrix, but the above results show the identified stiffness matrix is always unique
if the identified mass matrix is unique.
4.2
K = KT
(26)
Kφ = M φΩ 2
(27)
Mass Matrix Identification
As described in 4.1, only the mode orthogonality constraints are used in the proposed method, however it is
hard to express these constraints in formulation. Then
1447
modal masses are included into the constraints to simplify formulation. Mass matrix is identified by minimizing the Euclidean norm ∆M . Using the constrained
minimization theory, the Lagrange function is written by
+
i=1 j=1
+2
i=1 j=1
500
αij φT M φ − E ij
i=1 j=1
nA nA
6 6
500
βij M − M T ij
γij RT M R − M0 ij
500
nT
nT Y
(28)
Z
nT
nT Table 1 Parameters of test and analysis
i=1 j=1
−
γij Lij + LTij
(29)
i=1 j=1
where
Hij = φi φTj
(30)
Lij = Ri RjT
(31)
and ∗ shows a matrix that has the same structure connectivity
as the original mass matrix. That is, the matrix
∗ has a value of zero at all degrees of freedom with values of zero in the original matrix. Substitution of Eq.
(29) into Eqs. (21) and (23) yields
nT
nT αij Sij +
i=1 j=1
nT nT
i=1 j=1
6 6
γij Qij = φT MA φ − E (32)
i=1 j=1
αij Vij +
6 6
γij Uij = RT MA R − M0 (33)
i=1 j=1
where
T
Sij = φT Hij + Hij
φ
Qij = φT Lij + LTij φ
T
Vij = RT Hij + Hij
R
Uij = RT Lij + LTij R
X
Fig. 1 Truss structure
T
αij Hij + Hij
6 6
Z
Y
where αij , βij and γij are Lagrange multipliers. Differentiating Eq. (28) with respect to each element of ∆M
and equating the results to zero gives Eq. (29).
M = MA −
500
ψ = ∆M +2
500
X
(34)
(35)
(36)
(37)
Solving the simultaneous equations of Eqs. (32) and
(33), Lagrange multipliers αij and γij are obtained. Substitution of the Lagrange multipliers into Eq. (29) yields
the identified mass matrix.
We can find the identified stiffness matrix in a similar way to mass matrix identification. Therefore, we
omit the formulation for identifying stiffness matrix from
space limitations.
Parameters
Inner diameter
Outer diameter
Density
[×10−6 ]
Young’s modulus
Modulus of rigidity
∗1
∗2
∗1
∗2
Test
16.59 mm
19.05 mm
5.66 kg/mm3
5.66 kg/mm3
7000 kg/mm2
2600 kg/mm2
Analysis
16.59 mm
19.05 mm
4.53 kg/mm3
5.09 kg/mm3
7350 kg/mm2
2730 kg/mm2
Four bars at the tip
Other bars
5
NUMERICAL EXAMPLE
5.1
Model
The proposed method was applied to a truss structure
shown in Fig. 1. Table 1 shows parameters of the truss
structure used in the finite element analysis. In the analysis model, four bars at the tip of the truss structure have
a density which is different from the density of the other
bars. Different input data for the finite element analysis
were taken into account to construct the simulated test
and analytical models. We used the mass properties, and
the lower six frequencies and their modes as test results
for the system identification. A boundary condition is
that four points at the bottom of the truss structure are
fixed.
5.2
Mass Property
Mass matrix was identified for four cases in which different constraints are considered as
Case
Case
Case
Case
1;
2;
3;
4;
modal mass and mode orthogonality
mode orthogonality
mass property
mass property and mode orthogonality
Table 2 shows the mass properties. For mass property
1448
Table 2 Mass properties
Weight
[Kg]
Center of gravity
[mm]
Moment of inertia
[×106 kgmm2 ]
X
Y
Z
X
Y
Z
X
Y
Z
Test
9.64
9.64
9.64
0
0
810.7
8.55
8.66
0.90
Case 1
9.14
9.06
8.46
0
2.7
864.0
8.59
8.60
0.85
Case 2
8.46
8.45
8.45
0
3.4
853.2
7.65
8.01
0.79
constraints (cases 3 and 4), the corrected mass properties
agree well with the measured mass properties. However,
we cannot predict mass properties coincident with the
measured one for cases 1 and 2. It also can be seen that
case 1 generates physically unreasonable results. The
weight has different values in different directions, though
it should have the same value for all directions. Moreover, the center of gravity (C.G.) moves away from the
measured value. This indicates that the masses at points
closer to the tip of the truss structure increase. That
is, the larger the deflection is, the better the masses at
those points are corrected. For case 2, the weight has
no change from its initial weight. However, the mass
distribution is different from that of the initial model,
because C.G. in the Z direction and the moment of inertia around the Y-axis change.
Case 3
9.64
9.64
9.64
0
0
810.7
8.55
8.66
0.90
Natural frequencies [Hz]
Test
Case 1
Case 2
Case 3
Case 4
Analysis
150
100
50
0
1
5
6
7
Mode number
8
9
10
Case 1
Case 2
Case 3
Case 4
Analysis
Error [%]
5
(38)
0
-5
-10
-15
Figure 3 shows the frequency error expressed by
ftest − f
× 100
ftest
4
10
Therefore, frequencies considered in the identification
approach the test frequencies even if we use any normalized mode in case 1.
Error[%] =
3
15
Natural frequencies, calculated by using the identified
mass and stiffness matrices, are shown in Fig. 2 with the
test and initial analysis results. The lower six frequencies
appear to be almost the same as the test frequencies for
all cases. The natural frequencies are found by
φT Kφi
= Ti
φi M φi
2
Fig. 2 Natural frequencies
Natural Frequency
ωi2
Analysis
8.45
8.45
8.45
0
3.4
839.6
7.65
7.74
0.79
200
From the above results, we can say that the identified
mass matrix can give mass properties mating with the
measured ones by the introduction of the mass property
constraints.
5.3
Case 4
9.64
9.64
9.64
0
0
810.7
8.55
8.66
0.90
1
2
3
4
5
6
7
8
9
Mode number
Fig. 3 Frequency errors (six modes)
(39)
The errors are nearly equal to zero for the third to sixth
frequencies, however the error becomes large for lower
1449
10
frequencies, especially in case 3. This means the identified mass and stiffness matrices cannot satisfy both
the mode orthogonality and the dynamic equation. The
residuals of φT Kφ − φT M φΩ 2 in case 3 are indicated
in Fig. 4. There exist large residuals in the mode orthogonality constraints because the mode orthogonality
matrix φT M φ is not diagonal. Mode orthogonality between the first and fifth modes is extremely worse. Figure 5 shows the maximum residual errors of the dynamic
equation in case 3. For the first mode, the maximum
residual error increases after stiffness matrix identification. Therefore, the stiffness matrix is not identified so
that it may satisfy the dynamic equation. On the other
hand, the errors go down in the fourth to sixth modes.
These are reasons the frequency error of the first mode
has a large value as shown in Fig. 3. The frequency
errors are indicated in Fig. 6 when we use three test frequencies in stiffness matrix identification. The updated
lower three frequencies are the same as the test frequencies for all cases. It is hard to identify mass and stiffness
matrices mating with test results as the number of test
modes increases in case 3. This means mode orthogonality is important to the constraints in the mass matrix
identification.
12047
6217
6
5
Mo 4
de
3
num
2
ber
1
6
5
4
r
e
3
b
2 de num
o
M
1
Fig. 4 Residual of φT Kφ − φT M φΩ 2 in case 3
1
The updated frequencies outside the frequency band of
interest lose accuracy. However, Figs. 3 and 6 show a
tendency that they are close to test frequencies when
considering mass property constraints. It is found that
both mode orthogonality and mass properties are important constraints to identify mass and stiffness matrices.
0.8
( Kφι - MφιΩ2) / Kφι
Error
0.6
( KA φι - MφιΩ2) / KA φι
0.4
0.2
5.4
Uniqueness of Identified Mass and Stiffness
Matrices
We identified the mass and stiffness matrices by using
two different test modes (φ and 0.9φ). Figure 7 shows the
element ratios of two identified mass matrices. The lateral axis shows diagonal elements of the identified mass
matrix. The vertical axis indicates the ratio expressed
by
M
#
= M0.9 M1.0
0
2
3
4
Mode number
5
6
Fig. 5 Residual errors of the dynamic equation
in case 3
10
Case
Case
Case
Case
(40)
0
Error [%]
Mα is the identified mass matrix by using test mode αφ.
The operator defines the following operations:
#
Mij
= (Mα )ij / (Mβ )ij
1
(41)
It should be noted that the identified mass matrix is
unique when all of the ratios have a value of one. It is
seen from Fig. 7 that the mass matrix can be identified
uniquely for cases 2 to 4. Although the ratios of offdiagonal elements are not shown here, they also have a
value of one. Therefore, the mass matrix identified under the mode orthogonality and/or mass property constraints is always unique. On the other hand, all ratios
1
2
3
4
-10
-20
-30
1
2
3
4
5
6
7
8
9
Mode number
Fig. 6 Frequency errors (three modes)
1450
10
do not have a value of one for case 1. Therefore, the current method cannot give a unique identified mass matrix.
[7] Kabe, A. M., “Stiffness Matrix Adjustment Using
Mode Data,” AIAAJournal, Vol. 23, No. 9, pp. 1431–
1436, September 1985.
The element ratios of two identified stiffness matrices are
indicated in Fig. 8. We obtain the same result as the
identified mass matrix about the uniqueness of the identified stiffness matrix.
[8] Baruch, M., “Modal Data are Insufficient for
Identification of Both Mass and Stiffness Matrices,”
AIAAJournal, Vol. 35, No. 11, pp. 1797–1798, November 1997.
6
CONCLUSIONS
1.2
This paper proposes a method to obtain a unique solution in system identification. Mass and stiffness matrices
are identified by minimizing the Euclidean norm of these
matrices subject to some constraints. The mass properties are introduced into the constraints. The following
conclusions are obtained:
M#
1.1
1
(1) The identified mass matrix is not unique when modal
mass constraints are used. This mass matrix provides physically unreasonable results. Moreover, the
identified mass matrix dose not generate mass properties obtained by measurements. This problem
can be solved by introducing the mass properties of
structures into the constraints.
Case 1
Case 1
Case 1
Case 4
0.9
5
10
20
15
Node Number
(X)
(Y)
25
Fig. 7 Element ratios
of two identified mass matrices
(2) The identified stiffness matrix is always unique if the
identified mass matrix is unique.
1.001
Case 1 (X)
Case 1 (Y)
Case 1
Cases 2 to 4
(3) The identified mass and stiffness matrices can provide a highly accurate finite-element model mating
with both the vibration and mass property test results. The proposed method is effective for system
identification.
K#
1
REFERENCES
0.999
[1] Alvin, K. F., “Finite Element Model Update via
Bayesian Estimation and Minimization of Dynamic
Residuals,” AIAA Journal, Vol. 35, No. 5, pp. 879–
886, May 1997.
0.998
[2] Kuo, C. P., and Wada, B. K., “Nonlinear Sensitivity
Coefficients and Corrections in System Identification,”
AIAA Journal, Vol. 25, No. 11, pp. 1463–1468, November 1987.
5
10
20
15
Node Number
25
Fig. 8 Element ratios
of two identified stiffness matrices
[3] Baruch, M., “Optimal Correction of Mass and Stiffness
Matrices Using Measured Modes,” AIAAJournal, Vol.
20, No. 11, pp. 1623–1626, November 1982.
[4] Berman, A., and Nagy, E. J., “Improvement
of a Large Analytical Model Using Test Data,”
AIAA Journal, Vol. 21, No. 8, pp. 1168–1173, August
1983.
[5] Zhang, O., Zerva, A., and Zhang, D., “Stiffness Matrix Adjustment Using Incomplete Measured Modes,”
AIAA Journal, Vol. 35, No. 5, pp. 917–919, May 1997.
[6] Misawa, M., “Mass Matrix Identification Using Test
Data,” Proceeding of the 26th Space Structures Conference, pp. 184–187, 1984.
1451