Noise Robust Actuator Placement on Flexible Structures

Noise Robust Actuator Placement on Flexible Structures

Magnus Andersson

Department of Electrical Engineering Linkoping University, S-581 83 Linkoping, Sweden

www: http://www.control.isy.liu.se

email: magnusa@isy.liu.se

1997-09-01

REGLERTEKNIK

AUTOMATIC CONTROL

LINKÖPING

Technical reports from the Automatic Control group in Linkoping are available as UNIX-compressed Postscript les by anonymous ftp at the address 130.236.20.24 (ftp.control.isy.liu.se ).

Noise Robust Actuator Placement on Flexible Structures

Magnus Andersson

Department of Electrical Engineering Linkoping University, 581 83 Linkoping, Sweden

e-mail:

magnusa@isy.liu.se

Abstract

A novel criterion for placement of actuators on exible mechanical structures is presented. Using simulated

\measured modes" obtained from the model, the pro- posed criterion maximizes the correlation of the mea- sured modes and the normal modes. The measured modes deviate from the normal modes due to damp- ing, measurement noise and process noise. The sta- tistical properties of the criterion are investigated. In simulations the computed actuator locations on a small aircraft-like model shows increased robustness proper- ties against damping, for an acceptable loss of corre- lation. A computationally cheap actuator placement algorithm is proposed.

Key words:

actuator placement, exible structures, experimental modal analysis.

1 Introduction

The purpose of locating actuators (and sensors) on dynamical structures may be divided into three cat- egories: experimental modal analysis 5], identica- tion 6], and control 4]. Experimental modal analysis is the topic of analyzing the structure's modal prop- erties using measurements of the modal parameters:

resonance frequencies, mode shapes and damping.

Locating actuators on a exible mechanical structure in order to measure its behavior to applied (sinusoidal) forces is common in, e.g., aerospace applications. The objective of the measuring is to validate the designed structure's dynamic properties in the low-frequency area. In order to predict the structure's behavior to externally applied forces, it is of vital importance to measure/estimate the mode shapes accurate. There- fore, locating actuators in order to excite the target modes of the structure is of high industrial interest.

Using an initial model of the structure, a design of the actuator placement can be made in advance of the real measuring. Usually a nite element (FE) model of medium to high complexity is used in a typical appli- cation. This means that the model may have 10,000 to 100,000 degrees of freedom (dof). In order to cope with the complexity of the model, a candidate set ^Ac

of actuators must be specied for the actuator place- ment algorithms to become practical. Also, since the formulated optimization problems in general are non- convex, sub-optimal algorithms must be used in order to fulll the time and the memory requirements put on the actuator placement design. We will denote^Ar as the outcome from the actuator placement algorithm.

Complex structures have often several near multiple modes (modes with close resonance frequencies). A criterion for locating actuators for the purpose of ex- perimental modal analysis should therefore be able to excite the modes properly, without interference of near multiple modes.

2 Basic Theory

FE modeling of mechanical structures yields a mass matrix

M

, and a stiness matrix

K

of dimensions

l

^

l

. The normal modes



i and resonance frequencies

!

i of the model are determined via the generalized eigen- value problem

(

K

^;

!

_i²

M

)



i= 0

 

_Ti

M

i= 1

 i

=^f1

::: l

^g (1) where

l

denotes the number of degrees of freedom (dof).

The interpretation of the normal mode



i is as the shape of vibration (or relative amplitude) for the reso- nance frequency

!

i.

If displacements are measured, the IO-behavior of a mechanical structure may be written

M x

(

t

) +

L x

_(

t

) +

Kx

(

t

) =

Bu

(

t

)

y

(

t

) =

Cx

(

t

) (2) where

x

is the dof of the mechanical structure,

u

is the vector of

m

inputs, and

y

the vector of

p

outputs.

In (2)

B

(of dimension

l

^

m

) describes the location of the actuators, and

C

(of dimension

p

^

l

) the location of the sensors. Notice: the damping matrix

L

is in practice modeled separately from the FE modeling and is not further treated here, see 7].

The frequency response function (FRF) of (2) is dened as the transfer function

G

(

s

) =

Y

(

s

)

=U

(

s

) evaluated at the imaginary axis,

s

=

j!

,

G

(

j!

) =

C

(

K

+

j!L

^;

!

²

M

)^;1

B

+

D

(3)

where

!

denotes frequency. Dening  as

 = 



1

::: 

l] (4) a transformation

x

= 

q

, and a multiplication of ^T from the left on the upper equation in (2) yields:

q

(

t

) +

L q

_+ !

q

(

t

) =^T

B

1

u

(

t

)

y

(

t

) =

C



q

(

t

) (5) where

! = diag(

!

₂₁

::: !

²_l) (6) and

L

= ^T

L

. If

L

is modeled diagonal,

L

= diag(2

!

1



1

::: 

2

!

l



l) (7) the FRF of (5) can conveniently be written as

G

(

j!

) =^Xl

i=1

C

i



_Ti

B

!

_i^2;

!

²+

j

2



i

!

i

!

⁽⁸⁾ where



i is the damping ratio of mode



i. The as- sumption of a diagonal

L

is valid in practice as well as in theory for exible structures, see 3].

The Multivariate Mode Indicator Function (MMIF) in 8] is used in experimental modal analysis to indi- cate (multiple) modes of the structure. It is also used to determine the amplitudes and the phases (force pat- terns) of the sinusoidal input forces. The force pat- terns are computed in order to excite the structures for accurate mode estimations. This technique is called normal mode force appropriation (or phase resonance testing). Normal mode force appropriation is today a well-established technique for exciting a single mode of a structure normally using multiple actuators.

The frequency response output

Y

is dened as

Y

i=

G U

i



for

u

=

U

isin(

!

i

t

) (9) If a single mode



i is \purely" excited (at

!

=

!

i), will

Y

i in (8) equal^;

jC

i



_Ti

B U

i

=

2



i

!

²_i. Dropping the notation of

C

, an estimate ^



_di of



_i may be dened as



^_di= Im^f

Y

i^g (10) for an appropriate chosen force pattern

U

i. In (10), the superscript

d

denotes damping.

If a single mode



i is to be excited at a resonance fre- quency

!

_i, the real part of

Y

_iin (9) should vanish, due to (10). The objective is therefore to nd the force pattern

U

i minimizing the real part of

Y

i. This ob- jective corresponds to the solution of the minimization problem



imin= min_{Ui =1}^kRe^f

Y

i^gk2

Y

i ² (11)

where^k

Y

^k2=

Y

^H

Y

, and

Y

^His the complex conjugate of

Y

.

It follows that



imin in (11) is the smallest eigen- value



ik of the generalized eigenvalue problem

(

A

i^;



ik(

A

i+

B

i))

U

ik= 0

 k

= 1

::: m

(12) where

A

i= Re^f

Y

i^gTRe^f

Y

i^gand

B

i= Im^f

Y

i^gTIm^f

Y

i^g. The eigenvalue problem in (12) is of small dimen- sion,

m

^

m

, if

m

actuators are used. The set of

m

eigenvalues



ikin (12) can for each

i

be used to detect the single mode



i if



i1=



imin ^0, two near mul- tiple modes if also



i2 ^ 0, etc. Moreover, the eigen- vectors

U

ik in (12) are candidates as input force pat- terns

U

_i to the structure.

3 Noise Model

Assuming the entries of the measurement noise vec- tor



^mn_i are unbiased and independent between the sensors, it can be modeled as a normally distributed random vector of zero-mean, i.e.,



^^mn_i =^

::: x

^mn_ij

:::

^

 x

^mn_ij ^sN(0



_i^mn) (13a)

x

^mn_ij¹

 x

^mn_jj² independent ⁸

i

⁸

j

⁸

j

1⁶=

j

2

(13b) where the noise variance equals (

_i^mn)².

Moreover, if the locations of the process noise sources are known, the frequency response function

G

^pn from the

N

pn process noise sources to the

p

sensors, can be computed. Using

G

^pn together with a process noise source model, we arrive at the process noise model



^^pn_i =^NXpn

k=1



^^pn_ik=^NXpn

k=1



^pn_ik

x

^pn_ik

 x

^pn_ik^sN(0



_ik^pn) (14a)

x

^pn_ik1

 x

^pn_jk2 independent ⁸

i

⁸

j

⁸

k

1⁶=

k

2

(14b) where the process noise sources are assumed to be un- biased and independent. We also assume the measure- ment noise and the process noise to be independent,

x

^mn_ij

 x

^pn_kl independent ⁸

i

⁸

j

⁸

k

⁸

l

(15) The choice of noise variance levels may be seen as a design variable and has to be chosen properly. One approach is to set the measurement noise levels (

^mn_i )² and the process noise levels (

^pn_ik)² as a fraction of a (simulated) measured response.

For measurement noise variance: dene, ^Eias the mean

\power" measured by the sensors,

Ei(^At



^St) = 1

jAt^j

jAt^j

X

k=1

Eik



where

Eik(

a

k



^St) =^j



^_di^j2

jSt^j

 a

k^2At

(16)

where^j



^_di^j2 is the measured \power" for a single actu- ator

a

k in a specied test set^At of actuators, and for a test set^Stof sensors.

For process noise is the noise variance level design more simple. One approach is to let the process noise vari- ance be a fraction of the level of the commanded input force^j

U

^j2.

4 A Noise Robust Actuator Placement Criterion

The estimated mode ^



^nd_i from the structure, may be modeled



^^nd_i = ^



_di+ ^



^mn_i + ^



^pn_i (17) where superscript

d

denotes damping,

mn

measure- ment noise, and

pn

process noise.

A generalization of the modal assurance criterion (MAC) in 1], may be dened as

ndMACi = E



j



_Ti



^{^nd}_i ^j2

E^j



i^j2j



^^nd_i ^{j2 (18)} where E denotes expectation.

It can be veried that ndMACiin (18) equals:

ndMACi= ^{j Ti}^_di^j2+^j i^j2(^_i^mn)²+^P_k^j _{Ti pnik}^j2(^_ik^pn)²

j i^j2j^_di^j2+^pj i^j2(^mn_i )²+^j i^j2Pk^j pnik^j2(^_ik^pn)² (19) for the proposed noise model in (13)-(15), and (17), see Appendix A, where also other important properties of the criterion are shown.

A noise-robust actuator placement criterion may be for- mulated:

Ar= arg max

Ar^Ac

jAr^j=m

j

W

^Njq (20) where

W

is a diagonal weight matrix, 1 ^

q

^{ 1}, and^N the vector

N =^h

::: 

^;ndMACi^12

:::

^iT

 i

= 1

::: N

tm

(21)

The dMACi is dened as the ndMACi for zero noise levels, i.e.,

dMACi= ^j



_Ti



^{^}_di^j2

j



i^j2j



^_di^{j2 (22)} Notice that the criterion in 5] equals (18), (20), and (21) with

_i^mn = 0,

_i^pn = 0,

W

=

I

(identity matrix) and

q

= 2, i.e., with no noise model.

5 Actuator Placement Algorithm

Recall that a full search for^Aramong ^Ac is often not feasible due to the computational complexity of a crite- rion

P

, and the size of the model. Instead sub-optimal algorithms computing^Armust be used. Here an algo- rithm with found minor sub-optimality is proposed.

Algorithm for Actuator Placement Input:

^Ac

 m P

Output:

^Ar

1. Let

k

= 0,^Ar=and ^Ac=^Ac.

2. Let

k

=

k

+ 1. Determine

a

k= arg max

ak^2A_c

P

().

Let^Ar=^fAr

 a

k^gand ^Ac = ^Acⁿ

a

k

3. If

k < m

: goto 2.

4. Let

k

= 1.

5. Let ^Ar=^Arⁿ

a

_k. 6. Determine

a

k = arg max

ak^2Ac^nA_r

P

().

7. If

a

k ⁶=

a

k : ^Ar=^fAr

 a

k^g, goto 4.

Elseif

k

=

m

: end.

Else : let

k

=

k

+ 1, goto 5.

In 2], the termination of the proposed algorithm is shown.

6 Example

The measurement noise levels^f

^mn₀



^mn_l



^mn_h ^g(zero, low, and high) are here related to ^Ei as,

^mn₀ ^, ₍

_i^mnEi )² =0 ^$1dB SNR (23a)

^mn_l ^, ₍

_i^mnEi )² =10^{4 $}40 dB SNR (23b)

^mn_h ^, ₍

_i^mnEi )² =10^{2 $}20 dB SNR (23c)

The measurement noise variance levels in (23) is dened for

At=^St=^fall transl. dof^g

In Fig. 1 an aircraft like model is visualized. The model was constructed to have several near multiple modes, where denotes  a grid point representing 3 transla- tional and 3 rotational dof.

13 26 15

8

12 9 10

11 24

1 3 7

6 5

4 ¹⁴

25

16 17 18

19 23 27 28

20 21 22

2

z y

x 29

Figure 1:

The geometry of the model for actuator place- ment design. The grids are denoted^. Using the proposed actuator placement algorithm, the criterion

P

= ^jNj1 the measurement noise levels de-

ned in (23), zero process noise variance level, and the candidate set

Ac =^fall transl. dof^g

of actuators, the resulting sets of three actuator loca- tions ^Ar become as in Table 1. The ndMACi-values are decreasing with the increase of the measurement noise level. It is seen that for a higher level of noise variance, the actuators are moved towards the end of the wings and the tail. The subscripts

x y z

denotes the direction of the applied force.

The actuator locations in Table 1 are henceforth de-

ned as in Table 2.

In Table 3, the dMACi-values for the actuator locations is presented. Notice the small loss in the dMACi for an increasing level of noise variance.

A robustness test against the level of measurement noise levels is presented in Table 4. It is readily seen that an actuator placement design based on a low mea- surement noise variance has bad robustness properties against a higher levels of measurement noise.

From an engineering point of view, one may argue that the higher the noise variance levels are, the bet- ter the Signal-to-Noise-Ratio will be. Since, when the

Table 1:

The resulting actuator locations ^Ar, and the ndMACi-values for the nine low-frequency modes.

VarianceArModei/ndMACi mn123456789 mn 013z17z22x1111.997.997.998.9981 ^{mn l}11z12z21x1111.999.994.997.9981 ^{mn h}12z15y15z1.999.999.998.998.999.983.986.999

Table 2:

The dened actuator locations.

A0r 13z



17z



22x

Alr 11z



12z



21x

Ahr 12z



15y



15z

Table 3:

The resulting actuator locations ^Ar, and the dMACi-values for the nine low-frequency modes.

VarianceArModei/dMACi mn123456789 0

A0 r1111.997.997.998.9981 ^{mn l}A

l r

11111.999.985.986.999 ^{mn h}A

h r

11111.999.984.9871

Table 4:

Robustness properties of the actuator locations against dierent levels of measurement noise.

Robustness Criterion ^A0_r ^A_lr ^A_hr Mean(ndMACi)



_l^mn .997 .998 .996 Mean(ndMACi)



_h^mn .869 .962 .989 Min(ndMACi)



^mn_l .996 .993 .984 Min(ndMACi)



^mn_h .762 .883 .981

noise variance levels are increased, the actuators are placed such that the structure is more easily excited in order to yield good mode estimates. That is, for a

xed level of input force from the actuators, the ampli- tude of the measured displacements increases with the noise variance levels. That is why the actuators are placed farther away from the center of gravity when the noise variance levels are increased. Notice that the farther away from the center of gravity the actuators are placed, the less good the mode estimates will be for zero noise variance level. However, the price paid for this robustness against noise is low since the mode estimates are \good enough" when the actuators are placed with a high level of designed noise variance. In Table 5 the SNR for the noise level corresponding to

_l^mn is presented. It is clearly seen that the SNR is increased with the designed noise variance levels.

Table 5:

The Signal-to-Noise-Ratio for a xed level of noise variance level.

A0r ^Alr ^Ahr

-16.2 dB 0 dB 10.5 dB

7 Conclusions

A novel criterion for actuator placement for the pur- pose of experimental modal analysis is presented. The statistical properties of the criterion is good under some easily fullled additional conditions. Increased robust- ness properties are obtained for a minor loss of cor- relation. The proposed, computationally cheap actu- ator placement algorithm is in simulations found to yield actuator locations with good properties. In a multiple-mode aircraft-like model example the actuator locations have the desired properties for experimental modal analysis.

References

1] R. Allemang and D. Brown. A correlation co- e#cient for modal vector analysis. In Proc. 1st In- ternational Modal Analysis Conference, pages 110{116, 1982.

2] M. Andersson. Experimental design and updat- ing of nite element models. Lic. thesis, Automatic Control, Dept. of EE, Linkoping University, 581 83 Linkoping, Sweden, April 1997.

3] M. G%eradin and D. Rixen. Mechanical Vibrations - Theory and Application to Structural Dynamics. Wi- ley, 1994.

4] X. Guangqian and P. M. Bainum. Actuator placement using degree of controllability for discrete- time systems. Trans. of the ASME, 114:508{516, September 1992.

5] P. S. Holmes, J. R. Wright, and J. E. Cooper.

Optimum exciter placement for normal mode force ap- propriation using an a priori model. In Proc. 14th In- ternational Modal Conference, 1996.

6] P. C. Shah and F. E. Udwadia. A methodol- ogy for optimal sensor locations for identication of dy- namic systems. Journal of Applied Mechanics, 45:188{

196, March 1978.

7] M. Soni. Finite element procedure in damping design. In Proceedings of the 14th Inernational Seminar on Modal Analysis, pages 8.1.1{8.1.48, Leuven, 1989.

8] R. Williams, J. Crowley, and H. Vold. The mul- tivariate mode indicator function in modal analysis.

In Proceedings of the 3rd International Modal Analy- sis Conference, pages 66{70, 1985.

A Properties of the ndMAC

i

Proof of (19):

Evaluating (18), ndMACi equals

ndMACi= E^hj



_Ti



^{^}_di^j2+^j



_Ti



^{^mn}_i ^j2+^P_k



_Ti



^{^pn}_ik^2i E^hj



i^j2j



^_di^j2+^j



i^j2j



^^mn_i ^j2+^j



i^j2Pk



^^pn_ik^2i

(24) due to the independence properties of

x

^mn_ij and

x

^pn_kl, see (13)-(15). For instance,

E^hX

k



_Ti



^{^pn}_ik^2i=E^hX

k

X

l (



_Ti



^{^pn}_ik)(



_Ti



^{^pn}_il)

x

^pn_ik

x

^pn_ilⁱ

=^X

k

j



^T



^{^pn}_ik^j2(

_ik^pn)² (25) noticing that^P_k



_Ti



^{^pn}_ik is a scalar, and

E^j



i^j2j



^^mn_i ^j2= E^j



i^j2X

l (

x

^mn_il )^2=

p

^j



i^j2(

^mn_i )² (26) since there are

p

terms in the sum in (26). ² The ndMACimust fulll some criterions in order to be valid for actuator placement purposes:

C1: ndMACi^1 (27a)

C2a:

@

ndMACi

@

(

^mn_i )²

<

0



⁸

i

(27b) C2b:

@

ndMACi

@

(

^pn_ik)²

<

0



⁸

i k

(27c) C3a: _(mnlim

i )^2!1ndMACi=



^mn_i ^1



⁸

i

(27d) C3b: _(pnlim

ik)^2!1ndMACi=



^pn_ik^1



⁸

i k

(27d) C4: _(mnlim

i )^2!0 (_ik^pn)² 0 k

ndMACi= dMACi



⁸

i

(27e)

The formulated criterions in (27) can be motivated in- tuitively:

C1: The ndMACi-values should be less than 1.

C2: The ndMACi-values should not be favored by in- creased noise levels.

C3: For high noise levels, the ndMACi-values should be low.

C4: Low noise levels should not signicantly aect the ndMACi-values.

The criterions C1, and C4 is easily veried. The cri- terions C2a, C2b, C3a, and C3b can be veried under some additional conditions:

C2a:

^If

j



_Ti



^{^}_di^j2

j



i^j2j



^_di^j2

>

¹

p



1 +

Pk^j



^pn_ik^j2(

^pn_ik)²

j



^_di^j2



;

Pk^j



_Ti



^pn_ik^j2(

^pn_ik)²

j



i^j2j



^_di^{j2 (28)} the ndMACi is monotonically decreasing in

_i^mn.

C2b:

If

j



_Ti



^{^}_di^j2

j



_i^j2j



^{^}_di^j2

>

^j



_Ti



^pn_il^j2

j



i^j2j



^pn_il^j2



1 +

p

(

_i^mn)²

j



^_di^j2 +

Pk^j



^pn_ik^j2(

_ik^pn)²

j



^_di^j2

(29) then ndMACiis monotonically decreasing in

_il^pn.

C3a,b

: the ndMACi asymptotically converge to:

(^mn_ilim)^2!1ndMACi = 1

p

⁸

i

(30a)

(^pn_iklim)^2!1ndMACi = ^j



_Ti



^pn_ik^j2

j



i^j2j



^pn_ik^j2



⁸

i

⁸

k

(30b) compare with (19).

Proofs of C1-C4 can be found in 2].