Analyses of drives power reduction techniques for multi-axis random vibration control tests

Mechanical Systems and Signal Processing 135 (2020) 106395

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Analyses of drives power reduction techniques for multi-axis random vibration control tests Giacomo D’Elia a,⇑, Umberto Musella b,c, Emiliano Mucchi a, Patrick Guillaume b, Bart Peeters c a

University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy Vrije Universiteit Brussel, Pleinlaan 1, 1050 Brussels, Belgium c Siemens Industry Software NV, Interleuvenlaan 68, 3001 Leuven, Belgium b

a r t i c l e

i n f o

Article history: Received 22 March 2019 Received in revised form 24 September 2019 Accepted 25 September 2019

Keywords: Multi-axis vibration testing MIMO control Random vibration Minimum drives power Spectral density matrix

a b s t r a c t In multi-axis vibration control testing, the power required by the excitation system for replicating the user defined test specifications is a limiting factor which cannot be overlooked. Excessive power, on top of over-stressing the often expensive test equipment, could cause data acquisition overloads which inevitably interrupt the test even before the full level run. An accurate definition of the Multi-Input Multi-Output (MIMO) control target allows to perform the control test minimising the overall power required by the shakers. In this sense, in the recent years advanced procedures have been developed in this research direction. This paper analyses the available drives power reduction techniques, offering a detailed overview of the current state-of-the-art. Furthermore, this paper provides a novel solution to manage the cases where most of the power is required by a single drive of the multiple inputs excitation system. In order to point out the pros and cons of each procedure and to show the capabilities of the novel technique, the MIMO target generation algorithms are firstly theoretically explained and then experimentally compared by using a three-axial electrodynamic shaker. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Vibration Control (VibCo) tests are performed to replicate in the laboratory the vibration environment that a system needs to withstand in-service [1]. Historically, the replication of the multi-directional nature of a vibration environment has always been performed using single axis techniques and sequentially rotating the test article. This procedure is time consuming and the test item undergoes a stress state different from the in-service [2–5]. Significant advances in the test hardware and control vibration software [6–8], made the Multiple-Input Multiple-Output (MIMO) vibration control strategy the nowadays best solution to replicate real-life structural responses of test articles [9–13]. Moreover, the recent update to include tailoring guidelines for multi-exciter testing in the United States Military Standard [14] highlights the will of the environmental engineering community to turn the multi-axis control testing into the main reference procedure. This research focuses on a particular area of multi-axis control testing, i.e. the replication of the responses of a test specimen to broadband random Gaussian vibration environments. More specifically, the objective of this work is to investigate the so-called problem of meeting the minimum drive criteria [14]. The idea of minimising the drives power required to reach ⇑ Corresponding author. E-mail address: [email protected] (G. D’Elia). https://doi.org/10.1016/j.ymssp.2019.106395 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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the test specifications is an attractive and interesting solution, because the achievable power levels are often a limiting aspect for multi-exciter test. On the one hand, the drives power reduction allows to carry out the control test without over-stressing the excitation system, that means preserving the often expensive test equipment. On the other hand, it offers the possibility to fully exploit the excitation system capabilities preventing the data acquisition DACs overloads. For MIMO random control testing, at the authors’ knowledge, the available drives power reduction procedures are: i) the Extreme Inputs/Outputs Method – EIOM [15,16], ii) the Independent Drives Method – IDM [17] and iii) the Minimum Drives Method – MDM [18]. At the current state, although the three procedures have already been widely studied and accepted, there is still no work in which these procedures are theoretically and experimentally compared. The original contribution of this work is to analyse and compare the three state-of-the-art drives power reduction procedures. Moreover, this work proposes a novel MIMO target generation procedure based on the single drive power reduction, namely Minimum Single Drives Method – MSDM. This procedure wants to be an alternative solution when the power required by the multiple-exciter system is mostly exploited by a single shaker. In these cases, in order to safely push the excitation system and to avoid DACs overloads, it could be more advantageous to focus the method for minimising the power of the overloaded drive instead of reduce the overall power of the system. A test campaign with a three-axial electrodynamic shaker is carried out for pointing out the pros and cons of each procedure and for investigating the impact on the random control strategy of each target generation algorithm. The work is organized as follows. Section 2 briefly summarizes the features and the properties of the reference Spectral Density Matrix (the user-specified target for any MIMO control test). The first part of Section 3 critically reviews the three state-of-the-art drives power reduction procedures. The second part of Section 3 describes the proposed novel solution. Section 4 shows the experimental results. Section 5 concludes the work. 1.1. Nomenclature Since most of the derivations are in the frequency domain, all the arrays are functions of frequency f (in Hz), if not specified otherwise. Vectors are denoted by lower case bold letters, e.g. a, and matrices by upper case bold letters, e.g. A. Apex
0 is used to indicate the complex conjugate operation and the Hermitian superscript
H to indicate the complex conjugate transpose of a matrix, e.g. a0 and AH are the complex conjugate and the complex conjugate transpose of vector a and matrix ^ is used to A, respectively. Dagger symbol
y is used to indicate the Moore-Penrose pseudo-inverse of a matrix, whereas hat
b is the estimate of matrix A. emphasize the estimation of a quantity, e.g. A 2. Building the reference spectral density matrix To perform a VibCo test, the user specifies a target (test specification) and the control algorithm has to provide the input voltages to the shaker’s amplifier by which the excitation system is able to replicate the user defined reference. The power required by the exciter is strictly related to the target definition procedure. For MIMO random control testing, the generation of the target is nevertheless not straightforward and it is currently still being investigated in numerous publications [19–22]. When multiple outputs need to be controlled simultaneously, not only the required test level for each control channel is needed, but also the cross-correlation between each pair of control channels has to be defined. The control target is thus a full Spectral Density Matrix (SDM) in the frequency band of interest, where the diagonal of the matrix is composed by the reference Power Spectral Densities (PSDs), and the off-diagonal terms are the reference Cross Spectral Densities (CSDs). The reference PSDs could come directly from the field measured data and taken as test specifications after being averaged, smoothed and enveloped. Otherwise, especially for durability tests where it is impractical to replicate the real vibration environment due to cost and time reasons, the reference PSDs could be defined using Mission Synthesis procedures typically based on single axis spectral damage equivalence techniques [23]. Typical synthesized profiles, categorized according to the specific specimen, are provided by the Standards (e.g. MIL-STD-810 [1] and GAM.EG 13 [24]). Both in the case where the reference PSDs are obtained from measured field data and from Standards, the definition of the proper reference CSDs are very challenging. The case where the reference CSDs can be fully retrieved by the real environment has already been discussed in several works [9,13,22]. Unfortunately, the information about the cross-correlation between the channels could be unusable or even unavailable. For instance, the operational data could be unusable due to possible changes between the laboratory and the field environment: different boundary conditions, altered dynamic characteristics because of impedance mismatches or more environmental sources to be combined [25,12]. Moreover, the data may be unavailable if the test specifications are provided by the Standards. The standardisation of the reference CSDs is practically impossible because their definition is closely related to the specific test configuration and it is not extendible to different operational conditions. When no information about the cross-correlation is available, the reference CSDs can be defined starting from the respective reference PSDs and by specifying proper profiles of phases and coherences. For instance, the reference CSD between the i-th and the j-th reference channel can be computed as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   C SDij ¼ CS Dij ei/ij ¼ c2ij PS Di PS Dj ei/ij

ð1Þ

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where c2ij and /ij are the (ordinary) coherence and the phase angle between the two reference channels, respectively. Any SDM calculated from a set of random signals is positive (semi) definite [26]. It means that the definition of the phases and the coherences of the CDSs must guarantee the resulting reference SDM being positive (semi) definite. This algebraic constraint is a property of any successful MIMO target generation procedure because it ensures the physical meaning of the reference SDM [21,27]. A positive (semi) definite reference matrix with fixed PSDs is, however, not unique. The choice of defining the best phase and coherence profiles with respect to the specific control test is left to the environmental test engineer knowledge. In this case, the most commonly practised procedure to fill in the reference SDM is the Independent Ref  erences Method (IRM): the reference SDM is specified as a diagonal matrix by simply setting low coherence cjk ¼ 0:05 for all the cross terms [14]. Nevertheless, the possibility of defining different phase and coherence profiles for the CSD terms opens up the idea of providing a MIMO target generation procedure that, starting from the user specified reference PSDs, is able to complete the full reference SDM by meeting the aforementioned algebraic constraint and meanwhile by reducing the power required by the shakers to reach the test specifications. 3. Drives power reduction procedures The steps of a MIMO random control test are simplified in the block diagram in Fig. 1. The objective of the control system is to generate the drive signal vector, uðtÞ ¼ fu1 ðt Þ; . . . ; um ðtÞgT , which causes the output SDM (Syy 2 C‘‘ ) to match the user ‘‘ specified reference SDM (Sref ) within some acceptable tolerance limits. In the hypothesis that the dynamic system is yy 2 C linear time invariant, the input–output relation can be written in terms of SDMs [26]

Syy ¼ HSuu HH

ð2Þ

Suu ¼ ZSyy ZH mm

where Suu 2 C

ð3Þ ‘m

is the input SDM, H 2 C

m‘

represents the Frequency Response Functions (FRFs) matrix and Z 2 C

is the

system’s mechanical impedance matrix (Z ¼ H1 ). It is worth noting that there is no one to one correspondence between the m input drive signals and the ‘ output responses, therefore the number of inputs and outputs may be different. The case where ‘ > m is the so-called rectangular problem [27,28] and for computing the mechanical impedance matrix a pseudoinverse operation is needed (e.g. the Moore-Penrose pseudo-inverse, Z ¼ Hy ). Theoretically, for the inverse control problem, the reference SDM could be directly obtained driving the exciters with the input drives that return the computed input SDM [29]

h i bH b ref Z uðt Þ ¼ ifftðUÞ : E UUH ¼ Suu ¼ ZS yy

ð4Þ

b is the (pseudo) inverse of the FRFs matrix estimated during the System Identification (Sys. Id.) pre-test phase. Typwhere Z b ically, H is estimated via H1 estimator by performing a low level random pre-test with decorrelated inputs. However during the actual test, due to the inevitable presence of noise in the measurements and possible non-linear b – I) and an error would be introduced responses of the test item, the system could differ from the estimated one (H Z between the reference SDM and the measured output SDM

   H b Sref H Z b – Sref Syy ¼ H Z yy yy

ð5Þ

Fig. 1. MIMO random control testing block diagram.

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As a result, in order to reduce this error, a feedback control action is needed. The theoretical definition of the input SDM of Eq. (4) can thus be expressed in an extended form

2

Suu;11 6 . 6 . 4 . Suu;m1

3 2 Suu;1m Z^ 11 7 6 .. 7 ¼ 6 .. . 5 4 . b ‘1 . . . Suu;mm Z ... .. .

32 ref b 1m S Z 6 yy;11 .. 7 .. 76 . 56 4 . b ‘m Z Sref yy;‘1

... .. . ...

. . . Sref yy;1‘ ..

.

...

.. . Sref yy;‘‘

32 76 76 74 5

b0 Z 11 .. . b0 Z 1m

... .. . ...

3 b0 Z ‘1 .. 7 7 . 5 b0 Z m‘

ð6Þ

Eq. (6) evidences that all the elements of the reference SDM are involved on the computation of each diagonal term of the input SDM, i.e. the PSDs of the input drive signals. It follows that, if matrix Z has non-zero off diagonal elements, it can be possible to vary the drives power by changing the phases and the coherences of the reference CSDs without compromising the reference PSDs (test specifications). 3.1. State-of-the-art solutions 3.1.1. Extreme inputs/outputs method (EIOM). This MIMO target generation procedure has been the pioneer on the drive power reduction for multi-axis tests and it represents a milestone in this area. The first attempt has been proposed in the late 80’s [30]. The method has been finally developed around 20 years later [15]. Unfortunately, as explained in the following, the procedure contains a flaw and a solution to the problem has been suggested by the same author in a subsequent work [16]. The idea behind the EIOM is to take the sum of the drives power as an indicator for the overall power required by the exciters. In algebraic terms, this means to find the set of coherences and phases between the control channels that minimises the drives trace. In the general case of m inputs and ‘ outputs, by definition the trace of the input SDM is the sum of the diagonal terms [31]

P ¼ TrðSuu Þ ¼

m ‘ X ‘ X X i¼1

!

b ij Sref Z b0 Z yy;jk ik

ð7Þ

j¼1 k¼1

b H Z, b Eq. (7) can be written in terms of coherences and phases between the control channels By introducing matrix F ¼ Z [15]

P¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ‘ ‘1 X ‘ X X   ref   Sref cjk Sref yy;jj F jj þ 2 yy;jj Syy;kk F jk cos /jk  hjk j¼1

ð8Þ

j¼1 k¼jþ1

where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   i/ ref i/jk   jk ¼ c Sref Sref jk yy;jj Syy;kk e jk ¼ Sjk e   ih F jk ¼ F jk e jk

ð9Þ ð10Þ

Therefore, since the first term on the right hand side of Eq. (8) is always positive and fixed, drives trace (P) is the minimum if

(

/jk ¼ hjk þ p

c2jk ¼ 1

8j ¼ 1 : ‘; 8k ¼ 1 : ‘; j – k

ð11Þ

The reference phase condition of Eq. (11) guarantees that all the cosines are equal to 1, meaning that each cross term of the double summation will be subtracted from the positive and fixed term. Moreover, the coherences equal to 1 guarantee that all the cross terms will be subtracted with the maximum amplitude, meaning that the drives trace will be fully minimised. Unfortunately, this MIMO target generation procedure cannot be used to run a control test because it does not guarantee that the resulting reference SDM is positive (semi) definite. To meet the minimum drives trace condition of Eq. (11), all the cross-terms in the double summation of Eq. (8) are forced to assume the biggest negative values and they can eventually overcome in module the fixed and positive term, leading to a negative drives trace. This hypothesis is clearly not allowed: since the matrix trace is the sum of its eigenvalues, if the drives trace is negative then one eigenvalue is negative at least. For a matrix, one of the necessary and sufficient conditions in order to be positive (semi) definite is that all its eigenvalues must be (semi) positive. By definition, if the input SDM (Suu ) is negative definite, the respective reference SDM (Sref yy ) is negative definite too [31]. In order to overcome this limitation and to make the EIOM suitable for practical control testing, Smallwood [16] proposes the use of a constrained optimization algorithm. The idea is to fix the reference phases in accordance with the conditions of Eq. (11) and to find the coherence combination that minimises the drives trace with the additional constraint that the eigenvalues of the resulting reference SDM are all positive

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5

h  i h  i min P cjk : min eig Sref P0 yy

ð12Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ‘ ‘1 X ‘   X X ref   P cjk ¼ Sref F  2 c Sref jj jk yy;jj yy;jj Syy;kk F jk

ð13Þ

where

j¼1

j¼1 k¼jþ1

Loosely speaking the optimization algorithm decreases the coherence values iteration by iteration. The cross terms in the double summation of Eq. (13) are thus rescaled down until the drives trace is the minimum positive possible. Fig. 2 schematically represents the EIOM with the constrained optimization procedure. 3.1.2. Minimum drives method (MDM). This MIMO target generation procedure [18] provides an alternative solution to the previously discussed EIOM flaw. Starting from the drives trace formulation of Eq. (8), instead of modifying the reference coherence values, the MDM tackles the problem by adapting the reference phase condition of Eq. (11). The reason why the minimum drives trace conditions of Eq. (11) cannot be used to generate a physically realizable target is that the reference phase profiles cannot be independently set. In fact, in case of fully coherent control responses, the phases of the control channel pairs are mutually linked from a strong physical relation. Consequently, for ‘ fully coherent control outputs, just ‘  1 reference phases can be freely set following the minimum drives trace conditions. The remaining ones instead, must be retrieved following the phase pivoting principle [18] that respects the existing physical constraints. Therefore, by referring to Eq. (8), the MDM’s idea is to choose ‘  1 independent terms as the ones that contribute the  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ref most to the drives trace reduction, i.e. the ones that have the biggest amplitude F jk  Sref yy;jj Syy;kk and that simultaneously respect the phase dependencies. A fully coherent positive (semi) definite reference SDM with the maximum drives trace reduction is definitely guaranteed by applying the minimum drives trace conditions (Eq. (11)) just for the aforementioned terms and retrieving the phases of the remaining ðð‘  ‘Þ  ‘Þ=2  ð‘  1Þ terms following the phase pivoting principle. The steps of the MDM algorithm are schematically represented in Fig. 3. 3.1.3. Independent drives method (IDM). This MIMO target generation procedure [17] addresses the issue of reducing the drives power needed to perform a control test with fixed test specifications, by driving the excitation system with a set of uncorrelated inputs (independent drives). By referring to Eq. (6), the aim of the IDM is thus to find the set of phases and coherences for the reference CSDs which finally results in a diagonal input SDM. Considering the input-output relation of Eq. (2), in case of m inputs and ‘ outputs, the diagonal elements of the output SDM can be computed as

Syy;jj ¼

m X m X Suu;ik Hji H0jk

8j ¼ 1 : ‘

ð14Þ

i¼1 k¼1

If the input SDM (Suu ) is a diagonal matrix, Eq. (14) is reduced to

Fig. 2. EIOM algorithm block scheme.

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Fig. 3. MDM algorithm block scheme.

Syy;jj ¼

m X

Suu;ii jHji j2

8j ¼ 1 : ‘

ð15Þ

i¼1

Therefore the m independent drives, which are able to replicate the ‘ user defined test specifications, can be computed as

X ¼ Hy Y ref

ð16Þ

 where Y ref 2 R is the reference PSD vector and H 2 R is the (pseudo) inverse of H ¼ H:  H0 . Finally, by computing the input SDM as Suu ¼ diag ð X Þ, the full reference SDM with all the information about the cross correlation between the control channels can be easily defined as ‘1

H Sref yy ¼ HSuu H

y

m‘

ð17Þ

It is important noticing that it is not always possible to replicate the user defined test specifications using decorrelated inputs. In some cases, due to the dynamic behaviour of the unit under test or depending on the adopted sensors configuration, a correlation between the drives is strictly needed. It is thus possible that at some frequencies the input vector X, computed from Eq. (16), presents negative values suggesting the physical weakness of the process. In order to overcme this limitation, the negative values of X are set equal to zero and a new reference SMD (Sref yy new ) is obtained from the modified input SDM (Suu mod ¼ diag ðX mod Þ). Clearly, a variation of some values of X is translated into an alteration of the test specifications. Nevertheless it can be possible to rescale the reference PSDs up to the required level ref Sref yy res ¼ R Syy new R

ð18Þ

where R is a diagonal matrix and

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ref u Syy;jj Rjj ¼ t ref res Syy;jj new

ð19Þ

Sref yy res is the reference SDM with the user defined reference PSDs on the diagonal and with the CSD phases and coherences deriving from the modified input vector X mod (almost independent drives). The IDM algorithm is schematically shown in a block diagram in Fig. 4. 3.2. The proposed solution 3.2.1. Minimum single drive method (MSDM). The two previously discussed procedures (EIOM and MDM) take the sum of the drives power as an indicator for the overall power required by the system and therefore they focus the methods on the drives trace minimisation. In this way, these procedures try to avoid DACs overloads by equally reducing all drives. However, due to the dynamic of the unit under test, it is possible that the power required by the excitation system for replicating the desired test specifications is mostly exploited by a single drive. In these cases, the strategy of equally reducing all the drives is not effective for preserving the excitation

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Fig. 4. IDM algorithm block scheme.

system. It is more advantageous a method that better balances the overall power by pushing as much as possible the lightly stressed drives and relieving the overworked drive. The total power equally distributed between all the drives guarantees to fully exploit the excitation system capabilities in the safest way. Starting from the user defined test specifications, the aim of the MSDM is to find the proper values to be set for the cross-correlation between the reference channels that are able to minimise the power required by the overworked drive of the multi-input excitation system. In the general case of m inputs and ‘ outputs, the PSDs of the input SDM are [31]

Suu;ii ¼

‘ X ‘ X

b ij Sref Z b0 Z yy;jk ik

8i ¼ 1 : m

ð20Þ

j¼1 k¼1

According to the (i-th) overworked drive to be minimised, Eq. (20) can be rewritten as

Suu;ii ¼

‘ X ‘ X i Sref yy;jk Gkj

8i ¼ 1 : m

ð21Þ

j¼1 k¼1

where matrix Gi is defined as

b H ð:; iÞ Z b ði; :Þ Gi ¼ Z

8i ¼ 1 : m

Due to the Hermitian form of both

P¼

ð22Þ Sref yy

i

and G , Eq. (21) can be expressed as

‘ ‘1 X ‘ n o X X i i Sref Re Sref yy;jj Gjj þ 2 yy;jk Gkj j¼1

ð23Þ

j¼1 k¼jþ1

Therefore, the PSD of the overworked drive can be finally defined in terms of coherences and reference phases between the control channels as

Suu;ii ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ‘ ‘1 X ‘ X X   i ref  i  Sref cjk Sref yy;jj Gjj þ 2 yy;jj Syy;kk Gjk  cos /jk  ajk j¼1

ð24Þ

j¼1 k¼jþ1

where

    Gijk ¼ Gijk eiajk

ð25Þ

Hence, the PSD of the (i-th) overworked drive is the minimum if

(

/jk ¼ ajk þ p

c2jk ¼ 1

8j ¼ 1 : ‘; 8k ¼ 1 : ‘; j – k

ð26Þ

As previously explained, Eq. (26) does not guarantee that the resulting reference SDM is positive (semi) definite. To overcome this limitation, the MSDM can be completed following the phase pivoting principle proposed by the MDM [18]: the minqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ref imum (i-th) drive conditions are applied for the ð‘  1Þ independent terms with the biggest amplitude jGijk j Sref yy;jj Syy;kk and the remaining terms must be retrieved by respecting the existing phase dependencies.

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4. Test cases A test campaign is carried out by using the three-axial electrodynamic shaker Dongling 3ES-10-HF-500 at the University of Ferrara, shown in Fig. 5a. This advanced vibration test system adopts an hydraulic orthogonal decoupling bearings unit for connecting the three independent shakers with the Head Expander (HE). The HE is a magnesium alloy table of 14 kg. The Siemens SCADAS Mobile SCM202V (V8 input and DAC4 output modules) is used as data acquisition system and MIMO Random of Simcenter Testlab as vibration control software. The test campaign is performed on the bare HE with the sensor configuration shown in Fig. 5b, i.e. two tri-axial accelerometers named 1 and 2. In order to compare and to better highlight the capability of each procedure, three test cases are carried out considering different test set-up (different number of the control channels, different test specifications). The test specifications are provided in terms of PSDs only coming from the Standards and the reference SDM is fulfilled by applying the previously described methods. Furthermore, the test results are compared with the results obtained from the most commonly practised target generation procedure in MIMO random control testing: the Independent References Method (IRM). When no information about the CSD terms is known, the common option is to spec  ify the reference SDM as a diagonal matrix by simply setting low coherence cjk ¼ 0:05 for all the cross terms [14]. It is worth noticing that all the results are shown for normal end tests, i.e. full level successfully run for 1 min within the abort limits. 4.1. Test case A Test case A is a three-drives three-controls test case. The reference control channels are the X, Y and Z axes of the tri-axial accelerometer 1 shown in Fig. 5b. The PSD shape comes from single axis test specification [32] with the breakpoints reported in Table 1. The PSD profile has a frequency range of [10–32] Hz and a frequency resolution of 3.125 Hz. The same PSD profile is considered as test specification for all the reference control channels. A first analysis addresses the comparison of the state-of-the-art drives power reduction procedures (IDM, EIOM and MDM) with respect to the IRM. The drive traces resulting from the normal end tests are shown in Fig. 6. As expected, the IRM is the method that requires the highest power level to reach the test specifications. This procedure, by simply setting low coherence between the reference channels, very often turns to be overly conservative near frequencies dominated by a single mode. The result is a overly large drives trace in some frequency bands with the danger of over-stressing the test excitation system. The results of the drive traces in terms of V RMS give a useful insight of the drives power reduction capability of each procedures. The gap between the EIOM and the MDM is significant: a power reduction of 8% and 29% compared to the IRM, respectively. This difference derives from the different approaches adopted by the two procedures to fix the problem of obtaining a positive (semi) definite reference SDM starting from the minimum drives trace conditions of Eq. (11). The two different approaches are outlined in Fig. 7 that shows the reference coherences and the reference phases obtained from the two procedures. The EIOM fixes the reference phases in accordance with the conditions of Eq. (11) and decreases the coherence values with the constrained optimization routine until the reference SDM is positive (semi) definite. In some cases, in order to verify the constrain, the coherences are forced to low values that nullify the phases role. In these cases, the CSD terms have no beneficial effect on the drives power reduction. On the contrary, the MDM keeps fully coherent controls and adapts one of the three phases resulting from Eq. (11) by following the phase pivoting principle. For each spectral line, by properly choosing the phase to be adapted, the MDM is able to take full advantage of the minimum drives trace conditions and it guarantees the highest drives power reduction possible.

Fig. 5. (a) Three-axial electrodynamic shaker Dongling 3ES-10-HF-500 at the University of Ferrara. (b) Bare head expander sensor configuration.

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Table 1 Breakpoints of the PSD profile used for test case A. PSD Hz

g2 =Hz

10 100 300 500 2000

1:04e4 1:04e4 5e6 2:08e4 2:08e4 0:582 gRMS

Fig. 6. Test case A: drive trace results for the state-of-the-art solutions.

Fig. 7. Test case A: reference coherences and reference phases obtained from the MDM (black curves) and from the EIOM (gray curves).

Fig. 6 also highlights the excellent results obtained from the IDM: a power reduction of 25% compared to the IRM. The input SDM used to drive the excitation system is shown in Fig. 8. The input SDM corresponds to the drives from the MIMO controller to the amplifiers setted with the same gain factor. Fig. 8 highlights that at some frequencies it is not possible to replicate the test specifications with decorrelated inputs. The peaks in the coherence sub-plots underline that the correction of the negative values of input vector X (Eq. (18)) has been done (almost independent drives). However, it is worth underlining that the correction of the negative values of input vector X with zero, inevitably generates a diagonal matrix with at least one zero on the diagonal. This implies that the column vectors (or the row vectors) of the matrix are linearly dependent. The determinant of the input SDM is zero and the matrix is rank deficient. Then, the rank of the reference SDM computed via Eq. (17) is [31]

  rank Sref 6 min ½rankðHÞ; rankðSuu Þ yy

ð27Þ

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Fig. 8. Test case A: input SDM obtained from the IDM.

  Fig. 9 confirms that, at some frequencies, the reference SDM is rank deficient and positive semi-definite (det Sref ¼ 0). yy Finally, the proposed procedure on the single drive power minimisation (MSDM) is compared to the MDM. Fig. 10 shows the resulting RMS power levels for each drive and for the drives trace when the MSDM is applied for minimising the power of Drive X (dark-gray bars), of Drive Y (gray bars) and of Drive Z (light-gray bars). For this particular test case, the MSDM Z is certainly the most interesting application because about 60% of the total power required by the excitation system is spent for driving the Drive Z. In this sense, the MSDMZ that minimises the overworked drive by fairly spreading the drives power is more advantageous than the MDM that equally minimises the power of each drive. By comparing the two procedures, the power reduction on the Drive Z is 8%. Fig. 11 shows the control channel results for all the procedures. All the runs are normal end tests, i.e. no spectral line exceeds the abort thresholds (red lines) fixed at 6dB from the reference. 4.2. Test case B In order to better highlight the applicability field of the studied methods, test case B is intended to be an example where no MIMO target generation procedure has beneficial effects on reducing the drives power. It is a three-drives three-controls test case, where the reference control channels are the X, Y and Z axes of the triaxial accelerometer 2 shown in Fig. 5b. The test specifications come from single axis Standard [1] with the breakpoints reported in Table 2. The PSD profiles have a frequency range of [10–32] Hz and a frequency resolution of 3.125 Hz.

Fig. 9. Test case A: rank of the reference SDM obtained from the IDM (top); determinant of the reference SDM obtained from the IDM (bottom).

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Fig. 10. Test case A: RMS drives power comparison between the MDM and the MSDM.

Fig. 11. Test case A: MIMO random control results. The solid red lines and the dashed orange lines are the abort and the alarm thresholds fixed at 6 dB and 3 dB from the reference, respectively.

Table 2 Breakpoints of the PSD profiles used for test case B. PSDX

PSDY

PSDZ

Hz

g =Hz

Hz

g =Hz

Hz

g2 =Hz

10 20 120 121 200 240 240 500

6:5e3 6:5e3 2:0e4 3:0e3 3:0e3 1:5e3 3:0e5 1:5e4

10 20 30 78 79 120 500

1:3e4 6:5e4 6:5e4 2:0e5 1:9e4 1:9e4 1:0e5

10 40 500

1:5e2 1:5e2 1:5e4

2

0:74 gRMS

2

0:20 gRMS

1:04 gRMS

The control channels results for the normal end tests are shown in Fig. 12. The resulting drive traces and the RMS drives power comparison are reported in Fig. 13 and Fig. 14, respectively. It can be noted that there is no significant differences in terms of power reduction between the adopted procedures. The power required by the excitation system for replicating the test specifications is essentially the same, independent the applied method. This can be explained by looking at the amplitudes of the F matrix terms plotted in Fig. 15. Except for the narrowband around 300 Hz, in the entire bandwidth the diagb is almost onal terms of the matrix (black curves) are orders of magnitude above the off diagonal ones, meaning that H diagonal. Therefore, by referring to Eq. (8), the cross terms hold a negligible role and the overall power required by the exciters is not affected by the choice of coherences and phases.

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Fig. 12. Test case B: MIMO random control results. The solid red lines and the dashed orange lines are the abort and the alarm thresholds fixed at 6 dB and 3 dB from the reference, respectively.

Fig. 13. Test case B: drives trace results.

Fig. 14. Test case B: RMS drives power comparison.

4.3. Test case C Test case C is a three-drives six-controls test case. The reference control channels are X, Y and Z axes of the triaxial accelerometers 1 and 2 shown in Fig. 5b. A standard ‘‘blue-white noise” PSD profile, reported in Table 3, is considered as test specification for all the reference control channels. The frequency range is [10–32] Hz and the frequency resolution is 3.125 Hz. Fig. 16 shows the control results of the six reference channels for the normal end tests. It can be noted that no results are presented for the EIOM. Due to the high complexity of the six controls test case, the basic optimization routine adopted in the EIOM at some frequencies does not converge to a feasible solution and the resulting reference SDM cannot be used for the

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b b H Z. Fig. 15. Test case B: matrix F ¼ Z

Table 3 Breakpoints of the PSD profile used for the test case C. PSD Hz

g2 =Hz

10 50 2000

2:60e5 1:51e4 1:51e4 0:547 gRMS

Fig. 16. Test case C: MIMO random control results. The solid red lines and the dashed orange lines are the abort and the alarm thresholds fixed at 6 dB and 3 dB from the reference, respectively.

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Fig. 17. Test case C: results of the constrained optimization routine adopted in the EIOM at 600 Hz. a) cost function values; b) step size values; c) coherence values obtained at the 34th iteration; d) eigenvalues of the reference SDM obtained at the 34th iteration.

actual control test. A practical example is given in Fig. 17 that shows the results of the constrained optimization routine at 600 Hz. In particular, Fig. 17a) and b) illustrate the values of the cost function per iteration and the corresponding step size; Fig. 17c) and d) show the coherence values and the eigenvalues of the reference SDM obtained at the last iteration. As can be seen, after 34 iterations the optimization converges to a solution that does not satisfy the constraint, i.e. the minimum eigenvalue of the resulting reference SDM is negative. Since the sizes of the reference SDM quadratically increase with the number of controls, the constrained optimization algorithm has to manage a 6x6 matrix by matching the 15 coherence variables in countless combinations per spectral line. As also highlighted in Fig. 18, the minimum eigenvalue of the reference SDM (black curve) results negative for most of the bandwidth. Therefore, the control target is negative definite and it cannot be use for the actual control test. This test case clearly highlights the EIOM limitations when the number of control channels increases. The comparison with respect to the drives power reduction is illustrated in Fig. 19. The test results confirm the outcomes obtained in test case A. Considering the RMS values of the drives trace as an indicator for the global power reduction, both the IDM and the MDM have significant beneficial effects: a power reduction of 34% and 31% with respect to the standard IRM, respectively. It is also worth emphasising that, for this particular application, the MDM cannot fully exploit its potential. In case of fully coherent control responses, since the excitation system is a 3 DOF shaker, the control channels pairs along the

Fig. 18. Test case C: eigenvalues of the reference SDM obtained from the EIOM.

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Fig. 19. Test case C: comparison on the drives power reduction.

same axis need to be set in phase. This physical constraint on the reference phases setting, inevitably affects the choice of the independent terms on which the minimum drives trace conditions (Eq. (11)) can be applied. Concerning the proposed procedure on the single drive power minimisation, the MSDMZ guarantees an extra power reduction on the overstressed Drive Z of 9% and 14% compared to the IDM and the MDM, respectively. Moreover, since the Drive Z spends over half of the total power required by the excitation system, the MSDM Z also ensures remarkable results on the drives trace power reduction. 5. Conclusions This paper compares and analyses the currently available procedures for reducing the drives power in multi-axis random control testing. In addition, an alternative solution based on the power minimisation of the overstressed drive is proposed. A test campaign, carried out using a three-axial electrodynamic shaker, highlights significant differences on the capabilities of each MIMO target generation algorithm. The EIOM has inherent limitations due to the integration of a constrained optimization routine inside the algorithm. The strategy of lowering the coherence values to verify the constrain, results ineffective in terms of drives power reduction. Moreover, the convergence to a feasible solution is not always guaranteed and the user experience could be strictly needed to tune the converge parameters. Additionally, since an iterative routine is involved in the process, the computational costs and the computational time significantly increase with the number of variables in the process, making the EIOM not suitable when the number of control channels increases. The MDM overcomes the EIOM flaws. By keeping high coherent control responses and by adapting the reference phases, this fully automatic procedure quickly provides a reference SDM that guarantees high control performances and typically the best drives power reduction results. It is important to point out that the MDM can fully exploit its potential if there is one to one correspondence between the directions of the control responses and the DOFs of the excitation system.

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