Analyses of Target Definition Processes for MIMO Random Vibration Control Tests
12.3 Building the MIMO Random Vibration Control Reference Matrix
The reference matrixSrefyy to set as target of a multi-exciter random control test must be positive semi-definite to have a physical meaning. Beside the algebraic definition, there are some important properties to consider for practical applications.
Particularly, the following statements are equivalent toSrefyy being positive semi-definite [9,10]:
(a) all the eigenvalues ofSrefyy are semi-positive;
(b) Srefyy has a unique Cholesky Decomposition, meaning that it can be decomposed in the product of two triangular hermitian matrices, referred as the Cholesky Factors:Srefyy DLLH;
(c) theSylvester’s Criterionis respected, i.e. all theprincipal minorsofSrefyy have positive determinants. The principal minors are the square sub-matrices that share the diagonal with the full matrix.
Moreover, ifSrefyy is positive semi-definite then
(d) the trace ofSrefyy is real and semi-positive, being the matrix trace the sum of its eigenvalues;
(e) the determinant ofSrefyy is real and semi-positive, being the matrix determinant the product of its eigenvalues;
(f) alsoSuuDZSrefyyZHis positive semi-definite.
The diagonal terms ofSrefyy are usually known levels for the environmental test engineer, provided as test specifications.
Most of the MIMO vibco software have the possibility of defining element-wise the CSDs in terms of coherence and phase profiles. For computational reasons linked to the control process stability [1,8], coherences values of 0 and 1 are usually avoided. Typical values of low coherence and high coherence are0:05–0:08and0:95–0:98, respectively. No information is available to set this profiles and sometimes an open loop test is run to get a meaningful values. All the CSDs are then easily computed via
CSDijD jCSDijjexp.jij/Dij
pPSDiPSDjexp.iik/ (12.6)
where i is the imaginary unit andiandjare thei-th and thej-th control channels.
12.3.1 Phase Pivoting Method
Filling in the MIMO matrixelement by elementin terms of the`.`C1/=2`coherence and phase profiles could result in Srefyy being non positive semi-definite, following the fact that none of the properties (a), (b) or (c) have been taken into account in the completion process. Starting from the property (c), Peeters in his work [6] gives a rule to define meaningful coherence choices to get a physical realisable target in case of three control channels. The Sylvester Criterion is, in fact, fully fulfilled if (1) the coherence between two control channels is between 0 and 1 and (2) if the determinant of the full reference spectral matrix is semi-positive. Stated that (1) is the only physical option, to get a meaningful reference matrix is sufficient (and necessary) that
det.Srefyy/D12132 232 C2cos.1213C23/q
122 132 232 0 (12.7) where 1,2 and 3 are the control channels. For the special case of the phase between all the control channels being zero, it is possible to fix two of the three coherences and retrieve the allowed values for the last one. For instance, fixing12and13, Eq. (12.7) allows all the23values between the boundaries
23;minDq
12213q
.1122/.113/2 (12.8a)
23;max Dq
12213Cq
.1122/.113/2 (12.8b)
Fig. 12.2 Phase Pivotingprinciple. Given the phases between two pairs of control channels (e.g.13and12), the phase between the remaining pair (23) is automatically defined
The conditions (12.8) can be extended to a more general set of phases by noticing that the two limits (12.8a) and (12.8b) are valid for all the phases combinations that nullify the cosine argument. A general signal processing consideration can drive a meaningful choice of phases that will accomplish this requirement. In the generic CSD term is contained, in fact, the phase difference between pairs of control channels recordings spectra. These phase profiles cannot be independently set in case all the pairs are fully coherent. In Fig.12.2three control channels (1, 2 and 3) are taken as example: setting the phases12and 13means to set arelative constraintin the phase information carried by the recorded signals, i.e. that between control 1 and 2 and control 2 and 3 there are phase shifts of12 and13, respectively. Thus the phase shift between the control channels 2 and 3 is unequivocally defined as the difference between13 and12. Following this principle, in case of fully coherent control pairs, the user can independently set the`1element of the first row (but the same could be done by picking other elements) and automatically retrieve the remaining phases by using the first row as aphase pivot
i;jD1;j1;i (12.9)
To notice that, by following the phase pivoting principle, Eq. (12.7) returns always the conditions (12.8) and a positive semi-definite matrix could then always easily be obtained by choosing an appropriate coherences set. Then to obtain a positive semi-definite reference target for the three control channels case is sufficient to set two phases and two coherences and then retrieve the third phase with the relation (12.9) and the third coherence in the interval defined by (12.8). An extension to the general``reference matrix is not straightforward because on top of the condition (12.7),`.`3/=2C1additional conditions need to be included to get allowable coherence boundaries These will be given, according to (c), by the`2 remaining determinants. Even if a mathematical proof is still pending, in case all the control channels are fully coherent, the phase pivoting method has been proven via simulation to always lead to a physically realisable reference matrix. This can be intuitively explained by noticing that, on top of any mathematical demonstration, there is a physical consideration driving the choice of the phases between control channels.
12.3.2 Eigenvalues Substitution
According to the property (a), in case the phase and coherence profiles filled in the MIMO reference matrix will result in a non positive semi-definite matrix, it means that the eigendecomposition
Srefyy DQƒQH (12.10)
will return some eigenvalues that are negative in the frequency band of interest [9]. A quick solution would then be to force this matrix to be positive semi-definite substituting the negative eigenvalues with semi-positive ones.
Srefyy newDQƒOQH (12.11)
Fig. 12.3 Negative definite reference matrix (solid blue), positive semi-definite matrix obtained by substituting the negative eigenvalues with zero (dashed green) and its absolute value (dashed magenta). Positive semi-definite reference matrix obtained from thedashed magenta curvesby further replacing the PSDs with the required levels (dashed red)
whereƒO is obtained fromƒby replacing the negative eigenvalues. This operation would however corrupt the PSD levels that the user wants to achieve. In this sense, a minimal modification from the original matrix would be given by replacing with zeros the negative eigenvalues, paying a rank loss equal to the number of original negative eigenvalues. Another option to preserve the rank of the original matrix would be replace the negative eigenvalues with their absolute value. Regardless of any rank-related consideration, the strong limitation of this procedure is that after having modified the eigenvalues, the newly defined reference matrix has diagonal terms that differ from the starting ones. The only option available would be to replace the PSD values with the original reference PSD levels and check if the matrix is still positive semi-definite. If this is not the case, the process can be repeated but the convergence to a positive semi-definite matrix is not guaranteed. In Fig.12.3 it is shown an application example of the proposed procedure. In the figure the blue curve represents a target obtained by randomly selecting values of coherences and phases between the control channels. The matrix is non positive semi-definite as shown from in Fig.12.4where an eigenvalue is negative in the whole frequency range. The green and magenta curves represent the modified matrix by substituting the negative eigenvalue with a zero and its absolute value, respectively. It is worth to notice that substituting the eigenvalue with a zero means a minimal modification to the original matrix. This can be seen in Fig.12.3where the green levels are closer to the original target, compared with the magenta ones. However after substituting the PSD values with the required levels, only the substitution with the absolute value brings to a positive semi- definite target. This is shown in Fig.12.4where one the dashed green curve is negative in the whole frequency bandwidth.
12.3.3 Smallwood’s Extreme Inputs Method
A possible method to derive a Minimum (Maximum) Drives Target has been proposed by Smallwood in [5] and has been referenced as possible method do define the MIMO Random reference matrix in the US Multi-Exciter Test tailoring [1]. The idea of the Smallwood’sExtreme Inputs Methodis to find, with fixed PSD levels, the set of coherences and phases between the control channels that minimize the trace of the drives SDM, i.e.
given diag.Srefyy/ find Srefyy so that Tr.SuuDZSrefyyZH/ is minimum=maximum
Fig. 12.4 Eigenvalues of the matrices represented in Fig.12.3. Thedashed green curverefers to the reference matrix obtained by replacing the original negative eigenvalue with a zero and further replacing the PSDs with the required levels
By considering the basic equations for a linear time invariant system
Srefyy DHSuuHH (12.12)
SuuDZSyyZH (12.13)
it is possible to write the diagonal terms of Eq. (12.13) as Suu;iiDX`
jD1
X` kD1
ZjkSyy;jkZNik 8iD1Wm (12.14)
The trace of the drives SDM is the sum of the diagonal terms PTr.Suu;ii/D
Xm iD1
X` jD1
X` kD1
ZjkSyy;jkZNik
! DX`
jD1
X` kD1
Syy;jk
Xm iD1
ZjkZNik (12.15)
By defining the hermitian matrixFZHZand noticing thatSrefyy needs to be hermitian too, Eq. (12.15) can be rewritten as
PD X`
jD1
Syy;jjFjjC2 X`1
jD1
X` iDkC1
jSyy;jijjFjijcos.jiji/
Œ3pt D X`
jD1
Syy;jjFjjC2 X`1 jD1
X` iDkC1
ji
q.Syy;jSyy;i/jFjijcos.jiji/
(12.16) wherejiandjiare the phase angles (in radians) of theji-th off-diagonal terms ofSrefyy andF, respectively. Since the PSD terms are fixed, for a given structure, the first term on the right hand side of Eq. (12.16) is always positive and fixed. The second term contains the unknown quantitiesjiandji, and can be negative because of the cosines contained in the double sum. The theoretical minimum (maximum) trace, as pointed out in [5], is obtained when the second term is minimum (maximum), i.e. when the coherences are unitary and the cosines all equal1 (1). This observation leads to the following conditions that guarantee the theoretical minimum (maximum) drive traces
Pis minimum ()
jiD1
jiDjiC (12.17a)
Pis maximum ()
jiD1 jiDji
(12.17b) All the other possible combinations of coherences and phases will return drive traces that fall in the range between the minimum and the maximum value.