Multiple-input multiple-output random vibration control for a six-degree-of-freedom shaking table Pilotage aléatoire multi-entrées multi-sorties pour une table vibrante à six degrés de liberté Bart Peeters, Jan Debille LMS International – Leuven

Abstract To achieve military equipment qualification, modification and reliability test and analysis, the Hill Air Force Base (HAFB) Survivability and Vulnerability Integration Center (SVIC) has installed a unique six-degree-of-freedom shock and vibration system. Eight electrodynamic shakers, two in each horizontal axis and four in the vertical axis drive the system in the frequency range of 5 to 2000 Hz. The major design concern is the coupling of the shakers through a rigid table due to kinetic and dynamic effects and the behaviour of the structure. This has made the control task very challenging. To simulate a highly realistic six-degree-of-freedom vibration environment, a single power spectral density (PSD) clearly does not suffice as test specification. The PSDs along all axes as well as the intra- and inter-axes phase and coherence relations need to be specified before and controlled during the test. LMS International designed and implemented the multiple-input-multiple-output random control algorithm for the shaking table. This paper introduces the physical set-up and the control algorithm. Results from vibration control experiments will be discussed to assess the degree of control that can be achieved. Résumé Pour réaliser la qualification d'équipements militaires, les modifications et essais de fiabilité et les analyses, la Base Hill Air Force (HAFB) Survivability and Vulnerability Integration Center (SVIC) a installé un système unique, choc et vibration, à six degrés de liberté. Huit excitateurs électrodynamiques, deux dans chaque axe horizontal et quatre dans l'axe vertical pilotent le système dans la gamme de fréquence 5 à 2000 Hz. La préoccupation principale, pour la conception d’un tel système, est la prise en compte du couplage des excitateurs via la table rigide en raison des effets cinématiques, dynamiques et du comportement de la structure. Cette particularité a fait de la tâche de contrôle un challenge très stimulant. Pour reproduire un environnement vibratoire à six degrés de liberté fortement réaliste, une simple densité spectrale de puissance (PSD) n’est évidemment pas suffisante comme spécification de test. Les PSD sur tous les axes, de même que les phases intra et inter-axes, ainsi que les relations de cohérence doivent être spécifiées avant l’essai et contrôlées au cours de l'essai. LMS International a assuré le développement et la mise en oeuvre des algorithmes pour le contrôle en mode aléatoire multi-entrées multi-sorties pour la table vibrante. Cet article présente la mise en œuvre des paramètres physiques et les algorithmes de contrôle. Les résultats issus des essais vibratoires contrôlés seront discutés pour évaluer le degré de contrôle qui peut être réalisé.

1. INTRODUCTION In vibration qualification testing, it is common that the qualification requirements are specified as a power spectral density (PSD) that needs to be reproduced at a certain control location. Such a test is typically performed in a singleaxis setting, where the test article is subjected to vibrations in one direction only. If more than one direction is of interest, consecutive tests are performed after rotating the test article or using a slip table configuration. In the vibration control literature following typical drawbacks of sequential single-axis / single shaker testing are reported: •

The time to change the test set-up twice and repeating the control test can be excessive.

•

During the set-up changes, the risk of damage of the sometimes very expensive and fragile test articles exist.

•

There is definitely some lack of realism in single-axis testing, as the stress loading conditions will differ from the true three-dimensional stress situation.

In some cases it is therefore necessary to subject the test article to a three-dimensional, six degrees of freedom (DOF) vibration environment. A multi-axis / multi-shaker configuration has the following benefits : •

A highly realistic vibration environment is achieved during the qualification test. The fatigue failure predictions from such a test will be very accurate.

•

All structural modes are simultaneously excited. The qualification test data could be used for a complete modal analysis of the structure.

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Multiple-input mutiple-output random vibration control for a six-degree-of-freedom shaking table

x1 y1 z1

z2

z4

z3

y2 x2

Figure 1: Triaxial shaker table at Hill Air Force Base, driven by 8 shakers [8][9] and a schematic top view. •

For very heavy structures, often the excitation level from a single shaker is not sufficient. For very slender structures, a single shaker could damage the structure due to concentrated stresses. In both situations it is desirable to use multiple shakers to provide sufficient or distributed forces.

In the following of this paper, we will refer to single-axis / single shaker testing as single-input-single-output (SISO) testing. Multi-axis / multi-shaker testing will be called multiple-input-multiple-output (MIMO) testing. In the MIMO case, the qualification requirement cannot be a single PSD anymore. As random reference profile that needs to be reproduced during the test, a full spectral density matrix has to be specified instead. This matrix consists of the PSDs of all control channels and the cross spectral densities (CSDs) between these channels. The MIMO setting causes challenges to the control algorithm. Based on the reference spectral density matrix, multiple drive signals are generated that are sent to the shakers. During the control process, the drive signals are continuously updated. The drive corrections are computed from differences between the reference and the measured spectra. Dodds & Robson [1] performed some early work on MIMO random control. Smallwood and his co-workers [2][3][4][5] significantly contributed to and reported on the progress in MIMO random control. The algorithm presented in this paper is largely based on their work. Stroud & Hamma [6] are discussing open-loop, closed-loop and active MIMO vibration control. A very extensive overview of MIMO time waveform replication is provided in De Cuyper et al. [7]. In time waveform replication, which is in fact an open-loop control process, the test requirements are to reproduce deterministic time signals, instead of spectra. Shafer et al. [8] and Chen & Wilson [9] are presenting a unique triaxial shaker table at Hill Air Force Base that is driven by 8 shakers (Figure 1). Whiteman [10][11] provides a motivation to perform MIMO vibration qualification tests: to reduce testing time and to have more realistic stress loading conditions. Underwood & Keller [12] are also highlighting the benefits of vibration qualification testing using excitation from multiple actuators. Schedlinski & Link [13] and Füllekrug & Sinapius [14] are discussing the possibilities to perform a modal analysis on MIMO qualification test data. This paper first reviews some MIMO system theory, from which in next section a MIMO random control algorithm can be derived. The performance of the algorithm is verified by real-life experiments performed with its software implementation.

2. MIMO SYSTEM THEORY 2.1. From input to output In order to develop a vibration control algorithm, it is necessary to review some MIMO system theory. This section also clarifies the notation used throughout this paper. Assume that a MIMO system is characterised by its transfer function

H (ù ) ∈ ≤ l× m and consists of m inputs U (ù ) ∈≤ m and l outputs Y (ù ) ∈≤ l . Note that all quantities are specified in the frequency-domain as a function of the circular frequency ω [rad/s]. The output is obtained by multiplying the transfer function with the input : Astelab 2003

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Multiple-input mutiple-output random vibration control for a six-degree-of-freedom shaking table

Y ( ù ) = H (ù ) U (ù )

(1)

The purpose of random vibration control is to reproduce certain user-defined spectra at specific output locations. The input-output relation in terms of (power) spectra can be expressed as:

S yy = H S uu H H

(2)

where S yy (ù ) ∈≤ l×l is the output spectrum matrix; S uu (ù ) ∈ ≤ m× m is the input spectrum matrix; and • H denotes the complex conjugate transpose (Hermitian) of a matrix. The grey rectangular boxes give an idea about the dimensions of the involved matrices. For convenience of notation, the frequency dependency is omitted in equation (2).

2.2. From output to input In vibration control, the user specifies the output spectrum and the control algorithm has to produce input time series REF

that drive the shakers. The user-defined output spectrum is denoted as S yy

, the reference spectrum. From

equation (2), the corresponding input spectrum matrix S uu (ù ) ∈ ≤ m× m can be computed as:

S uu = H -1 S yyREF ( H -1 ) H

(3)

If H -1 (ω) cannot be computed due to singularity of the transfer function at frequency ω , it is replaced by the MoorePenrose pseudo-inverse [15], denoted as H † (ω) . Instead of the input spectrum matrix, the input time series are needed. Hereto the so-called Cholesky factorisation [15] REF

is applied to the reference output spectrum S yy

. The Cholesky factorisation splits a Hermitian, positive definite

matrix in a lower triangular matrix times its complex conjugate:

S REF = L LH yy

(4)

where L(ù ) ∈ ≤ l×l is a complex lower triangular matrix. Using (3) and (4), it can be shown that the Fourier transform of the input can be computed as:

U ( ù ) = H † (ù ) L( ù )W (ù )

(5)

where W (ù ) ∈ ≤ l represents the Fourier transform of a vector of l independent white noise sequences with unit correlation. The spectrum of these white noise sequences equals the identity matrix:

∀ω : S ww (ù ) = I l

(6)

Finally, by taking the inverse Fourier transform of (5) the desired input time series are obtained which can be sent to the shakers.

3. MIMO RANDOM CONTROL ALGORITHM In previous section, the input-output relations of a MIMO system were made clear and it was demonstrated how the input time series (the so-called drives) could be theoretically determined from the user-defined output reference spectrum matrix. In this section, the theory will be cast into a practical algorithm. The algorithm is given as a flow chart of which the elements will be discussed in some detail. Astelab 2003

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Multiple-input mutiple-output random vibration control for a six-degree-of-freedom shaking table Reference Spectrum

S REF yy ( ù )

System Identification

Hˆ ( ù ) Columnwise Randomisation

Cholesky Factor

L(ù ) Multisine

Decoupled Inputs

Time Data

Inputs

Structure

Outputs

Output Spectrum

W (ù )

U DC = Hˆ † LW

ukDC

uk

H

yk

Sˆ yy ( ù )

Error Correction

L → L+∆ Figure 2: Flow-chart of the MIMO random vibration control algorithm. The practical MIMO random vibration control algorithm, represented in Figure 2, consists of following steps: •

Compute the Cholesky factor L(ù ) of the user-defined reference output spectrum matrix.

•

Perform a low-level vibration test and apply non-parametric system identification to estimate the system transfer

ˆ (ù ) . Typically the transfer function estimate is computed as the function. This estimate is denoted by a hat : H least-squares estimator assuming a noise-free input (also called the MIMO H 1 estimator):

Hˆ ( ù) = Sˆ yu (ù ) Sˆuu−1 (ù )

(7)

where Sˆ yu ( ù ) ∈≤ l× m is the cross spectrum matrix between outputs and inputs ; Sˆ uu (ù ) ∈ ≤ m× m is the spectrum •

•

matrix between the inputs. Generate pseudo-random signals in frequency domain (a pseudo-random signal is a multisine of which all sine components have unit amplitude and random phase). Although these signals are not the same as, but only an approximation of white noise signals, they are still denoted by W (ù ) .

ˆ † (ù ) L(ù )W (ù ) . These are the so-called deCompute the input signals in frequency domain: U DC (ù ) = H coupled inputs. The decoupling, which is necessary for the randomisation procedure (see below), is achieved by stacking the white noise sources in a diagonal matrix instead of a column vector.

• •

Convert the inputs to time domain by applying the inverse DFT: u kDC ∈ϒ l× m . The subindex k denotes a discrete time instant. Apply time-domain randomisation. The randomisation procedure converts the line spectrum from the pseudorandom signals to a true continuous spectrum, which is required in a qualification test. The randomisation procedure also ensures a continuous input data stream to the shaker, without having to carry out intensive matrix computations. It is important to preserve the correlation between the inputs during randomisation. To meet these requirements, the time-domain randomisation procedure outlined in Smallwood [2] and Smallwood & Paez [4] is used. More details can also be found in Peeters & Debille [16].

•

Add the columns of u kDC together to yield the m drive signals u k that can be send to the shakers.

• •

Excite the structure. Measure the outputs y k .

•

Estimate the output spectrum matrix Sˆ yy (ù ) .

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Multiple-input mutiple-output random vibration control for a six-degree-of-freedom shaking table Compute an error correction ∆ (ù ) on the lower triangular matrix L(ù ) based on the difference between the

•

REF

reference S yy

( ù) and the estimated output spectrum Sˆ yy (ù ) . The error correction procedure is outlined in

Smallwood [2][3][5] and also in Peeters & Debille [16].

4.

PHYSICAL REALISABILITY OF THE SPECTRUM MATRIX

In MIMO random control not only the vibrations along each axis need to be controlled, but also the vibration relations that exist between each set of axes. In other words, next to reference power spectral densities, also reference cross spectral densities need to be specified by the user. For creating a test specification in the form of a PSD many procedures exist and a lot of experience is available in the test labs. Typically, simulated or measured PSDs are averaged, enveloped, smoothed, etc. to obtain the specification. More advanced methods use spectral damage equivalence techniques, such as fatigue damage spectrum and maximum response spectrum, to come to a synthetic PSD [17][18]. Procedures to create a CSD specification are not widely available. Moreover using the same approach as in case of PSDs may lead to test specifications that are not physically realisable. Before starting the vibration control experiment, it is therefore better to check this realisability. In the following we are mainly focussing on mathematical constraints. These mathematical constraints lead to interesting general conclusions on how a MIMO test specification should look like. It should be noted that there exist also purely physical constraints such as shaker, amplifier, and structural limitations, but these are not discussed in this paper.

4.1. Cross spectra, coherences and phases From the definition of the spectrum matrix, it is possible to prove that it is a positive semi-definite matrix [19]. By consequence, if the user specifies a spectrum matrix that is not positive semi-definite, it is not physically realisable. Loosely speaking, a positive semi-definite matrix has positive diagonal elements and off-diagonal elements that are “not too big”. The exact definition is found in [15]. A first way to obtain a reference cross spectrum matrix is simply taking a measured spectrum matrix. These measurements could for instance originate from a real-life experiment under representative loading conditions. Using such a matrix is always safe, since it can be proven that it is always positive semi-definite [19]. A second way to obtain a reference cross spectrum matrix is to define a profile of coherences and phases as a function of frequency. The cross spectra can be determined from power spectra, coherences and phases. For instance, the cross spectrum between x and y direction S xy equals:

S xy = ã 2xy S xx S yy e

jϕxy

(8)

where ã xy is the coherence value between x and y ; S xx and S yy are the power spectra in x and y direction1 ; ϕ xy 2

is the phase angle between x and y . The cross spectra between y and z direction and z and x direction are defined in a similar way. The advantage of using coherences and phases instead of cross spectra is that they have a clear physical meaning. For instance if the xy phase is 0°, the movement will preferably occur along a 45° straight line between the normalized x and y axes. Moreover if the xy coherence is 1, it will be a perfect straight line; otherwise the movement will somewhat deviate from the straight line. Defining the reference in this manner (8) improves insight into the specification but does not remove the burden from the user of determining if the cross-spectrum is realisable. An example of a set of coherences that leads to an unrealisable spectrum matrix is:

ã 2xy = 1 , ã 2yz = 1 , ã 2zx = 0

(9)

Indeed, if there exist a perfect linear model between x and y ( ã xy = 1 ), and a perfect linear model between y and z 2

( ã yz = 1 ), also a perfect linear model exists between z and x , and a coherence value ã 2zx = 0 is not possible (it 2

1

Note that the symbol

S yy

denotes the (scalar) power spectrum in y direction. In equation (2) the same symbol denoted the spectrum matrix between

all outputs (of which the outputs in x and y direction are two components).

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Multiple-input mutiple-output random vibration control for a six-degree-of-freedom shaking table should even be 1). For this set of coherences the situation is quite clear, but what about other sets of coherences? In the following, we try to answer this question.

4.2. Two output channels From the fact that a spectrum matrix should be positive semi-definite, conditions for the coherences and phases can be specified. To fix the ideas, we start the derivation for two channels. The cross spectrum matrix S ∈ ≤ 2×2 looks like:

[

]

 S xx X  S = E[   X * Y * ] =  * Y   S xy

S xy  S yy 

(10)

where •* denotes the complex conjugate. If a matrix is positive semi-definite its determinant is positive:

det(S ) ≥ 0 ⇔ S xx S yy − S xy* S xy ≥ 0 ⇔ γ

2 xy

=

S xy

2

S xx S yy

(11)

≤1

Elaborating this inequality leads to the very well known condition for the coherence value: it is a positive number that is always smaller than or equal to 1.

4.3. Three output channels For three channels, the cross spectrum matrix S ∈ ≤ 3× 3 looks like:

X  S = E[  Y  X *  Z 

[

Y*

S xx  * Z ] = S *xy S zx 

]

S xy S yy S *yz

S *zx   S yz  S zz 

(12)

From equation (12), the condition that the determinant is positive can be written as:

det(S ) ≥ 0 ⇔ S xx ( S yy S zz − S yz S *yz ) − S *xy ( S xy S zz − S zx* S *yz ) + S zx ( S xy S yz − S *zx S yy ) ≥ 0

(13)

In this expressions, the cross spectra can be substituted by their expressions as a function of power spectra, coherences and phases; see equation (8):

S xy = ã 2xy S xx S yy e S yz = ã 2yz S yy S zz e S zx = ã 2zx S zz S xx e

jϕxy jϕ yz

(14)

jϕzx

Inserting the expressions (14) into (13) and dividing the inequality by the 3 power spectra, yields following inequality:

1 − γ 2xy − γ 2yz − γ 2zx + 2 cos(ϕxy + ϕ xy + ϕ xy ) γ 2xy γ 2yz γ 2zx

≥ 0

(15)

This inequality is a necessary condition for this spectrum matrix to be positive semi-definite. By introducing the user’s choices for the 3D coherences and phases into inequality (15), it can be verified if these choices are physically meaningful. Next to warning the user that his choices are unfortunate, it would be most useful to provide corrected values. For this aim, we assume that the xy and yz values are fixed and that we possibly modify the zx values as to satisfy inequality (15). To simplify the derivation, we specialize to the case where all phases are 0°:

ϕ xy = 0 , ϕ yz = 0 , ϕ zx = 0

(16)

Inequality (15) simplifies to:

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(17)

Multiple-input mutiple-output random vibration control for a six-degree-of-freedom shaking table

Figure 3: Lower (left) and upper (right) limit for the third coherence (all

ϕ = 0 ).

As demonstrated in [19], basic calculus leads to following result: if γ xy + γ yz ≤ 1 , an upper limit for the third 2

2

coherence exists:

γ 2zx ≤  

γ 2xy γ 2yz + (1 − γ 2xy ) (1 − γ 2yz )  

2

(18)

In the other case, namely if: γ xy + γ yz > 1 , both a lower and an upper limit exist, and the conditions for the third 2

2

coherence become:

 γ 2 γ 2 − (1 − γ 2 )(1 − γ 2 )  ≤ γ 2 ≤  γ 2 γ 2 + (1 − γ 2 ) (1 − γ 2 )  xy yz zx xy yz  xy yz   xy yz  2

2

(19)

The lower and upper limits for the third coherence as a function of the two first coherences are graphically represented in Figure 3.

4.4. Cholesky factorisation The first step in the MIMO random control algorithm is the Cholesky factorisation of the reference spectrum matrix; see equation (4). A Cholesky factorisation can only be applied to positive (semi-) definite matrices. However, it is important to notice that the mathematical requirement for the spectrum matrix to be positive semi-definite is not at all restrictive. It is already contained in the physical requirement of a realisable spectrum matrix.

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Figure 4: Inside view on the mechanical system simulator. All modal parameters are adjustable.

Figure 5: Stabilisation diagram obtained by applying the LSCF method to the simulator FRFs.

5. MIMO RANDOM CONTROL RESULTS The MIMO random control algorithm outlined in previous sections is implemented in the LMS Test.Lab software [20] running under Windows NT. The drives are generated and the control and measurement channels measured using LMS Scadas III hardware. In order to verify the performance of the implementation, a real-life MIMO random vibration control experiment was carried out using a mechanical system simulator built with electronic components (Figure 4). The design of the simulator was (loosely) inspired by the shaking table set-up at Hill AFB (see Figure 1). The simulator has 8 inputs and 8 outputs. The number of modes was limited to 8. They are situated in the frequency range 0-2000 Hz. The first two input and output channels are labelled as x-channels, channels 3 and 4 as y-channels, and channels 5-8 as z-channels. LMS Cada-X Time Waveform Replication [21] was used to generate uncorrelated random signals that are sent to the 8 simulator inputs and to measure the system response at the 8 outputs. From the measured time signals, an 8 × 8 frequency response function (FRF) matrix can be estimated, see equation (7). A 3 × 3 submatrix of this FRF matrix involving all 3 directions x, y, z is represented in Figure 6. The columns correspond to the inputs; the rows correspond to the outputs. The recently developed LSCF modal parameter estimation method [22] is applied to the FRFs, resulting in the stabilisation diagram of Figure 5. The 8 modes are clearly visible in this diagram. It is now the task of the MIMO control algorithm to find suitable excitation signals that generate the desired responses. The reference PSDs at the 8 control locations are shown in Figure 7. The piecewise linear shapes of the reference PSDs are quite arbitrarily chosen. The top part of Figure 7 represents environmental testing results obtained with the first drives, i.e. before the first loop closure in which the error correction takes place. There are several spectral lines where the measured spectrum does not match the desired one. This shows that the initial FRF (7) is not perfect and that the L factor needs to be modified to compensate for the errors in the FRF; see equation (5) and Figure 2. The effectiveness of this procedure is demonstrated in the middle and bottom parts of Figure 7, in which the control results after 1 iteration and at the end of the test are shown: the spectra are well within the alarm (±3dB) and abort (±6dB) tolerances. In this control experiment, all intra-axes reference coherences are put equal to 1 at all frequencies. All inter-axes reference coherences are going2 from 0 at 15 Hz to 1 at 2000 Hz. The achieved coherences at the end of the test are shown in Figure 8. They correspond very well to the target coherences.

2

It is probably difficult to attribute a physical meaning to this inter-axes coherence behaviour. This fact however does not decrease the value of the control experiment.

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Figure 6: Some typical FRFs extracted from the 8 × 8 FRF matrix measured on the simulator. The columns are respectively related to the 1 st, 3 rd and 5 th input. The rows are respectively related to the 1 st, 3 rd and 5 th output.

6. CONCLUSIONS This paper presented a MIMO random vibration control algorithm. The flow chart was given and the elements were discussed in some detail. A lot of attention was paid to physical realisability aspects of the user-defined reference spectrum matrix. Finally the performance of the algorithm was verified by a real-life control experiment using an 8 input – 8 output mechanical system simulator. The MIMO random control results were excellent.

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Figure 7: MIMO random control test results using the mechanical system simulator (Figure 4). (Top) Results with the first drives, i.e. before the first loop closure. (Middle) Second drives results, i.e. after the first loop closure. (Bottom) Control achieved at the normal end of the test, i.e. after a lot of iterations. (Left) 2 PSDs along the x-axis. (Middle) 2 PSDs along the y-axis. (Right) 4 PSDs along the z-axis.

Figure 8: (Left) Intra-axes coherences at the end of the test. (Right) Inter-axes coherences at the end of the test.

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