Improved MIMO FRF estimation and model updating for robust Time Waveform Replication on durability test rigs B. Cornelis1, A. Toso1, W. Verpoest1, B. Peeters1 1 Siemens Industry Software NV Interleuvenlaan 68, B-3001, Leuven, Belgium e-mail:
[email protected]
Abstract A durability test on a multi-axial (MIMO) hydraulic test rig requires an accurate replication of a predefined load sequence. The so-called ‘Time Waveform Replication’ (TWR) iterative learning control method is the current state-of-the-art procedure for load replication. The TWR procedure relies on an experimentally identified MIMO Frequency Response Function (FRF) matrix model. As a more accurate FRF model would directly improve the TWR performance and robustness, this paper investigates two approaches to improve the FRF model quality. A first approach is to optimize the excitation signals which are used in the system identification. Alternatives such as pure random and pseudo-random excitation are compared. A second approach is to update the FRF model during the TWR iterations, i.e. the so-called ‘Adaptive Modelling’ procedure. Both approaches for improving the FRF accuracy are analyzed while taking into account the typical characteristics of the MIMO durability test rigs.
1
Introduction
The design of a new vehicle typically involves extensive tests on early prototypes. One of the key design goals is that a certain ‘target durability’ has to be reached. Namely, is has to be ensured that the vehicle will endure the operational loads that will occur during its total intended lifetime, without component failures [1]. Test drives on the road (either on public roads or on specialized proving grounds) have great value for testing the durability of a new vehicle design, but they also pose some drawbacks. Namely, the test procedures can become very time-consuming and expensive, and the results may be adversely affected due to the reliance on a driver and uncontrollable variability of traffic and weather conditions. As a consequence, it has become standard practice to partially replace the test drives by a so-called ‘service load simulation’, where the test drive is simulated in the laboratory on a hydraulic, multi-axial (MIMO) test rig [2-6]. In addition to reduced cost and better test repeatability, the service load simulation also allows for an acceleration of the test (i.e. a reduction of the required testing time). Namely, load analysis and editing techniques can be applied to the original measured signals so that the required testing time is decreased, while the damage potential is preserved [1, 6]. A successful service load simulation requires an accurate replication of the service loads (the ‘target signal definition’), so that the laboratory test is representative of the real-life loading during the test drive on the road. Prior to the actual durability test, it is required to perform a so-called ‘drive file generation’ procedure which generates the actuator drive signals, i.e. the input signals which drive the actuators of the test rig. These drive signals need to be such that the pre-defined target signals are indeed accurately reproduced on the test rig. The so-called ‘Time Waveform Replication’ (TWR) procedure is the current state-of-the-art solution for drive file generation [2-5]. The TWR procedure is an offline Iterative Learning Control (ILC) method [7], whereby an experimentally identified MIMO Frequency Response Function (FRF) matrix model is utilized in the iterations. However, as both the hydraulic test rig and the specimen under test (e.g. a car suspension with shock absorber and bushing elements) actually behave as nonlinear systems, the procedure can have convergence difficulties (or even diverge), so that a large amount of TWR iterations are required. As a consequence, the drive file generation can take up to 50% of the total
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duty cycle of the durability test rig [5], so that any improvements in the TWR convergence rate would have great value. Several research efforts were previously undertaken in order to improve the TWR convergence rate in the presence of nonlinearities. In [5], the offline TWR procedure is augmented with an online linear feedback controller. The authors of [8] propose a decoupling approach which is applicable to a square MIMO system. The full MIMO system is transformed into decoupled SISO systems, which are easier to control in a robust manner. A TWR extension which uses a nonlinear system model is considered in [9]. As the system model has to be inverted in the TWR procedure, the assumed nonlinear model requires a modified procedure where the model inversion is reformulated into an optimization problem. Finally, the authors of [10] propose a passive control strategy whereby a structural component is applied to the system in order to improve its controllability. Although these previous research efforts indeed improve the convergence rate and robustness of the TWR procedure in certain cases, they may require expert knowledge and additional setup efforts in order to be applied successfully in practice. For industrial applicability, the augmented TWR procedure should ideally be a full-automatic procedure, which does not require expert interventions. Also, if there is additional time required to properly set up the procedure, the time gained by improving the TWR convergence rate should outweigh the time lost because of more complicated setup efforts. This paper discusses two approaches to improve the convergence rate and robustness of the MIMO TWR method in a time-efficient manner:
First of all, it is clear that an improved system identification, which leads to better FRF models, will also directly improve the TWR performance. Several different excitation signals (e.g. pure random or so-called pseudo-random excitation) have been proposed in literature for (MIMO) system identification. An analysis and comparison of these alternative excitation strategies is made using experimental data obtained on a durability test rig. A second approach updates the FRF model during the TWR iteration phase, i.e. the so-called ‘Adaptive Modelling’ procedure. The performance of the Adaptive Modelling procedure is demonstrated on a virtual test rig consisting of a multi-body simulation model of a vehicle, whereby real-life measured spindle forces are used as target definition.
The remainder of the paper is organized as follows. Section 2 reviews the theoretical background of the TWR control method. Section 3 analyzes different excitation strategies which can be used for FRF system identification. Section 4 presents the Adaptive Modelling procedure which updates the FRF model during the TWR iterations. Finally, Section 5 presents the conclusions of the paper.
2
Time Waveform Replication (TWR)
The Time Waveform Replication (TWR) procedure is the current state-of-the-art solution for target simulation on a hydraulic test rig [2-5]. The target simulation process is illustrated in Figure 1. The goal is to replicate the so-called target signals, i.e. the load signals (e.g. spindle forces) which were measured during a test drive on the road. To achieve this, the proper actuator drive signals have to be found so that the responses are indeed equal to the targets. The drive signals are updated in an offline iterative procedure where the total error signals , i.e. the differences between the responses and target signals, are utilized to calculate updates to the drive signals. The system (between the drive signals and responses) includes the specimen under test (e.g. a suspension or full vehicle), the hydraulic test rig and the internal PID controllers which control each actuator. Some test setups have more responses than drive signals, so that in general is a non-square system. As the error signals are calculated at the response side of the system, a back-calculation using the inverse (or the pseudo-inverse in the non-square case) of the system transfer has to be performed in order to find the appropriate updates to the drive signals. A first phase of the TWR procedure therefore consists of a system identification stage where an experimental system model is calculated.
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Figure 1: Target Simulation on multi-axial durability test rig The standard TWR procedure identifies a linear frequency-domain system model (i.e. the FRF matrix) ̂ , using the -estimation technique [11, 12], i.e. ̂( )
[ ̂ ( )][ ̂ ( )]
(1)
̂ ( ) and ̂ ( ) are the estimated cross- and autospectrum matrices, respectively. For conciseness, the frequency-domain variable is omitted from now on. The system identification by Eq. (1) requires random, uncorrelated input signals so that ̂ is well-conditioned. Ideally, the input signals should also excite the specimen at a level comparable to the operational signal levels encountered in the road test, so that a good local linear approximation of the nonlinear system is obtained. These requirements can generally not be fulfilled at the same time, as large-amplitude random excitations may destroy the specimen. For displacement-controlled actuators, white-pink excitation signals (where high amplitude white noise is only applied in the low frequencies) are typically used as a compromise solution, as the system is still sufficiently excited at higher frequencies regardless of the decreasing amplitude. For forcecontrolled actuators, a flat (white) frequency spectrum has to be used in order to provide sufficient excitation to the structure. Section 3 will further discuss how the excitation time waveforms can finally be generated, given the Power Spectral Density (PSD) specifications (i.e. a white-pink or a white spectrum). The second phase of the TWR procedure is commonly referred to as the Target Simulation phase [5]. During this phase, the drive signals are iteratively updated in an offline manner until the responses sufficiently replicate the target signals. The TWR procedure achieves this by the following update equations [5]: ̂ (2) [ ] ( )
(3)
It has to be noted that Eqs. (2) and (3) are frequency-domain equations (where the frequency-domain variable is omitted as before), so that transformations to-and-from the time domain are required in order to obtain the time waveforms. The subscript in these equations indicates the iteration step. The signals and are the drive signals (input) and responses (output), respectively, while the signals are the target signals that have to be replicated. By this definition, the error signal in iteration is equal to . The matrix is a diagonal matrix which contains the so-called iteration gains , whereby . The procedure is initialized by setting . This implies that the first drive signals are obtained by applying gains on the target signals and then filtering the result with the inverse of the estimated FRF model, i.e. ̂ . Obviously, the optimal drive signals would be directly obtained in absence of modelling errors ( ̂ ) and when setting (i.e., equal to the identity matrix). In practice however, there will always be a modelling error so that the drive signals need to be further updated as in Eqs. (2) and (3). Moreover, the iteration gains typically have to be chosen more conservatively (e.g., ), in order to avoid divergence (cf. infra). Finally, it should be noted that Eq. (3) requires a playback of the drive signals on the actual test rig, whereby the total responses are measured. Each
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iteration of the Target Simulation phase thus consists of an off-line update step (Eq. (2)) followed by an acquisition on the test rig (Eq. (3)). This explains the time-consuming nature of the TWR procedure and the need to improve the TWR convergence rate. The impact of modelling errors on the TWR convergence rate and robustness can also be quantified. To allow for a tractable derivation, it is now assumed that the system is square, and that it can be approximated locally by an invertible FRF matrix . We then define the (inverse) modelling error as: ̂
(4)
By combining Eqs. (2)-(4), it can be demonstrated that the tracking error in iteration step is equal to: [
[
̂ ]
]
(5)
In the absence of modelling errors (i.e., so that ), it can be seen from Eq. (5) that the tracking error will decrease with a factor [ ], or a factor [ ] per channel. The convergence rate can thus be increased by choosing larger iteration gains (with as upper limit case). In presence of modelling errors ( ), the convergence rate will generally decrease. Moreover, there is a risk that the procedure will diverge if the modeling errors are too large. A necessary and sufficient condition for convergence of the TWR procedure is given (see [5]) by the following equation: ‖
[
̂ ]
‖
(6)
As discussed in [5], the convergence bound in Eq. (6) implies that the presence of modelling errors limits the allowable range of the iteration gains . The larger the modelling errors, the smaller the allowable iteration gains (and vice versa). If the modelling errors can be reduced, the TWR procedure can thus be made more robust (i.e., it is more likely that the convergence criterion of Eq. (6) is satisfied), while at the same time the convergence rate may also be increased (especially so if larger iteration gains can be applied).
3 3.1
Optimal excitation signals for MIMO FRF estimation Theory
There are several different types of excitation signals which can be used during the system identification phase [12-14]. In the following, a comparison will be made between two popular choices, i.e. pure random versus pseudo-random excitation. Pure random excitation is commonly used for system identification on durability test rigs. A broadband Gaussian excitation signal with a predefined PSD profile is hereby generated for each input drive channel. During the calculation of the -estimator, the drive and response signals are segmented into overlapping blocks and processed in the frequency domain through the Fast Fourier Transform (FFT). As a consequence, the result will be affected by spectral leakage artefacts due to the implicit periodicity assumptions of the FFT. Therefore, window functions (e.g. a Hanning window) are typically applied in order to reduce the leakage. However, even if a window function is applied, it is not possible to completely remove the spectral leakage. Moreover, the window itself may also introduce distortions in the FRF estimation. An alternative excitation type, which will be further investigated in this work, is the so-called pseudorandom excitation [12-15]. Essentially the pseudo-random excitation signal is a special case of a multisine [16-18], which is given by: ( )
∑
(
)
(7)
In Eq. (7), is the total number of sine components, and component (respectively), and is the fundamental frequency (with
are the amplitude and phase of ⁄ where is the period of
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Figure 2: Pseudo-random input drive signal generation (SISO case) the multi-sine). In the particular case of pseudo-random excitation, the amplitudes are deterministic (i.e. based on a pre-described PSD profile), while the phases are independent random variables drawn from a uniform distribution between and . Figure 2 illustrates how a pseudo-random signal is generated block-per-block using Eq. (7), for the SingleInput-Single-Output (SISO) case. In the figure, it is assumed that the signal has a white (flat) PSD amplitude. The deterministic amplitude is combined with a randomized phase and processed by an Inverse FFT, so that one block of the time waveform is generated. The block is then repeated a number of times, resulting in so-called ‘delay blocks’ and ‘capture blocks’ (in the figure, delay blocks and capture blocks are assumed). The delay blocks are discarded during the -estimation, in order to make sure that the system is in its periodic regime (i.e. all transients have died out). So, only the capture blocks of the drives and responses are effectively used in the -estimation. As illustrated in the figure, each capture block can also be repeated a number of times in order to average out additive noise (similar to Time Synchronous Averaging [12]). After a particular number of repeats, a completely new block will be ‘realized’ (by applying a new phase randomization) and again repeated as outlined before. Multiple realizations allow for averaging out nonlinearities, which is especially crucial for durability test rigs. When pseudo-random excitation is applied in the MIMO case, some care has to be taken so that the input signals are sufficiently decorrelated (i.e., so that the autospectrum matrix ̂ in Eq. (1) is wellconditioned). Therefore, the so-called Hadamard phase decorrelation method [15-18] is typically applied when generating MIMO pseudo-random excitation signals. The method makes use of the mathematical properties of Hadamard matrices, which are square matrices whose elements are either or , and whose rows are mutually orthogonal. Some examples of Hadamard matrices can be found using Sylvester’s construction, i.e.: [
]
From its definition, it follows that the
,
, with
Hadamard matrix
[
] and
(8)
satisfies following property: (9)
The Hadamard phase decorrelation procedure generates realizations for a MIMO system with actuators, starting from only realized block of the SISO procedure (Figure 2). This block (and its repeats) is send to each input actuator, with a scaling factor of or taken from the corresponding element of the Hadamard matrix of size . Each column of the Hadamard matrix can hereby be interpreted as a new realization, so that the procedure results in a total of realizations (when starting
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from SISO block). If more than realizations are required, the same procedure can be reapplied starting from different SISO blocks (generated by re-randomizing the phase). By virtue of the Hadamard matrix property of Eq. (9), it can be demonstrated [15-18] that the Hadamard phase decorrelation procedure results in a diagonal autospectrum matrix ̂ . It is however possible that the number of actuators is such that there does not exist a corresponding Hadamard matrix (e.g., ) Fortunately, an acceptable approximation can still be applied whereby the left upper submatrix of the nearest ‘larger’ Hadamard matrix is selected (cf. [17] for the case). For durability test rigs, which typically have a mix of different types of actuators (e.g. force-controlled actuators versus displacement-controlled actuators), there is a further complication that not all drive signals have the same PSD specification (e.g. white versus white-pink). An approximate solution would be to apply the Hadamard decorrelation for each actuator group separately. Alternatively, a multiplication with a frequency-dependent diagonal matrix can be applied in order to shape the spectrum of each drive signal, but then the block generation is more complicated (cf. [17, 18]) than the procedure outlined above. The main advantage of pseudo-random excitation is that the FRF estimation is not affected by spectral leakage, because the excitation is perfectly periodic within a (FFT) block. Therefore, it is also not required to apply a window function. From this point-of-view, the pseudo-random excitation has the potential to result in better quality FRFs, compared to pure random excitation. However, a disadvantage is that there are less blocks available for averaging (given a fixed measurement time), due to the fact that the delay blocks have to be discarded in the pseudo-random procedure. The goal of the experimental validation in Section 3.2 is to analyze the trade-off between the advantages and disadvantages of pure random and pseudo-random excitation, in particular when they are applied on durability test rigs.
3.2
Experimental validation
In this section, the discussed excitation strategies are experimentally validated. The experiments were performed on a suspension test rig (otherwise known as a ‘spindle-coupled’ test rig), whereby the actuators connect to the spindles of a front or rear suspension through dummy wheels (see e.g. [5]). Wheel force transducers are mounted between the dummy wheels and the spindles, which allows for a measurement of the imposed spindle forces and moments. The test rig in this study has two displacementcontrolled actuators and four force-controlled actuators. The two displacement-controlled actuators (one at the left side and one at right side) impose vertical displacements to the dummy wheels. The four forcecontrolled actuators (two at the left side and two at the right side) impose lateral and longitudinal motion. Together these six actuators thus allow controlling the vertical, lateral and longitudinal forces at the left and right spindle (6 spindle forces in total), so that a square FRF matrix model has to be identified. It is also possible to apply an additional braking torque to each spindle (the so-called ‘braking mode’) by using two additional force-controlled actuators, which then results in an FRF matrix. In this work, only the case (‘non-braking mode’) is considered. The signals are sampled at Hz. The force-controlled actuator signals are generated from a PSD profile which is white (flat) between and Hz. The displacement-controlled actuator signals are generated from a white-pink PSD profile which is flat from to Hz and has a dB drop from to Hz. The (SISO) pseudo-random excitation is generated block-per-block as in Figure 2, with only delay block and capture block per realization. This SISO signal is then processed by the Hadamard phase decorrelation procedure in order to generate an input for each actuator, whereby the force and displacement actuators are treated separately. To avoid sudden transitions between blocks of different realizations, a short linear fading window of seconds is applied at the start and end of each block. From simulations it was observed that this window does not have a significant impact on the quality of the estimated FRFs. The FRF matrix is estimated between and Hz, using the -estimator as in Eq. (1). For the estimation with pure random excitation, the acquired signals are segmented into blocks of samples with an overlap of between blocks and with a Hanning window applied to each block to reduce leakage. For the -estimation with pseudo-random excitation, the acquired data is segmented into non-overlapping blocks of samples, whereby no Hanning window is applied.
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Figure 3: Estimated FRFs (amplitude and phase) using pseudo-random excitation, with Hadamard decorrelation (left) vs. without Hadamard decorrelation (right) In a first experiment, the effect of the Hadamard phase decorrelation procedure (for pseudo-random excitation) is validated. Figure 3 shows examples of estimated FRFs (note that only four FRFs out of the FRF matrix are displayed). The results on the left are obtained when Hadamard decorrelation is applied, while the results on the right are obtained without applying Hadamard decorrelation. The drive signals were generated using the SISO procedure (Figure 2) for each channel independently in the latter case. The measured signals have a length of seconds (= non-overlapping blocks) in both cases. As the delay blocks are discarded, only blocks are available for averaging in the estimation. It can be observed that the Hadamard decorrelation results in smoother FRFs (both in the amplitude as in the phase response), which indeed confirms the usefulness of the Hadamard phase decorrelation. In the following, Hadamard decorrelation will therefore always be applied when using pseudo-random excitation. In a second experiment, the pseudo-random excitation (with Hadamard decorrelation applied) is finally compared to the pure random excitation. The measured signals again have a length of seconds in both cases (both for pseudo-random and pure random). In the case of pseudo-random excitation, blocks are available for averaging in the estimation (cf. supra). In comparison, the pure random excitation will have more blocks available for averaging (given the same measurement duration), firstly because the signals are segmented into overlapping blocks and secondly because no delay blocks have to be discarded. For pure random excitation with a total measurement duration a blocksize , and an overlap of , it can be shown that the total number of blocks available in the estimation is given by the following equation: ⌊
⌋
(10)
For the parameter settings in the current experiment, Eq. (10) results in blocks. With pure random excitation, the number of available blocks is thus larger by a factor of four compared to the pseudo-random excitation case (i.e., under the assumed parameter settings and given the same measurement duration in both cases). It should however be noted that the blocks are partly correlated in the pure random case (due to the overlap), so that in terms of ‘equivalent number of averages’ the factor will be slightly less than four. Figure 4 shows the estimated FRFs using the two excitation types, with pseudo-random excitation on the left and pure random excitation on the right. The first ‘row’ of the FRF matrix is displayed, with four force-to-force FRFs (i.e. force actuators at the input, spindle force at the output) displayed at the top and two displacement-to-force FRFs (i.e. displacement actuators at the input, spindle force at the output) displayed at the bottom. The pure random FRFs generally look smoother than the pseudo-random FRFs, so that the pure random excitation seems to be preferable in this application.
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Figure 4: Estimated FRFs (amplitude and phase); force-to-force (top), displacement-to-force (bottom); pseudo-random excitation + Hadamard (left) vs. pure random excitation (right) This observation can be explained by the fact that there are more blocks available for averaging when using pure random excitation (given the same measurement time as in the pseudo-random case). Assuming a SISO system without noise on the input, it can be demonstrated [11, 13] that the variance of the error on the estimated FRF amplitude can be approximated by: | ̂|
| |
(11)
In Eq. (11), are the number of equivalent (independent) blocks which are available for averaging, while is the coherence between input and output (in practice the coherence will be smaller than due to the presence of measurement noise and nonlinearities). Eq. (11) clearly shows the inverse proportional relation between the number of blocks and the error variance, which thus confirms the results in Figure 4. Besides the error which was defined in Eq. (11) (which is sometimes referred to as the ‘random’ or ‘stochastic’ error), there is a second type of error which is mainly caused by the limited FFT resolution and spectral leakage [19]. This error is a systematic error (‘bias’), and a third experiment is conducted to understand its importance. In contrast to the previous experiments, we now use a simulation model which is representative of aerospace structures in ground vibration testing [14]. The simulation system model is specified by a state-space representation, so that a ‘true FRF’ can also be calculated analytically.
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Figure 5: FRF estimation on aerospace structure model; True FRF (blue), FRF amplitude error with pseudo-random (green), FRF amplitude error with pure random (red) Figure 5 shows the amplitude errors versus the ‘true FRF’ (i.e. the analytical solution), both for pure random excitation (in red) and pseudo-random excitation (in green). Again, the total measurement time is the same in both cases, so that pure random has more blocks available for averaging (compared to the pseudo-random case). At frequencies which are far from the structural resonances, it can be seen that the pseudo-random excitation leads to larger errors compared to the pure random excitation. In these frequency regions, the ‘random errors’ are dominant, so that this observation is in accordance with the previous results in Figure 4 (for the durability test rig). If the experiment is redone using more blocks in the estimation (e.g. by increasing the total measurement time), it was also observed that these errors can be decreased, which is in accordance with Eq. (11). At frequencies which are close to the (sharp) resonances, it can however be observed that the pure random excitation leads to (very) large errors. These large errors are actually the systematic ‘bias’ errors which are caused by spectral leakage (hence they do not occur when using pseudorandom excitation, which is not affected by leakage). In accordance with [19], it was observed that these errors cannot be reduced by increasing the number of blocks, but rather by increasing the FFT blocksize. Furthermore, the bias errors are especially large when the resonances are lightly damped (i.e., as is the case for the aerospace structure of Figure 5). In summary, it can be argued that pseudo-random excitation is especially useful when the structure has lightly damped (sharp) resonances (e.g. the aerospace structure in Figure 5), as the bias errors then become important. In durability test rigs, there are typically no such sharp resonances (cf. Figures 3 and 4), so that the random errors (which mainly determine the smoothness of the FRF) can be considered as more important. The random errors can typically be reduced by increasing the number of blocks available for averaging (cf. Eq. (11)), so that pure random excitation seems preferable for this application, if the measurement time is fixed. A final remark is that a special sequence of pseudo-random excitation may allow for an assessment of the nonlinear distortions in the system [13, 16], in addition to estimating the FRF. From this point-of-view, the pseudo-random excitation could in fact be useful for durability test rigs, which indeed have significant nonlinearities. This option was however not further explored in the current work.
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4 4.1
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FRF model updating by Adaptive Modelling Theory
In this section, the so-called ‘Adaptive Modelling’ procedure [20] is reviewed. The Adaptive Modelling procedure improves the TWR convergence by updating the inverse FRF model ̂ after each Target Simulation iteration (i.e. Eqs. (2) and (3)). There are two alternative approaches which can be applied, namely the so-called ‘Forward Prediction’ and ‘Backward Prediction’ methods, as illustrated in Figure 6. The remainder of the paper will only consider the Forward Prediction method, but similar derivations can be made for the Backward Prediction method.
Figure 6: Adaptive Modelling: Forward Prediction (left) vs. Backward Prediction (right) The Forward Prediction method compares the measured responses (see Eq. (3)) with the ‘predicted responses’, i.e. the responses which would be acquired if the estimated FRF ̂ is used instead of the actual system . In presence of modelling errors, the measured responses ( ( )) and predicted responses ̂ ) will be different. The magnitude of this difference is thus an indicator for the quality of ̂ . The (̂ Adaptive Modelling procedure applies an -estimator to and ̂, which results in a square correction matrix i.e. ( ̂ ̂ )( ̂ )
(12)
Again, ̂ ̂ and ̂ are cross- and autospectrum matrices respectively. In principle, the full cross- and autospectrum matrices could be used in Eq. (12). However, this could lead to numerical problems when the responses are correlated (typically this is the case), as was discussed in Section 2. A practical approximation is therefore introduced: the -estimation is performed channel-per-channel (as if the system is completely decoupled). As a consequence, Eq. (12) results in a diagonal correction matrix . ̂ The diagonal correction matrix of Eq. (12) is used to update the inverse FRF model , i.e. ̂ ̂ (13) It can be seen from Eq. (13) that the updated inverse FRF model is obtained as the result of a matrix multiplication, so that all elements of the matrix are affected. However, not every possible modification can be applied using Eq. (13), because of the diagonal structure of (for example, it is not possible to ̂ permute two columns of ). The limitations imposed by the diagonal structure of can be explored by considering the (inverse) modelling error , which was previously defined in Eq. (4). To allow for a tractable derivation, it is
again assumed that the system is square, and that it can be approximated locally by an invertible FRF matrix . The (inverse) modelling error , before updating the FRF, is then given as: ̂
(14)
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The goal of the Adaptive Modelling procedure is to find a correction matrix ̂ . From Eqs. (13) and (14), it can be calculated that the ‘optimal’ ̂ , is given by the following expression: ̂
so that, using Eq. (13), , which indeed achieves (15)
It can be checked that this expression also occurred as a factor in Eqs. (5) and (6). If the modelling error is equal to zero, the optimal is obviously equal to the identity matrix, which by Eq. (13) means that the inverse FRF model is not updated. In presence of modelling errors, the second term ( ̂ ) will be non-zero, which means the inverse FRF model should be updated. It is noted that Eq. (15) cannot be calculated in practice, as the modelling error is unknown by definition. Instead, is estimated by the channel-per-channel -estimator, as indicated in the previous paragraph. The result of the channel-perchannel -estimation (i.e. Eq. (12)) yields a diagonal , whereas it can be observed from Eq. (15) that the optimal may be a full matrix (i.e. ̂ is not necessarily a diagonal matrix). However, if ̂ is a diagonal-dominant matrix, it is expected that the diagonal estimation may yield a good approximation. This would for example be the case for systems that are diagonal or weakly-coupled. For systems with strong cross-couplings (e.g. a suspension rig where movement of one actuator rod causes movements of other rods as well), the Adaptive Modelling procedure may still be able to reduce the modelling errors, but probably to a lesser extend compared to systems with weak cross-couplings.
4.2
Experimental validation
The performance of the Adaptive Modelling procedure is now validated experimentally. A multi-body simulation model [21] of a car is used as the system in these experiments. The model includes bushing elements which introduce nonlinear effects. The car body and rear axle are constrained so that only the front axle is allowed to move. The model is excited at the front wheels by vertical, lateral and longitudinal displacements. The responses of interest are the (left and right) vertical, lateral and longitudinal front spindle forces. This thus results in a system. The system identification (i.e. Eq. (1)) is performed between and Hz using pure random excitation. The (displacement) drive signals are generated from a white-pink PSD profile which is flat from to Hz and has a dB drop from to Hz. Real-life spindle forces, measured by a wheel force transducer on a representative predecessor vehicle, are used as the target definition for this experiment. The signals have a total length of seconds and are segmented into blocks of samples (sampled at Hz) with overlap. The frequency range of interest is between and Hz. The iteration gain factors (see Eq. (2)) are fixed to for all channels, frequencies, and target simulation iterations. The longitudinal channels are of particular interest, as it was observed that the ‘basic TWR’ approach (i.e. without Adaptive Modelling) diverged for these channels. For conciseness, only the results for the (left) longitudinal channel will thus be discussed. The time waveforms, which are obtained after TWR iterations (Eqs. (2) and (3)), are shown in Figure 7. The result for the basic TWR (without Adaptive Modelling) is shown on the left, the result for the TWR with Adaptive Modelling activated is shown on the right. It can be observed that a divergence occurred in the result without Adaptive Modelling, as there is a segment in the signal where the response (in red) overshoots the target signal (in blue). An overlay of the PSDs (calculated on the response signals) also provides useful insights, cf. Figure 8. The curve denoted by ‘iter. 1’ is the response PSD which is obtained after the first TWR iteration, i.e. the first acquisition on the test rig using the first drive signals. If the FRF is well-estimated, it is expected that this curve is dB below the target PSD curve (as the iteration gains are fixed to ). It can be observed that this is not the case in the frequency band Hz (where also an anti-resonance frequency occurs in the systen, cf. [20]). The curve denoted as ‘basic TWR’ shows the response PSD obtained after TWR iterations without applying Adaptive Modelling. It can be seen that the response overshoots the target, i.e. the procedure is diverging. This overshoot particularly occurs in the Hz band. In contrast, when Adaptive Modelling is activated, it can be observed that the response PSD matches the target PSD over the entire frequency range.
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Figure 7: Time waveforms (longitudinal spindle force) after 7 TWR iterations; (left): without Adaptive Modelling, (right): with Adaptive Modelling
Figure 8: PSD overlay Two commonly used performance metrics to assess the convergence rate of the TWR procedure are the so-called relative Root Mean Square (RMS) error and the RMS response [5]. The relative RMS error is the ratio of the RMS of the error signal over the RMS of the target signal. Ideally, the relative RMS error should converge towards after several TWR iterations. The RMS response is the ratio of the RMS of the response signal over the RMS of the target signal. The RMS response should converge towards (without overshooting in any iteration). The results for ‘basic TWR’ and ‘Adaptive Modelling’ are shown in Figure 9. Again, it can be observed that the ‘basic TWR’ diverges whereas ‘Adaptive Modelling’ yields a satisfactory result.
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Figure 9: Relative RMS error (left) and RMS response (right)
5
Conclusions
Nowadays, a large part of automotive durability testing is performed in the laboratory on multi-axial test rigs, whereby the real-life input loads are replicated by means of the TWR control method. A significant amount of testing time is currently lost due to the slow TWR convergence rate, which is directly related to the quality of the assumed FRF model. Therefore, this work investigated two approaches to improve the FRF quality:
First, a system identification of a suspension test rig was performed using different excitation strategies (i.e. pure random versus pseudo-random excitation). The results indicate that especially the so-called ‘random errors’ in the FRF estimation are of importance in durability test rigs, so that it is beneficial to increase the number of blocks used in the averaging. Therefore, the pure random excitation seems preferable over pseudo-random excitation, as pure random will yield more blocks in a given measurement time. Conversely, the pseudo-random excitation is clearly preferable if the ‘bias errors’ are significant. This is especially the case for structures with lightly damped (sharp) resonances (e.g. aerospace structures), but less so for the typical durability test rigs. Second, the so-called ‘Adaptive Modelling’ procedure can be applied in order to update the FRF model during the TWR iteration phase. The (inverse) FRF model is hereby multiplied with a diagonal correction matrix, which is obtained by a channel-per-channel -estimation between the acquired responses and the ‘predicted responses’ (i.e., the responses that would be obtained if the actual system is replaced by the FRF model). It was demonstrated that an ‘optimal’ update (i.e., an update which would completely eliminate all modelling errors) may not always be applicable in practice by the described procedure, due to the imposed diagonal structure of the correction matrix. However, it is expected that the procedure can find a good approximation for systems that are diagonal or weakly-coupled. The performance of the Adaptive Modelling procedure was also validated using a multi-body simulation model of a vehicle.
Acknowledgements Part of this work was supported by the IWT (Flemish Agency for Innovation by Science and Technology) in the frame of the IWT O&O project ADVENT (Advanced Vibration Environmental Testing).
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