Neural Network algorithms for magnetic diagnostics in the LTP

DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 1/20 Neural Network algorithms for magnetic diagnostics in the LTP Doc. Ref.: S2-IEC-TN-3052 Issue: 1.0 Date: 11-Mar-2009 CI number: L 32D0 Approval List Name Signature Date Prepared by: M. Diaz-Aguil´o E. Garc´ıa-Berro A. Lobo 11-March-2009 Revised by: A. Lobo 11-March-2009 Approved by: A. Lobo 11-March-2009 Authorised by: I. Lloro 11-March-2009 DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 2/20 (Intentionally blank) DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 3/20 Document Distribution List Name Position Company Ivan Lloro IEEC Stefano Vitale University of Trento Karsten Danzmann AEI Hannover Paul McNamara ESA Laurent Trougnou EMC and Magnetics Engineer ESA Bengt Johlander Senior Instrument System Engineer ESA C´esar Garc´ıa LPF Payload Manager ESA DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 4/20 Document Status Sheet Author Issue Date Page(s) Change description M DiazAguilo 1.0 11/03/2009 All First version DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 5/20 Table of Contents DocumentApprovalList.................................. 1 DocumentDistributionList ................................ 3 DocumentStatusSheet .................................. 4 TableofContents...................................... 5 ListofFigures ....................................... 6 ListofTables........................................ 6 Acronyms .......................................... 6 Applicabledocuments ................................... 7 Referencedocuments.................................... 7 1 Introduction and scope of this document 7 2 Detailed description of the problem 8 2.1 Interpolationtheory ................................. 10 2.2 Discussionsofar ................................... 12 2.3 Numericalsimulations ................................ 12 2.4 A more contrived interpolation scheme . . . . . . . . . . . . . . . . . . . . . . . 14 3 A novel approach: Neural Networks 14 3.1 Neuronmodel..................................... 15 3.2 Neural network architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Learning paradigms and learning/training algorithms . . . . . . . . . . . . . . . 17 3.3.1 Learningparadigms.............................. 17 3.3.2 Learningalgorithms ............................. 17 3.4 Performanceassessment ............................... 18 4 Results 18 4.1 Fieldestimationresults................................ 19 4.2 Field gradient estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Conclusion 20 DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 6/20 List of Figures 2.1 Schematic of the LPF Science-craft: The LCA is in the centre, surrounded by a double cylindrical shield, and outside it a number of electronic boxes are represented; most of them are sources of magnetic field. The four magnetometers are the white little boxes indicated by the arrows, and are mounted across the two cylindricalshells. ................................... 9 2.2 Qualitative behaviour of the magnetic field and gradient inside the LCA. Scales arenotreal. ..................................... 10 2.3 Averaged estimation errors in the components of the magnetic and of its modulus. They are reported in relative percent, i.e., 100(real −estimated value)/(real value). Colors correspond to each of the LTP TMs, respectively. . . . . . . . . 14 3.1 Schematics of the operations performed by an artificial neuron. . . . . . . . . . 15 3.2 Feed-forward neural network architecture. Magnetometers readings are the system inputs, and estimates of the field and gradient at the positions of the test masses are the outputs of the system. In this architecture, one only intermediate, or hidden layer is assumed. Each of the circles represents one neuron and corresponds to the model shown in figure 3.1. ................... 16 4.1 Probability density function of the error distributions for each field component at the position of test mass 1 (black trace) and test mass 2 (red): top left plot for Bx, top right for By, bottom left for Bzand bottom right for |B|. ...... 19 4.2 Probability density function of the errors distribution for the three components of ∇Bx. From top to bottom: ∂Bx/∂x,∂Bx/∂y and ∂Bx/∂z at the positions of the test masses. Errors are given in percents, black traces corresponding to TM1,andredonestoTM2. ............................. 20 List of Tables Acronyms AEI Albert-Einstein-Institut, Max-Planck-Institut fr Gravitationsphysik ASU Astrium Ltd., Astrium UK CI Configuration Item DC Direct Current DDS Data Management and Diagnostics Sub-System DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 7/20 DMU Data Management Unit ESA European Space Agency ICL Imperial College of London IEEC Institut d’Estudis Espacials de Catalunya IFAE Institut de F´ısica d’Altes Energies (Barcelona, Spain) LCA LTP Core Assembly LISA Laser Interferometer Space Antenna LPF LISA Pathfinder (formerly SMART-2) LTP LISA Technology Package LTPA LTP Architect NASA National Aeronautics and Space Administration NTE Nuevas Tecnolog´ıas Espaciales, S.A. (Lli¸c`a d’Amunt, Spain) PDF Probability Density Function TM Telemetry / Test mass Applicable documents Ref. Title Doc Number Issue Date AD1 Science Requirements and Toplevel Architecture Definition for the LISA Technology Package (LTP) on Board LISA Pathfinder (SMART-2) LTPA-UTN-ScRD 3.1 30-06-2005 Reference documents Ref. Title Doc Number Issue Date RD1 LTP Magnetic Field Interpolation S2-IEC-OTH-3026 1.0 25-11-2008 RD2 Vojislav Kecman, Learning and soft computing The MIT Press 1st edition 2001 DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 8/20 1 Introduction and scope of this document The Magnetic Diagnostics Subsystem of the LTP includes a set of four tri-axial fluxgate magnetometers, intended to sense with high precision the magnetic field at the positions they occupy in the LCA. Their readouts do not however provide a direct measurement of the magnetic field at the TMs, and an interpolation method must be implemented to obtain that information. However, such interpolation process faces serious difficulties: indeed, the size of the interpolation region, i.e., the LCA interior, is excessive for a linear interpolation to be reliable, but the number of magnetometer channels does not provide sufficient data to go beyond that (poor) approximation. This document sketches what could be a possible alternative to address the magnetic interpolation problem by means of neural network algorithms, and gives a few examples. At present, results look promising, hence the method is under exploration for improvement. The key point in this proposal is the ability neural networks have to learn from suitable training feedback. It appears that learning efficiency can be best improved by making use of data obtained in on-ground measurements prior to mission launch in all relevant satellite locations and real operation conditions. 2 Detailed description of the problem Magnetic noise in the LTP is budgetted to be a significant fraction of the total readout noise: 1.2×10−14 m s−2Hz−1/2out of 3×10−14 m s−2Hz−1/2—see Table 8.1 in [AD1]. This noise occurs because residual magnetisation and susceptibility in the proof masses couple to the surrounding magnetic field, giving rise to a force F=M+χ µ0 B·∇BV(2.1) in each of the TMs, where BMagnetic field in the TM MDensity of magnetic moment (magnetisation) of the TM VVolume of the TM χMagnetic susceptibility of the TM µ0Vacuum magnetic constant (4π×10−7m kg s−2A−2) h· · ·i TM volume avergage of enclosed quantity Magnetic noise is generated because the magnetic field and its gradient randomly fluctuate in the regions occupied by the TMs1. Quantitative assessment of magnetic noise in the LTP therefore requires real-time monitoring of the magnetic field, which in LPF is done by means of a set of four tri-axial fluxgate magnetometers. These devices have a high permeability magnetic core, which drives a design constraint to keep them somewhat far from the TMs. The price of 1Additional noise comes from TM susceptibility and magnetic remanence fluctuations, normally expected to be much less important. DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 9/20 Figure 2.1: Schematic of the LPF Science-craft: The LCA is in the centre, surrounded by a double cylindrical shield, and outside it a number of electronic boxes are represented; most of them are sources of magnetic field. The four magnetometers are the white little boxes indicated by the arrows, and are mounted across the two cylindrical shells. that is of course that an interpolation problem needs to be solved before the field in the TMs can be inferred. The sources of magnetic field are essentially due to components inside the spacecraft (S/C), as the interplanetary magnetic field is orders of magnitude weaker. There are no sources of magnetic field inside the LTP Core Assembly (LCA), all being placed within the S/C, outside the LCA walls —see figure 2.1. The number of identified sources is in the order of 50, and they can be modelled as magnetic dipoles in first approximation. Refinements may be useful, but this is not an issue for this document. What matters instead is that, the sources being outside the LCA, the magnetic field is smaller towards the centre of the LCA than it is in its periphery, where the magnetometers take measurements. Figure 2.2 shows a qualitative diagram of the situation. As we shall now see, there is no interpolation method which can give account of this circumstance on the basis of the information produced by the three vector magnetometers. DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 16/20 where the superindex Tstands for transpose matrix; in this case, wTis a row vector while xis a column vector, so that wTxis the scalar product of wand x. Finally, w0is the bias. •The above sum is used as the argument to the so called activation function,ϕ(Σ). The neuron’s output, also known as its activation, is thus o=ϕ(Σ) (3.2) In general, ϕ(Σ) can be selected in many different ways; here, differentiable activation functions will be used, which suit well the gradient descent back-propagation learning algorithm —see section 3.3 below. 3.2 Neural network architecture Artificial neural networks are software or hardware models inspired by the structure and behaviour of biological systems, and they are created by a set of neurons distributed in layers. There are many different types of neural networks in use today, but the architecture of a so called feed-forward network, where each layer of neurons is linked with the next by means of a set of weights, is the most commonly used, and will also be used here. In this preliminary study, the above architecture has been adopted, see figure 3.2: magnetometers’ data streams will be considered the system inputs, while magnetic field results and their gradients at the positions of the test masses will be the system outputs. Figure 3.2: Feed-forward neural network architecture. Magnetometers readings are the system inputs, and estimates of the field and gradient at the positions of the test masses are the outputs of the system. In this architecture, one only intermediate, or hidden layer is assumed. Each of the circles represents one neuron and corresponds to the model shown in figure 3.1. DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 17/20 3.3 Learning paradigms and learning/training algorithms The investigation of learning algorithms is currently an active field of research. The design and implementation of an adequate training scheme is the essential ingredient for a good quality estimation of the magnetic field and its gradient at the LTP TMs. 3.3.1 Learning paradigms There are two major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning and unsupervised learning. 1. Supervised learning. The idea of this is quite clearly suggested by its very name: a set of examples is filed, which consists in a number of vector of inputs (magnetometers’ readouts in this case) and the corresponding values of the magnetic field and its gradient at the TMs for a given (chosen) distribution of dipoles in the S/C. Let xrepresent a generic input vector, and ythe associated vector output. These two vectors constitute an example. The set of filed examples for supervised learning is thus a set of pairs (x, y), where x∈Xand y∈Y,Xand Ybeing some suitable sample spaces. The network is then fed the inputs xof one example and let it work out an output, o, say. This output is then compared with the correct one, y, and an error is calculated if o6=y. Iterations are then triggered to adjust the weighting factors such that the error is minimised. These will however vary as different examples are run, so a cost function is defined which enables the network to optimise the set of weights which works best for the set of examples analysed, based on some given criterion. 2. Unsupervised learning. In unsupervised learning a cost function is to be minimised as well, but this function can be any relationship between xand the network output, o. The cost function is determined by the task formulation. Unsupervised learning is thus a form of self-adaptive system, whose guide is not an a priori knowledge of the final result but knowledge gained from experience. In either case, the learning process is based on the architecture of the network, i.e., number of neurons and layers and their interconnections, as well as on the activation functions. These are parameters which, at least in the simplest cases, are tuned ab initio by the user based on observed performance of the network. In this study, supervised learning has been the implemented learning paradigm. Analysis has been done with the help of the Matlab software suite. 3.3.2 Learning algorithms There are many algorithms for training neural networks. When training feed-forward neural networks with supervised learning, a back-propagation algorithm is usually implemented. The error of the mapping at the output is propagated backwards in order to readjust the weights and improve the output error for the next iteration. The propagation can be implemented with different methods, the Ideal Gradient Descent being a classic which will also be used here. The method is widely used in the field of soft computing, and is a variant of the method of steepest descent. DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 18/20 Iterations on the weights of the different neurons at the different layers proceed according to the following algorithm: wn+1 =wn−η∂E ∂wn (3.3) where nlabels the current iteration step, and ηis the learning rate, adjustable by the user. E is the sum over the set of training examples of the square errors of the outputs: E=X examples (o−y)T(o−y) (3.4) where ois the (vector) output from the network, while yis the target, or correct output in the corresponding example. This Ecan only be defined in supervised learning, of course, and the idea of the above procedure is to find that point in weight space where Eis the minimum possible. Ecan therefore be considered the cost function to be minimised in this particular supervised training scheme, also known as batch mode as the analysis is done across the entire set of training patterns in a single block. There are a number of technical issues in pursuing the iterations in equation (3.3), such as the choice of the initial set of weights, identification of local minima of E, boundary effects, etc. which need to be addressed in each specific case. We skip any detailed discussion of these matters here. 3.4 Performance assessment In this last step, the trained network must be tested with examples which differ from those used in the learning process. This is needed to assess whether or not the trained neural network is able to generate the expected results when fed with previously unseen inputs, hence determine its usability for the specific purpose it is intended. 4 Results This section reviews the results obtained so far —preliminary. Training and testing have been done based on different field realisations, using information provided by ASU on 37 magnetic dipole sources within the spacecraft. Two different batches of examples, each including 1 000 realisations of a possible magnetic environment, have been generated following the directives in the bulletted list of page 13. The first batch has been used as the training set for a neural network with 12 inputs (3 inputs for each of the 4 vector magnetometers) and 6 outputs representing the field information at the position of the 2 test masses (3 field components for each test mass). The second batch has been used for validation to assess the performance of the net in front of unseen magnetometers readings. As a further extension, another network has been trained, including the magnetic field gradient values in the output vector. DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 19/20 −100 0 100 0 0.5 1 σ1=9.2% σ2=1.45% Interpolation error (%) Pdf (Bx) TM1 TM2 −100 0 100 0 0.5 1 σ1=7.91% σ2=1.98% Interpolation error (%) Pdf (By) TM1 TM2 −100 0 100 0 0.5 1 σ1=4.42% σ2=3.67% Interpolation error (%) Pdf (Bz) TM1 TM2 −100 0 100 0 0.5 1 σ1=4.28% σ2=0.83% Interpolation error (%) Pdf (|B|) TM1 TM2 Figure 4.1: Probability density function of the error distributions for each field component at the position of test mass 1 (black trace) and test mass 2 (red): top left plot for Bx, top right for By, bottom left for Bzand bottom right for |B|. 4.1 Field estimation results Figure 4.1 shows the probability density functions of the errors in the estimates of the components of the magnetic field at the positions of each TMs. The plot is based on the outcome of the 1 000 validation runs described in the previous section. Units are percentage errors. As can be observed, the order of magnitude of the errors of the estimated fields are now within much more acceptable margins (around 10-15%). This improvement represents the reduction of estimation errors by more than one order of magnitude in comparison with the former methods. 4.2 Field gradient estimation results The magnetic field gradient can and should also be estimated. An extension of the network has accordingly been developed to estimate it at the TMs. The 9 components ∂Bi/∂Bjof the gradient are not independent, since they must verify equations (2.2) which reduces their number to 5. Neural network estimates of the gradient components does not guarantee that those conditions are verified, but this is not a problem, as discrepancies are within tolerance errors. Neural network approaches can also generate other similar inconsistencies with rigorous analytic results but, again, these can be dealt with. A first batch of results on gradient estimation is shown in figure 4.2 for ∇Bxand at the DDS-LTP Ref. S2-IEC-TN-3052 Version 1.0 Neural Network algorithms for magnetic diagnostics in the LTP Date 11-Mar-2009 Page 20/20 −100 −50 0 50 100 0 0.5 1 σ1=3.6% σ2=0.97% Interpolation error (%) Pdf (dBx/dx) TM1 TM2 −100 −50 0 50 100 0 0.5 1σ1=4.18% σ2=1.05% Interpolation error (%) Pdf (dBx/dy) TM1 TM2 −100 −50 0 50 100 0 0.5 1σ1=9.39% σ2=6.76% Interpolation error (%) Pdf (dBx/dz) TM1 TM2 Figure 4.2: Probability density function of the errors distribution for the three components of ∇Bx. From top to bottom: ∂Bx/∂x,∂Bx/∂y and ∂Bx/∂z at the positions of the test masses. Errors are given in percents, black traces corresponding to TM1, and red ones to TM2. positions of both TMs. As may be observed, they are also within much more acceptable margins than the earlier interpolation approach could produce. 5 Conclusion The magnetic diagnostic sensor set in the LTP is such that to infer the magnetic field and gradient on the TMs based on their readouts is far from simple. More or less conventional interpolation schemes cannot generally go beyond the linear approximation, which grossly fails to produce reliable results, so Artificial Neural Network models are under investigation. The preliminary results reported herein are encouraging, as the network learning process is able to significantly improve estimation errors. The main problem is the adequacy of a training process to the set of data the magnetometers will deliver in flight. This underlines the need to characterise on ground to our best ability the magnetic properties of the S/C and LCA at all locations and working conditions, both regarding their DC and fluctuating values. Reliable information on that appears to be essential for a meaningful assessment of magnetic noise in the LTP.