Ibrahim Ayed

Nicolas Cedilnik

Patrick Gallinari

Maxime Sermesant

EP-Net: Learning Cardiac Electrophysiology Models for Physiology-based Constraints in Data-Driven Predictions

Keywords: Electrophysiology, Deep learning, Simulation

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Introduction

Mathematical modelling of the cardiac cell has been an active research area for the last decades. Cardiac electrophysiology models can accurately reproduce cardiac cells electrical behaviour. This provides a mathematical framework to simulate cardiac activity, however it remains challenging to achieve in 3D due to numerical issues and computational time. These difficulties become all the more apparent when trying to personalise such model by matching patient data. This matching can also be hampered by the approximations of the model.

The last decade has seen huge progress in data-driven approaches, with deep learning achieving impressive results in numerous tasks. It has been particularly successful in image and signal processing, where spatio-temporal correlations can be learned. However it suffers from robustness issues in healthcare due to its lack of physiological knowledge. More recently, physics-based learning proposed to use machine learning in order to solve physics equations [START_REF] Raissi | Hidden physics models: Machine learning of nonlinear partial differential equations[END_REF][START_REF] Long | PDE-Net: Learning PDEs from Data[END_REF][START_REF] Herzog | Data-Driven Modeling and Prediction of Complex Spatio-Temporal Dynamics in Excitable Media[END_REF]]. This has the potential to alleviate some numerical and computational time issues, and also to provide a framework to overcome model error. Additionally, it is a way to introduce physics-based prior knowledge in learning methods.

In this manuscript, we present a method to learn the dynamics from a database of cardiac electrophysiology simulations and use the trained network to forecast the behaviour of unseen simulations. A state estimation method is coupled with a forecasting network to produce predictions for a few iterations.

These preliminary results are promising in terms of learning the dynamical behaviour of cardiac electrophysiology models. Cardiac electrophysiology modelling is a very active research area, with a large variety in terms of model complexity. In this manuscript, we used the Mitchell-Schaeffer model [START_REF] Mitchell | A two-current model for the dynamics of cardiac membrane[END_REF] which is a two variables model that has been successfully used in patient-specific modelling [START_REF] Relan | Coupled personalization of cardiac electrophysiology models for prediction of ischaemic ventricular tachycardia[END_REF]. The variable v represents the transmembrane potential while the "gating" variable h controls the repolarisation:

Electrophysiological Modelling

∂ t v = ∆v + hv 2 (1 -v) τ in - v τ out + J stim (1)
∂ t h = 1-v τopen if v < v gate v τ close if v > v gate (2)
In order to use convolutional neural networks, it is much more efficient to work on a Cartesian grid. To this end, we used a Lattice Boltzmann method to solve the EP model [START_REF] Rapaka | LBM-EP: Lattice-Boltzmann method for fast cardiac electrophysiology simulation from 3d images[END_REF].

The domain is a slab of cardiac tissue of size 25 × 25 × 5 mm 3 , which we will denote Ω, discretised in voxels of 1 mm 3 . To initiate the propagation (J stim ), a current of 1 normalised transmembrane potential unit was applied on a single voxel for 10 ms. The Mitchell-Schaeffer model parameters are taken from the original paper, except h 0 which was set to 0.7. The simulation was conducted for 300 ms, and stored every ms with a discrete time step of 0.1 ms. The simulation database was created by stimulating the domain from every voxel. We therefore have 3 125 simulations in the database. Let V and H be the complete solutions in space and time of these equations.

Learning Method

In this section, we reformulate the problem we want to solve with learning and outline the features of the learning model we used.

Formulation

Combining equations ( 1) and ( 2), the studied dynamical system can be written in the form:

dX t dt = F (X t ) (3)
where X is a spatio-temporal three-dimensional vector field over the domain Ω ⊂ R 3 . In this particular case, we have:

X = V H
In other words, for any given time t and point of the domain x ∈ Ω, we have X t (x), a two-dimensional vector.

In practice, while the value of V can be measured, H is a hidden variable which is difficult to estimate. We model this fact by adding to our system an observation operator H defined in our case by:

H(X) = V (4)
which makes our model a partially observed one, as only some parts of the full state can be directly measured.

It is important to note that H can be used to model more general observation operators which can be any transformation of the state. Obviously, the more information is lost through the observation process, the harder it will be to reconstruct the dynamics of the system. Moreover, for practical purposes, we also require H to be differentiable. This might seem as a strong hypothesis but, in practice, the operators which are used can generally be smoothed without a significant departure from reality.

The general system that we model can thus be written:

       X 0 dX t dt = F (X t ) Y t = H(X t ) (5)

Approach

In the system (5), starting from observations Y , we need to estimate both the initial state and the dynamics it should follow. Obviously, from one observation alone, it is often impossible to estimate the full state but hopefully this becomes possible for most operators H when a long enough sequence of observations is used.

Our idea is to use an operator g θ to estimate X 0 from a sequence of past observations Y -k = (Y -k+1 , ..., Y 0 ) of length k and a second operator F θ to model the ODE governing the dynamics of X. Thus, θ is a variable summarizing the parameterization of both the dynamics and the initial state.

This means that the system can be written as:

       X 0 = g θ (Y -k ) dX t dt = F θ (X t ) Y t = H(X t ) (6)
We assume that F and g are parameterized so that the above system has a unique solution which will be denoted X θ . Defining a cost functional:

J (Y, Y ) = T 0 Y t -Y t 2 dt (7)
we can then frame the statistical learning problem as:

minimize θ E Y ∈Dataset J (Y, H(X)) subject to dX t dt = F θ (X t ), X 0 = g θ (Y -k ) (8)
This optimization problem is generally a difficult one and cannot be solved exactly for most choices of parameterization families for F and g. This makes necessary the use of gradient descent algorithms for which it is necessary to estimate the derivative w.r.t. θ of E Y ∈Dataset J (Y, X θ ) which justifies the use of differentiable parameterizations.

In our case, starting from a random initialization of θ, we follow the steps:

1. Solve the forward state equation ( 6) to find X θ with an explicit differentiable solver; 2. Get the gradient of θ -→ E Y ∈Dataset J (Y, X θ ) with automatic differentiation tools; 3. Update θ in the steepest descent direction.

In our case, we parameterize g and F as neural networks so that the corresponding automatic differentiation tools can be used. The simplest example for explicit solvers is a Euler scheme but more complicated numerical methods can also be used. As shown in [START_REF] Ayed | Learning dynamical systems from partial observations[END_REF], this framework can also be extended to the modelling of stiff equations by using implicit solvers through the more general adjoint method.

Experiments

In this section we present the experiments we undertook. Even though we are still at a preliminary stage regarding the choice of architectures and hyperparameters, the results are already encouraging. Operator g is implemented as a three-dimensional U-Net inspired from [START_REF] Abdulkadir | 3d u-net: Learning dense volumetric segmentation from sparse annotation[END_REF] with 25 filters at the initial stage, figure 2 shows the general structure of this architecture. F is implemented as a Residual Network [START_REF] He | Deep residual learning for image recognition[END_REF] with three-dimensional convolutions, two downsampling initial layers, three intermediary blocks (figure 3 shows the structure of one) and an inner dimension of 24. In order to solve the forward equation, we use an explicit Euler scheme.

Implementation details

To construct a training dataset, we have randomly selected 500 simulations, temporally downsampled three times. A validation set of 300 simulations and a test set of 200 were also randomly selected from the remaining simulations and downsampled likewise. In particular, this means that test and training simulations have different initial conditions so that the model is tested with data it has never seen. Only the V variable was used and we have taken k = 4 and a training horizon of 8 time-steps.

The optimization over θ uses the ADAM optimizer [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] with a learning rate of 10 -3 . We also use exponential scheduled sampling [START_REF] Bengio | Scheduled sampling for sequence prediction with recurrent neural networks[END_REF] with parameter 0.9999 during training and start with a reweighted orthogonal initialization for the parameters of F .

Results

We present here results on the forecast over 8 time frames after assimilating the first 4 frames (see Fig. 4). We can observe very good agreement with the ground truth on this forecast, which represents an important part of cardiac dynamics within this virtual slab of tissue, from early depolarisation to full depolarisation.

Table 1 shows MSE results for our algorithm for different forecasting horizons: for a horizon K, this means that we feed the model with three initial measurements then let it predict K steps forward without any additional information. We can see that up to a reasonable number of steps, here corresponding to 51 ms, our model does reasonably well which is very encouraging given that we are still at an early stage of experimenting with it. Figure 4 visually confirms those numerical results.

It has to be noted that for such cardiac model, predicting the propagation between time 10 and 60 ms on such a slab of tissue represents the whole depolarisation wave, which is the most important part of cardiac electrophysiology dynamics.

Moreover, it is important to notice that, while the model takes a few hours to train, once this is done the inference is very quick and does not require any recalibration.

Discussion

We have presented a learning model able to learn the considered dynamics and thus showing promise regarding the automated learning of cardiac electrophysiology models. Moreover, while we only presented results which are completely unsupervised regarding the full state of the studied system, which makes us unable to reconstruct H for example, it is possible with our approach to add in stronger physical priors by pre-training the model over fully observed states, adding some information about the equation into F . The model we present is very versatile and can thus be used with different types of dynamics, degrees of supervision... which could open the way for many other applications where quick simulation is needed.

However, this is still a first step and we need to experiment with our model in more challenging settings i.e., with a more realistic H which is not a simple and dense linear projection in real-world applications. Another shortcoming of our model is the fact that uncertainty is not taken into account, at input as well as at output level, while it is an important feature of many real-world applications especially in healthcare. This should be an essential direction of research for our future work. Finally, it is also important to note that, while Mean Squared Error was chosen for training as well as for testing, it is not necessarily the best metric for all applications: depending on the use case, it might be useful to design a different one, more adapted to the problem.

Conclusion

We proposed in this manuscript an approach to learn the dynamics of a cardiac electrophysiology model along with a data assimilation procedure, so that limited measurements can be used to estimate an initial state and predict the future evolution of the electrophysiology model.

In clinical practice, it is challenging to completely map cardiac activation, especially in case of arrhythmia. This method could be used to complete in space and time limited clinical measurements.

Such coupling between simulation and learning opens up many possibilities, especially in order to make such approach more clinically applicable and patientspecific. It has the potential to tackle both computational as well as robustness issues that appear when using computer models or machine learning separately.

Fig. 1 .

 1 Fig. 1. Cardiac electrophysiology model simulation. (Left) v and h values along time for a given point of the domain, (Right) spatial propagation of v at a given time point.

Fig. 2 .

 2 Fig. 2. Figure illustrating the general form of the U-Net architecture. Illustration taken with a slight modification from [12].

Fig. 3 .

 3 Fig. 3. Figure showing a typical residual block for the ResNet. Illustration taken from [4].

Fig. 4 .

 4 Fig. 4. Transmembrane potential ground-truth (top) and forecasted (bottom) for one slice of the tissue slab.

Table 1 .

 1 Relative mean-squared error of normalised transmembrane potential per time-step for different forecasting horizons.

	Mean Squared Error 0.005	0.012	0.048	0.060	0.14	0.58

Time horizon K 2 (6 ms) 5 (15 ms) 8 (24 ms) 11 (33 ms) 14 (42 ms) 17 (51 ms)


Eric Bavu

Aro Ramamonjy

Hadrien Pujol

Alexandre Garcia

Éric Bavu

Timescalenet: a multiresolution approch for raw audio recognition

Keywords: Machine hearing, Learnable Biquadratic filters, Deep Learning, Time domain modelling

INTRODUCTION

Machine hearing can involve the use of hand-crafted features, time-frequency representations, or raw audio. The use of the latter representation has emerged as an active area of research in the last few years : the unprocessed, time-domain audio signals contain all the information to be extracted for machine hearing tasks. In previously published studies [START_REF] Dieleman | End-toend learning for music audio[END_REF][START_REF] Dai | Very deep convolutional neural networks for raw waveforms[END_REF][START_REF] Tüske | Acoustic modeling with deep neural networks using raw time signal for LVCSR[END_REF][START_REF] Sainath | Learning the speech frontend with raw waveform CLDNNs[END_REF][START_REF] Lee | Samplecnn: End-to-end deep convolutional neural networks using very small filters for music classification[END_REF][START_REF] Sainath | Multichannel signal processing with deep neural networks for automatic speech recognition[END_REF], the models mostly used large filters, which can model passband filters approximating cochlear filter estimates. These studies, along with recent advances in machine learning for onedimensional signals [START_REF] Van Den Oord | Wavenet: A generative model for raw audio[END_REF][START_REF] Kaiser | Depthwise separable convolutions for neural machine translation[END_REF][START_REF] Rethage | A wavenet for speech denoising[END_REF] have motivated the present work, which aims at showing the benefits of a multi-resolution approach for machine hearing. The proposed approach avoids using large convolutional kernels, by introducing a new form of recurrent neural cell, directly inspired by IIR digital signal processing.

METHODS

TimeScaleNet's architecture is detailed in 2.1. This architecture can be split in two subnets, acting at two different timescales. The architecture and the detailed implementation of these two subnets are explained in 2.2 and 2.3. The training procedure is also detailed in 2.4.

Global neural network architecture

As shown on Fig. 1, the first subnet of TimeScaleNet's architecture, BiquadNet (see 2.2), acts at the sample level, and aims at encoding the information for time scales in the range of [100 µs ; 20 ms], thus allowing to compute a timefrequency-like representation, that is fed to the next subnet of our architecture. The proposed "biquadratic" RNN filter can be thought as a set of infinite impulse-response (IIR) filters, expressed as a biquadratic filterbank [START_REF] Lawrence | Theory and application of digital signal processing[END_REF] with learnable coefficients, that self-adapts to the audio dataset that has to be classified. This strategy allows a computationally-efficient IIR bandpass filtering using only two learnable parameters for arbitrarily long receptive fields, rather than 1-dimensional convolutional neural networks with wide kernels reported in previous studies [START_REF] Sainath | Learning the speech frontend with raw waveform CLDNNs[END_REF][START_REF] Sainath | Multichannel signal processing with deep neural networks for automatic speech recognition[END_REF][START_REF] Hoshen | Speech acoustic modeling from raw multichannel waveforms[END_REF].

The obtained time-frequency-like representation at the output of BiquadNet is then fed to the second subnet, referred in the following as "FrameNet" (see 2.3), because it acts at the frame level, in order to efficiently encode the time fluctations in the range of [20 ms ; 200 ms]. This second scale of analysis aims at extracting the relevant relationships between time fluctuations in different learnt pooled frequency channels, with a large receptive field. For this purpose, we propose the use of residual networks of one-dimensional depthwise separable atrous convolutions, which allow to operate on channel-wise frames in a computationally efficient way.

The output of FrameNet is then flattened, and fed to two full-connected layers with Selu activations, in order to compute a vector of dimension N classes representing the probability of belonging to the classes of the dataset.

BiquadNet : raw waveform processing

From a machine-learning point of view, the first layer of Bi-quadNet is a non-conventional recurrent neural network cell. From a digital signal processing however, this RNN cell is directly derived from the widely used infinite impulse response (IIR) biquadratic filters, which are very efficient types of filters to implement, because they require less computation and memory than FIR filters in order to perform similar filtering operations. It is also useful to note that passband biquadratic filters (also referred as two-poles two-zeros in the litterature) have been demonstrated to be good numerical models of auditory filterbanks [START_REF] Slaney | An efficient implementation of the Patterson-Holdsworth auditory filter bank[END_REF][START_REF] Richard F Lyon | Cascades of two-pole-two-zero asymmetric resonators are good models of peripheral auditory function[END_REF]. In the present work, we implemented a bidirectional biquadratic RNN cell, which allows to achieve forward-backward filtering [START_REF] Smith | Introduction to Digital Filters with Audio Applications[END_REF], in order to perform a perfect zero-phase filtering in the time domain. In order to prevent a potential numerical instability we only use second-order IIR filters. This is the main reason why most digital signal processors implement stacks of biquadratic IIR filters. This kind of topology can be easily transposed to machine learning, where deep neural network topologies often use stacking of similar layers. Using the Z-transform, biquadratic filters exhibit two zeros and two poles, and can be represented as in Figure 2 :

x[n] b (0) i y i [n] z -1 z -1 b (1) i -a (1) i z -1 z -1 b (2) i -a
(2) i

Fig. 2. Flow graph of the learnable biquadratic IIR filters

Since we aim at obtaining a "time-frequency"-like representation at the output of BiquadNet, we restrict the possible values of the coefficients of the learnable IIR filterbank to correspond to passband versions of a biquadratic IIR filter. This allows to simplify the stability properties of the learnt filters and ensures that the learnt biquadratic filters are stable, since a (1) and a (2) are constrained to stay inside the "stability triangle" during the whole learning process. Each biquadratic bandpass filter of the learnable filterbank can be fully determined using only two parameters,

K (i) = tan πf (i) c /f s and Q (i) , where f s is the sample frequency, f (i)
c is the central frequency of the i th bandpass filter, and Q (i) is the quality factor of the i th bandpass filter . The five coefficients represented in Figure 2 can be expressed directly using these two meaningful variables K (i) and Q (i) , that are chosen to be the learnable variables in TimeScaleNet. The corresponding custom RNN cell has been implemented using high order operations of the Tensorflow open source software library that allow to recursively scan functions over arbitrarily long sequences and to unfold dynamically the computational graph at runtime. This implementation is compatible with a backpropagation-through-time process, in order to compute the derivative chain rule and to update the neural network parameters at each iterations of the machine learning process. The expression of the custom biquadratic bidirectional RNN is fully differentiable, which allows to be compatible with the proposed machine learning approach for audio recognition, while being directly linked to standard digital audio signal processing approaches.

As shown on Figure 1, the 128 time-domain outputs of the biquadratic RNN layer are then fed to a deterministic module that allows to compute a framed log-energy, in order to obtain a time-frequency-like representation, with overlapping windows of 23.2 ms and a stride of 5.8 ms. This non-learnable module consists in the computation of a stabilized logarithmic compression of a sliding mean quadratic value over successive overlapping timeframes. This module is followed by layer normalization [START_REF] Lei Ba | Layer normalization[END_REF], which allows to compute layer-wise statistics and to normalize the Selu [START_REF] Klambauer | Self-normalizing neural networks[END_REF] nonlinear activations across all summed inputs within the layer, instead of within the batch. The features are then pooled across the whole frequency channels using pointwise convolutions, in order to obtain a representation that has the ability to encode well phonemes such as vowels formants and consonants, by aggregating relevant learnt frequency channels together.

FrameNet : large-scale time relationship learning

FrameNet acts at the time frame level, in order to efficiently encode the relevant relationships between time fluctuations in different pooled frequency channels, with a large time receptive field, thanks to one-dimensional atrous convolutions [START_REF] Van Den Oord | Wavenet: A generative model for raw audio[END_REF][START_REF] Kaiser | Depthwise separable convolutions for neural machine translation[END_REF][START_REF] Rethage | A wavenet for speech denoising[END_REF]. The stacked residual dilated convolutions therefore allow the network to operate on multiple time scales in the range of [20ms; 200ms] without impacting too much the computational efficiency (Figure 3). The convolutions are performed independently over every pooled channel (depthwise separable convolutions). This approach has been motivated by preliminary analysis of the energy fluctuations in different frequency channels using classical spectrogram representations. These computed depthwise convolutions are then projected onto a new channel space for each layer using a pointwise convolution, and residual connections are used between each layer, in order to offer increased representation power, by circumventing some of the learning difficulties introduced by deep layers.

Training procedure

In our experiments, TimeScaleNet is trained with one-hot encoded labels, therefore allowing to compute the cross-entropy loss between estimated labels and ground truth labels. The learning and backpropagation of errors through the neural network is optimized using the Adaptive Moment Estimation (Adam) [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] algorithm. The models have been implemented and tested using the Tensorflow open source software library, and computations were carried out on four Nvidia GTX 1080Ti GPU cards, using mini-batches of 70 raw waveforms for each training steps. All the weights involved in layers followed with Selu activations were initialized using the He initialization [START_REF] He | Delving deep into rectifiers: Surpassing humanlevel performance on imagenet classification[END_REF]. K (i) and Q (i) were initialized using clipped random values matching the range of the equivalent rectangular bandwidth cochlear model introduced by Patterson [START_REF] Roy D Patterson | Auditory models as preprocessors for speech recognition[END_REF], for central frequencies spanning from 40 Hz to f s /2.1.

Dataset and evaluation metrics

In the present paper, TimeScaleNet is evaluated for raw audio recognition, using the publicly available Google speech commands dataset v2 [START_REF] Warden | Speech commands: A dataset for limited-vocabulary speech recognition[END_REF]. This dataset has recently served a competition hosted by Kaggle, which consisted in recognizing the ten words "Yes", "No", "Up", "Down", "Left", "Right", "On","Off", "Stop", and "Go" along with the "silence" class (i.e. no word spoken) and "unknown" class, which is randomly sampled from the remaining 25 keywords from the dataset. The dataset is split into training, validation and test sets in the ratio of 80:10:10 while making sure that the audio clips from the same person stays in the same set, using the exact procedure detailed by the maintainer of the dataset in [START_REF] Warden | Speech commands: A dataset for limited-vocabulary speech recognition[END_REF]. In order to analyze precisely the performances of the proposed TimeScaleNet for the task of supervised multiclass classification, several evaluation metrics will be used in the following : The class recall R i , the class precision P i , and their macro-averaged versions R M and P M . The macroaveraged F 1 score is also derived, in order to evaluate the relations between data's positive labels and those given by the classifier, which allow full understanding of the overall classification task achieved by the neural network.

RESULTS AND DISCUSSION

For the Speech Commands Dataset, the learning process has been performed using TimeScaleNet during 45 epochs, without dropout regularization. These 45 epochs correspond to a total of 1200 hours of audio waveforms processed by the proposed model. Table 1 shows the obtained evaluation metrics on the Speech Commands dataset.

The evaluation metrics shown on Table 1 show that for speech recognition, TimeScaleNet appears to classify the 12 classes with a very high accuracy (94.87% for the evaluation set, 94.78% for the testing set, after 45 epochs of learning), with a very good homogeneity for all the classes.

For reference, we also evaluated the performances of TimeScaleNet on the Speech Commands dataset with a frozen BiquadNet, using a deterministic (non-learnable) biquadratic filterbank matching the Patterson's cochlear model with Glasberg and Moore parameters, which achieved 92.4% accuracy over the testing set. A similar experiment has also been per- formed using a log-mel-spectrogram as an input to FrameNet, which achieved 89.7% accuracy over the testing set. These two preliminary experiments mainly motivated the development of the BiquadNet part of TimeScaleNet, because this time domain approach allows to achieve a significant performance boost over handcrafted time-frequency features representations. The 94.78% accuracy achieved on the testing set using the proposed TimeScaleNet matches some of the highest values found in [START_REF] Zhang | Hello edge: Keyword spotting on microcontrollers[END_REF]. To the best authors knowledge, the only published models that significantly outperforms TimeScaleNet on this particular dataset are DS-CNN in [START_REF] Zhang | Hello edge: Keyword spotting on microcontrollers[END_REF] and res15 [START_REF] Tang | Deep residual learning for small-footprint keyword spotting[END_REF], which exhibits the best results to date with a mean accuracy of 95.8 %.

Analysis of the learnt representation from raw waveforms using BiquadNet

In this subsection, we analyse the variables learnt by Biquad-Net, in order to give further insight on the learning process involved. The architecture of BiquadNet has been specifically developed to automatically build a 2D map that can be interpreted as an energy-like representation in 128 pooled frequency channels. Figure 4 shows the computed magnitude response at the output of BiquadNet, sorted by ascending order of frequency at which the maximum occurs for each filters. BiquadNet learns to build a selective filterbank which pools several frequency bands together, in order to pass them to FrameNet, which then encodes the time fluctuations in those pooled frequency bands at the frame level. Some of the channels shown on Figure 4 exhibit frequency patterns that could be linked to vowels or nasals, whereas the last channels exhibit a frequency patterns that could serve the purpose of encoding fricatives or plosives only, with wideband, high frequency content. It is also interesting to note that the pooled frequency channels representation build by BiquadNet for speech recognition further increases the density of activations for frequencies between 200 Hz and 1000 Hz, and may explain why TimeScaleNet allows a better accuracy than with a frozen version of BiquadNet with the Patterson's cochlear model using the parameters of Glasberg and Moore, shown as a dotted line on Figure 4. Interestingly, for the 100 first channels, which may mainly encode vowels and nasals, the learnt channels follow a very similar evolution than the Mel scale of 128 filters between 150 Hz and 5000 Hz. This is a really interesting property, since there was initially no intent to use the mel scale in the present study. However, for the highest channel numbers depicted on Figure 4, where the frequencies at which the maximum magnitude occurs at a larger frequency than 2500 Hz, the learnt filterbanks switches back to a Glasberg model, and clusters high frequencies together, which could help in recognizing consonants.

CONCLUSION

In this paper, we presented a machine learning approach of multiresolution modelling of unprocessed, time domain audio waveforms. The proposed deep neural network (TimeScaleNet) aims at merging digital signal processing techniques with new machine learning techniques. We show that this whole process allows to achieve speech recognition with a very high accuracy, which matches the performances of the best models to date on the Speech Commands dataset. By analyzing the learnt parameters in BiquadNet for this particular task and by deriving the equivalent filterbank magnitudes from the frozen model after convergence, we give further interpretability of the proposed machine hearing process. We also show that on this particular task, the proposed neural network builds a representation that both encodes the frequency content between 200 Hz and 3000 Hz with a pattern matching the mel-scale, and encodes higher frequency content with a pattern matching the Patterson's model. The approach also allows to pool frequency bands together, which can efficiently encode nasals, vowels, fricatives, and plosives for speech recognition. These results allow to interpret the machine learning task in light of cognitive models of audition, while standing on both machine learning and digital signal processing solid basis.

Fig. 1 .

 1 Fig. 1. Schematical representation of TimeScaleNet

Fig. 3 .

 3 Fig. 3. Schematics of one of the two stacks of depthwise separable atrous layers used in FrameNet, from data point of view. Only the depthwise convolution is shown here, with arrows showing the frame indexes involved in dilated convolutions for the computation of the output at frame index k 0 .

Fig. 4 .

 4 Fig. 4. Magnitude response of the equivalent filterbank at the output of BiquadNet, after convergence for the Speech Commands Dataset [20].

Table 1 .

 1 Evaluation metrics obtained after convergence (45 epochs of learning), for the Speech Commands Dataset[START_REF] Warden | Speech commands: A dataset for limited-vocabulary speech recognition[END_REF].

	Data	Cardinality	Accuracy	Precision M	Recall M	F 1,M
	Speech Evaluation Set Speech Testing Set	4916 5157	94.87 ± 0.24% 94.91 ± 0.22% 94.88 ± 0.26% 94.78 ± 0.26% 94.87 ± 0.25% 94.87 ± 0.25% 94.87 ± 0.25% 94.9 ± 0.24%


Simon Cailhol

Paul-Etienne Vidal

Senior Member, IEEE Frédéric Rotella

A Generic Method of Pulsewidth Modulation Applied to Three-Phase Three-Level T-Type NPC Inverter

Keywords: Circuit modeling, dc-ac power conversion, energy conversion, pulsewidth modulation (PWM), pulsewidth modulated inverters

This paper illustrates the application of a solving method of linear systems to a multilevel inverter. This study applies a generic pulsewidth modulation (PWM) method to a multilevel topology. This method finally highlights some specific modulation schemes. First, the study describes the mathematical model of the inverter leg. The leg is made of three level T-type structure. The model is extended to the three-phase inverter. The generalized inverse is used to generate the solution set of the conduction rates. Among the modulation strategies possible, three specific modulation schemes are proposed. The modulation schemes are applied to simulate a PWM three-phase inverter. Finally, some experimental results complete the analysis.

I. INTRODUCTION

E LECTRIFIED systems are of growing importance in many applications developed today. Among them, some widespread applications such as pipeline pumps [START_REF] Rossmann | Retrofit of 22 pipeline pumping stations with 3000-hp motors and variable-frequency drives[END_REF], fans [START_REF] Menz | Replacement of a wound rotor motor with an adjustable speed drive for a 1400 kW kiln exhaust gas fan[END_REF], water pumping stations [START_REF] Schmitt | Retrofit of fixed speed induction motors with medium voltage drive converters using NPC three-level inverter high voltage IGBT based topology[END_REF], and traction applications [START_REF] Bernert | Recent developments of high power converters for industry and traction applications[END_REF] concern medium voltage converters. New trends for converter design, emerging from eco-friendly applications, are commonly studied. Grid integration of renewable-energy sources [START_REF] Alepuz | Interfacing renewable energy sources to the utility grid using a three-level inverter[END_REF], reactive-power compensation [START_REF] Dixon | Reactive power compensation technologies: State-of-the-art review[END_REF] in order to maximize the system efficiency and so on, belong to this category. It is noted that for such applications, the converter control achieves the exchanged energy monitoring. Consequently, the converter architecture and its control are the key issues for the electric power conversion applications.

The availability of widebandgap semiconductors, combined with their increase reliability leads to new interest for developing original inverter topologies. Indeed, studies work on the correlation of a given architecture and its dedicated modulation strategy with an application. Among these studies, the multilevels architecture and particularly the neutral point clamped (NPC) were widely studied. First, dedicated to high power application, a de-(Corresponding author: Simon Cailhol.)

The authors are with the Laboratoire Génie de Production, Université Fédérale de Toulouse, Tarbes Cedex 65016, France (e-mail:, simon.cailhol@ enit.fr; paul-etienne.vidal@enit.fr; frotella@enit.fr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIA.2018.2829468 rived topology, named three level T-type converter (3LT 2 C) is under interest. Effectively, this topology benefits from its ability at being used in high power photovoltaic systems [START_REF] Xuan | Design and implementation of a high power three-level T-type inverter for a photovoltaic system[END_REF], as well as variable speed drives for low voltages applications, [START_REF] Schweizer | High efficiency drive system with 3-level T-type inverter[END_REF]. Moreover, it has been demonstrated that a 3LT 2 C increases the converter efficiency compared to similar multilevel inverters, and more precisely compared to a three-level NPC, with respect to power losses [START_REF] Schweizer | Design and implementation of a highly efficient three-level T-type converter for low-voltage applications[END_REF], [START_REF] Shin | Power loss comparison with different PWM methods for 3L-NPC inverter and 3L-T type inverter[END_REF].

Studies demonstrated that the modulation strategy of a threelevel NPC can also be applied to the 3LT 2 C topology. Indeed, usual carrier based pulsewidth modulation (PWM) associated with two carrier waveforms [START_REF] Nguyen | The carrierbased PWM method to reduce common-mode voltage for three-level Ttype neutral point clamp inverter[END_REF], or space vector modulation (SVM) which allows harmonic selection [START_REF] Choi | New modulation strategy to balance the neutral-point voltage for three-level neutral-clamped inverter systems[END_REF], have been successfully tested. Nevertheless, the major drawbacks of such an inverter topology concern the control of the common mode voltage or the unbalance voltage of dc input sources. Most of the modulation strategies aim to balance these drawbacks.

It appears to our knowledge that it does not exist any generic modulation model of 3LT 2 C converter. The aim of this study is to apply the PWM modeling technique proposed in [START_REF] Vidal | Generalized inverses applied to pulse width modulation for static conversion: A first study[END_REF], in order to establish a generic inverter modulation model of such a structure, which can match the following criteria:

1) to be applied to carrier-based PWM; 2) to be extended to SVM principles; 3) to highlight how to implement a generic modulation scheme; 4) to define admissible modulation solutions. The contribution of this paper is twofold, which is given as follows:

1) the modeling of the T-type 3LT 2 C converter in a generic framework; 2) the determination of modulation strategies using a generic mathematical method. This paper is organized as follows: first, the considered structure is described and a mathematical model is proposed in Section II. Then, based on the defined model, the considerations to generate modulation strategies are provided. Three modulation strategies are described in Section III. Simulation results are provided in Section IV-C. Finally, some experimental results allow us to recover the theoretical and simulation data. A discussion section analyze the efficiency of the generic model as well as its ability to produce the efficient PWM schemes.

II. INVERTER MODELING

From a practical point of view, a three-phase 3LT 2 C inverter is made of three legs based on a similar switching cell structure. In this section, we will first detail the model of a leg of such an inverter and then extend this model to a complete three-phase inverter.

A. T-Type Switching Cell

The switching cell can be built from two technological achievements, as illustrated in Fig. 1. This figure depicts the electrical equivalent circuits of available commercial modules. The inverter cell of the leg i considered is made of three switches, named K i1 , K i2 , and K i3 . Due to the switches connection, to avoid short circuit of the input voltage sources, only one switch can be turned ON at once. Thus, K i1 , K i2 , and K i3 are elementary switching cells. Every switch state corresponds to an output line voltage v io , as given in Table I. The other possibilities are not allowed.

As far as the modulation model is concerned, the switching cell can be considered as three elementary ideal switches, associated with two independent and perfect input voltage sources, as illustrated in Fig. 2.

The Fig. 3 gives an example of a switching sequence and the corresponding line voltage during the period T sw , where

f sw = 1
T sw is the switching frequency. In such example, it is assumed that every switch can be turned ON and OFF at every T sw .

As exemplified in Fig. 3, t i1 is the time when K i1 is closed. Then, K i2 is switched ON from t i1 to t i2 , and finally K i3 between t i2 and T sw . To recover the carrier-based PWM schemes, let us define, τ i,k , k ∈ {1; 2; 3} as the conduction rates, or duty cycles, of switch K ik over T sw

τ i1 = t i1 T sw τ i2 = t i2 -t i1 T sw τ i3 = T sw -t i2 T sw . ( 1
)
The mean value of the line voltage over the switching period, V io , is computed as

V io = E 2 (τ i1 -τ i3 ) . ( 2
)
Unfortunately, these conduction rates, associated with a carrierbased PWM scheme, does not allow us to produce the line voltage depicted in Fig. 3. Then, new variables are defined. As K i3 state is dependent of K i1 and K i2 states, let us express a i1 and a i2 as a function of the switching times t i1 and t i2 as

⎧ ⎪ ⎨ ⎪ ⎩ a i1 = t i1 T sw a i2 = t i2 T sw .
(

) 3
As illustrated in Fig. 4, an advantage of such a model appears: a carrier-based modulation scheme is used, associated with a single carrier easily generated. Finally, the switching times t i1 and t i2 are used to generate every control signal for K ik . They are obtained, thanks to the Boolean function of the comparison of a i to a sawtooth signal defined between 0 and 1 at frequency f sw , as illustrated in Fig. 4. Finally, the mean value of the line voltage, V io , is deduced

V io = E 2 (a i1 + a i2 ) - E 2 . ( 4
)
Defining V * io = V io + E 2 leads to the relationship V * io = E 2 Aa i ∀i, i ∈ {1, 2, 3} (5)
where A = 1 1 and a i = a i1 a i2 T . The control strategy vector a i appears as the solution of the linear system (5).

B. 3LT 2 C Inverter

The model of the three-phase inverter is obtained by duplicating [START_REF] Alepuz | Interfacing renewable energy sources to the utility grid using a three-level inverter[END_REF], thanks to the Kronecker product ⊗

V * lo = E 2 (I 3 ⊗ A) a (6)
where V * lo is the vector the three line voltages V * io such as

V * lo = V * 1o V * 2o V * 3o
T and a = a 1 a 2 a 3 T . I 3 is the 3 × 3 identity matrix. The Kronecker product ⊗ performs a weighted duplication of the matrix A on each component of I 3 [START_REF] Graham | Kronecker Products and Matrix Calculus: With Applications[END_REF], [START_REF] Brewer | Kronecker products and matrix calculus in system theory[END_REF].

With a three-phase balanced load, the phase voltage vector V ln is

V ln = MV lo ( 7
)
where M is defined

M = 1 3 ⎡ ⎢ ⎣ 2 -1 -1 -1 2 -1 -1 -1 2 ⎤ ⎥ ⎦ .
Moreover, the neutral point voltage introduced for the phase voltage calculation allows simplifying V ln such as

V ln = MV lo = MV * lo . ( 8
)
Due to M (I 3 ⊗ A) = M ⊗ A and [START_REF] Van Loan | The ubiquitous Kronecker product[END_REF], the inverter model can be expressed as a linear problem

V ln = E 2 (M ⊗ A) a (9)
where the unknown are the a ij .

III. CONTROL STRATEGIES

For an elementary switching cell, the solution expressions were expressed and detailed in [START_REF] Cailhol | A generic method of pulse width modulation applied to 3-level T-type NPC inverter[END_REF]. The aim of the following section is to extend these methods to a three-phase inverter.

A. Generic Solutions

Let us observe that rank(M ⊗ A) = rank M ⊗ A V ln is fulfilled for the sake of necessity. Thus, the linear system ( 9) is consistent. The solutions are using (M ⊗ A) † , which is the pseudoinverse of (M ⊗ A) [START_REF] Ben-Israel | Generalized Inverses: Theory and Applications[END_REF]

a = 2 E (M ⊗ A) † V ref + I 6 -(M ⊗ A) † (M ⊗ A) z ( 10
)
where z is an arbitrary six-dimensional vector.

Then, the solutions are split in two summed up parts as follows.

1) A fixed solution: 2

E (M ⊗ A) † V ref , which is noted a ref . 2) An independent solution: I 6 -(M ⊗ A) † (M ⊗ A) z,
which is in the kernel of M ⊗ A. Looking at the independent solution, some degrees of freedom can be deduced from

(I 6 -(M ⊗ A) † (M ⊗ A))z.
The degree of freedom n λ , which is the dimension of the kernel of M ⊗ A, is given by

n λ = n -rank(M ⊗ A) = 4 ( 11
)
where n is the number of columns of M ⊗ A.

A simple way to exhibit a n λ -dimensional vector λ is to use a maximal rank factorization algorithm. Namely there exists F , a full column rank (n × n λ )-dimensional matrix and G, a full row rank (n λ × n)-dimensional matrix, such as

F G = I 6 -(M ⊗ A) † (M ⊗ A) . ( 12
)
Defining λ = Gz, the control vector a is obtained

a = (M ⊗ A) † V ref + F λ = a ref + F λ ( 13
)
where λ = λ 1 λ 2 λ 3 λ 4 is the tuning parameters vector. The matrix F is chosen in order to ease the tuning parameters manipulation (see the Appendix)

F = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -1 0 0 1 1 0 0 1 0 -1 0 1 0 1 0 1 0 0 -1 1 0 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . ( 14
)
Furthermore, the line voltage V io defined in (4) combined with the expression of a in [START_REF] Shin | Power loss comparison with different PWM methods for 3L-NPC inverter and 3L-T type inverter[END_REF] allows us to compute the neutral point voltage V no as

V no = Eλ 4 - E 2 . ( 15
)
Therefore, λ 4 appears as the common mode tuning parameter, whereas λ 1 , λ 2 , and λ 3 are legs tuning parameters.

B. Implementation Constraints

Among the solution set expressed in [START_REF] Shin | Power loss comparison with different PWM methods for 3L-NPC inverter and 3L-T type inverter[END_REF], some solutions are not physically acceptable. Indeed, components of a have to be in [0, 1]. Moreover, among each inverter leg, they have to be ordered. These two constraints are summarized as 0 ≤ a ij ≤ 1 ( 16)

a i1 ≤ a i2 . ( 17
)
Due to the structure of the matrix (M ⊗ A) † , the fixed elements a ref i1 and a ref i2 of the solution are equal and are denoted a a ref i .

Considering the independent solution, two aspects of the tuning parameters can be stated as follows.

1) Common Mode Tuning Parameter: λ 4 acts on every element of the vector a. Thus, it must be tuned first according to [START_REF] Brewer | Kronecker products and matrix calculus in system theory[END_REF]. Indeed, 0

≤ a ij ≤ 1, i ∈ {1, 2, 3} , j ∈ {1, 2} leads to 0 ≤ a ref i + λ 4 ≤ 1 which defines bounds for λ 4 -min(a ref ) ≤ λ 4 ≤ 1 -max(a ref ). ( 18
)
These bounds will be denoted by λ 4 low and λ 4 high in the following.

2) Per Leg Tuning Parameter: The λ i i ∈ {1, 2, 3} are dedicated to phases i, respectively. Once λ 4 is defined, the other components of λ are set accordingly to ( 16) and [START_REF] Van Loan | The ubiquitous Kronecker product[END_REF].

First, (17) leads to

λ i ≥ 0 . ( 19
)
Second, ( 16) can be split in two parts. 1) A lower bound

λ i ≤ a ref i + λ 4 . ( 20
)
2) A upper bound inequality

λ i ≤ 1 -(a ref i + λ 4 ) . ( 21
)
Finally, for λ i i ∈ {1, 2, 3}, the three conditions ( 19)-( 21) are summed up as

0 ≤ λ i ≤ min(a ref i + λ 4 , 1 -(a ref i + λ 4 )) . ( 22
)
The upper bound appearing in (22) will be denoted as λ i high for i ∈ {1, 2, 3}.

IV. PARTICULAR SOLUTIONS

A. Common Mode Tuning Parameter

In this study, the common mode tuning parameter λ 4 is chosen at the middle point, namely

λ 4 = λ 4 low + λ 4 high 2 = -min(a ref i ) + 1 -max(a ref i ) 2 = 1 2 + med(a ref i ) 2 (23)
where med(a ref i ) is the fixed part of a expressed from V med definition. As illustrated in Fig. 5, λ 4 is set between λ 4 high Fig. 5. λ 4 and its margins.

V no = V med 2 . ( 24
)

B. Per Leg Tuning Parameters

In this study, three λ i i ∈ {1, 2, 3} are considered as follows. 1) λ i = 0 defines t i1 = t i2 , leading to a two-switches leg inverter. 2) λ i = λ i high alternates between t i1 = 0 and t i2 = T sw , leading to a reduction of the switching losses. Indeed, K i1 or K i3 are not switched for half a period, and E 2 edges on V lo are insured. 3)

λ i = λ i low + λ i high 2
maximizes the time between switching while all the switches are used during each switching period.

C. Simulation

A Scilab [START_REF] Enterprises | Scilab: Logiciel open source gratuit de calcul numrique[END_REF] test-bench based on the switching cell given in Fig. 2 has been developed to test the three strategies proposed. The cell is connected to a E = 50-V dc bus. The switching frequency is set at f sw = 1 kHz. The three modulation strategies have been tested by modulating 90% of a 25-Hz 50-V peak sinusoidal waveform.

The proposed strategies generate a signal, which fundamentally corresponds to the desired reference voltage.

1) λ i = 0: This strategy introduces no shift between a i1 and a i2 , as illustrated in Fig. 6. Consequently, the switch K i2 is not used. As depicted in Fig. 7, the switching cell is working as a usual two-levels/two-switches cell. The voltages and switches states are plotted for half a period of the sine reference waveform. On one hand, K i2 never switches, which reduces the number of switching, and consequently the switching losses, at each switching period. On the other hand, this strategy increases the harmonic distortion of the load current, due to a lower number of switching.

2) λ i = λ i high : This strategy introduces a difference between a i1 and a i2 , as illustrated in Fig. 8. The three switches are thus used as displayed in Fig. 9. However, this figure points out that K i2 is switched all along the period of V ref i , whereas K i1 and K i3 are used on half a period only. Such a strategy leads to clamp alternatively the time rates to 0 for a i1 , and 1 to a i2 . Indeed, K i1 and K i3 are only used, respectively, during the positive and negative parts of V ref . Similarly to the previous strategy, the switching losses are reduced, but higher harmonic distortion will be experienced.

3)

λ i = λ i low + λ i high 2 :
This strategy involves all the switches of the cell at each switching period. As shown in Fig. 10, a i1 and a i2 are not set to 0 and 1, respectively, for more than one switching period. Thus, compared to the previous strategies, it leads to an increase of the switching losses. Three switching occur at each switching period, as illustrated Fig. 11. To conclude, these results highlight that the modeling scheme given in Fig. 4 is fulfilled.

V. EXPERIMENTAL RESULTS

The three-phases inverter of the experimental test-bench is made up of three normal T-type switching cells modules, APTGLQ40HR120CT3G from Microsemi in Fig 12 . A classical scheme will be used for the control/command layer: the command signals will be send to the switches trough a DSPACE device embedded with FPGA in order to format the signals (i.e., inclusion of the dead time). The signals applied are similar to those defined in Section IV-C. For every modulation strategy described above, the leg voltage v lo are displayed, as shown in Fig. 13.

A. λ i = 0
It is noticed that the high magnitude of the switching edges Δv lo = E seems to be an important factor, and also a drawback, concerning resonance phenomena as well as overall quality of currents.

B. λ i = λ i h ig h

Let us notice that this strategy does not suffer from the same drawback encountered in the previous strategy. Effectively, the voltage edge is smaller

Δv lo = E 2 . C. λ i = λ i lo w + λ i h ig h 2
A first advantage is to obtain a low distortion signal. Indeed, as displayed in Fig. 13, both Δv lo = E and Δv lo = E are encountered leading to mitigated voltage edges. Let us remark that the rising edges can be reduced by using a triangle carrier.

Consequently, it will produce only Δv lo = E 2 voltage edges.

VI. DISCUSSION

The main advantage of our contribution is to embed the design of modulation schemes in a generic framework. Indeed, the generic modeling process described in Section II can be used to any inverter topology. In addition, the modulation strategy definition in Section III is based on a generic mathematical tool. Moreover, software already provide stable algorithms for the calculus of our pseudoinverse matrices.

To highlight the efficiency of our method, we focus our discussion on the general solution provided in [START_REF] Shin | Power loss comparison with different PWM methods for 3L-NPC inverter and 3L-T type inverter[END_REF]. The generic PWM scheme described is associated with the boundaries of the solution set. Indeed, free specific examples are illustrated. Following the criteria to optimize, power losses or voltage edges, for instance, one scheme must be applied. The modeling method allows us to implement a generic modulation scheme. Indeed, a simple if statement makes possible to switch at any time from one strategy to another one, where only the λ i i ∈ 1, 2, 3 will be changed. Moreover, it is highlighted that every expression can be defined, as it fits the maximal λ i high , an minimal 0, values. The nonlinear behavior of the system while functioning is not compromised. Effectively, the average model is only used to generate the time rate as references, before being com-pared to a time representative signal. Finally, binary gate signals are generated.

More precisely in the solution expression [START_REF] Shin | Power loss comparison with different PWM methods for 3L-NPC inverter and 3L-T type inverter[END_REF], the fixed part, (M ⊗ A) † V ref is directly computed from the desired reference voltages. It corresponds to averaged steady-state voltages. For every time rates a ij , the independent part of the solution F λ is made of the common component λ 4 and a per leg tuning parameter λ i , i ∈ {1; 2; 3}. λ 4 was demonstrated to be proportional to the common mode voltage v no (24). Also, the proposed method exhibits easily the boundaries for the λ i . It allows us to define immediately any expression for λ i , which fulfills the constraints (22). For instance, when λ i = λ i low +λ i high 2 , i ∈ {1, 2, 3}, the time rates solution is obvious, whereas specific waveforms of the independent part F λ are not. Indeed, based on the oneselfexperience, it would have been difficult to suggest such a waveform, as displayed in Fig. 14.

As far as the method is concerned, it is pointed out that a single carrier is needed while all possible solutions are highlighted. Because the modulating signal obtained is consistent to a time rate, but also related to a switching time, it can be inserted into an SVM scheme. This is due to the fact that the modulation strategies modeling method is generic.

VII. CONCLUSION

Based on a generic framework, the contribution exposed in this paper is twofold as follows.

1) A model of the T-type 3LT 2 C converter is exhibited.

2) Control strategies for the T-type 3LT 2 C are provided. The mathematical model of the inverter considered paves the way to the modeling of any inverter topology.

Furthermore, the obtained model is a linear system easy to handle with mathematical linear system resolution techniques. This is used to involve the pseudoinverse theory when designing the control strategies. Such a tool is used to exhibit the degree of freedom of the linear solution set. The tuning parameters are then used to address the physical constraints of the inverter.

Simulation and experimental results have proven the feasibility of the proposed approach.

APPENDIX MAXIMAL RANK FACTORIZATION

The independent solution is

I 6 -(M ⊗ A) † (M ⊗ A)z = 1 6 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ z. ( 25
)
In such an expression, the components of z are linked through the matrix I 6 -(M ⊗ A) † (M ⊗ A). To highlight the four tuning parameters [START_REF] Choi | New modulation strategy to balance the neutral-point voltage for three-level neutral-clamped inverter systems[END_REF], a maximal rank factorization is performed. F is chosen to dispatch one tuning parameter per leg, and the last one is distributed over all the legs

F = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -1 0 0 1 1 0 0 1 0 -1 0 1 0 1 0 1 0 0 -1 1 0 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
.

Fig. 1 .

 1 Fig. 1. T-type switching cells [14]. (a) Normal. (b) RB-IGBT based.

Fig. 2 .

 2 Fig. 2. Elementary cell structure.

Fig. 3 .

 3 Fig. 3. Example of a switching sequence.

Fig. 4 .

 4 Fig. 4. Command orders generation principle.

Fig. 6 .

 6 Fig. 6. λ i = 0 i ∈ {1; 2; 3}, time rates, and bounds.

Fig. 7 .

 7 Fig. 7. λ i = 0 switches sequence.

Fig. 8 .

 8 Fig. 8. λ i = λ i high time rates and bounds.

Fig. 9 .

 9 Fig. 9. λ i = λ i high switches sequence.

Fig. 10 .

 10 Fig. 10. λ i = λ i low + λ i high 2

Fig. 11 .

 11 Fig. 11. λ i = λ i low + λ i high 2

Fig. 12 .

 12 Fig. 12. Microsemi T-type module. (a) Supplied module (b) Electrical scheme.

Fig. 13 .

 13 Fig. 13. Output voltages v lo for the three described strategies.

Fig. 14 .

 14 Fig. 14. Independent part for time rates of the second leg.

F

  6×4 and G 4×6 are defined so that I 6 -(M ⊗ A) † (M ⊗ A)z = F Gz = F λ where λ components are the four tuning parameters.

TABLE I OUTPUT VOLTAGES

 IVOLTAGES

He started his research activity with an internship on power converters modelization and control in the Laplace Laboratory, Toulouse, France. From 2015, he was engaged with the LGP, ENIT, Universit de Toulouse, Toulouse, France, and changed his research topic to step back and focus on power electronics. He worked first on the multiphysics modelization of the materials of power electronics modules, from 2015 to 2016. From 2016, he works on the design of power converters for aeronautics.

Paul-Etienne