Member, IEEE Damien Querlioz

Olivier Bichler

Member, IEEE Philippe Dollfus

Christian Gamrat

Immunity to Device Variations in a Spiking Neural Network with Memristive Nanodevices

Keywords: spiking neural networks, memristors, memristive devices, spike timing dependent plasticity, unsupervised learning, neuromorphic

Memristive nanodevices can feature a compact multi-level non-volatile memory function, but are prone to device variability. We propose a novel neural network-based computing paradigm, which exploits their specific physics, and which has virtual immunity to their variability. Memristive devices are used as synapses in a spiking neural network performing unsupervised learning. They learn using a simplified and customized "spike timing dependent plasticity" rule. In the network, neurons' threshold is adjusted following a homeostasis-type rule. We perform system level simulations with an experimentally verifiedmodel of the memristive devices' behavior. They show, on the textbook case of character recognition, that performance can compare with traditional supervised networks of similar complexity. They also show that the system can retain functionality with extreme variations of various memristive devices' parameters (a relative standard dispersion of more than 50% is tolerated on all device parameters), thanks to the robustness of the scheme, its unsupervised nature, and the capability of homeostasis. Additionally the network can adjust to stimuli presented with different coding schemes, is particularly robust to read disturb effects and does not require unrealistic control on the devices' conductance. These results open the way for a novel design approach for ultra-adaptive electronic systems.

I. INTRODUCTION

M EMRISTIVE nanodevices provide fantastic opportuni- ties for microelectronics. They can indeed provide a compact multi-level non-volatile memory function [START_REF] Strukov | The missing memristor found[END_REF]. However, they are often subject to strong variability [START_REF] Alibart | High precision tuning of state for memristive devices by adaptable variation-tolerant algorithm[END_REF], [START_REF] Agnus | Two-terminal carbon nanotube programmable devices for adaptive architectures[END_REF], [START_REF] Kim | A functional hybrid memristor crossbar-Array/CMOS system for data storage and neuromorphic applications[END_REF], so that fully exploiting their potential would be easier with architectures offering a strong immunity to device variations. Spiking neural networks could provide a serious lead since the brain itself relies on variable neurons and synapses [START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF] and manages computational efficiency that outperforms manmade systems. This idea takes particular meaning in that many groups (constituting the "neuromorphic" community) already imitate the brain with electronics, using Complementary Metal Oxide Semi-conductor (CMOS) circuits to model its spiking neurons and synapses. These works, however, are often limited by the number of implementable synapses: implementing plastic synapses requires many transistors [START_REF] Arthur | Learning in silicon: Timing is everything[END_REF], [START_REF] Indiveri | A VLSI array of low-power spiking neurons and bistable synapses with spike-timing dependent plasticity[END_REF]. Memristive nanodevices could provide the compact synapses required to advance neuromorphic circuits. In recent years several classes of them have indeed emerged, as e.g. resistive RAMs and memristors [START_REF] Strukov | The missing memristor found[END_REF], [START_REF] Alibart | High precision tuning of state for memristive devices by adaptable variation-tolerant algorithm[END_REF], or adaptive transistors [START_REF] Alibart | An organic nanoparticle transistor behaving as a biological spiking synapse[END_REF], [START_REF] Agnus | Two-terminal carbon nanotube programmable devices for adaptive architectures[END_REF]. It has been suggested [START_REF] Perez-Carrasco | On neuromorphic spiking architectures for asynchronous STDP memristive systems[END_REF], [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF], [START_REF] Afifi | Implementation of biologically plausible spiking neural network models on the memristor crossbarbased CMOS/nano circuits[END_REF], [START_REF] Querlioz | Simulation of a memristorbased spiking neural network immune to device variations[END_REF], and shown experimentally [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF], [START_REF] Seo | Analog memory and spiketiming-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF], [START_REF] Yu | An electronic synapse device based on metal oxide resistive switching memory for neuromorphic computation[END_REF], [START_REF] Kuzum | Nanoelectronic programmable synapses based on phase change materials for brain-inspired computing[END_REF], [START_REF] Bichler | Visual pattern extraction using energy-efficient "2-PCM synapse" neuromorphic architecture[END_REF], that such devices could reproduce a learning rule of biological synapses -spike timing dependent plasticity (STDP) [START_REF] Markram | Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs[END_REF], [START_REF] Bi | Synaptic modification by correlated activity: Hebb's postulate revisited[END_REF] -that is believed to be a foundation of learning in the brain [START_REF] Dan | Spike timing-dependent plasticity of neural circuits[END_REF]. A system consisting of nanoscale synapses and CMOS neurons could be a major breakthrough in computing, allowing cognitive-type tasks with high efficiency.

This idea is currently receiving considerable interest [START_REF] Snider | From synapses to circuitry: Using memristive memory to explore the electronic brain[END_REF], [START_REF] Modha | Cognitive computing[END_REF], [START_REF] Pershin | Neuromorphic, digital and quantum computation with memory circuit elements[END_REF], [START_REF] Cantley | Hebbian learning in spiking neural networks with nanocrystalline silicon TFTs and memristive synapses[END_REF]. However, its sustainability is still to be demonstrated, especially with regards to the variability issue that is common to all memristive technologies [START_REF] Borghetti | Memristive' switches enable 'stateful' logic operations via material implication[END_REF], [START_REF] Yu | On the stochastic nature of resistive switching in metal oxide RRAM: physical modeling, monte carlo simulation, and experimental characterization[END_REF], [START_REF] Kim | A functional hybrid memristor crossbar-Array/CMOS system for data storage and neuromorphic applications[END_REF]. It is also not clear if biological STDP is the best approach for electronics, the constraints of which differ strongly from Biology's.

In this paper, system simulations introduce quantitative results in terms of computing performance and robustness to variability. We exploit a simplified and customized STDP scheme for memristive devices, which is key to achieve effective learning with extreme robustness to memristive devices' variability. It is associated with the use of unsupervised learning, and of a homeostasis-type mechanism. We describe the required technology and architecture (section II) and perform system-level simulations on a standard database of machine learning [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF] that show the potential of the approach and its robustness. The system consists of an unsupervised layer that extracts features of the inputs using a simplified spike timing dependent plasticity (section III). The network performance compares favorably with traditional -but supervisednetworks with similar numbers of adjustable parameters and achieves excellent tolerance to various memristive devices' parameters variability. Finally, the robustness of the network to other device nonidealities (read disturb and limited resolution effects) is established (section IV).

Partial and preliminary results have appeared in [START_REF] Querlioz | Simulation of a memristorbased spiking neural network immune to device variations[END_REF]. This papers adds new discussions and results, especially the plausibility of the device model with regards to measurements of real devices, adaptations of the programming scheme, and the impact of diverse device nonidealities. Other proposals have been made to exploit variable adaptive devices in the context of nanotechnological implementations. Most proposed architectures rely on reconfigurable logic [START_REF] Heath | A defecttolerant computer architecture: Opportunities for nanotechnology[END_REF] or on state-based supervised neural networks [START_REF] Lee | Defect-tolerant nanoelectronic pattern classifiers[END_REF], [START_REF] Chabi | Robust neural logick block (NLB) based on memristor crossbar array[END_REF]. In the first approach variability is addressed through error mapping and redundancy, in the second case through a standard neural network approach using supervised learning based on error gradient descent.

Our approach of using unsupervised learning with asynchronous spiking neural networks to tackle the variability issue of nanodevices is original and takes inspiration from recent ideas in computational neuroscience and neural networks [START_REF] Guyonneau | Neurons tune to the earliest spikes through STDP[END_REF], [START_REF] Brader | Learning real-world stimuli in a neural network with spike-driven synaptic dynamics[END_REF], [START_REF] Nessler | STDP enables spiking neurons to detect hidden causes of their inputs[END_REF]. Different works have been published that go into that direction. As mentioned above, several proposals exist to use memristive devices for STDP [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF], [START_REF] Perez-Carrasco | On neuromorphic spiking architectures for asynchronous STDP memristive systems[END_REF], [START_REF] Afifi | Implementation of biologically plausible spiking neural network models on the memristor crossbarbased CMOS/nano circuits[END_REF], [START_REF] Yu | A neuromorphic visual system using RRAM synaptic devices with sub-pJ energy and tolerance to variability: Experimental characterization and large-scale modeling[END_REF]. In this paper we use a simplified STDP scheme that is easier to implement. Additionally, we propose to give to the neurons a homeostasis property and that is shown to be essential for the robustness of the scheme to variations. One work had already shown that memristive devices with STDP could allow the emergence of receptive fields in a variability-compatible unsupervised approach and synchronous neurons [START_REF] Snider | Self-organized computation with unreliable, memristive nanodevices[END_REF]. Our work uses asynchronous designs, like the ones used in the neuromorphic community [START_REF] Arthur | Learning in silicon: Timing is everything[END_REF], [START_REF] Indiveri | A VLSI array of low-power spiking neurons and bistable synapses with spike-timing dependent plasticity[END_REF], and performs full learning on a standard dataset. Finally, an alternative way to allow learning with memristive devices in a variability-compatible way can be to use all digital designs [START_REF] Snider | From synapses to circuitry: Using memristive memory to explore the electronic brain[END_REF]. This requires more devices per synapses [START_REF] Snider | Instar and outstar learning with memristive nanodevices[END_REF]. In this paper we show that variation-tolerance can be retained by using nanodevices with continuous variation of the conductance.

II. ARCHITECTURE AND IMPLEMENTATION OF THE NETWORK

We first introduce the architecture that we propose for our classifier system. CMOS input and output "neurons" are connected by the nanodevices that act as synapses. It is natural to lay out the nanodevices in the widely studied crossbar as illustrated on Fig. 1, where CMOS silicon neurons and their associated synaptic driving circuitry are the dots, the squares being the nanodevices. The synapses indeed act as adaptive resistors. With the crossbar layout, if several synapses are active at the same time (i.e. receiving voltage spikes), the output receives directly the sum of the currents flowing through the synapses. In a more futuristic design, the system could also be laid out in a CMOL architecture where nanodevices crossbar is fabricated on top of the CMOS neurons and driving circuits [START_REF] Lee | Defect-tolerant nanoelectronic pattern classifiers[END_REF]. When an input neuron spikes, it applies an Input pulse (a) to the nanodevices to which it is connected. When an output neuron spikes it applies an Output pulse (b) to the nanodevices to which it is connected. When the voltage applied on the device (difference between the voltages applied at the two ends (c) or (d)) reaches V T + or V T -, its conductance is increased or decreased, respectively.

This kind of connectivity corresponds to a feed-forward architecture in a neural network. Of particular interest, in the case of our network, it limits the sneak path issue because both programming and reading are performed in parallel. This issue usually limits the competitiveness of crossbars [START_REF] Liang | Cross-point memory array without cell Selectors-Device characteristics and data storage pattern dependencies[END_REF] and requires complex counter-measures like complementary resistive switches [START_REF] Linn | Complementary resistive switches for passive nanocrossbar memories[END_REF], or the use of nonlinear devices [START_REF] Yang | Engineering nonlinearity into memristors for passive crossbar applications[END_REF].

As a replacement of memristive devices, the architecture may also exploit phase change memories associated in "2-PCM" circuits as evidenced experimentally in [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF], [START_REF] Bichler | Visual pattern extraction using energy-efficient "2-PCM synapse" neuromorphic architecture[END_REF], [START_REF] Suri | Physical aspects of low power synapses based on phase change memory devices[END_REF], which has the advantage of technological maturity.

The input neurons present the stimuli as asynchronous voltage spikes using several possible coding schemes described in section III-C. Spiking rate is proportional to stimulus intensity. These stimuli may originate for example directly from a spiking retina [START_REF] Lichtsteiner | A 128x 128 120 dB 15 mus latency asynchronous temporal contrast vision sensor[END_REF] or cochlea [START_REF] Chan | AER EAR: a matched silicon cochlea pair with address event representation interface[END_REF] designed in the neuromorphic community that present data as asynchronous spikes, similarly to their biological counterparts.

As a result of learning, the output neurons should become selective to the different stimuli classes that are presented in a fully unsupervised manner: the output neurons should develop selectivity to specific features contained in the input patterns. The learning rule of the nanodevices needs to be fully local to be implementable in the crossbar architecture. The behavior of the neurons needs to be simple to make it easy to implement in a compact way. We now describe how this can be achieved.

The next subsections describe the different elements into more details. The synapses learn using a simplified STDP scheme (section II-A1). The output neurons behave as leaky integrate-and-fire neurons (section II-B1) and have a homeostasis property (section II-B3). They are also connected by inhibitory connections using diffuser networks (section II-B2). A. Synaptic behavior 1) Simplified STDP learning: In this work, the synapses are acting in two ways. They are variable resistors and thus transmit spikes with a variable conductance (which plays the exact role of a synaptic weight). Additionally, they adapt their conductance depending on the activity of the neurons to which they are connected, which provides the foundation of learning by the system.

The memristive nanodevices are programmed as follows. When being applied a positive voltage pulse higher than a threshold V T + , they increase their conductance. When applied a voltage pulse smaller than a negative threshold V T -, they decrease their conductance [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF]. Previous works have shown that memristive devices can implement spike timing dependent plasticity (STDP), a learning rule used by brain synapses [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF], [START_REF] Seo | Analog memory and spiketiming-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF], [START_REF] Perez-Carrasco | On neuromorphic spiking architectures for asynchronous STDP memristive systems[END_REF], [START_REF] Afifi | Implementation of biologically plausible spiking neural network models on the memristor crossbarbased CMOS/nano circuits[END_REF], [START_REF] Snider | Spike-timing-dependent learning in memristive nanodevices[END_REF], [START_REF] Yu | An electronic synapse device based on metal oxide resistive switching memory for neuromorphic computation[END_REF], [START_REF] Kuzum | Nanoelectronic programmable synapses based on phase change materials for brain-inspired computing[END_REF], [START_REF] Yu | A neuromorphic visual system using RRAM synaptic devices with sub-pJ energy and tolerance to variability: Experimental characterization and large-scale modeling[END_REF]. These works focused on faithful imitation of Biology. In this work, we focus on proposing simpler scheme, targeted toward pattern extraction, as illustrated in Figure 2.

• When an input neuron spikes, it applies a long voltage pulse to its synapses (Input pulse, Figure 2(a)). This voltage is high enough to drive some current into a memristive device, but not enough to reprogram it. This current is integrated by the output neurons (resistor role of the synapse). If several synapses connected to the same output neuron are active at the same time their currents are summed. • When an output neuron spikes, it applies a pulse that is a succession of a negative bias and of a positive bias (Output pulse, Figure 2(b)). If no Input pulse is being applied to the device, only the second part reaches a threshold and the conductance of the synapse is decreased by δG m (Figure 2(d)). However, if the input neuron had spiked recently, and the Input pulse is still being applied on the other end of the device, the voltage applied on the device actually increases its conductance by δG p ((Figure 2(c)). This simple learning rule, easily implemented with nanodevices, is the ground for learning. Compared with the purely bioinspired and more complex scheme introduced in [START_REF] Perez-Carrasco | On neuromorphic spiking architectures for asynchronous STDP memristive systems[END_REF], no delay matching is necessary between the Input and Output synaptic waveforms, which should make the driving circuitry much easier to design.

The way this simple learning rule works is straightforward. When an output neuron declares a spike (at time t spike ), it increases by δG p the conductance of the synapses connected to input neurons that spiked recently (from t spike -t P RE to t spike , if t P RE is the duration of the Input pulse), and decreases by δG m the conductance of the synapses that did not. This increases the sensitivity of the neuron to the specific pattern that activated it, making it more likely to spike for a similar (correlated) pattern in the future. This process -that works surprisingly well in practice, as we show in this paper -has been partially theorized in [START_REF] Nessler | STDP enables spiking neurons to detect hidden causes of their inputs[END_REF]. A comparison with the traditional biological STDP scheme is presented in Figure 4.

A disadvantage of this rule is the long Input pulse, which drives current for a long time, and thus increases power consumption. If it becomes significant, a low power version of the learning rule is possible and illustrated in Figure 3, for the cost of limited added complexity. The input neuron does not apply a voltage pulse during the whole time it is active, but only at the beginning to drive current into the output neuron. Additionally, as soon as one of the output neuron becomes active, a signal is sent back to the input neurons, which apply a short Input pulse again if they are still active. This lower power version of the learning scheme is more realistic with most current technologies like [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF].

2) Memristive devices modeling: To model the conductance increments and decrements in our system simulations, we use the model introduced in [START_REF] Querlioz | Learning with memristive devices: how should we model their behavior?[END_REF]. It takes inspiration from experimental memristive devices measurements [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Seo | Analog memory and spiketiming-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF]. An increase in the conductance is modeled by the equation:

δG p = α p e -βp G-G min Gmax-G min . (1)
Similarly, a decrease is modeled by:

δG m = α m e -βm Gmax-G Gmax-G min . (2)
The exponential factor expresses the fact, observed in most memristive technologies, that a given voltage pulse has a reduced effect on the device conductance if applied several times [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Seo | Analog memory and spiketiming-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF], [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF]. Agreement with the experimental data of [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF] is presented in Figure 5. The parameters α p , α m , β p , β m depend heavily on the Input and Output pulse voltages that are chosen. These parameters, as well as minimum and maximum conductances G min and G max are subject to device variability in real devices.

B. Output neurons

1) Output neurons' dynamics: Exploiting the devices requires connecting them to processing units -silicon neurons able to process and generate spikes in a bioinspired manner by integration of their input. We call X the state variable (a current or a voltage, equivalent of the biological "membrane potential") of the neuron (expressed in normalized unit where the maximum value of the state variable X is 1). Neurons are leaky integrate-and-fire type, which is meant to solve the simple following equation. :

τ dX dt + gX = γI input (3)
where τ is a leak time constant, and g and γ are constants. I input represents the current flowing through the line of the crossbar connected to the neuron:

I input = j I j ( 4
)
where I j are the currents through each memristive device j connected to output neuron.

The neuron declares a spike if X reaches a given threshold X th , in which case X is reset to zero.

An approach widely studied in the neuromorphic community is to use analog circuits (generally with transistors operating in the sub-threshold regime) able to receive and generate asynchronous spikes [START_REF] Arthur | Learning in silicon: Timing is everything[END_REF], [START_REF] Indiveri | A VLSI array of low-power spiking neurons and bistable synapses with spike-timing dependent plasticity[END_REF] to design such neurons. This kind of CMOS design is particularly low power, because most transistors operate in the subthreshold regime, and thanks to the use of asynchronous computation. Though smaller than in nanodevices, variability is also a problem for such transistor designs. It is a challenge for any neuromorphic design [START_REF] Chan | AER EAR: a matched silicon cochlea pair with address event representation interface[END_REF] and will become even more crucial when scaling to modern technology. Digital designs may also be used that do not suffer from variability directly but may have higher area and power requirements [START_REF] Merolla | A digital neurosynaptic core using embedded crossbar memory with 45pJ per spike in 45nm[END_REF].

2) Output neurons' lateral inhibition : When an output neuron spikes, it sends inhibitory signals to the other output neurons of the layer that prevent them from spiking during the inhibition time and resets their potential to zero. With this inhibition, the network is reminiscent of a Winner-Takes-All topology [START_REF] Nessler | STDP enables spiking neurons to detect hidden causes of their inputs[END_REF].

More precisely, when an output neuron spikes, the state variable X of the other output neurons is reset to zero during a time t inhibit .

X = 0 if t spike < t < t inhibit . (5)
In hardware, this inhibition between the neurons can be implemented in a compact way through diffuser networks as in [START_REF] Arthur | Learning in silicon: Timing is everything[END_REF], which require a minimum number of transistors.

3) Homeostasis : A final issue for the architecture is the adjustment of the neurons' threshold. Simple algorithms exist for traditional (non spiking) artificial neuron networks, but they do not work directly for spiking neurons. A bioinspired original route is homeostasis [START_REF] Marder | Variability, compensation and homeostasis in neuron and network function[END_REF]. A target activity is defined for the neurons (i.e. a number of times an output neuron should spike over an extended period of time, like 100 digits presentation). Regularly the threshold of the neuron is increased if the average activity of the neuron is above the target, and decreased if it is below.

dX th dt = γ (A -T ) , (6)
where A is the mean activity (or firing rate) of a neuron, T is the target activity, and γ is a multiplicative positive constant.

This ensures that all the output neurons are used and adjust the neurons' thresholds to the stimuli for which they become specialized. In neuromorphic hardware, homeostasis has been implemented with analog memories like in [START_REF] Bartolozzi | Implementing homeostatic plasticity in VLSI networks of spiking neurons[END_REF] or could be implemented digitally. The advantage inherent in this technology is evidenced in this paper in section III-B.

C. Simulations of the System

In this paper, all simulations are system-level and are based on a C++ special purpose code ("Xnet") [START_REF] Bichler | Extraction of temporally correlated features from dynamic vision sensors with spike-timing-dependent plasticity[END_REF], [START_REF] Querlioz | Simulation of a memristorbased spiking neural network immune to device variations[END_REF]. The code is event-based for simulation performance and runs on traditional central processing units (CPUs). Simulation parameters as introduced above are τ = 100 ms, g = 1, X th = 0.5 (normalized unit), inhibition time t inhibit = 10 ms, α p = 10 -2 , α m = 5 • 10 -3 , G min = 10 -4 , G max = 1 (normalized units where the mean of the maximum conductance value is 1), β p = 3.0, β m = 3.0. The width of the Input pulses is 25 ms. Parameter variations are introduced around all the parameters using Gaussian random numbers (the value of their standard deviation is given in Section III). The initial conductances are selected randomly around mid-range (0.5). The stimuli are applied using the coding schemes described in section III-C.

The two variations of voltages pulses (Figures 2 and3) gave identical performance in terms of recognition rate. All the results presented in the paper use the pulses of Figures 2.

For demonstration of the concept, in this paper we use the widely studied case of handwritten number recognition. The MNIST database is used, which consists in handwritten number of 28 × 28 pixels by 250 writers [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF].

In order to achieve learning, we present the full MNIST training database (60, 000 digits) three times to the system. Each input neuron is connected with one pixel of the image. It emits spikes with a jittered rate that is proportional to the pixel intensity (maximum rate is 22 Hz) as illustrated in Figure 11(b). The initial phase is random. Input neurons present spikes corresponding to a given digit during a period of 350 ms, after which they present spikes corresponding to another digit. No kind of preprocessing on the digits is used and the set is not augmented with distortions. The network is then tested on the MNIST test database, which consists in 10,000 digits that have not been presented during training. Simulation time was about 8 hours per run on an AMD Opteron 2216 CPU.

Figure 6 plots the synaptic conductances (or weights) learned by the system in a configuration with only 10 output neurons. It is remarkable that without any supervision and using only our local custom STDP rule, the system has identified 9 (out of 10) different numbers, the real features of the input. Moreover it has learned the distinctive features of the digits (and not just the most likely handwriting): it has learnt the loop of the digit two, the bottom of the six, or the horizontal parts of three and eight.

To evaluate the capability of the system, we can define a recognition rate. For that purpose, we associate output neurons with the digit for which they spike the most frequently a posteriori, using a subset of 1000 well identified numbers. In hardware this association could be performed with complex digital circuitry. An alternative can be to associate the unsupervised network with a supervised one [START_REF] Querlioz | Bioinspired networks with nanoscale memristive devices that combine the unsupervised and supervised learning approaches[END_REF]. All the simulations were repeated ten times, and the recognition rate given is averaged on the ten runs.

In order to evaluate quantitatively the network's recognition rate, Figure 7 plots the final recognition rate on the test database. With ten output neurons the recognition rate reaches 60%. To improve the recognition rate, we can introduce additional output neurons, in which case some output neurons respond to different handwritings of the same digit. All the output neurons respond to some kind of digits, as explained in section II-B3. With 50 output neurons, the recognition rate reaches 81%, and with 300 output neurons 93.5%. A s/m=10% 25% 50% 100%

Figure 9. Conductance as a function of pulse number (plotted similarly to Figure 5), for 100 devices with different relative standard dispersion σ /µ on the parameters αp, αm (from left to right 10%, 25%, 50% and 100%).

traditional artificial neural network with back-propagation and 300 hidden neurons (obtained with the same number of adjustable parameters) reaches 95% [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], which compares to our rate of 93.5%. In the literature, the best algorithm has a largely superior 99.7% recognition rate, but using 12 million adjustable parameters (vs. 235, 200 here) and a largely augmented training set [START_REF] Ciresan | Deep big simple neural nets excel on handwritten digit recognition[END_REF]. Though our numbers are clearly more modest, the interest here is that the network is fully unsupervised, with simple local learning rules, and variability immunity as is seen in the next section.

Thanks to the unsupervised nature of learning, the complicated "labeling" step does not need to be performed right away. Lots of unknown data may be presented for training, and the labeling can be performed from a limited subset of well identified data. This is a strong advantage for many problems involving natural data. In many cases, a lot of data is available, but that was not classified, and the system can here perform this analysis by itself.

III. VARIABILITY IMMUNITY A. Synaptic variability

We now exploit our system-level simulations to analyze the robustness of our approach. We first study the impact of nanodevices (synaptic) variability on the recognition rate of the network. For this study, we simulated the network with 50 output neurons, introducing different kinds of variability. Results are reported in Figure 8.

In the top curve of Figure 8, we evaluate the impact of variability of the initial conductance (i.e. the initial conductance of the nanodevices, before learning starts). We can see it has no impact on the final recognition rate, even when the variability is extreme. This means that we do not need to control the initial conductance precisely. This is fortunate since this would be complex: controlling the exact state of memristive devices organized in a crossbar is a difficult issue, due to the device variations and the sneak paths.

In the middle curve of Figure 8, we evaluate the impact of the variations on the learning increments and decrements (the parameters α p , α m of equations 1 and 2). These variations can be caused by dispersion of the device thresholds and of their programming characteristics. We can see that the network is immune to variations up to 25% of the mean value of these parameters (i.e. σ /µ = 25%). This is already an extremely high level of variation for an electron device, but typical for research nanodevices. With 50% of variation, there is a small decrease of recognition rate (79% instead of 82% with no variability). With an extreme variability of 100% on the synaptic parameters, the recognition rate decreases significantly, but interestingly the functionality of the network is not challenged. We should note that with a variability of 50%, as many as 4% of the nanodevices cannot be programmed in at least one direction (i.e. they have an α value of 0). The latter figure becomes 30% when parameter variability is 100%. This is an indication of the overall tolerance to defects. To give a feel of the magnitude of these dispersions, we plotted on Figure 9, the conductance as a function of pulse number (plotted similarly to Figure 5), for 100 devices with different relative standard dispersion σ /µ on the parameters α p , α m .

The bottom curve of Figure 8 adds variations of G min , G max . In that case some nanodevices overpower the others due to increased maximum conductance. Results are however similar to the previous case, and extreme robustness is preserved: with 50% variability, the recognition rate is lowered from 81.9% to a reasonable 77.2% . The most sensitive parameter is the maximum conductance. It should be noted that memristive currently devices developed in academic groups may have relative standard variations in G min , G max in the range between 50% or 100% ( [START_REF] Kim | A functional hybrid memristor crossbar-Array/CMOS system for data storage and neuromorphic applications[END_REF], [START_REF] Borghetti | Memristive' switches enable 'stateful' logic operations via material implication[END_REF] suppl. info.). The situation is expected to improve dramatically when the technology becomes mature. However, some significant degree of variability might be intrinsic to their physics [START_REF] Yu | On the stochastic nature of resistive switching in metal oxide RRAM: physical modeling, monte carlo simulation, and experimental characterization[END_REF].

This degree of robustness to device variation is exceptional in electronic systems and constitutes one of the strongest points of the approach. It takes root in the unsupervised nature of learning: output neurons learn features for which they are naturally suited. Synaptic variability determines how a neuron reacts initially, and thus what it is going to learn. This means that, in a way, variability is not a real problem, but a feature to initiate learning.

The simplicity of the voltages pulses is also fundamental to ensure this robustness.

B. neuron variability

We now study the impact of the variability of the CMOS neurons. The impact of the variability of their threshold is seen in Figure 10. We can see that without homeostasis the impact of the variability is dramatic. The more excitable neurons (the Figure 11. Illustration of the three stimulus coding schemes. Stimulus spikes timing for ten input neurons with the same pixel value in the three schemes. All three lead to a recognition rate between 81% and 82%, without any parameter needing to be adjusted. ones with lower thresholds) spike predominantly and the other neurons do not specialize efficiently. In a typical simulation with a threshold variation of 25 %, the most excitable neuron spikes 51 % of the time, and in a simulation with a threshold variation of 50 %, it spikes 91% of the time. Homeostasis however fully compensates this issue (all the neurons spike between 1.5 and 3% of the time, most neurons spiking around 2%). And the same recognition rate as without threshold variation is achieved as evidenced in Figure 10. Homeostasis appears as particularly valuable in this application. This also stresses the importance to evaluate the robustness of the system not only to nanodevices' variations, but also to the CMOS'.

C. Stimulus encoding

Another remarkable point is the insensitivity to stimulus encoding scheme. Different schemes are possible to encode the stimulus into spikes as illustrated in Figure 11. In the first two schemes (Figures 11(a) and 11(b)), the input neurons spike periodically, the firing rate being proportional to the input quantity. The input neurons can spike either in phase (Figure 11(a)) or out of phase (Figure 11(b)). This is a natural scheme based on how spiking retinas currently work [START_REF] Lichtsteiner | A 128x 128 120 dB 15 mus latency asynchronous temporal contrast vision sensor[END_REF]. In the last case (Figure 11(c)), input neurons spike in a Poissonian way, the time constant being inversely proportional to the input. All three coding styles lead to a recognition rate between 81% and 82%. The only condition for this is that the window for potentiation in the STDP scheme is long enough with regards to the typical interspike interval. In this work, the average interspike interval is 45 ms for a black pixel, and insensitivity to stimulus encoding scheme is achieved if the window for potentiation is longer than 25 ms (while 15 ms is sufficient in the jittered periodic case). This shows again the robustness of the approach.

Additionally, noise can be inserted in the input with no impact. Adding 10% of perfectly random spikes distributed on all the input neurons has no impact on the system's recognition rate. This kind of robustness to spikes' precise timing is consistent with computational neuroscience works on STDP [START_REF] Guyonneau | Neurons tune to the earliest spikes through STDP[END_REF]. This result is not surprising since the precise timing of the spikes does not play a major role neither for the neurons nor for the simplified STDP learning rule.

IV. DEVICE NON-IDEALITIES

Finally, we check that our approach is not sensitive to two crucial non-idealities of real memristive devices: the limited resolution of the conductance and the read disturb effect.

A. Limited Resolution of the Memristive Devices

The nanodevices that we are considering have an "analog memory " property. It has been evidenced in many real-life devices [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF], [START_REF] Seo | Analog memory and spiketiming-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF], [START_REF] Yu | An electronic synapse device based on metal oxide resistive switching memory for neuromorphic computation[END_REF], [START_REF] Kuzum | Nanoelectronic programmable synapses based on phase change materials for brain-inspired computing[END_REF]. In these real devices, however, the conductance cannot be programmed with an arbitrary precision [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF]. It is thus important to clarify which precision is required for our application. To that purpose, we performed simulations with different increments α p of equation (1) (the decrement α m of equation ( 2) is kept proportional to α p ). The results in terms of recognition rate are plotted in Figure 12 (α p is expressed in normalized units where the maximum conductance is 1). We see that recognition rate is maximum for α p until 0.05, which is achievable by several experimental devices. For example the best measurements in [START_REF] Seo | Analog memory and spiketiming-dependent plasticity characteristics of a nanoscale titanium oxide bilayer resistive switching device[END_REF] achieve α p ∼ 0.01, or α p ∼ 0.02 in [START_REF] Suri | Phase change memory as synapse for ultra-dense neuromorphic systems: Application to complex visual pattern extraction[END_REF] and α p ∼ 0.07 in [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF]. For higher values of α p , the recognition rate decreases and the devices' resolution is not sufficient.

B. Impact of Read Disturb

Another significant device non-ideality can materialize if the Input pulses (which are subthreshold) have an impact on the devices' conductance. This corresponds to the Read Disturb phenomenon for memory devices. To assess the impact of this phenomenon, we performed simulations where the Input pulses increase the device conductance by a fraction of the increase of programming threshold:

δG subthreshold = δG p . (7)
The results are presented in Figure 12. It appears that readdisturb parameter as high as 0.1 (meaning that a read pulse has 10% of the impact of a write pulse) may be tolerated, since it is fully compensated by learning. Such a value is a lot higher than what is seen in real devices [START_REF] Borghetti | Memristive' switches enable 'stateful' logic operations via material implication[END_REF], [START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF]. Read distrub should thus not be an issue for our approach. Figure 12. Impact of the conductance increment on the recognition rate for a system with 50 outputs. Inset: Impact of the read disturb parameter on the recognition rate for a system with 50 outputs (as defined by equation ( 7)).

Every simulation was repeated five times.

V. CONCLUSION

In this work, using system-level simulations, we have shown how, by using a simple custom Spike Timing Dependent Plasticity scheme, memristive devices associated with CMOS neuromorphic circuits could perform unsupervised learning in a way that is extremely robust to variability. (A relative standard dispersion σ /µ of 50% is tolerated on all device parameters). Non-supervision is the foundation for this tolerance and provides extreme adaptability to the system. Homeostasis was also introduced as an essential component to retain insensitivity to CMOS variability. Additionally, the system is particularly tolerant to the read disturb effect, and does not require unrealistic control of the memristive devices' conductance.

This design approach could be the groundwork for future circuits that can process natural data in a compact and low power way. Thanks to their unsupervised learning capabilities, such circuits will be able to adapt to various environments. Future work should focus on the experimental demonstration of these concepts beyond single devices, and to demonstrate its scaling to more complex multi-layer networks, and to other kinds of sensory stimuli like video, auditory or olfactory data.

Figure 1 .

 1 Figure 1.Circuit topology. Wires originate from CMOS input layer (horizontal black wires) and from the CMOS output layer (vertical gray wires). Memristive nanodevices are at the intersection of the horizontal and vertical wires.

Figure 2 .

 2 Figure 2. Pulses for simplified STDP (voltage pulses as a function of time).When an input neuron spikes, it applies an Input pulse (a) to the nanodevices to which it is connected. When an output neuron spikes it applies an Output pulse (b) to the nanodevices to which it is connected. When the voltage applied on the device (difference between the voltages applied at the two ends (c) or (d)) reaches V T + or V T -, its conductance is increased or decreased, respectively.

Figure 3 .

 3 Figure 3. Pulses for simplified STDP (voltage pulses as a function of time).

Figure 4 .

 4 Figure 4. The simplified STDP rule, compared with the standard biological STDP rule.

Figure 5 .

 5 Figure 5. Evolution of the conductance for the devices from[START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF], fitted with equations 1 and 2. (a) device conductance (measured at 1V ) after each pulse in a serie of potentiating (V = -3.2V ) pulses. (b) same with depressing (V = 2.8V ) pulses. Diamond: experimental data, reproduced from[START_REF] Jo | Nanoscale device as synapse in neuromorphic systems[END_REF]. Full line: equations 1 and 2.

Figure 6 .

 6 Figure 6. Conductances (weights) learned in a simulation with 10 output neurons. Red is maximum conductance, blue is minimum conductance.

Figure 7 .

 7 Figure 7. Recognition rate on the test dataset as a function of the number of output neurons (simulations were repeated ten times, error bar is one standard deviation).

Figure 8 .

 8 Figure 8. Recognition rate on the MNIST testing set with different kinds of nanodevice (synaptic) variability. All the simulations were repeated ten times, the error bar is one standard deviation. All the simulations have homeostasis.

Figure 10 .

 10 Figure 10. Impact of the neurons' threshold X th variability, with and without homeostasis on the recognition rate. All the simulations were repeated ten times.

This work was supported by the European Union through the FP7 Project NABAB (Contract FP7-216777) and by CNRS/INSIS (PEPS Synapse).


D Dochain
email: denis.dochain@uclouvain.be

A Rapaport
email: alain.rapaport@inra.fr

An asymptotic observer for batch processes with single biogas measurement

Keywords: State observer, observability, biological systems, biosystem

In this paper we propose an observer for an example of systems for which the boundary of its domain is unobservable, and all trajectories converge to this boundary. The proposed case study, a bioreactor in batch operating conditions with one simple microbial growth reaction and gas production is standard and largely encountered in practical situations. Comparison of the proposed observer with a classical Luenberger observer has been performed: it shows that this later does not guarantee a convergence with no estimation error.

INTRODUCTION

The state observation of nonlinear systems is a wide and active research area resulting in a large scientific publications on the subject. Let us point out a few (e.g. Gauthier & Kupka [1998] [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF], [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]) as well as the interesting and nicely survey paper by Krener [START_REF] Krener | Nonlinear Observers[END_REF]). In the present paper, we would like to address a state observation issue that, as far as we know, has yet been addressed, i.e. the problem of the on-line reconstruction of the variables of nonlinear systems for which there is a loss of observability on the boundary of its domain. we shall indeed concentrate on a biological system example with one biomass x, one substrate s and gaseous outflow rate y measured on-line when the initial condition (x 0 , s 0 ) is unknown. Incidentally the design of the state estimate takes advantage of the fundamental reaction invariant property of reaction systems [START_REF] Gavalas | Nonlinear Differential Equations of Chemically Reacting Systems[END_REF]).

The paper is organized as follows. Section 2 introduces the dynamical model of the biological system. Section 3 provides an analysis of the observability properties of the system under study. An asymptotic observer is derived in Section 4 while Section 5 gives a Luenberger observer for the purpose of comparison. Section 6 provides numerical simulation results to illustrate the performance of the proposed observer as well as its comparison with the Luenberger observer designed in Section 5. Section 7 studies the behavior of our proposed observer for specific growth rate models (Hill, Haldane) that do not satisfy Hypothesis 2.

DYNAMICAL MODEL

Let us consider the dynamical model of a simple microbial growth reaction in a batch reactor (see e.g. [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF]):

ẋ = µ(s)x ṡ = -µ(s)x (1)
where x and s stand for the concentrations in biomass and substrate, respectively. Without loss of generality, we assume that the yield coefficient of the transformation of the substrate into biomass is equal to 1. The specific growth rate function µ(•) satisfies the usual assumption:

Hypothesis 1. The function µ(•) is Lipschitz continuous on R + , positive on (0, +∞) with µ(0) = 0.
For convenience, we denote the number μ = max s>0 µ(s)

This number could be finite or not. In the present work, we consider that a gaseous by-product flow rate, such as biogas (e.g. [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]), is measured on-line as a quantity proportional to the output variable y = µ(s)x. It often happens in batch biorocesses that the initial quantities of reactants (x 0 , s 0 ) are not well kwnon while the biogas production is the only available measurement during the operation of the process. As the solutions of (1) clearly satisfy lim s→+∞ s(t) = 0, the initial quantity of substrate can be recovered as

s 0 = lim t→+∞ t 0 y(τ )dτ
However the main interest is usually to estimate the total production of biomass, that is lim t→+∞ x(t). The purpose of the present work is the present a simple and reliable methodology to estimate the x concentration.

OBSERVABILITY ANALYSIS

Equivalenty to the set of differential equations (1), one can consider the dynamics in the (z, s) coordinates with

z = x + s d dt z s = f (z, s) = 0 -µ(s)(z -s) (2) y = h(z, s) = µ(s)(z -s) (3)
on the positive cone

C := {(z, s) ∈ R 2 | z > s > 0}
Here we require a stronger assumption on the function µ. Hypothesis 2. The function µ is C 2 , concave and increasing on R + , with µ(0) = 0.

Such an assumption is fulfilled for the well-known Monod function (see Figure 1) µ(s) = µ max s K s + s the most largely specific growth rate model used in biotechnology and biological systems.

L f h(z, s) = -[µ (s)(z -s) -µ(s)]µ(s)(z -s) = - µ (s) µ(s) h(z, s) -µ(s) h(z, s). (4)
For a given non-negative number y, we define the function

ϕ y (s) = µ(s) - µ (s) µ(s) y whose derivative is ϕ y (s) = µ (s) - µ (s) µ(s) - µ (s) 2 µ(s) 2 y.
Under Hypothesis 2, ϕ y (s) is positive for any s, and therefore the inverse ϕ -1

y is well defined from ϕ(R + ) to R + . As h(z, s) > 0 on C, one can write        s = ϕ -1 h(z,s) L f h(z, s) h(z, s) z = h(z, s µ(s) + s
and conclude that the map

z s → h(z, s) L f h(z, s)
is injective on C, that is the system is differentially observable on C.

On the boundary of C, one has L f i h ≡ 0 for any integer i and therefore the system is not differentially observable on ∂C.

One can note that the boundary z = s (that is x = 0) of the cone C is invariant but repulsive for the dynamics, while the boundary s = 0 is attractive. Therefore for any initial condition with z 0 > s 0 (that is x 0 > 0), the solution converges to the boundary s = 0 where the system is no longer observable. This feature prevents classical constructions of nonlinear observers, such as the high-gain observer, that requires the (differential) observability on a compact invariant set (which here has to contain s = 0...).

AN ASYMPTOTIC OBSERVER

In this section, we first show that the system is detectable from any positive initial condition, without requiring Hypothesis 2. Proposition 4. Assume that one as y(0) > 0, then one has

lim t→+∞ s(t) = 0, lim t→+∞ x(t) = z where z = y(t) µ +∞ t y(τ )dτ + +∞ t y(τ )dτ (5)
for any t ≥ 0.

Proof. When y(0) > 0, one has x 0 > 0 and s 0 > 0.

Clearly, the solution of (1) fulfills (x(t), s(t)) → (z, 0) when t tends to +∞, where z = x 0 +s 0 . Therefore, one can write

s(t) = +∞ t y(τ )dτ
at any time t ≥ 0, and then one also

x(t) = y(t) µ +∞ t y(τ )dτ .
Finally, one obtains

z = x(t) + s(t) = y(t) µ +∞ t y(τ )dτ + +∞ t y(τ )dτ
One can then consider the following asymptotic observer. Proposition 5. For any initial condition (x 0 , s 0 ) with x 0 > 0 and s 0 > 0, the following observer

   v(t) = y(t), v(0) = 0 x(t) = y(0) µ(v(t)) + v(t), (t > 0) (6) fulfills lim t→+∞ x(t) -x(t) = 0
Moreover, the error x -x is decreasing with time.

Proof. Note that for t > 0, the solution of ( 6) is given by the expression

x(t) = y(0) µ t 0 y(τ )dτ + t 0 y(τ )dτ
The convergence of the observer is then a simple consequence of Proposition 4. The time derivtive of the x-error is determined straightforwardly as

d dt (x -x)(t) = -y(t 0 ) µ t t0 y(τ )dτ µ t t0 y(τ )dτ 2 y(t) < 0
which shows the monotonic behavior of the error. Remark 6. The internal variable v(•) of the observer ( 6) is well defined for any t ≥ 0, while the estimation x(•) as "output" of this system is defined only for t > 0, which is quite unusual in the observer constructions. More precisely, one has lim t→0 -x(t) = +∞, but in practice the estimation x(•) drops down very quickly from large values at small times t > 0, as it can be seen on numerical simulations in Section 6. This is not related to a "peaking phenomenon" in the dynamics (there is no "high-gain"), but is simply due to the fact that x is not defined at t = 0. Remark 7. The fact that the error x -x is decreasing with time guarantees that x(t) is an upper estimation of x(t) at any time t and that the estimator does not oscillate as it could happen with high-gain observers.

A LUENBERGER OBSERVER

In this section, we consider functions µ(•) that satisfy Hypothesis 2, so that the differential observability is fulfilled during the transient. We study the behavior of an Luenberger observer when time tends towards infinity. As we aim at reconstructing the biomass x, and the output y is not a state variable of the original dynamics, we write the system (1) in (x, y) coordinates as follows ẋ = y ẏ = φ(x, y)y (7) with φ(x, y) = -µ • µ -1 y x x + y x Note that µ -1 is well defined on [0, μ] under the Hypothesis 2. However, the map φ has a singularity at x = 0, but solutions of (1) satisfy x(t) ≥ x 0 at any t > 0 and therefore avoid this singularity for any initial condition with x 0 > 0.

For the derivation of an observer, one has to extend this dynamics for values of x that are non positive or such that y/x is larger than μ. Let be a positive number, and consider the map φ(x, y) = φ(max(x, , y/μ), y)

which coincides with φ along any solution of (7) with

x 0 ≥ . Moreover φ is Lipschitz continuous on R × R + .
We can then consider a Luenberger observer in the (x, y) coordinates for the dynamics with φ replaced by φ:

ẋ = y + G 1 (ŷ -y(t)) ẏ = φ(x, y(t))y(t) + G 2 (ŷ -y(t)) (8)
Let us write the error equation as follows:

d dt e x e y = 0 G 1 δ(t)y(t) G 2 A(t)
e x e y where δ(t) = φ(x(t), y(t)) -φ(x(t), y(t))

x(t) -x(t) if x(t) = x(t) ∂ x φ(x(t), y(t)) if x(t) = x(t)
Under Hypothesis 2, the derivative µ is bounded on R + and then the map x → φ(x, y) has linear growth for any y ∈ R + . As y(•) is bounded, δ(•) is thus bounded whatever the solution x(•) is.

Therefore, as one has lim t→+∞ y(t) = 0, we obtain

lim t→+∞ A(t) = Ā = 0 G 1 0 G 2
which is not a Hurwitz matrix. Therefore, the asymptotic convergence of the error towards 0 is not guaranteed and the choice the gains G 1 , G 2 does not allow to assign the speed of convergence.

NUMERICAL SIMULATIONS

For the Monod function µ(s) = s 1 + s with the initial condition (s 0 , b 0 ) = (1.1, 1.5) we have compared the asymptotic observer with the Luenberger one initialized with (x, ŷ) = (1, y(0)) and various gains G 1 , G 2 . We found systematically a biased asymptotic error of the Luenberger observer, as depicted on Fig. 2 (for G 1 = G 2 = -20 that gave the best result). These simulations show that the innovation ŷ -y of the Luenberger observer reaches zero while the x error has not yet converged to zero, which explains its non null asymptotic error. We have Fig. 2. Comparison of our proposed observer with the Luenberger observer in absence of noise also performed simulations with measurements randomly disturbed by a white noise proportionate up to 10% of the signal (see Fig. 3). It shows the good behavior of the asymptotic observer with respect to measurement noise. Indeed, the dynamics of x in equations ( 6) use integrals of the output, and not directly the output (apart the value of y at time 0) as classical observers, which filters the noise. The asymptotic observer is mainly affected by the error on the initial measurement. We have also tested the observer with a louder noise (up to 25% of the signal). Then, a large error on the initial measurement leads, as expected, to an asymptotic bias, as for the Luenberger observer (see Fig. 4). In this Section we consider two different growth functions µ(•) that do not satisfy Hypothesis 2. This does not prevent the convergence of the asymptotic observer of Proposition 5, which requires Hypothesis 1 only to be fulfilled. The system remains detectable in the sense of Proposition 4, but the observability analysis of Section 3 can no longer be conducted. As a matter of comparison, we have simulated the Luenberger observer anyway.

Hill function

The Hill function for the specific growth rate writes as follows (see Figure 5): The Haldane model for the specific growth rate (see Figure 7) is given by the following expression:

µ(s) = µ max s α K α s + s α
µ(s) = µ 0 s K s + s + s 2

Ki

This function is non monotonic (see Fig. 7). Therefore the inverse of µ is not uniquely defined for the construction of the Luenbeger observer in (x, y) coordinates. However, for each positive value m of the function µ, one has µ

-1 (m) = {s -(m), s + (m)} with 0 < s -(m) ≤ √ K s K i ≤ s + (m)
, and as solutions of system (1) converge to s = 0, we have considered µ -1 (m) = s -(m) in the Luenberger observer (8) (which gives the right inverse a soon as t satisfies s(t) ≤ √ K s K i ). Simulations of Fig. 8 have been ran for µ 0 = 1, K s = 1 and K i = 0.2. These simulations show that the proposed observer works satisfactorily in simulation for a large variety of growth

CONCLUSIONS

We have proposed a new observer for systems that are not observable on a subset that attracts the dynamics. This observer guarantees an asymptotic convergence, while a classical Luenberger observer produces an asymptotic bias. Usually, one requires from a smooth observer to have tuning parameters for assigning an exponential speed of convergence. Here, as the trajectories of the system converge to a subset of non observability, one cannot expect to obtain simultaneously an exact convergence and an assignable speed of convergence of an observer. The observer we propose does not have a adjustable speed of convergence but it guarantees an exact convergence, and numerical simulations show its good robustness with respect to measurement noise. However, the asymptotic error of the proposed observer relies mainly on the initial measurement of the output. Robust extensions of this observer will be the matter of a future work.

Fig. 1 .

 1 Fig. 1. The Monod model Lemma 3. Under Hypothesis 2, the system (2)(3) is differentially observable on C, but not on its boundary. Proof. On the open cone C, one has µ(s) > 0 and can write

Fig. 3 .

 3 Fig. 3. Comparison of our proposed observer with the Luenberger observer with noise

Fig. 5 .Fig. 6 .

 56 Fig. 5. The Hill modelThis function is increasing but not concave (see Fig.5). Simulations of Fig.6have been ran for µ max = 1, K s = 0.5 and α = 2.

Fig. 8 .

 8 Fig. 7. The Haldane model


D Querlioz

J Saint-Martin

V.-N Do

A Bournel

P Dollfus

Fully quantum self-consistent study of ultimate DG-MOSFETs including realistic scattering using a Wigner Monte-Carlo approach

A new self-consistent quantum simulator based on the Monte Carlo solution of Wigner transport equation is used to analyze the operation of 6 nm-long DG-MOSFETs. By comparison with other simulation approaches, the work emphasizes the important role of scattering and quantum effects on the electrical characteristics of such nano-devices. The results are confronted to ITRS specifications and the various effects of aggressive oxide thickness thinning on device performance are discussed.

Introduction

To capture the physics occurring in nano-scaled devices, we have developed a fully quantum simulator based on the Wigner formalism. Using this new tool, this work analyzes the performance of ultra-short DG-MOSFETs. The study emphasizes the importance of scattering and of quantum effects on the device operation and electrical characteristics regarding ITRS requirements for the HP 16 nm node [START_REF]ITRS[END_REF]. The simulator is based on the Wigner Transport Equation (WTE) which is solved in the mode-space approximation. This approach is fully self-consistent and includes all relevant scattering mechanisms as accurately as in standard Ensemble Monte Carlo (EMC) simulation. Our particle technique to solving the Wigner transport equations is inspired by an approach suggested by Shifren et al [START_REF] Shifren | A Wigner function-based quantum ensemble Monte Carlo study of a resonant tunneling diode[END_REF], with some major improvements on the particle injection technique [START_REF] Querlioz | An Improved Wigner Monte-Carlo Technique for the self-consistent Simulation of RTDs[END_REF][START_REF] Querlioz | A study of quantum transport in end-of-roadmap DG-MOSFETs using a fully self-consistent Wigner Monte Carlo approach[END_REF], which allows us to accurately describe the quantum transport effects, as shown by simulation of RTDs [START_REF] Querlioz | An Improved Wigner Monte-Carlo Technique for the self-consistent Simulation of RTDs[END_REF]. Compared to the technique of Ref. [START_REF] Sverdlov | Quantum transport in ultra-scaled double-gate MOSFETs: A Wigner functionbased Monte Carlo approach[END_REF][START_REF] Nedjalkov | Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices[END_REF] this one can be implemented into selfconsistent EMC simulation. In comparison with direct Schrödinger's equation [START_REF] Gilbert | Efficient quantum three-dimensional modeling of fully depleted ballistic silicon-on-insulator metal-oxidesemiconductor field-effect-transistors[END_REF] or the well-known non-equilibrium Green's function (NEGF) approach [START_REF] Venugopal | A simple quantum mechanical treatment of scattering in nanoscale transistors[END_REF][START_REF] Bescond | 3D quantum modeling and simulation of multiple-gate nanowire MOSFETs[END_REF][START_REF] Svizhenko | Two-dimensional quantum mechanical modeling of nanotransistors[END_REF][START_REF] Khan | Self-consistent treatment of quantum transport in 10-nm FinFET devices using CBR method[END_REF][START_REF] Seonghoon | A threedimensional simulation of quantum transport in silicon nanowire transistor in the presence of electron-phonon interactions[END_REF], this Wigner EMC technique offers the advantage of taking easily into account all scattering mechanisms (phonons, impurity, surface roughness) and of being naturally a time simulation of quantum transport which may include transient effects.

Presentation of the model

This work presents the first self-consistent simulation of MOSFET using a particle description of quantum transport based on the Wigner function [START_REF] Jacoboni | Quantum transport and its simulation with the Wigner-function approach[END_REF].

In this formalism, for each confined sub-band of the device E n (x), the Wigner function f w (x,k) evolves according to

w w w w f f k Q f C f t m x ∂ ∂ + = + ∂ ∂ (1)
where the quantum evolution term of the Wigner function is given by

( ) ( ) ( ) ( ) 1 , ' , ' , ' 2 w w w Qf x k dk V x k k f x k π = - ∫ . ( 2
)
The Wigner potential ( )

, w V x k is defined by ( ) ( ) , s i n 2 2 w n n x x V x k dx kx E x E x ′ ′       ′ ′ = + - -             ∫ (3)
and Cf w is the collision term with 2DEG scattering rates. In this technique, we consider the Wigner function f w as a sum of pseudo-particles localized in both x and k space weighted by a parameter called affinity (2) that contains the quantum information on the particles. The x and k coordinates evolve exactly as that of classical particles, while the affinity is updated according to the Qf w term of the WTE. Scatterings are included using the standard MC algorithm (Figs. [START_REF]ITRS[END_REF][START_REF] Shifren | A Wigner function-based quantum ensemble Monte Carlo study of a resonant tunneling diode[END_REF], by making use of 2DEG scattering rates. The calculation of these inter and intra sub-band scattering rates is described in [START_REF] Saint-Martin | Multi sub-band Monte Carlo simulation of an ultra-thin double gate MOSFET with 2D electron gas[END_REF], and is similar to [START_REF] Lucci | Multi-subband monte carlo modeling of nano-mosfets with strong vertical quantization and electron gas degeneration[END_REF][START_REF] Fischetti | Monte Carlo study of electron transport in silicon inversion layers[END_REF]. The simulated device is meshed into vertical slices in which the 1D Schrödinger equation is solved to compute the subband profiles E n (x) and wave-functions n ξ [START_REF] Saint-Martin | Multi sub-band Monte Carlo simulation of an ultra-thin double gate MOSFET with 2D electron gas[END_REF].

Simulation results on ITRS 16nm devices

A. Electrical results

We have studied two 6 nm-long DG-MOSFETs (described in Fig. 2), differing in their gate equivalent oxide thickness (EOT).

Unless otherwise stated the gate work function is assumed to be that of a midgap gate minus 0.2 eV. Figs. 3a and3b show typical density and potential mappings and exhibit oxide repulsion typical of quantization effects. Subband profiles are plotted in Fig. 4, together with cartographies of the squared wave function of each type of subbands considered in the simulation. In subthreshold regime the Wigner results are very close to quantum ballistic ones but the current is significantly higher than that obtained using semi-classical simulation. This shows that scattering has a weak impact in this regime but that I off is strongly enhanced by S-D tunneling through the channel barrier. In contrast, the ballistic approach strongly overestimates the I on current due to the significant influence of scattering in both the channel and the highly doped access regions [START_REF] Saint Martin | On the ballistic transport in nanometer-scaled DG MOSFETs[END_REF]. As shown in Fig. 7, the drain current in saturation regime is perceptibly reduced by including quantum transport effects, which suggests the arising of quantum reflections in the channel under high drain bias as a consequence of the steep potential drop (4).

(eV)

B. Discussion on the impact of scattering in the channel

The effect of scattering in the channel is well illustrated in Fig. 8 where we compare the results of Wigner simulation using standard scattering rates (solid lines) with that obtained by switching the scattering mechanisms off in the channel (dashed lines). An effective ballisticity B eff = I on /I onball of 86% is obtained as a result of an I on enhancement by 16% in the case with no scattering in the channel. The electron velocity profile along the channel is plotted in Fig. 9 in both cases of Wigner and Boltzmann transport simulations. Results are very similar in both cases. As a consequence of high ballisticity, they exhibit a strong overshoot velocity of 4.5×10 7 cm/s near the drain-end of the channel. The distribution of scattering mechanisms occurring at different positions in the channel is displayed in Fig. 10. As a consequence of carrier heating, the relative contribution of intervalley scattering increases near the drain. The part of roughness scattering decreases near the drain-end. However this diffusion mechanism remains predominant in this ultra-thin body device. More surprisingly, for an EOT of 0.5 nm (recommended by ITRS), ITRS targets are not fully reached. The main electrical characteristics obtained for both devices using Wigner simulation are summarized in Table I and compared with some ITRS specifications for HP 16 nm node.

The EOT thinning reduces significantly I off , which appears on the delay-I off plot of Fig. 12. However, the resulting enhancement of gate capacitance does not allow the device to meet the ITRS requirements regarding the delay. The I on improvement is actually limited because of high access resistance in the narrow source and drain. Novel technological advances with reduced access resistances, e.g. metallic source/drain, might be required to reach these objectives.

As a consequence of this limited I on improvement and because of gate capacitance enhancement, oxide thinning does not provide any benefit in terms of delay. For the same I off , the delay is actually more important for a 0.5 nm EOT than for a 1.0 nm EOT. For I off =0.11µA/µm, the delay is 0.15 ps for an EOT of 1 nm and 0.19 ps for an EOT of 0.5 nm.

Conclusion

Thanks to a powerful new quantum simulator, this work shows the difficulty to meet ITRS specifications for the HP 16 nm node due to the importance of both scattering and quantum effects in ultimate DG-MOSFET. Additionally, it is shown that for this aggressive nanoscale design, reducing the EOT from 1 nm to 0.5 nm can be detrimental to the delay CV/I.

Fig. 1 .

 1 Fig. 1. MC Algorithm used for the Wigner simulation

Fig. 2 .

 2 Fig. 2. Simulated DG-MOSFET structure. EOT is 1.0 nm or 0.5 nm. The highly-doped S/D regions are 15 nm-long.

Fig. 3 .Fig. 4 .

 34 Fig. 3. a)Electron density n(x,z) at V GS =0.45V and V DS =0.7V b) Bottom of conduction band E C (x,z)

Figs. 5

 5 Figs. 5, 6 and 7 show the I-V characteristics obtained for EOT = 1 nm. They are compared with that obtained from semi-classical Boltzmann MC simulation including multisubband description (14) (dashed lines) and from ballistic NEGF simulation (17) (dotted lines).In subthreshold regime the Wigner results are very close to quantum ballistic ones but the current is significantly higher than that obtained using semi-classical simulation. This shows

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. Logarithmic and linear plots of the drain current I D versus V GS using Wigner, Boltzmann and ballistic NEGF simulations, at high V DS.

6 IFig. 8 .Fig. 9 .

 689 Fig. 8. I D (V GS ) for high V DS , with and without scattering in the channel. Inset: I D (V DS ) for high V GS , with and without scattering in the channel

Fig. 10 .Fig. 11 .

 1011 Fig. 11. C G V DD /I on (I off ) for various gate workfunctions (indicated through the difference ∆Φ with a midgap gate).

Fig. 12 .

 12 Fig. 12. Intrinsic gate delay C G V DD /I on as a function of I off for DG with EOT = 0.5 nm.

Acknowledgments

This work was partially supported by the European Community 6th FP, through NoE SINANO (IST 506844) and IP PULLNANO (IST 4-026828), and by the French ANR, through project MODERN.