Jochen Senger
email: jochen.senger@kit.edu

IZBS Daniel Weygand

Christian Motz

Peter Gumbsch

Oliver Kraft

Evolution of mechanical response and dislocation microstructures in small scale specimens under slightly different loading conditions

Keywords: dislocation dynamics, dislocation structures, simulation supplied): micro samples, boundary conditions dislocation structures, micro samples, simulation, boundary conditions, dislocation dynamics mc.manuscriptcentral.com/pm-pml

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Evolution of mechanical response and dislocation microstructures in small scale specimens under slightly different loading conditions

Introduction

In the last few years, compression [1][2][3][4][5][6][7] and tension experiments [8] on micron-and submicron-scale metallic samples have drawn quite some attention. It was found in these experiments that the flow stress of the tested metallic specimens is sizedependent. Flow stress increases with decreasing sample size. From a continuum mechanical point of view this is somewhat surprising since the loading condition is nominally uniaxial and free of imposed strain gradients. Therefore, even advanced continuum descriptions like strain gradient plasticity can not capture the phenomenon. Details of the dislocation behaviour must be considered to explain the phenomenon. Several qualitative explanations have been put forward, including dislocation starvation [5,7] or truncation of dislocation sources [9]. Furthermore, discrete dislocation dynamics (DDD) simulations schemes have been applied to study these mechanisms in more detail [10,11,12]. Parthasarathy et al. [10] confirmed that higher flow stresses can be attributed to smaller source lengths in smaller samples.

Similarly, Senger et al. [11] found that dislocation reactions take place in larger samples such that larger and therefore weaker dislocation sources are formed through dislocation reactions. This trend has been also observed by Motz et al. [12] for relaxed, initially pinning point free dislocation structures.

Apart from these mechanistic explanations, other aspects concerning the experimental set-up and the impact of focused ion beam (FIB) damage on the size effect are still under debate [13][14][15]. The central question in this debate is how the flow stress and the evolution of the dislocation microstructure depend on experimental aspects like misalignment leading to non-uniform loading or constraints imposed by the boundary conditions [16].

To address misalignment effects, DDD simulations for uniaxial loading of micrometre-sized samples with superimposed torsion were performed in this study.

Such torsion moments can occur in experiments due to alignment imperfections between sample and loading gadget [8] and is, of course, highly undesired but typically beyond control for experiments at the micrometer scale.

Simulation method

Simulations of deformation experiments on small-scale columns are performed with the 3-dimensional discrete dislocation dynamics simulation tool described in [17,18].

Apart from the variations in the loading conditions, the simulations reported here are similar to those in [11]. Briefly, the crystallographic orientation is taken from the systems such that an initial dislocation density of 2.1×10 13 m -2 is achieved for all column sizes. The columns have a square cross-section and given column sizes reflect the length of the edge of the square. All initial Frank-Read sources are placed within a cylinder around the tensile axis with a diameter equal to the square size. However, the presented dislocation densities are related to the complete volume. The investigated column sizes are between 0.5 and 2.0 µm with a fixed aspect ratio of 1:3. Eight simulations with random position of the initial Frank-Read sources are performed for specimens with a diameter of 0.5 µm and five simulations for 1 µm samples for each condition. Only two simulations are performed for the largest columns, due to the high computational costs. This can be justified since statistical variations are smaller in the larger sample as shown in [11] where for exactly this sample size the yield strength and the stresses in the plastic regime were within a few MPa for different initial configurations.

Columns with identical initial dislocation structures are loaded under three different boundary conditions. In all simulations, the displacement of the bottom of the column is fixed. Tensile loading along the long column axis is the main loading component, on which torsion as a secondary component is superimposed. Within the small strain assumption of our DDD model, tensile and compression simulations are statistically equivalent. For the first loading condition, pure tension (PT), the displacement at top surface of the column, is prescribed in the tensile direction only and the in-plane displacements are free allowing sideward motion. This corresponds to no friction between a pillar and the flat punch in a compression test or to an experimental setup, which is laterally very compliant both in tension or compression.

For the second boundary condition, constrained tension (CT), the additional constrain is that the in-plane displacements at the top surface are fixed during the entire test. This condition corresponds to tests with a laterally stiff experimental setup where both ends of the sample are clamped in tension or high friction in compression.

In the third boundary condition, the tension-torsion test (TT), a torsion moment is superimposed and the column top displacement is laterally fixed. The torsion is generated by a rotation of the top surface around the tensile axis. This loading condition is meant to mimic a non-ideal situation due to initial misalignment within the experimental setup. Of course, other superimposed constraints can be imagined, e.g. additional bending [19]. Table 1 summarizes the imposed constraints.

All simulations are performed with displacement control with a constant strain rate & ε of 5000 s -1 . Variation of the strain rate between 100 s -1 and 5000 s -1 in PT-tests shows no significant change in flow stress level and size effect. Therefore, we argue that strain rate effects do not play a significant role in our simulations. In the simulations, it is assumed that the linearly increasing torsion occurs only during the initial loading. This is meant to reflect the self-alignment of the column or tensile sample with respect to the loading device in a real experiment. For this, the torsional displacements components are superimposed and continuously increased until the total tensile strain reaches 0.1%. This tensile strain corresponds to a tensile stress of about 72 MPa which is just below the activation stress of the initial Frank-Read sources. Two torsion rates are chosen so that torque angles of φ = 0.5° and 2.0° are reached at 0.1% tensile strain. This corresponds to shear stresses caused by the torsion angle of 40 and 160 MPa, respectively, at the surface of a cylinder around the tensile axis with a diameter equal to the square size. These values are independent of the sample size because the shear stresses depend only on the shear angle according to Hooke's law. Then, for reaching tensile strains larger than 0.1%, the torsion angle reached at 0.1% tensile strain is kept constant and only the tensile strain is increased further.

In the subsequent part, yield strength R p0.01% is measured at 0.01% plastic tensile strain and the discussed flow stress Rp0.2% R p0.2% are taken at 0.2% plastic tensile strain.

Results

Samples with 0.5 µm diameter

The averaged stress-strain curves of the simulations with eight different initial Superimposed torsion generally leads to larger dislocation densities as in CT-tests. In the 2° twisted samples, dislocation density increases already during the deformation to 0.1% tensile strain thereafter the torsion rate is set equal to zero but dislocation density still increases significantly.

Fig. 2 shows superimposed snapshots of the simulated identical dislocation structure at various time intervals of the simulation under different boundary conditions. Fig. 2a shows that active slip occurs mostly on one plane (blue plane marked by an arrow). With the CT-boundary condition and the very same initial dislocation structure, the same glide plane is active but more dislocation activity on a parallel glide plane as well as on other glide systems is observed (Fig. 2b).

In the simulation with an additional twist to a torsion angle of 0.5° using boundary condition TT, it is clearly seen that beside the marked slip plane more slip systems were activated compared to the PT-and CT-conditions. In this particular simulation, the yield strength at 0.01% is not reduced compared to the tension tests.

However, the flow stress for this configuration is reduced compared to the PT case for small plastic strains. This can be attributed to the multiaxial stress state that increases the activation of more slip systems as confirmed by the observed dislocation structure shown in Fig. 2c.

The superimposed structure of the 2.0° twisted sample is split in the part where both, torsion angle and tensile displacement, are linearly increased (Fig. 2d) and in the part where the torsion angle is constant (Fig. 2e). Long dislocation segments (light blue) are formed in the beginning along the torsion axis which cannot easily leave the specimen due to the torsion component of the stress field (Fig. 2d). This is in contrast to dislocations activated on other slip systems where they can leave In the presented sample, the flow stress in the TT-test (2.0°) is slightly raised with work-hardening for tensile strains larger than 0.65% compared to the PT-test since dislocation on less preferred glide systems have to be activated. In other simulations, where flow stresses in PT-and TT-test are similar, the same source was found to be active under both conditions.

Samples with 1.0 µm diameter

For samples with side length of 1.0 µm, simulations with the same boundary conditions as above were conducted in order to assess whether the size effect on plasticity [11] is influenced by the boundary conditions. For both the PT-and CTboundary conditions (Fig. 3) a rather similar averaged overall behaviour without any appreciable work-hardening is observed. The yield drop can be explained with the generation of new dislocation sources which are longer than the initial Frank-Read sources as reported in [11]. Increasing sample diameter results in a reduced variation of flow stress for all boundary conditions confirming previous results [11]. In the strain range from 0.4-0.8%, the spread of flow stress is on a similar level within each boundary condition. The individual stress-strain curves of the samples with The dislocation density is increasing more strongly for the CT-tests compared to PT-tests. In both cases, several sources on parallel glide planes are active at the same time (see example of dislocation structure in Fig. 4), but for PT-tests, glide is concentrated on fewer planes. In the CT-condition, there are sources active on planes which cut the top or bottom surface of the specimen and dislocations pile-up occur on these planes. This causes a major contribution to the increase in dislocation density which is observed in these simulations. However, for the used aspect ratio of 1:3, dislocation sources away from the constrained ends of the sample continue to be active and, thus, hardening does not occur. In the TT-test with the angle of 0.5°, activity on planes with different glide plane orientations compared to the active planes in PT-and CT-tests occurs. However, activity on these secondary planes is low compared to the primary planes (Fig. 4c).

For twisting to an angle of 2°, a stronger activity on all four glide planes is observed. Even at small strain of 0.1%, the dislocation density has increased by about a factor of 2.5 from the initial value (superimposed structure from 0 to 0.1% total strain in Fig. 4d). Again, long dislocation segments (light blue) are formed by strain gradients in the initial plastic regime. With increasing tensile strains, sources on glide planes with a high Schmid factor are activated. In Fig. 4e parallel planes (blue) can

Samples with 2.0 µm diameter

In 2.0 µm samples, yield strength is further decreased in TT-tests (Fig. 3). As expected, averaged flow stresses are lower than in 1.0 µm. For this diameter, extreme softening behaviour in PT-and CT-tests is related to short initial sources compared to the sample diameter. In plastic regime, longer, weaker sources are generated [11].

Again, superimposed torsion results in a higher dislocation density.

Discussion

The aim of this study is to elucidate the influence of the boundary conditions on the dislocation behaviour in micrometre-sized samples during tensile loading. This is important because in experiments, pile-ups can occur, e.g. by hindered dislocation escape due to the indenter [3] or by low aspect ratios [20]. We have found that for tensile loading a constraint of the lateral deformation at the top and bottom of the specimen does not significantly affect the stress-strain behaviour for the deformation that can be reached in our simulations. If the displacement of the top surface perpendicular to the tensile axis is prohibited (like in CT-tests) in a 0.5 µm specimen, slip on several parallel planes of the primary slip system as well as on planes with lower Schmid factors, occurs. In 1.0 µm specimens more sources on parallel glide planes are activated in the CT-condition compared to PT. The concentration of slip on distinct glide planes has also been observed in experiments [8]. Some of these additional glide planes cut the top or bottom surface resulting in dislocation pile-ups.

Due to the aspect ratio of 1:3, however, glide planes in the middle of the sample remain active and, thus, the constraint effect is minimized in agreement to experimental observations [3]. Also, no work-hardening is observed in the presented PT-and CT-tests.

In TT-condition, the yield strength, determined from the tensile load, decreases for all sample sizes with increasing torsion angle. While in small samples, yield strength scatters and can also be equal to tension tests, the onset of plasticity in large samples is always reduced without a variation of yield strength. Furthermore, only in the thin specimens flow stress follows a near-elastic range when the first, temporary active source is shut-down in a zone of low stress. For tension and torsion, contrary to tensile loading, the resolved shear stress on a dislocation source depends on the exact source position and the angle to the torsion axis [START_REF] Hirth | Theory of Dislocations[END_REF]. The maximum shear stress depends only on the torsion angle, but due to the constant aspect ratio not on the specimen size. This leads to two zones on each glide system where the resolved shear stress caused by the torsion is equal to zero. In large samples with many sources, the probability is large for an initial Frank-Read source to be near the surface in a zone of high stress. Therefore, in contrast to the small ones, no scatter in yield strength is observed for these samples. The initial FR sources are distributed over the entire cross-section. For large samples there is a probability close to one to find a source in the highly stressed zone, leading to a small scatter in the yield strength which is not the case in small samples. Here yield stresses similar to the PT and CT loading conditions are also observed.

An important influence of the boundary conditions can be seen in the evolution of the dislocation density. In 0.5 µm specimens under tensile loading, dislocation density is at a constant level, apart from the temporary increases due to source activation, as dislocation can leave the sample without multiplication [5,7]. Due to strain gradients introduced by a superimposed small torsion angle, the dislocation density increases significantly (Fig. 1). The larger the torsion angle, the stronger the increase of the dislocation density. This is in agreement with the argument that geometrically necessary dislocations are formed as a function of strain gradients [START_REF] Fleck | [END_REF]. According to [START_REF] Fleck | [END_REF], the dislocation density is calculated which is necessary to compensate the plastic torsion angle reached at 0.1% tensile strain. The calculated values are found to be lower than the measured one. The difference increases with increasing torsion rate and sample size. However, due to the chosen crystallographic orientation, the formed long dislocations (light blue) have no pure screw character and reside on an inclined glide plane with respect to the torsion axis.

Furthermore, dislocation reactions in larger samples contribute to the higher density.

Due to the torsion, source activation may occur on other glide systems than without torsion since the resolved shear stresses become a strong function of position in the sample. This leads to multi-slip behaviour and the observed occurrence of long dislocation lines almost parallel to the sample axis. These long dislocation segments (light blue lines in Fig. 2d-ef and 4d-ef) occur due to source activation in the sector with a high resolved shear stress generated by the imposed torsion angle. When these dislocations reach a zone with a small shear stress near the centre or the sector with low resolved shear stresses, they stop and may act as obstacles for gliding dislocations on other slip systems. Further motion is only possible by the long range interaction with other dislocations. In the investigated strain range, however, the flow stress is not significantly affected by these forest dislocations as active sources are hardly pinned especially in the highly stress regions near to the surface. Hardening due to the increase in dislocation density in tension tests with superimposed torsion is less pronounced as in bending tests [19]. Fig. 5 summarizes the mean values of flow stress R p0.2% at 0.2% plastic strain for all simulations. The size effect that has been reported before [11] is not significantly influenced by the loading condition even if the dislocation microstructure and densities are quite different. This is a quite surprising observation since strain gradient effects as present in the twisted samples were expected to have a more pronounced effect. Compared to both tensile boundary conditions, however, the superimposed torsion angle of 2.0° leads only to an increase of the standard deviation.

to some extent. This may be attributed to the inhomogeneous stress state and source statistics in small samples.

Conclusion

In the presented simulations, it is found that the evolution of the same initial dislocation microstructure is strongly affected by the boundary conditions. Yield strength R p0.01% is reduced with superimposed torsion because of higher shear stresses at low tensile strains. A transition from single to multiple slip with the occurrence of forest dislocations in the twisted samples has been observed. However, the flow stress R p0.2% in the investigated strain regime and the influence of sample size are not significantly affected by the boundary constraints and the resulting forest dislocations.

F

= 0 u x = u y = u z = 0 y = h u x = unconstrained u y = h & ε t u z = unconstrained u x = 0 u y = h & ε t u z = 0 u x = x (cosϕ & t -1) -z sinϕ & t u y = h & ε t u z = x sinϕ & t + z (cosϕ & t -1)
Table 1: Summary of the displacements u at bottom and top surface for the three different boundary conditions. The nodal degree of freedoms, were no prescribed displacements are assumed, are set traction free. Figure 2: Superimposed dislocation structure in a 0.5 µm specimen: (a) pure tension, (b) constrained tension, (c) tension and torsion with a torsion angle of 0.5°. Sample in (d) shows the superimposed structure in the tensile strain range from 0-0.1% where torsion angle and tensile strain are linearly increased (TT: 2.0°). In (e), the superimposed structure from tensile strain 0.1% up to 0.8% is presented where only tensile strain is increased. The dislocation structure of the sample loaded with TT (2.0°) at 0.8% tensile strain is shown in (f). Active glide plane under tension and 0.5° torsion is marked with a black arrow.

  initial dislocation structure consists of Frank-Read sources of random orientation with a uniform length of 220 nm. These sources are equally distributed on the 12 glide

  dislocation configurations and the corresponding evolution of the dislocation density in 0.5 µm samples for different loading conditions are shown in Fig. 1. Eight identical initial structures have been used in each case to illustrate the effect of the different boundary conditions.In pure tension tests, the periodic activation of one single source is usually observed. This leads to plastic deformation without strain hardening. The flow stress is constant and no overall increase in dislocation density is observed.In contrast to pure tension, individual configurations show either a constant flow stress or hardening behaviour under CT-conditions. Dislocations pile up at the top and bottom surface and dislocation density increases continuously. With a superimposed torsion angle, only flow stress is similar to both tensile tests. In the TT-average yield strength is reduced with an increased torsion angle.

  sample through the free surfaces and, thus, not lead to an increase of the dislocation density. At higher tensile strains, activity starts on glide planes which are preferred under tension. The dislocations which are activated by the torsion are still in the sample and can act as obstacles. The dislocation microstructure at 0.8% tensile strain is presented in Fig.2f. With the twist angle of 2.0°, a reduced activity of the most active source in the other simulations (marked by the arrow in Fig.2a-c) is observed. Instead, several other sources on different glide planes are activated. They produce more cross-slip events. The cross-slipped dislocation often leaves such segments along the specimen axis as shown in Fig.2f.

  are found to be systematically below the PT-and CTcurve at low strains, reach or exceed the stresses of the PT-and CT-curves for larger strains. The average flow stress of the 2.0° twisted samples reaches the flow stress of both tension tests only for strains larger than 0.35%. Here, yield strength R p0.01% is systematically reduced in specimens with superimposed torsion. The scatter of the flow stress in 1.0 µm specimens is less pronounced than in the 0.5 µm samples for all tested boundary conditions.

  are mainly generated after 0.1% tensile strain. Fig 4f highlights the complexity of the final microstructure. In the samples presented in Fig.4, flow stress in the 2.0° twisted sample is lower compared to PT-tests for tensile strains smaller than 0.6%.

Figure 1 :

 1 Figure 1: Averaged stress-strain curves (thick lines) and evolution of dislocation density (thin lines) in 0.5 µm thin samples under different loading conditions. Error bars indicate the standard deviation of the flow stresses at 0.1 and 0.5% tensile strain.

Figure 3 :

 3 Figure 3: Averaged stress-strain curves (thick lines) and evolution of dislocation density (thin lines) in 1.0 µm thin specimens and 2.0 µm samples (dashed lines) under different loading conditions. Error bars indicate the standard deviation of the flow stresses in the 1.0 µm samples at 0.1 and 0.5% tensile strain.

Figure 4 :

 4 Figure4: Superimposed dislocation microstructure in a 1.0 µm sample: (a) pure tension, (b) constrained tension (CT), (c) tension and torsion (TT) with a torsion angle of 0.5°. The microstructure of the 2.0° twisted sample is decomposed in the tensile strain (d) from 0-0.1% and (e) for strains from 0.1-0.8%. Current dislocation structure of the TT (2.0°) sample at 0.8% tensile strain is illustrated in (f).

Figure 5 :

 5 Figure 5: Size effect in micron samples. Mean values of flow stresses at 0.2% plastic strain are given by the symbols. Error bars indicate the standard deviation. Two straight lines described by a power-law are given. For clarity, stress values are presented abreast.

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgement

The financial support of the European Commission NANOMESO Project under contract number NMP3-CT-2006-016710 and Deutsche Forschungsgemeinschaft (DFG) under project Gu367/18-2 is gratefully acknowledged.


M.D Nima Etminan
email: etminan@uni-duesseldorf.de

M.D Kerim Beseoglu

M.D Hans-Jakob Steiger

M.D. Daniel Hänggi

The impact of hypertension and nicotine on the size of ruptured intracranial aneurysms

Keywords: subarachnoid hemorrhage, cerebral aneurysm, risk factors, size

Background and Purpose

The goal of the present study was to analyze retrospectively the impact of risk factors for subarachnoid hemorrhage (SAH) on the size of ruptured intracranial aneurysms in order to identify variables which might influence the discrepancy between average sizes of ruptured and unruptured aneurysms.

Methods

The records and angiographies of 373 patients treated at our center due to aneurysmal SAH between 2004 and December 2008 were retrospectively analyzed. Modifiable and non-modifiable risk factors for SAH were correlated with exact measurement of aneurysm size using 3-dimensional rotational digital subtraction angiographies (3D-DSA).

Results

Average maximum aneurysm diameter in patients with combined history of hypertension and cigarette smoking was 5.47±3.22 mm [95% CI 4.71-6.24], thus significantly smaller (p<0.001) than in patients with hypertension only (6.27±3.28 mm, 95% CI 5.75-6.78), with cigarette smoking only (7.61± 4.29 mm, 95% CI 6.43-8.79) and patients with no history for risk factors (8.08±4.73mm, 95% CI 6.96-9.21). Odds ratio for aneurysm size less than 7mm in patients with combined hypertension and cigarette smoking was 3.63 [95% CI 1.78-7.42], 3.09 [95% CI 1.95-4.92] in patients with hypertension only and 1.02 [95% CI 0.64-1.62] in patients with cigarette smoking only.

Conclusions

The present analysis demonstrates a distinct correlation between hypertension, smoking and the size of ruptured aneurysms in SAH patients. Arterial hypertension and cigarette smoking appear to destabilize cerebral aneurysms growth. Our data strongly suggests that these factors should also be considered, when treatment indications for small unruptured aneurysms are discussed.

INTRODUCTION

Despite extensive, multimodal development of treatment strategies, aneurysmal subarachnoid hemorrhage (SAH) remains a disease with high morbidity and mortality, causing case fatality rates between 25 and 50%. [START_REF] Hop | Case-fatality rates and functional outcome after subarachnoid hemorrhage: a systematic review[END_REF] Due to the poor prognosis of aneurysmal SAH, the natural history of unruptured intracranial aneurysms (UIAs) and their treatment indications remain important questions.

Prospective studies on UIAs have identified aneurysm size and location as key risk factors among others such as Japanese/Finnish descent, age and female gender. [START_REF] Juvela | Natural history of unruptured intracranial aneurysms: probability and risk factors for aneurysm rupture[END_REF][START_REF] Rinkel | Natural history, epidemiology and screening of unruptured intracranial aneurysms[END_REF][START_REF] Wiebers | Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgical and endovascular treatment[END_REF] Furthermore, retrospective studies have previously analyzed the impact of risk factors on the severity of SAH and it was found that hypertension and cigarette smoking (i.e. nicotine use) as well as age and gender are significant risk factors for SAH and correlate with unfavorable outcome after SAH. [START_REF] Eden | Gender and ethnic differences in subarachnoid hemorrhage[END_REF][START_REF] Feigin | Risk factors for subarachnoid hemorrhage: an updated systematic review of epidemiological studies[END_REF][START_REF] Juvela | Natural history of unruptured intracranial aneurysms: risks for aneurysm formation, growth, and rupture[END_REF][START_REF] Juvela | Prehemorrhage risk factors for fatal intracranial aneurysm rupture[END_REF][START_REF] Nahed | Hypertension, age, and location predict rupture of small intracranial aneurysms[END_REF][START_REF] Rosengart | Prognostic factors for outcome in patients with aneurysmal subarachnoid hemorrhage[END_REF] The goal of the present study was to retrospectively analyze the correlation of age, gender, location of the aneurysm, history of hypertension and/or cigarette smoking with the size of ruptured aneurysms in a large patient cohort.

ABBREVIATIONS:

ACA=

MATERIAL AND METHODS

We collected retrospectively data from patients suffering from SAH who were admitted to the Department of Neurosurgery at Düsseldorf University Hospital (Germany) between January 2004 and December 2008. The inclusion criteria comprised: 1) aneurysmal SAH and 2) digital subtraction angiography (DSA) to allow exact measurements. The exclusion criteria were defined as; 1) nonaneurysmal SAH, 2) inadequate diagnostic quality, 3) fusiform, traumatic or mycotic aneurysms, and 4) arteriovenous malformation associated aneurysms. SAH was verified by computerized tomography (CT), magnetic resonance imaging (MRI) or spinal tap.

Aneurysm size, configuration and location were assessed by reviewing the initial 3-dimensional rotational digital subtraction angiographies (3D-DSA). Exact aneurysm sizes were measured by an independent investigator and defined as the largest diameter, whereas aneurysm configuration was defined as even or lobulated. Aneurysm size was categorized in accordance to the classification of the International Study on Unruptured Intracranial Aneurysms (ISUIA) into A) less than 3 mm B) 3-7 mm C) 8-12 mm D) 13-25 mm and E) greater than 25 mm. [START_REF] Wiebers | Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgical and endovascular treatment[END_REF] If patients harbored multiple intracranial aneurysms, the most probable ruptured aneurysm with regards to SAH distribution, aneurysm size and configuration was included for further analysis. [START_REF] Nehls | Multiple intracranial aneurysms: determining the site of rupture[END_REF][START_REF] Suzuki | Aneurysms arising from the proximal (A1) segment of the anterior cerebral artery. A study of 38 cases[END_REF] Furthermore, the major known risk factors for SAH, such as the history for hypertension and nicotine use, as well as the grading according to the Fisher and World Federation of Neurological Surgeons (WFNS) classification, were retrieved from the records. [START_REF]Report of World Federation of Neurological Surgeons Committee on a Universal Subarachnoid Hemorrhage Grading Scale[END_REF][START_REF] Fisher | Relation of cerebral vasospasm to subarachnoid hemorrhage visualized by computerized tomographic scanning[END_REF] For statistical analysis patients were categorized into 4 different age groups: A) less than 30 years B) 31-50 years C) 51-70 years D) greater than 70 years.

Information regarding the long-term history for hypertension and cigarette smoking was obtained from the patient or a surrogate.

In accordance to previous studies, hypertension was defined positive in patients with a definite longterm history of hypertension with systolic blood pressures (BP) greater than 160 mmHg and/or diastolic blood pressure greater than 95 mmHg. There was no differentiation between patients with medically treated primary or secondary hypertension or those with untreated hypertension. Patients with a history of borderline hypertension (160/95mmHg>BP>140/95mmHg) were not considered as hypertensive. [START_REF] Juvela | Prehemorrhage risk factors for fatal intracranial aneurysm rupture[END_REF][START_REF] Chobanian | The Seventh Report of the Joint National Committee on Prevention, Detection, Evaluation, and Treatment of High Blood Pressure: the JNC 7 report[END_REF] Cigarette smoking was included for further risk analysis only for those patients who were frequent and current smokers at the time of admission to our hospital. The individual amount as well as the duration of cigarette smoking were not differentiated due to the difficulty to obtain the history within this patient population and therefore patients were classified as ever or never smokers.

Patients were categorized according to their individual risk profile in one of the following risk groups: 1) no risk factor 2) nicotine alone 3) hypertension alone and 4) nicotine and hypertension.

The statistical analysis was performed using SPSS software v. 15.0.1 (© SPSS Inc. Chicago/Illinois, USA). Differences between groups were analyzed using Student's t-test for metric variables and nonparametric tests for ordinal or nominal scaled variables. Analysis of variance (univariate ANOVA) was followed by Bonferroni post-hoc test to further determine individual differences between analyzed groups. Statistical significance was defined as p<0.05 (CI 0.95).

RESULTS

Patient and vascular characteristics

Between January 2004 and December 2008 a total of 456 patients with aneurysmal subarachnoid hemorrhage were admitted to our department. Eighty-three (83) patients of this patient population were excluded retrospectively from further analysis due to the aforementioned exclusion criteria. Gender-specific risk factor distribution did not differ significantly among males and females (p=0.355). There was no significant difference regarding the distribution of initial WFNS (p=0.078)

and Fisher (p=0.338) grading between the risk groups.

The correlation of aneurysm size, location and shape, with systemic risk factors revealed the aneurysm size being the most dependent parameter. Mean aneurysm size in patients with a combined history of hypertension and cigarette smoking (i.e. nicotine) was 5.47±3.22 mm, [95% CI 4.71-6.24], thus significantly smaller (p<0.001) than in patients with hypertension only (6.27±3.28 mm, 95% CI 5.75-6.78). Patients with a history for cigarette smoking had only slightly smaller ruptured aneurysms (7.61± 4.29 mm, 95% CI 6.43-8.79), compared to patients with no risk factors (8.08±4.73mm, 95% CI 6.96-9.21). The difference between average aneurysm sizes in patients with no risk factors and smoking history was not significant (p=0.56) (Figure 1). The correlation analysis of aneurysm location and size demonstrated a significant difference in mean aneurysm size only for the AComA as compared to BA, ICA and MCA (Figure 2).

Figure 2

Stratification of patients in age groups and correlation with risk factors demonstrated a possible trend:

Within the age groups, patients without risk factors had larger aneurysms than patients with hypertension. This difference was even more pronounced in patients with additional smoking history.

Furthermore, within each risk group, older patients had larger ruptured aneurysms than their younger counterparts. Due to small group sizes and therefore large variance within each risk/age group, statistical significance with regard to the subgroups could not be obtained (data not shown).

DISCUSSION

The recommendation for treatment of UIAs currently relies to a great extent on the results of the ISUIA trial, which demonstrated that aneurysm size and location are the major predictive factors of rupture risk. [START_REF] Wiebers | Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgical and endovascular treatment[END_REF] For example, the rupture risk of UIAs in the anterior circulation smaller than 7mm was estimated as 0.2 percent per year. The risk of rupture and average sizes of UIAs are often challenged by retrospective studies on ruptured intracranial aneurysms, which illustrate that the majority of ruptured intracranial aneurysms in large SAH cohorts are significantly smaller than 7mm. [START_REF] Joo | What Is the Significance of a Large Number of Ruptured Aneurysms Smaller than 7 mm in Diameter?[END_REF] Although prospective and retrospective studies must be compared carefully, the discrepancy between average sizes of unruptured and ruptured aneurysms remains striking and unexplained. Despite the currently accepted treatment strategies following the ISUIA trial controversies about the influence of systemic risk factors for rupture of UIAs such as age, gender, hypertension and cigarette smoking still exist. [2,6,10,14,15] The goal of the present analysis was to provide data on the influence and importance of systemic risk factors for aneurysm growth and rupture.

On average, the ruptured aneurysms of the actual sample measured 6.86±4.26 mm (range 0.7-40mm).

73% of all ruptured aneurysms measured less than 7mm in diameter, and 88% of the anterior circulation aneurysms measured less than 7mm. Similar results have been described before. [START_REF] Forget | A review of size and location of ruptured intracranial aneurysms[END_REF] With regard to risk factors for aneurysm rupture, mainly patient age and female gender have been reported in the literature as non-modifiable risk and smoking status and hypertension as additional modifiable risk factors. [START_REF] Rinkel | Natural history, epidemiology and screening of unruptured intracranial aneurysms[END_REF] Our analysis investigated possible correlations of these factors with the size of ruptured aneurysms. Multivariate correlation analysis did not show such a correlation for age or gender, but a significant correlation for cigarette smoking and hypertension. Hypertensive patients had significantly smaller ruptured intracranial aneurysms (6.27±3.28 mm) than normotensive, nonsmoking patients (7.61±4.29 mm). Furthermore cigarette smoking in combination with hypertension significantly amplified the effect of hypertension on aneurysm size (5.47±3.22 mm).

Consequently, the probability of harboring an aneurysm less than 7mm was significantly higher in smoking patients suffering from hypertension (OR 3.63, 95% CI 1.78-7.42) and also non-smoking hypertensive patients (OR 3.09, 95% CI 1.95-4.92) in comparison with normotensive smoking patients (OR 1.02, 95% CI 0.64-1.62) or normotensive non-smokers (OR 1). The amplifying tendency of concomitant hypertension and nicotine consumption on the reduction of aneurysm size was observed also within the age groups. However, due to the small subgroups and large variation statistical analysis did not reach significance level.

This finding is in contrast with a previous report demonstrating that that smoking is associated with the development of large middle cerebral artery aneurysms. [START_REF] Qureshi | Factors associated with aneurysm size in patients with subarachnoid hemorrhage: effect of smoking and aneurysm location[END_REF] On the basis of the current available data, it cannot be excluded that middle cerebral aneurysms have different dynamics than aneurysms of other locations, as a corresponding subgroup analysis is impossible because of insufficient numbers of data.

The prevalence of hypertension of 45.2 % in our patient population is very similar to the figures of comparable studies. [START_REF] Kwak | The correlation between hypertension in past history and the incidence of cerebral aneurysms[END_REF][START_REF] Lanzino | Age and outcome after aneurysmal subarachnoid hemorrhage: why do older patients fare worse?[END_REF] However, the impact of hypertension on aneurysm size has not been further characterized in these previous analyses.

Although deduction from our results to UIAs must be made cautiously, one could conclude that UIAs in hypertensive patients and smokers might have an increased risk of rupture below the critical size as defined by ISUIA, which could suggest that treatment of UIAs smaller than 7mm in hypertensive patients and smokers may be beneficial. Whether medical treatment of hypertension and/or cessation of smoking could also lead to restabilization of UIAs is currently a matter of speculation. The pathophysiology of an increased rupture risk of cerebral aneurysms due to risk factors such as hypertension and/or nicotine remains incompletely understood at present. A hypertension-induced damage of the aneurysmal wall has been described due to occlusion of the vasa vasorum and a disturbance in the synthesis of elastin and collagen. [START_REF] Inci | Intracranial aneurysms and arterial hypertension: a review and hypothesis[END_REF] Furthermore, hypertension was described to cause a misbalance of the arterial vessel wall homeostasis due to impaired nitric oxide and prostacyclin production resulting in a loss of elasticity and increased exposition of the aneurysm to wall shear stress. [START_REF] Kassam | Altered arterial homeostasis and cerebral aneurysms: a molecular epidemiology study[END_REF] Smoking, on the other hand, affects vascular homeostasis and structural integrity.

Inactivation of α-1-antitrypsin and subsequent impact on aneurysm formation may be a possible mechanism. [START_REF] Peters | Functional polymorphism in the matrix metalloproteinase-9 promoter as a potential risk factor for intracranial aneurysm[END_REF][START_REF] Schievink | Alpha 1-antitrypsin deficiency in intracranial aneurysms and cervical artery dissection[END_REF][START_REF] St Jean | Alpha-1-antitrypsin deficiency in aneurysmal disease[END_REF] We observed that older patients had larger aneurysms, which could be explained through a selection effect: On average, aneurysms appear to grow during lifetime. [START_REF] Wermer | Follow-up screening after subarachnoid haemorrhage: frequency and determinants of new aneurysms and enlargement of existing aneurysms[END_REF][START_REF] Wiebers | The significance of unruptured intracranial saccular aneurysms[END_REF] It cannot be excluded, however, that aneurysms in older patients enlarge faster. [START_REF] Wiebers | The significance of unruptured intracranial saccular aneurysms[END_REF] Lastly, our data demonstrated that AComA aneurysms were significantly smaller than aneurysms of other parent arteries. This phenomenon has been previously described in other studies. [START_REF] Kassell | Size of intracranial aneurysms[END_REF][START_REF] Weir | Unruptured intracranial aneurysms: a review[END_REF] There is no generally accepted explanation. One could only speculate that the average aneurysm diameter at the time of rupture is related to the diameter of the parent artery.

The major limitation of the current study to be mentioned is its retrospective nature. In addition, we did not differentiate between medically treated and untreated hypertension. [START_REF] Juvela | Prehemorrhage risk factors for fatal intracranial aneurysm rupture[END_REF][START_REF] Stratos | Ascending aorta distensibility abnormalities in hypertensive patients and response to nifedipine administration[END_REF] Despite these limitations the present study has, to the best of our knowledge, demonstrated a correlation of the size of ruptured aneurysms and systemic risk factors for the first time.

SUMMARY

The present retrospective analysis demonstrates that hypertension and smoking have a significant influence on the size of ruptured aneurysms in patients suffering from SAH. For patients presenting with a ruptured aneurysm, the size of that ruptured aneurysm is almost 4 times more likely to be less than 7 mm, in those patients that have a history of hypertension and for smoking. As a consequence our data strongly suggests that patients with these risk factors have a lower threshold of aneurysm rupture than patients without risk factors. Therefore the history of hypertension and cigarette smoking should be considered in the assessment of treatment of small unruptured intracranial aneurysms.

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  Anterior cerebral artery; AComA= Anterior communicating artery; BA= Basilar artery; ICA= Internal carotid artery; MCA= Middle cerebral artery; PericA= Pericallosal artery; PComA= Posterior communicating artery; PICA= Posterior inferior cerebellar artery; VA= Vertebral artery

Figure 1

 1 Figure 1

Figure 1 Figure 2

 12 Figure 1 Impact of risk factors on ruptured aneurysm size. (** p<0.001) Figure 2 Ruptured aneurysm size variation

Table 1

 1 Patients characteristics 8% e) larger than 25mm: 0.5%. Eighty-eight percent (87.7%) of the aneurysms smaller than 7 mm were localized in the anterior and 12.3% in the posterior circulation. Of the aneurysms larger than 7mm, 79.1% were localized in the anterior circulation versus 20.9% in the posterior circulation. Detailed distribution of aneurysm with regard to parent vessel and size is shown in

	Patients characteristics for n=373 patients
	Age mean	53.55 ± 12.7 years, range 16 -89 yrs
	Age groups (years)
	<30	10 (2.7%)
	31 -50	150 (40.2%)
	51 -70	169 (45.3%)
	>70	44 (11.8%)
	Gender
	M : F 143 (38.3%) : 230 (61.7%)
	Grading upon admission for n=373 patients
	WFNS
	I	130	34.9%
	II	82	22.0%
	III	25	6.6%
	IV	47	12.6%
	V	89	23.9%
	Fisher
	0	12	3.2%
	1	28	7.5%
	2	33	8.9%
	3	127	34.1%
	4	173	46.4%
	Risk factor incidence for n=352 †
	No risk factors	70	19.9%
	Nicotine only	53	15.0%
	Hypertension only	159	45.2%
	Nicotine + Hypertension	70	19.9%
	†= 21 patients excluded due to incomplete risk factor profile

Table 2 .

 2 Aneurysm shape was lobulated in 59.1 %, whereas 40.9 % of the aneurysms were even. Fifty-six percent (55.6 %) of the patients were treated surgically and 44.4 % by-endovascular technique.

Table 2

 2 Aneurysm size distribution

	Aneurysm size	<7mm	>7mm	N=
	Localisation
	AComA	107 (77.0%)	32 (23.0%)	139
	BA	7 (25.9%)	20 (74.1%)	27
	ACA	5 (100%)	0	5
	PComA	29 (69.0%)	13 (31.0%)	42
	PERICA	8 (57.1%)	6 (42.9%)	14
	VA	6 (75%)	2 (25%)	8
	ICA	15 (53.6%)	13 (46.4%)	28
	MCA	51 (58%)	37 (42%)	88
	PICA	14 (73.7%)	5 (26.3)	19
	Else	2	1	3
	Anterior circulation	214 (87.7%) 102 (79.1%) 316 (84.7%)
	Posterior circulation	30 (12.3%)	27 (20.9%)	57 (15.3%)
	N=	244	129	373

Table 3

 3 Odds ratio for ruptured aneurysm size lesser than 7 mm

	Odds ratio for ruptured aneurysm size < 7mm	OR	95% CI
	Nicotine + hypertension	3.63	1.78 -7.42
	Hypertension alone	3.09	1.95 -4.92
	AComA	2.37	1.48 -3.79
	Male	1.08	0.69 -1.67
	Nicotine alone	1.02	0.64 -1.62
	OR=Odds ratio; CI= Confidence interval
	Aneurysm configuration (p=0.304) and localization (p=0.536) was not significantly correlated to the
	presence of risk factors.

COMPETING INTERESTS:

None declared


S Aubouet

L Dugard
email: luc.dugard@inpg.fr

O Sename

S A S Soben

Experimental results of an H ∞ -observer for an industrial semi-active suspension

Keywords: H ∞ -observer, automotive, suspension

In this paper, an H ∞ -observer to be used in suspension control application is addressed to estimate a quarter car model equipped with an industrial SOBEN damper, using a reduced number of sensors. Vehicle estimation is one the main challenges in suspension control since many control strategies developped in past studies use variables that are nonmeasurable in practice. This observer is designed in the H ∞ framework in order to minimize the effect of the unknown road disturbance on the estimated states and uses a single, cheap and reliable acceleration sensor mounted on the unsprung mass. The estimated variables are the vertical velocities and positions of the sprung and unsprung masses of the quarter car. Some experiments done on a testing car highlight the performances of the observer. This observer could be used in a suspension control application in order to reduce the number of sensors and the cost of the control system.

INTRODUCTION

This paper is concerned with the implementation of an embedded state observer for a quarter car equipped with a semi active suspension. Experimental results obtained by SOBEN on a testing car are presented here to highlight the performance of the synthesis method detailed in a previous paper [START_REF] Aubouet | H ∞ /lpv observer for a semi active suspension[END_REF].

Suspension control based on quarter vehicles has been widely explored in the past few years to improve vertical movements. Active control laws have been developed [START_REF] Choia | Field test results of a semi-active suspension system associated with skyhook controller[END_REF]; [START_REF] Gaspar | Active suspension design using LPV control[END_REF]; [START_REF] Fialho | Road adaptive active suspension design using linear parameter varying gain scheduling[END_REF], and semi-active control laws [START_REF] Spelta | Design and applications of semi-active suspension control systems[END_REF]; [START_REF] Canale | Semiactive suspension control using "fast" model-predictive techniques[END_REF]; [START_REF] Giorgetti | Hybrid model predictive control application toward optimal semi-active suspension[END_REF]; [START_REF] Sammier | Skyhook and H ∞ control of active vehicle suspensions: some practical aspects[END_REF]. Active suspensions provide excellent performances but are not realistic in an industrial context because of the excessive cost of the actuators and their huge energy consumption. Semi-active suspensions provide satisfying performances and can be adopted in mass-produced vehicles if the number and the cost of the sensors required by the control strategy is low, which has not always been the case in the past studies. Furthermore, many control strategies assume a full-state measurement [START_REF] Tseng | A comparison of alternative semi-active control laws[END_REF]; [START_REF] Yi | A robust semiactive suspension control law to improve ride quality[END_REF], or require at least two sensors as in the well-known Skyhook control strategy [START_REF] Spelta | Design and applications of semi-active suspension control systems[END_REF]; [START_REF] Sammier | Skyhook and H ∞ control of active vehicle suspensions: some practical aspects[END_REF]. Therefore the state estimation problem is very important if we wish to reduce the number of sensors, i.e. reduce the cost and improve the reliability of the system. Unknown input observers have been studied by many authors [START_REF] Koenig | Observer design for unknown input nonlinear descriptor systems via convex optimization[END_REF][START_REF] Koenig | Unknown input proportional multipleintegral observer design for linear descriptor systems: Application to state and fault estimation[END_REF]; M. Darouach (2000Darouach ( , 1994)); [START_REF] Yi | Design of disturbance decoupled bilinear observers[END_REF]; [START_REF] Tsui | A new design approach to unknown input observers[END_REF], and also applied to automotive systems affected by road disturbances J.K. [START_REF] Hedrick | Observer design for electronic suspension applications[END_REF]; [START_REF] Yi | Observer design for semi-active suspension control[END_REF]. In [START_REF] Yi | Observer design for semi-active suspension control[END_REF], a distur-bance decoupled quarter car observer is designed using the vertical accelerations of the sprung and unsprung masses, but two expensive sensors are required, and this observer is too sensible to measurement noise to be implemented in a real car, where the noise is very important due to the vibrations of the engine. This paper deals with an observer that estimates the state of the vertical quarter car model using a single reliable accelerometer. The observer is designed in the H ∞ framework in order to minimize the effect of the unknown road disturbance on the estimated states. This work completes some previous results on the same project bringing experimental results obtained on a real testing car. Indeed in [START_REF] Aubouet | H ∞ /lpv observer for a semi active suspension[END_REF], the synthesis equations of the observer were detailed and the observer performance was emphasized in simulation only. Here some experiments show the performances of the observer in the embedded application. The industrial SOBEN damper considered in the application has been described in detail in [START_REF] Aubouet | Performance analysis and simulation of a new industrial semi-active damper[END_REF]. This paper is organised as follows: Section 2 presents the system to be observed, Section 3 recalls the estimation problem considered in this paper, Section 4 deals with the synthesis of the H ∞ /LP V observer and Section 5 gives some experimental results. This paper is finally concluded in Section 6 and some future developments are proposed.

VEHICLE MODEL

In this section, the system to be observed is presented. This vertical linear quarter car model represented on Figure 2 is classical and suits for vehicle estimation since its structure is simple and represents the interesting degrees of freedom of a real quarter car. Figure 2 also presents a picture of the new SOBEN damper installed on the testing car used for the experiments. If the linear model of the damper is considered, the equations of this model are given by (1).

{ m s zs = k(z us -z s ) + c • ( żus -żs ) m us zus = k(z s -z us ) + c • ( żs -żus ) + k t (z r -z us )
(1) where c represents the linear damping rate of the suspension. This quarter car model will be used in the synthesis of the observer and can be formulated as a linear time invariant system given by (2).

{ ẋ = A • x + D • v y = C • x (2)
where v = żr ∈ R d is the unknown road disturbance, x = (z def , żs , z us -z r , żus ) T ∈ R n are the state variables of the quarter car model, y ∈ R m is the vertical acceleration of the unsprung mass measured by an accelerometer and

A ∈ R n,n , D ∈ R n,d and C ∈ R m,n are given by A =        0 1 0 -1 - k m s - c m s 0 c m s 0 0 0 1 k m us c m us - kt m us - c m us        , C = 1 mus •    k c -k t -c    T , D =    0 0 -1 0   
This model will be used in the observer synthesis in Section 4.

Remark:

In (2), no control signal u is considered. This choice will be detailed in section 5.

PROBLEM STATEMENT

The system to be observed is the quarter car model presented in Section 2 and given in (2). The full-order observer synthesized in this paper has the general structure given by (3).

{ ż = N • z + L • y x = z -E • y (3)
Where z ∈ R n is the state variable of the observer and x ∈ R n the estimated state variables. n,m are matrices to be designed. Then considering ( 2) and ( 3), the estimation error can be expressed as e = x -x = (I n + EC) • x -z (4) and then the dynamics of the estimation error is:

N ∈ R n,n , L ∈ R n,m , E ∈ R
ė = ẋ - ẋ ė = Ax + Dv -N z -Ly + EC(Ax + Dv) (5)
By using ( 4), ( 5) leads to ė = N e + (A -

N (I n + EC) -LC + ECA)x + (D + ECD)v (6) Let us define K = N E + L and P = I n + EC, then (6) turns into ė = N e + (P A -(N + KC))x + P Dv (7)
The state x is an asymptotic estimate of x for any x(0) and x(0) if and only if N is Hurwitz and

{ N = P A -KC P D = 0 (8)
The design of the observer involves the calculation of 8). A method to solve this problem is proposed in Section 4.

N ∈ R n,n , L ∈ R n,m , E ∈ R n,m satisfying (

OBSERVER DESIGN

In this section, a method is proposed to synthesize a road disturbance decoupled H ∞ full-order observer based on the unsprung mass vertical acceleration measurement. The problem formulated in Section 3 is solved. Some previous works on this topic have been used and completed, like M. Darouach (1994Darouach ( , 2000)).

Road disturbance decoupling

The first condition of ( 8) is equivalent to

z • ψ = A (9)
where z ∈ R n,(n+2m) and ψ ∈ R (n+2m),n are defined by

     z = ( N K E ) ψ = ( I n C -CA ) (10)
There exist a solution z of (9) if

rank(ψ) = rank ( ψ A ) (11)
Since condition ( 11) is satisfied, the solution exists and is of the form z = α + Yβ where

{ α = A • ψ + β = I n+2m -ψ • ψ + (12)
Y is any matrix with appropriate dimensions and ψ + is any generalized inverse matrix of ψ. The matrix Y will be determined later. From z = α + Yβ, (4) turns into ė

= N • e + (I n + EC)D • v (13) Let us define Ñ ∈ R (n+2m),n and Ẽ ∈ R (n+2m),m such that Ñ = ( I n 0 m,n 0 m,n ) Ẽ = ( 0 n,m 0 m,m I m,m ) Therefore we have N = z • Ñ E = z • Ẽ Finally, (13) can be expressed as ė = A 0 • e + B 0 • v (14) with A 0 = (α + Yβ) Ñ and B 0 = (I n + (α + Yβ) ẼC)D.
The equation ( 14) that rules the estimation error is affected by the unknown road disturbance v. If P D = O, the effect of the road disturbance on the estimated states is cancelled. This condition is equivalent to (I n +EC)D = O, where E has to be determined. This equation is solvable if and only if rank(CD) = rank(D). Otherwise the disturbance effect has to be minimized and the problem is to find Y such that A 0 is stable and the effect of v on e is minimized.

Proposition 1. There exists a full-order observer ensuring

|| e/v || ∞ < γ ∞ if there exists X = X T ≻ 0, Ỹ and a scalar γ ∞ that solve the LM I (15).   Q 1 + Q T 1 Q 2 I n * -γ ∞ I d O d,n * * -γ ∞ I n   ≺ 0 (15)
Where X = X T ≻ 0, Ỹ = XY are the decision variables and

{ Q 1 = (Xα + Ỹβ) Ñ Q 2 = (X + (Xα + Ỹβ) ẼC)D (16)
Remark: γ ∞ has to be minimized in order to minimize the road disturbance effect on the estimated variables.

Proof 1. This proof has already been detailed in [START_REF] Aubouet | H ∞ /lpv observer for a semi active suspension[END_REF].

Filtering

In this paragraph, a weighting filter has been added to the system to focus the interesting frequency range where the disturbance effect minimization has to be done. Indeed, it is important that the bandwidth of the observer suits the bandwidth of the system to be observed. This filter allows the designer to choose the frequency range where the observer has to provide the best results. For suspension control applications, the interesting frequency range where the damper is controlled is [0 -20Hz], therefore the bandwith of the weighting filter will be chosen equal to 20Hz. The new estimation variable to be considered in this section is the filtered estimation variable e f . Therefore the problem is now to minimize γ ∞ such that

|| e f /v || ∞ < γ ∞ ( 17
)
Proposition 2. There exist a full-order observer ensuring (17) if there exist

X 1 = X T 1 ≻ 0, X 2 = X T 2 ≻ 0, Ỹ and a scalar γ ∞ that solve (18).    A T 0 X 1 + X 1 A 0 O n X 1 B 0 O n X 2 B f X 2 A f O n,d I n * * -γ ∞ I d O n,d * * * -γ ∞ I n    ≺ 0 (18)
Where X = X T ≻ 0 is defined such that

X = ( X 1 O n O n X 2 ) ( 19
)
A f ∈ R n,n and B f ∈ R n,n determine a given weighting filter.

Proof 2. This proof has already been detailed in [START_REF] Aubouet | H ∞ /lpv observer for a semi active suspension[END_REF].

Pole placement

This method ensures the stability of the observer and the minimization of the disturbance effect, but the poles of the observer may be very high or comprise high imaginary parts. Such poles may render the observer oscillating and sensible to measurement noises. In order to avoid such a behavior that may lead to implementation problems and bad estimation performances, a pole placement method [START_REF] Chilali | Robust pole placement in lmi regions[END_REF] using LM I regions has been introduced. Therefore the poles of the observer will be chosen automatically to minimize the disturbance effect, but they will be chosen in an appropriate region chosen by the designer. The poles of the observer can be placed in the intersection of a cone D 1 , given by the LM I region (20) and a half plane D 2 , given by ( 21). The cone is defined with apex at the origin and inner angle 2θ to ensure that the observer is stable and has poles with moderate imaginary parts compared to real parts. The half plane is delimited by a vertical straight line to ensure that the poles have real parts higher than -p m .

D 1 = { z ∈ C : ( sin θ(z + z) cos θ(z -z) cos θ(z -z) sin θ(z + z) ) ≺ 0 } (20) D 2 = {z ∈ C : -z -z -2p m ≺ 0} (21)
Then some LM I constraints can be added to the synthesis equations to ensure that the poles of the observer are in these regions. These LM I are given in [START_REF] Aubouet | H ∞ /lpv observer for a semi active suspension[END_REF], and the pole placement method in LM I regions is detailed in [START_REF] Chilali | Robust pole placement in lmi regions[END_REF].

Therefore to summarize 4.1, 4.2 and 4.3, the method to design the proposed LT I observer can be formulated as follows:

• Choose the weighting filter A f and B f appropriate to the system • Choose D 1 and D 2 according to the desired poles real and imaginary parts bound • Solve LM I (18) to find X 1 and Ỹ

• Calculate Y = X -1 1 Ỹ, z = α + Yβ using (12) • Deduce N , K, E, L = K -N E

Remark

The upper bound p max has also to be chosen by the designer. It determines the bandwith of the observer. The observer must be faster than the system to be observed to get accurate results. However for noise filtering, this bound has to be less than the frequencies of the noise. For the application considered in this paper, the noise is located in the frequency range [400 -1000Hz]. Therefore p max = 100 allows the observer to filter the noise.

EXPERIMENTAL RESULTS

In this section, numerical results are given and different experimental results are presented to evaluate the observer performances.

Synthesis results

In this paragraph, the numerical values of the calculated observer are given. The chosen filter is described by the diagonal structure proposed in paragraph 4.2 where ω f = 2π • 20 and Gf = ω f . The LM I regions ( 20) and ( 21) are respectively determined by θ = π 4 and p m = 100. The minimal γ ∞ obtained solving the LM I is γ ∞ = 0.34. The poles of the observer and the variables to be computed are given bellow: -88.63 -54.42 -13.16 -1.62 3.41 -2.21 -7.15 -0.23 -78.18 50.72 163.92 5.29 -0.27 0.17 -15.6125 2.6223 31.9974 -0.3860 -149.5006 -65.4088 -95.8086 13.6066 28.6836 1.8587 -76.7823 -2.0463 1.3160 -4.5023 -1.0315 -0.0384

P oles =   
  , L =    -0.0036 0.0880 -0.0244 1.0006   , P A-(N +KC) = 10 -5 •   
  , cond(N ) = 188
These numerical results show that the disturbance effect has been reduced, since P D is small and the condition P A -(N + KC) = 0 has been respected. The Bode diagrams of the transfer functions between the ground disturbance v and the estimation error e on each state, given on Figure 2, emphasize the attenuation of the ground disturbance effect on the estimation error.

Furthermore the pole placement constraints have been respected. Figure 3 shows that the poles have been placed in the specified cone and their real part are smaller than 100. The cone constraint corresponds to a negative real part with an imaginary part smaller than half the real part.

Here the computed poles are real. Both he conditioning number cond of the matrix N and the pole placement are correct, therefore this observer can be implemented.

Description and set-up of the experiment

Four semi active damper prototypes have been build by SOBEN and mounted on a testing car. These dampers can be controlled by the mean of a servomechanism installed on each damper to control the internal oil flow, and therefore, the damping properties. The Figure 4 represents the damper prototype mounted on SOBEN testing car.

Fig. 4. Mounted semi active SOBEN damper

The vehicle is equipped with a deflection sensor, a vertical unsprung mass accelerometer and a vertical sprung mass accelerometer for each suspension. The acquisition of these data is done by a set of five electronic boards developped by SOBEN. Each damper has a small acquisition board (see Figure 5) that converts the analog measurements into CAN frames(Controller Area Network).

Fig. 5. Damper control board

A central board where the observer is implemented receives the frames through the CAN network and computes the estimated states for each damper, using the four unsprung masses accelerations. The other sensors (deflections and sprung masses accelerations) are not used in the observer. These sensors have only been installed to validate the observer. This architecture is described on Figure 6. The four damper boards are operational, but the central board has not been programmed yet. Therefore a Dspace card has been used for the experiments presented in this paper. The implementation of the observer is done automatically by the software provided by Dspace. However, the implementation can not be done if the observer is illconditionned. Therefore the poles of the observer have to be properly chosen. A PC can be connected to analyse and to record on-line the different signals.

Experimental results

This section gives the experimental results obtained with the observer designed in section 7. The estimated deflection velocity of the suspension has been compared to the measured velocity on Figure 7 (top). To validate the observer, a potentiometric sensor has been installed on the damper. The velocity measurement is very useful in a suspension control application, but this sensor can not be used in mass-produced vehicles because of its price and very small life time. Furthermore, the quality of the measured signal is very bad because of the noise. This noise could be reduced with a filter, but the introduced delay would damage the control performances if this filtered signal were used in a controller. Here the proposed observer reduces the noise witout introducing any delay.

Figure 7 also shows the sprung mass vertical acceleration.

A second accelerometer has been installed on the sprung mass to validate the estimated variables. The measured sprung mass acceleration has been compared to the estimated one, wich is given on Figure 7 (middle). This variable is also well estimated, and well filtered. The unsprung mass acceleration (measured and estimated), which is the input of the observer, is represented on Figure 7 (bottom).

The estimation and measurement are very similar.

Many control strategies have been developped to control the suspensions using the sprung mass acceleration. It would have been possible to install the accelerometer used by the observer on the sprung mass, but in this case, the installation of semi active dampers on an existing car would take a very long time and be very expensive. Therefore this is not possible on mass-produced vehicles.

If the accelerometer is installed directly on the damper, installing semi active dampers is easier because it simply consists in changing the dampers. Then the unsprung mass acceleration is measured and the observer can be used to estimate and filter the sprung mass acceleration.

Remark:

In the observer synthesis, no control input of the damper has been taken into account. In the observer, the damper has been considered linear, with a constant damping rate, which is not realistic since SOBEN damper is nonlinear and semi active, therefore its damping rate is variable and controlled. Different experiments have been done with the observer proposed here, for different control signals, which corresponds to different damping rates. The damping rate does not influence the accuracy of the estimations, therefore taking these variations into account in the observer is not useful.

CONCLUSION

In this paper, a method to synthesize an observer for a suspension control application has been presented. This observer is based on a single reliable sensor providing the unsprung mass acceleration measurement. The estimation is decoupled from the unknown road disturbance through an H ∞ minimization and some ponderation filters are introduced to focus the accuracy of the observer on the interesting frequency range. The proposed synthesis method also includes a pole placement in LM I regions to avoid inadapted dynamics that may preclude the implementation and damage the estimation accuracy in the real embedded application. Finally, this observer has been implemented and embedded on a testing car. The experiment results emphasize the observer performance, and the industrial interest for an automotive suspension control application. Future works will consist in designing such an observer for a full car. Then this observer will be included in the static state feedback control strategy detailed in Aubouet et al. (2008a) and implemented by SOBEN on the testing car in the near future. The objective is to design a global attitude control strategy using the four suspensions with only one acceleration sensor for each suspension. A reduced-order observer version of this observer could also be developped.

Fig. 1 .

 1 Fig. 1. Quarter car model and SOBEN damper This simple vehicle model is made up of a sprung mass, a spring, a damper, an unsprung mass and a tire modelled by a spring. The parameters of this model are given in the Table1.Table1. Quarter car parameters and variables

Fig. 2 .Fig. 3 .

 23 Fig. 2. Transfer || e/v || -Bode diagrams

Fig. 6 .

 6 Fig. 6. Control architecture of the four dampers

Fig

  Fig. 7. Experimental results

Table 1 .

 1 Table 1. Quarter car parameters and variables

	ms, mus	Sprung, unsprung mass
	k, kt	Suspension, tire stiffness
	zr	Ground vertical position
	zs, zus	Sprung, unsprung mass acceleration
	zs, zus	Sprung, unsprung mass position
	z def = zs -zus	Suspension deflection
	Fs	Damping force