J Kighelman

H Cerda

L Arurault

Gilbert Teyssedre

Laurent Boudou

P Combes

J Tschiember

Dielectric and Optical Characterizations of Insulating Coatings Prepared by Micro-Arc Oxidation Process on Aluminum Space Parts

Keywords: Aluminum oxide, MAO process, dielectric and optical properties

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INTRODUCTION

In orbit, satellites are exposed to significant thermal variations. To ensure reliable operation of their on-board systems and equipment, a thermal control of the space-craft is necessary using cold, neutral or warm coatings. For very specific missions, like solar orbiter, cold coatings, also called "white coatings", are used. They exhibit specific optical properties (solar absorbance < 0.2 and IR emissivity > 0.8). Furthermore, to prevent any electrostatic discharge problems, due to implanted charge accumulation under space radiations, coatings also need to have specific electrical properties, such as low resistivity. The aluminum oxides obtained from MAO process, which are typically used in the fields of wear resistance as well as thermal and electrical insulation, could exhibit thermo-optical properties compatible with space environment specification.

MATERIALS

AND CHARACTERISATION TECHNIQUES Samples were prepared using MAO process on aluminum substrate. MAO is an electromechanical surface treatment consisting of thick crystalline oxide growing on a metallic substrate [START_REF] Yerokhin | Plasma electrolysis for surface engineering[END_REF][START_REF] Yerokhin | Oxide ceramic coatings on aluminium alloys produced by a pulsed bipolar plasma electrolytic oxidation process[END_REF]. Different coatings obtained with different process conditions, were optically and electrically tested. The main process parameters are summarized in Table 1.

Electrical measurements DC characterization

Figure 1 shows typical current transients obtained when applying voltage steps in the range 0 to 100 V followed by short-circuit steps. The polarization time was fixed to 10 minutes and can be considered as leading to a representative conduction current. The volume resistivity values, estimated from the value of the current at the end of the polarization step, and obtained from a linearization of the I(V) data are given in Table 3, along with surface resistivity results. Table 3 shows that the surface and volume resistivities of sample #1 are more than two orders of magnitude higher when compared to the three others. Thereafter the dynamic dielectric characterization results are presented for samples exhibiting the highest and lowest volume resistivity values, i.e. samples #1 and #4.

Dynamic characterization

Figures 2 and3 show the modulus of the impedance, |Z|, along with the phase angle between applied voltage and current, as a function of frequency, for samples #1 and #4. For an ideal capacitor, the impedance varies linearly as 1/f and the phase angle is -90°C. For sample #4, the variation of |Z| shows a resistor behavior for low frequency and high temperature. In the high frequency domain and for all the temperatures investigated a capacitive behavior is revealed. This behavior allows us to estimate a volume resistivity that can be compared to DC measurement previously presented, and the dielectric permittivity in the region of capacitive behavior. The dielectric permittivity varies from 14 to 15 for temperatures in the range -100°C to 200°C. The volume resistivity calculated at 20°C and 0.1 Hz (Table 4) is about 6.5 10 7 .m and is significantly lower than the value calculated from DC measurements at room temperature (Table 3). We assume that this difference can be due to the presence of water within the coating during the measurement. Indeed, DC measurements were carried out under vacuum whereas a nitrogen flux is injected for dynamic measurements without prior outgassing. In order to study this point a thermal treatment was applied prior to dynamic dielectric measurements. The results are presented in the next section. For sample #1, the frequency range explored is not large enough to see a resistive behavior in the low frequency domain, but the resistivity can be estimated to a value higher than 3 10 9 .m. From the high frequency domain the dielectric constant was calculated. It varies from 26 to 30 for temperatures varying from -100°C to 200°C. Sample #1 shows a higher volume resistivity, as expected from DC data and a higher dielectric constant compared to sample #4. In both cases, the values are higher than that of bulk alumina (typically r =8-9). The porosity of coatings, defective regions in the structure (cf. discussion) and possibly water inclusion, along with some uncertainty regarding coating thickness might explain such difference. Thermal treatment effect on optical and dielectric properties Sample #4 is the most appropriate sample for a space application as cold coating, because of its good optical properties and low conductivity compared to the other tested samples. In the following, we investigate the effect of a thermal treatment on both optical and electrical properties.

For dielectric measurements, the thermal treatment consists of an in situ annealing at 200°C for 1 hour.

Figure 4 shows the corresponding results. The peak observed at around 100°C on |Z| for the untreated sample can be due to the presence -and possibly release-of water within the coating. After the thermal treatment, this peak disappears, and the value of the impedance increases, clearly indicating an increase in the resistivity of the material. For optical properties, the thermal treatment consisted in annealing samples under air atmosphere at 200°C or 400°C for 5 hours. The thermo-optical properties were measured at room temperature under vacuum to avoid water absorption along with after storage in air at room temperature. The results shown in Table 5 do not reveal any significant variation of the optical properties after heating or after storage in air, pointing towards stable optical properties of the coatings.

DISCUSSION

Changes in dielectric properties of the coatings with process conditions may have several origins, resorting to the porosity, to the water uptake and to composition of the coatings. Various structural characterizations have been implemented to sort out the possibilities. First, X-ray diffraction patterns indicate that the layers are constituted of several crystalline phases. To sum-up, an increase in silicate content of the electrolyte favors the formation of silicate phases in the coatings: the ratio of alumina (Al2O3) to silicate (Al2SiO5 -called mullite-) is around 90:10 in sample #1 and 50:50 in the others. An important point is that large silicate content induces an increase in the amorphous phase fraction of the coatings (cf. broad band in the region 15-30° in Fig. 5). Voigt and co-authors [START_REF] Voigt | Correlation of chemical composition and electrical properties of rf sputtered alumina films[END_REF] studied the correlation between chemical composition and electrical properties of rf sputtered alumina films. They found that the leakage current increases with the concentration of OH groups, which act as defect sites. As OH groups are presumably located preferentially in the amorphous phase, the way of investigating it consisted in combining TGA and DTA. Table 6 shows the weight losses in the sample interpreted as arising from free water release, and from the known thermodynamic stability of aluminas given, in increasing order of stability, as:

Al(OH)3  AlOOH  -Al2O3  -Al2O3  -Al2O3  -Al2O3
The lower losses of water associate with OH groups in sample #1 appear to reflect the higher crystalline phase content for this material. It can be seen that electrical resistivity decreases as the concentration of Al(OH)3 and AlOOH groups increases, being the higher for sample #4. Therefore, the presence of OH groups could explain the lower electrical resistivity of coatings. Finally, the good thermal stability of the coatings deserves to be mentioned: the weight losses for temperature increase up to 1400°C is in the range 0.6 to 2.3%, about 20% of it corresponding to loss of adsorbed water.

CONCLUSION

We have correlated the operational conditions of the Micro-Arc Oxidation (MAO) process to the dielectric and thermo-optical properties of coatings grown onto Al substrates. The main outputs of the study are as follows: i/ Strong variation in electrical conductivity of the coatings were obtained depending on process conditions. Relatively low electrical resistivity materials can be processed, which appears promising for space application.

ii/ Low resistivity appears to go with appropriate thermo-optical properties; the low resistivity could be associated to the presence of OH groups in the coatings.

iii/ However, resistivity appears to increase significantly with thermal treatment of the coatings. This feature has to be investigated in more detail, considering water release and possible hydroxyl groups removal with thermal treatments.

Figure 1 :

 1 Figure 1: Transient currents for sample #4 at room temperature.

Figure 2 :Figure 3 :

 23 Variation of the impedance modulus (a) and phase angle between voltage and current, (b), as a function of frequency for different temperatures for sample #1. Data are given in temperature steps of 20°C. Cite as: J. Kighelman, H. Cerda, L. Arurault, G. Teyssedre, L. Boudou, P. Combes, J. Tschiember, "Dielectric and optical characterizations of insulating coatings prepared by micro-arc oxidation process on aluminum space parts", 9th Int. Conf. on Properties and Applications of Dielectric Materials (ICPADM), Harbin, China, 19-23 July 2009. Proc. ICPADM-09, pp. 1110-1113, 2009. DOI: 10.1109/ICPADM.2009.5252307 Variation of the impedance modulus (a), and phase angle between voltage and current (b), as a function of frequency for different temperatures for sample #4. Data are given in temperature steps of 20°C.

Figure 4 :

 4 Figure 4: Impedance (solid symbols) and phase angle (open symbols) measured at 0.1Hz as a function of temperature prior to (triangles) and after (circles) thermal treatment on sample #4.

1 Figure 5 :

 15 Figure 5: X-Ray diffraction patterns of the samples

Table 1 : Process parameters for coatings preparation. (f=frequency, J=current density, t= process duration, Cs=silicate concentration) d = substrate thickness. J and Cs are given in relative value for confidentiality purpose.

 1 The comparison between table 1 and 2 indicates that high current density and high silicate concentration are required to obtain good optical properties. This observation is confirmed by complementary results on the impact of process parameters on the final optical properties of the coatings, not presented here for space saving.

	Sample	J (r.u.)	t (min)	Cs (r.u.)	d (m)
	#1	J1	150	C1	200
	#2	2.4xJ1	50	6xC1	2000
	#3	1.5xJ1	30	6xC1	1000
	#4	2.8xJ1	30	6xC1	1000
	Structural characterization of the coatings was achieved
	by a set of techniques ranging from XRay diffraction,
	thermogravimetric analysis -TGA-, differential thermal
	analysis -DTA-and Electron Probe MicroAnalyser
	(EPMA).
	To compare the optical properties of different coatings
	the solar absorbance  and the thermal emissivity ε
	were measured using an absorptiometer and an
	emissiometer (EL510 and EL520 Elan Informatique). 
	quantifies the amount of solar rays absorbed by the
	coating and  the coating capacity to convert its thermal
	energy into ray.
	For DC current measurements, an aluminum top
	electrode, 12mm in diameter, was deposited on the
	sample surface by CVD (chemical vapor deposition)
	while the ground electrode was the aluminum substrate.
	Current measurements were carried out under vacuum
	(10 -1 Pa) at room temperature by means of a Keithley
	617 ammeter. The surface resistivity was measured using
	the concentric ring probe configuration. The inner
	electrode and the guarded ring were deposited by CVD.
	Impedance measurements were performed using a

Cite as: J. Kighelman, H. Cerda, L. Arurault, G. Teyssedre, L. Boudou, P. Combes, J. Tschiember, "Dielectric and optical characterizations of insulating coatings prepared by micro-arc oxidation process on aluminum space parts", 9th Int. Conf. on Properties and Applications of Dielectric Materials (ICPADM), Harbin, China, 19-23 July 2009. Proc. ICPADM-09, pp. 1110-1113, 2009. DOI: 10.1109/ICPADM.2009.5252307

Table 2 :

 2

	Sample	Coating Thickness (m)	 (±0.04)	ε (±0.04)	Cold coating specification
	#1	95	0.47	0.77
	#2	100	0.10	0.83	<0,2
	#3	50	0.27	0.76	ε>0,8
	#4	100	0.17	0.77

Thickness, solar absorbance  and thermal emissivity  of the coatings.

Table 3 : DC electrical characteristics of coatings.

 3

	Sample	Volume resistivity (.m)	Surface resistivity (/)
	#1	4.1.10 12	1.10 16
	#2	6.4.10 9	5.10 11
	#3	8.6.10 9	2.10 12
	#4	2.0.10 9	3.10 11

Table 4 : Dielectric parameters extracted from the variation of |Z| vs. frequency and temperature.

 4

	Sample	Volume resistivity (at 20°C)	Relative permittivity (-100 to 200°C)
	#1	> 3 10 9 .m	26-30
	#2	6.5 10 7 .m	14-15

Table 5 : Impact of thermal treatment and of subsequent storage in air on thermo-optical properties of sample #4.

 5

	Sample #4	 (±0.04)	ε(±0.04)
	untreated	0.17		0.77
		vacuum	7days / air	vacuum	7days / air
	5h at 200°C	0.10	0.12	0.85	0.89
	5h at 400°C	0.17	0.14	0.73	0.88

Cite as: J. Kighelman, H. Cerda, L. Arurault, G. Teyssedre, L. Boudou, P. Combes, J. Tschiember, "Dielectric and optical characterizations of insulating coatings prepared by micro-arc oxidation process on aluminum space parts", 9th Int. Conf. on Properties and Applications of Dielectric Materials (ICPADM), Harbin, China, 19-23 July 2009. Proc. ICPADM-09, pp. 1110-1113, 2009. DOI: 10.1109/ICPADM.2009.5252307

Table 6 : Weight losses (in weight %) attributed to water of the various samples as obtained by TGA. Temperature intervals were defined according to the range of thermal stability of the corresponding groups.

 6

	Sample Adsorbed water	Al(OH)3	AlOOH
		(<=100°C)	(100-400°C)	(400-700°C)
	#1	0.12	0.22	0.13
	#2	0.35	0.83	0.54
	#3	0.22	0.50	0.25
	#4	0.57	1.12	0.60

ACKNOWLEDGEMENTS

We are imbedded to Region Midi-Pyrenees for support to this work. GIT and ASTRIUM acknowledge the European Spatial Agency for their support.


M-Q Hoang

L Boudou

S Le Roy

G Teyssèdre

Electrical characterization of LDPE films using Thermo-Stimulated Depolarization Currents method: measurement and simulation based on a transport model

Thermo-stimulated depolarization current (TSDC) measurements have been performed on low density polyethylene (LDPE), a non polar polymer material. The aim is to develop the understanding of the conduction mechanisms at play in this material, and particularly to understand the effect of temperature and antioxidant presence. Variations of the TSDC experimental protocol (charging voltage, pre-heating temperature before quenching) are investigated. Furthermore, original measurements are also performed, aiming at driving the carriers outside the dielectric during the reheating rate. To do so, a relatively small voltage, positive or negative, is applied during the temperature ramp. Experimental results obtained with LDPE samples are interpreted with the help of a bipolar transport model.

INTRODUCTION

LDPE is a low polar polymer which has been widely used as electrical insulating material due to its properties such as low permittivity and losses [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. It is a semicrystalline polymer composed of lamellar crystals, which are typically arranged as spherulites when cooled from the melt. The polymer crystal formation and the spherulitic structure in bulk polymers have been extensively described in the literature [START_REF] Bassett | [END_REF]- [5], and this microstructure widely influences the final electrical properties of the dielectric. Numerous studies have been performed on organic solid dielectrics in order to have access to information related to space charge transport and trapping. Thermo-Stimulated Depolarization Current (TSDC) measurements are likely to give information on the trapping levels for carriers. In this communication, TSDC measurements have been used with two original protocols of application of the voltage and temperature to develop the understanding of the charging mechanisms at play in this material.

SAMPLES AND TSDC PROTOCOLS

Test samples were films of LDPE with Santonox-type antioxidant. It was obtained from pellets in 100µm film form by hot pressing. 32mm-diameter circular electrodes were gold coated on each face. TSDC experiments were carried out in a thermally controlled chamber following the cycle depicted in Fig. 1. Samples were polarized with a DC voltage V P at a fixed temperature T P , for a fixed time t P = 30 min. Then they were rapidly cooled to a temperature close to the boiling point of nitrogen. The voltage was then removed, and the depolarization current was measured, using a Keithley-617 electrometer, under short circuit conditions from -100 to 100°C and for a constant heating rate β = 6 °C/min. All the TSDC measurements were done under controlled helium atmosphere.

EXPERIMENTAL RESULTS

Relaxation peaks in polyethylene

In the temperature range from -100 to 100°C, LDPE shows three relaxations processes located, in increasing temperature order, around -30°C (β), 50°C (α1) and 65°C (α2). Fig. 2 and Fig. 3 show respectively the plot of the TSDC at several applied polarization voltages for two polarization temperatures 25 and 50°C. In the TSDC spectra at T P = 25°C (Fig. 2), two relaxation peaks are observed. The first, called β is located at -30°C. The second peak, called α 1 , is located at 50°C and can be compared with the α peak revealed by thermomechanical spectroscopy [START_REF] Yang | Dynamic mechanical and thermal analysis of aligned vapor grown carbon nanofiber reinforced polyethylene[END_REF]. One of the likely mechanisms responsible for the α peak is molecular motions in crystalline or paracrystalline regions of the polymer. This leads to the release of trapped charges within the interface between the crystalline and amorphous regions [START_REF] Lewin | The measurement of charge trapping characteristics in high voltage insulation materials[END_REF]. Furthermore, the β and α peaks are located at the same temperatures as those labelled as C3 and C4 in the work of Mizutani et al [START_REF] Mizutani | Anomalous Transient Currents in High-Density Polyethylene around 50-70°C[END_REF]. These peaks are assigned to the presence of defects, acting as deep traps (0.8 to 1.4 eV) and located in the amorphouscrystalline interface [START_REF] Mizutani | Anomalous Transient Currents in High-Density Polyethylene around 50-70°C[END_REF], or in the crystalline region [START_REF] Fleming | Simultaneous thermally stimulated luminescence and conductivity in low-density polyethylene[END_REF]. Moreover, the β and α relaxations are only observed in the case of LDPE with antioxidant [START_REF] Tebani | Thermally stimulated depolarization current and space charge measurements on low-density polyethylene: Influence of the presence of antioxidant[END_REF], so the antioxidant most probably adds deep trapping levels and is also responsible for the polymer morphological change.

The TSDC spectra at T P = 50°C (Fig. 3) shows a second peak of relaxation α 2 with an opposite sign, corresponding to an anomalous depolarization current. Usually, removing the applied voltage causes a discharge current of opposite sign compared to the charging current. This anomalous depolarization current has already been observed by Mitzutani et al. [START_REF] Mizutani | Anomalous Transient Currents in High-Density Polyethylene around 50-70°C[END_REF]. Fig. 3 TSDC responses of LDPE at T P = 50°C: evolution of the β, α 1 and α 2 peaks with the applied voltage V P . In the configuration used, positive currents are anomalous discharge currents.

In a recent study on the impact of antioxidant on the electrical properties of LDPE, Tebani et al. [START_REF] Tebani | Thermally stimulated depolarization current and space charge measurements on low-density polyethylene: Influence of the presence of antioxidant[END_REF] have shown, using Pulse Electro-Acoustic (PEA) method, that negative charges are accumulated at both cathode and anode, after removing the applied voltage (Fig. 4).

The α 1 and α 2 peaks were then interpreted as charge detrapping. The negative charges accumulated near the cathode, when evacuated from the bulk under short circuit conditions and temperature increase, are responsible for a normal discharge current (α 1 peak), while the negative charges accumulated near the anode and extracted through the anode are responsible for a positive, i.e. anomalous, discharge current (α 2 peak). ) Fig. 4 Space charge profiles for LDPE-AO with antioxidant after different times in Volt off [START_REF] Tebani | Thermally stimulated depolarization current and space charge measurements on low-density polyethylene: Influence of the presence of antioxidant[END_REF]. Polarization and measurements were achieved at 50°C.

TSDC with two temperature ramps

In order to separate the α 1 and α 2 peaks, TSDC measurements were conducted according to a partial heating method [START_REF] Kazuyuki | Thermally Stimulated Current in Low-Density Polyethylene/MgO Nanocomposite on the Mechanism of Its Superior Dielectric Properties[END_REF] and according to the protocol described in Fig. 5(a). After the polarization period at T P and V P followed by a rapid cooling, sample is heated a first time, from -100°C to 51°C. The upper temperature was selected according to curve (1) in fig. 5(b) and corresponds to the temperature at which the depolarization current sign is changing from negative to positive, after α 1 peak. The sample was then cooled to -100°C and reheated to 100°C with the same heating rate β = 6 °C/min.

During the first heating cycle, charges accumulated near the cathode, and responsible for the α 1 peak, were evacuated. During the second heating, only the positive α 2 peak is recorded with higher amplitude when compared to the α 2 peak recorded by classical TSDC measurement (dash line Fig. 5(b)). This last result seems to confirm that negative charges accumulated at the cathode and anode are responsible respectively for α 1 and α 2 peaks. The idea is now to drive theses charges by the application of a small voltage during heating. A positive voltage application on the electrode that was previously anode should tend to drive negative charges from (pre-)cathode to (pre-)anode and as a consequence should increase the α 2 peak amplitude, while a negative voltage should drive negative charges from (pre-)anode to (pre-)cathode and then decrease the α 1 peak amplitude. Indeed, it can be shown [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] that the external current essentially corresponds to the space-averaged transport current. If one hypothesizes that charges are trapped in a narrow region close to an electrode, then, their extraction through this electrode produces only a very small current (depending only on their distance from the electrode). Conversely, it is their transport in the dielectric bulk that produces a significant current.

Driving voltage during temperature ramp

In this study, samples were first polarized with V P = 2.5 kV at 50°C. After cooling to -100°C under applied voltage, the sample was reheated using a constant heating rate β = 6 °C/min. The originality here is to apply a small positive or negative driving voltage during the temperature rate. Hence, three different tests have been performed following voltage protocols described in Fig. 6(a). Test 1 corresponds to the classical TSDC protocol where the sample is under short circuit during heating ramp. In Test 2, a positive voltage of 50 V is applied during heating, while in Test 3 the driving voltage is -50V.

As expected, a positive driving voltage leads to an increase of α 2 peak magnitude (curve (2), Fig 6). Furthermore the magnitude of the α 1 peak in this case is near zero. When the driving voltage is negative, an increase in α 1 peak amplitude and a decrease in α 2 peak amplitude are observed (curve (2) Fig 6 .). These results can be interpreted and explained by a negative charge detrapping process, where a positive driving voltage enhances the transport and extraction of negative charges from the pre-cathode, and a negative driving voltage enhances charge transport and extraction from the pre-anode. At this stage however, we cannot fully conclude on the reason for having two peaks in depolarization current: either it is linked to the specific position of the charges in the dielectric and related induced field settled by poling prior to TSDC measurement, or there are two populations of charges of different nature, being e.g. injected electrons at the cathode and anionic charges adjacent to the anode. If a single kind of carriers were at the origin of the double peak, then it should be reproduced by modelling charge transport during a TSDC run. Besides, this task would also help in exploiting TSDC for extracting relevant transport parameters. In the following, we present preliminary results obtained in an attempt to model TSDC based on electronic carriers transport.

SIMULATION RESULTS

Model

The model used to simulate TSDC measurements has been widely described in [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] [START_REF] Roy | Thermo-Stimulated Depolarization Currents in Polyethylene Films. Numerical Simulations and Experiments[END_REF]. We only briefly recall the temperature-dependent processes. Two kinds of carriers are considered in the model: electrons and holes. They can be either trapped or mobile. Mobile carriers are associated with an effective mobility µ that we consider of hopping type:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) ( 2 sinh ) ( . exp ) , ( 2 ) , ( ,
, t T k eEd t T k w e t x E d t x B B h µe h e ν μ (1)
where d is the distance between traps, E the electric field, an e the elementary charge. ν is the attempt to escape frequency, k B the Boltzmann constant, w µ the hopping barrier height and T the temperature.

Deep trapping is described through a single trapping level for each kind of carrier. Charge carriers have a given probability to escape from traps by overcoming a potential barrier, w tr . This is accounted for by the detrapping coefficients D i of the form, for each kind of carrier:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ) ( . . exp . ) ( , , t T k w e t D B tr h e h e ν ( 2)
The recombination of carriers is accounted for considering different coefficients S i for the several electron-hole pairs. Carriers are provided by charge injection at the electrodes and are extracted without any barrier. We have supposed that charges are injected at the electrodes according to a modified Schottky law:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = 1 4 ) ( exp ) ( exp ) ( ) ( , 2 , πε elect B B h e h e eE t T k e t T k w t AT t j ( 3
)
where A is the Richardson constant, w e,h are the injection barriers for electrons and holes and ε is the dielectric permittivity. The time and space dependent equations to solve and the numerical schemes used are given in [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF].

Simulation results

The objective of this simulation is to further understand the TSDC profiles obtained during a temperature ramp when a voltage is applied. In that goal, the three experimental tests described in Fig 6(a) are reproduced in the simulation. The parameters used are those given in [START_REF] Roy | Thermo-Stimulated Depolarization Currents in Polyethylene Films. Numerical Simulations and Experiments[END_REF]. No further attempt has been made to 'fit' the present TSDC results, and we are aware that this set of parameters mainly gives positive charges inside the dielectric [START_REF] Roy | Thermo-Stimulated Depolarization Currents in Polyethylene Films. Numerical Simulations and Experiments[END_REF]. The simulated TSDC profiles should however give the same dynamics as what is observed experimentally, the only difference being the sign of the current.

Fig 7 shows the simulated TSDC profile as a function of temperature obtained for the short circuit protocol. The current density is negative, as we are dealing with positive charges. Anyway, the TSDC profile shows a single peak, centered at around 64°C. No TSDC peak is observed in simulation at temperatures less than 0°C, as the model does not include polarization processes. Fig. 7 TSDC spectrum predicted by the model under short circuit conditions for Tp=50°C, Vp=2.5kV with model parameters given in [START_REF] Roy | Thermo-Stimulated Depolarization Currents in Polyethylene Films. Numerical Simulations and Experiments[END_REF]. Fig 8(a) shows the simulated TSDC spectra for the three experimental protocols of Fig. 6(a). The application of a negative driving voltage enhances the transport and extraction of positive charges at the anode. As a result, the absolute value of the current density will be increased compared to the short-circuit case. This is what is observed for the TSDC of Test 3 compared to the one of Test 1. On the contrary, the application of a small positive driving voltage enhances the transport and extraction of positive charges towards the cathode. In this case, the current density related to the application of a positive driving voltage is of opposite sign compared to the one of Test 1, hence decreasing the TSDC, which tends to zero and becomes positive with the temperature ramp. The absolute values of the simulated current densities are always increasing with temperature, whereas the experimental TSDC shows a maximum in absolute value. An explanation is that the model enables an injection of charges during TSDC measurements, depending on the electric field at the electrodes. This 'conduction part' to the TSDC is possibly over-estimated in the model. However, when adding the TSDC of Tests 2 and 3 (Fig. 8(b)), the conduction part due to charge injection is nearly annihilated, and the result is roughly twice the simulated TSDC of Test 1, which is an expected result if one suppose that the area under the TSDC is proportional to the stored charge. Though results are promising, refinements are necessary, especially as regards the temperature dependence of model parameters.

Fig. 1

 1 Fig. 1 Voltage and temperature protocol of TSDC measurements. Dotted and solid lines represent respectively the set point temperature and the measured temperature along the cycle.

1 Fig. 2

 12 Fig.2TSDC responses of LDPE at T P = 25°C: evolution of the β and α 1 peaks with the applied voltage V P .

Fig. 5 (

 5 Fig.5(a) Voltage and temperature protocol for partial heating method (b) TSDC responses without partial heating (dashed line) and after partial heating to 51°C (solid line).

Fig. 6

 6 Fig. 6 TSDC measurements under driving voltage applied to LDPE at T P = 50°C and V P = 2.5 kV: (a) Voltage protocols; (b) TSDC spectra for different driving voltages.

Although the simulated space charge pattern is positive at the beginning of the ramp-up, the model is able to give the same trend as what is observed experimentally.

CONCLUSION

By using different variants of TSDC experiments, namely partial heating and superposition of a driving voltage during TSDC measurements, we have shown, in the light of previous space charge measurements, that the α 1 and α 2 depolarization peaks observed in the range 30 to 70°C in TSDC of LDPE actually correspond to the release of negative charges, located respectively close to the electrodes that were cathode and anode during polarization. It is not possible at this time to decide if the carriers are of the same nature or not. TSDC modelling using a transport model has been initiated in order to unravel the actual nature of this double population of charges.


Mai Quyen Hoang
email: maiquyen.hoang@laplace.univ-tlse.fr

Laurent Boudou

Séverine Le Roy

Gilbert Teyssedre

M-Q Hoang

Paul Sabatier

G Teyssèdre Paul

Electrical characterization of PEN films using TSDC and PEA measurements

Keywords: PEN, TSDC, PEA, relaxation, polarization, effect of temperature, discharge current

INTRODUCTION

Numerous studies have been performed on organic solid dielectrics in order to have access to information related to space charge transport and trapping. Thermo-Stimulated Depolarization Current (TSDC) measurements are likely to give information on the trapping levels for carriers. It may also contain a response from oriental polarization. Charge release and dipolar effects may both be related to molecular motions appearing in the material. [START_REF] Belana | Tsc study of the polar and free charge peaks of amorphous polymers[END_REF]- [START_REF] Belana | Behavior of amorphous poly(ethylene-terephthalate) annealed at T-Less-Than-TG by Thermally Stimulated Currents[END_REF]. Therefore, discriminating responses related from space charge relaxation from dipolar processes can be critical. Poly(ethylene naphthalene 2,6dicarboxylate) (PEN) has been chosen for this study as it is an aromatic polar polymer, and is likely to give a significant response from dipolar mechanisms in TSDC measurements. In this study, TSDC measurements have been used with two protocols of application of the voltage and the temperature to develop the understanding of the charging and polarization mechanisms at play in this material. Space charge measurements have also been performed using the Pulsed Electroacoustic method (PEA) in order to confirm the space charge behavior in the PEN samples.

II. SAMPLES AND MEASUREMENT PROTOCOLS

PEN films were commercial TEONEX ® Q51 films provided by DuPont Teijin Films Co. The melting point of these semicrystalline materials is at 269°C and the glass transition temperature at 121°C [3]. The film thickness is 188 µm. For all measurements, 20mm-diameter circular gold electrodes were deposited on each face. TSDC measurements were carried out in a thermally controlled chamber following the cycle depicted in Fig. 1. Samples were first polarized with a DC voltage VP (1 to 2 kV) at a constant temperature TP (130 or 170°C), for a fixed time tP = 15 min. After a fast cooling under applied voltage, samples were reheated from 25 to 220°C, using a constant heating rate υ= 3°C/min, under short circuit conditions. The corresponding discharge current, due to charges thermally detrapped or to dipolar processes, was recorded, using a Keithley-617 electrometer. All the TSDC measurements were done under controlled helium atmosphere.

Space charge measurements were performed using the Pulsed Electroacoustic (PEA) method in order to check the charge distribution within the dielectric before (cf. step PEA-1 on Fig. 1) and after (PEA-2 on Fig. 1) the TSDC measurement. In that goal, PEN samples were submitted to the voltage and temperature protocols in the TSDC chamber, as the PEA set-up does not withstand temperatures higher than 90°C. Samples were removed from the TSDC set-up at PEA-1 and PEA-2 (Fig. 1), and space charge measurements were carried out at room temperature under short circuit conditions.

III. TSDC RESULTS

PEN samples were polarized at two temperatures TP: 130°C and 170°C for different voltages VP. Fig. 2 presents the TSDC profiles obtained for PEN samples polarized at 130°C. Two relaxation peaks were observed. The first one, called β* is located at 65°C. It has been ascribed to the relative motion of the two naphthalene rings present in the polymer chain [START_REF] Krause | Comportement électrique et physico-chimique de films de poly(ethylene-2,6-naphtalene dicarboxylate) soumis à des contraintes électriques et thermiques[END_REF]- [START_REF] Dortliz | Molecular mobility in poly(ethylene-2,6naphthalene dicarboxylate) as determined by means of deuteron NMR[END_REF]. The mechanisms associated with this relaxation process are complex and depend upon the measurement frequency [START_REF] Martinez-Vega | Molecular mobility in amorphous and partially crystalline PEN after isothermal crystallization[END_REF].

The second peak, called α, is located at 140°C. It has been related to the glass transition of this material, involving longrange cooperative segments motion [START_REF] Belana | Tsc study of the polar and free charge peaks of amorphous polymers[END_REF]. It has a characteristic dielectric loss peak, which is broad, asymmetrical in the Cole-Davidson representation [START_REF] Williams | Dielectric relaxation behavior of amorphous polymers and related materials[END_REF]. The magnitude of the β * and α peak vary almost linearly with the applied voltage. The area under the α peak increases also linearly with the voltage, although it is more difficult to observe as the two peaks overlap. The TSDC profiles of PEN samples polarized at TP = 170°C are presented in Fig. 3. As for a polarization temperature of 130°C, a β* and a α relaxation peak can be observed, at 65 and 140°C respectively. A third peak is observed for TP = 170°C, called ρ, and located at 180°C. This relaxation peak has already been observed by Canãdas et al [START_REF] Canãdas | Relaxational study of poly(ethylene-2,6naphthalene dicarboxylate) by t.s.d.c., d.e.a. and d.m.a[END_REF] and appear related to charge detrapping within the material. Several TSDC measurements were carried out at TP = 170°C for different applied voltages.. As for TP = 130°C, an increase in the applied field implies a linear increase in the β* and α peak area. This is however not the case for the ρ peak area, which clearly shows a non-linear increase. This non-linearity goes in the direction of a non-polar phenomenon at the origin of the ρ peak.

IV. PEA MEASUREMENTS

In order to confirm the mechanism responsible of the observed ρ peak, space charge measurements were performed on PEN samples which were already polarized with a DC voltage (2kV) at temperatures of 130°C or 170°C.. Fig. 4 shows the space charge profiles obtained under short-circuit conditions (a) after polarizing at 130°C and freezing the polarization to room temperature (PEA-1) and (b) after subsequent TSDC measurement (PEA-2), meaning after temperature increase to 220°C in depolarization followed by cooling to room temperature. No charges are detected within the bulk at the end of the TSDC polarization protocol (Fig. 4a), even after 15 min under short-circuit. However, influence charges are observed at each electrode, which do not change with time. These charges could be due to heterocharges within the dielectric in the narrow region close to the electrodes, hence not detected by the set-up. These image charges could also be linked to polarization phenomena of the PEN sample. A rough calculation of the area under the influence charge peak of Fig. 4a gives a surface charge of 2.1 10 -5 C/m 2 . When calculating the area under the α current peak of Fig. 2, for 2kV and a heating rate of 3°C/min, and taking into account the fact the β* and a α current peaks overlap, the surface charge obtained is 2.5 10 -5 C/m 2 . The two values are consistent, meaning that dipolar processes dominate the influence charge peaks observed in the PEA measurements. At the end of the TSDC scan (PEA-2), no charges are detected within the PEN sample (Fig. 4b). It is to note that the image charge on the anode seems to be an artifact, as no more charge are observed on the cathode, and the integrated charge over the sample should be void. This could be due to a bad matching impedance between the PEN sample and the semiconductor at the top electrode of the PEA set-up. As long as this is true, no more polarization exists within the sample at the end of the TSDC protocol. Fig. 5 shows the space charge profiles obtained under short-circuit conditions (a) before (PEA-1) and (b) after (PEA-2) the TSDC scan, for a polarization temperature of 170°C. At the end of the TSDC polarization (PEA-1), negative charges are observed in the bulk of the sample. This is a remarkable behavior given the relatively low field (10kV/mm) applied in these measurements. These negative charges do not move after 15 min of short-circuit at room temperature. At the end of the TSDC scan (PEA-2), a negligible amount of charges are detected within the PEN sample (Fig. 5b), and no surface charges are observed. Negative charges have been evacuated during the ramp-up, and polarization mechanisms have also been released. The charge profiles for a polarization temperature of 130°C and 170°C are then really different, and the presence of charges at PEA-1 for TP = 170°C explains the difference in the TSDC profiles. Hence, the ρ peak is certainly due to negative charges present within the dielectric, and released during the reheating rate. It is to note that the negative charges within the bulk alone cannot explain the magnitude of the influence charges at each electrode. As for TP = 130°C (Fig. 4), there should also be a contribution of a polar phenomenon in the influence charges.

V. TSDC WITH PARTIAL HEATING

To further investigate the behavior of the the ρ-peak, TSDC measurements were conducted according to a partial heating method [START_REF] Kazuyuki | Thermally Stimulated Current in Low-Density Polyethylene/MgO Nanocomposite on the Mechanism of Its Superior Dielectric Properties[END_REF] following the protocol described in Fig. 6a. After the polarization period at TP = 170°C and VP (2 kV in this case) followed by a rapid cooling, sample is heated a first time, from 25°C to 150°C. The upper temperature was selected according to Fig. 3 and corresponds to the temperature between the α and the ρ peak. The sample was then cooled to 25°C and reheated up to 220°C with the same heating rate υ = 3°C/min. TSDC profiles are presented in Fig. 6b. The current recorded during the first partial heating, up to 150°C, is consistent with what has been already presented in Fig. 3 for an applied voltage of 2 kV. The β* and α relaxations can be observed, at 65 and 140°C respectively. During the second heating, these peaks are not present anymore, and only the ρ peak is measured, with a current maximum at around 180°C. We suppose that during the first partial heating rate, only the polar groups responsible of β* and α peaks relaxed. Thus, trapped charges still present are released at a temperature higher than 150°C during the second heating. However, care has to be taken, as the ρ-peak could be due to a combination of space charge and orientation polarization phenomena (due to charge or other). Hence, space charge measurements after the first partial heating at 150°C need to be performed, in order to verify if trapped charge are still present within the bulk.

VI. CONCLUSION

TSDC and PEA methods at various temperatures reveal that the voltage and especially the temperature play an important role in the electrical properties of PEN films. An increase in the polarization temperature gives rise to different TSDC results, as only polarization peaks are observed for a polarization temperature of 130°C, while a third peak at 180°C is detected when the polarization temperature is higher. This peak has been attributed to space charge present within the bulk of the PEN samples. Space charge measurements, performed at different points on the TSDC protocol, have confirmed the presence of negative space charges in the bulk of PEN when the sample is polarized at 170°C, while no charges are detected for a lower temperature. It has also been possible to observe a polarization phenomenon with the PEA method. However, more work needs to be done in order to check if the ρ peak in TSDC can be attributed only to space charge, or if polar phenomenon also contribute to this current peak.

Fig. 1 .Fig. 3 .

 13 Fig. 1. Voltage and temperature protocol of TSDC measurements. Dotted and solid lines represent respectively the set point temperature and the measured temperature along the cycle.

Fig. 2 .Fig. 5 .

 25 Fig. 2. TSDC profiles of PEN polarized at TP = 130°C: evolution of the β * and α peaks with the applied voltage VP.

Fig. 4 .

 4 Fig. 4. Space charge profiles by PEA in Volt off for PEN previously polarized at 130°C: (a) Before reheating (PEA-1) of TSDC protocol; (b) After reheating (PEA-2) of TSDC protocol; C: Cathode, A: Anode during pre-polarization..

Fig. 6 .

 6 Fig. 6. (a) Voltage and temperature protocol for partial heating method; (b) TSDC responses without partial heating (solid line), after the first partial heating to 150°C (dashed line) and after the second heating rate (dash-dotted line).

ACKNOWLEDGMENT

The authors thank DuPont Teijin Films Co. for providing PEN samples. This work is supported by the French National Research Agency (ANR) within the ModElec project ANR-11-JS09-009-01.


Leonardo B Groff
email: leonardo.groff@ufrgs.br

Giorgio Valmorbida
email: giorgio.valmorbida@centralesupelec.fr

Joao M Gomes Da Silva

An implicit representation for the analysis of piecewise affine discrete-time systems

Keywords: piecewise affine systems, discrete-time systems, stability, piecewise quadratic Lyapunov functions

This paper presents a new framework for stability assessment of discrete-time piecewise affine systems. An implicit representation for piecewise functions based on ramp nonlinearities is proposed. Instead of the usual sector inequalities adopted in the study of Lurie systems, we directly exploit properties of the ramp function, which are given by a particular set of identities and inequalities. These properties are then used to derive stability conditions associated to piecewise quadratic Lyapunov functions. These functions are implicitly defined from quadratic forms involving ramp functions. These conditions can be cast as LMIs and their numerical solution is illustrated by examples highlighting the advantages of the proposed method over existing stability analysis methods for PWA systems.

Introduction

Models for Piecewise Affine (PWA) systems have been proposed to study engineered systems such as nonlinear circuits [START_REF] Julian | Highlevel canonical piecewise linear representation using a simplicial partition[END_REF][START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF], where simple piecewise affine nonlinearities may lead to complex behavior, or in the context of hybrid systems [START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF]. Moreover, systems presenting some static nonlinearities, such as saturation or deadzone, can also be studied in this framework since these functions are indeed piecewise affine [START_REF] Gomes Da | Polyhedral regions of local stability for linear discretetime systems with saturating controls[END_REF][START_REF] Lathuilière | Periodic orbits in planar linear systems with input saturation[END_REF]. The practical interest on PWA continuous functions in discrete-time systems also appears in the context of Receding Horizon Optimal Control (ROHC) [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], in which multi-parametric linear or quadratic programs can be solved offline to obtain PWA control laws associated to a specific partition of the state space, termed explicit Model Predictive Control.

Studies on this class of systems trace back to [START_REF] Sontag | Nonlinear regulation: the piecewise linear approach[END_REF], where the following explicit representation was introduced for a PWA function f :

R n → R n f , f = f j , ∀x ∈ Γ j ⊂ R n , f j (x) = A j x + b j , (1)
This work was supported by STIC-AmSud 18-STIC-01. This research is funded in part by ANR via project HANDY, number ANR-18-CE40-0010. j = 1, . . . , n s , with disjoint sets Γ j defining a partition of R n , i.e. ∪ ns i=j Γ j = R n .

The stability analysis of PWA continuous-time systems using the explicit representation (1) has been studied with continuous piecewise quadratic Lyapunov functions (LF) [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF][START_REF] Iervolino | Conecopositive piecewise quadratic Lyapunov functions for conewise linear systems[END_REF][START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF]. In [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF], the sets Γ j are described by the intersections of half spaces. In [START_REF] Iervolino | Conecopositive piecewise quadratic Lyapunov functions for conewise linear systems[END_REF] the particular case of conewise linear systems is addressed. In [START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF], a representation of the polyhedral regions of the state space partition by vertices and cone rays is considered. In this case, to guarantee the positivity of the Lyapunov function and the negativity of its derivative along the trajectories of the system, sufficient conditions based on the cone rays and the vertices that represent each region have to be tested.

On the other hand, the stability analysis of discretetime PWA systems has been studied for instance in [START_REF] Feng | Stability analysis of piecewise discretetime linear systems[END_REF][START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF][START_REF] Hovd | Relaxing PWQ Lyapunov stability criteria for PWA systems[END_REF][START_REF] Rubagotti | A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains[END_REF]. Similar to [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF], polyhedral partitions described by the intersections of half spaces are considered in these works. Differently from the continuous-time case, discontinuous piecewise quadratic Lyapunov functions can be studied for discrete-time systems. However, a drawback of the results using a different quadratic function for each set Γ j in the partition appears when assessing the decrease of the LF during a transition. Indeed, one needs to enumerate all possible transitions between sets in the partition and evaluate the associated decrease of the LF.

The goal of this paper is to investigate stability conditions for discrete-time PWA systems. With this aim we introduce an implicit representation of PWA functions. The proposed representation allows to avoid some shortcomings of the explicit representation [START_REF] Adegbege | A framework for multivariable algebraic loops in linear anti-windup implementations[END_REF]. In particular, by adopting the proposed representation we show that it is possible to parametrize continuous piecewise quadratic Lyapunov functions by considering generalized quadratic forms involving ramp functions. The stability of PWA systems can thus be assessed by evaluating Lyapunov stability conditions through linear matrix inequalities (LMI) tests, and does not require the enumeration of transitions between the sets of the partition.

This paper is organized as follows: Section 2 presents the proposed implicit representation of PWA systems, based on vector-valued ramp functions, which is the basis of the stability analysis framework. Properties of vector-valued ramp functions and conditions to certify the positivity of generalized quadratic forms involving such functions are introduced in Section 3. In Section 4 we apply the positivity verification to formulate conditions for assessing the stability of discrete-time PWA systems using PWQ Lyapunov functions. Finally we illustrate the obtained results with numerical examples in Section 5 and present concluding remarks and perspectives in Section 6.

Notation For a vector v ∈ R n , v i denotes its ith entry, for a matrix M ∈ R n×m , M i,j denotes its (i, j) entry. Define

D n = {M ∈ R n×n | M i,j = 0, i = j}, P n×m = {M ∈ R n×m | M i,j ≥ 0, ∀i, j}, and S n = M ∈ R n×n | M = M .
For M ∈ R n×n we define He(M ) := M + M . The set of non-negative real numbers is denoted R ≥0 . The absolute value of a scalar is denoted | • | and for M ∈ R m×n , M denotes the largest singular value of matrix M . We use 0 (boldface) to denote a matrix of zeros of suitable dimensions. For a vector z ∈ R d , diag(z) corresponds to the diagonal matrix of dimension d × d with its diagonal given by vector z.

Problem Statement

In this paper we study the stability of piecewise affine discrete-time dynamical systems given by

x + = f (x), ( 2
)
where x is the state, f (x) is a continuous piecewise affine vector function, and x + denotes the next time-step value for the discrete-time system. With this aim, we introduce now an implict representation for a generic piecewise affine vector function f : R n → R n f as follows:

f (x) = F 1 x + F 2 φ(y(x)) (3a) y(x) = F 3 x + F 4 φ(y(x)) + f 5 (3b) where x ∈ R n , y ∈ R ny , F 1 ∈ R n×n , F 2 ∈ R n×ny , F 3 ∈ R ny×n , F 4 ∈ R ny×ny , f 5 ∈
R ny and the vector function φ : R ny → R ny is defined elementwise by the ramp function as

φ i (y) = r(y i ) := 0 if y i < 0 y i if y i ≥ 0 , i = 1, . . . , n y (4)
as depicted in Figure 1.

y i r(y i ) Fig. 1. Ramp function r(yi).
Using ( 3)-( 4) as a model for continuous PWA functions avoids the explicit definition of partitions and the corresponding affine functions, as in the standard representation [START_REF] Adegbege | A framework for multivariable algebraic loops in linear anti-windup implementations[END_REF]. In fact, with (3)-( 4), it is the vector function φ(y(x)) and the regions where its arguments are not negative that define the PWA partition of R n . For this reason, we refer to (3)-( 4) as an implicit representation of PWA functions. Examples of this representation and its relation with the explicit representation (1) are given in Section 5.

Note that in this implicit representation (3b) is, in general, an implicit equation (see Appendix A for a discussion on the well-posedeness and solution of this equation). However, for particular structures of matrix F 4 , explicit solutions to (3b) can be obtained. Henceforth we assume that (3b) is well posed, i.e. for each x ∈ R n there exists a unique solution y(x) for the equation. We should also observe that f (x) is continuous thanks to the continuity of φ.

The main feature of (3) that will be exploited in the formulation of stability conditions for PWA discrete-time systems is the characterization of the vector-valued ramp function φ in terms of identities and inequalities, presented below in Section 3. With these relations we follow a similar path to the one adopted to study stability of systems with sector bounded nonlinearities using sector inequalities. These properties will be key to obtain numerically tractable conditions for the verification of Lyapunov inequalities.

Differently from approaches based on the explict representation (1) (e.g. [START_REF] Feng | Stability analysis of piecewise discretetime linear systems[END_REF][START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF]), the proposed implicit representation also simplifies the stability analysis since the partition and possible transitions between sets do not have to be explicitly accounted for in the piecewise quadratic Lyapunov inequalities. Another important aspect is that handling uncertainties in the partition induced by (3) can be simpler than with an explicit representation since these uncertainties can be cast as uncertainties on the matrices F 3 , F 4 and f 5 . These uncertainties can be described by matrix sets such as polytopic or norm-bounded ones [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF].

3 Conditions for Positivity of Generalized Quadratic Forms Involving Vector-Valued Ramp Functions

Note that φ is a sector bounded nonlinearity as φ i (y) = r(y i ) belongs to the sector [0, 1], ∀i = 1, . . . , n y . Several results to verify the positivity of generalized quadratic forms involving sector nonlinearities rely on sector inequalities that hold either globally or locally [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF][START_REF] Hu | Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions[END_REF]. These standard sector inequalities cover a broad class of nonlinearities lying in the considered sector. Hence, stability conditions based on these inequalities are conservative if we are interested in the study of a specific nonlinearity. In the following, we provide a set of relations that are verified only for the vector-valued ramp function φ. These relations are based on an exact characterization of the ramp function (4), by using some identities and inequalities.

Lemma 1 For any matrix T ∈ D ny the function φ in (4) satisfies the identity

s 1 (T, y) := φ (y)T φ(-y) = 0, ∀y ∈ R ny . (5)
Proof. Since for θ < 0 we have r(θ) = 0, and for θ ≥ 0 we have that r(-θ) = 0, the relation

r(θ)r(-θ) = 0. ( 6
)
holds for all θ ∈ R. Since the elements of φ are defined as ramp functions we have

s 1 (T, y) = ny i=1 T i,i r(y i )r(-y i )
which, using ( 6) and (4), gives (5).

Lemma 2 For any vector ζ ∈ R n ζ and matrix R ∈ R n ζ ×ny the function φ in (4) satisfies the identity

s 2 (R, ζ, y) := ζ R (y -(φ(y) -φ(-y))) = 0, (7)
∀y ∈ R ny .

Proof.

Let us first note that the ramp function r satisfies the identity

θ -(r(θ) -r(-θ)) = 0. (8)
Indeed, if θ < 0, we have r(θ) = 0 and r(-θ) = -θ. Thus, for θ < 0, θ -r(θ)+r(-θ) = θ -0-θ = 0. If θ ≥ 0 we have r(θ) = θ and r(-θ) = 0, thus, for θ ≥ 0, θr(θ)+r(-θ) = θ -θ +0 = 0. Since the elements of vector y -(φ(y) -φ(-y) are given by y i -(r(y i ) -r(-y i ))) we have that [START_REF] Feng | Stability analysis of piecewise discretetime linear systems[END_REF] follows from [START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF].

Lemma 3 For any matrix M ∈ P (1+2ny)×(1+2ny) the vector function φ in (4) satisfies the inequality

s 3 (M, y) :=     1 φ(y) φ(-y)     M     1 φ(y) φ(-y)     ≥ 0. ( 9
)
∀y ∈ R ny .

Proof. As the ramp function satisfies

r(θ) ≥ 0, ∀θ ∈ R (10)
from ( 4) one has that φ i (y) = r(y i ) ≥ 0 and φ i (-y) = r(-y i ) ≥ 0, ∀i = 1, . . . n y . Thus, if all entries of M are nonnegative, it follows that s 3 (M, y) is a nonnegative scalar.

We now use the above lemmas to set conditions to verify the positivity of generalized quadratic forms of the type

h(x) = χ(x) Hχ(x). (11)
with χ(x) = 1 x φ (y(x)) φ (-y(x)) and φ defined as in (4).

Proposition 1 Given a generalized quadratic form h as in [START_REF] Hu | Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions[END_REF], if there exist matrices

T ∈ D ny , R ∈ R (1+n+2ny)×ny , M ∈ P (1+2ny)×(1+2ny) such that h(x) + s 1 (T, y(x)) + s 2 (R, χ, y(x)) -s 3 (M, y(x)) ≥ 0 (12) then h(x) ≥ 0 ∀x ∈ R n . ( 13
)
Proof. From Lemmas 1 and 2, which hold for all y(x), if ( 12) is satisfied it follows that

h(x) ≥ s 3 (M, y(x)), ∀x ∈ R n .
With Lemma 3 we conclude that h(x) ≥ 0, ∀x ∈ R n .

Remark 1 Setting conditions to verify the nonnegativity of a generalized quadratic form as [START_REF] Hu | Stability and Performance for Saturated Systems via Quadratic and Nonquadratic Lyapunov Functions[END_REF] by solving the inequality (12) makes possible the solution to the Lyapunov inequalities related to the stability of PWA systems. These inequalities are studied in the next section.

Since we consider an expression h with terms φ(y) and φ(-y) the use of the relations in Lemmas 1, 2 and 3 is important to restrict the validity of the test for the non-negativity of h in Proposition 1 only to functions φ of which the components are ramp functions. Thus, by using Lemmas 1, 2 and 3, we restrict the elements of φ in the generalized quadratic forms treated in Proposition 1 to ramp functions and not a broad set of functions.

Remark 2 It is possible to use the Karush-Kuhn-Tucker (KKT) optimality conditions to implicitly characterize nonlinearities in terms of identities and inequalities. Such an idea was detailed in [START_REF] Primbs | Kuhn-Tuckerbased stability conditions for systems with saturation[END_REF] for the saturation nonlinearity. We illustrate this approach for the ramp function, which can be expressed as the solution to the optimization problem parameterized in θ as follows minimize

r 1 2 (r -θ) 2 subject to r ≥ 0. ( 14
)
With the Lagrangian associated to the optimization problem, L(r, λ) = 1 2 (r -θ) 2 -λr, we obtain the KKT conditions (r -θ) -λ = 0; λr = 0; r ≥ 0; λ ≥ 0 which are necessary for optimality. Note that these relations offer a characterization in terms of linear and quadratic identities and inequalities in three variables (θ, r, λ). To obtain a description in the variables (θ, r) one can use λ = (r -θ) above to obtain

(r -θ)r = 0 (16a) r ≥ 0 (16b) (r -θ) ≥ 0. ( 16c
)
These relations can also be obtained using ( 6), ( 8) and [START_REF] Hovd | Relaxing PWQ Lyapunov stability criteria for PWA systems[END_REF]. Indeed, note that (16a) corresponds to (6), since, from (8), r(θ) -θ = r(-θ). Also, (16b) corresponds to [START_REF] Hovd | Relaxing PWQ Lyapunov stability criteria for PWA systems[END_REF], and again, using r(θ) -θ = r(-θ) we have that [START_REF] Hovd | Relaxing PWQ Lyapunov stability criteria for PWA systems[END_REF], gives (16c).

Stability Analysis of PWA Systems with PWQ Lyapunov Functions

In this section we apply the results for the verification of non-negativity of generalized quadratic forms presented in the previous section to study the stability of the origin of a discrete-time systems (2) with f (x) defined by the implicit representation (3), with n f = n.

We assume that φ(y(0)) = 0 and thus the origin is an equilibrium point, since f (0) = 0 in this case. If φ(y(0)) = 0, we have that (3b) leads to y(0) = f 5 . Since φ(y(0)) = 0 implies y i (0) ≤ 0, we have that f 5i ≤ 0, i = 1, . . . n y .

The stability of the origin of system ( 2) is studied with a continuous piecewise quadratic Lyapunov function, given by a generalized quadratic form on x and the function φ(y(x)). Hence, differently from previous approaches, the definition of an explicit quadratic form on x for each set of the partition is not required. More precisely, we consider Lyapunov candidate functions V : R n → R ≥0 , V (0) = 0 given by

V (x) = x φ(y(x)) P x φ(y(x)) . ( 17
)
with P = P1 P2 P 2 P3 , P 1 ∈ S n , P 2 ∈ R n×ny and P 3 ∈ S ny .

We obtain below a quadratic upper bound for V (x) which will be used in the proof of exponential stability of the origin of (2). To this end, we first compute an upper bound for φ 2 . With y := y -f 5 , (3b) gives

y = F 3 x + F 4 φ(y + f 5 ). (18)
Since for f (0) = 0 we have f 5i ≤ 0 and 0 ≤ r(y i + f 5 ) ≤ r(y i ), for i = 1, . . . , n y , we thus obtain φ(y

+ f 5 ) = ∆y with ∆ ∈ D = {∆ ∈ D ny |∆ i,i ∈ [0, 1]}.
From the wellposedness assumption (see Appendix A), we have that (I -F 4 ∆) is invertible for all ∆ ∈ D, thus using [START_REF] Lathuilière | Periodic orbits in planar linear systems with input saturation[END_REF] we obtain y = (I -

F 4 ∆) -1 F 3 x and φ(y) = φ(y + f 5 ) = ∆y = ∆(I -F 4 ∆) -1 F 3 x, yield- ing φ(y(x)) ≤ σ x , with σ = max ∆∈D ∆(I -F 4 ∆) -1 F 3 . Hence, from (17)
, it follows that

V (x) ≤ P 1 x 2 + 2 P 2 x φ + P 3 φ 2 ≤ P 1 + 2σ P 2 + σ 2 P 3 x 2 = 2 x 2 . ( 19
)
The theorem below presents conditions for the global exponential stability of the origin of (2) using (17) as a Lyapunov function candidate.

Theorem 1 If there exist matrices P ∈ S (n+ny) , T ∈ D ny , R ∈ R (1+n+2ny)×ny , M ∈ P (1+2ny)×(1+2ny) and a positive scalar 1 such that

(V (x) -1 x x) + s 1 (T, y) + s 2 (R, χ, y) -s 3 (M, y) ≥ 0 (20)
and matrices T ∈ D 2ny , R ∈ R 1+n+4ny×2ny , M ∈ P (1+4ny)×(1+4ny) and a scalar η ∈ (0, 1) such that

-(V (x + ) -(1 -η)V (x)) + s 1 ( T , ỹ) + s 2 ( R, χ, ỹ) -s 3 ( M , ỹ) ≥ 0 ( 21
)
with ỹ = y y + and χ = 1 x φ(ỹ) φ(-ỹ) then the origin of (2) is globally exponentially stable.

Proof. From Proposition 1 and ( 19), if [START_REF] Rubagotti | A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains[END_REF] and [START_REF] Sontag | Nonlinear regulation: the piecewise linear approach[END_REF] hold it respectively follows that

1 x 2 ≤ V (x) ≤ 2 x 2 (22a) V (x + ) ≤ (1 -η)V (x). ( 22b
)
Thus, [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] allows to conclude that x(k) ≤ ce δk x(0)

with c = ( 2 1 ) 1 2 , δ = ln( √ 1 -η), ∀x(0) ∈ R n . Moreover, (22a) implies that V (x) is radially unbounded.
The generalized quadratic form involving the the state and a nonlinearity as in [START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF] has been studied in the context of stability analysis of continuous-time linear complementarity systems [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF]. Here, the generic formulation presented in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] is used considering ramp functions, that also satisfy complementarity condition (16a) as discussed in Remark 2. In that paper the authors suggest that their stability conditions could benefit from a numerical formulation exploiting co-positivity conditions. The co-positivity is here accounted for by considering the inequalities of Lemma 3 in Theorem 1. The numerical formulation we propose is detailed in the next section.

LMI conditions

The relations ( 20) and ( 21) can be written in the generic quadratic form given by ( 11)- [START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF], where the corresponding matrices H present an affine dependence on the elements of matrix P . Hence, conditions in LMI form can be obtained to ensure [START_REF] Rubagotti | A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains[END_REF] and [START_REF] Sontag | Nonlinear regulation: the piecewise linear approach[END_REF]. This is formalized in the following Corollary to Theorem 1.

Corollary 1 If there exist matrices P 1 ∈ S n , P 2 ∈ R n×ny , P 3 ∈ S ny , T ∈ D ny , T ∈ D 2ny , symmetric matrices M ∈ P (1+2ny)×(1+2ny) and M ∈ P (1+4ny)×(1+4ny) , matrices R ∈ R (1+n+2ny)×ny and R ∈ R (1+n+4ny)×2ny and positive scalars η < 1 and 1 such that the following LMIs are verified

H + 1 2 He{Ψ + RΞ} -I M I ≥ 0 (23a) H + 1 2 He{ Ψ + RΞ } -Ĩ M Ĩ ≥ 0 (23b)
where

H =        0 0 0 0 0 P 1 -1 I n P 2 0 0 P 2 P 3 0 0 0 0 0        , I =        1 0 0 0 0 0 0 I ny 0 0 0 I ny        , Ψ =        0 0 0 0 0 0 0 0 0 0 0 T 0 0 0 0        , Ξ =        f 5 F 3 F 4 -I ny I ny        , H = -        0 0 0 0 0 N 1 N 2 0 0 N 2 N 3 0 0 0 0 0        , Ĩ =        1 0 0 0 0 0 0 I 2ny 0 0 0 I 2ny        , Ψ =        0 0 0 0 0 0 0 0 0 0 0 T 0 0 0 0        , Ξ =        f 5 F 3 F 4 -I 2ny I 2ny        with F3 = F 3 F 3 F 1 , F4 = F 4 0 F 3 F 2 F 4 , f5 = f 5 f 5 , N 1 = F 1 P 1 F 1 -(1 -η)P 1 , N 2 = F 1 P 1 F 2 -(1 -η)P 2 F 1 P 2 , N 3 = F 2 P 1 F 2 -(1 -η)P 3 F 2 P 2 P 2 F 2 P 3
.

then the origin of (2) is globally exponentially stable.

Proof. Consider V (x) defined as in [START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF]. We show now that LMIs (23a) and (23b) are equivalent to the satisfaction of conditions ( 20) and ( 21) in Theorem 1.

Recalling that χ = 1 x φ(y) φ(-y) , it follows that χ Hχ = V (x) -1 x x, χ Ψχ = s 1 (T, y) and χ I M Iχ = s 3 (M, y). Moreover, taking into account (3b), one has that that χ RΞχ = s 2 (R, χ, y). Hence, by left and right multiplying (23a) respectively by χ and χ, it follows that (20) holds.

Considering now χ = 1 x φ(ỹ) φ(-ỹ) , with ỹ = y y + , it follows that χ H χ = -(V (x + ) -

(1 -η)V (x))
. Furthermore, as y + = F 3 x + + F 4 φ(y + ) + f 5 = F 3 (F 1 x + F 2 φ(y)) + F 4 φ(y + ) + f 5 , one has ỹ = F3 x + F4 φ(ỹ) + f 5 and it follows that χ Ψ χ = s 1 ( T , ỹ), χ Ĩ M Ĩ χ = s 3 ( M , ỹ) and χ RΞ χ = s 2 ( R, χ, ỹ). Thus by left and right multiplying (23b) respectively by χ and χ, it follows that (21) holds, which concludes the proof.

Remark 3 Differently from results in the literature considering the description of the polyhedral partitions by hyperplanes (e.g. [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF][START_REF] Feng | Stability analysis of piecewise discretetime linear systems[END_REF][START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF]) or by vertices and cone rays (e.g [START_REF] Iervolino | Conecopositive piecewise quadratic Lyapunov functions for conewise linear systems[END_REF][START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF]), in our approach only two LMIs have to be tested to assess the stability of the PWA system. Note that in the aforementioned works, an LMI is associated to each possible state transition from a region Γ j to a region Γ i . These inequalities are required to enforce the strictly decrease of the LF. Moreover, for each region Γ j , an LMI constraint is needed to ensure the positivity of the piecewise quadratic Lyapunov function. Thanks to the proposed representation, all possible transitions are implicitly taken into account and no enumeration is needed.

Remark 4

The LMIs in Corollary 1 correspond to the expressions of the V and its one-step variation ∆V with the additional terms from Lemma 1 to Lemma 3 as in [START_REF] Rubagotti | A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains[END_REF] and [START_REF] Sontag | Nonlinear regulation: the piecewise linear approach[END_REF]. Note that, as detailed in Section 3 and discussed in Remark 2, the ramp function is specified when using the relations r(θ)r(-θ) = 0 from Lemma 1, θ -(r(θ) -r(-θ)) = 0 from Lemma 2, and r(θ) ≥ 0, r(-θ) ≥ 0 from Lemma 3. These Lemmas introduce respectively matrices (T , R and M ) and ( T , R and M ) in the LMIs in Corollary 1. Except for very particular cases, all these matrices need to be free variables whenever solving [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF]. This way, we are restricting the analysis of generalized quadratic forms described only by the vector valued ramp functions φ, and not another nonlinear function that satisfies a subset of the relations in Section 3. Moreover, we stress that matrices T and T , related to the complementarity condition detailed in Lemma 1 are sign-free variables. This is in contrast to the sector condition where a set of nonlinearities is described using the same expression as (5) however with an inequality and a positive semi-definite matrix T (without the terms from Lemmas 2 and 3). Whether the sign of the terms in these T and T matrices from the solutions to ( 23) is positive or negative will depend on the problem data (mainly on the matrix F 4 ).

Similar LMIs, without matrix R as in Corollary 1, can be obtained if the algebraic relation θ -(r(θ) -r(-θ)) = 0 in Lemma 2 is used in both Lemma 1, and Lemma 3.

Namely by using r(-θ) = θ -r(θ) thus allowing to replace χ by vector 1 x φ(y) . On the other hand, the formulation of Corollary 1 avoids products between matrices T and M and the matrices F 3 , F 4 and f 5 defining y. This feature will be instrumental to study polytopic uncertain system with parameter dependent multipliers T and M .

Numerical Examples

In this section, we illustrate the results of Theorem 1 with four numerical examples. In the first, we demonstrate the global stability (of the origin) of a generic piecewise linear system. In the second one, we analyze the global stability of a linear system subject to actuator saturation. A third example treats a benchmark example borrowed from the explicit MPC literature. In all examples we have used the values for parameters η = 10 -3 and 1 = 10 -5 .

Example I. Consider a piecewise linear system given by (2) with f (x) described by the implicit representation (3) with

F 1 = 0.5 0.1 + κ -1 0.5 , F 2 = κ 1 1 0 0 F 3 = -1 -1 1 -1 , F 4 = 0 -2 3 -1 0 , f 5 = 0 0 .
The corresponding partition of R 2 defined from equation (3b) is given by the following sets

Γ 1 = {x ∈ R 2 |φ 1 (y(x)) = 0, φ 2 (y(x)) = 0} = {x ∈ R 2 | -x 1 ≤ x 2 ; x 1 ≤ x 2 }, Γ 2 = {x ∈ R 2 |φ 1 (y(x)) ≥ 0, φ 2 (y(x)) = 0} = {x ∈ R 2 |x 1 ≤ 0; x 2 ≤ -x 1 }, Γ 3 = {x ∈ R 2 |φ 1 (y(x)) = 0, φ 2 (y(x)) ≥ 0} = {x ∈ R 2 |x 2 ≤ x 1 ; x 2 ≥ -5x 1 }, Γ 4 = {x ∈ R 2 |φ 1 (y(x)) ≥ 0, φ 2 (y(x)) ≥ 0} = {x ∈ R 2 |0 ≤ x 1 ; x 2 ≤ -5x 1 }.
and is depicted in Figure 2. The sets in the partition do not satisfy Γ i ∩ Γ j = ∅ for i = j. However, f is uniquely defined since the functions are continuous and coincide on the boundary of the sets.

x 1 x 2 Γ 1 Γ 4 Γ 3 Γ 2 Fig. 2. Example I -partition of R 2
Note that a corresponding explicit representation for f (x) as in (1) can be obtained as follows:

x + = A j x = (F 1 + F 2 ∆ j (I -F 4 ∆ j ) -1 F 3 )x, if x ∈ Γ j , for j = 1, . . . , 4 with ∆ 1 = diag((0, 0)), ∆ 2 = diag((1, 0)) ∆ 3 = diag((0, 1)), ∆ 4 = diag((1, 1)).
Applying the conditions of Theorem 1 through the LMI formulations in Corollary 1, we can show that the origin of the system is globally stable for κ = 0.699, and ( 17) is a Lyapunov function for the system with

P =       
2.2172 -0.0151 -0.4494 0.0094 -0.0151 1.6462 0.0094 0.3570 -0.4494 0.0094 -1.2060 -0.8242 0.0094 0.3570 -0.8242 -0.4758

       .
Note that the matrix P is not positive definite. Indeed the positive definiteness of matrix P is not imposed by the conditions in Theorem 1. However, since (20) holds we have that the Lyapunov function is guaranteed to be positive definite. Some trajectories of the system are shown in Figure 3, along with the level sets of the decreasing Lyapunov function. For comparison, the dual problem presented in [7, Section II] demonstrate that there does not exist a quadratic Lyapunov function, that is V (x) = x P 1 x, with P 1 ∈ R n×n , that certifies the stability for κ ≥ 0.357, and through simulation, we find that the origin of the system is stable for -0.35 < κ < 0.7. It should also be pointed out that with the method proposed in [START_REF] Feng | Stability analysis of piecewise discretetime linear systems[END_REF], using a piecewise quadratic Lyapunov function, it is not possible to certify the stability of the system for κ ≥ 0.51 (considering known all the admissible transitions between regions), which shows that our conditions lead to less conservative results.

As pointed out in Remark 4, testing the LMIs [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF] for this example without all variables, for instance by imposing R = 0 and R = 0 resulted in no feasible solution for any value of κ. This highlights the importance of the term introduced by s 2 in Lemma 2.

Example II. Consider the following system taken from [START_REF] Drummond | Generalized absolute stability using Lyapunov functions with relaxed positivity conditions[END_REF], discretized with a sampling period of 100ms, and subject to asymmetric actuator saturation

x + = Ax + Bsat [-1,15] (Kx) (24)
with A = 0.9464 0.0957 -0.9568 0.9033 , B = 0.0049 0.0959 and K = 9.9000 0.4950 .

First, note that the saturation function is a piecewise affine function defined as follows:

sat [-1,15] (Kx) =        -1 if Kx ≤ -1 Kx if -1 ≤ Kx ≤ 15 15 if Kx ≥ 15. (25)
From (25), it folllows that (24) can be cast as a PWA system (2) with an implicit representation (3) defined with:

F 1 = A + BK, F 2 = -B B F 3 = K -K , F 4 = 0, f 5 = -15 -1
leading to the following partition of R 2 in terms of φ

Γ 1 = {x ∈ R 2 |φ 2 (y(x)) ≥ 0} = {x ∈ R 2 |Kx ≤ -1}, Γ 2 = {x ∈ R 2 |φ 1 (y(x)) = φ 2 (y(x)) = 0} = {x ∈ R 2 | -1 ≤ Kx ≤ 15}, Γ 3 = {x ∈ R 2 |φ 1 (y(x)) ≥ 0} = {x ∈ R 2 |15 ≤ Kx}
which is depicted in Figure 4.

It can be shown (see [7, Section II]) that there does not exist a common quadratic Lyapunov function for the

x 1 x 2 Γ 2 Γ 1 Γ 3 Fig. 4. Example II -partition of R 2 .
linear systems defined by A and (A + BK). Since the quadratic global stability of a linear system subject to a saturating linear state feedback imposes the existence of a common Lyapunov function for the open-loop system and the closed-loop system without saturation, we conclude that there is no quadratic function to assess the global stability of the origin of system [START_REF] Burgat | Stability and control of saturated linear systems[END_REF]. However, considering a piecewise quadratic Lyapunov function as in [START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF] and applying Theorem 1, we can certify that the origin is globally exponentially stable with This matrix was obtained from the solution to the LMIs described in Corollary 1.

P =        0.
In Figure 5, a trajectory of the system and the level sets of the decreasing Lyapunov function are depicted. , with u given by the explicit MPC law computed in [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF] leading to the explicit PWA representation in Table 1.

Region i Control ui       
-5.9220 -6.8883

5.9229 6.8883 -1.5379 6.8296

1.5379 -6.8296

       x ≤        2
    x ≤     2.6341 -0.0353 -0.0267     -2
Table 1 Explicit MPC law: inequalities defining the sets of the partition and the corresponding affine control law.

The closed-loop system can therefore be described by a PWA system (2), with f (x) in (3) given by the following matrices:

F 1 = A + BK 1 , F 2 = B 1 -1 1 -1 φ(y) F 3 =        K 2 -K 1 K 1 -K 2 -K 1 K 1        , F 4 =        0 0 0 0 1 0 0 0 -1 1 0 0 1 -1 1 0        , f T 5 = -0.6423 -0.6423 -2 -2 K 1 = -5
.9220 -6.8883 , K 2 = -6.4159 -4.6953 .

By applying Theorem 1, we can find a quadratic Lyapunov function that certifies the global stability of the origin. A quadratic function is a particular case of the generic form [START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF], in which we consider P 2 = 0 and P 3 = 0, in this case V (x) = x P 1 x with P 1 = 0.9262 0.4674 0.4674 1.0815

.

A trajectory and the level sets of the obtained Lyapunov function are shown in Figure 6. As observed in Remark 3, since in this case there are 7 sets in the partition, if we consider the results in references [START_REF] Feng | Stability analysis of piecewise discretetime linear systems[END_REF][START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF], which are based on the explicit representation (1), we have to test 7 2 = 49 LMIs, regarding all the possible transitions between the sets of the partition, plus 7 LMIs to ensure the positivity of the PWQ Lypaunov function on each set. This number of LMIs can be reduced if we perform a reachability analysis, that requires the solution of 49 linear programming feasibility problems. In contrast, our conditions requires only the solution of two LMIs. It can also be shown that the number of variables is smaller with our approach.

Conclusion and future work

We have presented a new framework for the stability analysis of discrete-time PWA systems. To this end we introduced a novel implicit representation of PWA functions based on the use of ramp functions. By exploiting some properties of ramp functions as a set of identities and inequalities, we obtain Lyapunov inequalities related to piecewise quadratic Lyapunov functions candidates. These inequalities are then expressed as LMIs.

In the proposed framework: a) there is no need of defining the quadratic function associated to each set of the partition since this is implicitly obtained with a generalized quadratic form; b) there is no need to enumerate possible transitions between regions; c) the conditions for stability are cast in two LMIs and can be efficiently tested with standard optimization packages; d) the use of properties associated to ramp functions applies only this class of function and therefore are less conservative than generic sector bounded conditions; e) from the novel proposed representation, the matrices describing the partition of the system appear affinely on the stability LMI conditions, which allows to directly consider uncertainties in the partition.

Future work includes the extension of the framework to study local (regional) stability, including efficient ways of estimating the region of attraction of the origin of PWA systems, and the synthesis of stabilizing feedback control laws also in PWA form. The proposition of stability conditions for continuous-time system is also being studied.

A Conditions for well-posedness

Note that in Example II above, the solution to equation (3b) is explicit since F 4 = 0, giving y = F 3 x + f 5 . It is then straightforward to compute f (x) using the value of y. Explicit solutions can also be obtained in case matrix F 4 is structured, for instance for a strictly lower triangular structure as in Example III.

In general, with F 4 = 0, as in Example I above, (3b) becomes an implicit equation and the existence of a unique solution y for all x ∈ R n must be ensured. With this aim, below we provide a condition for the well-posedness of (3b), that is, the existence and uniqueness of solutions to f (y(x)) = F 3 x + f 5 ∀x ∈ R n , (A.1) with f (y(x)) = y(x) -F 4 φ(y(x)). In [24, Proposition 2] it is shown that for a locally Lipschitz function f (y) such that the Jacobian satisfies J y f (y) ∈ M ⊂ R ny×ny for almost all y ∈ R ny , where M is a compact, convex set, with each of its elements being non-singular, there exists a unique globally Lipschitz function y(ξ) satisfying f (y(ξ)) = ξ. Such a result is used in [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF] to obtain a condition for the well-posedness of an algebraic loop involving saturation and deadzone functions.

Using the definition of the ramp function in (4), we have that the Jacobian f (y) in (A.1) with respect to y of is given by J y f (y) = (I -F 4 ∆) with ∆ ∈ D = {∆ ∈ D ny |∆ (i,i) ∈ [0, 1]}, which is a compact and convex set of matrices. Thus, following [24, Proposition 2] a unique solution to (A.1) exists if (I -F 4 ∆) is non-singular for all ∆ ∈ D. A condition for the well-posedness is an LMI (see [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF][START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF]) as in the proposition below. For implementation purposes, for instance when the PWA function has to be computed to generate a control

2 Fig. 3 .

 23 Fig. 3. System trajectories and Lyapunov function level sets for Example I.

2 Fig. 5 .

 25 Fig. 5. System trajectory and Lyapunov function level sets for Example II.

Fig. 6 .

 6 Fig. 6. System trajectory and Lyapunov function level sets for Example III.

Proposition 2 ([ 24 ,

 224 Proposition 1]) If there exist a matrix W ∈ D ny , W > 0 such that -2W +W F 4 +F 4 W < 0 then (I -F 4 ∆) is non-singular ∀∆ ∈ D.

input, a well posed equation (A.1) can be solved from the solution of a Linear Complementarity Problem [START_REF] Cottle | The Linear Complementarity Problem[END_REF].

Note that from ( 16) and (4), we have

. . , n y . Setting now ξ = F 3 x+f 5 in equation (3b), and using y i = (F 4 φ + ξ) i in the above expressions one obtains respectively