Valentin Berry
email: valentin.berry@univ-poitiers.fr

P Leblanc

S Holé

T Paillat

Space charge measurement at solid/liquid interface by PWP method

Keywords: Space Charges, PWP method, Electrical Double Layer, Interface

The presence of electric charges within liquid is relatively unexplored despite its role in electrochemical energy storage and industrial safety. Quantifying charge distribution in liquids and at solid/liquid interfaces is therefore crucial to the efficiency and longevity of devices. Over time, metrologies using stimulus to measure space charges in solids have been transposed to liquids. The stimulus disturbs electrostatic equilibrium to induces electrical response. This study builds upon measuring electric charges in the Electrical Double Layer (EDL) using the Pressure-Wave-Propagation (PWP) method. It presents how the pressure wave amplitude, the liquid conductivity and the liquid nature impact the electrical response.

I. INTRODUCTION

When a solid and a liquid come into contact, poorly understood electrochemical phenomena electrically polarize the interface to form a charge distribution known as the electrical double layer (EDL). This distribution is generally divided into two layers of charge: a first layer located within the solid and a second layer of opposite polarity located inside the liquid, while maintaining electroneutrality. Several models have been developed to describe the EDL, including the Stern model [START_REF] Stern | Zur Theorie Der Elektrolytischen Doppelschicht[END_REF], which remains one of the most commonly accepted. This model is based on a charge creation mechanism through an adsorption process related to the chemical nature of solid and liquid. Even a few parts per million of impurities in the liquid are sufficient to significantly alter the properties of the EDL.

In the field of electrical engineering, particularly for power components such as power transformers cooled by a flow of a dielectric liquid, the EDL is a major concern for industrial professionals as it can lead to electrostatic hazard. Indeed, when the liquid is set in motion, it carries one part of the electric charges present in the liquid and disrupts the electrical double layer balance. To restore its electrostatic equilibrium, new charges are generated at the interface, which can lead to localized increases in electrical potential, such as in the case of metallic conductors or dielectrics. In some cases, these local potential increases can reach the dielectric breakdown of the material which is at risk for the equipment.

Quantifying and determining the distribution of charges within liquids and at solid/liquid interfaces are therefore essential to prevent such risks.

In the field of energy, especially in energy storage, both supercapacitors and electrochemical batteries play a key role in facilitating the integration of intermittent renewable energy sources. These systems operate based on the transfer of electrical charges at solid-liquid interfaces and on the Electric Double Layer (EDL). The liquids employed are conductive fluids or electrolytes. A thorough comprehension of the organization of electric charge distribution within the liquid is equally crucial to optimize and ensure the reliability of these devices.

Various experimental techniques have been used to study the EDL, such as flow electrification [START_REF] Moreau | Space charge density in dielectric and conductive liquids flowing through a glass pipe[END_REF][START_REF] Paillat | Space charge density at the wall in the case of heptane flowing through an insulating pipe[END_REF] or using the Kerr effect [START_REF] Shi | Space charge dynamics investigation based on Kerr electro-optic measurements and processing of CCD images[END_REF]. However, these techniques indirectly provide information about the charge distribution of the EDL through modeling that relies on various unverified assumptions. Recently, methods used for measuring space charges in solids have been adapted for charge measurement in liquids, such as the thermal step method (TSM) [START_REF] Sidambarompoule | A Non-destructive Thermal Stimulus Method as a Tool for studying the Electrical Double Layer[END_REF], the pulsed electro-acoustic (PEA) method [START_REF] Wu | Space charge behavior in the sample with two layers of oilimmersed-paper and oil[END_REF] and the pressure wave propagation (PWP) method [START_REF] Ndour | Direct observation of electric charges at solid/liquid interfaces with the pressure-wave-propagation method[END_REF]. This work builds upon previous research conducted on PWP method [START_REF] Ndour | Direct observation of electric charges at solid/liquid interfaces with the pressure-wave-propagation method[END_REF][START_REF] Lewiner | Pressure wave propagation methods: a rich history and a bright future[END_REF]. Its objective is to present the latest advancements in measurement techniques and the analysis of electrical measurement signals for interfaces between conductive solids and more and less conductive liquids.

II. THEORETICAL APPROACH

A. Electrical double layer description

The electrical double layer can be likened to an electrical capacitor, with the solid acting as one electrode and the liquid as the other electrode. According to Stern (Fig. 1), the charges in the solid are uniformly distributed on the first atomic layers, while the charges in the liquid undergo the diffusion and are distributed deeper into the liquid. At equilibrium, the space charge density is highest at the interface (ρ w ) and decreases exponentially towards the liquid bulk until it becomes zero. The thickness of this charge distribution is conventionally referred to as the Debye length δ 0 (1) [START_REF] Touchard | Flow electrification of liquids[END_REF], which is determined only by the electrical properties of the liquid including the permittivity ε, the conductivity σ and the average diffusion coefficient of charged species in the liquid D 0 . According to this model, 95% of the charges in the liquid are located near to the interface in a thickness of 3δ 0 . For conductive liquids (aqueous solutions), ρ w is approximately equal to 10 4 C. m -3 with δ 0 of a few angstroms, while for dielectric liquids (mineral oils, heptane), its order of magnitude is around 10 -4 C. m -3 with δ 0 that can reach millimeters [START_REF] Paillat | Space charge density at the wall in the case of heptane flowing through an insulating pipe[END_REF].

δ 0 = ( εD 0 σ ) 1 2
(1)

B. PWP method

The Pressure Wave Propagation (PWP) method was developed by Laurenceau in 1976 for measuring space charges in solids [START_REF] Laurençeau | New Principle for the Determination of Potential Distributions in Dielectrics[END_REF]. It involves generating a planar pressure wave that propagates through the studied material, destabilizing the electrostatic equilibrium of charges and inducing a measurable electrical response (current or voltage) through a set of electrodes (Fig. 2) [START_REF] Cheng | A Piezo-PWP method for charge measurement of the cell suspension[END_REF][START_REF] Takada | Acoustic and optical methods for measuring electric charge distributions in dielectrics[END_REF]. The pressure wave can be generated either by a laser pulse (Laser-Induced Pressure Pulse, LIPP) [START_REF] Alquie | Stress-Wave Probing of Electric Field Distributions in Dielectrics[END_REF] or by a piezoelectric actuator (Piezoelectrically-Induced Pressure Pulse, PIPP) [START_REF] Eisenmenger | Observation of charge compensated polarization zones in polyvinylindenfluoride (PVDF) films by piezoelectric acoustic step-wave response[END_REF].

In planar geometry, the generated current response i PWP (t) (2) [START_REF] Holé | Influence of divergent electric fields on space-charge distribution measurements by elastic methods[END_REF] depends on the properties of the pressure stimulus P(x, t) through the strain wave S(x, t) , the Fig. 2. Principe of the PWP method.

capacitance of the excited part of the sample C 0 , the physical properties of the liquid such as permittivity ε, electrostrictive coefficients a, and, the charge distribution ρ(x) in the medium through the distribution of the electric field E(x) via Maxwell-Gauss equation [START_REF] Paillat | Space charge density at the wall in the case of heptane flowing through an insulating pipe[END_REF].

i PWP (t) = -C 0 ∫ (1 - a ε ) E(x) ∂S(x, t) ∂t dx (2) d(ε(x)E(x)) dx = ρ(x) (3)
The strain S(x, t) is the derivative over space of the matter displacement of position x and time t. As the derivative over time of the displacement is the speed, it is interesting to replace ∂S (x, t) ∂ ⁄ t in (2) by ∂V (x, t) ∂ ⁄ x where V(x, t) is the speed of which the matter displaces in other words. Using the Maxwell-Gauss equation (3) and assuming a uniform material, the i PWP (t) current is defined as a function of the charge distribution ρ(x) by (4).

i PWP (t) = C 0 ε (1 - a ε ) ∫ ρ(x)V(x, t)dx (4)

III. PRINCIPLE OF PWP MEASUREMENT IN LIQUIDS

A. Experimental setup

The measurement cell (Fig. 3) comprises an electrical control unit coupled with a high DC voltage power supply (Stanford PS35) that supplies the level of the excitation voltage V ex . This configuration enables the generation of a rapid voltage drop (Fig. 4) across a piezoelectric actuator resulting in the induction of a planar pressure pulse. The electrical control unit consists of a gate driver ICs for highvoltage transistor GaN power switches (CoolGaN TM ) driven in a hard-switched mode. The GaN used enables ultra-fast switching for the excitation voltage V ex up to 600 V. The print circuit board have been designed to ensure the good noise immunity and minimize the parasitic inductances at high frequency. The piezoelectric transducer is made of PZ24 ceramic [START_REF]Piezoelectric ceramic Pz[END_REF], and has a circular shape with a diameter of 1.5 cm. With a thickness of 200 m, it is capable of generating short-duration pressure pulses, leading to improved measurement resolution, while maintaining mechanical durability. Figures 4 and5, which are the result of averaging over 1000 consecutive excitations at a frequency of 100 Hz, illustrate the electrical excitation V ex generated by the electronic board for three different levels of excitation voltage, along with the corresponding induced pressure waves. The control voltage transitions from V ex to 0 V in 30 ns, with a constant fall time t ex independent of the amplitude (in the voltage range studied the variation of t ex is smaller than 3 ns due to the variation of temperature of electronic components). Because the variation of the voltage applied on the piezoelectric actuator is decreasing, this actuator generates negative pressure wave. This pressure wave (Fig. 5) exhibits a relatively symmetrical shape during its decay and growth phases, with a slight rebound observed at the end. This bounce is probably attributed to oscillations around 0 V in the applied voltage to the piezoelectric actuator (Fig. 4). The amplitude of the wave increases proportionally with the excitation level, reaching 6 MPa, while its duration  remains completely independent of the excitation level as confirmed by dimensionless study (Fig. 6). Calculated at 50 % of the maximum pressure value, the duration of the pressure wave is 34.88 ns. S. Holé [START_REF] Holé | Single transducer generation of unipolar pressure waves[END_REF], who pioneered this principle of stimulus generation for measurement in solids, theoretically established the shape of the pressure wave based on the electrical excitation and verified its consistency with experimental measurements. It was also demonstrated that the measurement resolution is linked to the propagation velocity of the pressure wave v and its bandwidth (f max ) according to Equation ( 5) [START_REF] Dagher | A preliminary study of space charge distribution measurements at nanometer spatial resolution[END_REF].

R = v 2f max (5)
According to S. Holé [START_REF] Hole | Resolution of direct space charge distribution measurement methods[END_REF], two types of resolutions should be considered. The positioning R p and discerning R D resolutions. The first emphasizes the positioning of the charge within the sample, while the second reflects the ability to discriminate between two separate charges. For an elastic pressure wave, they are given respectively by:

R p = 0.98vτ SNR (6) R D = 1.68vτ √SNR ( 7
)
where SNR is the signal-to-noise ratio. In this paper, the resolution calculated is R D which is similar to R keeping SNR to the one of measurements.

The generated pressure wave propagates through the aluminum waveguide (3), which also serves as the excitation electrode, then into the liquid volume (4), and finally into the brass measurement electrode [START_REF] Sidambarompoule | A Non-destructive Thermal Stimulus Method as a Tool for studying the Electrical Double Layer[END_REF]. The voltage signal V PWP (t) obtained between the two electrodes is amplified by 60 dB (Analog Modules 322-12-50) and is digitized using a Tektronix® MSO64B oscilloscope. The voltage signal is directly related to the current i PWP (t) through the 50 Ω input impedance of the amplifier. The electrical excitation V ex can be adjusted in amplitude from 30 to 200 V and modulated in frequency up to 1 kHz. The interelectrode spacing (d) can be adjusted from 0 to 3 mm and is controlled by a micrometer screw within a 1 µm uncertainty. Fig. 6. Normalized pressure wave profile generated for the three levels of excitation in Figure 4 (normalization established with the maximum value of the pressure wave).

B. Electric properties of the different liquids

Different liquids with varying levels of conductivity have been examined. They are either of different chemical nature in their pure state (water, glycerol and ethanol), or modified by the addition of additives. KOH 0.001M corresponds to a base of water with 0.001M of potassium hydroxide (KOH). In the case of pure glycerol, a saltwater solution with 32 ppm of NaCl is added at different concentrations to vary the conductivity of the mixture. The conductivity and permittivity properties of each solution were measured using impedance spectroscopy (AMETEK® Modulab XM CHAS 08) and summarized in Table I. To conduct the studies when the electrical double layer is at the equilibrium, fully developed [START_REF] Maatschappij | Electrostatics in the Petroleum Industry: The Prevention of Explosion Hazards[END_REF], the relaxation time of electric charges in the liquid is determined based on the liquid properties:

τ r = ε σ (8)
For the estimation of the Debye length (1), the diffusion coefficient D 0 has also been measured for various liquids using impedance spectroscopy [START_REF] Sidambarompoulé | Study of electrical properties and estimation of average mobility and diffusion coefficients in several insulating liquids by dielectric spectroscopy[END_REF] and compared to bibliography data [START_REF] Hills | Diffusion coefficients in ethanol and in water at 298K: Linear free energy relationships[END_REF][START_REF] Agmon | The Grotthuss mechanism[END_REF][START_REF] Ternström | Mutual Diffusion Coefficients of Water + Ethylene Glycol and Water + Glycerol Mixtures[END_REF].

C. Typical recording of the signal

Before conducting the measurements, empty tests (without liquid) were performed to qualify the cell. Regardless of the inter-electrode spacing and excitation voltage level, the measured signal present only a background noise below 0.1 mV.

The measured signals are the result of averaging over 1000 consecutive acquisitions at a frequency of 100 Hz. This frequency is sufficiently low to allow the EDL to reequilibrate between two successive excitations, considering the relaxation time (Table I). Each measurement is repeated 5 times for an estimated reproducibility error of less than 8 %. Figure 7 shows a typical recording of the electrical voltage signal V PWP (t) obtained in the cell for ethanol when V ex = 110V. It is characterized by two sets of transients. Fig. 7. Electrical response obtained at aluminum/Gly0 and at Gly0/brass interfaces for V ex = 110V.

The first consists of a negative peak followed by a positive peak, appearing 4.75 µs after the excitation of the piezoelectric element. This represents the electrical response induced by the first interface once the pressure pulse has traveled through the waveguide. The second part of the signal occurs 600 nanoseconds later and corresponds to the second interface, once the pressure has crossed the sample. This time, it consists of a positive peak followed by a negative peak. This sign inversion in the electrical responses is consistent. Indeed, the first interface is excited by the pressure wave from the solid (aluminum) to the liquid (glycerol), while the second interface is excited from the liquid to the solid (brass). The differences in amplitude between the two signatures are likely due to the differences in nature between the two interfaces as well as the pressure wave is partially reflected at the back electrode. The time interval between these two sets of signals precisely corresponds to the propagation velocity of the pressure wave in glycerol ( v = 1998m. s -1 ) through the 1 mm inter-electrode spacing.

IV. RESULTS AND DISCUSSIONS

In the following sections of this paper, the analysis of the electrical response is exclusively focused on the transient of the first interface (solid/liquid). At this interface, the electrical signal is probably not disturbed by any reflections of the pressure wave. The analysis concedes the amplitude of the first and second peaks pic (V peak-1 and V peak-2 ) as well as the duration of the signal signature t sig obtained between 5 % of V peak-1 and 5 % of V peak-2 as shown on Fig. 7. The peaks following these two peaks are not considered since they are replicas. A. Variation of the pressure wave amplitude.

Figure 8 shows the variation of V PWP (t) as a function of the potential V ex for the Gly6 liquid at 20°C. Increasing the voltage amplitude results in an increase in the stimulus without altering its temporal behavior. This observation is particularly evident when analyzing the amplitude of the peaks V peak-1 and V peak-2 , which appear to vary linearly and in the same proportions as V ex (Fig. 9). This suggests that when V ex increases linearly, both the pressure wave amplitude and the voltage V PWP (t) increase in the same way. t sig (Fig. 10) appears to be independent of the excitation voltage within the studied range, with a relatively significant measurement uncertainty (6 %) due to the noise in the signal. The behavior of the impact on the variation of pressure wave amplitude on the electrical response is confirmed by a numerical study [START_REF] Berry | Numerical study of the electric charge measurement by the Pulse Wave Propagation method: Application to the Electrical Double Layer[END_REF].

Furthermore, considering the shape of the pressure wave induced by the piezoelectric and the results of the numerical study [START_REF] Berry | Numerical study of the electric charge measurement by the Pulse Wave Propagation method: Application to the Electrical Double Layer[END_REF], this signature of two peaks in the signal, negative for the first and for positive the second, can only be attributed to an electrical double layer composed of negative charges in the solid and positive charges in the liquid. V ex (V)

V ex = 64 V 4, 4,6 4,8 5,0 5,2 -2 -1 0 1 2
V ex = 76 V 4, 6 4,8 5,0 5,2 -2 -1 0 1 2

B. Impact of the liquid conductivity

Figure 11 shows the temporal evolution of V PWP (t) for different conductivities at V ex = 110V . An increase in conductivity σ leads to an increase of the voltage amplitude without affecting significantly its temporal behavior.

Fig. 11. Electrical response at solid/liquid interface for « Gly0 », « Gly2 », « Gly4 » and « Gly6 » solutions for V ex = 110V.

The amplitudes of the peaks V peak-1 and V peak-2 exhibit an increasing exponential progression and are influenced in the same proportions (Fig. 12). The increase in conductivity results in a significant increase in the amplitudes of the different peaks, on the order of 175% between 10 -7 and 5 • 10 -6 S. m -1 . Above this range, the amplitudes of the peaks tend to approach a constant value. This overall behavior is likely due to the antagonistic effects of increasing conductivity [START_REF] Berry | Numerical study of the electric charge measurement by the Pulse Wave Propagation method: Application to the Electrical Double Layer[END_REF]. On one hand, increasing the liquid conductivity, increases the space charge density of the EDL [START_REF] Sidambarompoule | A Non-destructive Thermal Stimulus Method as a Tool for studying the Electrical Double Layer[END_REF] and promotes V PWP , while on the other hand, it reduces the thickness of the diffuse layer and tends to decreases V PWP [START_REF] Berry | Numerical study of the electric charge measurement by the Pulse Wave Propagation method: Application to the Electrical Double Layer[END_REF]. In the studied conductivity range, especially at low values, the increase in space charge density appears to dominate. Fig. 12. Amplitude of the first voltage peaks as a function of solution conductivity.

Furthermore, as long as the thickness of the diffuse layer is higher than the spatial resolution R D (Table I), t sig evolves according to δ 0 as predictable by numerical study [START_REF] Berry | Numerical study of the electric charge measurement by the Pulse Wave Propagation method: Application to the Electrical Double Layer[END_REF]. This behavior is visible on the first points in Fig. 13. On the last points, when the value of δ 0 is similar or smaller than the spatial resolution, t sig seems to be constant. Then, when the spatial resolution is similar or higher than δ 0 some loss of information distorts the measurement. C. Impact of the liquid nature.

Figure 14 illustrates the temporal evolution of V PWP (t) for four different liquids excited by a pressure wave generated for V ex = 75V with an inter-electrode spacing fixed at 1 mm. Regardless of the studied liquid, V PWP presents a specific temporal shape, with amplitude and signal duration unique to each liquid. Depending on the nature of the liquid, the signature of V PWP evolves. The first peak is positive, and the second peak is negative for KOH 0.001M and water, while it is the opposite for ethanol and glycerol.

This behavior can mainly be attributed to a polarization inversion at the solid/liquid interface and the electrochemical reactions which occur at the interface. According to (4) and because the pressure wave is negative the electric charge distribution would be negative in the solid and positive in the liquid for the aluminum/KOH and aluminum/water couples, and vice versa for the other couples.

In line with the previous findings, the signal amplitude is significantly influenced by the liquid's conductivity, increasing with it (Fig. 15). However, the signal duration is less sensitive to these variations (Fig. [START_REF]Piezoelectric ceramic Pz[END_REF][START_REF] Holé | Single transducer generation of unipolar pressure waves[END_REF] as the ratio between the extent of the pressure wave and δ 0 is similar for the different liquids. Regardless of the liquids studied, the thickness of the diffuse layer (3δ 0 ) remains well higher than the measurement resolution (Table 1).

Finally, the signal-to-noise ratio appears to be quite dependent on the liquid. However, given the current understanding of the measurement, no interpretation can be proposed at this stage. V. CONCLUSION This adaptation of space charge measurement using PWP method to solid/liquid interfaces appear particularly promising. For the range of conductive liquids used, a consistent signal has always been measured. It is specific to the interface properties, exhibiting different amplitudes and dynamics. All these measurements are limited by the sensitivity and resolution of the measurement. Sensitivity can be improved by increasing the excitation voltage. As for resolution, it is more delicate to enhance as it depends on the choice of the piezoelectric material and thickness, as well as the speed of voltage drop.

Fig. 1 .

 1 Fig. 1. Stern model description.

Fig. 3 .

 3 Fig. 3. Geometry of the measurement cell (1: Electrical control unit, 2: Piezoelectric actuator, 3: Waveguide electrode, 4: Liquid volume, 5: Measurement electrode).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Applied voltage on piezoelectric actuator for different excitation voltages V ex .

Fig. 8 .

 8 Fig. 8. Electrical signal at the first interface for different values of V ex (64, 76, 102.5 and 110 V).

Fig. 9 .Fig. 10 .

 910 Fig. 9. Amplitude of the first voltage peaks as a function of V ex .

Fig. 13 .

 13 Fig. 13. Duration of the signal at the first interface as a function of solution conductivity.

Fig. 14 .

 14 Fig. 14. Electrical response at aluminum and different liquid interfaces (KOH solution, Water, Glycerol, Ethanol) at V ex = 75V.

Fig. 15 .

 15 Fig. 15. Absolute value of the two first peaks of the electrical response at the aluminum/liquid interface for the different solutions studied at V ex = 75V.

Fig. 16 .

 16 Fig. 16. Duration t sig of the signal at the first interface as a function of the different liquids studied.

Fig. 17 .

 17 Fig. 17. Spatial extent of the signal at the first interface as a function of the different liquids studied.

Table I

 I Electrical properties of the different liquids. The « Gly0 » solution corresponds to pure glycerol, while the others are glycerol-based additive solutions.

	Sol.	ε r	σ (S.m -1 )	v (ms -1 )	D 0 (m 2 s -1 )	δ 0 (µm)	τ r (µs)	R D (µm)
	KOH 0.001M	75.09 1.83•10 -2 1.55•10 3 1.64•10 -2 24.43	0.04	9.79
	Water	102.07 4.11•10 -4 1.45•10 3 1.70•10 -4 19.34	2.20	20.26
	Ethanol	27.53 7.81•10 -6 1.54•10 3 4.76•10 -7 14.86	31.19	19.89
	Gly0	53,24 6,54•10 -7 1.99•10 3 3.95•10 -6 53.25 721.02	31.91
	Gly1	51,69 7,54•10 -7 1.97•10 3 4.73•10 -6 53.52 606.56	30.69
	Gly2	53,85 9,74•10 -7 1.94•10 3 3.43•10 -6 40.92 489.33	27.19
	Gly3	52,64 9,93•10 -7 1.95•10 3 1.79•10 -5 91.71 469.32	25.24
	Gly4	53,30 1,71•10 -6 1.96•10 3 5.67•10 -6 39.57 276.27	25.42
	Gly5	54.59 2.52•10 -6 1.96•10 3 7.13•10 -6 36.95 191.36	23.75
	Gly6	59.72 1.55•10 -5 1.98•10 3 2.84•10 -4 98.48	34.15	20.39

ACKNOWLEDGEMENTS

This work is supported by the "Investissement d'Avenir" program (LABEX INTERACTIF, reference ANR-11 LABX-0017-01), "Region Nouvelle-Aquitaine" and the PLATABAT project (2022-23222510).


Matteo Della
email: matteo.dellarossa@uclouvain.be

Thiago Alves Lima
email: thiago.alves-lima@univ-lorraine.fr

Marc Jungers
email: marc.jungers@univ-lorraine.fr

Raphaël M Jungers
email: raphael.jungers@uclouvain.be

Graph-Based Conditions for Feedback Stabilization of Switched and LPV Systems *

This paper presents novel stabilizability conditions for switched linear systems with arbitrary and uncontrollable underlying switching signals. We distinguish and study two particular settings: i) the robust case, in which the active mode is completely unknown and unobservable, and ii) the modedependent case, in which the controller depends on the current active switching mode. The technical developments are based on graph-theory tools, relying in particular on the path-complete Lyapunov functions framework. The main idea is to use directed and labeled graphs to encode Lyapunov inequalities to design robust and mode-dependent piecewise linear state-feedback controllers. This results in novel and flexible conditions, with the particular feature of being in the form of linear matrix inequalities (LMIs).

Our technique thus provides a first controller-design strategy allowing piecewise linear feedback maps and piecewise quadratic (control) Lyapunov functions by means of semidefinite programming. Numerical examples illustrate the application of the proposed techniques, the relations between the graph order, the robustness, and the performance of the closed loop.

Introduction

In this paper, given matrices A 1 , . . . , A M ∈ R n×n and B 1 , . . . , B M ∈ R n×m , we consider the discrete-time switched control system

x(k

+ 1) = A σ(k) x(k) + B σ(k) u(k), (1)
where σ : N → ⟨M ⟩ := {1, . . . , M } is a switching signal. Switched systems are a popular model for hybrid or cyber-physical systems, with many applications in modern engineering; see e.g., [START_REF] Liberzon | Switching in systems and control[END_REF]; [START_REF] Hespanha | Switching between stabilizing controllers[END_REF]; [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF]. We are interested in the feedback stabilization problem of (1) under arbitrary switching, i.e. we aim to design a feedback control stabilizing policy for (1), no matter the underlying switching rule σ : N → ⟨M ⟩. In recent years, stabilization of switched systems has been tackled from the perspective of designing a stabilizing switching sequence, i.e. the signal σ : N → ⟨M ⟩ is considered a control input for the "autonomous" system (1) with u(k) ≡ 0. In this context, see [START_REF] Geromel | Stability and stabilization of discrete time switched systems[END_REF]; [START_REF] Hu | Stabilization of switched systems via composite quadratic functions[END_REF]; [START_REF] Lin | Stability and stabilizability of switched linear systems: A survey of recent results[END_REF]; [START_REF] Fiacchini | Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: A set-theory approach[END_REF]; [START_REF] Jungers | On feedback stabilization of linear switched systems via switching signal control[END_REF] and references therein, among many other results. The problem where both the switching signal and the input u(k) are available for design has also been tackled [START_REF] Fiacchini | Control co-design for discrete-time switched linear systems[END_REF]. Instead, in this manuscript, we study the design of feedback maps u(k) = Φ(x(k)) for which the switched closed-loop system (1) is asymptotically stable, without any further assumption on the external switching policy. The underlying switching rule σ : N → ⟨M ⟩ cannot be designed/modifiable by the user, and it can thus be seen as an external disturbance. Two cases can be highlighted, supposing, respectively, that the value of the switching signal is:

• Un-observable by the designer/controller, in this case we aim to construct a robust feedback controller Φ : R n → R m ,

• Observable at the current instant of time, and in this case the goal is to design mode-dependent feedback controllers Φ 1 , . . . , Φ M : R n → R m .

The stabilization problem has been intensively studied in a close and related context, that is, in the framework of polytopic linear parameter-varying (LPV) systems, see [START_REF] Blanchini | Stabilization of LPV systems: State feedback, state estimation, and duality[END_REF] and references therein for a formal introduction. While the class of LPV systems provides a more general model than (1), the corresponding stabilization problems are strongly related, and in some remarkable cases they are proved to be equivalent, see [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF]. Several results have been established in the LPV framework, both for the robust and mode-dependent (in this literature, also called gain-scheduling) cases. First, it has been proved that a quadratic Lyapunov function approach, while leading to numerically-appealing conditions, is conservative, see [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF]; [START_REF] Blanchini | Nonquadratic Lyapunov functions for robust control[END_REF]. In order to have less conservative conditions in studying stability and stabilizability of (1), approaches based on piecewise-defined (control) Lyapunov functions (and, thus piecewise continuous feedback maps) have been proposed: maxima and minima of quadratic functions in Goebel, Teel, et al. (2006); Goebel, Hu, & Teel (2006), polyhedral functions in [START_REF] Blanchini | Stabilization of LPV systems: State feedback, state estimation, and duality[END_REF]; [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF]; [START_REF] Blanchini | Set-theoretic methods in control[END_REF]. For an equivalence result between these two approaches, see [START_REF] Hu | Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions[END_REF]. While these methods provide exhaustive characterizations of the robust and mode-dependent stabilizability properties [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF][START_REF] Hu | Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions[END_REF], from a numerical point of view they are affected by two main limitations; first of all, the number of quadratics composing the candidate piecewise Lyapunov function (or, equivalently, the number of vertices of the candidate polyhedral Lyapunov function) to reach necessary conditions is theoretically unbounded. Secondly, the arising conditions are in the form of bilinear matrix inequalities (BMI), which are known to be NP-hard in general [START_REF] Toker | On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback[END_REF]. Other works have tackled related stabilization problems, with results leading to linear matrix conditions, for example considering static output feedback with polynomial Lyapunov functions (see [START_REF] Chesi | Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach[END_REF]; Chesi (2013) and references therein), or designing dynamic output feedbacks with quadratic Lyapunov functions, such as [START_REF] Blanchini | A separation principle for linear switching systems and parametrization of all stabilizing controllers[END_REF]. Herein, we focus on piecewise linear controllers.

In a slightly different setting, in studying stability of (1) (i.e., considering

B 1 = • • • = B M = 0)
, the concept of path-complete Lyapunov functions (PCLFs) has been introduced in [START_REF] Ahmadi | Joint spectral radius and pathcomplete graph Lyapunov functions[END_REF], in order to provide flexible conditions based on multiple Lyapunov functions. This framework involves a combinatorial component given by a directed graph that describes the set of Lyapunov inequalities to be verified and that has to be path-complete in the sense that it captures every finite switching sequence. The complexity of this underlying graph can be increased by the user to reduce the conservatism of the arising stability conditions. For recent developments on this topic and more discussion regarding the relationship between graphs and conservatism, we refer to [START_REF] Debauche | Necessary and sufficient conditions for templatedependent ordering of path-complete Lyapunov methods[END_REF]. In this setting, in (Philippe et al., 2019, Theorem III.8), it has been proved that any path-complete Lyapunov function induces/can be used to construct a common Lyapunov function in the form of max-min of quadratics, thus building a bridge between the PCLF framework of [START_REF] Ahmadi | Joint spectral radius and pathcomplete graph Lyapunov functions[END_REF] and the piecewise quadratic functions approach of [START_REF] Johansson | Computation of piecewise quadratic lyapunov functions for hybrid systems[END_REF]; Goebel, Hu, & Teel (2006); Goebel, Teel, et al. (2006). Summarizing, it is shown in [START_REF] Philippe | On path-complete Lyapunov functions: Geometry and comparison[END_REF] that the PCLF framework provides a compressed representation of common Lyapunov functions, which allows for faster computation. In addition, the directed graph defining the PCLF provides a discrete design parameter for the user, which can be optimized in order to mitigate the numerical computational effort.

In this manuscript, we propose a novel approach to robust and mode-dependent stabilization of (1). Our method relies on path-complete Lyapunov functions theory, and, more specifically, we adapt and generalize this framework from stability to stabilizability analysis. The arising sufficient stabilization results, while depending on an underlying combinatorial structure (a path-complete graph), lead to linear matrix inequalities (LMI) conditions, thus bypassing the numerical limitations of previous results [START_REF] Blanchini | Set-theoretic methods in control[END_REF][START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF]Goebel, Hu, & Teel, 2006;Goebel, Teel, et al., 2006;[START_REF] Hu | Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions[END_REF]. From an analytic point of view, our conditions lead to the construction of piecewise quadratic control Lyapunov functions, in the form of minimum of quadratics, for both the robust and mode-dependent cases; moreover, the resulting feedback controllers are in a piecewise linear form. We thus build a connection with existing literature on piecewise quadratic (control) Lyapunov functions and piecewise defined controller, see for example [START_REF] Johansson | Computation of piecewise quadratic lyapunov functions for hybrid systems[END_REF]; Goebel, Hu, & Teel (2006); [START_REF] Hu | Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions[END_REF]; [START_REF] Legat | Piecewise semi-ellipsoidal control invariant sets[END_REF].

We point out that we recover, as a particular case of our conditions, the techniques proposed in [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF]; [START_REF] Lee | Uniform stabilization of discrete-time switched and Markovian jump linear systems[END_REF]; [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF]; [START_REF] Essick | Control of linear switched systems with receding horizon modal information[END_REF]; however, on top of being more general, our approach does not require the observation and memorization of past/future values of the switching signal (as in the memory/horizon-based control of [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF]; [START_REF] Lee | Uniform stabilization of discrete-time switched and Markovian jump linear systems[END_REF]; [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF]; [START_REF] Essick | Control of linear switched systems with receding horizon modal information[END_REF]). Indeed, the core of our proof technique is based on the above-mentioned results from path-complete Lyapunov functions [START_REF] Philippe | On path-complete Lyapunov functions: Geometry and comparison[END_REF], which shows that a path-complete graph describes implicitly a common Lyapunov function, thus leading to a closed expression for the feedback maps (in a piecewise-linear form). This in particular allows us to tackle robust stabilization of LPV systems, for which the underlying time-varying parameter is un-observable. Moreover, our stabilization techniques are valid for a class of graphs that goes beyond the memory structures used in [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF]; [START_REF] Lee | Uniform stabilization of discrete-time switched and Markovian jump linear systems[END_REF]; [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF]; [START_REF] Essick | Control of linear switched systems with receding horizon modal information[END_REF], thus allowing for a larger class of stabilization certificates. For a discussion of the benefits of such generalization in the stability context, we refer to [START_REF] Philippe | On path-complete Lyapunov functions: Geometry and comparison[END_REF]; [START_REF] Debauche | Necessary and sufficient conditions for templatedependent ordering of path-complete Lyapunov methods[END_REF][START_REF] Della Rossa | Interpretability of path-complete techniques and memory-based Lyapunov functions[END_REF]. Our theoretical developments are then illustrated, both in the robust and mode-dependent cases, with the help of numerical examples already introduced in the literature, thus allowing the comparison between our technique and existing results.

The rest of the manuscript is organized as follows: in Section 2 we recall the necessary definitions and base results for switched systems and graph theory. In Section 3 we derive our main results concerning robust stabilizability, while in Section 4 we present the stabilizability statement in the mode-dependent case. Section 5 closes the manuscript with some concluding remarks and possible directions for future research.

Notation: We denote by N the set of natural numbers including {0}, by Z + the set of natural numbers excluding {0}. Given M ∈ Z + , we define ⟨M ⟩ := {1, . . . , M }. Given n, m ∈ Z + , C 0 (R n , R m ) is the set of continuous functions from R n to R m . We denote by S n×n the set of the n × n symmetric matrices, and by S n×n + the set of n × n positive definite matrices. A function U : R n → R is said to be positive definite if U (0) = 0 and U (x) > 0 for all x ̸ = 0. It is said to be radially unbounded if lim λ→+∞ U (λx) = +∞ for all x ̸ = 0.

Preliminaries

This section introduces the studied setting and recalls the necessary definitions and tools.

Stabilization Notions and Characterizations

Given M ∈ Z + , and f 1 , . . . , f M : R n → R n such that f j (0) = 0 for all j ∈ ⟨M ⟩, consider the switched system

x(k + 1) = f σ(k) (x(k)), x(0) = x 0 , k ∈ N, (2)
where σ : N → ⟨M ⟩ is an external switching signal and x 0 ∈ R n is the initial condition. We denote by Φ σ (k, x 0 ) the solution of (2) starting at x 0 ∈ R n and with respect to the signal σ : N → ⟨M ⟩, evaluated at time k ∈ N.

Definition 1 System (2) is said to be uniformly globally asymptotically stable (UGAS) if there exists a KL function1 β such that, for all σ : N → ⟨M ⟩ and for all x 0 ∈ R n , we have

|Φ σ (k, x 0 )| ≤ β(|x 0 |, k).
If the function β is of the form β(r, k) := Cγ k r for some C > 0 and γ ∈ [0, 1) the system is said to be uniformly exponentially stable (UES), and γ is said to be the decay rate of the system.

Let us consider M ∈ Z + and a set

F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩}.
We want to study the discrete-time control switched system defined by

x(k + 1) = A σ(k) x(k) + B σ(k) u(k), k ∈ N, (3)
where σ : N → ⟨M ⟩ is an external switched signal and u : N → R m is a control input.

Remark 1 (Related models: LPVs) As far as the general class of switching signals is considered, i.e., with no further assumption on the feasible signals σ ∈ S := {θ : N → ⟨M ⟩}, system (3) can be equivalently represented in different frameworks. Notably, we can consider the formalism of linear parameter-varying (LPV) systems, i.e. the case of

x(k + 1) = A(w(k))x(k) + B(w(k))u(k), (4)
where w : 4) provides a more general framework with respect to (3), since it allows the state to "follow" directions obtained as convex combination of sub-systems in F. Under some assumption on the proposed design methods and on the systems matrices, the stabilization techniques for (3) are also effective for (4), as discussed in what follows.

N → Λ M := {w ∈ R M + | M i=1 w i = 1} is a time-varying parameter taking values in Λ M , the standard simplex of dimension M -1, and A(w) = M i=1 w i A i and B(w) = M i=1 w i B i . System (
△

Definition 2 (Stabilization Notions) System (3) is said to be 1. Robust Feedback Stabilizable (RFS) if there exists a Φ : R n → R m such that the closed-loop system given by x(k + 1) = A σ(k) x(k) + B σ(k) Φ(x(k)) (5)
is UGAS.

2. Mode-Dependent Feedback Stabilizable (MDFS) if there exist Φ 1 , . . . , Φ M : R n → R m such that the closed-loop system given by

x(k + 1) = A σ(k) x(k) + B σ(k) Φ σ(k) (x(k)) (6)
is UGAS.

In order to present the main equivalence results concerning robust and mode-dependent stabilization, we introduce a tailored notion of piecewise linear functions, inspired by (Boyd & Vandenberghe, 2004, Problem 3.29).

Definition 3 (Piecewise Linear Functions) A function Ψ : R n → R m is said to be piecewise linear if:

• It is homogeneous of degree 1, i.e., for any λ ∈ R + and any x ∈ R n , it holds that Ψ(λx) = λΨ(x);

• For some c ∈ Z + , there exist c closed convex cones C 1 , . . . C c ⊂ R n and c linear maps Ξ 1 , . . . , Ξ c : R n → R m such that j∈⟨c⟩ C j = R n , Int(C i ) ∩ Int(C j ) = ∅ for all i ̸ = j and Ψ(x) = Ξ j (x) for all x ∈ Int(C j ) for all j ∈ ⟨c⟩.
We note that Definition 3 does not imply continuity, therefore the piecewise linear feedback maps that we consider are possibly discontinuous, on the null measure set defined by the union of the boundaries of the cones C 1 , . . . C c in the definition. In what follows, we recall a characterization result proved in [START_REF] Blanchini | Stabilization of LPV systems: State feedback, state estimation, and duality[END_REF]; [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF]; [START_REF] Blanchini | Set-theoretic methods in control[END_REF] and reviewed here for the sake of completeness.

Proposition 1 (Blanchini et al., 2007, Proposition 2)

Given M ∈ Z + and a set F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩}, system (3) is:
(A) RFS if and only if there exists a piecewise linear feedback control Φ : R n → R m such that the closedloop (5) is UGAS.

(B) MDFS if and only if there exist piecewise linear feedback controls Φ 1 , . . . , Φ M : R n → R m such that the closed-loop (6) is UGAS.

The proof of this proposition contained in [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF] relies on the construction of control-invariant polyhedral sets and, thus, polyhedral Lyapunov functions. Note that the key result in the proposition is that the existence of piecewise linear feedback maps is both necessary and sufficient for the stabilizability of (3), in both robust and mode-dependent cases. We also note that by homogeneity (see (A.Bacciotti & L.Rosier, 2005, Section 5.3) for the details), when a stabilizing piecewise linear feedback is provided, the closed loop will be uniformly exponentially stable (UES).

Remark 2 While Proposition 1 completely characterizes the (robust and mode-dependent) stabilizability problem, the underlying conditions are unsatisfactory from a numerical point of view. Indeed, since the proof of [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF], Proposition 2) relies on the construction of polyhedral control Lyapunov functions, the following limitations can be highlighted:

• The number of vertices of the polyhedral level sets of the candidate Lyapunov functions has to be fixed a priori. To reach necessary conditions, the aforementioned number of vertices could be arbitrarily large.

• Even when the number of vertices is fixed, the arising matrix conditions turn out to be bilinear matrix inequalities (BMIs) which are, in general, NP-hard to solve, see the discussion in (Blanchini & Miani, 2008, Section 7.3.2).

Another classic approach (see [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF] and references therein) relies on quadratic control Lyapunov functions and linear feedbacks, i.e. supposing the existence of K 1 , . . . , K M ∈ R n×m (a unique K ∈ R n×m in the robust case) and a P ∈ S n×n + such that

(A i + B i K i ) ⊤ P (A i + B i K i ) -P ≺ 0, ∀ i ∈ ⟨M ⟩,
which are nonlinear matrix inequalities. Re-parametrizing, considering the variables Q = P -1 and R i = K i P -1 , and finally pre-and post-multiplying the latter inequality by Q, we obtain

(QA ⊤ i +R ⊤ i B ⊤ i )Q -1 (A i Q+ B i R i ) -Q ≺ 0,
which can be rewritten, via Schur complement, as

Q ⋆ A i Q + B i R i Q ≻ 0, ∀i ∈ ⟨M ⟩,
leading to LMI conditions (the robust case, with a constant K ∈ R n×m , is similar). The drawback of this well-known design technique are the following:

• The existence of quadratic common Lyapunov function (and linear feedbacks) is only sufficient for stability/stabilizability of switched systems, as highlighted by Proposition 1.

• The change-of-variables technique is not applicable, in general, for broader contexts, for example involving multiple-(control)-Lyapunov functions.

For the reasons mentioned above, in the following sections, we propose a novel feedback design technique based on graph theory, which will lead to LMI-based conditions.

Preliminaries on graph theory and path-complete Lyapunov functions

In the following, we collect some graph-theory notions which will be used in our formal statements. Moreover, we recall the basic ideas of the sufficient conditions for stability of (2) based on path-complete Lyapunov functions, for an overview see [START_REF] Philippe | On path-complete Lyapunov functions: Geometry and comparison[END_REF]; [START_REF] Ahmadi | Joint spectral radius and pathcomplete graph Lyapunov functions[END_REF]; [START_REF] Philippe | Stability of discrete-time switching systems with constrained switching sequences[END_REF]. A (labeled and directed) graph G = (S, E) on ⟨M ⟩ is defined by a finite set of nodes S and a subset of labeled edges E ⊂ S × S × ⟨M ⟩.

Definition 4 (Path-Complete Graphs) A graph G = (S, E) is path-complete for ⟨M ⟩ if, for any l ≥ 1 and any "word" (j 1 . . . j l ) ∈ ⟨M ⟩ l , there exists a path

{(s k , s k+1 , j k )} 1≤k≤l such that (s k , s k+1 , j k ) ∈ E, for each 1 ≤ k ≤ l.
Intuitively, a graph is path-complete if any possible switching sequence can be reconstructed by walking through the edges of the graph. In our setting, path-complete graphs will represent and encode the structure of our stability certificates, or in other words, the structure of the inequalities among the candidate Lyapunov functions we aim to verify, as formalized in what follows.

Definition 5 (Path-complete Lyapunov functions) Given a family

F = {f 1 , . . . , f M } ⊂ C 0 (R n , R n ), a path-complete Lyapunov function (PCLF) for F is a pair (G, V ) where G = (S, E) is a path-complete graph, and V = {V s | s ∈ S} ⊆ C 0 (R n , R
) is a set of positive definite and radially unbounded functions such that the following inequalities are satisfied:

∀ e = (a, b, i) ∈ E, ∀x ∈ R n \ {0} : V b (f i (x)) < V a (x).
We present, in what follows, the main stability result we will use in our stabilizing feedback design.

Proposition 2 Consider a switched system (2) defined by

F = {f 1 , . . . , f M } ⊂ C 0 (R n , R n ). If there exists a path-complete Lyapunov function (G, V ) for F , then (2) is UGAS.
The proof of Proposition 2 can be found in [START_REF] Ahmadi | Joint spectral radius and pathcomplete graph Lyapunov functions[END_REF]; [START_REF] Philippe | Stability of discrete-time switching systems with constrained switching sequences[END_REF][START_REF] Philippe | On path-complete Lyapunov functions: Geometry and comparison[END_REF]. This result establishes, given any path-complete graph, sufficient conditions for stability of (2). For a discussion of the conservatism and comparison among different graph structures for stability, see [START_REF] Debauche | Necessary and sufficient conditions for templatedependent ordering of path-complete Lyapunov methods[END_REF]. In order to adapt path-complete Lyapunov stability techniques from stability to stabilization problem of (3), we need to refine Definition 4, introducing additional properties on the considered graphs, as defined in what follows.

Definition 6 (Complete graphs) A graph G = (S, E) on ⟨M ⟩ is complete if for all a ∈ S, for all i ∈ ⟨M ⟩, there exists at least one node b ∈ S such that the edge e = (a, b, i) ∈ E.

We note that any complete graph2 is in particular path-complete; for a graphical visualization, we point out that the graphs in Figure 1 are complete. In what follows, we introduce a hierarchical family of complete graphs firstly defined in the seminal paper [START_REF] De Bruijn | A combinatorial problem[END_REF], used in what follows as a tool for numerical verification of the proposed conditions.

Definition 7 [START_REF] De Bruijn | A combinatorial problem[END_REF])) Given M ∈ Z + , l ∈ N the (primal) De Bruijn graph of order l (on ⟨M ⟩) denoted by H l (M ) = (S p,l , E p,l ) is defined as follows: S p,l := ⟨M ⟩ l and, given any node î = (i 1 , . . . , i l ) ∈ S p,l , we have (î, ȷ, h) ∈ E p,l for every ȷ of the form ȷ = (h, i 1 , . . . , i l-1 ), for any h ∈ ⟨M ⟩.

It can be seen that the graph H l (M ) is complete ∀ M ∈ Z + , l ∈ N; see Fig. 1 for a graphical representation.

Robust Feedback via Path-complete Graphs

In this section, we provide an LMI formulation for the design of piecewise linear feedback for the robust stabilization problem. The arising conditions will generalize existing approaches (e.g. [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF][START_REF] Blanchini | Stabilization of LPV systems: State feedback, state estimation, and duality[END_REF][START_REF] Hu | Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions[END_REF]), via the use of multiple, or more precisely, path-complete Lyapunov functions. The theoretical developments are then illustrated by numerical examples. Figure 1: The graphs H 1 (2) and H 2 (2), respectively the primal De Bruijn graphs of order 1 and 2 on ⟨M ⟩ = {1, 2}.

Piecewise Linear Robust Feedback

In this subsection we provide the first stabilization result, making use of the path-complete framework to provide min of quadratics control Lyapunov functions, together with piecewise linear robust feedback control map for system (3).

Proposition 3 Consider M ∈ Z + and a set

F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩}, and a complete graph G = (S, E) on ⟨M ⟩. Suppose there exist {P s } s∈S ⊂ S n×n + , {K s } s∈S ⊂ R m×n such that (A i + B i K a ) ⊤ P b (A i + B i K a ) -P a ≺ 0, ∀e = (a, b, i) ∈ E, (7)
then the map Φ(x) := K γ(x) x, x ∈ R n , robustly exponentially stabilizes system (3), where γ : R n → S is any function satisfying

γ(x) ∈ argmin s∈S {x ⊤ P s x}, ∀x ∈ R n , (8)
and for which Φ : R n → R m is piecewise linear (recall Definition 3).3

Proof. The proof is inspired by proof of (Philippe et al., 2019, Proposition III.1). Consider V s :

R n → R defined by V s (x) = x ⊤ P s x, function W (x) = min s∈S {V s (x)} and f is : R n → R n defined by f is (x) = (A i + B i K s )
x, for any s ∈ S and any i ∈ ⟨M ⟩. Since P s ≻ 0 for all s ∈ S, the function W : R n → R is positive definite and radially unbounded. Condition (7) can be rewritten as

V b (f ia (x)) < V a (x), ∀x ∈ R n \ {0}, ∀e = (a, b, i) ∈ E. (9)
Consider any x ∈ R n , x ̸ = 0 and any i ∈ ⟨M ⟩. Suppose γ(x) = a ∈ argmin s∈S {x ⊤ P s x}, and thus W (x) = V a (x). By completeness of G, there exists a b ∈ S such that e = (a, b, i) ∈ E, and thus we have

W (f ia (x)) = min s∈S {V s (f ia (x))} ≤ V b (f ia (x)) < V a (x) = W (x).
We have thus proved that the function W : R n → R is a Lyapunov function for the closed-loop system

x(k + 1) = f σ(k) (x(k)) := A σ(k) + B σ(k) K γ(x(k)) x(k)
for any σ : N → ⟨M ⟩. Since the closed loop system is homogeneous of degree 1 (i.e. f j (λx) = λf j (x) for any j ∈ ⟨M ⟩, for any x ∈ R n , for any λ ∈ R) it is in particular UES, concluding the proof. □ Thanks to the completeness of G and the particular structure of (7), we can provide, in what follows, necessary and sufficient LMI conditions for (7).

Lemma 1 Conditions (7) are satisfied if and only if there exist

{P s } s∈S ⊂ S n×n , {K s } s∈S ⊂ R m×n such that P b (A i P a + B i K a ) ⋆ P a ≻ 0, ∀ e = (a, b, i) ∈ E. ( 10
)
Matrices {P s } s∈S ⊂ S n×n + , {K s } s∈S ⊂ R m×n satisfying (7) are then given by defining P s = P -1 s and K s = K s P -1 s , for any s ∈ S.

Proof. Sufficiency: Suppose (10) holds; it is easy to prove that it implies P s ≻ 0 for all s ∈ S. For any e = (a, b, i) ∈ E we pre-and post-multiply (10) by diag(P -1 b , P -1 a ), obtaining

P -1 b P -1 b (A i + B i K a P -1 a ) ⋆ P -1 a ≻ 0, ∀ e = (a, b, i) ∈ E.
By defining changes of variables P s = P -1 s and K s = K s P -1 s for any s ∈ S we obtain

P b P b (A i + B i K a ) ⋆ P a ≻ 0, ∀ e = (a, b, i) ∈ E, (11)
which, by Schur complement, is equivalent to

P a -(A i + B i K a ) ⊤ P b P -1 b P b (A i + B i K a ) ≻ 0, (12)
∀ e = (a, b, i) ∈ E, leading to (7).

Necessity: Suppose that ( 7) holds. Rewrite it as ( 12) and apply Schur complement again to obtain (11). Next, apply a congruence transformation with diag(P -1 b , P -1 a ) followed by the same changes of variables as before to obtain (10). Therefore, feasibility of (7) implies satisfaction of (10) with P s = P -1 s and

K s P -1 s = K s .
□ Proposition 3 and the subsequent LMI characterization in Lemma 1 are stated for generic complete graphs. In Definition 7, we introduced the class of De-Bruijn graphs, denoted by H l (M ). In the following statement, as a corollary, we explicitly write the conditions of Lemma 1 associated to H l (M ), proving the numerical scheme used in the following sections.

Corollary 1 (De Bruijn conditions, Robust Case)

Consider M ∈ Z + , a set F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩},
and any l ∈ N. Suppose there exist {P î} î∈⟨M ⟩ l ⊂ S n×n , {K î } î∈⟨M ⟩ l ⊂ R m×n such that, ∀î = (i 1 , . . . , i l ) ∈ ⟨M ⟩ l and ∀ h ∈ ⟨M ⟩, the inequalities

P (h,î -) (A h P î + B h K î) ⋆ P î ≻ 0 (13)
hold, where î-:= (i 1 , i 2 , . . . , i l-1 ) ∈ ⟨M ⟩ l-1 . Then the piecewise linear map Φ(x) := K γ(x) x, x ∈ R n , where K î = K îP -1 î , robustly exponentially stabilizes system (3) where γ : R n → ⟨M ⟩ l is a function satisfying

γ(x) ∈ argmin î∈⟨M ⟩ l {x ⊤ P îx} , ∀x ∈ R n ,
with P î = P -1 î and for which Φ : R n → R m is piecewise linear. Moreover, the function W (x) := min î∈⟨M ⟩ l {x ⊤ P îx} is a Lyapunov function for the closed loop system.

Proof. Conditions in ( 13) are the specification of conditions (10) when considering the graph H l (M ) introduced in Definition 7; by Proposition 3 we conclude. □ Remark 3 (Robust Stabilization of LPVs) It is important to point out that the stabilization techniques proposed in Proposition 3 and then numerically tackled in Lemma 1 and Corollary 1 for (3) are effective also for the more general class of LPV systems as in (4). Indeed, suppose that a piecewise linear control Φ : R n → R m as in Proposition 3 is given, it can be shown, by a convexity argument, that the convex difference inclusion x(k + 1) ∈ co {A i x(k) + B i Φ(x(k)) | i ∈ ⟨M ⟩} is exponentially stable, for a direct proof, we refer to (Blanchini et al., 2007, Proposition 2). This, in particular, implies the exponential stability of (4). △

Concluding this section, we briefly illustrate an open direction of research. Corollary 1 provides sufficient conditions for the robust feedback stabilizability of (3). It can be seen that, if the conditions related to the l = 0 l = 1 l = 2 l = 3 l = 5 max α 0.6646 0.6684 0.6856 0.6984 0.7032 Table 1: Relation between maximum α and graph order l for Example 2. graph H l (M ) are feasible, then also the conditions related to graphs H l ′ (M ) are feasible, for any l ′ ≥ l. Thus, by increasing the order l ∈ N one can reduce the conservatism of the proposed conditions (as we will illustrate in the next subsection). Such a hierarchy is known to provide non-conservative conditions (by increasing the order l ∈ N) for the stability problem, see [START_REF] Ahmadi | Joint spectral radius and pathcomplete graph Lyapunov functions[END_REF]. On the other hand, the question of whether this is also the case for the robust stabilization problem studied here is still open. More precisely, future research will investigate the following claim:

Conjecture 1 Consider M ∈ Z + and a set F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩},
and suppose that system (3) is robust feedback stabilizable, in the sense of Definition 2. Then, there exists a l ∈ N (large enough) for which the conditions in Corollary 1 are feasible.

Numerical Examples: Robust Case

In this subsection, we provide some numerical examples in order to illustrate the results presented in Subsection 3.1, and the advantages of the proposed techniques with respect to existing literature.

Example 1

Consider system (1) with matrices

A 1 = 0.1 0.9 0 0.1 , A 2 = α 0 1 α , B 1 = 0 0 , B 2 = α -1 .
where α > 0 is a free parameter. The goal is to find the maximal α for which conditions of Corollary 1 are feasible, thus providing feedback gains for which the closed-loop system is UES. The stabilization of this system was also studied in (Lee, 2006, Example 2), where the authors design "finite-path dependent state feedback controllers", that is, state-feedback controllers that depend on the history of the switching signal σ.

The maximum value of α for which stability is achieved in [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF] is α = 0.6667 by using controllers with memory. If no memory is employed, the maximum obtained value in [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF] is α = 0.6173. By using the conditions of Corollary 1, we obtain that the maximum value obtained is α = 0.5872 for the case of the De Bruijn graph of order l = 0, and α = 0.6667 for all l ≥ 1. Thus, with the strategy developed here, we can robustly stabilize the system for an α that is as large as the maximum α obtained in [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF] by using memory-dependent controllers, which require knowledge/measurement of switching signals. In conclusion, our robust feedback strategy is as conservative as the memory-dependent strategy in [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF]. This was expected since the underlying conditions of [START_REF] Lee | On uniform stabilization of discrete-time linear parameter-varying control systems[END_REF] are substantially equivalent to the ones in Corollary 1.

Our graph approach allows us to provide a feedback control map in a closed form as defined in Proposition 3, without the necessity of storing the past active modes.

Example 2

We modify the system from Example 1 by setting B 1 = [1 0] ⊤ . In this case, the maximum α obtained for which the system can be stabilized as a function of the order l of the chosen De Bruijn graph is illustrated in Table 1. As demonstrated by the table, an increase in l tends to improve the robustness to the uncertainty α.

Given l = 2 and α = 0.6856, for any î ∈ ⟨2⟩ 2 , we obtain the following Lyapunov matrices P î and controllers K î:

Example 3

This example illustrates the relationship between the De Bruijn graph order l and the closed-loop performance, given by the minimum γ ∈ [0, 1) that satisfies W (x(k + 1)) ≤ γ 2 W (x), where W is the Lyapunov function defined by W (x) := min î∈⟨M ⟩ l {x ⊤ P îx} given in Corollary 1. This γ represents an upper bound for the exponential decay rate of the solutions of the closed loop, recall Definition 1. The conditions of Corollary 1 can be straightforwardly modified to tackle this problem by multiplying the (2, 2)-block of the matrix in (13) by γ 2 ; then a line search is performed to minimize the parameter γ. Consider the system defined by the following matrices:

A 1 = 0 1 -1 0 , A 2 = -1 0 0 -0.95 , B 1 = B 2 = 1 0 .
In Table 2, we summarize the relationship between the graph order l and minimum decay rate γ achieved. Moreover, Fig. 2 illustrates the sub-level set of the Lyapunov function W given by E W (x) = {x ∈ R n : W (x) ≤ 1} and its relationship with the graph order l. The level sets of W (x) are marked in red in the figure, while the sub-level sets defined of each function V s (x) are illustrated in black.

Example 4

Consider a system defined by

B 1 = B 2 = B 3 = [1 0] ⊤ , A 1 = 1 0.2 0 0.5 , A 2 = 1.1 0.2 0 -0.5 , A 3 = 0.5 0.8 1.1 0.5 .
This example illustrates the geometric intuition behind the min of quadratics strategy that selects the linear controller at each instant of time. By choosing l = 2, the conditions of Corollary 1 are feasible and thus we find the Lyapunov function W (x) = min î∈⟨3⟩ 2 {x ⊤ P îx}, which guarantees a decay rate of γ = 0.7536, defined by the matrices P î, for î ∈ ⟨3⟩ 2 in (14). The obtained controller gains are given in (15).

P
In Fig. 3 we have plotted the 1-level set of W (x), and one closed-loop trajectory, starting at the initial condition x 0 = [0.4 1.4] ⊤ and following a periodic switching sequence σ = {1, 2, 3, 1, 2, 3 . . . }. As previously illustrated, the level set of W (x) correspond to the union of the level sets of the multiple V î, î ∈ ⟨3⟩ 2 . Recalling Definition 3, the cones defining the partition associated to the piecewise linear feedback map Φ in Proposition 3 are defined by the argmin among a set of quadratic functions, see (8). That is, in such cones, the minimum is attained for the same s ∈ S = ⟨3⟩ 2 , i.e. W (x) = V s (x) is achieved. For example, the black cone in Fig. 3 represents the cone where the value of W (x) coincides with V 33 (x). Whenever the state x belongs to this cone, the controller gain K 33 is activated. For the plotted trajectory, only three controllers have been active, namely K 13 , K 21 , and K 33 . However, the state can fall within different cones where other controller gains will be activated for different initial conditions and switching signals. In conclusion, from a geometric perspective, an increase in the graph order l tends not only to refine the level set of W (x) but also to produce more cones so that different controller gains better adapted to certain regions of the state space are used, thus producing an improved control law.

Mode-Dependent Case

In this section we provide our main stabilization results in the mode-dependent case.

Piecewise Linear Mode-Dependent Feedbacks

In this subsection, adapting the proof technique of Proposition 3, we propose conditions depending on a graph structure, leading to piecewise-linear mode-dependent feedback gains. To this aim, we require an additional property on the underlying graphs, and we thus introduce the following definition.

Definition 8 (Deterministic Graph) A graph G = (S, E) on ⟨M ⟩ is said to be deterministic if, for all a ∈ S and i ∈ ⟨M ⟩ there exists at most one b ∈ ⟨M ⟩ such that e = (a, b, i) ∈ E.

Proposition 4 Consider M ∈ Z + , a set F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩}
and a complete and deterministic graph G = (S, E) on ⟨M ⟩. Suppose there exist {P s } s∈S ⊂ S n×n + , {K s,j } (s,j)∈S×⟨M ⟩ ⊂ R m×n such that

(A i + B i K a,i ) ⊤ P b (A i + B i K a,i ) -P a ≺ 0, (16)
∀e = (a, b, i) ∈ E. Then, the piecewise linear maps Φ i (x) := K γ(x),i x, for i ∈ ⟨M ⟩, exponentially stabilize system (3)(in the mode-dependent sense of Definition 2) where γ : R n → S is any function satisfying

γ(x) ∈ argmin s∈S {x ⊤ P s x}, ∀ x ∈ R n ,
and for which the functions Φ i : R n → R m are piecewise linear.

Proof. First of all, since G is complete and deterministic, for any a ∈ S and any i ∈ ⟨M ⟩, there exists a unique b ∈ S such that e = (a, b, i) ∈ E and thus the notation K s,i is well posed; then the proof fundamentally follows the structure of the proof of Proposition 3. Define

V s : R n → R by V s (x) = x ⊤ P s x, function W (x) = min s∈S {V s (x)} and f is : R n → R n by f is (x) = (A i + B i K s,i )
x, for any s ∈ S and any i ∈ ⟨M ⟩. Again, (16) implies that W : R n → R is positive definite and radially unbounded, since P s ≻ 0 for all s ∈ S. With this notation, condition (16) implies inequalities (9), and thus, following the reasoning of proof of Proposition 3, we prove that the function W : R n → R is a Lyapunov function for the closed-loop system x(k + 1) = A σ(k) + B σ(k) K γ(x(k)),σ(k) x(k) for any σ : N → ⟨M ⟩, concluding the proof. □ Next, we present necessary and sufficient LMI conditions ensuring (16).

Lemma 2 Conditions (16) are satisfied if and only if there exist {P s } s∈S ⊂ S n×n , {K s,i } s∈S,i∈⟨M ⟩ ⊂ R m×n such that the LMIs P

b (A i P a + B i K a,i ) ⋆ P a ≻ 0, ∀ e = (a, b, i) ∈ E, (17)
are feasible. Matrices {P s } s∈S ⊂ S n×n , {K s } s∈S ∈ R m×n satisfying (7) are then given by defining P s = P -1 s and K s,i = K s,i P -1 s .

Proof. The proof follows the same steps of the proof of Lemma 1 and is thus omitted. □ Similar to the Section 3, we are going to introduce below the equivalent LMI conditions for the class of De-Bruijn graphs of order l ∈ ⟨M ⟩, denoted by H l (M ), which are complete and deterministic, recall Definition 7.

Corollary 2 (De Bruijn

: Mode-Dependent Case) Consider M ∈ Z + , a set F = {(A i , B i ) ∈ R n×n × R n×m | i ∈ ⟨M ⟩} and any l ∈ N. Suppose there exist {P î} î∈⟨M ⟩ l ⊂ S n×n , {K î,h } î∈⟨M ⟩ l ,h∈⟨M ⟩ ∈ R m×n such that, ∀î = (i 1 , . . . , i l ) ∈ ⟨M ⟩ l and ∀ h ∈ ⟨M ⟩, the inequalities P (h,î -) (A h P î + B h K î,h ) ⋆ P î ≻ 0, (18)
hold, where î-= (i 1 , . . . , i l-1 ) ∈ ⟨M ⟩ l-1 . Then, the feedbacks maps Φ h (x) := K γ(x), h x, x ∈ R n , h ∈ ⟨M ⟩, where K î,h = K î,h P -1 î exponentially stabilize system (3) where γ : R n → ⟨M ⟩ l satisfies γ(x) ∈ argmin î∈⟨M ⟩ l {x ⊤ P îx} with P î = P -1 î and for which the functions Φ h : R n → R m are piecewise linear. Moreover, W (x) := min î∈⟨M ⟩ l {x ⊤ P îx} is a Lyapunov function for the closed-loop system (6).

Proof. The proof is obtained by applying Proposition 4 and Lemma 2 to the graph H l (M ) introduced in Definition 7.

□

Remark 4 (Relations with Gain-Scheduling Stabilization of LPV) As highlighted in Remark 3, the robust stabilization problem for switched systems studied in Section 3 is equivalent to robust stabilizability for LPV systems (4). Unfortunately, this is not the case for the modedependent stabilization problem studied in this section. Indeed, in (Blanchini et al., 2007, Example 4.1) it is shown that, given M ∈ Z + and a set

F = {(A i , B i ) ∈ R n×n ×R n×m | i ∈ ⟨M ⟩}, mode-dependent stabilizability of (
3) is strictly weaker than mode-dependent (in this literature, a.k.a. gain-scheduling) stabilizability of (4).

Then, Proposition 4 cannot be applied directly for (4). On the other hand, in the particular case

B 1 = • • • = B M ∈ R n×m i.e
. when the input matrix does not depend on the mode, mode-dependent stabilizability of (4) and mode-dependent stabilizability of (4) are indeed equivalent (see [START_REF] Blanchini | Stability results for linear parameter varying and switching systems[END_REF], Proposition 2)), and thus Proposition 4 can be applied to the (more general) class of LPV systems. △

Remark 5 (Comparison with Existing Results)

The LMIs conditions presented in Corollary 2 already appeared in [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF], in a slightly different setting. Indeed, the authors of [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF] related the feasibility of the conditions with the existence of controllers relying on the knowledge of past switching sequences, and thus it is required that the controller store the past active modes. On the other hand, we generalize the results in [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF] through Proposition 4 and Lemma 2, where we rely only on the assumption that a graph is complete and deterministic to obtain LMI conditions assuring the existence of stabilizing piecewise linear controllers. Thus, more general graphs other than De Bruijn ones employed in Corollary 2 can be used too, potentially leading to more efficient results, given a particular system. On top of recovering the results of [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF] in a more general setting, the graph-theory approach used here allows us to provide piecewise linear controllers (along with min-of-quadratic Lyapunov functions) in a closed form without the necessity of observing and storing the past active modes. For a more formal discussion on the relations between graph-based stability conditions and conditions relying on past/future switching sequences, we refer to [START_REF] Della Rossa | Interpretability of path-complete techniques and memory-based Lyapunov functions[END_REF]. We point out that in [START_REF] Lee | Detectability and stabilizability of discrete-time switched linear systems[END_REF] a negative result is proved: there exist mode-dependent feedback stabilizable systems of the form (6) for which there does not exist a l ∈ N large enough such that the conditions in Corollary 2 are feasible. We do not report the proof here; it can be found (Lee & Khargonekar, 2009, Theorem 27 & Example 28) and references therein. △

Example 5

Consider a four-mode, third-order system studied in [START_REF] Blanchini | Stabilization of LPV systems: State feedback, state estimation, and duality[END_REF] In [START_REF] Blanchini | Stabilization of LPV systems: State feedback, state estimation, and duality[END_REF], by using techniques based on polyhedral Lyapunov functions, a stabilizing mode-dependent controller is provided, also providing an upper-bound on the decay rate for the closed loop, given by γ = 0.96. Here we apply Corollary 2 with different graph orders l to relate to the minimum upperbound on the decay rate by using a similar modification to the LMIs (18) as in Example 3. The results are illustrated in Table 3, showing that an increase in the order graph l allows smaller values of the decay rate γ, i.e., improving the convergence speed of the arising closed-loop. The decrease in theoretical conservatism comes at the cost of increasing the computational complexity, as an increase in l also generates an increase in the number of variables of the LMIs in Corollary 2. l = 0 l = 1 l = 2 l = 3 l = 4 min γ 0.9156 0.9058 0.9023 0.9009 0.9003 Table 3: Relation between minimum upper-bound on the decay rate γ and graph order l for Example 5.

Conclusions

We presented a graph-based construction of piecewise linear feedback controllers for discrete-time switched linear systems. The chosen graph structure influences the resulting LMI conditions, providing the user with flexible conditions in order to manage both numerical complexity and theoretical conservatism. The proposed techniques were demonstrated both in the robust case, where no knowledge of the switching signal is available, and in a mode-dependent case, where partial knowledge is utilized. The general graph framework also allowed to recover and generalize several results that already appeared in the literature. Future work will examine the conservatism introduced by different graph structures and broaden the scope of the method to handle a wider range of switched or hybrid systems. This will include exploring nonlinear dynamics and more complex switching situations.

  Relation between minimum upper bound on the decay rate γ and graph order l for Example 3. -0.3162 -0.3347 , K 12 = -0.3342 -0.3596 , K 21 = 0.6279 -0.4343 , K 22 = -0.0714 -0.4110 .

K

  11 = -0.4816 -0.3583 , K 12 = -0.4660 -0.3460 , K 13 = -0.4861 -0.3628 , K 21 = -0.3591 -0.5802 , K 22 = -0.3652 -0.5817 , K 23 = -0.4011 -0.6044 , K 31 = -0.4124 -0.5761 , K 32 = -1.0548 -0.4989 , K 33 = -0.9970 -0.5116 .

Figure 2 :Figure 3 :

 23 Figure 2: Level sets for the obtained Lyapunov functions W (x) for different graph orders l in Example 3.

  defined by B 1 = B 2 = B 3 = B 4

A continuous function β : R + × R + → R + is of class KL if β(0, s) = 0 for all s, β(•, s) is strictly increasing and unbounded for all s, β(r, •) is decreasing and β(r, s) → 0 as s → ∞, for all r.

The considered notion of completeness, arising from automata theory, should not be confused with the notion of graph completeness imposing the existence of any possible edge, i.e. E = S × S × ⟨M ⟩, more common in classic graph theory.

If the node set S is an ordered set, we can simply choose γ(x) = min S (argmin s∈S {x ⊤ Psx}), where min S (S ′ ) denotes the minimal element of S ′ ⊆ S, with respect to the order of S.

* This paper was not presented at any conference. This study was partially financed by the European Research Council (ERC) under the European Union's Horizon 2022 research and innovation program under grant agreement No 864017 -L2C and by the ANR project HANDY 18-CE40-0010.