Stefania Maria Collura

Damien Guilbert
email: damien.guilbert@univ-lorraine.fr

Gianpaolo Vitale

Massimiliano Luna

Francesco Alonge

Angel Filippo D'ippolito

Scipioni

Filippo D'ippolito

Angel Scipioni

Design and experimental validation of a high voltage ratio DC/DC converter for proton exchange membrane electrolyzer applications

Keywords: Electrolyzer, DC/DC converter, Wind energy, High conversion ratio, Control technique, Modeling

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Over the last decades, reserves of fossil fuels (e.g. oil, natural gas) have been depleting due to the growing demand [START_REF] Höök | Depletion of fossil fuels and anthropogenic climate change-A review[END_REF]. Indeed, fossil fuels are widely used for diverse applications, especially in transportation and electricity production [START_REF]AR6 Climate Change 2021: Impacts, Adaptation and Vulnerability -IPCC[END_REF]. Their increasing use all over the world leads up to the large release of greenhouse gases, responsible for global warming and hazardous meteorological phenomena [START_REF] Höök | Depletion of fossil fuels and anthropogenic climate change-A review[END_REF]. Based on the current reserves and fossil fuels production [START_REF] Bp | [END_REF], oil and natural gas should be depleted during the 21 st century (forecast exploitation time around 50 years). Hence, new energy solutions must be developed to face the depletion of fossil fuels and global warming.

Among the envisaged solutions, hydrogen is considered a promising solution for a sustainable future by providing a clean and efficient energy carrier [START_REF] Zoulias | A review on water electrolysis[END_REF]. Hydrogen can supply or store energy. Furthermore, it has a very high energy content (i.e. 120 MJ/kg) and can be used in fuel cells (FCs) to generate electricity or power and heat as byproducts.

Although hydrogen is not naturally present in nature, it can be found in fossil fuels, biomass, and water. At the present time, hydrogen can be produced in many ways such as by thermochemical, electrolytic, direct solar water splitting, and biological processes [START_REF]DOE Hydrogen and Fuel Cells Program[END_REF]. Among these different hydrogen production processes, water electrolysis is a promising option. The operation of water electrolysis is mainly based on the use of electricity to split de-ionized, pure or distilled water into hydrogen and oxygen. The chemical reaction occurs in a system called electrolyzer (EL).

On one hand, given that electricity is needed to start hydrogen production, different energy sources can be used. On the other hand, in order to drastically minimize environmental impact, the use of renewable energy sources (e.g.

wind, solar) is mandatory [START_REF] Panwar | Role of renewable energy sources in environmental protection: A review[END_REF]. Besides, over the last years, wind energy has gained a growing interest in hydrogen production, as demonstrated in the literature [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF]- [START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pinarbaşi-Kayseri[END_REF]. In particular, micro-wind energy conversion systems (µWECSs), involving power levels under 10 kW, are attractive for their low cost and easy installation.

Different types of ELs can be distinguished by their electrolyte and the charge carrier: (1) alkaline ELs; (2) proton exchange membrane (PEM) ELs; and (3) solid oxide (SO) ELs [START_REF] Koponen | Review of water electrolysis technologies and design of renewable hydrogen production systems[END_REF], [16]. For this reason, a PEM EL has been considered for carrying on this work.

Like for FCs, DC/DC converters are mandatory to interface the DC voltage grid and the EL. Generally, the EL required a very low DC voltage in order to generate hydrogen. Indeed, at nominal power, the cell voltage of an EL is equal around to 2.5 V [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF]. With the aim to optimize the reliability of the EL, the number of cells has to be limited.

As emphasized in a previous work [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF], DC/DC converters for EL applications must meet several requirements from voltage ratio, energy efficiency, and low output current ripple point of view. Among these issues, the most important requirement expected from the DC/DC converter is a high conversion ratio. Indeed, for hydrogen production via water electrolysis from WECSs, the DC bus voltage is very high (i.e. between a hundred and a thousand volts) [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF].

Given that the voltage of the electrolyzer is quite low (i.e. around ten volts) [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF], DC/DC converters for EL applications must have a high conversion ratio ability.

Based on the current literature survey carried out in a previous work [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF], three types of DC/DC converters are mainly used for these applications: (1) buck converter; (2) half-bridge converter (HBC); and (3) full-bridge converter (FBC). On one hand, classic buck converters are widely used within hydrogen production systems based on WECS [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF]- [START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pinarbaşi-Kayseri[END_REF]. In [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF], Şahin et al. have proposed a synchronous buck converter by replacing the diode with a power switch.

Therefore, the diode reverse recovery issue can be removed. In the other related papers [START_REF] Sarrias-Mena | Electrolyzer models for hydrogen production from wind energy systems[END_REF][START_REF] Zhou | Power flow control in different time scales for a wind/hydrogen/super-capacitors based active hybrid power system[END_REF][START_REF] Zhou | Real-Time Emulation of a Hydrogen-Production Process for Assessment of an Active Wind-Energy Conversion System[END_REF][START_REF] Jacobs | DC/DC Converter for a Small Scale Wind Hydrogen System[END_REF][START_REF] Zhou | Modeling and control design of hydrogen production process for an active hydrogen/wind hybrid power system[END_REF][START_REF] Muyeen | Electrolyzer switching strategy for hydrogen generation from variable speed wind generator[END_REF][START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pinarbaşi-Kayseri[END_REF], a simple buck converter is used since the study is mainly focused on energy management and cost analysis. By comparison, in [START_REF] Monroy-Morales | Modeling and control design of a Vienna rectifier based electrolyzer[END_REF],

Monroy-Morales proposes a system composed of a rectifier and a classic buck converter to supply an electrolyzer.

The rectifier is used to convert an AC grid to a DC grid; whereas the buck converter is used to adjust the required voltage to supply the electrolyzer. On the other hand, classic buck converters present several disadvantages, particularly from voltage ratio and output current ripples reduction point of view. These topologies must be particularly used for applications requiring a low voltage ratio as reported in the literature (i.e. for very small, low voltage wind turbines) [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF]- [START_REF] Genç | Cost analysis of wind-electrolyzer-fuel cell system for energy demand in Pinarbaşi-Kayseri[END_REF]. On the other side, isolated DC/DC converter topologies (e.g. HBC, FBC) provide more perspective, especially in terms of voltage conversion ratio due to the use of a transformer. In [START_REF] Andrijanovitš | Three-Level Half-Bridge ZVS DC/DC Converter for Electrolyzer Integration with Renewable Energy Systems[END_REF], [START_REF] Andrijanovitš | Analysis of State-of-the-Art Converter topologies for Interfacing of Hydrogen Buffer with Renewable Energy Systems[END_REF],

Andrijanovitš et al. have developed a zero-voltage switching half-bridge DC-DC converter to interface the DC grid and the electrolyzer within a stand-alone power supply based on renewable energy sources. By comparison, in [START_REF] Blinov | New DC/DC Converter for Electrolyser Interfacing with Stand-Alone Renewable Energy System[END_REF],

Blinov and Andrijanovitš have developed a half-bridge phase-shifted active rectifier for electrolyzer applications.

Finally, full-bridge DC-DC converters have been proposed in [START_REF] Chandrasekhar | Performance of Soft-Switched DC-DC Resonant converter for Electrolyzer[END_REF][START_REF] Viswamohan | Soft-Switching Techniques for DC-to-DC Converters in Electrolyzer Application[END_REF][START_REF] Gautam | A Comparison of Soft-Switched DC-to-DC Converters for Electrolyzer Application[END_REF][START_REF] Rajesh Babu | A comparative Analysis of DC-DC Converters for Renewable Energy System[END_REF][START_REF] Pittini | Isolated full-bridge boost DC-DC converter designed for bidrectional operation of fuel cells/electrolyzer cells in grid-tie applications[END_REF][START_REF] Cavallaro | Electrolyser in H2 Self-Producing Systems Connected to DC Link with Dedicated Phase Shift Converter[END_REF][START_REF] Shandrasekhar | Design of LCL Resonant Converter for Electrolyser, The Annals of[END_REF]. The advantages and disadvantages of each topology have been provided in a previous review work [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF]. If the transformer is well designed for a given application, these topologies are particularly suitable for hydrogen production via water electrolysis including WECS. However, so far, they have not been considered in the literature for these applications [START_REF] Guilbert | DC/DC converter topologies for electrolyzers: state-of-the-art and remaining key issues[END_REF].

Accordingly, the main purpose of this paper is to present a suitable high voltage ratio DC/DC converter for EL applications within a hydrogen production system based on a WECS. In this application, the DC bus voltage is quite high (i.e. around a hundred volts) and is very representative of the usual DC bus grid voltage met in wind power systems; consequently, a DC/DC converter with high conversion ratio is required. Another important aspect treated in the paper is the control of the DC/DC converter. In this paper, model-based control techniques are used. To this end, a mathematical model is obtained, assuming the behavior of the transformer as ideal. Using this model, two controllers are designed, one consisting of integral action, and the other consisting of an integral action together with a proportional action. Experimental tests have been carried out in order to validate the concept of the developed converter and its control for applications requiring a high conversion ratio.

This paper is divided into five sections. After this Introduction presenting the current state-of-the-art and motivations to carry out this work, Section 2 describes the investigated hydrogen production system (i.e. from the wind turbine to the EL), and particular attention is given to the proposed DC/DC converter. Then, in Section 3, details about the design and control laws for the DC/DC converter are provided. Finally, Section 4 presents the experimental test bench used to validate the proposed DC/DC converter and its control, as well as the obtained results.

THE INVESTIGATED SYSTEM AND THE CHOSEN CONVERTER TOPOLOGY

Within the framework of this work, a hydrogen production system based on a µWECS is investigated and is shown in Fig. 1. This system is composed of the following components: a horizontal axis wind turbine, a three-phase bridge rectifier, the proposed DC/DC converter, and finally a PEM electrolyzer and metal-hydride hydrogen storage. In the following, further details are provided for each component being part of the studied hydrogen production system.

Figure 1: Synopsis of the investigated hydrogen production system.

Wind turbine

In this work, the features of a horizontal axis three-phase wind turbine manufactured by FORTIS company were used. The wind turbine consists of the following parts: frame/generator/mast adapter; rotor blades/hub and finally a tail with the tail vane. The technical data of the wind turbine are provided in Table III. The wind turbine is located on the campus of the IUT (i.e. University of Technology) in Longwy (France), a section of the University of Lorraine.

The trend of turbine power with wind speed is provided in Fig. 2.

Figure 2: Power curve of the wind turbine according to the wind speed.

PMSG and Three-phase diode rectifier

The FORTIS wind turbine is connected to a brushless permanent magnet synchronous generator (PMSG), that outputs a three-phase AC voltage, whose frequency and amplitude depend on wind speed. The PMSG output is connected to a three-phase diode rectifier and a capacitive filter to obtain a DC voltage. The variation of the DC voltage with wind speed is provided in Fig. 3. According to this figure, the DC bus voltage range is from 75 V (cut in wind speed) to 500 V (rated wind speed).

Based on the wind speed measurements carried out on the campus, the considered wind speed range is from 5 to 7 m.s -1 . As a result, the DC bus voltage at the output of the rectifier is comprised between 150 V and 220 V. In order to store hydrogen produced by the PEM EL, three metal hydride storage systems (MHS 800 shown in Fig. 1)

from the HELIOCENTRIS company are used. These tanks are equipped with a low-temperature metal alloy on a titanium (Ti) and magnesium (Mg) base. They present several benefits for storage as defined below:

• Absorption of the hydrogen in the alloy lattice after the adsorption at the surface;

• can store hydrogen at high volume and low weight density (fit for stationary applications);

• low thermal conductivity.

The technical data of the MHS 800 metal hydride storage are provided in Table V.

Proposed and developed DC/DC converter

In the proposed application, the electrolyzer is supplied by a wind turbine connected to a PMSG and, in turn, to a three-phase bridge rectifier (Fig. 1). On the one hand, in the operating range, the voltage at the output of the three-phase diode rectifier (DC bus voltage) varies from 150 V to 220 V. On the other hand, the electrolyzer requires a rated power of 400 W with a voltage of 8 V. Therefore, a step-down DC/DC converter is required. Unfortunately, such a strong voltage reduction cannot be obtained with a traditional single-stage buck converter since a very low duty cycle (ranging between 0.036 and 0.053) would be necessary; in such case, the low conduction time of the power switch would require a very fast on and off time of the device, and the current supplied by the source would have a high di/dt. Moreover, at constant input-output power, a strong voltage reduction implies a very high value of the output current; thus, the inductor, the power switch, and the free-wheeling diode would be subjected to high current levels degrading the efficiency due to Joule losses. The second disadvantage could be partially solved by an interleaved step-down converter in which the current is shared among more inductors [START_REF] Şahin | Implementation of an electrolysis system with DC/DC synchronous buck converter[END_REF]. As a matter of fact, this circuit operates as more parallel connected converters and the total current is divided. Hence, the current ratings of power electronics devices could be reduced. Anyway, to assure that all currents are equally divided among inductors, and to properly control and improve the reliability of interleaved topologies, several current sensors must be used [START_REF] Mayo-Maldonado | A novel PEMFC power conditioning system based on the interleaved high gain boost converter[END_REF], [START_REF] Garrigós | Interleaved, switched-inductor, multi-phase, multi-device DC/DC boost converter for non-isolated and high conversion ratio fuel cell applications[END_REF]. However, the more current sensors are used, the more the control is complex. Anyway, the classic interleaved step-down converter does not help to solve the first disadvantage (i.e., the need for very fast devices and the related high di/dt) because it still has a limited voltage ratio.

For the above-mentioned reasons, a different topology has been chosen for the proposed DC/DC converter. The synopsis of the chosen topology is shown in Fig. 5; it encompasses a buck converter stage connected to an FBC, operating at a constant duty cycle, a transformer (2) with a suitable reduction ratio, a bridge rectifier (3), an LC filter (4). Further details regarding the design of the converter are provided in Section 3. In the following, the reasons that led to choosing this topology will be explained. A strong voltage reduction can be obtained easily only by using a transformer with a suitable voltage ratio. The working frequency of such a component is chosen as a compromise: a value lower than 100 Hz implies a heavy and bulky transformer; on the other hand, significant switching losses are exhibited when the frequency exceeds 50 kHz. Therefore, a working frequency of 25 kHz has been chosen for the transformer. Being the transformer an AC device, an FBC is required to convert the rectified DC wind turbine voltage into a 25 kHz square wave that is fed into the high side of the transformer (high voltage, low current). In addition, a classic buck converter is placed between the three-phase rectifier and the input of the FBC to have one degree of freedom (i.e., the control variable) on the voltage/current combination at each power level. The higher current appearing at the low side of the transformer is simply rectified using a diode bridge. Finally, a low-pass LC filter reduces the current and voltage ripple to extend the life of the EL. The present study focuses specifically on ELs supplied by µWECS, i.e., with power levels lower than 10 kW. As for the high-side section of the circuit, in the chosen case study a maximum current of 1.8 A is obtained at the considered power and voltage levels (400 W and 220V); in the worst case (i.e., for a typical 10 kW, 400 V three-phase wind turbine), the current is lower than 15 A.

Hence, the choice of the classic buck converter placed between the three-phase rectifier and the input of the FBC is adequate, and no current sharing is required. Specifically, the buck converter is used to obtain a moderate voltage reduction ratio, which is then further increased by the transformer. As for the low-side section of the circuit, in the chosen case study a maximum current of 50 A is obtained at the considered power and voltage levels (400 W and 8

V); in the worst case (i.e., for a 10 kW wind turbine), the current is 1250 A. However, these higher current levels are not a problem since the related circuit is completely passive (i.e., diodes are used instead of active switching devices). Even if several parallel-connected diodes are used, being the circuit passive no current sharing issues arise.

Therefore, all the electric components required by the proposed topology are readily available on the market (diodes)

or can be simply designed and built (high-frequency transformers with high secondary current). It is worth noting that, for power levels higher than 10 kW, alternative DC/DC converter topologies may be more suitable, but the analysis of this scenario falls outside the scope of the present paper. In general, it is possible to size the output filter of the FBC ((4) in Fig. 4) considering that the output waveforms of an FBC with unity turn ratio, controlled with duty ratio D at a switching frequency F sw , are the same as those of a buck converter operating at 2D and 2F sw [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF]. In the case under study, the FBC is operated at a constant duty ratio equal to 0.5. As a result, the output waveforms should have no ripples at all. However, the leakage inductance of the high-frequency transformer (HFT) and the inductance of the output filter of the FBC significantly delay the commutation between the diode paths corresponding to a positive and negative current, and the voltage at the secondary side of the HFT is zero during the commutation. Hence, the actual duty ratio of the equivalent buck converter is lower than unity and not constant. Two SEMISTACK-IGBT from SEMIKRON company will be used for implementing the proposed converter, as discussed in Section 4; hence, no sizing is required for IGBTs and diodes. On the other hand, the different passive components to design for our system are shown in Fig. 6. Actually, there are 6 passive components to design (3 inductors, 3 capacitors): one LC couple at the output of the buck converter, and two LC couples both at the input and the output of the full-bridge DC-DC converter.

Figure 6: Overview of passive components to design.

The HFT aims at reducing the voltage from about 80 V to about 8 V (EL rated voltage) and increasing the current from 5 A to 50 A. Therefore, a turn ratio equal to N=10 is required. The HFT has been built on purpose and exhibits a magnetizing inductance of 0.4 mH, a leakage inductance of 24 µH, and parasitic resistance of 5.4 mΩ. The inductor of the output filter is sized imposing an output current ripple of 1% in the worst case (i.e. when the equivalent buck converter is controlled with a duty cycle equal to D=0.5). Taking the turn ratio of the HFT into account, we obtain the minimum value for such an inductor [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF]:

= / • • , • / = 50 (1)
The chosen commercial inductor has an inductance of 50 µH and parasitic resistance of 4.5 mΩ.

As for the output capacitor, it is designed to obtain a voltage ripple (r v ) of 1% in the worst case (i.e. with a low duty ratio). Considering D=0.165 for the HFB (i.e. D=0.33 for the buck converter), the maximum corner frequency f c,max of the low pass filter can be determined as follows [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF]:

∆ = ! = " # (1 -2() * + , - (2)
. /,012 = 23 45 6 ! • " # • 7 8 = 3112 :

(

Therefore, the required minimum capacitance value is:

; ,0<= = > ? @ "+ ,, A = 53 3 (4)
The chosen commercial capacitor has a capacitance of 56 µF and parasitic resistance of 28 mΩ.

The input current of the FBC has a large and spiky ripple superimposed on the DC component. If this ripple is handled only by a capacitor, the required capacitance value is 10 mF. However, this choice would lead up to very slow dynamics of the converter. Hence, an LC filter is preferred as the input filter of the FBC. The related components will be chosen after sizing the passive components of the input buck converter. This converter aims at reducing the rectified output of the wind generator (about 200 V, 400 W) to the voltage and current levels required at the input of the FBC. Due to the non-ideal commutation and reduced equivalent duty ratio, the input voltage must be slightly higher than 80 V; actually, 92.5 V and 4.32 A are required at the input of the FBC (i.e. at the output of the buck converter). Using the usual formulas for sizing the output inductor of the buck converter [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF] and imposing a current ripple of 25% in the worst case, we obtain:

BC/D,0<= = • • , • / = 1.16 G (5)
The chosen commercial inductor has an inductance of 1.2 mH and parasitic resistance of 33.33 mΩ.

As for the output capacitor, it is designed to obtain a voltage ripple of 1%. Considering D=92.5/200=0.4625, the maximum corner frequency f c,max of the low pass filter can be determined as follows [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF]:

. /,012 = 3 45 6 ! • " # • 78 = 1737 :

Therefore, the required minimum capacitance value is:

; BC/D,0<= = > I @ "+ ,, A = 7 3 (7)
The chosen commercial capacitor has a capacitance of 20 µF and parasitic resistance of 16 mΩ. The corner frequency of the obtained LC low-pass filter is 1027 Hz. The subsequent LC low-pass filter (i.e. the input filter of the FBC) is designed to have a slightly lower corner frequency (735 Hz) and a sufficiently large input impedance compared to the output impedance of the preceding LC filter (0.4 Ω). Consequently, the chosen commercial inductance and capacitance values are 100 µH and 470 µF, respectively. The related parasitic resistances are 9 mΩ and 110 mΩ, respectively. In Fig. 6, these two passive components are named L and C.

Control laws for the proposed DC/DC converter a) Mathematical model

The electrical scheme of the overall system is shown in Fig. 7, where the buck converter drives a full-bridge whose output is connected to the primary of a transformer; the secondary of the transformer supplies the EL through a rectifier and an LC filter. The electrolyzer is physically modeled by means of resistance, R elec , and a DC generator, E elec , series connected. The voltage V DC is the DC bus voltage at the output of the three-phase rectifier (see Fig. 1), assumed equal to 200 V, which represents the input of the buck converter. The output of the latter supplies the full-bridge which operates at a duty cycle of 50% and produces an alternate voltage. Such a waveform is applied to the transformer, whose output is, firstly, rectified and then applied to the electrolyzer through the filter L 3 -C 3 . The parasitic resistances of this filter are neglected. The transformer will be assumed ideal for the development of the mathematical model, whereas for simulation its leakage inductance and resistance and magnetization inductance and resistance will be included in the physical model.

In these hypotheses, since the full-bridge input voltage, J , and the corresponding input current K are positive whatever the operative conditions, the following useful considerations for modeling the system of Fig. 7 can be made:

1. When the switch S 1 is in the ON state and IGBT1 and IGBT4 are in the ON state, the voltage J L = J at the primary of the transformer is positive; the voltage J L = MJ L at the secondary is positive too, and consequently the diodes D 6 and D 9 are ON and then the current K L = K L M ⁄ , where N is the inverse of the turns ratio (J L J L ⁄ ).

2. When the switch S 1 is in the ON state and IGBT2 and IGBT3 are in the ON state, the voltage J L = -J ; the voltage J L = MJ L at the secondary is negative, and consequently the diodes D 7 and D 8 are ON and then the current

K L = K L M ⁄ .
3. When the switch S 1 is in the OFF state and IGBT1 and IGBT4 are in the ON state, the same considerations in 1 hold. It follows that the model is the same of that relative to the operating conditions as in 1, assuming

O P/ = 0.
4. When the switch S 1 is in the OFF state and IGBT2 and IGBT3 are in the ON state, the same considerations in 2 hold. It follows that the model is the same of that relative to the operating conditions as in 2.

The above considerations clearly show that the model corresponding to the operation detailed in 1 is valid also in the operations discussed in 2, and in those discussed in 3-4 assuming V dc =0.

Regarding the development of the model describing the operation in 1, it is worth noting that the assumption of ideal transformer allows obtaining a linear and time-invariant model in which all the variables are continuous-type, in the sense that the set full-bridge, transformer, and rectifier is crucial for obtaining the desired operations in alternating current (AC), but this set does not contribute to the construction of the model itself.

In order to develop the model of the system depicted in Fig. 7, relative to the condition where S 1 is in the ON state, note that the input variables are O P/ and Q RSR/ , the state variables are the currents flowing through the inductances , and T , namely K, K and K U , respectively, and the voltages on the capacitances ; , ; and ; T , namely J / , J / and J /T . It is assumed that the current K U flowing through the inductance T is the output variable. This current coincides practically with the input current of the electrolyzer K RS and, consequently, current control of the electrolyzer is performed. The use of the current K U as output simplifies the model of the system together with the control problem, because it is a state variable, maintaining the same objective of controlling the input current of the electrolyzer.

For the operation case detailed in 1, the state-space model of the physical model shown in Fig. 7, from the DC bus voltage O P/ at the output of the three-phase rectifier to the electrolyzer, is provided below:

VW = X YZ V + \ YZ ] (8)
^= _ YZ `V (9)
where V = aK K J / J / K U J /T b `, ] = aO P/ Q RSR/ b `, ^= J _3 ,

X YZ = c d d d d d d d d e -fI g I > I > -> I > # -I g # g ? > # > # I -I 0 0 0 0 -> # Z ? > # 0 0 0 0 0 # 0 0 Z ? > ? 0 0 0 0 0 - Z # 0 Z > ? - Z # ? > ?
-> ? 0 ? h I ? i j j j j j j j j k

, \ YZ = c d d d d d e > 0 0 0 0 0 0 0 0 0 0 h I ? i j j j j j k _ YZ `= a0 0 0 0 1 0 b.
The model corresponding to l in the OFF state, is given by:

VW = X Y V + \ Y ] (10)
^= _ Y `V (11)
where X Y = X YZ , _ Y `= _ YZ `, and:

\ Y = c d d d d e 0 0 0 0 0 0 0 0 0 0 0 h I ? i j j j j k .
Starting from the developed models ( 8)-( 9) and ( 10)- [START_REF] Jacobs | DC/DC Converter for a Small Scale Wind Hydrogen System[END_REF], and assuming that the buck operates in continuous conduction mode, the average state space model is given by:

VW = XV + \] ( 12
) ^= _ `V (13)
where:

X = X Y + (X YZ -X Y )m = X Y _ `= _ Y `+ (_ YZ `-_ Y `)m = _ Y `
In which m is the duty cycle (i.e. the control input of the whole system). With these definitions, equation ( 12) can be put under this form:

VW = XV + \ Y ] + (\ YZ -\ Y )]m (14)
In order to bring back again the model ( 14) in the standard form, we define the equilibrium state of the model ( 14), corresponding to the constant input ] and a duty-cycle m * , as the state V * which satisfies the equation:

0 = XV * + \ Y ] + (\ YZ -\ Y )]m *
This state surely exists since the matrix X is invertible and it is given by:

V * = X 7 a\ Y ] + (\ YZ -\ Y )]m * b (15)
Denoting by V o = V -V * the state variable relative to the equilibrium state, the model of the system can be put under the standard form:

V o W = XV o + pm q (16) ô = _ `V o (17)
in which m q is the control variable, p = (\ YZ -\ Y )] and ô is the output relative to the equilibrium output _ `V * .

b) Controller design

The electrolyzer current to duty cycle transfer function of model ( 16)-( 17) is given by: As it can be observed, : T ≅ ˆ and, consequently, there is a pole-zero cancellation in the transfer function r s . It follows that the model is not controllable and observable at the same time. Instead, it can be easily verified that it is controllable, but not observable from the chosen output. However, it is known that the canceled pole is on the left half plane of the Laplace domain.

The bode diagrams of G p (jω) are given in Fig. 8. An analysis of these diagrams shows that the closed-loop system, realized with the transfer function G p (s) as an open-loop transfer function and unitary feedback, is unstable. Then, a controller has to be designed so that the closed loop system is stable and satisfies the following additional design requirements: With the aim to design the above controller, control theory is applied [START_REF] Mohan | Power Electronics. Converters, applications and design[END_REF], [START_REF] Ogata | Modern Control Engineering[END_REF]. First of all, an integral control G I (s)=K i1 ⁄s is added to the plant G p (s), obtaining the following open-loop transfer function:

r < (y) = { < r s (y) y ⁄ ( 19
)
where a gain { < = 0.62 is chosen in order to obtain a crossover frequency of 100 rad/s. The Bode diagrams of the open-loop transfer function [START_REF] Andrijanovitš | Three-Level Half-Bridge ZVS DC/DC Converter for Electrolyzer Integration with Renewable Energy Systems[END_REF] are given in Fig. 9. The closed-loop system is stable and the requirements of gain and phase margins are satisfied. for the second test, the DC bus voltage is modified from 150 V to 200 V while the converter operated in closed-loop.

The obtained results are shown in Fig. 11. As it can be observed in Fig. 11a, the designed DC/DC converter, although controlled by an integral type controller based on the EL current, is able to supply the electrolyzer with a good 11b, the sudden variation of the DC bus voltage leads up to an overshoot of the buck current converter around 6 A. This overshoot has to be as low as possible in order to protect the power switch against high current stress. Besides, the current ripples are higher due to the DC bus voltage increase. Regarding the EL current, a low overshoot can be noticed (around 2 A) before reaching its steady-state operation (around 10 A) quickly.

The steady-state operation of the buck converter and electrolyzer is shown in Fig. 12, in both the above-mentioned experiments, is in the range of continuous conduction mode, with the duty-cycle greater than 0.3. Besides, the EL current ripple is close to zero due to the use of an LC filter between the full-bridge and the EL, whereas the EL voltage is perfectly continuous. Therefore, by minimizing this ripple, the performance of the whole system DC/DC converter-electrolyzer has been enhanced.

CONCLUSION

The main objectives of the work were to design and develop a high-voltage ratio DC/DC converter based on buck and full-bridge DC/DC converters. Indeed, based on the current literature, it was demonstrated that current hybrid renewable energy systems with a hydrogen buffer storage are limited to low-power applications. This can be explained by the use of classical DC/DC converters (buck converters for electrolyzer applications) which present several drawbacks from voltage ratio and output current ripple points of view. Therefore, in order to move towards medium and high-power applications, DC/DC converters must feature high conversion ratio ability.

The design of the whole system including the different filters has been carried out with the aim to minimize the output current ripple Besides, the control has been developed to ensure good dynamic performances in case of sudden variations of the operating conditions. Experimental tests have allowed validating the proposed high-voltage ratio converter and the control as well. The obtained experimental results demonstrate the good performances from steady-state error and stability points of view.

Furthermore, the current ripple at the input of the electrolyzer is close to zero, as expected, and allows optimizing the electrolyzer performance from efficiency and hydrogen production points of view. Future works will be carried out in order to develop other robust controls with the aim to enhance the performance of the proton exchange membrane electrolyzer.

  full-bridge converter (V)vc1voltage across C1 (V).

  vc2voltage across C2 (V).

  vc3voltage across C3 (V).

  v1ttransformer input voltage (V).

Figure 3 :

 3 Figure 3: Variation of DC bus voltage at the output of the rectifier according to wind speed.

Figure 4 :

 4 Figure 4: Operating principle of the PEM EL.

Figure 5 :

 5 Figure 5: Synopsis of the proposed DC/DC converter.

Figure 7 :

 7 Figure 7: Physical model of the set controlled buck, full-bridge, transformer, rectifier, and electrolyzer.

  _ `(yz -X)7 p = { s (47| I )(47| # )(47| ? ) (47s I )(47s # )(47s ? )(47s } )(47s ~)(47s • ) 4.85 × 10 ƒ , : = -3.125 × 10 " , : = -1.193 × 10 † , : T = -2.857 × 10 ‡ , ˆ = -2.845 × 10 ‡ , ˆ = -640 -‰23680, ˆT = -640 + ‰23680, ˆ † = -1147, ˆ ‡ = -104 -‰1311, ˆ" = -104 + ‰1311

Figure 8 :

 8 Figure 8: Bode diagram of Gp (jω).

1 . 3 .

 13 Bandwidth: \ ≥ 20 Hz, 2. Type of the system: 1, Phase margin: G OE ≥ 60°, 4. Attenuation at 1310 rad/s (first resonance frequency): X L= ≥ 10 dB, 5. Gain margin: G Ž ≥ 6 dB.

Figure 9 :

 9 Figure 9: Bode diagrams of function r < (‰•).

4 .

 4 EXPERIMENTAL TEST BENCH AND VALIDATION 4.1. Developed experimental test benchIn order to validate the high voltage ratio converter and its control, experimental tests have been performed. The experimental test bench set up for this work is shown in Fig.10. It is composed of an auto-transformer which substitutes the wind generator, a 3-phase diode rectifier combined with a buck converter and LC filters, a full-bridge converter followed by a transformer, a diode bridge rectifier including an LC filter, a PEM EL, a metal-hydride hydrogen storage system, de-ionized water tanks, a DS1104 dSPACE board, an interface board, and finally a computer. The 3-phase rectifier, buck, and full-bridge DC/DC converters have been realized by means of two SEMISTACK-IGBT from SEMIKRON company (as shown in Fig.10). The control of the buck converter, and consequently of the whole DC/DC converter, is closed-loop (as reported in Section 3), whereas that of the full bridge converter is open-loop (based on a duty cycle of 50%), and both have been first carried out in Matlab/Simulink environment, and then implemented into a DS1104 dSPACE board. The control of the buck converter is based on the measurement of the EL current, which is acquired by a PAC10 current clamp from Chauvin Arnoux company. The PWM gate control signals to control the buck and full-bridge DC/DC converters are generated by the dSPACE board.However, the voltage levels of the generated PWM signals from the dSPACE board, 0-5 V, are not suitable to drive the SEMIKRON driver boards SKHI 22, 0-15 V. For this reason, an interface board is used between the dSPACE board and the driver boards to convert the control signals (0-5V) to (0-15V). Unlike the first SEMISTACK-IGBT, used as a diode rectifier and buck converter, the interface board is implemented on the second SEMISTACK-IGBT, used as full bridge converter. The system specifications are summarized in TableVI.

Figure 10 :

 10 Figure 10: Developed experimental test bench in laboratory: 1) autotransformer (input), 2) dSPACE control desk,

  Obviously, the static behavior is of type 1. By comparison, in Fig.

Figure 11 :

 11 Figure 11: (a) first dynamic test, channel 1: buck current (5 A.div-1), channel 2: EL voltage (10 V.div-1). (b)

Figure 12 :

 12 Figure 12: Steady-state operation, channel 1: buck current (10 A.div-1), channel 2: EL current (10 A.div-1),

  Table I indicates the main features of each technology; whereas Table II presents the advantages and disadvantages of each technology. From Tables I and II, alkaline and PEM ELs are currently the two main technologies, which are commercially available. Alkaline ELs are the most mature and widespread compared to PEM ELs (under development). As highlighted in Table II, alkalineELs have a longer lifespan than PEM ELs. However, PEM ELs have several advantages over alkaline ELs, such as compactness, fast system response, wide partial load range and high flexibility in terms of operation. As a result, this technology is an attractive option for integration into the grid including renewable power generating systems[16].


Jin Zhang
email: zhangjin1116@126.com

Wen Kang
email: kangwen@amss.ac.cn

Emilia Fridman
email: emilia@tauex.tau.ac.il

Alexandre Seuret
email: aseuret@laas.fr

Delayed stabilization of parabolic PDEs via augmented Lyapunov functionals and Legendre polynomials

Keywords: Parabolic PDEs, Time-delays, Lyapunov functionals, Bessel-Legendre inequality, LMIs

We first study stabilization of heat equation with globally Lipschitz nonlinearity. We consider the point measurements with constant delay and use spatial decomposition. Inspired by recent developments in the area of ordinary differential equations (ODEs) with time-delays, for the stability analysis, we suggest an augmented Lyapunov functional depending on the state derivative that is based on Legendre polynomials. Global exponential stability conditions are derived in terms of linear matrix inequalities (LMIs) that depend on the degree N of Legendre polynomials. The stability conditions form a hierarchy of LMIs: if the LMIs hold for N , they hold for N + 1. The dual observer design problem with constant delay is also formulated. We further consider stabilization of Korteweg-de Vries-Burgers (KdVB) equation using the point measurements with constant delay. Due to the third-order partial derivative in KdVB equation, the Lyapunov functionals that depend on the state derivative are not applicable here, which is different from the case of heat equation. We suggest a novel augmented Lyapunov functional depending on the state only that leads to improved regional stability conditions in terms of LMIs. Finally, numerical examples illustrate the efficiency of the method.

Introduction

Control of partial differential equations (PDEs), e.g. heat equation, becomes an active research topic [START_REF] Curtain | An introduction to infinitedimensional linear systems theory[END_REF][START_REF] Pisano | On the ISS properties of a class of parabolic DPS' with discontinuous control using sampled-inspace sensing and actuation[END_REF]. It is of interest to design a control law for PDEs using delayed inputs/outputs. Constructive conditions in terms of LMIs for delayed control of PDEs were presented in [START_REF] Fridman | Sampled-data distributed H∞ control of transport reaction systems[END_REF][START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]. The derived conditions allow to give an upper bound on the delay preserving the performance (e.g. exponential decay rate). Moreover, KdVB equation has been derived as the governing evolution equation for waves propagating in fluid-filled elastic or viscoelastic tubes incorporating the effects of dispersion, dissipation and nonlinearity [START_REF] Demiray | A travelling wave solution to the KdV-Burgers equation[END_REF]. The objective of the present work is the derivation of less conservative L-MI conditions for the stability analysis of heat and KdVB equations with time-delay. In application to chain of subobservers as used in [START_REF] Ahmed-Ali | Observer design for a class of parabolic systems with large delays and sampled measurements[END_REF], such conditions will allow to reduce the order of the chain.

For the stability analysis of ODEs with time-delay, Jensen's inequality [START_REF] Gu | An integral inequality in the stability problem of timedelay systems[END_REF][START_REF] Gu | Stability of time delay systems[END_REF] and Wirtinger-based integral inequality [START_REF] Seuret | Wirtinger-based integral inequality: application to time-delay systems[END_REF] were usually employed. Several contributions to derive less conservative integral inequalities for time-delay systems were provided in [START_REF] Park | Auxiliary functionbased integral inequalities for quadratic functions and their applications to time-delay systems[END_REF][START_REF] Zeng | Free-matrix-based integral inequality for stability analysis of systems with timevarying delay[END_REF]. Recently, a novel integral inequality so-called Bessel-Legendre (B-L) inequality that encompasses Jensen's inequality and Wirtinger-based integral inequality as particular cases was introduced in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF] by using Legendre polynomials. The latter presented a hierarchy of LMI conditions that are competitive with [START_REF] Gu | An integral inequality in the stability problem of timedelay systems[END_REF][START_REF] Gu | Stability of time delay systems[END_REF][START_REF] Park | Auxiliary functionbased integral inequalities for quadratic functions and their applications to time-delay systems[END_REF][START_REF] Seuret | Wirtinger-based integral inequality: application to time-delay systems[END_REF][START_REF] Zeng | Free-matrix-based integral inequality for stability analysis of systems with timevarying delay[END_REF] in terms of conservatism and of complexity. In [START_REF] Baudouin | Stability analysis of a system coupled to a heat equation[END_REF], stability analysis of a coupled ODE-heat equation was presented via a new B-L inequality.

In the present work, we consider stabilization of heat and KdVB equations in the presence of constant output de-lay. Note that in the case of constant delay, input delay can be always moved to output by changing the time. We first study stabilization of heat equation under the point measurements with constant delay by using spatial decomposition approach (as introduced in [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]). For the stability analysis of the closed-loop system, we suggest an augmented Lyapunov functional depending on the state derivative and that is based on Legendre polynomials. Such functionals extend the Lyapunov constructions of [START_REF] Ahmed-Ali | Observer design for a class of parabolic systems with large delays and sampled measurements[END_REF][START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]. Sufficient stability conditions are derived in terms of LMIs that are parameterized by the degree N of the polynomials. The same LMIs are also applicable to the case of stabilization under the spatially averaged measurements with constant delay. As a by-product, for the stability analysis of heat equation with constant state delay, we present the LMIs that appear to guarantee the exponential stability of delayed ODEs and that in the numerical example recover the analytical upper bounds on delay with a finite degree of the polynomials. We also formulate the dual observer design problem.

We further consider stabilization of KdVB equation under the point measurements with constant delay as studied in [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF]. Due to the third-order partial derivative in KdVB equation, the Lyapunov functionals that depend on the state derivative are not applicable here, which is different from the case of heat equation. We suggest a novel augmented Lyapunov functional encompassing the Lyapunov functional introduced in [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF] as a particular case that leads to improved regional stability conditions.

As in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF][START_REF] Seuret | Complete quadratic Lyapunov functionals for distributed delay systems[END_REF], the derived stability conditions form a hierarchy of LMIs: if the LMIs hold for N , they hold for N + 1. By solving the LMIs, improved upper bounds on delay that preserve the stability are found. Finally, numerical examples illustrate the efficiency of the method. Some preliminary results for the scalar heat equation were presented in [START_REF] Zhang | Improved observer design for heat equation with constant measurement delay via Legendre polynomials[END_REF].

Notation and Preliminaries

Throughout the paper, R n denotes the n-dimensional Euclidean space with the vector norm |•|, R n×m is the space of all n × m real matrices. The notation P > 0, for P ∈ R n×n means that P is symmetric and positive definite. 0 n×m (0 n ) stands for the matrix in R n×m (R n×n ) whose entries are zero. For any square matrix X, He{X} = X + X T . L 2 (0, l) stands for the Hilbert space of square integrable vector (or scalar for n = 1) functions z : (0, l) → R n with the norm

z L 2 (0,l) = l 0 z T (x)z(x)dx. H i (0, l) (i = 1, 2) are the Sobolev space: H i (0, l) = {z : d j z dx j ∈ L 2 (0, l) ∀0 ≤ j ≤ i} with the norm z H i (0,l) = i j=0 d j z dx j 2 L 2 (0,l) .
The notation ( k i ) refers to the binomial coefficients given by

k! (k-i)!i! . Let L k (s) (k ∈ N 0 ) be the shifted Legendre polynomials over interval [-h, 0]: L k (s) = k i=0 (-1) i+k ( k i ) k+i i s+h h i . (1.1)
These polynomials satisfy the following properties:

Property 1.1 (i) Orthogonality: ∀k, i ∈ N 0 , 0 -h L k (s)L i (s)ds = 0, k = i, h 2k+1 , k = i. (1.2)
(ii) Boundary conditions:

∀k ∈ N 0 , L k (0) = 1, L k (-h) = (-1) k . (1.3) (iii) Differentiation: Lk (s) = 0, k = 0, k-1 i=0 2i+1 h (1 -(-1) k+i )L i (s), k ≥ 1.
(1.4)

We will employ extended Bessel-Legendre inequality that is obtained by integration in x ∈ [0, l] of the inequality of Lemma 3 of [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]:

Lemma 1.1 Consider a function z ∈ L 2 ([-h, 0]; L 2 (0, l)),
and scalars h > 0 and l > 0. Then for any n × n matrix R > 0, the following inequality holds for all N ∈ N 0 :

l 0 0 -h z T (x, t + s)Rz(x, t + s)dsdx ≥ N k=0 2k+1 h l 0 (Θ k (x, t)) T RΘ k (x, t)dx, (1.5)
where

Θ k (x, t) (k ∈ N 0 ) correspond to the projection of z(x, t + s) over L k (s)
given by (1.1):

Θ k (x, t) = 0 -h L k (s)z(x, t + s)ds, k ∈ N 0 . (1.6)
2 Stabilization of heat equation with an output delay

Problem formulation

Consider the following semilinear diffusion equation:

z t (x, t) = D z(x, t) -βz x (x, t) + Az(x, t) +φ(z(x, t), x, t) + B N j=1 b j (x)u j (t), t ≥ 0, x ∈ [0, l], l > 0 (2.1)
under the Dirichlet boundary conditions

z(0, t) = z(l, t) = 0. (2.2)
Here z(x, t) = [z 1 (x, t), . . . , z n (x, t)] T ∈ R n is the state, u j (t) ∈ R r (j = 1, . . . , N ) are the control inputs, A ∈ R n×n and B ∈ R n×r are constant matrices and β ∈ R n×n is the diagonal matrix of convection coefficients. The diffusion term is given by

D z(x, t) = ∂ ∂x (d 1 (x)z 1 x (x, t)), . . . , ∂ ∂x (d n (x)z n x (x, t)) T (2.3) with d i (x) ∈ C 1 satisfying 0 < d i 0 ≤ d i (x) (i = 1, . . . , n) for x ∈ [0, l].
Following [START_REF] Am | Network-based H∞ filtering of parabolic systems[END_REF], we assume that for some positive definite [START_REF] Curtain | An introduction to infinitedimensional linear systems theory[END_REF] for φ(z, x, t) = φ M z).

Ψ ∈ R n×n , function φ ∈ C 1 satisfies φ T (z, x, t)φ(z, x, t) ≤ z T Ψz (2.4) for all z ∈ R n , x ∈ [0, l], t ≥ 0.
As in [START_REF] Fridman | Sampled-data distributed H∞ control of transport reaction systems[END_REF][START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF], the control inputs u j (t) enter (2.1) through the shape functions

b j (x) = 1, x ∈ Ω j , 0, otherwise, j = 1, . . . , N, (2.5)
where Ω j = [x j-1 , x j ) (j = 1, . . . , N ) divide the domain [0, l] into N sub-intervals. Here points 0 = , j = 1, . . . , N.

x 0 < x 1 < • • • < x N = l satisfy x j -x j-1 = ∆ j ≤ ∆,
(

In addition, the measurement is affected by a time-delay which is assumed to be constant, denoted hereafter by h > 0. For the control design, our method works also for unknown but constant h that belongs to a prescribed interval [h 1 , h 2 ]. Then, point measurements are provided by N sensors distributed over the whole domain [0, l]:

y j (t) = 0, t ≤ h, z(x j , t -h), t > h, j = 1, . . . , N. (2.7)
Note that our model (2.1) may present a metal bar of the length of l that is heated along its length. In this case we measure the temperature in the spatial points along the bar (see e.g. Example 1.1.2 in [START_REF] Curtain | An introduction to infinitedimensional linear systems theory[END_REF]). Another application of spatially sampled measurements was recently given in application to multi-agents in [START_REF] Terushkin | Network-based deployment of nonlinear multi agents over open curves: a PDE approach[END_REF], where the measurements are provided by leaders placed in the points xj .

We design for system (2.1) a stabilizing controller

u j (t) = -Ky j (t), t ≥ 0, j = 1, . . . , N (2.8)
with the gain K ∈ R r×n . Denote by the errors between the delayed state z(x, t -h) and point measurements z(x j , th):

f j (x, t -h) ∆ = z(x, t -h) -z(x j , t -h) = x xj z ξ (ξ, t -h)dξ, x ∈ Ω j , j = 1, . . . , N.
(2.9)

Then the closed-loop system (2.1), (2.7), (2.8) has the form:

z t (x, t) = D z(x, t) -βz x (x, t) + Az(x, t) +φ(z(x, t), x, t) -BK(1 -χ [0,h] (t)) ×[z(x, t -h) -f j (x, t -h)], t ≥ 0, (2.10)
where x ∈ Ω j , j = 1, . . . , N and χ [0,h] (t) denotes the characteristic function of the time interval [0, h].

Now we consider the well-posedness of system (2.10). Let H 1 0 (0, l) = {θ ∈ H 1 (0, l) : θ(0) = θ(l) = 0} and

D(A) = {θ ∈ H 2 (0, l) : θ(0) = θ(l) = 0}.
By using the step method (see e.g. [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]), it can be shown that for any initial condition z(•, 0) ∈ H 1 0 (0, l), there exists a unique strong solution of (2.10) such that

z ∈ L 2 (0, ∞; D(A)) ∩ C([0, ∞); H 1 0 (0, l)), z t ∈ L 2 (0, ∞; L 2 (0, ∞))
and equation (2.10) hold almost everywhere on [h, ∞).

Improved stability conditions

For the stability analysis of system (2.10), we suggest the following augmented Lyapunov functional (that extends the Lyapunov constructions of [START_REF] Ahmed-Ali | Observer design for a class of parabolic systems with large delays and sampled measurements[END_REF][START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]): (2.11) where N ∈ N 0 , and

V N (t) = V P N + V P2 + V S1 + V S2 + V R , t ≥ h,
V P N = l 0 ζ T N (x, t)P N ζ N (x, t)dx, V P2 = n i=1 l 0 P i 2 d i (x)(z i x (x, t)) 2 dx, V S1 = l 0 t t-h e -2α(t-s) z T (x, s)S 1 z(x, s)dsdx, V S2 = l 0 t t-h e -2α(t-s) z T x (x, s)S 2 z x (x, s)dsdx, V R = h l 0 t t-h e -2α(t-s) (s -t + h)z T s (x, s)Rz s (x, s)dsdx (2.12) with ζ N (x, t) = col{z(x, t), Θ 0 (x, t), . . . , Θ N -1 (x, t)} (2.13) and with (N + 1)n × (N + 1)n matrix P N , n × n matri- ces P 2 = diag{P 1 2 , . . . , P n 2 } > 0, S 1 > 0, S 2 > 0, R > 0, a scalar α > 0 and Θ k (x, t) (k = 0, . . . , N -1)
given by (1.6). The terms V S1 and V R are the PDE extensions of the standard Lyapunov functionals for delay-dependent analysis [START_REF] Fridman | Introduction to time-delay systems: analysis and control[END_REF][START_REF] Fridman | Exponential stability of linear distributed parameter systems with time-varying delays[END_REF]. The term V S2 is introduced to compensate l 0 z x (x, t -h)dx (instead of Halanay's inequality employed in [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF] for the case of time-varying delays). Note that z(•, s) for s ∈ [0, h] in (2.12) is defined as solution of (2.10), (2.2) with the initial condition z(•, t) ∈ H 1 0 (0, l).

Theorem 2.1 For given positive scalars h, l, ∆, α and a positive integer N , assume that there exist (N +1)n×(N + 1)n matrix P N , n × n matrices P 1 = diag{P 1 1 , . . . , P n 1 },

P 2 = diag{P 1 2 , . . . , P n 2 } > 0, S 1 > 0, S 2 > 0, R > 0, W > 0 and a scalar λ φ > 0 that satisfy PN > 0, (2.14)
Υ N + Λ N + U N ≤ 0, (2.15)
W -e -2αh S 2 ≤ 0, (

where Proof : Consider Lyapunov functional V N (t) given by (2.11). Lemma 1.1 gives a lower bound as follows

PN = P N + 1 h e -2αh S 1 diag{0, 1, . . . , 2N -1}, Υ N = He{G T N P N H N } + 2αG T N P N G N -e -2αh N k=0 (2k + 1)Γ T N (k)RΓ N (k), G N = I n 0 n×5n 0 n×N n 0 N n×n 0 N n×5n hI N n , H N = F T N Γ T N (0) . . . Γ T N (N -1) T , F N = 0 n I n 0 n×(N +4)n , Γ N (k) = I n , 0 n×2n , (-1) k+1 I n , 0 n×2n , γ 0 N k I n , . . . , γ N -1 N k I n , γ j N k = -(2j + 1)(1 -(-1) k+j ), if j ≤ k, 0, if j > k, U N = π 2 l 2 diag{0 3n , W -e -2αh S 2 , 0 (N +2)n } (2.17) and Λ N = {Λ ij } is symmetric matrix composed of Λ 11 = He{P 1 A} + S 1 + λ φ Ψ, Λ 12 = -P 1 + A T P 2 , Λ 14 = -Λ 16 = -P 1 BK, Λ 15 = P 1 , Λ 22 = -2P 2 + h 2 R, Λ 23 = -P 2 β, Λ 24 = -Λ 26 = -P 2 BK, Λ 25 = P 2 , Λ 33 = S 2 + He{D 0 (αP 2 -P 1 )}, Λ 44 = -e -2αh S 1 , Λ 55 = -λ φ I n , Λ 66 = -π 2 ∆ 2 W, Λ 77 = 0 N n (2.18) with Ψ given by (2.4) and D 0 = diag{d 1 0 , . . . , d n 0 }. Then there exists M 0 > 0 such that a unique strong solution of (2.10), (2.2) initialized with z(•, t) ∈ H 1 0 (0, l) for t ∈ [0, h] satisfies the inequality z(•, t) 2 H 1 0 (0,l) ≤ M 0 e -2α(t-h) h 0 [ z(•, s) 2 H 1 0 (0,l) + z s (•, s) 2 L 2 (0,l) ]ds, t ≥ h, (2.19
V N (t) ≥ l 0 ζ T N (x, t)P N ζ N (x, t)dx + l 0 t t-h e -2α(t-s) z T (x, s)S 1 z(x, s)dsdx, ≥ l 0 ζ T N (x, t)P N ζ N (x, t)dx +e -2αh l 0 t t-h z T (x, s)S 1 z(x, s)dsdx ≥ l 0 ζ T N (x, t) PN ζ N (x, t)dx.
Thus, the positivity of V N (t) results from PN > 0 given by (2.14).

For t ≥ h, differentiating V P N along (2.10) yields

VP N = 2 l 0 ζ T N (x, t)P N ζN (x, t)dx, (2.20)
where

ζN (x, t) = col{z t (x, t), Θ 0 t (x, t), . . . , Θ N -1 t (x, t)}.
Via (2.10) and (2.13), the definitions of G N and F N given by (2.17) yield

ζ N (x, t) = G N η N ,j , z t (x, t) = F N η N ,j , (2.21)
where

η N ,j = col{z(x, t), z t (x, t), z x (x, t), z(x, t -h), φ(z(x, t), x, t), f j (x, t -h), 1 h Θ 0 (x, t), . . . , 1 h Θ N -1 (x, t)} with f j (x, t -h) defined by (2.9).
Then, an integration by parts ensures that

Θ k t (x, t) = L k (s)z(x, t + s)| 0 s=-h - 0 -h d ds L k (s) z(x, t + s)ds = Γ N (k)η N ,j , k = 0, . . . , N , (2.22)
where we applied (1.3) and (1.4) with Γ N (k) given by (2.17). From (2.20), (2.21) and (2.22), it follows that

VP N = 2 N j=1 xj xj-1 η T N ,j G T N P N H N η N ,j dx, (2.23)
where H N is given by (2.17).

We have

VP2 = 2 n i=1 l 0 P i 2 d i (x)z i x (x, t)z i xt (x, t)dx, (2.24)
VS1 + 2αV S1 = l 0 z T (x, t)S 1 z(x, t)dx -e -2αh l 0 z T (x, t -h)S 1 z(x, t -h)dx, (2.25) VS2 + 2αV S2 = l 0 z T x (x, t)S 2 z x (x, t)dx -e -2αh l 0 z T x (x, t -h)S 2 z x (x, t -h)dx.
(2.26)

Note that z i xt (i = 1, . . . , n) in VP2 are well-defined as in Remark A.1 of [START_REF] Fridman | Sampled-data distributed H∞ control of transport reaction systems[END_REF].

Further by using Lemma 1.1 and taking into account (2.22), we obtain

VR + 2αV R = h 2 l 0 z T t (x, t)Rz t (x, t)dx -h l 0 t t-h e -2α(t-s) z T s (x, s)Rz s (x, s)dsdx ≤ h 2 l 0 z T t (x, t)Rz t (x, t)dx -e -2αh N k=0 (2k + 1) × N j=1 xj xj-1 η T N ,j Γ T N (k)RΓ N (k)η N ,j dx. (2.27)
To cancel the term of the right hand side of (2.24), we employ the descriptor method [START_REF] Fridman | Introduction to time-delay systems: analysis and control[END_REF][START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF], where the right-hand side of

0 = 2 l 0 [z T (x, t)P 1 + z T t (x, t)P 2 ][-z t (x, t) + D z(x, t) -βz x (x, t) + Az(x, t) + φ(z(x, t), x, t) -BKz(x, t -h)]dx + 2 N j=1 xj xj-1 [z T (x, t)P 1 +z T t (x, t)P 2 ]BKf j (x, t -h)dx (2.28) with some n × n matrix P 1 = diag{P 1 1 , . . . , P n 1 } is added to VN (t) + 2αV N (t).
An integration by parts and substitution of the Dirichlet boundary conditions (2.2) lead to

2 l 0 [z T (x, t)P 1 + z T t (x, t)P 2 ] D z(x, t)dx = 2 n i=1 l 0 [P i 1 z i (x, t) + P i 2 z i t (x, t)] ∂ ∂x (d i (x)z i x (x, t))dx = -2 n i=1 l 0 P i 1 d i (x)(z i x (x, t)) 2 dx -VP2 , (2.29)
l 0 z T (x, t)P 1 βz x (x, t)dx = l 0 z T x (x, t)P 1 βz(x, t)dx. Thus, by noting that matrix P 1 β is diagonal, we have 2 l 0 z T (x, t)P 1 βz x (x, t)dx = 0.

(2.30)

To "compensate" the cross terms in (2.28) with f j (x, t -h) given by (2.9), for any n × n matrix W > 0 application of Wirtinger's inequality (see e.g. (3.177) in [START_REF] Fridman | Introduction to time-delay systems: analysis and control[END_REF]) yields

xj xj-1 f T j (x, t -h)W f j (x, t -h)dx = xj xj x xj z T ξ (ξ, t -h)dξ W x xj z ξ (ξ, t -h)dξ dx + xj xj-1 x xj z T ξ (ξ, t -h)dξ W x xj z ξ (ξ, t -h)dξ dx ≤ ∆ 2 π 2 xj xj-1 z T x (x, t -h)W z x (x, t -h)dx, j = 1, .
. . , N. Thus, the following inequality holds:

N j=1 xj xj-1 z T x (x, t -h)W z x (x, t -h) -π 2 ∆ 2 f T j (x, t -h)W f j (x, t -h) dx ≥ 0.
(2.31)

From (2.4) we have 0 ≤ λ φ l 0 z T (x, t)Ψz(x, t) -φ T (z, x, t)φ(z, x, t) dx (2.32)
with some scalar λ φ > 0.

Condition Υ N + Λ N + U N ≤ 0 in (2.15) implies S 2 + He{D 0 (αP 2 -P 1 )} ≤ 0, thus, αP 2 -P 1 ≤ 0. Taking into account d i 0 ≤ d i (x) (i = 1, . . . , n) we have 2 n i=1 l 0 d i (x)(αP i 2 -P i 1 )(z i x (x, t)) 2 dx ≤ l 0 z T x (x,
≤ N j=1 xj xj-1 η T N ,j (Υ N + Λ N )η N ,j dx - l 0 z T x (x, t -h)(e -2αh S 2 -W )z x (x, t -h)dx ≤ N j=1 xj xj-1 η T N ,j (Υ N + Λ N + U N )η N ,j dx, t ≥ h, (2.34
) where we applied Wirtinger's inequality with (2.16). Here Υ N and U N are given by (2.17) and Λ N is composed of (2.18). Thus, from (2.15) we have VN (t) + 2αV N (t) ≤ 0 for all t ≥ h implying V N (t) ≤ e -2α(t-h) V N (h) for all t ≥ h. Due to the positivity of V N (t), there exists some M 1 > 0 such that

V N (t) ≥ M 1 z(•, t) 2 H 1 0 (0,l) , t ≥ h. (2.35)
Moreover, V N (h) can be upper bounded by

V N (h) ≤ M 2 h 0 [ z(•, s) 2 H 1 0 (0,l) + z s (•, s) 2 L 2 (0,l) ]
y j (t) =    0, if t ≤ h, 1 ∆j xj xj-1 z(ξ, t -h)dξ, if t > h, j = 1, . . . , N,
we obtain (2.10) with

f j (x, t -h) ∆ = 1 ∆j xj xj-1 [z(x, t -h) -z(ξ, t -h)]dξ.
(2.37)

Since xj xj-1 f j (ξ, t -h)dξ = 0 and d dx f j (x, t -h) = z x (x, t - h), for any n × n matrix W > 0 the Poincaré's inequality [16] leads to xj xj-1 f T j (x, t -h)W f j (x, t -h)dx ≤ ∆ 2 π 2 xj xj-1 d dx f T j (x, t -h)W f j (x, t -h) 2 dx ≤ ∆ 2 π 2 xj xj-1 d dx f T j (x, t -h) W d dx f j (x, t -h) dx, = ∆ 2 π 2 xj xj-1 z T
x (x, t -h)W z x (x, t -h)dx, j = 1, . . . , N. Then, we obtain (2.31) with f j (x, t -h) given by (2.37). Thus, LMIs of Theorem 2.1 are applicable to the case of spatially point or averaged measurements with time-delay.

Consider next d i (x) ≡ d i 0 (i = 1, . . . , n), φ(z, x, t) ≡ 0, β = 0 and the measurement y(t) = 0, if t ≤ h, z(x, t -h), if t > h.
(2.38)

Thus, system (2.1) becomes

z t (x, t) = D z(x, t) + Az(x, t) + Bu(t), t ≥ 0, x ∈ [0, l], l > 0 (2.39)
that is stabilized by a state-feedback

u(t) = -Ky(t), t ≥ 0 (2.40)
with y(t) given by (2.38) and K ∈ R r×n . Then the resulting closed-loop system has the form

z t (x, t) = D 0 z xx (x, t) + Az(x, t) -BK(1 -χ [0,h] (t))z(x, t -h), t ≥ 0 (2.41)
with D 0 = diag{d 1 0 , . . . , d n 0 } and constant matrices A ∈ R n×n , B ∈ R n×r , K ∈ R r×n and χ [0,h] (t) defined below (2.10). Since the diffusion term in (2.41) is constant, we choose P 1 in (2.28) as a non-diagonal matrix (see e.g. [START_REF] Solomon | Stability and passivity analysis of semilinear diffusion PDEs with time-delays[END_REF]):

l 0 z T (x, t)He{P 1 D 0 }z xx (x, t)dx = - l 0 z T x (x, t)He{P 1 D 0 }z x (x,
ż(t) + (D 0 -A)z(t) + BK z(t -h) = 0.
(2.43)

Note that system (2.43) corresponds to the first modal dynamics (with k = 1) in the modal representation of the Dirichlet boundary-value problem (2.2), (2.41) with l = π

żk (t) + (k 2 D 0 -A)z k (t) + BK zk (t -h) = 0, k = 1, 2, . . . (2.44)
projected on the eigenfunctions of the operator D 0 ∂ 2 ∂x 2 (this operator has eigenvalues -k 2 D 0 , see e.g. [START_REF] Wu | Theory and applications of partial functional differential equations[END_REF]). The stability of (2.2), (2.41) implies the stability of (2.44). Thus, LMIs of Corollary 2.1 are tight [START_REF] Fridman | Exponential stability of linear distributed parameter systems with time-varying delays[END_REF].

The dual observation problem

Consider the semilinear diffusion equation

z t (x, t) = D z(x, t) -βz x (x, t) + Az(x, t) + u(t) +σ(z(x, t), x, t), t ≥ 0, x ∈ [0, l], l > 0
(2.45) under the Dirichlet boundary conditions (2.2) with the state z(x, t) ∈ R n and the control input u(t) ∈ R n . The diffusion term is given by (2.3) and σ is a known function of class C 1 satisfying σ T z σ z ≤ Ψ with some positive definite Ψ ∈ R n×n , where σ z denotes the partial derivative of a function σ(z, x, t) with respect to z.

We suggest a nonlinear observer of the form ẑt (x, t) = D ẑ(x, t) -β ẑx (x, t) + Aẑ(x, t) + u(t)

+σ(ẑ(x, t), x, t) + L N j=1 b j (x)[y j (t) -ẑ(x j , t -h)], t ≥ 0, x ∈ [0, l], l > 0 (2.46) under the Dirichlet boundary conditions ẑ(0, t) = ẑ(l, t) = 0, (2.47)
where y j (t) (j = 1, . . . , N ) are given by (2.7), L ∈ R n×n is an observer gain and ẑ(x, t) = 0 for t ∈ [-h, 0].

Then using function χ [0,h] (t) defined below (2.10), the estimation error ê(x, t) = z(x, t)-ẑ(x, t) satisfies the Dirichlet boundary value problem êt (x, t) = D ê(x, t) -βê x (x, t) + Aê(x, t) + φ(ê(x, t), x, t)

-L N j=1 b j (x)(1 -χ [0,h] (t))ê(x j , t -h), t ≥ 0, (2.48)
where φ(ê, x, t) = σ(z, x, t) -σ(ẑ, x, t)

= 1 0 σ z (ẑ + ξê, x, t)dξê with φ T (ê, x, t)φ(ê, x, t) ≤ êT 1 0 σ T z (ẑ + ξê, x, t)σ z (ẑ + ξê, x, t)dξ ê ≤ êT Ψê.
Note that in the latter we applied Jensen's inequality [START_REF] Gu | An integral inequality in the stability problem of timedelay systems[END_REF][START_REF] Gu | Stability of time delay systems[END_REF]. Thus, Theorem 2.1, where BK = L, gives sufficient conditions for the exponential stability of (2.48) under the Dirichlet boundary conditions.

Hierarchy of LMIs

Following arguments for Hierarchy of LMIs in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF][START_REF] Seuret | Complete quadratic Lyapunov functionals for distributed delay systems[END_REF], we find that the stability conditions of Theorem 2. ) with N = 0) and Jensen's inequality [START_REF] Gu | An integral inequality in the stability problem of timedelay systems[END_REF][START_REF] Gu | Stability of time delay systems[END_REF].

3 Regional stabilization of KdVB equation with an output delay

Problem formulation

Consider the following KdVB equation:

z t (x, t) = -z(x, t)z x (x, t) + λz(x, t) + βz xx (x, t) -z xxx (x, t) + N j=1 b j (x)u j (t), t ≥ 0, x ∈ (0, l), l > 0 (3.1) under the periodic boundary conditions z(0, t) = z(l, t), z x (0, t) = z x (l, t), z xx (0, t) = z xx (l, t), (3.
2) where z(x, t) ∈ R is the state, u j (t) ∈ R (j = 1, . . . , N ) are the control inputs that enter (3.1) through the shape functions b j (x) (j = 1, . . . , N ) given by (2.5), λ and β are positive constants and the initial condition is given by z(x, 0) = z 0 (x). Note that for λ = 0, the open-loop system (3.1) has constant solutions, whereas for λ > 0 the openloop system may become unstable.

We study the KdVB equation (3.1) under the delayed stabilizing controller (2.8) with K ∈ R that leads to the closedloop system for t ≥ 0

z t (x, t) = -z(x, t)z x (x, t) + λz(x, t) + βz xx (x, t) -z xxx (x, t) -K(1 -χ [0,h] (t))[z(x, t -h) -f j (x, t -h)],
(3.3) where x ∈ Ω j , j = 1, . . . , N , f j (x, t -h) is given by (2.9) and χ [0,h] (t) is defined below (2.10). Now we consider the well-posedness of system (3.3). The solution of system (3.3) under the periodic boundary conditions (3.2) should be understood in the weak sense. Namely, we define the space

   H 1 per (0, l) = {g ∈ H 1 (0, l) : g(0) = g(l)}, g 2
H 1 per = P 11 l 0 g 2 (x)dx + P l 0 [g (x)] 2 dx, (3.4)
where P 11 and P are positive constants that are related to the Lyapunov functional (see (3.7) below). Given T > 0, a weak solution of (3.

3) on [0, T ] is a function z(x, t) ∈ C(0, T ; H 1 per (0, l)) such that z t ∈ L ∞ (0, T ; L 2 (0, l)) ∩ L 2 (0, T ; H 1 per (0, l)) and d dt l 0 z(ξ, t)φ(ξ)dξ = - l 0 z(ξ, t)z x (ξ, t)φ(ξ)dξ +λ l 0 z(ξ, t)φ(ξ)dξ + β l 0 z ξ (ξ, t)φ ξ (ξ)dξ - l 0 z ξξ (ξ, t)φ ξ (ξ)dξ -K N j=1 xj xj-1 (1 -χ [0,h] (t)) ×[z(ξ, t -h) -f (ξ, t -h)]φ(ξ)dξ
(3.5) hold for any φ ∈ H 1 per (0, 1) and almost all t ∈ [0, T ].

The condition (3.5) is motivated via the integrationby-parts formula. Based on the Galerkin approximation method (see, e.g. [START_REF] Pisano | On the ISS properties of a class of parabolic DPS' with discontinuous control using sampled-inspace sensing and actuation[END_REF]), one can show that (3.3) has a unique weak solution for all T > 0 provided the initial value z 0 ∈ H 3 (0, l) ∩ H 1 per (0, l) satisfies the compatible conditions [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF]:

z 0 (0) = z 0 (l), z 0 (0) = z 0 (l).
(3.6)

Improved regional stability conditions

For the stability analysis of system (3.3), an augmented Lyapunov functional depending on the state z(x, t) only (rather than the state derivative z t (x, t)) was introduced in [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF] to derive the stability conditions in terms of LMIs that allow to find an upper bound on the delay that preserves regional stability. To obtain improved regional stability conditions, we here consider a novel augmented Lyapunov functional via Legendre polynomials:

VN (t) = VP N + VP + 2 i=1 ( VSi + VRi ), N ∈ N 0 , (3.7)
where

VP N = l 0 ζT N (x, t)P N ζN (x, t)dx, VP = P l 0 z 2 x (x, t)dx, VS1 = S 1 l 0 t t-h e -2α(t-s) z 2 (x, s)dsdx, VS2 = S 2 l 0 t t-h e -2α(t-s) z 2 x (x, s)dsdx, VR1 = R 1 l 0 t t-h e -2α(t-s) (s -t + h)z 2 (x, s)dsdx, VR2 = R 2 l 0 t t-h e -2α(t-s) (s -t + h)z 2 x (x, s)dsdx with ζN (x, t) = [z(x, t), Θ 0 (x, t), . . . , Θ N (x, t)] T ,
and with (N +2)×(N +2) matrix

P N = {P ij }, scalars P > 0, S i > 0, R i > 0 (i = 1, 2
) and α > 0. Here Θ k (x, t) (k = 0, . . . , N ) are given by (1.6). For N = 0, VN (t) coincides with the Lyapunov functional introduced in [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF].

Since the solution of (3.3) does not depend on the values of z(x, t) for t < 0 [START_REF] Liu | Delay-dependent methods and the first delay interval[END_REF], we redefine the initial condition to be a function

z(x, t) = z 0 (x), t ≤ 0.
Thus, we have

VN (0) = [P 11 + 2P 12 h + P 22 h 2 + S 1 h + R1h 2 2 ] l 0 z 2 0 (x)dx +(P + S 2 h + R2h 2 2 ) l 0 [z 0 (x)] 2 dx
, where we applied (1.2). Note that the latter coincides with that for the case of N = 0 considered in [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF]. Thus, we employ the following results borrowed from [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF] that guarantee simple bounds on VN (h) and z(x, t), and that present a solution of (3.3): Lemma 3.1 [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF] Consider system (3.3) and the functionals VN (t) and V(t) respectively given by (3.7) and

V(t) = P 11 l 0 z 2 (x, t)dx + P l 0 z 2 x (x, t)dx (3.8)
with scalars P 11 > 0 and P > 0 that are related to the functional VN (t). Denote

M max (P 11 + 2P 12 h + P 22 h 2 + S 1 h + R1h 2 2 )P -1 11 , (P + S 2 h + R2h 2
2 )P -1 + e 2α1h -1, where S i (i = 1, 2) are from functional VN (t), h and α are positive scalars, and α 1 is a positive tuning parameter. Given positive tuning parameters C and C 1 .

(i) Assume that along (3.3) V(t) -2α 1 V(t) ≤ 0, t ∈ [0, h], (3.9)
VN (t) + 2α VN (t) -2α 1 V(t) ≤ 0, t ∈ [0, h], (3.10)
and M C 2 < C 2 1 , (3.11)
hold. Then

VN (h) ≤ M P 11 l 0 z 2 0 (x)dx + P l 0 [z 0 (x)] 2 dx < C 2 1 if z 0 2 H 1 per = P 11 l 0 z 2 0 (x)dx + P l 0 [z 0 (x)] 2 dx < C 2 .
Assume additionally that along (3.3)

VN (t) + 2α VN (t) ≤ 0, t > h (3.12)
holds. Then the solution of (3.3) satisfies

VN (t) ≤ M e -2α(t-h) P 11 l 0 z 2 0 (x)dx + P l 0 [z 0 (x)] 2 dx (3.13) for all t ≥ h.
(ii) If, in addition to the conditions of (i), there exist scalars P > 0, ρ > 0 and > 0 such that

VN (t) ≥ ρ l 0 z 2 (x, t)dx + P l 0 z 2 x (x, t)dx, (3.14)
≥ 1 + ρ, -P 1 * -ρ < 0. (3.15) Then z(x, t) ∈ (-C 1 , C 1 ) ∀x ∈ [0, 1], t ≥ 0. (3.16)
We now present improved conditions in terms of LMIs to guarantee (

Theorem 3.1 Given positive scalars h, l, ∆, α, K > λ, and positive tuning parameters α 1 > λ, C and C 1 , let there exist scalars (N + 2) × (N + 2) matrix P N = {P ij } and scalars P > 0, S i > 0, R i > 0 (i = 1, 2), W > 0, W 1 , ρ > 0 and > 0 such that (2.16), (3.15) and

PN > 0, (3.17)
Ξ N (-C 1 ) ≤ 0, Ξ N (C 1 ) ≤ 0, (3.18)
Υ N 1 + Λ N (-C 1 ) ≤ 0, Υ N 1 + Λ N (C 1 ) ≤ 0, (3.19)
Υ N 2 + Φ N (-C 1 ) ≤ 0, Υ N 2 + Φ N (C 1 ) ≤ 0, (3.20)
where

PN = P N + diag -ρ, S 1 e -2αh h , . . . , S 1 e -2αh (2N +1) h , Ξ N (z) =     -P 11 (α 1 -λ) 0 1 2 W 1 * -P 11 β -P (α 1 -λ) + W 1 1 2 P z * * -P β     , Υ N i = He{ ḠT N i P N HN i } + 2α ḠT N i P N ḠN i , i = 1, 2, ḠN i = 1 0 1×3 0 1×(N +1) 0 1×(N +i) 0 (N +1)×1 0 (N +1)×3 hI N +1 0 (N +1)×(N +i)
,

HN i = 0 (2N +i+5)×1 ΓT N i (0) . . . ΓT N i (N ) T , ΓN i (k) = 1 (-1) k+1 0 1×2 γ 0 N k . . . γ N N k 0 1×(N +i) (3.21)
with γ j N k (j = 1, . . . , N ) given by (2.17 x (x, t)dx, which, together with ρ > 0 and P > 0, implies the positivity of VN (t). In the next step, the objective is to show that LMIs (3.18)-(3.20) guarantee (3.9), (3.10) and (3.12) respectively.

For any t ≥ 0, differentiating VP N along the trajectories of (3.3) yields

VP N = 2 l 0 ζT N (x, t)P N ζN (x, t)dx, = l 0 ηT N ῩN 1 ηN dx + 2P 11 l 0 z(x, t)z t (x, t)dx +2 N +2 k=2 P 1k l 0 Θ k-2 (x, t)z t (x, t)dx
(3.23) where (3.5) has been used with φ = z, and with ῩN 1 given by (3.3) and the augmented vector ηN = [z(x, t), z(x, t -h), z x (x, t), z xx (x, t),

= l 0 ηT N (x, t) Ῡη N (x, t)dx +2P 11 λ l 0 z 2 (x, t)dx -2P 11 β l 0 z 2 x (x, t)dx +2 N +2 k=2 P 1k λ l 0 z(x, t)Θ k-2 (x, t)dx -2 N +2 k=2 P 1k l 0 z(x, t)z x (x, t)Θ k-2 (x, t)dx -2 N +2 k=2 P 1k β l 0 z x (x, t)Θ k-2 x (x, t)dx +2 N +2 k=2 P 1k l 0 z xx (x, t)Θ k-2 x (x, t)dx -2K(1 -χ [0,h] (t)) N j=1 xj xj-1 [P 11 z(x, t) + N +2 k=2 P 1k Θ k-2 (x, t)][z(x, t -h) -f j (x, t -h)]dx.
(3.25) Concerning the other terms of the Lyapunov functional, we have VP = 2P

l 0 z x (x, t)z xt (x, t)dx = -2P l 0 z xx (x, t)z t (x, t)dx = 2P λ l 0 z 2 x (x, t)dx -2P β l 0 z 2 xx (x, t)dx +2P l 0 z(x, t)z x (x, t)z xx (x, t)dx +2P K(1 -χ [0,h] (t)) N j=1 xj xj-1 z xx (x, t) ×[z(x, t -h) -f j (x, t -h)]dx, (3.26) and VS1 + 2α VS1 = S 1 l 0 z 2 (x, t)dx -S 1 e -2αh l 0 z 2 (x, t -h)dx, (3.27)
VS2 + 2α VS2 = S 2 l 0 z 2 x (x, t)dx -S 2 e -2αh l 0 z 2 x (x, t -h)dx.

(3.28)

Then the application of Lemma 1.1 leads to

VR1 + 2α VR1 = R 1 h l 0 z 2 (x, t)dx -R 1 l 0 t t-h e -2α(t-s) z 2 (x, s)dsdx ≤ R 1 h l 0 z 2 (x, t)dx -R 1 e -2αh N k=0 2k+1 h l 0 [Θ k (x, t)] 2 dx, (3.29) and VR2
+ 2α VR2 = R 2 h l 0 z 2 x (x, t)dx -R 2 l 0 t t-h e -2α(t-s) z 2 x (x, s)dsdx ≤ R 2 h l 0 z 2 x (x, t)dx -R 2 e -2αh N k=0 2k+1 h l 0 [Θ k x (x, t)] 2 dx. ( 3
≤ N j=1 xj xj-1 ηT N ,j [Υ N 2 + Φ N (z)]η N ,j dx -(S 2 e -2αh -W ) l 0 z 2 x (x, t -h)dx ≤ N j=1 xj xj-1 ηT N ,j [Υ N 2 + Φ N (z)]η N ,j dx, t > h, (3.33
) where we applied (2.16). Here Υ N i (i = 1, 2) and ηN are respectively given by (3.21) and (3.24), ηN ,j = [η T N , f j (x, th)] T , and Λ N (z) and Φ N (z) are symmetric matrices composed of (3.22).

Similarly, differentiating V(t) in (3.8) along the trajectories of (3.3) yields This model accounts for an activator temperature z 1 that undergoes reaction, advection and diffusion, and for a fast inhibitor concentration z 2 , which may be advected by the flow [START_REF] Selivanov | Distributed event-triggered control of diffusion semilinear PDEs[END_REF].

V(t) -2α 1 V(t) = 2 l 0 κ T Ξ N (z)κdx, t ∈ [0, h], (3.34) where κ = [z(x, t), z x (x, t), z xx (x, t)] T and Ξ N (z) is given by (3.21). Since Ξ N (z), Λ N (z) and Φ N (z) are affine in z, Ξ N (z) ≤ 0, Υ N 1 + Λ N (z) ≤ 0 and Υ N 2 + Φ N (z) ≤ 0 hold for all z ∈ (-C 1 , C 1 )
Assume that there are N = 20 in-domain sensors transmitting measurements (2.7) at xj = 2j-1 2N , j = 1, . . . , N (the centers of Ω j = [ j-1 N , j N )) implying ∆ = 0.5. LMIs of Theorem 2.1 with various values of N and α = 0 lead to the maximal value of h (see Table 4.1) that preserves the stability. As expected, better results are obtained as the degree of the polynomial N increases, but at the cost of additional decision variables. Moreover, the maximal value of h remains Note that the analytical upper bound for the asymptotic stability of the latter ODE is h = π (see Chapter 2.3.2 in [START_REF] Fridman | Introduction to time-delay systems: analysis and control[END_REF]). It is clear that our method recovers the analytical upper bound with a finite degree N = 4 of the polynomials. 4.3) that preserves the exponential stability of the closed-loop system for any initial values satisfying z 0 H 1 per < 0.044. It is clear that the improvement on the upper bound of delay is achieved as the degree N of the polynomial increases and the maximal value of h remains as 2.4442 × 10 -3 when N ≥ 3. We find also the feasible solutions of the LMIs, e.g. The latter allows to choose a larger initial condition (comparatively to z 0 (x) = 0.0025sin(2πx), 0 ≤ x ≤ 1 in [START_REF] Kang | Distributed stabilization of Korteweg-de Vries-Burgers equation in the presence of input delay[END_REF]), e.g. z 0 (x) = 0.0037sin(2πx), 0 ≤ x ≤ 1 satisfying In this paper, stabilization of heat equation and KdVB equation under constant output delay has been studied. By constructing two augmented Lyapunov functionals that respectively depend on the state derivative and the state only, sufficient conditions in terms of LMIs that preserve the exponential stability have been derived. The resulting LMI conditions show improvements in numerical examples. The suggested augmented Lyapunov functionals can be used for delayed control of various PDEs via spatial decomposition method: Kuramoto-Sivashinskii equation (as considered in [START_REF] Kang | Distributed sampled-data control of Kuramoto-Sivashinsky equation[END_REF]), damped wave and beam equations (see e.g. [START_REF] Terushkin | Network-based control of a semilinear damped beam equation under point and pointlike measurements[END_REF][START_REF] Terushkin | Network-based deployment of nonlinear multi agents over open curves: a PDE approach[END_REF]).

  ), Λ N (z) = {Λ ij } and Φ N (z) = {Φ ij } are symmetric matrices composed ofΦ 11 = 2P 11 λ + S 1 + R 1 h, Λ 11 = Φ 11 -2α 1 P 11 , Φ 12 = -Φ 17 = -P 11 K, Φ 14 = Λ 14 = W 1 , Φ 15 = Λ 15 = [ P 12 . . . P 1(N +2) ]hλ, Φ 22 = Λ 22 = -S 1 e -2αh , Φ 24 = -Φ 47 = P K, Φ 25 = -Φ T 57 = -[ P 12 . . . P 1(N +2) ]hK, Φ 33 = -2P 11 β + 2P λ + 2αP + S 2 + R 2 h + 2W 1 , Λ 33 = Φ 33 -2α 1 P, Φ 34 = Λ 34 = P z, Φ 35 = Λ 35 = -[ P 12 . . . P 1(N +2) ]hz, Φ 36 = Λ 36 = -[ P 12 . . . P 1(N +2) ]hβ, Φ 44 = Λ 44 = -2P β, Φ 77 = -W π 2 ∆ 2 , Φ 46 = Λ 46 = [ P 12 . . . P 1(N +2) ]h, Φ 55 = Λ 55 = -R 1 he -2αh diag{1, . . . , 2N + 1}, Φ 66 = Λ 66 = -R 2 he -2αh diag{1, . . . , 2N + 1}, (3.22)other blocks are zero matrices. If (3.11) holds, then for any initial state z 0 ∈ H 3 (0, l) ∩ H 1 per (0, l) satisfying the compatible conditions (3.6) with z 0 H 1 per < C, system (3.3) under the periodic boundary conditions (3.2) possesses a unique weak solution in the sense that for any T > 0, z(x, t) ∈ C(0, T ; H 1 per (0, l)). Moreover, the solution of (3.3) satisfies (3.13) for all t ≥ h. Proof : First, Lemma 1.1 ensures the following inequality VP N + VS1 ≥ l 0 ζT N (x, t) PN ζN (x, t) + ρ l 0 z 2 (x, t)dx, where matrix PN > 0 is given in (3.21) and guarantees VN (t) ≥ VP N + VS1 + P l 0 z 2 x (x, t)dx ≥ ρ l 0 z 2 (x, t)dx + P l 0 z 2

2 4

 2 given by (3.16) if LMIs (3.18)-(3.20) are feasible. From (3.32)-(3.34), it follows that LMIs (3.18)-(3.20) guarantee (3.9), (3.10) and (3.12) respectively. Finally, assertion of Theorem 3.1 follows from Lemma 3.1. Numerical examples Example 4.1 Consider the chemical reactor model from [18,22] governed by heat equation (2.1) under the Dirichlet boundary conditions (2.2) with the measurements (2.7), where n = 2, r = 1, l = 10, D 0 = diag{0.01, 0.005}, β = diag{0.011, 1.1}, φ = [φ 1 (z 1 ), 0] T , Ψ = diag{10 -4 , 0} and

Example 4 . 3

 43 Consider KdVB equation (3.1) under the periodic boundary conditions (3.2) with λ = 15, β = 0.5 and l = 1. Assume that there are N = 10 in-domain sensors transmitting point measurements (2.7) at xj = 2j-1 2N , j = 1, . . . , N implying ∆ = 0.1. By using the point measurements (2.7), we study system (3.1) under the control law (2.8) with K = 20. LMIs of Theorem 3.1 with various values of N , α = 1, α 1 = 20, C = 0.044 and C 1 = 0.05 lead to the maximal value of h (see Table

N = 3 :

 3 P 11 = 189.4785, P = 2.1978.

Table 4. 2 1 . 9 × 10 - 3 Theorem 3 . 1 with N = 0 1 . 9 × 10 - 3 N 3 N 3 N

 219103311910333 Maximal value of h via Corollary 2.1 (Example 4.2). N = 0 N = 1 N = 2 N = 3 N ≥ 4 hmax 1.9999 3.0344 3.1362 3.1414 3.1415 Table 4.3 Maximal value of h via Theorem (3.1) (Example 4.3). hmax Kang & Fridman [13] = 1 2.4431 × 10 -= 2 2.4437 × 10 -

  t)He{D 0 (αP 2 -P 1 )}z x (x, t)dx.

	(2.33)
	Finally, in view of (2.23)-(2.27), (2.29), (2.30) and (2.33),
	adding the right-hand side of (2.28) to VN (t) + 2αV N (t)
	and applying S-procedure with (2.31) and (2.32), we have
	VN (t) + 2αV N (t)

  L 2 (0,l) that cannot be compensated under the Dirichlet boundary conditions (2.2). Therefore, the descriptor method (e.g. in (2.28)) allows to overcome this difficulty. For the case of the averaged measurements

	ds
	(2.36)
	for some M 2 > 0, where the bound is finite due to the well-
	posedness. Then (2.19) follows from (2.35) and (2.36) with
	M 0 = M2 M1 .
	Remark 2.2

The feasibility of the strict inequalities (2.14)-(2.16) with α = 0 implies the feasibility of (2.14)-(2.16) with the same decision variables and with small enough α = α 0 > 0, and thus guarantees a small enough decay rate. 2

Remark 2.1 Note that direct substitution of z t (x, t) by the right-hand side of (2.10) leads to the quadratic variable diffusion term D z(x, t) 2

  t)dx. Then based on Theorem 2.1, we easily obtain the following stability result with an arbitrary N ∈ N: Corollary 2.1 Given postive scalars h, l, ∆ and α, let there exist (N + 1)n × (N + 1)n matrix P N , n × n matrices P 1 , P 2 = diag{P 1 2 , . . . , P n 2 } > 0, S 1 > 0, S 2 > 0 and R > 0 that satisfy (2.14) and ΛN = [Λ ij ] and ŨN are obtained from Υ N , Λ N and U N by setting W = 0 and Λ 11 = He{P 1 A+ π 2 l 2 D 0 (αP 2 -P 1 )} + S 1 + π 2 l 2 S 2 and taking away the third, fifth and sixth block-columns and block-rows respectively. Then system (2.41) under the Dirichlet boundary conditions (2.2) is exponentially stable with a decay rate α > 0. Moreover, if the strict inequalities (2.14) and (2.42) are feasible with α = 0, then system (2.41) under the Dirichlet boundary conditions (2.2) is exponentially stable with a small enough decay rate α = α 0 > 0. LMIs of Corollary 2.1 with l = π guarantee the exponentially stability of the vector ODE with delay

	ΥN + ΛN + ŨN ≤ 0, where ΥN , Remark 2.3 (2.42)

Table 4 .

 4 1 Maximal value of h via Theorem 2.1 (Example 4.1). Consider heat equation (2.41) under the Dirichlet boundary conditions (2.2) with l = π, D 0 = I 2 and LMIs of Corollary 2.1 with various values of N and α = 0 lead to the maximal value of h shown in Table 4.2. As explained in Remark 2.3, the latter results correspond also to the stability of the following ODE with time-delay

			hmax		No. of decision variables
	N = 0 0.5139 5n 2 +9n+2 2
	N = 1 0.5904 4n 2 + 5n + 1
	N ≥ 2 0.5909 (N 2 +2N +5)n 2 +(N +9)n+2 2
	as 0.5909 when N ≥ 2. Clearly, an improvement close to
	15% is achieved.
	Example 4.2 A =	1 1	, BK =	0 0	.
			-1 0		0 1
	ż(t) +	0 -1	z(t) +	0 0	z(t -h) = 0.
		1 1			0 1

Preprint submitted to Systems & Control Letters

This work was supported by Israel Science Foundation under Grant 673/19, C. and H. Manderman Chair on System Control at Tel Aviv University, the Planning and Budgeting Committee (PBC) Fellowship from the Council for Higher Education, Israel, Joint Research Project HetCPS, Ministry of Science & Technology of Israel and CNRS, France, National Natural Science Foundation of China under Grant 61803026, and Fundamental Research Funds for the Central Universities under Grant FRF-TP-20-039A2Z.