Jérôme Puech

L Hermozo

Hélène Brogniez

Rémy Roca

Thomas Fiolleau

J.-P Chaboureau

F Auguste

Dominique Bouniol

Julien Delanoë

V Cipolla

J.-C Orlhac

Drogo S Le

The C2Omodo Concept: A Tandem of New Generation High Resolution All-Sky Atmospheric Sounders in the Frame of the AOS Mission

Keywords: microwave radiometer, new generation, time-derivative, convective mass flux, instrumental synergy

come

INTRODUCTION

Vertical structures, processes and interactions between various components of the atmosphere are much more complex to measure and monitor in cloudy or precipitating skies. Their understanding is crucial as such processes are responsible for global energy, water and air transfers from the lower atmosphere to the free troposphere. Tropical atmospheric convection for instance plays a fundamental role in the water cycle, meteorological monitoring and prediction, and for climate understanding. Even though convection has been extensively studied for the past century [START_REF] Houze | 100 years of research on mesoscale convective systems[END_REF], measurements of the vertical processes that they involve are scarce, if not available, due to their high variability in time and space, and to the complex vertical processes involved, which occur in deep convection. The C²OMODO (Convective Core Observations through MicrOwave Derivatives in the trOpics) concept would make it possible to measure deep convection's life cycle, the energy processes and transfers, and to better explain the water cycle by exploiting the temporal evolution of the brightness temperatures at frequencies sensitive to the cloudy ice particles. Two microwave radiometers, the new generation multi-spectral, high-resolution, all-sky atmospheric sounders, SAPHIR NG [2], form the C²OMODO tandem. These instruments inherit from the SAPHIR instrument [2] on-board the French-Indian Megha-Tropiques mission [START_REF] Malassingne | The SAPHIR-NG instrument: a high resolution microwave sounder[END_REF] and benefit from technological improvements since the design of SAPHIR, which data is successfully used more than ten years after its launch [START_REF] Brogniez | Coupling of Clouds and Tropospheric Relative Humidity in the Tropical Western Atlantic: Insights from Multisatellite Observations[END_REF][5] [START_REF] Roca | The Megha-Tropiques mission: A review after three years in orbit[END_REF][7] [START_REF] Duruisseau | Assimilating cloudy and rainy microwave observations from SAPHIR on board Megha Tropiques within the ARPEGE global model[END_REF]. The C²OMODO concept is currently part of the NASA AOS (Atmosphere Observing System) mission, which addresses the following objectives: low and high clouds, convective storms, cold clouds and precipitation, air quality and aerosol attribution, aerosol redistribution and processing, and aerosols direct and indirect effects.

THE C²OMODO CONCEPT IN THE FRAME OF THE AOS MISSION

The AOS mission

The C²OMODO concept is the French contribution to the AOS mission lead by NASA, providing measurements of the atmospheric vertical structure that will describe interactions and feedback processes between aerosols, clouds, atmospheric convection and precipitation. With an international collaboration between NASA centers, JAXA, CSA and CNES, AOS includes several instruments such as lidars, radars, radiometers, polarimeters and stereocameras organized in two orbital planes : the polar and inclined (55°) orbits, at an altitude of ~430km.

The C²OMODO concept

C²OMODO is a tandem of two microwave radiometers separated by a few minutes time gap. Each instrument is onboard one platform of the inclined orbit: the AOS-I platform, operated by NASA and the PMM platform operated by JAXA. With its wide swath of ~800 km and high resolution measurements (<=5km), C²OMODO will enable to map the small scale vertical mass fluxes of cloud ice within the convective core of tropical and sub-tropical convective systems. This will be obtained by providing temporal derivatives of Brightness Temperature (dBT/dt) measurements [START_REF] Brogniez | Time-delayed tandem microwave observations of tropical deep convection: Overview of the C2OMODO mission[END_REF][9] at high frequencies within the 183.31 GHz and 325,15 GHz absorption lines. It thus addresses two of the eight AOS objectives: high clouds and convective storms.

Instrumental synergy

Thanks to the high variety of instruments over the different orbits, the instrument synergy is the key stone of the AOS mission as it offers new perspectives to improve our understanding of convective dynamics and processes, enabling more accurate predictions of rain and severe weather, and thus to better constrain meteorological and climatic models. For instance, [START_REF] Brogniez | Time-delayed tandem microwave observations of tropical deep convection: Overview of the C2OMODO mission[END_REF] underlined the high potential of combining Ka Doppler radar high vertical resolution measurements and C²OMODO observations to derive upward motion of ice particles over a wide horizontal extent thanks to C²OMODO's wide swaths.

TRADE-OFFS FOR THE C²OMODO CONCEPT

Towards an optimized tandem of instruments

In order to capture the geophysical variations of the signal that are directly linked to the vertical hydrometeor mass fluxes within the different parts of the convective cells at various stages of their development, both radiometers of the tandem are separated by a specific time gap of 30s to 3min, along the same orbit. Scientific studies lead by the C²OMODO Science Team are ongoing to define the optimal time gap and tolerance to address C²OMODO scientific objectives. This optimal is a compromise between: 1) temporal scale of convective cells' evolution ; 2) instrument's radiometric performances (the smallest BT variation measured by the instrument), related to technological constrains ; and 3) manoeuver constrains of the AOS inclined orbits for time gap stability maintaining.

Towards optimized spectral configuration and radiometric performances

The SAPHIR-NG instruments enable measurement of the atmospheric electromagnetic field at frequencies ranging in the [0.2 GHz ; 11 GHz] band within the 183.31 GHz and 325.15 GHz water vapor absorption lines. Complementary information on the column-integrated and vertical profile of cloud ice content is thus provided over a wider altitude range, and a wider ice particle size range than with the single 183.31 GHz absorption band, as sensitivity of 325 GHz measurements to higher altitudes and smaller cloud ice particles are exploited. Moreover, the channels within the [0.2 GHz; 11 GHz] band within both absorption lines show different sensitivities of to ice change in time [START_REF] Auguste | Deep Convection as Inferred From the C2OMODO Concept of a Tandem of Microwave Radiometers[END_REF]. The C²OMODO Science Team is currently undertaking multispectral sensitivity studies of the retrieved dIWP/dt, vertical ice momentum and velocity to the channel configuration , according to the type and development stage of the convective cells. Various spectral configurations are evaluated: SAPHIR-like (with varying channel bandwidths), AWS-like ICI-like spectral configurations, and a combination of such spectral configurations, and illustrated in Figure 1. Such sensitivity studies result in a trade-off between:

-the vertical sampling of the information content (driven by the number of channels and their respective bandwidths), -the quality of the information content (driven by the channel bandwidths directly impacting radiometric performance). The use of high performance elements in the 325 GHz Front-End sub-system is currently evaluated and tested to improve radiometric sensitivity (see section 4.1.2). -the instrumental constraints (weight/consumption/size budgets and technological maturity)

INSTRUMENT RECENT TECHNICAL ADVANCES

Instrument architecture innovations

Several innovations were integrated to the SAPHIR-NG instrument, the driver being the radiometric performances improvement. Two sub-assemblies are considered in the following section of the paper: the antenna sub-system, that emulates the radiation pattern for the 3 frequencies, and the Front-End at 325 GHz, which translates the RF signal into the intermediate frequency (IF).

Quasi-optical network

The Quasi-optical network (QON) enables co-location and co-registration of the measurements at the different frequencies. Its aims and functions for SAPHIR NG are described in [2]. The QON design must follow accommodation constraints to feed the 3 different receivers and to fit in a reduced allocated instrumental volume.

A design study of the whole assembly has enabled to define the architecture of the optimal solution in terms of radiometric performances and accommodation, and is illustrated in Figure 2. The measured RF losses in reflection for D1 for the 2 bandwidths of interest within 89 GHz and 183 GHz demonstrated very low values, below 0,1 dB over the whole incidence angle range.

Such performances confirm its functions of transmission of the 325 GHz signal and of reflection of the 89 and 183 GHz signals. They are quite stable for a wide range of incidence angles. This means that the quasi-optical network (QON) will be robust regarding the position of D1 towards the main incident beam. The next step of the development will be to perform the detailed RF design of the whole assembly.

325 GHz Front-End

A high performance 325 GHz mixer has been designed by the LERMA laboratory [12] in order to improve the noise performance of the 325 GHz Front-End. Three different topologies have been extensively studied: the 325 GHz Sub-Harmonic Mixer (SHM) MMIC (Monolithic Microwave Integrated Circuit) design, the 325 GHz Fundamental Balanced Mixer (FBM) MMIC design, and a 325 GHz subharmonic discrete diode mixer. All these topologies integrate anti-parallel Schottky diode structures on GaAs substrate. The main difference between SHM and FBM lies on the needed power coming from the local oscillator sources, for which the required level is higher in the FBM case at 325 GHz.

A discrete diode mixer was first designed. C2N laboratory has produced many discrete diodes with different anode size for the sub-harmonic type mixer. These components can be easily mounted (flip-chip mounted on the quartz surface) on the existing mixer mechanical blocks. This was manufactured and tested together with an IF (Intermediate Frequency) Low Noise Amplifier (LNA) (NF (Noise Figure) of 1.16 dB, see Figure 4). As a second step, the sub-harmonic MMIC mixers with different anode size of the diodes were designed and fabricated by the C2N laboratory with their mechanical block. For this topology, the anti-parallel diode circuit is included on the same substrate with the other RF functions.

The noise temperature of the complete chain including the IF LNA (with the same NF as in the previous case) was measured, using a 155-185 GHz source. In order to get the results over the whole frequency range, an automatic test bench was set-up at LERMA, to assess the receiver noise temperature versus the frequency, as shown in Figure 5. The noise performances of the DSB receiver are quite stable versus frequency with a value below 1400 K in the bandwidth of interest for both blocks and more particularly with values of 1320K@325 GHz for the first one and of 1280K@325 GHz for the second one, in accordance with retro-simulation. The last simulation curve in grey demonstrates the potential improvement in case of the use of a high performance LNA (NF=0.8 dB).

CONCLUSION

The C²OMODO concept is a tandem of high resolution multi-spectral all-sky atmospheric sounders which provides measurements of convective systems' vertical structure and dynamics during their life cycle. It addresses two of the eight objectives of the NASA AOS mission, a fruitfull collaboration between NASA agencies, JAXA, CSA and CNES, and providing global measurements of the aerosols and cloud/precipitation interactions and feedbacks. The contribution of C²OMODO to the AOS mission is twofold, both thanks to the high added value of the time-derived observations of the tandem; and thanks to the potential of AOS instrumental synergy (by combining C²OMODO's wide swath observations and KuDPR high vertical resolution measurements, for instance). To best fit the C²OMODO scientific objectives, various sensitivity studies to the information content are ongoing, to define an optimized time gap of the tandem, and spectral configuration of both instruments of the tandem . The SAPHIR NG sounders also benefit from several technical improvements since the SAPHIR instrument, both to ensure time and spatial co-location of measurements at the different frequencies, though the advances made on the QON subsystem, and to improve radiometric performances, through the system temperature decrease.

Figure 1 :

 1 Figure 1: Channels within the [0.2 GHz; 11 GHz] band with respect to intermediate frequency, in GHZ, for the various spectral configurations analyzed.

Figure 2 :

 2 Figure 2: QON architecture Taking into account the performances of the dichroïcs versus frequency and their behavior either in transmission or in reflection, the optimized solution was to select a high pass scheme for the 89-183/325 GHz dichroïc D1 design. Wide band performances of the D1 dichroïc, over the total bandwidths of the three central frequencies at 89 GHz, 183 GHz and 325 GHz were demonstrated by modelling by Cardiff University [11]. The measured values in transmission for the whole bandwidth of interest [315-335] GHz are given in Figure 3 for D1 versus the incidence angle.

Figure 3 :

 3 Figure 3: Transmission Losses (dB) versus frequency (GHz) for incidence angles of 2.5 (red), 12.5 (orange), 22.5 (green), 32.5 (blue) and 42.5 (Indigo) degrees

Figure 4 :

 4 Figure 4: (a) Discrete diodes made by LERMA/C2N. (b) Mounting of the discrete diodes inside the 325 GHz subharmonic discrete diode mixer block.

Figure 5 :

 5 Figure 5: Noise temperature measurement versus frequency of the 325 GHz sub-harmonic MMIC mixer with 2 different blocks (SN01 and SN02 with different anode size) at 300K room temperature for raw and corrected measurements at an optimal input power with dedicated LO unit.


Angel Mercado-Uribe

Jesús Mendoza-Ávila
email: jesus.mendoza-avila@inria.fr

Denis Efimov
email: denis.efimov@inria.fr

Johannes Schiffer

A Control Leonov Function Guaranteeing Global ISS of Two Coupled Synchronverters

The synchronverter control algorithm is a highly promising option for operating AC power inverters in future low-inertia power systems. Yet, as with conventional synchronous generators, the standard synchronverter algorithm only provides limited robustness guarantees. Therefore, we propose an additional control that confers the closed-loop system with global robustness in the Input-to-State Stability (ISS) sense. In this paper, such a control law is derived for the case of two identical synchronverters interconnected over dynamic power lines by using the Control Leonov Function (CLeF) framework. The control is illustrated by simulations.

I. INTRODUCTION

.

.

.

Unfortunately, as with SGs in classical power systems, the standard synchronverter algorithm only provides limited

robustness properties of the closed-loop system with respect to perturbations [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF]. By recognizing this fact, an additional control termed virtual friction is considered in [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF], which has been proven to be effective to (locally) improve the system behavior and, in particular, its robustness [START_REF] Blau | Synchronverters used for damping inter-area oscillations in two-area power systems[END_REF], [START_REF] Brown | Using synchronverters for power grid stabilization[END_REF]. The present paper is dedicated to addressing the same challenge by designing an additional control for the synchronverter, which confers the closed-loop system with the ISS property.

To achieve this objective, we use the concept of Leonov functions [START_REF] Efimov | A relaxed characterization of ISS for periodic systems with multiple invariant sets[END_REF], [START_REF] Efimov | On boundedness of solutions of state periodic systems: A multivariable cell structure approach[END_REF], [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Schiffer | Strong and weak Leonov functions for global boundedness of state periodic systems[END_REF]. This is motivated by the fact that the dynamics of AC power systems are periodic with respect to a part of the state vector. This periodicity of the dynamics implies that the power system exhibits multiple equilibria. Therefore, the construction of Lyapunov functions becomes very complicated. For instance, an application of existing approaches [START_REF] Forni | The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: An overview[END_REF] in a state periodic system would require to construct a Lyapunov function, which respects the system periodicity. These strong requirements can often not be met in power systems applications and, consequently, the majority of stability results are merely local [START_REF] Liu | Stability and control of power grids[END_REF]. By using a Leonov function instead, we explicitly exploit the periodicity of the system dynamics to relax the (definiteness and periodicity) requirements compared to standard Lyapunov functions. Recently, the Leonov function framework has been extended to a universal formula for ISS and iISS stabilization via the concepts of an ISS-and iISS-Control Leonov Function (ISS-and iISS-CLeF, respectively) [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF].

Using the Leonov function method, global stability and robustness properties are derived for a SG connected to a infinite bus in [START_REF] Barabanov | Conditions for almost global attractivity of a synchronous generator connected to an infinite bus[END_REF], [START_REF] Schiffer | An input-to-state stability approach to verify almost global stability of a synchronous-machine-infinite-bus system[END_REF]. Likewise, first extensions to provide almost global stability properties of power systems and microgrids are given in [START_REF] Schiffer | Almost global synchronization in radial multi-machine power systems[END_REF], [START_REF] Schiffer | Global synchronization analysis of droop-controlled microgrids: A multivariable cell structure approach[END_REF].

While the above results use the Leonov function framework concern and aiming on the global analysis of system properties, this work is focused on control design. More precisely, our main contribution is to apply the ISS-CLeF concept from [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF] to confer the standard synchronverter algorithm from [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] with additional robustness properties in the ISS sense. The control is derived and the global ISS properties are rigorously established for the case of a microgrid consisting of two synchronverters feeding a common load. Compared to this, the analysis of stability and robustness is only locally proven in [START_REF] Rosso | Robust stability analysis of synchronverters operating in parallel[END_REF], [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF]. The efficacy of the proposed control is illustrated in a simulation example.

This paper is structured as follows. Some preliminaries on practical ISS-and iISS-CLeFs are presented in Section II. The description of the system and its respective model is developed in Section III. The main result is provided in Section IV. A simulation example is shown in Section V. Finally, conclusions and future work are given in Section VI.

II. PRELIMINARIES A. Notation

We denote the sets of natural, real and integer numbers by N, R and Z, respectively. Let R + = {s ∈ R : s ≥ 0} and

Z + = Z ∩ R + . The variable v ∈ R k (v ∈ Z k )
denotes a vector of real (integer) numbers with k ∈ N components, whose infinity norm is represented by ∥v∥ ∞ = max (|v 1 | , ..., |v k |) and its Euclidean norm is denoted by

∥v∥ 2 = v 2 1 + • • • + v 2 n .
The distance from a point p ∈ R n to the set S ⊂ R n is defined as ∥p∥ S = inf a∈S ∥p -a∥ 2 . The identity matrix is represented by I k ∈ R k×k . Let i ∈ Z, then the notations i n and i n×m , n ∈ N, m ∈ N, represent a column vector of n elements and a matrix with n rows and m columns, respectively, with all its elements being identical to i. The variable v ∈ L k ∞ with k ∈ N denotes a function whose components belong to the set of Lebesgue ∞-space.

A function α : R + → R + belongs to the class K if it is continuous, strictly increasing and α (0) = 0. A function α : R + → R + belongs to the class K ∞ if α ∈ K and it is unbounded. A continuous function β : R + × R + → R + belongs to the class KL if, for each fixed t ∈ R + , β (•, t) ∈ K ∞ and for each fixed s ∈ R + , β (s, •) is non-increasing and it tends to zero for t → ∞.

B. Practical ISS-and iISS-Control Leonov Functions

Consider a system of the form

θ = f 1 (x, d), ż = f 2 (x, d) + g(x)u, (1)
where x ∈ R n is the state vector, x = (z, θ) ∈ R n with z ∈ R k and θ ∈ R q , n = k + q for k, q > 0, u ∈ L p ∞ is an admissible control input, and d ∈ L m ∞ is a vector of perturbation signals. Moreover, f 1 : R n × R m → R q , f 2 : R n × R m → R k and g : R n → R k×p are assumed to be locally Lipschitz continuous vector functions. Furthermore, define the vector

ξ j = [0 ⊤ k 2πj ⊤ ] ⊤ ∈ R n , j ∈ Z q .
Let us introduce the following assumptions for the system [START_REF] Barabanov | Conditions for almost global attractivity of a synchronous generator connected to an infinite bus[END_REF].

Assumption 1: The vector fields f 1 , f 2 and g in (1) are 2π-periodic with respect to θ.

Assumption 2: Let the set W ⊂ R n contain all αand ω-limit sets of (1) for d = 0 m and u = 0 p . Its projection W, on M = R k × [0, 2π) q , is compact and decomposable (in the sense of [START_REF] Forni | The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: An overview[END_REF]).

Then, we introduce the following definition.

Definition 1 ([14]): A C 1 function V : R n → R is said to be a practical ISS-CLeF for the system (1) if ∂V (x)
∂ z is 2πperiodic in θ, and there exist functions ϑ, ϑ, σ, σ, ϖ, υ ∈ K ∞ and scalars g, ḡ, ζ ≥ 0, such that for all x = (z, θ) ∈ R n :

ϑ(∥x∥ W′ ) -σ(∥ θ∥) -g ≤ V (x) ≤ ϑ(∥x∥ W′ + ḡ) -σ(∥ θ∥), (2)
and for all

x ∈ {x ∈ R n | V (x) ≥ 0} and all d ∈ R m : inf u∈R p {a(x, d) + β(x)u} ≤ -ϖ(V (x)) + υ(∥d∥) + ζ, (3) where a(x, d) = ∂V (x) ∂ θ f 1 (x, d) + ∂V (x) ∂ z f 2 (x, d) and β(x) = ∂V (x) ∂ z g(x)
, and W ′ (the projection of W′ on M ) is formed by atoms W i of the decomposition of W, such that

(W i + {ξ j }) ⊂ Ω 1 ∪ Ω 2 for some j ∈ Z q , where Ω 1 = {x ∈ R n | V (x) < 0}, Ω 2 = {x ∈ R n | β(x) = 0}. Additionally, if (3)
holds for ζ = 0, then V is said to be an ISS-CLeF. Moreover, if (3) holds for ζ = 0 and a non-negative function ϖ : R + → R + , then V is said to be an iISS-CLeF. Now, following [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF], we introduce the functions

ā (x) = sup d∈R m a (x, d) -v (∥d∥) -ζ , (4)
with v (s) ≥ v (s) for all s ≥ 0 and ζ ≥ ζ, and

ϕ (∥x∥ W ) = sup σ(∥ θ∥)≤ϑ(∥x∥ W′ +ḡ) ā (x) + 1 2 ϖ (V (x)) .
(5) Likewise, in [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF] the following variant of the small control property is introduced: for any ε > 0, there is δ > 0 such that sup

0<∥x∥Ω≤δ ϕ (∥x∥ W′ , z) ∥β(x)∥ 2 < ε, (6)
where

Ω = {x ∈ R n | V (x) = 0; β(x) = 0}.
We write the well-known Sontag's universal formula [START_REF] Sontag | A 'universal' construction of Artstein's theorem on nonlinear stabilization[END_REF] as follows

K (ϕ(•), β(•)) = ϕ+ √ ϕ 2 +∥β∥ 4 ∥β∥ 2 if β ̸ = 0, 1 if β = 0. (7)
Finally, we define the set

W -= ∪ l - i=1 W i , (8)
such that (W i + {ξ j }) ⊂ Ω 1 \ Ω 2 for some j ∈ Z q , W i are atoms of the decomposition of W and ϵ ∈ (0, ϵ], where ϵ > 0 is the distance between the sets {x ∈ R n | V (x) = 0} and W-. Then, the following theorem can be stated. Theorem 1 ( [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF]): Let the system (1) satisfy Assumptions 1 and 2, and the set Wcl , which contains all invariant solutions of the closed-loop system, be decomposable. If there exists a (practical) ISS-CLeF (iISS-CLeF) for the system (1) satisfying the inequality (6), then the controller

u (x) = -min 1, ∥x∥ W- ϵ K (x) β ⊤ (x) , (9)
with K (x) ≥ K (ϕ(∥x∥ W ), β(x)), which is continuous for all x ∈ R n \ Ω 1 , provides the (practical) ISS (iISS) property to the system (1) with respect to W cl .

III. CONSIDERED SYSTEM AND PROBLEM STATEMENT A. Model of two synchronverters coupled via dynamic power lines in abc-coordinates

We consider a small power system composed of two inverters connected in parallel to a load. We assume that both inverters are operated with the standard synchronverter algorithm [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. Similarly to [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF], we assume that the load is purely resistive. So, we obtain the scheme of Figure 1, where for k = 1, 2; the parameters R k > 0 and L k > 0 are the line resistance and inductance, respectively, while the variables ω k (t) ∈ R, i k,abc (t) ∈ R 3 , e k,abc (t) ∈ R 3 represent frequency, currents and voltage, respectively. Likewise, R L > 0 represents the load resistance and i L abc (t) ∈ R 3 is the load current and

V L,abc (t) = R L 2 k=1 i k,abc (t) ∈ R 3 (10)
is the load voltage 1 .

- Following [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF], the electromotive forces are given by

+ i1 abc R1 L1 RL iL abc L2 R2 - + i2 abc Tm 1 Tm 2 D1 J1 e1 abc D2 J2 e2 abc
e k,abc = 2 3 b k ω k   sin (δ k ) sin δ k -2 3 π sin δ k + 2 3 π   , b k = 3 2 M f k i f k , for k = 1, 2
, where δ k are the phase angles, M f k is the mutual inductance and i f k denotes the rotor current of the k-th synchronverter. Then, the dynamics of the k-th synchronverter are given by [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] δk =ω k ,

J k ωk = -D k ω k + T m k -T e k + ρ m k , L k di k,abc dt = -R k i k,abc + e k,abc -V L,abc + ρ e k , (11)
where the electrical torque T e k can be expressed as

T e k = ω -1 k i ⊤ k,abc e k,abc , (12)
which is well-defined for ω k = 0, since the term ω -1 k cancels out with ω k in e k,abc . The positive parameters J k as well as D k are the virtual inertia and damping coefficients, respectively. The signal T m k (t) ∈ R represents a virtual torque, which is usually taken as control input for active power and frequency control [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. The signal ρ m k (t) ∈ R represents perturbations arising in the implementation of the synchronverter algorithm, such as measurement noise or oscillations pertaining to harmonics. Finally, ρ e k (t) ∈ R 3 is an external perturbations vector, which can represent, e.g., additional, unmodeled load variations. We aim at designing a control law that confers the synchronverter dynamics [START_REF] Kroposki | Benefits of power electronic interfaces for distributed energy systems[END_REF] with improved robustness properties. Similarly to [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF], we write the control input T m k as

T m k = Tm k + u k , (13)
where Tm k is a positive constant representing the nominal torque, and u k is a free control input for k = 1, 2. Remark 1: The current i f could be used as an additional control input for reactive power and voltage control [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. In the present work, we follow [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF] and assume that i f is constant, see also [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. 1 In the sequel, the time-dependency of signals is omitted, whenever it is clear from the context.

B. Model of two synchronverters coupled via dynamic power lines in dq-coordinates

For the subsequent derivations, it is convenient to represent the system [START_REF] Kroposki | Benefits of power electronic interfaces for distributed energy systems[END_REF] in dq-coordinates [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF]. Thus, let ϕ(t) ∈ R and consider the dq-transformation matrix defined as follows

T dq (ϕ) = 2 3 cos (ϕ) cos ϕ -2 3 π cos ϕ + 2 3 π sin (ϕ) sin ϕ -2 3 π sin ϕ + 2 3 π , (14)
as well as the rotation matrix

T r (ϕ 1 -ϕ 2 ) = cos (ϕ 1 -ϕ 2 ) -sin (ϕ 1 -ϕ 2 ) sin (ϕ 1 -ϕ 2 ) cos (ϕ 1 -ϕ 2 ) , (15)
which allows to transform dq-variables from a dq-frame with ϕ = ϕ 1 to another dq-frame with ϕ = ϕ 2 , see [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF].

By applying [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF] with ϕ = δ k to the k-th synchronverter, the model ( 11) can be written in its local dq-coordinates as δk =ω k ,

J k ωk = -D k ω k + Tm k + u k -T e k + ρ m k , L k di k,dq dt = [L k J (ω k ) -R k I 2 ] i k,dq + e k,dq - T dq (δ k )V L,abc + µ e k , (16)
where µ e k ∈ R 2 is the transformation of ρ e k and

J (ω k ) = 0 -ω k ω k 0 , e k,dq = b k ω k 0 1 .

So, electrical torque and load voltage result in

T e k = b k i k,q and T dq (δ k )V L,abc = R L 2 j=1 T r (δ k -δ j ) i j,dq .
To combine both synchronverter models ( 16) in a single system, we define the difference between angles θ = δ 1 -δ 2 , and the state vectors

z k = ω k i k,d i k,q ⊤ , for k = 1, 2.
Therefore, the model for the whole system of Figure 1 can be written compactly as

Γ 1 ż1 =A 1 (ω 1 ) z 1 -B (θ) z 2 + ê1 T m1 + µ 1 , Γ 2 ż2 =A 2 (ω 2 ) z 2 -B (θ) ⊤ z 1 + ê1 T m2 + µ 2 , θ =ω 1 -ω 2 , (17)
where ê1 = 1 0 0 ⊤ , µ k = ρ m k µ e k ⊤ , and

A k (ω k ) =   -D k 0 -b k 0 -(R k + R L ) -L k ω k b k L k ω k -(R k + R L )   , B (θ) =R L   0 0 0 0 cos (θ) -sin (θ) 0 sin (θ) cos (θ)   , Γ k =   J k 0 0 0 L k 0 0 0 L k   .
Following [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF], we assume that the two synchronverters have identical parameters, such that Γ k = Γ. Likewise, recall that T m k = Tm k + u k . Hence, we obtain

Γ ż1 = Â (ω 1 ) z 1 -B (θ) z 2 + ê1 Tm + ê1 u 1 + µ 1 Γ ż2 = Â (ω 2 ) z 2 -B (θ) ⊤ z 1 + ê1 Tm + ê1 u 2 + µ 2 θ =ω 1 -ω 2 , (18)
where

Â(ω k ) =   -D 0 -b 0 -(R + R L ) -Lω k b Lω k -(R + R L )   , k = 1, 2.
C. Equilibrium set Now, the equilibria of the unforced nominal system (18) (i.e., with u 1 = u 2 = 0 and µ 1 = µ 2 = 0 2 ) are given by

ω 1 -ω 2 = 0, Â (ω 1 ) z 1 -B (θ) z 2 + ê1 Tm = 0 3 , Â (ω 2 ) z 2 -B (θ) ⊤ z 1 + ê1 Tm = 0 3 . ( 19
)
Based on [START_REF] Groß | On the steady-state behavior of a nonlinear power system model[END_REF], [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF], we consider the following assumption. Assumption 3: All solutions of [START_REF] Schiffer | Strong and weak Leonov functions for global boundedness of state periodic systems[END_REF] belong to the set

S E = S E * ∪ S E * * , (20)
where

SE * = [z1, z2, θ] ∈ R 7 | z1 = z2 = z * , θ = 2jπ , SE * * = [z1, z2, θ] ∈ R 7 | z1 = z2 = z * * , θ = (2j -1) π ,
for j ∈ Z. Moreover, S E * corresponds to the locally stable equilibrium points and S E * * , contains the unstable ones.

Remark 2: Assumption 3 is reasonable, because we are considering identical parameters as [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF]. In general, the system may also possess other equilibria.

Under Assumption 3, we denote (z * , 0) as the reference equilibrium and the shifted state vector is defined as follows

x = z1 z2 θ ⊤ = z 1 -z * z 2 -z * θ ⊤ . (21)
Then, the shifted system can be written as

Γ ż1 = Ã (z1,1) z1 -B( θ)z2 -B( θ)z * + ê1u1 + µ1, Γ ż2 = Ã (z2,1) z2 -B( θ) ⊤ z1 -B( θ) ⊤ z * + ê1u2 + µ2, θ =ê ⊤ 1 (z1 -z2) , (22)
where B( θ) = B( θ) -B (0) and

à (z k,1 ) =   -D 0 - b -Li * q -(R + RL) -L (z k,1 + ω * ) b + Li * d L (z k,1 + ω * ) -(R + RL)   .
Hence, all invariant solutions of the system (22) are contained in the set

W = W1 ∪ W2, W2 ={x ∈ R 7 | z1 = z2 = z * * -z * , θ = (2j + 1)π}, W1 ={x ∈ R 7 | z1 = z2 = 03, θ = 2jπ}, j ∈ Z. (23)
Now, we formulate the problem addressed in this work. Problem 1: Consider the system [START_REF] Schiffer | Almost global synchronization in radial multi-machine power systems[END_REF]. Design a feedback control law for u k , k = 1, 2, such that the closed-loop set Wcl ⊃ W1 is globally ISS with respect to µ 1 and µ 2 .

IV. ISS-CONTROL LEONOV FUNCTION DESIGN

In this section, we present a control law, which solves Problem 1. Since the system dynamics ( 22) are periodic with respect to the variable θ, Problem 1 is hard to solve using ISS methods based on conventional Lyapunov theory, see also the discussion in [START_REF] Efimov | A relaxed characterization of ISS for periodic systems with multiple invariant sets[END_REF], [START_REF] Efimov | On boundedness of solutions of state periodic systems: A multivariable cell structure approach[END_REF], [START_REF] Schiffer | Strong and weak Leonov functions for global boundedness of state periodic systems[END_REF]. Thus, to design the controller, we follow our recent work [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF].

For the sake of a compact presentation of the subsequent derivations, we introduce the following variables

σ 1 = (2 (R + R L ) -λ) p 2 -R L ∥ (z * 2 , z * 3 ) ∥ 2 , σ c =2p 2 R L cos( θ), σ s = 2p 2 R L sin( θ), σ * 1 =σ 1 + 0.5λp 2 , σ * k,2 = p k,1 b -p 2 (Li * d + b) , σ * * k = (2D -0.5λ) p k,1 -8ϵλ -1 , (24)
for p 1,1 > 0, p 2,1 > 0, p 2 > 0, λ > 0, ϵ > 0 and k = 1, 2.

A. Preliminary lemmata

In order to verify the required properties of a ISS-CLeF for the system [START_REF] Schiffer | Almost global synchronization in radial multi-machine power systems[END_REF], we introduce the matrices

MS = M S,1 0 4×1 0 1×4 λϵ -4p 2 2 R L , MS,1 = σ 1 I 2 Mσ M ⊤ σ σ 1 I 2 , (25)
M * S,1 = σ * 1 I 2 Mσ M ⊤ σ σ * 1 I 2 , Mσ = σ c -σ s σ s σ c , (26)
and

M * =   8ϵ λ ê1 0 3×2 0 3×3 ϵê 1 0 3×3 8ϵ λ ê1 0 3×2 -ϵê 1 ϵê ⊤ 1 -ϵê ⊤ 1 λϵ 2 -4p 2 2 R L   , (27)
to present the following three lemmata. Lemma 1: The matrix MS is positive definite for any θ ∈ S if and only if Due to space limitations, the previous proofs are omitted. However, they are based on Schur's complement.

ϵ > 4λ -1 p 2 2 R L , λ > p -1 2 2p 2 R -R L ∥ (z * 2 , z * 3 ) ∥ 2 . (

B. Main Result

To present the main result, we define

K( Φ, β) = Φ+ √ Φ2 +∥β∥ 4 ∥β∥ 2 if ∥β∥ ̸ = 0 1 if ∥β∥ = 0 , (30)
where

β(z 1 , z2 ) =2 p 1,1 z1,1 p 2,1 z2,1 , Φ(z 1 , z2 ) = -z1 z2 ⊤ M * z1 z2 , (31)
with

M * =        σ * * 1 Lp2i * q σ * 1,2 0 0 0 Lp2i * q σ * 1 0 0 σc -σs σ * 1,2 0 σ * 1 0 σs σc 0 0 0 σ * * 2 Lp2i * q σ * 2,2 0 σc σs Lp2i * q σ * 1 0 0 -σs σc σ * 2,2 0 σ * 1        . (32)
In line with Problem 1, we are interested in robustifying the set of stable equilibria, i.e., W1 . This is a particular application of the approach presented in [START_REF] Mendoza-Ávila | A universal formula for ISS and integral ISS stabilization of multistable periodic systems[END_REF], which leads to our main result of this work. We stress that the control may lead to new, additional equilibria of the closed-loop system and, thus, Wcl ⊃ W1 .

Theorem 2: Choose p 1,1 > 0, p 2,1 > 0, ϵ > 0, λ > 0 and p 2 > 0 in [START_REF] Sontag | A 'universal' construction of Artstein's theorem on nonlinear stabilization[END_REF], such that (28) and (29) are satisfied. Assume that the set W cl is decomposable. Then, the control law

u = u 1 u 2 ⊤ = -K(z 1 , z2 ) β ⊤ (z 1 , z2 ). (33)
ensures that the system ( 22) is ISS with respect to the closedloop set Wcl ⊃ W1 and to the external inputs µ 1 and µ 2 .

where βµ = β ⊤ µ , 0 6×1 ⊤ and

M =         D λ 1 Lp2i * q σ * 1,2 0 0 0 ϵ Lp2i * q σ1 0 0 σc -σs 0 σ * 1,2 0 σ1 0 σs σc 0 0 0 0 D λ 2 Lp2i * q σ * 2,2 -ϵ 0 σc σs Lp2i * q σ1 0 0 0 -σs σc σ * 2,2 0 σ1 0 ϵ 0 0 -ϵ 0 0 ϵ λ         , with D λ1 = (2D -λ) p 1,1 , D λ2 = (2D -λ) p 2,1 , ϵ λ = λϵ -4p 2
2 R L , and the auxiliary variables defined in [START_REF] Sontag | A 'universal' construction of Artstein's theorem on nonlinear stabilization[END_REF]. To show that condition (3) of Definition 1 is verified, we consider the set, where β in (37) is zero, i.e., the set

S = x ∈ R 7 | ∥β(z 1 , z2 )∥ = 0 . (38)
Then, we have from (37) that

a(x, μ)| x∈S ≤ -λ V (x)| x∈S + υ∥μ∥ 2 -υ μ⊤ m μm - xS μe ⊤ MS βS β⊤ S υI 4 xS μe , (39)
where xS = z1,2 , z1,3 , z2,2 , z2,3 , θ ⊤ , MS is defined in [START_REF] Weiss | Stability analysis for coupled synchronous generators with virtual friction[END_REF], μm = ρ m1 , ρ m2 ⊤ , μe = µ ⊤ e1 , µ ⊤ , we conclude that condition (3) of Definition 1 is satisfied. Hence, (34) is an ISS-CLeF for the system [START_REF] Schiffer | Almost global synchronization in radial multi-machine power systems[END_REF].

The last step of the proof is to show that the controller (33) provides the ISS property in closed-loop with the system [START_REF] Schiffer | Almost global synchronization in radial multi-machine power systems[END_REF] with respect to W1 . For this, we use Theorem 1 and the fact that W′ = W1 . It, thus, remains to verify condition [START_REF] Efimov | On boundedness of solutions of state periodic systems: A multivariable cell structure approach[END_REF].

Consider the following function

Φ(z 1 , z2 ) = sup V ≥0 ā(x) + λ 2 V (x) ≤ sup V ≥0 -x ⊤ M * x ,
where ā = sup μ∈R 6 a(x, μ) -υ∥μ∥ 2 , and M * = M * + M * , with M * and M * defined in ( 27) and (32), respectively. By Lemma 2, the matrix M * is positive semidefinite if (29) is satisfied. Therefore, the previous inequality is reduced to

Φ(z 1 , z2 ) ≤ Φ(z 1 , z2 ). ( 40
)
The functions Φ(z 1 , z2 ) and β(z 1 , z2 ) given in (31) are substituted in the formula (30) to define the controller (33). Clearly the 2π-periodicity follows trivially. So, we only have to check the continuity of the formula of (30) for all x ∈ R 7 .

We have that Φ(z 1 , z2 ) and β(z 1 , z2 ) are continuous by construction. Hence, (30) is also continuous for

Fig. 1 :

 1 Fig. 1: Two parallel synchronverters feeding a resistive load.

1 2L 2 p 2 2 4λmin

 12 Lp 2 I 4 0 1×4 ⊤ . By invoking Lemma 1, under the required conditions by the theorem, the matrix MS is positive definite. Thus, using the Schur complement for υ ≥

  ∥β(z 1 , z2 )∥ ̸ = 0. To analyze the case ∥β(z 1 , z2 )∥ = 0, recall the property √ a + b ≤ √ a + √ b for any a, b ≥ 0. Hence, K( Φ, β) ≤ Φ + Φ ∥β∥ -2 + 1.

  28) Lemma 2: The matrix M * is positive semidefinite for any θ ∈ S if and only if

	ϵ ≥ 16λ -1 p 2 2 R L . Lemma 3: The matrix M * S,1 is positive definite for any (29) θ ∈ S if (28) is satisfied.

The control was designed by using the novel theory of ISS-CLeFs. In comparison to the classic control Lyapunov function, this new methodology relaxes the restrictions to design the control function by exploiting the periodicity of the system dynamics. This allows us to obtain the controller by analyzing a reduced system instead of the whole system.

Future work is geared towards extending the presented control design approach to other settings in low-inertia power systems, such as multiple, non-identical synchronverters, different types of loads and other type of inverter controls. Also, we plan to implement and validate our proposed control in our power-hardware-in-the-loop laboratory [10].

on Energy Infrastructures and Geothermal Systems IEG, (e-mail: schiffer.b-tu.de). This work is partially supported by the ANR-DFG project SyNPiD -446182476.

Proof: We propose the following ISS-CLeF candidate

where the matrices P 1 = diag (p 1,1 , p 1,2 , p 1,3 ) > 0 and P 2 = diag (p 2,1 , p 2,2 , p 2,3 ) > 0 and ϵ ∈ R + are design parameters. Since all its parameters are positive, we can readily see that the function (34) satisfies condition [START_REF] Beck | Virtual synchronous machine[END_REF] with, see [START_REF] Schiffer | Global synchronization analysis of droop-controlled microgrids: A multivariable cell structure approach[END_REF],

Now, defining μ⊤ = µ ⊤ 1 , µ ⊤ 2 , the derivative of (34) along the trajectories of the system ( 22) is given by

where

Analyzing the terms in (36) that do not depend on u, for any positive constant υ > 0, we have

For the next step, it is important to stress that, in general, ÃP ksym (z k,1 ) depends on zk,1 , k = 1, 2. This dependency generates cross terms of order 3, which cannot be compensated by the quadratic terms. To avoid this issue and taking advantage of the structure of the system ( 22), we choose p 1,3 = p 1,2 and p 1,3 = p 1,2 , so that all cross terms of order 3 are canceled. Likewise, we fix p 1,2 = p 2,2 = p 2 to reduce some terms and using trigonometric properties, we obtain

we have

Using Young's inequality and the triangle inequality, we can bound a(x, μ) in (36) as follows

Recall the set S in (38). On S, we have from (31) that

where M * S is presented in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. By using Lemma 3, under the conditions of the theorem, M * S is positive definite. So, ∥β(z 1 , z2 )∥ = 0 =⇒ Φ(z 1 , z2 ) ≤ 0. Moreover, ∥β(z 1 , z2 )∥ and Φ(z 1 , z2 ) are zero simultaneously only at the origin and Φ(z 1 , z2 ) and ∥β(z 1 , z2 )∥ 2 are quadratic polynomiallike functions, so ( 6) is satisfied for some ε > 0 and for all x ∈ R 7 . From (41), we can see that K( φ, β) S ≤ 1 + ε for all x ∈ R 7 . So, function (30) is continuous for all x ∈ R 7 .

Finally, since W1 = W′ ⊂ Wcl , the controller (33) vanishes at every element of the set W1 . Thus, the set W1 is kept and the function ( 34) is an ISS-Leonov function for the closed-loop system ( 22), (33) with respect to Wcl .

V. SIMULATION RESULTS For the system in Figure 1, we consider identical piecewise constant perturbations, see Figure 2a. We compare the behavior of the system with and without the controller (33) under initial condition as 0 7 .

Based on [START_REF] Natarajan | Synchronverters with better stability due to virtual inductors, virtual capacitors, and anti-windup[END_REF], we consider two identical synchronverters [START_REF] Rosso | Robust stability analysis of synchronverters operating in parallel[END_REF], whose parameters are

By numerical computation, we can determine that the equilibria of the unforced system [START_REF] Rosso | Robust stability analysis of synchronverters operating in parallel[END_REF] are given by In Figure 2b, the norm of the state x is shown. We see that in both cases, the trajectories converge to zero but the trajectories for the system with controller (33) converge much faster. Likewise, we can verify that in presence of perturbations, the final norm of the state is by far smaller, when the controller (33) is used.

VI. CONCLUSIONS AND FUTURE WORK

We designed an ISS controller, which provides robustness to two identical synchronverters connected in parallel with a resistive load. The closed-loop system is ISS against external perturbations in the frequency and current dynamics.


Jesus Mendoza-Avila
email: jesus.mendoza-avila@inria.fr

Denis Efimov
email: denis.efimov@inria.fr.

Ángel Mercado-Uribe

Johannes Schiffer
email: schiffer@b-tu.de.

Design of Controls for Boundedness of Trajectories of Multistable State Periodic Systems

In this paper, the concept of control Leonov functions is introduced, and it is shown that their information is enough to design continuous and periodic controllers that provide boundedness of the state for a class of multistable stateperiodic systems. These feedback control laws are based on a mild adaptation of Sontag's universal formula, and a kind of small control property. The proposed method is illustrated via application in a microgrid.

I. INTRODUCTION

The control Lyapunov function framework has been found very useful for analysis of stabilizability and stabilization of nonlinear dynamical systems with a unique equilibrium at the origin [START_REF] Artstein | Stabilization with relaxed controls[END_REF]- [START_REF] Efimov | A condition of clf existence for affine systems[END_REF]. Merely, in many cases the existence of a control Lyapunov function is a necessary and sufficient condition to conclude stabilizability of the system under study. Then, explicit expressions of control feedback laws are determined by using the so-called universal Sontag's formula [START_REF] Sontag | A 'universal' construction of Artstein's theorem on nonlinear stabilization[END_REF]. These approaches have been extended for systems with exogenous inputs by means of the concepts of ISS-and integral ISS-control Lyapunov functions [START_REF] Liberzon | Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation[END_REF].

Many engineering applications concern systems with multiple equilibria or invariant sets, e.g., power or biological systems. However, the study of stability and boundedness of solutions for multistable systems requires significant modifications and relaxations of the employed Lyapunov stability notions, see for example [START_REF] Angeli | Detection of multistability, bifurcations, and hysteresis in a large class of biological positivefeedback systems[END_REF]- [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF]. In this context, extensions of the control Lyapunov function framework for multistable systems have been presented in [START_REF] Barroso | Universal formula for robust stabilization of affine nonlinear multistable systems[END_REF], [START_REF]Control lyapunov function method for robust stabilization of multistable affine nonlinear systems[END_REF].

In this note, we are interested in a special class of such multistable systems being periodic with respect to a part of the state. Some examples of such a class of systems include the forced nonlinear pendulum [START_REF] Forni | Differential analysis of nonlinear systems: Revisiting the pendulum example[END_REF], [START_REF] Efimov | Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids[END_REF], power systems [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF], [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF], microgrids [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF], [START_REF] De Persis | Bregman storage functions for microgrid control[END_REF], and phase-locked loops [START_REF] Leonov | Nonlinear analysis of phase-locked loop (PLL): Global stability analysis, hidden oscillations and simulation problems[END_REF]. Recently, a fitting tool for the analysis of boundedness of solutions for multistable state-periodic systems has been proposed in [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF], [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF] through the concept of Leonov functions. These Leonov functions are useful to establish attractiveness of certain sets of the state space, and boundedness of solutions is concluded by exploiting the periodicity of the system's dynamics [START_REF] Efimov | A new criterion for boundedness of solutions for a class of periodic systems[END_REF]- [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF]. These results are based on the ideas of [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF] (see also [START_REF] Gelig | Stability of nonlinear systems with non-unique equilibrium state[END_REF], [START_REF] Noldus | New direct Lyapunov-type method for studying synchronization problems[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF]), who have proposed the so-called cell structure approach. Moreover, the concepts of ISS-Leonov functions and iISS-Leonov functions have been presented in [START_REF] Efimov | A relaxed characterization of ISS for periodic systems with multiple invariant sets[END_REF] and [START_REF] Mendoza-Avila | On relaxed conditions of integral ISS for multistable periodic systems[END_REF], respectively, as relaxed characterizations of the corresponding properties.

An extension of the Leonov function framework for ISS and iISS stabilization of multistable state-periodic systems has been presented in [START_REF] Mendoza-Avila | Design of controls for iss and integral iss stabilization of multistable state periodic[END_REF] through the concepts of ISSand iISS-control Leonov functions. Unfortunately, the latter do not imply also boundedness of periodic variables by its definition [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF], [START_REF] Forni | Characterizations of integral input-to-state stability for systems with multiple invariant sets[END_REF]. Therefore, in this paper we further develop the results of [START_REF] Mendoza-Avila | Design of controls for iss and integral iss stabilization of multistable state periodic[END_REF] in order to design continuous and periodic feedback laws providing the boundedness of all states for multistable state-periodic systems. To this end, we introduce the concept of control Leonov function (CLeF). The main advantage of the CLeF's approach is that the designed controllers are capable to ensure global boundedness of the trajectories of the controlled plant, while the most of the reported methods provide just local results, see for example [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF], [START_REF] De Persis | Bregman storage functions for microgrid control[END_REF]. This is illustrated by applying the proposed method for the design of a controller for global boundedness of solutions in a small microgrid.

The remainder of the paper is structured as follows. Section II presents the considered class of systems and some useful definitions. Section III contains our main results. Then, Section IV presents its application in a small microgrid. Finally, the conclusions are given in Section V.

II. PRELIMINARIES A. Notation

Z and R stand for the sets of integer and real numbers, respectively. Moreover, R + represents the set of non-negative real numbers, i.e., R

+ = {x ∈ R : x ≥ 0}. Similarly, Z + = {x ∈ Z : x ≥ 0}
A function α : R + → R + belongs to the class K if it is continuous, strictly increasing and α(0) = 0. The function α : R + → R + belongs to the class K ∞ if α ∈ K and it is unbounded. A continuous function β : R + × R + → R + belongs to the class KL if, for each fixed t ∈ R + , β(•, t) ∈ K ∞ and, for each fixed s ∈ R + , β(s, •) is non-increasing and it tends to zero for t → ∞.

The distance from a point x ∈ R n to the set S ⊂ R n is defined as |x| S = inf a∈S |x -a|, where |x| = |x| {0} for x ∈ R n is the usual Euclidean norm for a vector x ∈ R n . Furthermore, we introduce the vector norm

|x| ∞ = max 1≤i≤n |x i |. Then, |x| ∞ ≤ |x| ≤ √ n|x| ∞ .
Besides, for a signal d : R → R m the essential supremum norm is defined as ∥d∥ ∞ = ess sup t≥0 |d(t)|, and the set of such inputs with a finite norm is denoted by L m ∞ .

B. Multistable state periodic system

Consider the system

ẋ = f (x), (1)
where x ∈ R n is the state, and f : R n → R n is a map of class C 1 (continuously differentiable with respect to its arguments), satisfying f (0) = 0. For any x ∈ R n , we denote by X(t, x) the uniquely defined solution of (1) at time t fulfilling X(0, x) = x.

A set S ⊂ R n is invariant for (1) if X(t, x) ∈ S for all t ∈ R and for all x ∈ S; for x ∈ R n the point y ∈ R n belongs to its ω-limit (α-limit) set if there is a sequence t i , lim i→+∞ t i = +∞, such that lim i→+∞ x(t i , x) = y (lim i→+∞ x(-t i , x) = y); for any x ∈ R n its αand ω-limit sets are invariant [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems, and bifurcations of vector fields[END_REF]. Now, we say that the system (1) is periodic in a part of the state [START_REF] Gelig | Stability of nonlinear systems with non-unique equilibrium state[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] if it satisfies the following assumption.

Assumption 1: Let x = (z, θ) ∈ R n , where z ∈ R k and θ ∈ R q are two subsets of the state vector, n = k + q, k > 0 and q > 0. The vector field f in (1) is 2π-periodic with respect to θ, i.e., for all

x = (z, θ) ∈ R n f (x) = f (x + ξ j ), ξ j = [0 ⊤ k 2πj ⊤ ] ⊤ ∈ R n ,
where 0 k is a zero vector of dimension k and j ∈ Z q .

In other words, we suppose that the system (1) can be embedded in a manifold M = R k × S q , where S is the unit sphere, by a simple projection of the variables θ on S q .

C. Boundedness of solutions of periodic systems

The so-called cell structure approach was introduced in [7], [START_REF] Leonov | On the boundedness of the trajectories of phase systems[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] to establish boundedness of solutions for periodic systems, satisfying Assumption 1 for q = 1. This approach can be seen as a fine handling of attractiveness and periodicity, which leads to boundedness of trajectories. That is, the admissible behavior of the trajectories is bounded by a cell covering of the domains of the periodic variables, which is formed by isolated, periodically distributed, attractive, and forward invariant sets.

More precisely, for the system (1), with q = 1, the works in [START_REF] Gelig | Stability of nonlinear systems with non-unique equilibrium state[END_REF], [START_REF] Yakubovich | Stability of stationary sets in control systems with discontinuous nonlinearities[END_REF] show that the existence of a sign-indefinite function V 0 (x), whose time derivative satisfies V0 (x) ≤ -ϱV 0 (x) for all t ≥ 0, where ϱ > 0, implies attractiveness and forward invariance of the set Ω 0 = {x ∈ R n : V 0 (x) ≤ 0}, i.e., X(t, x 0 ) ∈ Ω 0 for all t ∈ [0, +∞] provided that x 0 ∈ Ω 0 . Then, by exploiting the periodicity of the system (1), the functions V j (x) = V j ((z, θ)) = V 0 ((z, θ + 2πj)) are defined for all j ∈ Z, such that, the sets Ω j = {x ∈ R n : V j (x) ≤ 0} are also attractive and forward invariant for [START_REF] Artstein | Stabilization with relaxed controls[END_REF]. Finally, it is concluded that the intersections of the sets Ω j form a kind of cell cover of the domain of the variable θ, where each cell is bounded, isolated and forward invariant.

In [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF], a generalization of the cell structure approach is developed for systems whose dynamics are periodic with respect to multiple state variables, i.e., for the case when the system (1) satisfies Assumption 1 for q > 1. To this end, [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF] introduced the notion of Leonov functions. This concept has been further developed in [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF].

Following [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF], consider the sets:

U = ∪ r∈Z+ U r , where U r = {x = (z, θ) ∈ R n : |θ| ∞ = 2rπ, f (x) ≡ 0}, and W ⊂ R n , such that for some c ∈ (0, 2π) and c ∈ (π, 2π), with c < c, we have {x ∈ R n : c ≤ |θ| ∞ ≤ c} ⊆ W.
Note that the set U includes all equilibria of (1) obtained by shifting the one at the origin by using the property that f is 2π-periodic in θ. However, in general, the system (1) may possess other equilibria. Hence, let us define the equilibrium set of (1) as D = {x ∈ R n : f (x) ≡ 0}. Clearly, U ⊆ D. On the other hand, the set W represents a barrier in R n between the origin and the rest of the elements in U.

Definition 1: [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF]. A C 1 function V : R n → R is a strong Leonov function for (1) if there exist a constant g ≥ 0, functions α ∈ K ∞ , ψ ∈ K and a continuous function λ : R → R satisfying λ(0) = 0, and λ(s)s > 0 for all s ̸ = 0, such that

α(|z|) -ψ(|θ|) -g ≤ V (x) ∀x = (z, θ) ∈ R n , inf x∈W V (x) > 0, sup x∈U V (x) ≤ 0, (2)
and the following dissipation inequality holds:

∂V (x) ∂x f (x) ≤ -λ(V (x)) ∀x ∈ {x ∈ R n : V (x) ≥ 0}.
(3) Theorem 1: [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF]. If for the system (1) under Assumption 1, there exists a strong Leonov function as in Definition 1, then for all x 0 ∈ R n , the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0. By exploiting the periodicity of the system (1), the proof of Theorem 1 in [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF] consists of two steps: first, it is shown that the sets Ω j = {x ∈ R n : V j (x) ≤ 0}, where j = [j 1 , . . . , j q ] ∈ Z q and V j (x) = V (x -ξ j ), are globally attractive and forward invariant for [START_REF] Artstein | Stabilization with relaxed controls[END_REF]. Note that U ∈ Ω j for any j. Then, it is proven that the intersection of the sets Ω j is composed by compact and isolated "cells". For this, the condition inf x∈W V (x) > 0, plays a crucial role. So, for any x 0 ∈ R n , the solution X(t, x 0 ) asymptotically enters and remains in one of those cells. Thus, for any x 0 ∈ R n , the corresponding solution X(t, x 0 ) is bounded for all t ≥ 0.

Corollary 1: [START_REF] Schiffer | Strong and weak leonov functions for global boundedness of state periodic systems[END_REF]. If for the system (1), under Assumption 1, there exists a Leonov function as in Definition 1 with

V (0) ≤ 0, inf x∈U \U0 V (x) ≤ 0, inf x∈W V (x) > 0,
and with the following dissipation inequality

∂V (x) ∂x f (x) < 0 ∀x ∈ R n \ {D ∪ {x ∈ R n : V (x) ≤ 0}},
then for all x 0 ∈ R n the corresponding trajectories X(t, x 0 ) are bounded for all t ≥ 0.

III. MAIN RESULTS

Consider a system of the form

θ = f 1 (x), ż = f 2 (x) + g(x)u, (4)
where x ∈ R n is the state vector, x = (z, θ) ∈ R n with z ∈ R k and θ ∈ R q , n = k + q for k, q > 0, and u ∈ L p ∞ is an admissible control input. Moreover, f 1 : R n → R q , f 2 : R n → R k and g : R n → R k×p are supposed to be locally Lipschitz continuous vector functions, with f 1 (0) = 0 and f 2 (0) = 0. Assumption 2: The vector fields f 1 , f 2 and g in ( 4) are 2π-periodic with respect to θ.

Now, the problem is to determine conditions that ensure the existence of a continuous and periodic feedback u(x), such that the trajectories X(t, x 0 ) of the system (4) are bounded for all t ≥ 0. Let us define the sets D, U and W for the system (4), with u ≡ 0, as in Subsection II-C by taking

f (x) = [f ⊤ 1 (x), f ⊤ 2 (x)] ⊤ .
Then, we present the following definition of CLeF's.

Definition 2: A C 1 function V : R n → R is said to be a CLeF for the system (4) if there exist functions φ, ϑ, σ 1 , σ 2 ∈ K ∞ , scalars g 1 , g 2 ≥ 0, and a continuous function ϖ : R → R satisfying ϖ(0) = 0, and ϖ(s)s > 0 for all s > 0, such that for all x = (z, θ) ∈ R n :

φ(|z|) -σ 1 (|θ|) -g 1 ≤ V (x) ≤ ϑ(|x| U ) -σ 2 (|θ|) + g 2 , inf x∈W V (x) > 0, sup x∈U V (x) ≤ 0, (5) and for all x ∈ R n \ Ω with Ω = {x ∈ R n : V (x) < 0}: inf u∈R p {a(x) + b(x)u} ≤ -ϖ(V (x)), (6)
where a(x)

= ∂V (x) ∂θ f 1 (x) + ∂V (x) ∂z f 2 (x) and b(x) = ∂V (x)
∂z g(x), with a function b(x) being 2π-periodic in θ, and satisfying b(0) ≡ 0.

Note that the restriction (5) for CLeF's is stronger than the condition (2) for Leonov functions. However, this stronger restriction is needed for the design of a controller u periodic with respect to θ. Hence, we can see from (6) that, under a proper design of the control input u, a CLeF is indeed a Leonov function for the controlled system.

In general, CLeFs are non-periodic, hence the partial derivative of V with respect to θ may contain non-periodic terms of θ, which appears in the function a(x). From (5), we have σ 2 (|θ|) ≤ ϑ(|x| U ) + g 2 for all x ∈ R n \ Ω, then we can define a function

ϕ(|x| U , z) = sup σ2(|θ|)≤ϑ(|x| U )+g2 {a(x) + 1 2 ϖ(V (x))}, (7)
which is 2π-periodic with respect to θ since |x| U = |x-ξ j | U .

On the other hand, the function b(x) is 2π-periodic with respect to θ by construction. Now, following [START_REF] Sontag | A 'universal' construction of Artstein's theorem on nonlinear stabilization[END_REF], consider the so-called "universal formula" given by

K(ϕ, |b|) = ϕ+ √ ϕ 2 +|b| 4 |b| 2 if |b| ̸ = 0, 1 if |b| = 0, (8)
where the arguments of the functions ϕ(|x| U , z) and b(x) have been omitted to get a shorter expression. Note that since these functions are supposed to be 2π-periodic with respect to θ, then (8) inherits this property as well. Now, let us check the continuity of the function [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] in the set R n \ Ω, i.e., wherever V (x) ≥ 0. By construction the functions ϕ(|x| U , z) and b(x) are continuous everywhere, then the function ( 8) is continuous for |b(x)| ̸ = 0. Recall the property √ a + b ≤ √ a + √ b, for any a, b ≥ 0. So, we have

K(ϕ(|x| U , z), |b(x)|) ≤ ϕ(|x| U , z) + |ϕ(|x| U , z)| |b(x)| 2 + 1. Clearly, if ϕ(|x| U , z) < 0 then K(ϕ(|x| U , z), |b(x)|) ≤ 1,
hence this condition is sufficient to ensure the continuity of (8) for b(x) ≡ 0. From ( 6), it follows that b(x) ≡ 0 leads to a(x) ≤ -ϖ(V (x)) for all x ∈ R n \ Ω. Consequently, we obtain that b(x) ≡ 0 implies

ϕ(|x| U , z) ≤ sup σ2(|θ|)≤ϑ(|x| U )+g2 {-1 2 ϖ(V (x))}, (9)
for all x ∈ R n \ Ω. Hence, for b(x) ≡ 0: if V (x) > 0 then we have ϕ(|x| U , z) < 0, as it is needed. However, if V (x) = 0, we may have ϕ(|x| U , z) = 0. In this case, we need to guarantee that the term ϕ(|x| U , z) vanishes "faster" than b(x) around the points of the set Γ = {x ∈ R n : V (x) = 0; |b(x)| = 0}. For this, let us suppose that for each constant ε > 0 there is a δ > 0 such that

sup 0<|x|Γ≤δ ϕ(|x| U , z) |b(x)| 2 < ε. (10)
We call this condition a small control property in the framework of Leonov functions (SCP-LeF). Hence, if the condition (10) is verified, then the formula (8) satisfies

K(ϕ(|x| U , z), |b(x)|) ≤ 2ϕ(|x| U , z) |b(x)| 2 + 1 ≤ 2ε + 1,
for all 0 < |x| Γ ≤ δ. Since we can make ε arbitrarily small then we conclude that the condition (10) ensures the continuity of ( 8) in all points of Γ. Consequently, we can say that if the SCP-LeF (10) is satisfied, then the function

(8) is continuous in R n \ Ω.
Due to the continuity of the function [START_REF] Angeli | Characterizations of input-to-state stability for systems with multiple invariant sets[END_REF] in the set R n \ Ω, there exists a continuous periodic function K(x), such that

K(ϕ(|x| U , z), |b(x)|) ≤ K(x) (11)
for all x ∈ R n \ Ω. For x ∈ Ω, the latter property cannot be ensured but the function K(x) keeps its continuity globally.

Note that if we have ϕ(|x| U , z) < 0 wherever b(x) ≡ 0 for all x ∈ R n and the SCP (10) is satisfied, then the function ( 8) is continuous everywhere. Thus, we can set [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF], i.e., we can directly apply the Sontag's formula for the design of the controller as in the Lyapunov framework. Finally, we propose the following feedback control law

K(ϕ(|x| U , z), |b(x)|) = K(x) in
u(x) = -K(x)b ⊤ (x), (12)
which is continuous in R n , and 2π-periodic in θ. Hence, the closed-loop system (4),( 12) inherits these properties.

Moreover, the controller (12) vanishes at every element of the set U since by definition b(0) ≡ 0. Therefore, U is contained in the equilibrium set of the closed-loop system (4), [START_REF] Barroso | Universal formula for robust stabilization of affine nonlinear multistable systems[END_REF]. Finally, we are able to establish the following results. Theorem 2: Let the system (4) satisfy Assumption 2. If there exists a CLeF as in Definition 2 satisfying the SCP-LeF [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF], then the periodic feedback control law ( 12) is continuous on R n , and for all x 0 ∈ R n the trajectories X(t, x 0 ) of the closed-loop system (4), [START_REF] Barroso | Universal formula for robust stabilization of affine nonlinear multistable systems[END_REF] are bounded for all t ≥ 0.

Corollary 2: Let the system (4) satisfy Assumption 2. If there exists a CLeF as in Definition 2 with

V (0) ≤ 0, inf x∈U \U0 V (x) ≤ 0, inf x∈W V (x) > 0, ( 13
)
and with the dissipation inequality

inf u∈R p {a(x)+ b(x)u} < 0, ∀x ∈ R n \{cl(Ω) ∪D}, ( 14
)
with a(x) and b(x) as in Definition 2, besides the SCP-LeF ( 10) is satisfied with

Γ = {x ∈ R n : V (x) = 0; |b(x)| = 0} ∪ {x ∈ D : V (x) > 0; |b(x)| = 0}
, then the feedback control law ( 12) is continuous on R n , and for all x 0 ∈ R n the trajectories X(t, x 0 ) of the closed-loop system (4), [START_REF] Barroso | Universal formula for robust stabilization of affine nonlinear multistable systems[END_REF] are bounded for all t ≥ 0. Corollary 2 is a counterpart of Corollary 1 that relaxes the conditions of Definition 1. This simplifies the construction of CLeF's. The proofs of these results are omitted due to space limitation.

IV. APPLICATION TO CONTROLLER DESIGN FOR A

MICROGRID

Let us follow the modeling of droop-controlled microgrids presented by [START_REF] Schiffer | Global synchronization analysis of droop-controlled microgrids-a multivariable cell structure approach[END_REF], (see also [START_REF] Schiffer | A survey on modeling of microgrids-from fundamental physics to phasors and voltage sources[END_REF]). So, consider a microgrid of two nodes and one line, where the phase angle and the frequency at the i-th node are represented by θ i and ω i , respectively, with i = 1, 2. Then, under some standard assumptions like voltage amplitudes are positive real constants for both nodes, and line impedance is purely inductive, the active power flow at the i-th node, with i = 1, 2, is given by

P i = G ii V 2 i +|B ik |V i V k sin(θ i -θ k ), with i, k = 1, 2; i ̸ = k, where V 1 , V 2 >
0 denote the voltage amplitudes, G 11 , G 22 > 0 denote the active power loads, and B 12 = B 21 < 0 represents the susceptance of the power line connecting the nodes. Now, by introducing the variable η = θ 1 -θ 2 and the parameter A = |B 12 |V 1 V 2 , we define the potential function U (η) = -A cos(η), and its gradient ∇U (η) = A sin(η). In addition, we denote the active power set-point by

P d = [P d 1 , P d 2 ] ⊤ ∈ R 2
, and the incidence matrix of the network by B = [1, -1] ⊤ . Therefore, we can write P -P d = B∇U (η) -P net , where P = [P 1 , P 2 ] ⊤ ∈ R 2 , and

P net = [P d 1 -G 11 V 2 1 , P d 2 -G 22 V 2 2 ] ⊤ ∈ R 2 .
Hence, a model for synchronization of the considered microgrid is given by

η = Bω, M ω = -B∇U (η) + P net + u, (15)
where ω = [ω 1 , ω 2 ] ⊤ ∈ R 2 are frequencies, M = mI 2 > 0 is the inertia matrix (I 2 is an identity matrix of order two), and u ∈ R 2 is the control input.

Let the left psuedo-inverse matrix of B be given by B + = (B ⊤ B) -1 B ⊤ . For |A -1 B + P net | < 1, the system [START_REF] Efimov | Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids[END_REF], with u = 0, has equilibrium points at [η * + 2jπ, ω * , ω * ] ⊤ ∈ R 3 and [(1 + 2j)π -η * , ω * , ω * ] ⊤ ∈ R 3 for any j ∈ Z, where ω * is the nominal frequency of the network, such that the condition ω 1 = ω 2 = ω * is desired, and η * is the solution of the equation B∇U (η * ) = P net , satisfying -π 2 ≤ η * ≤ π 2 . Then, let us define the error variables η = η -η * ∈ R and ω = ω -ω * ∈ R 2 . So, we rewrite the system (15) as

η = B ⊤ ω, M ω = -Bχ(η) + u, (16)
where χ(η) = ∇U (η + η * ) -∇U (η * ). Clearly, the system [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF], with u = 0, has an equilibrium point at the origin, and in general, at all points [2jπ, 0, 0] ⊤ ∈ R 3 and [(1 + 2j)π -2η * , 0, 0] ⊤ ∈ R 3 for any j ∈ Z. Now, consider the function

V (η, ω) = 1 2 (m + ϱ 2 )ω ⊤ ω + ϱχ(η)B ⊤ ω + χ 2 (η) -1 2 κAη 2 -|B + P net | 2 + 2A(U (η + η * ) -U (η * )), (17)
where ϱ, κ > 0 are constant parameters to be determined.

Since -π 2 ≤ η * ≤ π 2 , then U (η * ) > 0. So, we have -A(1 -cos(η * )) ≤ U (η + η * ) -U (η * ) ≤ A(1 + cos(η * )).

Moreover, by applying Young's inequality we obtain

2ϱ∥χ(η)B∥∥ω∥ ≤ ϱ 2 ∥ω∥ 2 + 2χ 2 (η) (since ∥B∥ 2 = 2), then 1 2 m∥ω∥ 2 ≤ 1 2 (m + ϱ 2 )ω ⊤ ω + ϱB ⊤ ωχ(η) + χ 2 (η) ≤ 1 2 (m + 2ϱ 2 )∥ω∥ 2 + 2χ 2 (η)
. Thus, we can conclude that for all η ∈ R and ω ∈ R 2 ,

1 2 m∥ω∥ 2 -1 2 κAη 2 -g ≤ V (η, ω) ≤ 1 2 (m + 2ϱ 2 )∥ω∥ 2 + 2χ 2 (η) -1 2 κAη 2 + ḡ, (18)
where κ, m, A > 0, ḡ = A(1 + cos(η * )) -|B + P net | 2 , and

g = |B + P net | 2 + A(1 -cos(η * )).
Now, recall the sets U and W introduced in Subsection II-C. Particularly, for the system [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF] we have U = [2jπ, 0, 0] ⊤ ∈ R 3 : j ∈ Z . Note that χ(2πj) = 0 and U (2πj + η * ) -U (η * ) = 0, for any j ∈ Z. So, by evaluating the function [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF] at the points of the set U, we obtain

V (2πj, [0, 0] ⊤ ) = -2κAπ 2 j 2 -|B + P net | 2 < 0, (19)
for any j ∈ Z.

On the other hand, for a sufficiently small κ > 0 we can establish 1 2 κAη 2 ≤ |B + P net | 2 , for all -2π ≤ η ≤ 2π. Moreover, let us evaluate the function [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF] in the points

[πi, 0, 0] ⊤ for i ∈ {-1, 1}, this is V ([πi, 0, 0]) =4A 2 sin 2 (η * ) + 4A 2 cos(η * ) -|B + P net | 2 -1 2 κA(πi) 2 ≥4A 2 sin 2 (η * ) + 4A 2 cos(η * ) -2|B + P net | 2 ≥2A 2 sin 2 (η * ) + 4A 2 cos(η * ) ≥ 0, since by definition -π 2 ≤ η * ≤ π 2 , such that 0 ≤ cos(η * ) ≤ 1.
Thus, by continuity we can say that the function [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF] is positive in a (maybe small) neighborhood of the points where |η| = π. Accordingly, we can define a set {[η, ω] ⊤ ∈ R 3 : c 1 ≤ |η| ≤ c 2 } ⊆ W, with some 0 < c 1 ≤ π ≤ c 2 < 2π, and W as given in Subsection II-C, so we can guarantee that inf

[η,ω] ⊤ ∈W V (η, ω) > 0. (20)
Now, the derivative of ( 17) along the trajectories of the system ( 16) is given by V (η, ω) = a(η, ω)+b(η, ω)u, where

a(η, ω) = -(1 + ϱ 2 m -1 -2A)ω ⊤ Bχ(η) -κAηB ⊤ ω -2ϱm -1 χ 2 (η) + 2A|B + P net |B ⊤ ω + (ϱω ⊤ B + 2χ(η))∇ 2 U (η + η * )B ⊤ ω, and b(η, ω) = (1 + ϱ 2 m -1 )ω + ϱm -1 Bχ(η).
Note that ∥b(η, ω)∥ = 0 implies ω = -ϱ(m + ϱ 2 ) -1 Bχ(η).

Hence, since

|∇ 2 U (η + η * )| ≤ A, 2|η||χ(η)| ≤ η2 + χ 2 (η), and
2|B + P net ||χ(η)| ≤ |B + P net | 2 + χ 2 (η), we have a | ∥b∥=0 = -4ϱ(m + ϱ 2 ) -1 (A + m(m + ϱ 2 ) -1 ∇ 2 U (η + η * ))χ 2 (η) + 2κϱA(m + ϱ 2 ) -1 ηχ(η) -4ϱA(m + ϱ 2 ) -1 |B + P net |χ(η) ≤ -Aϱ(m + ϱ 2 ) -1 (2 -4m(m + ϱ 2 ) -1 -κ)χ 2 (η) + ϱA(m + ϱ 2 ) -1 (κη 2 -2|B + P net | 2 ),
On the other hand, let us evaluate the function [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF] in the surface where ∥b(η, ω)∥ = 0, this is

V | ∥b∥=0 = m(m + ϱ 2 ) -1 χ 2 (η) -1 2 κAη 2 -|B + P net | 2 + 2A(U (η + η * ) -U (η * )). Thus, V | ∥b∥=0 (η, ω) ≥ 0 implies m(m + ϱ 2 ) -1 χ 2 (η) + 2A(U (η + η * ) -U (η * )) ≥ 1 2 κAη 2 + |B + P net | 2 . Assumption 3: For some ϱ > 0, the inequality m(m + ϱ 2 ) -1 χ 2 (η) > 2A(U (η + η * ) -U (η * )) is satisfied wherever V | ∥b∥=0 (η, ω) ≥ 0.
Therefore, under Assumption 3, we have 2m(m + ϱ

2 ) -1 χ 2 (η) ≥ 1 2 κAη 2 + |B + P net | 2 > 0, wherever V | ∥b∥=0 (η, ω) ≥ 0 Thus, for A > 1, we obtain a | ∥b∥=0 ≤ -Aϱ(m + ϱ 2 ) -1 (2 -8m(m + ϱ 2 ) -1 -κ)χ 2 (η),
and, for some sufficiently large ϱ > 0 and some sufficiently small κ > 0, we conclude that

a | ∥b∥=0 (η, ω) < 0, (21)
wherever V | ∥b∥=0 (η, ω) ≥ 0. Finally, recall Definition 2 and Corollary 2, then note that the properties (18), [START_REF] De Persis | Bregman storage functions for microgrid control[END_REF] and [START_REF] Leonov | Nonlinear analysis of phase-locked loop (PLL): Global stability analysis, hidden oscillations and simulation problems[END_REF] are equivalent to the conditions (5) and ( 13), while the property ( 21) is equivalent to the condition [START_REF] Forni | Differential analysis of nonlinear systems: Revisiting the pendulum example[END_REF]. Thus, if Assumption 3 is satisfied with some sufficiently large ϱ > 0, then we can say that the function ( 17) is a CLeF for the system [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF] in the sense of Corollary 2, provided that κ > 0 is sufficiently small. Now, we can follow the procedure developed in Section III to design a controller for the system [START_REF] Ortega | Transient stabilization of multimachine power systems with nontrivial transfer conductances[END_REF]. First, from [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF] we can easily see that

1 2 (m+ϱ 2 )ω ⊤ ω+ϱχ(η)B ⊤ ω+χ 2 (η)+ 2A(U (η + η * ) -U (η * )) > 1 2 κAη 2 + |B + P net | 2 , wherever V (η, ω) > 0. Then, similarly to (7) we define ϕ(|(η, ω)| Ũ , ω) = sup V (η,ω)>0 {a(η, ω)} ≤ sup V (η,ω)>0 -(1 + ϱ 2 m -1 -2A)ω ⊤ Bχ(η) + (1 + 1 2 κ)Aω ⊤ ω -2ϱm -1 χ 2 (η) + (ϱω ⊤ B + 2χ(η))∇ 2 U (η + η * )B ⊤ ω +A(2|B + P net | 2 + κη 2 ) ≤ -(1 + ϱ 2 m -1 -2A)ω ⊤ Bχ(η) + χ 2 (η) + (1 + 1 2 κ)Aω ⊤ ω -2ϱm -1 χ 2 (η) + (ϱω ⊤ B + 2χ(η))∇ 2 U (η + η * )B ⊤ ω + 2A( 1 2 (m + ϱ 2 )ω ⊤ ω + ϱχ(η)B ⊤ ω + 2A(U (η + η * ) -U (η * ))) = φ(|(η, ω)| Ũ , ω).
where Young's inequality was applied to split the terms 2∥|B + P net |B∥∥ω∥ ≤ 2|B + P net | 2 + ω⊤ ω, and 2∥B η∥∥ω∥ ≤ 2η 2 + ω⊤ ω.

Clearly, b(η, ω) and φ(|(η, ω)| Ũ , ω) are 2π-periodic with respect to η. Then, from the formula (8) we obtain

K(ϕ, b) ≤ φ + φ2 + |b| 4 |b| 2 , (22)
where we have omitted the arguments of the respective function due to the space limitation. Next, let us check the continuity of the function [START_REF]On boundedness of solutions of state periodic systems: a multivariable cell structure approach[END_REF] for V (η, ω) ≥ 0. Clearly, this property is obtained wherever ∥b(η, ω)∥ ̸ = 0, by construction. Therefore, consider Assumption 3 and the constraint ∥b(η, ω)∥ = 0, which implies ω = -ϱ(m + ϱ 2 ) -1 Bχ(η):

φ | ∥b∥=0 ≤ -4ϱA(m + ϱ 2 ) -1 (1 -m(m + ϱ 2 ) -1 -1 2 ϱ( 1 2 κ + 1)(m + ϱ 2 ) -1 -mρ -1 )χ 2 (η).
Thus, we can see that for some sufficiently large ϱ > 0 and some sufficiently small κ > 0,

φ | ∥b∥=0 (|(η, ω)| Ũ , ω) ≤0, (23)
wherever V | ∥b∥=0 (η, ω) ≥ 0. Therefore, we can ensure that the function ( 22) is continuous, wherever ( 23) is satisfied with strict inequality. So, it remain us to check the points where the equality occurs, this is at every element of the set {[2jπ, 0, 0] ⊤ , [(1 + 2j)π -2η * , 0, 0] ⊤ : j ∈ Z}. Recall that we have already proven that V ([2jπ, 0, 0]) < 0 for any j ∈ Z. On the other hand, since -1 2 π < η * < 1 2 π, then 2A(U ((1 + 2j)π -η * ) -U (η * )) = 4A 2 cos(η * ) > 0. Hence, by Assumption 3 we can establish that V ([(1 + 2j)π -2η * , 0, 0]) < 0 for any j ∈ Z, as well. Thus, we can ensure that the function ( 22) is continuous, wherever V (η, ω) ≥ 0, as it is required. Note that the SCP-LeF [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF] is straightforwardly satisfied for the function [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF], since the function φ | ∥b∥=0 (|(η, ω)| Ũ , ω) is strictly negative at every element of the set Γ given in Corollary 2.

Thus, since the function χ(η) is bounded, then we can verify that there exists a constant K 1 > 0 such that Finally, we propose the controller u(η, ω) = -K((1 + ϱ 2 m -1 )ω + ϱm -1 Bχ(η)),

which is continuous in R 3 , 2π-periodic with respect to η, and vanishes at every equilibrium point of the system ( 16), since all of them lay in th surface ∥b(η, ω)∥ = 0. Hence, the closed-loop system η = B ⊤ ω, M ω = -(1 + ϱ 2 m -1 ) K ω -(1 + ϱm -1 K)Bχ(η), is continuous in R 3 , 2π-periodic with respect to η, and has equilibria at all points [2jπ, 0, 0] ⊤ ∈ R 3 and [(1 + 2j)π -2η * , 0, 0] ⊤ ∈ R 3 for any j ∈ Z. Moreover, boundedness of solutions of this closed-loop system is supported by the results of [START_REF] Schiffer | Global synchronization analysis of droop-controlled microgrids-a multivariable cell structure approach[END_REF] through Corollary 1 and a Leonov function slightly different to [START_REF] Schiffer | On stability of a distributed averaging PI frequency and active power controlled differential-algebraic power system model[END_REF].

V. CONCLUSIONS

In this paper, we addressed the design of continuous and periodic in θ feedback laws for boundedness of trajectories of the system (4). To this end, the concept of CLeF was introduced, with the corresponding SCP-LeF condition. Two variants of control synthesis were proposed based on Sontag's formula. Finally, the advantage of this methodology was illustrated by an application in a small microgrid.

  lim |b(η,ω)|→0 K( φ(|(η, ω)| Ũ , ω), b(η, ω)) = K 1 ,On the other hand, note that the function φ(|(η, ω)| Ũ , ω) is quadratic in the variables ω, hence we can also claim that there exist a constant K 2 > 0, such thatlim ∥ω∥→∞ K( φ(|(η, ω)| Ũ , ω), b(η, ω)) = K 2 , Hence, we have K( φ(|(η, ω)| Ũ , ω), b(η, ω)) ≤ max{K 1 , K 2 } = K, wherever V (η, ω) ≥ 0.

This work is partially supported by ANR-DFG project SyNPiD -Project number 446182476.