Michael Von Cotta-Schönberg

Silvio Enea

Piccolomini

After his accession to the Papacy in August 1458, Pius II received a number of embassies from the rulers of Europe coming to present the declaration of their lord's obedience to the Apostolic See and the new pope. In February 1459, the pope thus received the ambassadors of the Duke of Savoy. In his oration to the ambassadors, the "Quotiens nova", the pope asserted papal supremacy in all matters, spiritual and temporal, by virtue of the unlimited mission entrusted to the Apostle Peter by Jesus Christ. This mission and the power of the apostolic office is passed on from pope to pope, through apostolic succession, and had now reached the present pope, Pius II.

Editor and translator

Michael v. Cotta-Schönberg Mag. Art. (University of Copenhagen) Bachelier en Philosophie (Université de Louvain) Emeritus Deputy Director / The Royal Library, Copenhagen Emeritus University Librarian / University of Copenhagen ORCID identity: 000-0001-8499-4142 e-mail: typsita@gmail.com

Foreword

In 2007, I undertook a project of publishing the Latin texts with English translations of the orations of Enea Silvio Piccolomini / Pope Pius II. Altogether 80 1 orations (including papal responses to ambassadorial addresses) are extant today, though more may still be held, unrecognized, in libraries and archives.

At a later stage the project was expanded to include ambassadors' orations to the pope, of which about 40 are presently known.

I do not, actually, plan to publish further versions of the present volume, but I do reserve the option in case I -during my future studies -come across other manuscripts containing interesting versions of the oration or if important new research data on the subject matter are published, making it appropriate to modify or expand the present text. It will therefore always be useful to check if a later version than the one the reader may have previously found via the Internet is available.

I shall much appreciate to be notified by readers who discover errors and problems in the text and translation or unrecognized quotations.

12 September 2019 MCS 1 81 orations, if the "Cum animadverto" is counted is a Piccolomini-oration, see oration "Quam laetus" [18], Appendix Federigo of Urbino came to see the pope and received a very warm welcome. Ambassadors from Duke Ludovico of Savoy also came to make their submission to the bishop of Rome, according to ancient custom. 3

Themes

In his reply to the ambassadors of the Duke of Savoy, Pius spoke on three themes which would recur in all his replies to the ambassadors of the princes coming to declare their lord's obedience to the pope.

The first and most important theme was the supremacy of the papal office in all matters, spiritual and temporal, by virtue of the unlimited mission entrusted to the Apostle Peter by Jesus Christ. This mission and the power of the apostolic office is passed from pope to pope, through apostolic succession, and has now reached the present pope, Pius II, to whom all Christians owe absolute obedience. Those who disobey him, including the princes, do so at the peril of their soul: In the present oration, Pius motivated the supremacy of the pope not on the apostolic succession, which is the "mechanism" of transmission, but on the basis principle of the unity of God and the Church which only allows for one ruler of the Church:

The body of Jesus Christ hanging on the Cross is One, and the blood that was shed for our sake is One. The bread broken for all is One, and the chalice of whole the Church is One. There is but One bishop. For God the Father, the Unborn, is One. And God the Son, the only begotten, Word and man, is One. And the Spirit of Truth, the Paraclete, is One. There is only One Preaching, One Faith, and One Baptism. There is only One Church, founded by the holy apostles, from end to end, with the blood of Christ, and with sweat and toil. In that Church there is one bishop, whom all must obey, and that he is the Bishop of Rome no Christian can deny. [Sect. 4] The second theme was the pope's -quite traditional -declaration that he is personally unworthy of his high office and does not merit the extravagant praises of the ambassadors. But since God has chosen him for his own inscrutable purposes, he is now God's Vicar on earth2 :

In Ourselves there is nothing to praise. So, eminent ambassadors, though you have praised Us beyond measure, there is nothing to admire and honour in Us, except that We have been placed on that throne which is the first and greatest of all. We do not know nor do We try to understand why this has happened, for human reason is incapable of grasping the designs of God. [Sect. 1]

The third theme was praise of the ambassador's master: the pope praised Duke Louis of Savoy and expressed his goodwill towards him, his house and his subjects. The audience would have been quite aware that the Duke was the son of the former antipope, Felix V (Duke Amédée VIII of Savoy), and personally known to Pius from the period when he served as secretary to the antipope. This may be a supplementary reason why Pius in his reply stresses the need for all Christians to be in communion with the pope in Rome.

Date, place, audience and format

The oration was given on 9 February 1459, when Pius visited Perugia.1

The audience presumably consisted of the cardinals who were travelling with him to the Congress in Mantua, members of the Papal Court, ambassadors present in Perugia, and Perugian notabilities.

The format was a formal papal response given by the pope personally to the oration and declaration of obedience by the ambassadors of the Duke of Savoy, Louis I.

Text 2

The oration is included in a special Collection of Pius II's Responses from 1459 and 1460), largely consisting of responses to addresses by ambassadors,3 which is extant in at least three manuscripts. The Chisianus is the eldest of the three and identical with or very close to the oration as delivered by Pius, but actually there are only small differences between the texts in the five manuscripts. Mansi's edition is based on the Lucca manuscript (G).

Manuscripts

Editions

Present edition

For principles of edition (incl. orthography) and translation, see Collected Orations of Enea Silvio Piccolomini / Pope Pius II, vol. 1, ch. 9-10.

Text:

The present edition is based on the manuscripts from Lucca, the Archivio Segreto Vaticano and the Chisianus, with the Chisianus as the lead text.

Pagination:

Pagination is from the Lucca manuscript.

Sources 1

In this short oration, 9 direct and indirect quotations from various sources have been identified, the large majority from the Bible, and none from classical sources.

Biblical: 7 Classical: 0 Patristic and medieval: 2 Contemporary: 0 All: 9

Biblical sources: 7

Old Testament: 1

• Wisdom: 1

New Testament: 6

• Matthew: 1

To the ambassadors of Savoy

[1] Whenever declarations of obedience are brought [to Us] from the princes, it behooves Us to say something about the authority and supreme power of the First See. This We do not do in order to praise Ourselves, but in order to glorify the name of God. For everything that is said about the Roman See is really about Christ, Our Saviour, who consecrated that See with the martyrdom of His holy apostles Peter and Paul, In Ourselves there is nothing to praise. So, eminent ambassadors, though you have praised Us beyond measure, there is nothing to admire and honour in Us, except that We have been placed on that throne which is the first and greatest of all. We do not know nor do We try to understand why this has happened, for human reason is incapable of grasping the designs of God. It suffices that it has been so decided by the Greatest and Best God1 who often chooses the weak of this world to confound the strong,2 who gave Moses to his people as their leader, who ordained Aaron as High Priest, and who took David from his herds to make him king.

[ [2] Its is He who has made Us the head of the Christian Church and charged Us with steering the Bark of Saint Peter, for the God of the Old Testament is the same as the God of the New. He, the Spirit of consolation, who has spoken through the prophets, 1 is the same who enflamed the hearts of the apostles and gave them to speak in the tongues of all men. 2 In the Old Law the holy and undivided Trinity gave its salutary doctrine to humankind through Moses the Lawgiver according to the conditions at the time. But in the New Law, the same Trinity has acted more benignly: at its command, the incarnate Son of God has quite clearly shown [us] the way of life, and He did not hesitate to give His life for His sheep. 3 And since He had to return to Heaven and did not want his flock to be attacked by ravenous wolves, he appointed Saint Peter as His Vicar when He said: Feed my sheep, 4 and I will give to thee the keys of the kingdom of heaven 5 etc. 6 This means that supreme power was vested in Peter. And what was granted to Peter has been passed on to Peter's successors, and now the authority of this great office 7 has come to Us. Though We are far inferior to Peter in merit, We are nonetheless equal to him in power. We tremble with fear because of Our sinfulness, but nonetheless We have now been elevated to this lofty position.

[3] We declare that all who audaciously and obstinately oppose Our commands issued according to the sacred canons are the servants of the Devil and cannot obtain salvation. In this connection it is highly appropriate to quote the words of Saint Ignatius who in his time had heard Saint John the Evangelist. This is what he says: Let governors be obedient to Caesar; soldiers to those that command them; deacons to the presbyters, as to high-priests; the presbyters and deacons and the rest of the clergy together with all the people, and the soldiers and the governors and even Caesar to the bishop; the bishop to Christ, even as Christ to the Father. And thus unity is preserved throughout. 8 In spite of this, there are some who dare to spurn the commands of the Apostolic See and appeal its judgments to a higher tribunal, as if there was one. [Obviously,] they are not aware that the First See may be judged by no one, 9 and that he who disobeys the command of the High Priest incurs the death penalty. And anyone who ignores the decrees of the Head of Church, that is the Roman Pontiff, perverts the Gospel, tears the seamless tunic of Christ, 10 and distresses his bride, the Church. [4] The body of Jesus Christ hanging on the Cross is One, and the blood that was shed for our sake is One. The bread broken for all is One, and the chalice of whole the Church is One. There is but One bishop. For God the Father, the Unborn, is One; and God the Son, the Only Begotten, Word and man, is One; and the Spirit of Truth, the Paraclete, is One. There is only One Preaching, One Faith, and One Baptism. There is only One Church, founded by the holy apostles, from end to end, 1 with the blood of Christ, and with His own sweat and toil. In that Church there is one bishop, whom all must obey, and that he is the Bishop of Rome no Christian can deny. Who does not understand that the gate to salvation must be sought at the Apostolic See and that all who follow their own designs and try to avoid or nullify the Roman ordinances are in reality building up Hell.

[5] This is known to Our beloved son, Louis, Duke of Savoy, 2 famous for his power and family. Following the common way of his ancestors, he declares his obedience as owed to Us and to the Apostolic See. He promises to obey our commands and to behave as a good son ought to. Mighty is this prince, and vast are the lands that he rules. His forefathers were of Saxon blood, and some authors claim that the Saxons themselves descend from the Macedonians. The family of Savoy is related to the French and Burgundian nobility. It unites noble fame with noble manners. And Louis himself is a most gracious prince, easily approachable and devoted to religion. He upholds justice and serves equity. We praise his devotion and piety. We gratefully accept his offers, especially with regard to the expedition against the Turks, which he highly approves of. Indeed, he may contribute much to that cause, and We do no doubt that he will fulfil his promises. We and Our venerable brethren, the cardinals of the Holy Roman Church, will show exceptional favour and honour to him, to his illustrious family and noble offspring, and to all who are dear to him. What else you will tell us, in private, We shall hear benignly and gladly.

1 Wisdom, 8, 1: a fine usque ad finem 2 Louis I (1413-1465): Duke of Savoy from 1440 until his death

Table of volumes in Collected Orations of Pope Pius II. 12 vols. Edited and translated by Michael von Cotta-Schönberg Table of contents

  Soon after the coronation of Pius II on 3 September 1458, embassies began arriving in Rome to present the declaration of obedience of their masters to the Apostolic See and to the newly elected pope. The envoys of the Duke of Milan were received on 4 October, and the envoys of Florence and Avignon on 10 October.2 On 22 January 1459, the pope left Rome in order to go to Mantua for the Congress on the War against the Turks, summoned by him. The Congress was to open on 1 June, and the pope's travel there became a veritable papal progress through Italy.

	I. INTRODUCTION 1. Context 1
	I.	INTRODUCTION
	1.	Context
	2.	Themes
	3.	Date, place, audience and format
	4.	Text
		4.1. Manuscripts
		4.2. Editions
		4.3. Present edition
	5.	Bibliography
	6.	Sigla and abbreviations
	II. TEXT AND TRANSLATION
		6

Pius arrived in Perugia on 1 February and stayed in that city for almost three weeks. During his stay there, he received the declarations of obedience of Count Federigo of Urbino, who came in person, and of the Duke of Savoy. In his Commentarii, the pope wrote:

  And since He[the Lord Jesus] had to return to Heaven and did not want his flock to be attacked by ravenous wolves, he appointed Saint Peter as His Vicar when He said: Feed my sheep, and I will give to thee the keys of the kingdom of heaven etc.

	This means that supreme
	power was vested in Peter. And what was granted to Peter has been passed on to Peter's
	successors, and now the authority of this great office has come to Us. Though We are far
	inferior to Peter in merit, We are nonetheless equal to him in power. We tremble with fear
	because of Our sinfulness, but nonetheless We have now been elevated to this lofty
	positione. We declare that all who audaciously and obstinately oppose Our commands
	issued according to the sacred canons are the servants of the Devil and cannot obtain
	salvation. [Sect. 2-3] 1

  Ed. M. Meserve and M. Simonetta. Vols. 1 ff. Cambridge, MA, 2003 ff. Orationes politicae et eclesiasticae. Ed. Giovanni Domenico Mansi. 3 vols. Lucca: Benedini, 1755-1759 • Collected Orations of Pope Pius II. Ed. and transl. by Michael v. Cotta-Schönberg. 12 vols. 2019-2020 Strnad, Alfred A.: Johannes Hinderbachs Obedienz-Ansprache vor Papst Pius II -Päpstliche und kaiserliche Politik in der Mitte des Quattrocento. In: Römische historische Mitteilungen, 10 (1966/1967) 41-183 Voigt, Georg: Enea Silvio de' Piccolomini als Papst Pius der Zweite und sein Zeitalter. 3 vols. 1] {173r} Quotiens nova principum affertur oboedientia, totiens nobis 1 de primae sedis auctoritate summaque potestate dicere aliquid necessitas ingruit. Quod nequaquam nostrae laudis causa, sed pro 2 divini nominis gloria facimus. Quidquid 3 enim 4 de Romana sede profertur, totum Christi est salvatoris nostri, qui eam beatorum apostolorum suorum Petri et Pauli martyrio consecravit, in nobis nihil est, quod magnificare possimus. Etsi verbis vestris, praestabiles oratores, supra modum collaudati sumus, nihil est in nobis, quod admirari quispiam aut venerari queat, nisi quod in eo solio collocati sumus, quod est omnium primum ac maximum. Quod cur factum sit, nec scimus neque investigamus quidem 5 , quia non est divini consilii humana ratio capax. Satis est Deum maximum atque optimum 6 ita constituisse, qui saepe infima hujus mundi eligit, ut fortia quaeque confundat, qui Moysen populo suo ducem dedit, qui sacerdotem summum Aaron ordinavit, qui David regem ex gregibus assumpsit.

	Patristic and medieval sources: 2
	• Decretum Gratiani: 1
	• Ignatius of Alexandria: 1 1
	Contemporary sources: 0
	6. Bibliography
	Berlin,
	1856-63
	7. Sigla and abbreviations
	• Luke: 1
	• John: 3
	• Acts: 1 Redaktion ed. von Julia Knödler. Teil 2: 2./3, ed. Martin Wagendorfer. 2 vols. Hannover,
	2009. (Monumenta Germaniae Historica. Scriptores Rerum Germanicarum. Nova Series;
	Classical sources: 0 24)
	HB = Aeneas Silvius Piccolomini: Historia Bohemica. Herausg. J. Hejnic & H. Rothe. 2 vols. Köln, 1 omit. MA 2005. (Bausteine zur slavischen Philologie und Kulturgeschichte. Neue Folge. Reihe B; 20) 2 omit. MA
	3 quicquam J
	4 etiam MA
	5 quod J

1 On Piccolomini's use of sources in general, see Collected orations of Pope Pius II, ch. 8.

Annales ecclesiastici ab anno MCXCVIII ubi

Card. Baronius desinit. Auct. Odoricus Raynaldus. Tom. XVIII-XIX. Roma: Varesius, 1659-1663 2 Cotta-Schönberg, Michael von & Anna Modigliani: Nicholas V's only surviving oration the Nihil est of 24 March 1447. In: Roma nel Rinascimento, 2016, pp. 271-288 Cronache e storie inedite della città di Perugia dal MCL al MDLXIII … . Part la. Eds. F. Bonaini, A. Fabretti & F.-L. Polidori (eds.). Firenze, 1850 Decretum magistri Gratiani. Ed. Lipsiensis Secunda. Edd. A.L. Richter & A. Friedberg. 2 vols. Leipzig, 1879. Pastor, Ludwig: Geschichte der Päpste im Zeitalter der Renaissance. 16 vols. Freiburg i.B., 1886-1933 Pius II: Commentarii rerum memorabilium quae suis temporibus contigerunt [1464] • Commentarii rervm memorabilivm que svis temporibus contigervnt. Ed. A van Heck. 2 vols. Città del Vaticano, 1984 (Studi e testi; 312-313) • Commentaries. 1 Letter to the Philadelphians 2 References to the Annales is usually given in this form: (e.g.) Rainaldus, ad ann. 1459, nr. 67 (without reference to a specific edition). In the bibliographies of the individual orations, the first edition is usually listed, but the Theiner edition is actually the preferred one since it is the easiest available on the web Pius II: Orationes. [1436-1464] • Pius II: G = Lucca / Biblioteca Capitolare Feliniana / 544 H = Roma / Biblioteca Apostolic Vaticana / Chisianus J.VII.251 J = Roma / Achivio Segreto Vaticano / Arm. XXXII 1 Abbreviations CO = Pius II: Commentarii rerum memorabilium quae suis temporibus contigerunt. [1464] HA = Eneas Silvius Piccolomini: Historia Austrialis. Teil 1: Einleitung von Martin Wagendorfer. 1. MA = Pius II: Orationes politicae et ecclesiasticae. Ed. Giovanni Domenico Mansi. 3 vols. Lucca: Benedini, 1755-1759 Ad oratores ducis Sabaudiae [6 maximum atque optimum : optimum atque maximum G, MA

  2] Ille nos Christianae praefecit ecclesiae, et beati Petri navigium gubernare mandavit, quia non est alius veteris testamenti {173v} Deus, alius novi. Qui per prophetas locutus est consolator spiritus, idem apostolorum inflammavit pectora et linguis omnium loqui dedit. In veteri lege sancta et individua trinitas 1 per legiferum Moysen juxta dispositionem temporum, doctrinam humano generi tribuit 2 salutarem. In nova vero eadem trinitas benignius egit, cujus imperio incarnatus Dei filius vitam vitae manifestius demonstravit, et animam suam pro suis ovibus ponere non dubitavit. Is autem rediturus in caelum, ne gregem suum invaderent lupi rapaces, vices suas commisit beato Petro 3 , dicens: Pasce oves meas, et tibi dabo claves regni caelorum, et cetera 4 , per quae ei summa potestas credita designatur. Quod Petro, idem et successoribus traditum est, et ad nos usque tantae dignitatis auctoritas derivata, qui etsi meritis longe inferiores dignoscimur esse quam Petrus, potestate tamen pares sumus -et quamvis contremiscimus, peccatis obnoxii, hunc tam sublimem locum ascendisse.[3] Fatemur tamen salvari non posse et diaboli prorsus 5 esse mancipium, qui jussionibus nostris secundum sacros canones emanantibus temere ac pertinaciter adversatur. Ad quam rem perpulchre 6 accomodari possunt verba sanctissimi viri Ignatii, qui beati Johannis evangelistae auditor fuit. Sic enim ait: Principes subditi estote Caesari, milites principibus, diaconi presbyteris, presbyteri vero et diaconi simul cum omni populo et militibus atque principibus, sed et Caesar oboediat 7 episcopo; episcopus vero Christo, sicut Christus patri, et ita unitas per omnia conservetur. Quae cum ita sint, audent tamen aliqui apostolicae sedis mandata contemnere et tamquam majus sit aliquod 8 tribunal ab ejus jussionibus provocare non advertentes, quia prima sedes non judicatur a quoquam, et quod morte dignus est, qui summi sacerdotis imperio non paret. Evangelium quippe corrumpit et inconsutilem Christi tunicam scindit 9 et sponsam ejus lacerat ecclesiam, qui capitis sui, hoc est Romani praesulis, decreta contemnit.

	1 trinitatis J
	2 praebuit J
	3 commisit beato Petro : beato Petro commisit G, MA
	4 etc. G, MA
	5 omit. MA 21

  Una est caro domini Jesu in cruce suspensa, et unus ejus sanguis, qui pro nobis effusus. Unus etiam 1 panis pro omnibus confractus, et unus calix totius ecclesiae, et unus episcopus, nam et unus est ingenitus Deus pater, et unus unigenitus 2 filius Deus, verbum et homo, et unus paraclytus spiritus veritatis. Una etiam praedicatio, et fides una, et unum baptisma. Si una ecclesia est, quam fundaverunt sancti apostoli a fine usque ad fines in sanguine Christi propriis sudoribus ac 3 laboribus, et in ea unus episcopus, cui parere omnes oportet, atque hic Romanus, quod nemo Christianus inficiari 4 potest, quis non intelligit apud apostolicam sedem 5 salutis portam esse quaerendam et omnes ad Gehennam aedificare, qui suis utentes sensibus Romanas sanctiones evadere atque divellere nituntur? [5] Novit haec dilectus filius noster Ludovicus, Sabaudiae dux, potentia clarus et genere, qui vulgatam progenitorum suorum incedens viam ac debitam nobis et apostolicae sedi oboedientiam exhibens, mandatis nostris obtemperaturum et boni filii officium repromittit. Praepotens hic princeps est et latissimis regionibus imperat. Majores ejus ex Saxonum sanguine 6 orti. Saxones quidam auctores ex Macedonibus originem ducere tradunt 7 . Francorum et Burgundiorum 8 nobilitas Sabaudiensi familiae mixta est; huic claritudini et mores optimi juncti sunt. Est enim Ludovicus mansuetissimus princeps, aditu facilis, religionis cultor, justitiae tenax, et servantissimus aequi. Laudamus devotionem et pietatem ejus. Quae nobis offert, et praecipue in expeditione contra Turcos, quam tantopere probat, gratissimo complectimur animo: multum enim in ea re conferre 9 potest, nec dubitamus quin opere compleat, quod verbis pollicetur. Nos ac venerabiles fratres nostri sanctae Romanae ecclesiae cardinales cum 10 personam suam, tum illustrem prosapiam ejus ac praecipue nobilissimam sobolem, et omnes sibi caros singularibus favoribus atque honoribus prosequemur 11 . Reliqua, quae semotis arbitris relaturi estis, benigne et gratanter audiemus.

	[4] 1 est MA
	1 "Qui locutus est prophetas": from the Nicene Creed 2 unigenitus corr. from ingenitus H; ingenitus J
	2 Cf. Acts, 2, 4 3 et J
	3 John, 10, 11 4 inficiare J
	4 John, 21, 17 5 apostolicam sedem : sedem apostolicam G, MA
	5 Matthew, 16, 19 6 Saxonum sanguine : sanguine Saxonum G, MA
	6 Two of the biblical texts traditional used to prove the supremacy of the Papacy 7 ducere tradunt : tradunt ducere G [NB: not MA]
	7 "dignitas" 8 Burgundorum G, MA
	8 St. Ignace of Antioch: Letter to the Philadelphians, 4 9 conferri MA
	9 Decretum, C.9.3.13 (col. 610). 10 tum J
	10 John, 19, 23 11 prosequimur G, MA

CO, II, 19, 1 (Meserve, pp.

276-277); Rainaldus, ad ann. 1459, nr. 24; Pastor, II, pp. 16-17, 36; Strnad, pp. 74-76; Voigt, III, p. 37 2 The orations of the ambassador of Milan, poet Laureate Thomasus de Rheate, is still extant (Pastor, p. 17, n. 1) as are the oration of the ambassador of Florence, Archbishop Antonino (Strnad, p. 54) and others (see Collected Orations of Pope Pius II, vol. 12) 3 CO,II, 19 (Meserve, I,

Pius had actually heard his processor, Nicolaus V, make the same assertion in his oration "Nihil est" to the ambassadors of King Alfonso V, presenting his obedience to the then newly elected pope, in March 1447: Non enim pro sua sapientia ignorat neminem, qui ab hac sede alienat et deviat, in statu salutis esse posse, ita ut quemadmodum, qui in archa Noe fuerint, salvati sint; qui vero extra illam, omnes perierint. Ita qui a Petri sede alieni interire necesse sit, see Cotta-Schönberg & Modigliani

Pius had also heard such a declaration of humility made by his predecessor in the aforementioned oration Nihil est: Quod vero ad nos et ea, quae de nobis multa dixistis, attinet, absit omnino, ut tanto pontificatu dignos nos esse iudicemus. Sumus enim testes et conscii fragilitatis debilitatisque nostrae et quales viros, quam probatos, quam sanctos haec excelsa sedes exquirit certe non ignoramus, quorum numero nos omnino non esse cognoscimus

Cronache, p. 634, quoted by Strnad, p. 96

Concerning the textual transmission of Pius II´s orations, see Collected orations of PopePius, vol. 1, ch. 5

See Collected Orations of Pope Pius II, cf. 5.1.2

Manuscripts for which an orthographical profile is given in Collected orations of Pope Pius II, vol. 11, are marked with an asterisk

Traditional title of the Roman God Jupiter, "Jupiter Optimus Maximus", applied by Piccolomini to the Christian God

Cf. Luke, 1, 52

omit. G, MA

obediant G, H

aliquid G, MA

omit. MA


Bernard Brogliato
email: bernard.brogliato@inria.fr

Andrey Polyakov

Denis Efimov
email: denis.efimov@inria.fr

The implicit discretization of the super-twisting sliding-mode control algorithm

Keywords: Implicit Euler discretization, sliding mode control, super-twisting algorithm, Lyapunov stability

This paper deals with the analysis of the time-discretization of the super-twisting algorithm, with an implicit Euler method. It is shown that the discretized system is well-posed. The existence of a Lyapunov function with convex level sets is proved for the continuoustime closed-loop system. Then the global asymptotic Lyapunov stability of the unperturbed discrete-time closedloop system is proved. The convergence to the origin in a finite number of steps is proved also in the unperturbed case. Numerical simulations demonstrate the superiority of the implicit method with respect to an explicit discretization with significant chattering reduction.

I. INTRODUCTION

The sliding mode algorithms found their place in the theory and practice of control and estimation [START_REF] Fridman | Sliding Mode Enforcement after 1990: Main Results and Some Open Problems[END_REF]. Among them, the super-twisting algorithm is one of the most popular higher-order sliding mode (HOSM) tools [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Derafa | Super twisting control algorithm for the attitude tracking of a four rotors UAV[END_REF]. The time discretization of set-valued sliding-mode control (SMC), has been recognized as being a major issue as it leads to input and ouput chattering (so-called digital or numerical chattering) in the case of an explicit discretization [START_REF] Galias | Euler's discretization of single input sliding-mode control systems[END_REF], [START_REF] Du | Robustness analysis of a continuous higher order finite-time control system under sampled-data control[END_REF]. Another method has been introduced in [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], see also [START_REF] Kikuuwe | Proxy-based sliding mode control: A safer extension of PID position control[END_REF], [START_REF] Aung | A new parabolic sliding mode filter augmented by a linear low-pass filter and its application to position control[END_REF] for a similar approach. It is based on an implicit discretization of the continuous-time system. It has been shown theoretically and experimentally to provide significant chattering alleviation [START_REF] Wang | Experimental comparisons between implicit and explicit implementations of discrete-time sliding mode controllers: Toward input and output chattering suppression[END_REF], [START_REF] Huber | Implicit discrete-time twisting controller without numerical chattering: Analysis and experimental results[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Huber | Experimental results on implicit and explicit timediscretization of equivalent-control-based sliding-mode control[END_REF], without modifying the controller's structure. The implicit discretization applies to various systems and controllers: first-order SMC for linear time-invariant systems with matching disturbances [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] or parameter uncertainties [START_REF] Miranda-Villatoro | Set-valued sliding-mode control of uncertain linear systems: continuous and discrete-time analysis[END_REF], nonlinear Lagrangian systems with matching disturbances and parameter uncertainties [START_REF] Miranda-Villatoro | Multivalued robust tracking control of Lagrange systems: continuous and discrete-time algorithms[END_REF], twisting algorithm [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], [START_REF] Huber | Implicit discrete-time twisting controller without numerical chattering: Analysis and experimental results[END_REF]. Another very interesting feature of the implicit discretization is that it yields global Lyapunov finitetime stability [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Brogliato | Globally stable implicit Euler time-discretization of a nonlinear single-input sliding-mode control system[END_REF], [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], contrarily to the explicit method which guarantees local stability only [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF] which may be quite problematic in applications, in addition to digital chattering.

It is noteworthy that all the above cited articles, show that the so-called emulation of a feedback controller yields poor results if an explicit Euler method is used to discretize SMC, whereas most of the performances and stability properties are preserved if the controller is emulated with an implicit method. In addition the input converges to its set-valued continuous-time counterpart [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF]Propositions 3 and 4].

The aim of this note is to present the analysis of the implicit discretization of the super-twisting algorithm using the implicit Lyapunov function approach. It is organised as follows: section II introduces the supertwisting scheme, and shows the existence of a strict Lyapunov function with convex level sets; section III proposes an implicit discretization of the continuoustime scheme, and shows its well-posedness (nonanticipativity, existence and uniqueness of the controller); section IV is dedicated to the stability analysis of the discrete-time closed-loop system; the results of simulation are presented in section V; conclusions end the paper in section VI.

a) Notations and definitions: The set-valued sign function is defined as sgn

(x) = 1 if x > 0, -1 if x < 0, and [-1, 1] if x = 0. Let K ⊆ IR n be a closed non empty convex set. Its normal cone at x ∈ K is defined as N K (x) = {w ∈ IR n | w, y -x ≤ 0 for all y ∈ K}. In particular N [-1,1] (s) =    IR -if s = -1 {0} if s ∈ (-1, 1) IR + if s = 1 . It follows from Convex Analysis that N [-1,1] (s) is the inverse of the sign set-valued function, that is: x ∈ sgn(y) ⇔ y ∈ N [-1,1] (x)
, for all reals x and y. Let G : IR n ⇒ IR m be a set-valued mapping, and f : IR n → IR m be a single-valued mapping. Then 0 ∈ f (x) + G(x) is a generalized equation.

II. THE SUPER-TWISTING ALGORITHM

A. The control algorithm

The closed-loop system we are dealing with is given as [START_REF] Koch | Discrete-time equivalents of the super twisting algorithm[END_REF], [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF]:

ẋ1 (t) = -λ 1 |x 1 (t)|sgn(x 1 (t)) + x 2 (t) ẋ2 (t) ∈ -λ 2 sgn(x 1 (t)) + ∆(t). (1)
It stems from the application of the super-twisting controller given by:

u(t) = -λ 1 |x 1 (t)|sgn(x 1 (t)) + ν(t), ν(t) ∈ -λ 2 sgn(x 1 (t)) (2)
applied to the plant

ẋ1 (t) = u + ϕ(t), φ(t) = ∆(t) (3)
with the definition of the (unmeasurable) state x 2 ∆ = ν +ϕ, and sup t≥0 |∆(t)| ≤ L for some known constant L. Notice that in the undisturbed case ϕ(t) ≡ 0 one has x 2 (t) = ν(t) for all t ≥ 0.

B. Global asymptotic Lyapunov stability

The finite-time stability of the closed-loop system (1) has been analysed in [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF], [START_REF] Seeber | Stability proof for a well-established super-twisting parameter setting[END_REF], where different Lyapunov functions have been exhibited. A crucial property to transport the Lyapunov stability properties from the continuous-time system to the discretized system, is that the (continuous-time) Lyapunov function has convex level sets. It is not obvious that the Lyapunov functions proposed in [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF], [START_REF] Seeber | Stability proof for a well-established super-twisting parameter setting[END_REF] satisfy such a property (though some numerical calculations suggest that this could be the case). In addition, our further constructions also require a smoothness of Lyapunov functions. This is not the case for Lyapunov functions obtained in [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via Lyapunov function designing[END_REF], [START_REF] Seeber | Stability proof for a well-established super-twisting parameter setting[END_REF].

Lemma 1. If L ≥ 0, λ 1 > 4 √ 2L and L < λ 2 < λ 2 1 2 √ 2 -L, then the system (1) admits a strict Lyapunov function V ∈ C(IR n , [0, +∞) ∩ C 1 (IR n \{0}, (0, +∞))
, with ellipsoidal level sets.

Proof. I. Let us study initially the case

0 ≤ L < 1 2 , L < λ 2 < 1 -L and λ 1 = 2 √ 2.
Let us consider the implicit Lyapunov function (ILF) candidate V (•) for the super-twisting system given by the equality Q(V, x) = 0 where (see e.g., [START_REF] Polyakov | Finite-time and Fixed-time Stabilization: Implicit Lyapunov Function Approach[END_REF] for details)

Q(V, x) = x D(V -1 )P D(V -1 )x -1, where x = (x 1 , x 2 ) ∈ IR 2 , D(V -1 ) = V -2 0 0 V -1 , P = 1+ε -1 -1 1
is a positive definite matrix for any ε > 0. The considered Lyapunov candidate is welldefined and positive definite if the following LMI holds [START_REF] Polyakov | Finite-time and Fixed-time Stabilization: Implicit Lyapunov Function Approach[END_REF] P G D + G D P 0,

where G D = ( 2 0 0 1 ). The latter matrix inequality holds if ε > 1 8 . The partial derivative of V (•) is given by

∂V ∂x = 2V x D(V -1 )P D(V -1 ) x D(V -1 )(P G D + G D P )D(V -1 )x ,
Let us denote

F (x) = -λ 1 |x 1 | sgn(x 1 ) + x 2 -λ 2 sgn(x 1 ) + [-L, L] ,
where λ 1 , λ 2 and L are defined in [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF]. Note that the autonomous system ẋ ∈ F (x) includes the timevarying system (1) as particular case since |∆(t)| ≤ L [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]. Let us estimate the time derivative of the ILF candidate along the trajectories of the system (1)

sup ξ∈F (x) ∂V ∂x ξ = 2V sup ξ∈F (x) x D(V -1 )P D(V -1 )ξ x D(V -1 )(P G D + G D P )D(V -1 )x = 2 sup z∈F (D(V -1 )x) x D(V -1 )P z x D(V -1 )(P G D + G D P )D(V -1 )x ,
where x D(V -1 )P D(V -1 )x = 1. Obviously, due to condition (4), the latter fraction is negative if

sup z∈F (y)
y P z < 0 for y P y = 1.

If y 1 = 0, then y P y = 1 implies y 2 2 = 1 and

F (y) = y2 [-λ2,λ2]+[-L,L]
. Hence, sup z∈F (y) y P z = -1 + (λ 2 + L) < 0. Let us consider now the case y 1 > 0 (the case y 1 < 0 can be treated similarly). One has sup z∈F (y)

y P z = sup L∈[-L,L] y P -λ1 √ y1+y2 -λ2+ L = sup L∈[-L,L] -λ 1 (1 + ε)y 3 2 1 + λ 1 y 2 √ y 1 + (1 + ε)y 1 y 2 -y 2 2 +(y 1 -y 2 )(λ 2 -L).
The identity y P y = 1 gives y 2 2 -2y 1 y 2 +(1+ε)y 2 1 = 1 and

y 2 = y 1 -1 -εy 2 1 if y 1 -y 2 ≥ 0, y 2 = y 1 + 1 -εy 2 1 if y 1 -y 2 < 0, with y 1 ∈ 0, 1 √ ε . Hence, for y 1 -y 2 ≥ 0 we derive sup z∈F (y) y P z ≤ q 1 (y 1 ) -(1 -λ 2 -L) 1 -εy 2 1 where q 1 (y 1 ) = -λ 1 εy 3 2 1 -λ 1 √ y 1 1 -εy 2 1 + 2εy 2 1 +(1 -ε)y 1 1 -εy 2 1 + 1 -εy 2 1 -1. For y 1 -y 2 < 0 we have sup z∈F (y) y P z ≤ q 2 (y 1 ) -(λ 2 -L) 1 -εy 2 1 where q 2 (y 1 ) = -λ 1 εy 3 2 1 + λ 1 √ y 1 1 -εy 2 1 + 2εy 2 1 -1 +(ε -1)y 1 1 -εy 2 1 . If ε = 1 and λ 1 = 2 √ 2 then q 1 (0) = 0, q 1 (y 1 ) < 0 for y 1 ∈ (0, 1] , q 2 1 √ 2 = 0, q 2 (y 1 ) < 0 for y 1 ∈ 0, 1 √ 2 ∪ 1 √ 2 , 1 .
The detailed proof of the latter inequalities is omitted for shortness. The graphs of the functions over the interval [0, 1] are depicted in Fig. 1. Therefore, Q with Fig. 1. Graphs of q 1 (•) (solid line) and q 2 ( • ) (dashed line) for

ε = 1, λ 1 = 2 √ 2. P = 2 -1 -1 1
is the implicit Lyapunov function for (1) with λ 1 = 2 √ 2 and L < λ 2 < 1 -L, i.e., there exists a positive definite function V :

IR n → [0, +∞) that is strict Lyapunov function to (1) (the notion of strict Lyapunov function is given in Appendix) and Q(V (x), x) = 0 for all x ∈ IR n . The level sets of V (•), Ω(r) = {x ∈ IR 2 | V (x)
≤ r}, are given by (see e.g., [START_REF] Polyakov | Finite-time and Fixed-time Stabilization: Implicit Lyapunov Function Approach[END_REF])

Ω(r) = {x ∈ IR 2 | x D(r -1 )P D(r -1 )x ≤ 1}.
Obviously, Ω(r) is an ellipsoid for any r > 0. II. Now, let λ 1 , λ 2 and L satisfy the conditions of the lemma. The linear change of coordinates

x new = 2 √ 2 λ 2 1 x, transforms (1) to ẋnew 1 = -2 √ 2 |x new 1 | sgn(x new 1 ) + x new 2 ẋnew 2 = -2 √ 2λ2 λ 2 1 sgn(x new 1 ) + 2 √ 2∆ λ 2 1 , where 2 √ 2|∆(t)| λ 2 1 ≤ L := 2 √ 2L λ 2 1 < 1 2 , and L < 2 √ 2λ2 λ 2 1
< 1-L by assumptions of the lemma. As it has been proven above, the transformed system has a strict Lyapunov function

V new : IR n → [0, +∞) with ellipsoidal level sets. Then, V (x) = V new 2 √ 2 λ 2 1 x , x = (x 1 , x 2 )
defines the strict Lyapunov function for the original system. Finally, notice that the linear transformation of coordinates keeps the level sets of the Lyapunov function ellipsoidal.

For any L > 0, we can always select parameters λ 1 and λ 2 satisfying Lemma 1. In other words, the supertwisting system always admits a smooth Lyapunov function with ellipsoidal level sets, under some proper selection of parameters. The restrictions on these parameters are more conservative compared to [START_REF] Moreno | Strict Lyapunov functions for the super-twisting algorithm[END_REF], due to the requirements of convexity and smoothness of the corresponding Lyapunov function.

III. THE SUPER-TWISTING ALGORITHM IMPLICIT

DISCRETIZATION

The controller is discretized as follows:

u k = -λ 1 |x 1,k+1 |sgn(x 1,k+1 ) + ν k+1 ν k+1 ∈ ν k -λ 2 h sgn(x 1,k+1 ) (5)
where t k+1 -t k = h > 0 is the time-step (or sampling period), and we denote f (t k ) = f k . The variable x1,k+1 is an intermediate variable, used to calculate the input, because x 1,k+1 is not available due to the unknown disturbance. It will be defined precisely later. The input in [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF] is the control to be applied at t = t k and on the whole time interval [t k , t k+1 ). The plant discretized model is chosen as:

x 1,k+1 = x 1,k + hu k + h φk ϕ k+1 = ϕ k + h ∆k , (6)
where φk and ∆k are the discrete-time counterparts of ϕ(t) and ∆(t) in ( 3), which depend on the used method. For instance, the simple explicit Euler method yields φk = ϕ k = ϕ(t k ) and ∆k = ∆ k = ∆(t k ), while the exact discretization (zero-order hold) yields from the mean value theorem, assuming that ∆(•) is continuous: φk = ϕ k + (τ -t k )∆(γ) and ∆k = ∆(s), for some s, τ ∈ (t k , t k+1 ), γ ∈ (t k , τ ). This may be extended to piecewise continuous perturbations. Notice that when ∆(t) ≡ 0, then all methods yield the same discretization. Let us set:

x1,k+1 = x 1,k + hu k , (7)
which plays the role of an unperturbed nominal plant, so that x1,k is a dummy variable which can be interpreted as the state of the unperturbed plant, and x1,k+1 is available at t = t k as shown next. Inserting ( 5) into [START_REF] Du | Robustness analysis of a continuous higher order finite-time control system under sampled-data control[END_REF] we obtain:

x1,k+1 = x 1,k -hλ 1 |x 1,k+1 |sgn(x 1,k+1 )+hν k+1 ν k+1 ∈ ν k -λ 2 h sgn(x 1,k+1 ), (8)
which is the unperturbed "virtual" closed-loop system. We rewrite (8) equivalently as:

x1,k+1 ∈ x 1,k -hλ 1 |x 1,k+1 |sgn(x 1,k+1 ) + hν k -h 2 λ 2 sgn(x 1,k+1 ). (9)
Let us recall that sgn(0) = [-1, 1] so that indeed the right-hand side of ( 9) is set-valued, and ( 9) is a generalized equation with unknown x1,k+1 . Let us introduce the variable (a selection): ξ k+1 ∈ sgn(x 1,k+1 ). The controller u k to be calculated at t = t k and to be applied on [t k , t k+1 ) is given as follows:

u k = -λ 1 |x 1,k+1 | ξ k+1 + ν k+1 ν k+1 = ν k -λ 2 h ξ k+1 , (10)
Notice that ( 10) is just a rewriting of (5). Rewriting [START_REF] Facchinei | Finite-Dimensional Variational Inequalities and Complementarity Problems, volume I of Operations Research[END_REF] we also obtain the generalized equation:

g(x 1,k+1 ) ∈ -λ 2 h 2 sgn(x 1,k+1 ), (11)
with g(x)

∆ = x + a |x|sgn(x) + b k , a ∆ = hλ 1 > 0, b k ∆ = -x 1,k -hν k .
The variable ξ k+1 is the solution of the generalized equation

f (-λ 2 h 2 ξ k+1 ) ∈ N [-1,1] (ξ k+1 ), (12)
where the function f (•) is defined as:

f (y) = g -1 (y) =        -a+ √ a 2 -4(b k -y) 2 2 if y≥b k - a- √ a 2 +4(b k -y) 2 2 if y < b k .
(13) Let us prove that the inclusion [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] follows from [START_REF] Galias | Euler's discretization of single input sliding-mode control systems[END_REF] by inversion of set-valued mappings: we have [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF]. The inversion of the function g(•) can be calculated from the fact that

g(x 1,k+1 ) = -λ 2 h 2 ξ k+1 ∈ -λ 2 h 2 sgn(x 1,k+1 ). Now ξ k+1 ∈ sgn(x 1,k+1 ) ⇔ x1,k+1 ∈ N [-1,1] (ξ k+1 ). Let β k+1 be a selection of N [-1,1] (ξ k+1 ), then we can write g(β k+1 ) = -λ 2 h 2 ξ k+1 , hence β k+1 = f (-λ 2 h 2 ξ k+1 ). Consequently f (-λ 2 h 2 ξ k+1 ) ∈ N [-1,1] (ξ k+1 ) which is
g(x) = x + a √ x + b k if x > 0, g(x) = x -a √ -x + b k if x < 0.
Remark 1. The controller u k in ( 10) is nonanticipative since both ξ k+1 and x1,k+1 are functions of known quantities at t = t k : h, λ 1 , λ 2 , ν k and x 1,k , both being solutions of the generalized equations [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] and [START_REF] Galias | Euler's discretization of single input sliding-mode control systems[END_REF].

The following holds: Lemma 2. The generalized equations in [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] and ( 11) have a unique solution for any data h, λ 1 , λ 2 , ν k and x 1,k .

Proof. Notice that both f (•) and g(•) are continuous. It readily follows from [9, Corollary 2.2.5] that [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] has at least one solution, and the set of solutions is compact. Let now ) have a unique intersection, see Fig. 2. Using the fact that both mappings in [START_REF] Galias | Euler's discretization of single input sliding-mode control systems[END_REF] are the inverse of their counterparts in [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], their intersection is also always unique.

ζ k+1 ∆ = -λ 2 h 2 ξ k+1 , then (12) is equivalent to 0 ∈ f (ζ k+1 ) + N [-λ2h 2 ,λ2h
This means that the controller in [START_REF] Fridman | Sliding Mode Enforcement after 1990: Main Results and Some Open Problems[END_REF] is always uniquely defined from the available data at t = t k . Let us recall that b k = -x 1,k -hν k . Fig. 2 is read as follows:

1) case 1, b k < -h 2 λ 2 : ζ k+1 = -h 2 λ 2 , ξ k+1 = 1, x1,k+1 > 0, 2) case 2, b k ∈ [-h 2 λ 2 , h 2 λ 2 ]: ζ k+1 = b k ∈ [-h 2 λ 2 , h 2 λ 2 ], ξ k+1 = b k -h 2 λ2 ∈ [-1, 1], x1,k+1 = 0, 3) case 3, b k > h 2 λ 2 : ζ k+1 = h 2 λ 2 , ξ k+1 = -1,
x1,k+1 < 0. The value of x1,k+1 can be computed easily in cases 1 and 3 by solving g(x 1,k+1 ) = ±λ 2 h 2 . Using [START_REF] Fridman | Sliding Mode Enforcement after 1990: Main Results and Some Open Problems[END_REF] we deduce the following algorithm for the calculation of the control input at time t k : [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF].

• data: x 1,k and ν k , a = hλ 1 , h > 0, λ 1 > 0, λ 2 > 0. • if b k < -h 2 λ 2 , then |x 1,k+1 | = -a+ √ a 2 -4(b k +λ2h 2 ) 2 ; ν k+1 = ν k -hλ 2 ; u k = -λ 1 |x 1,k+1 | + ν k+1 , ❝❛s❡ ✶ ❝❛s❡ ✷ ❝❛s❡ ✸ ❜ ❦ ❤ ✷ ✕ ✷ ⑦ ① ✶❀❦✰✶ ❜ ❦ ❤ ✷ ✕ ✷ ❜ ❦ ✏ ❦✰✶ ❢✭✏ ❦✰✶ ✮ ◆ ❬ ✕ ✷ ❤ ✷ ❀✕ ✷ ❤ ✷ ( ✭✏ ❦✰✶ ✮ ⑦ ① ✶❀❦✰✶ Fig. 2. Graphical interpretation of
• else if b k ∈ [-h 2 λ 2 , h 2 λ 2 ], then u k = ν k+1 = - x 1,k h , • else if b k > -h 2 λ 2 , then |x 1,k+1 | = -a+ √ a 2 +4(b k -λ2h 2 ) 2 ; ν k+1 = ν k + hλ 2 ; u k = λ 1 |x 1,k+1 | + ν k+1 , • terminate

IV. ANALYSIS OF THE DISCRETE-TIME CLOSED-LOOP SYSTEM

In the previous section, we have proved that the applied controller allows to generate a sequence of states x1,k . Notice that in the absence of perturbation, we have x1,k = x 1,k for all k ≥ 0 if x1,0 = x 1,0 . Definition 1. The discrete-time sliding surface is defined as

Σ d = {(x 1,k , ν k ) ∈ IR 2 | x1,k = 0, ν k = 0}.
Let us characterize the fixed points of the unperturbed closed-loop system which is obtained from [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], setting ϕ k = 0 (hence x 1,k = x1,k for all k). Lemma 3. Let ϕ k = 0 for all k. The unique fixed point of the unperturbed (or virtual) closed-loop system ( 8) is (x 1 , ν ) = (0, 0).

Proof. Recall that in this case x1,k = x 1,k . From ( 8) we obtain 0 = -hλ 1 |x 1 |sgn(x 1 ) + hν and 0 ∈ -λ 2 h sgn(x 1 ) ⇔ x 1 = 0. Thus ν = 0.

Let us now analyse what happens on

, i ≤ k + 1. Then x 1,k = h 2 λ 2 ξ 1 k+1 for some ξ 1 k+1 ∈ [-1, 1], while x 1,k+1 = h 2 λ 2 ξ 2 k+1 + h φk for some ξ 2 k+1 ∈ [-2, 2]. If x1,k+2 = 0 and ν k+1 = 0 then x 1,k+1 will also satisfy x 1,k+1 = h 2 λ 2 ξ 1 k+2 for some ξ 1 k+2 ∈ [-1, 1]
Proof. We have x1,k+1 = 0 and ν k = 0. From ( 9) it follows 0 ∈ x 1,k -h 2 λ 2 sgn(0), which proves the first part for some selection ξ 1 k+1 ∈ [-1, 1]. The second part follows using (6):

x 1,k+1 ∈ x 1,k -h 2 λ 2 sgn(0) + h φk so that x 1,k+1 = h 2 λ 2 ξ 1 k+1 -h 2 λ 2 ξ 3 k+1 + h φk for some (possibly different from ξ 1 k+1 ) selection ξ 3 k+1 ∈ [-1, 1], and noting that ξ 1 k+1 -ξ 3 k+1 ∈ [-2, 2]
. The last statement is a corollary of the first one.

An important point is that Lemma 4 says nothing neither about the invariance of Σ d , nor about its asymptotic or finite-time reachability: it just says that in case the system remains in Σ d , then the closedloop system has an accuracy h 2 at x 1 . The results in [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] are not applicable to the super-twisting algorithm, which does not belong to the class of systems analysed in that article. They also require the knowledge of a Lyapunov function for [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], whose level sets are convex, thus Lemma 1 is used. We now state the following result: Lemma 5. Let F : IR n ⇒ IR n be an upper semicontinuous map such that F (x) is nonempty convex and compact for any x ∈ IR n . Let the differential inclusion

ẋ(t) ∈ F (x(t)) (14)
have the unique globally asymptotically stable equilibrium x = 0. If its strict Lyapunov function V ∈ C(IR n , [0, +∞)) ∩ C 1 (IR n \{0}, (0, +∞)) has convex level sets, then any sequence {x k } +∞ i=0 generated by the inclusion 0 ∈ x k+1 -x k -hF (x k+1 ) converges to zero as k → +∞, for any bounded initial data.

Proof. Let us denote the level set of V (•) as follows

Ω(λ) = {x ∈ IR n |V (x) ≤ λ},
and define δ : (0, +∞) → (0, +∞) as δ(λ) = inf x∈∂Ω(λ) W (x), where W (•) is defined by [START_REF] Levant | On fixed and finite time stability in sliding mode control[END_REF], and ∂Ω(λ) corresponds to boundary of Ω(λ). Notice that the set Ω(λ) (as well as its boundary) is compact, due continuity and radial unboundedness of the Lyapunov function V (•). Moreover, the multivalued function

Ω : [0, +∞) ⇒ IR n (resp. ∂Ω : [0, +∞) ⇒ IR n )
is continuous in the Hausdorff metric. Hence, continuity/positivity of the function δ(•) follows from continuity/positivity of the function W (•). From ( 17) we obtain:

sup x∈∂Ω(λ) sup ξ∈F (x) ∂V ∂x ξ ≤ -δ(λ) < 0 if λ > 0.
If ξ ∈ F (x k+1 ), then for any x k = x k+1 -hξ, one has

∂V (x k+1 ) ∂x ξ = ∂V (x k+1 ) ∂x x k+1 -x k h ≤ -δ(V (x k+1 )).
Since the level set Ω(V (x k+1 )) is convex, and x k+1 ∈ ∂Ω(V (x k+1 )), then the last inequality immediately implies the point x k is separated from the convex compact set Ω(V (x k+1 )) by the tangential plane at the point x k+1 , as shown in Fig. 3 Hence, the distance Fig. 3. Illustration of proof of Lemma. 5 from x k to Ω(V (x k+1 )) is greater than the distance from x k to the tangential plane (see Fig. 3). The corresponding distance n (x k -x k+1 ) can be estimated from below as follows:

n (x k -x k+1 ) = ∂V (x k+1 ) ∂x ∂V (x k+1 ) ∂x (x k -x k+1 ) = = h - ∂V (x k+1 ) ∂x (x k+1 -x k ) h ∂V (x k+1 ) ∂x ≥ d(V (x k+1 )),
where

d(λ) = hδ(λ) inf x∈∂Ω(λ) ∂V (x) ∂x -1 > 0 if λ > 0.
Since ∂V (•) ∂x : IR n → IR is a continuous positive definite function then d : (0, +∞) → (0, +∞) is continuous as well. The latter means that

V (x k+1 ) -V (x k ) < -ε k ,
where ε k > 0 is separated from zero while V (x k+1 ) is separated from zero. Therefore, we derive that the sequence V (x k ) is monotone decreasing to zero as k → +∞.

Corollary 1. Suppose that the perturbation ∆(t) ≡ 0,

λ 1 > 0 and 0 < λ 2 < λ 2 1 2 √ 2 .
The origin of the discretetime closed-loop system [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF], is globally asymptotically stable in the sense of Lyapunov.

Proof. Recall that in such a case x1,k = x 1,k , provided the initial values coincide. Thus, [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] is the discretization of (1) since also x 2 = ν. Taking into account Lemma 1, we conclude from Lemma 5 that the state of the system (8), which rewrites

x 1,k+1 = x 1,k -hλ 1 |x 1,k+1 |sgn(x 1,k+1 )+hν k+1 ν k+1 ∈ ν k -λ 2 h sgn(x 1,k+1 ), , (15)
converges asymptotically to zero, i.e., the implicit scheme provides a globally asymptotically stable discretization.

Remark 2. We have not yet proved that x1,k attains the origin in a finite number of steps, even in the unperturbed case. For the moment only asymptotic convergence holds. In [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] it has been possible to prove that the state x1,k of the unperturbed virtual system, converges to zero in a finite number of steps, even in the presence of a disturbance. This was done either by direct calculations, or from a suitable Lyapunov function analysis. In our case, the closed-loop system with perturbation is given by:

x 1,k+1 = x 1,k -hλ 1 |x 1,k+1 |ξ k+1 + hν k+1 + h φk ϕ k+1 = ϕ k + h ∆k ν k+1 = ν k -λ 2 hξ k+1 0 ∈ f (-λ 2 h 2 ξ k+1 ) -N [-1,1] (ξ k+1 ) 0 ∈ g(x 1,k+1 ) + λ 2 h 2 sgn(x 1,k+1 ) g(x 1,k+1 ) = x1,k+1 + hλ 1 |x 1,k+1 |sgn(x 1,k+1 ) -x 1,k -hν k . (16)
It is not clear how to prove that x1,k reaches zero in a finite number of steps when φk = 0 for all k ≥ 0. The so-called equivalent controller is obtained from ( 16) by setting x 1,k+1 = x 1,k = 0 and ν k+1 = ν k . It gives u eq k = -ϕ k , which means that the equivalent controller compensates for the disturbance with a one-step delay. However it is not implementable since ϕ k is unknown. Nevertheless we can now state the following: Corollary 2. Assume that ∆(t) ≡ 0. The sliding surface Σ d is attained in a finite number of steps and is invariant.

Proof. We know from Corollary 1 that for any > 0, there exists 0 < n < +∞ such that |x 1,k | < and |ν k | < for all k ≥ n. Choosing n large enough (but finite for h > 0) it follows that -x 1,khν k ∈ [-h 2 λ 2 , h 2 λ 2 ] for all k ≥ n. As shown above, in such a case (named case 2), one has u k = -x 1,k h . We can now use [START_REF] Du | Robustness analysis of a continuous higher order finite-time control system under sampled-data control[END_REF] (which is equivalent to (6) since the disturbance is null) to deduce that x 1,k+1 = x 1,k + hu k = 0. From (8), we infer that

ν k+1 = ν k -λ 2 h -x 1,k -hν k -h 2 λ2 = - x 1,k
h . Therefore, -x 1,k+1 -hν k+1 = -hν k+1 = x 1,k and taking n large enough (or equivalently small enough) we deduce that -

x 1,k+1 -hν k+1 ∈ [-h 2 λ 2 , h 2 λ 2 ]. Consequently, u k+1 = - x 1,k+1 h = 0, and ν k+2 = ν k+1 -hλ 2 ξ k+2 = ν k+1 -hλ 2 -x 1,k+1 -hν k+1 -h 2 λ2
= 0. Repeating the same argument, it follows that x 1,k = ν k+1 = 0 for all k ≥ n.

It is noteworthy that in the absence of perturbation, the discrete-time system not only attains Σ d in a finite number of steps, but then stays on it with zero input and zero oscillations (suppression of the digital chattering). This is coherent with the results in [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF], [START_REF] Huber | Experimental results on implicit and explicit timediscretization of equivalent-control-based sliding-mode control[END_REF].

V. SIMULATIONS For simulation the plants is discretized as in ( 6) using the explicit Euler method, and the explicit Euler discretization of the controller reads as u k = -λ 1 |x 1,k |sgn(x 1,k ) + ν k , and ν k+1 = ν kλ 2 h sgn(x 1,k ). Let us select the following parameters: λ 1 = 10, λ 2 = 6, which satisfy all needed restrictions, and initial conditions x 1 (0) = ν(0) = 10, then let us compare the explicit discretization Euler method, and the implicit one proposed in this note. 5, where we can notice that the control in the implicit case is converging exactly to zero in a finite time, while that obtained by the explicit discretization, has a constant amplitude after a transient. The plot for ∆(t) = 2(1 + sin(2t 2 )), h = 0.001s and ϕ(0) = 0 are given in Fig. 6, and the corresponding control amplitudes of both methods, can be compared in Fig. 7: again the control generated by the implicit method has smaller amplitude, and much less chattering. The numerical results confirm the theoretical findings, and a much better accuracy for the proposed implicit scheme (though we have no proof for the latter property). See also [START_REF] Du | Robustness analysis of a continuous higher order finite-time control system under sampled-data control[END_REF]Theorems 1,[START_REF] Acary | Chattering-free digital sliding-mode control with state observer and disturbance rejection[END_REF] for the analysis and simulation of the explicit method for HOSM controllers.

VI. CONCLUSIONS

In this article the properties of the time-discretization of the super-twisting algorithm are analysed. An implicit Euler scheme is used to define the discretetime system. First it is shown that the continuous-time super-twisting controller, yields a closed-loop system which admits a strict Lyapunov function whose level sets are convex . Then it is proved that the discrete-time controller can be uniquely computed from available data, as the solution of two generalized equations. Finally it is shown that the existence of a Lyapunov function for the continuous-time closed-loop system, with convex level sets, implies the global asymptotic stablity of the discrete-time closed-loop system. Convergence Fig. 6. Regulation error e k with perturbation for h = 0.001s; explicit (solid red line) and implicit Euler method (dashed blue line) Fig. 7. Control amplitudes with perturbation for h = 0.001s; explicit (solid red line) and implicit Euler method (dashed blue line) to the origin in a finite number of steps is proved in the unperturbed case. Numerical simulations confirm that the implicit discretization, supersedes the explicit one. Future investigations should focus on robustness and accuracy properties, and extensions towards HOSMC.

A. Strict Lyapunov functions

Let us recall that a function V ∈ C(R n , [0, +∞)) ∩ C 1 (R n \{0}, (0, +∞)) is said to be a strict Lyapunov function to the system (14) if V (0) = 0, V (x) → +∞ as x → +∞ and there exists W ∈ C(R n \{0}, (0, +∞)) such that

sup ξ∈F (x) ∂V ∂x ξ ≤ -W (x) ∀x ∈ R n \{0}. (17)
Such a strict Lyapunov function always exists for globally asymptotically stable systems [START_REF] Huber | Experimental results on implicit and explicit timediscretization of equivalent-control-based sliding-mode control[END_REF], see [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF].

  First, let ∆(t) = ϕ(0) = 0, then the behavior of a homogeneous norm e k = |x 1,k | + ν 2 k for both methods (solid red line corresponds to the explicit Euler method, and dashed blue line represents the implicit one) is shown in Fig. 4 in a logarithmic scale for two values of constant discretization step (h = 0.1s and h = 0.001s). The corresponding values of the norm of control |u k | are presented in Fig.

Fig. 4 .

 4 Fig. 4. Regulation error e k without perturbation for h = 0.001s (left) and h = 0.1s (right).

Fig. 5 .

 5 Fig. 5. Control amplitudes in the case without perturbation for h = 0.001s (left) and h = 0.1s (right).

  2 ] (ζ k+1 ). It can be checked that df

dy (y) > 0 for all y = b while df dy (b k ) = 0. Therefore applying [9, Proposition 2.3.2] and [9, Theorem 2.3.3], it follows that the generalized equation (12) always has a unique solution. This means that the graphs of the set-valued mapping ζ k+1 → N [-λ2h 2 ,λ2h 2 ] (ζ k+1 ) and of the single-valued mapping ζ k+1 → -f (ζ k+1

  Σ d , in the presence of a perturbation. Assume that the discrete-time state belongs to Σ d for all t i

	Lemma 4.

This work was supported by the ANR project Digitslid ANR-18-CE40-0008-01.