An encrypted model predictive control strategy for resilience operations

An enc yp ed model p edic i e con ol s a egy o esilience ope a ions Giuseppe F anz` eSenio Membe , IEEE, Vicenc¸ Puig and F ancesco Tedesco Senio Membe , IEEE Abs ac — In his pape , a esilien model p edic i e con ol a chi ec u e is p oposed o cons ained cloud-based ne wo ked con ol sys ems subjec o alse da a injec ions on bo h he con olle - o-ac ua o and senso - o-con olle channels. The basic idea consis s in exploi ing he capabili y o he enc yp ion p ocess o hide he da a s uc u e, sha ed be ween he con olle and plan sides, o any hi d-pa y. Then, by adequa ely coupling he la e wi h he esilien na u e o he eceding ho izon con ol philosophy, an a ay o a ack coun e measu es is de e mined o he on-line ope a ions. Besides his, in o de o secu e da a packe ansmissions, cloud compu ing ope a ions a e pe o med by adop ing an addi i e homomo phic c yp osys- em so ha enc yp ed model p edic i e con ol sequences a e ob ained. Finally, a pla oon o ehicles is used o alida e he whole a chi ec u e in simula ion. I. INTRODUCTION In ecen yea s, ad ancemen s in In e ne o Things (IoT), Cybe -Physical Sys ems (CPSs), and sma g ids ha e opened doo s o deploymen lexibili y and e iciency im- p o emen s ac oss a ious ields. Howe e , his p og ess has heigh ened he demand o sophis ica ed con ol a chi ec u es o mee new enginee ing equi emen s. Ne wo ked con ol sys ems inc easingly ely on cloud and dis ibu ed compu - ing, whe e da a ulne abili y on cloud se e s is a c i ical conce n. Enc yp ed con ol, enabled by homomo phic enc yp ion (HE) me hods, ensu es con iden iali y o sys em s a es and con ol ac ions [1]. This app oach is pa icula ly ele an o cloud-based ne wo ked con ol sys ems, whe e h ea s like alse da a injec ions and ea esd opping jeopa dize no mal ope a ions [2]. The concep o using HE o ne wo ked con ol sys- ems’ p i acy was in oduced in [3], wi h subsequen e- sea ch explo ing ad anced con ol schemes wi hin enc yp ion amewo ks [4]. No ably, model p edic i e con ol (MPC) app oaches ha e gained in e es [5], add essing challenges such as handling enc yp ed da a o cloud-based sys ems. This wo k was in pa suppo ed by he esea ch p ojec - ID:20222N4C8E ”Resilien and Secu e Ne wo ked Mul i ehicle Sys ems in Ad e sa y En i onmen s” g an ed by he I alian Minis y o Uni e si y and Resea ch (MUR) wi hin he PRIN 2022 p og am and Eu opean Union - Nex Gene a ion EU and in pa by he esea ch p ojec - ID:PE00000014 “SEcu i y and RIgh s in he Cybe Space - SERICS” g an ed by he I alian Minis y o Uni e si y and Resea ch (MUR), and Eu opean Union wi hin he Nex Gene a ionEU p og am. Giuseppe F anz` e is wi h DIMEG, Uni e si ` a della Calab ia, Via Pie o Bucci, Cubo 42-C, Rende (CS), 87036, ITALY, [email p o ec ed] Vicenc¸ Puig is wi h CS2AC Uni e si a Poli ´ ecnica de Ca alunya (UPC), Rambla San Neb idi, 22, 08222 Te assa and also wi h CSIC-UPC, Ca e Llo ens A igas, 4-6, 08028 Ba celona, Spain, [email p o ec ed] F ancesco Tedesco is wi h DIMES, Uni e si ` a della Calab ia, Via Pie o Bucci, Cubo 42-C, Rende (CS), 87036, ITALY, [email p o ec ed] To deal wi h hese issues, a cloud-based model p edic i e con ol a chi ec u e is de eloped by in eg a ing in o a no el esilien amewo k cloud compu ing echnology and homo- mo phic c yp og aphy. In his con ex , he in oduc ion o he cloud compu ing on enc yp ed da a has a wo old aim: p o ec he ansmi ed packe s in e ms o con iden iali y and educe he chance o in ude s o al e ing hei in eg i y. F om a con ol poin o iew, his is ansla ed in o de ining a sequence o se -con ainmen condi ions, complying wi h a se - heo e ic wo s -case app oach, ha unequi ocally iden i y any admissible a ack occu ence and p o ide a easible command inpu a each ime ins an . A second con ibu ion elies on he cha ac e iza ion o he compu a ions among enc yp ed da a pe o med on he cloud. To his end, an ad-hoc a ian o he adi ional MPC algo i hm is conside ed in [5]. Al hough easible, bo h app oaches p esen he ollowing weaknesses: he con olle s uc u e mus be a ailable on he ac ua o side; hey lead o conse a i e con ol ac ions due o he exploi a ion o a single P oximal G adien Me hod (PGM) i e a ion [9]. The p oposed scheme subs an ially educes (o e en o e - comes) hese de imen al e ec s in i ue o he ollowing easons: 1) he ull con olle s uc u e is no manda o y on he ac ua o , in ac ew p elimina y in o ma ion a e exploi ed; 2) con ol pe o mance losses only occu when enc yp ed ope a ions a e in oked. PRELIMINARIES AND NOTATION Th ough his pape , we use he ollowing no a ions. Le ( +k| ), k( ) = ˆ kbe he k−s eps s a e ahead p edic ion o a gene ic sys em a iable om onwa d. De ini ion 1: Gi en he se s A,E ⊂ IRn,A ∼E := {a∈ A:a+e∈ A,∀e∈ E} is he Pon yagin-Minkowski Di e ence.2 Gi en a symme ic ma ix P∈IRn×n, P > 0 (P≥0) means ha i is posi i e de ini e (semide ini e). Homomo phic Enc yp ion [10] Le IR,ZZ,S,Kp,Ks,Mand Cbe he se o eal numbe s, in ege s, secu i y pa ame e s, public keys, secu i y keys, plain ex s (message space), and ciphe ex s. Mo eo e , conside he se s ZZ+:= {z∈ZZ |z≥0}and ZZn:= {z∈ ZZ |0≤z≤n}. Then, a public-key enc yp ion scheme is a iple (GEN, ENC, DEC)whe e GEN :S → Kp× Ks:k7→ (pk, sk)is a key gene a ion algo i hm, ENC :Kp× M → C: (pk, m)7→ cis an enc yp ion algo i hm, DEC : Ks× C → M : (sk, c)7→ mis a dec yp ion algo i hm, kis he key leng h (secu i y pa ame e ) and (pk, sk) = GEN(k) a pai o public and sec e keys. The ope a o s ENC and DEC pe o m componen -wise ac ions on ec o s and ma- ices. Public-key enc yp ion schemes sa is y he ollowing p ope y: P ope y 1 - DEC(sk, ENC(pk, m)) = m, ∀m∈ M and ∀(pk, sk) = GEN(k) De ini ion 2: The scheme (GEN, ENC, DEC)is an ad- di i e homomo phic enc yp ion i DEC(sk, c ⊕c0) = m+ m0,∀m, m0∈ M and c, c0∈ C such ha ENC(pk, m) = m and ENC(pk, m0) = m0,wi h ⊕:C × C → C a bina y ope a ion o e C. The Paille scheme [11] is an addi i e homomo phic enc yp- ion scheme. As mos o c yp osys ems, i is based on a subse o in ege s, whose ca dinali y depends on he chosen key o he enc yp ion. The key gene a ion complies wi h he ollowing ule: GEN :k7→ (pk, sk)=(a·b, lcm(a−1, b −1)) whe e aand ba e wo la ge p ime numbe s o he same leng h, i.e. a, b ∈(2k−1,2k), o some k∈ZZ+,such ha gcd(ab, (a−1)(b−1)) = 1.No ice ha gcd(·,·) and lcm(·,·)is he g ea es common di iso and he leas common mul iple ope a o s, espec i ely. Fo he enc yp ion o any plain ex m∈ZZpk,a andom in ege ∈ZZ∗ pk := {m∈ZZpk |gcd(m, pk) = 1}is chosen and he ciphe ex is ob ained as ENC : ( , m)7→ c= ((pk + 1)m pk mod pk2) Con e sely, o any ciphe ex c∈ZZpk2, he co esponding plain ex esul s om he dec yp ion DEC : (sk, c)7→ Lpk(csk mod pk2)µmod pk wi h Lpk(γ) := (γ−1)/pk and µ:= sk −1mod pk he so-called modula mul iplica i e in e se. I can be shown ha enc yp ion ollowed by dec yp ion p o ides an in ege equi alen o he plain ex . Mo e o mally, o e e y m∈ZZpk one has ha DEC(ENC(m, )) = m, ∀ ∈ZZ∗ pk .Mo eo e , no ice ha o e e y m1, m2∈ZZpk such ha m1+m2∈ZZpk, he ollowing p ope y holds ENC(m1, )⊕ENC(m2, s) = ENC(m1, )ENC(m2, s)mod pk2= ENC(m1+m2, ·s),∀ , s ∈ZZ∗ pk (1) Acco ding o (1), a semi-enc yp ed p oduc can be also compu ed. In ac , o e e y m1, m2∈ZZpk such ha m1·m2∈ZZpk,one has m2ENC(m1, s) =ENC(m1, )m2mod pk2 =ENC(m1·m2, m1 ),∀ ∈ZZ∗ pk (2) II. PROBLEM FORMULATION Conside he class o Cloud-based Ne wo ked Con ol Sys ems (Cb-NCS) depic ed in Fig. 1 whose physical plan s a e desc ibed by he ollowing disc e e- ime linea in a ian s a e space models: x( + 1) = Ax( ) + Bu( ) + Bdd( )(3) P Ac ua o Senso Con olle u( ) x( ) z( ) u( ) c u( ) a CLOUD Fig. 1. Cloud-based ne wo ked con ol sys em subjec o ad e sa y a acks whe e x( )∈IRnxdeno es he s a e, u( )∈IRnu he command inpu and d( )∈ D ⊂ IRnx,∀ ∈ZZ+:= {0,1, . . .},an exogenous dis u bance. Mo eo e , he ollow- ing cons ain s a e p esc ibed: u( )∈ U, x( )∈ X (4) wi h Uand Xcon ex and compac subse s o IRnuand IRnx, espec i ely, and 0nu∈ U,0nx∈ X. In he sequel, i is hypo hesized ha plan /con olle ope a- ions may be comp omised because o he ollowing easons: •exchanged da a - malicious ex e nal agen s mod- i y command inpu and senso measu emen s when sen / ecei e ope a ions a e pe o med h ough a com- munica ion medium: - con olle - o-ac ua o link: u( ) := uc( ) + ua( ) - senso - o-con olle link: z( ) := x( ) + xa( ) whe e ua( )∈IRnuand xa( )∈IRnxa e unknown and unbounded malicious signals, while u( )∈IRnuand z( )∈IRnxaccoun o he esul ing co up ed con ol signals and s a e measu emen s, espec i ely. •cloud laye - s eal hy malwa es (see [12] and e e ences he ein), in en ionally designed o cause dis up ion o se e capabili ies, ope a e on he con olle uni by a wo old ac ion: 1) ea esd opping on he compu ed con ol inpu s; 2) eco d and eplay ope a ions on he ansmi ed da a. Then, he p oblem o be sol ed is s a ed as ollows: Gi en he Cb-NCS o Fig. 1 desc ibed by (3)-(4) and subjec o •FDI a acks on bo h command and measu emen com- munica ion channels, •malwa e h ea s on he emo e side, de elop a con ol a chi ec u e enjoying a ack de ec ion capabili ies and da a con iden iali y p ope ies, such ha he esul ing con olle , based on he ully a ailabili y o he s a e ec o u(·) = g(z(·)), egula es he s a e ajec o y in a Uni o mly Ul ima e Bounded (UUB) sense [13] despi e any admissible dis u bance ealiza ion and a ack occu ence. III. THE PROPOSED SOLUTION:AN OVERVIEW In he sequel, wo me hodologies a e exploi ed: model p e- dic i e con ol philosophy and c yp osys ems. In pa icula , wo MPC s a egies, namely Nominal-MPC and Resilien - MPC a e designed o comply wi h esilien asks, while enc yp ion/dec yp ion ac ions a e pe o med o make sa e he da a ansmission be ween plan and con olle sides. he p oposed solu ion is summa ized in he scheme o Fig. 2 and he ea e discussed. The s a ing poin consis s in cha ac- e izing he emo e side acco ding o enc yp ion/dec yp ion a gumen s. Speci ically, he equi ed compu a ions usually pe o med on he so-called Physical Laye a e spli as ollows: • he Nominal-MPC con olle is used du ing a ack- ee condi ions bu pu aside as soon as he a ack is ecognized, he De ec o and a Con olle Bu e (used o s o e he las admissible a ack- ee s a e measu e- men , he ea e deno ed as z−1) in cha ge o e eal he p esence o a acks; •a second con olle Resilien -MPC, ac i a ed unde a ack scena ios, is compu ed on he cloud whe e high compu a ional esou ces a e a ailable. No ice ha z−1is cons an ly upda ed du ing he a ack- ee ope a ions, while, unde an a ack scena io, he las s o ed da a is used o ini ialize he suppo con olle Resilien - MPC. Acco ding o his amewo k, enc yp ion/dec yp ion ope a ions ake place only when a acks a e unde way. Speci ically he ollowing ac ions a e pe o med on he emo e side. Du ing he a ack- ee ope a ions (Label No), he plan Pis exclusi ely egula ed by means o he com- mand inpu uc( ) ha is ne e enc yp ed. Once he De ec o e eals he a ack occu ence (Label Yes), he communica ion medium is no longe eliable and he plan p oceeds in an open-loop ashion by using i s uc( )(consecu i ely usable o a ini e numbe o s eps) hen he Nleng h esilien sequence uMP C ( )as soon as i has been ecei ed. Since he a ack du a ion is no a-p io i known, a each N ime s eps uMP C (·)is upda ed in o de o enjoy esilien capabili ies. He e, his is add essed by mainly ope a ing on he cloud and by p ese ing he con iden iali y o he da a sen along he communica ion ne wo k. The la e is achie ed hanks o enc yp ion/dec yp ion ope a ions o he s o ed measu emen z−1 ha allows o a oid om one hand ea esd opping phenomena and om he o he hand, when he enc yp ed da a [[uMP C ( )]] is ansmi ed, he oppo uni y o he in ude o p ope ly in e he sha ed in o ma ion. On he plan side, one has he ollowing easoning. Unde a ack- ee scena ios, he Sma Dec yp ion ecognizes ha a single ec o , namely uc( ),has been ecei ed and he e o e no dec yp ion ope a ions a e equi ed. Then, he Sma Ac u- a o selec s u eas( ) = u( −1) ( ha is admissible because i has been designed o be usable o se e al consecu i e ime ins an s): such an ac ion is manda o y since i is unknown i he cu en ecei ed inpu u( )has been ins an aneously modi ied (u( ) = uc( ) + ua( )). Hence, he command u( ) is i s checked by using he win model Σand hen applied a he nex ime ins an . Con e sely, he Sma Dec yp ion ac i a es he dec yp ion phase when a sequence o con ol mo es is ecei ed, while he Sma Ac ua o applies he same k− h mo e o uMP C ( )bo h o Pand Σ,i.e., u( ) = u eas( )=(uM P C ( ))k. No e ha , wi hin his amewo k, a each ime ins an he De ec o ecei es he pai (z( ), zΣ( )) in o de o iden i y he cu en ope a ing condi ion o he plan P, and e en ually o eco e he a ack- ee mode. Σ Sma Ac ua o Senso Ac ua o P u( ) u ( ) eas u( -1) Sma Dec yp ion a u ( ) Nominal-MPC Con olle z -1 De ec o Enc yp ion Resilien -MPC Enc yp ed ope a ions a x ( ) c u ( ) [ | [ | z -1 z( ) No Yes u( ) x( ), x ( ) ( ) Σ z( ), z ( ) ( ) Σ [ | [ | MPC u ( ) Fig. 2. Resilien enc yp ed-based con ol a chi ec u e IV. DETECTION,COUNTERMEASURES AND ON-LINE OPERATIONS Acco ding o he discussion on he a chi ec u e o Fig. 2, he design o he Nominal-MPC and Resilien -MPC mus be join ly pe o med in o de o ensu e he easibili y e en- ion. In he sequel, a se - heo e ic app oach will be pu sued by exploi ing he a gumen s o [7], [6] and [8] p ope ly adap ed o he p oposed amewo k. Fi s , an admissible obus posi i ely in a ian (RPI) egion E0 o he closed-loop s a e e olu ions (6) is de i ed by conside ing he ollowing s a e- eedback con ol law uc( ) = K x( −τ( )) (5) which sa is ies he p esc ibed cons ain s (4) and ensu es ha he egula ed s a e ajec o y x( + 1) = Ax( ) + B K x( −τ( )) + Bdd( )(6) is UUB i espec i e o any delay occu ence τ( )≤ τmax, τmax ≥1.Then, he uppe bound ¯ Non he con ol ho izon leng h pe aining o he Resilien -MPC con olle is gi en by: ¯ N:= max k{k∈ZZ+|E0∼ k−1 X i=0 AiBdS 6=∅},(7) As a consequence, le N≤¯ Nbe gi en, he RPI egion Ξ0 o he closed-loop s a e e olu ions is de e mined along he same lines exploi ed o E0wi h N−1 X =0 A BdD.Hence, he sequence o obus one-s ep s a e ahead con ollable se s {Ξi}is compu ed acco ding o he ollowing ecu sions: Ξi={x∈IRn:∃u∈ U | Ax +Bu ∈˜ Ξi−1}(8) wi h ˜ Ξi:=Ξi∼ N−i−1 X =0 A BdD, i = 1, . . . , N −1,(9) Con e sely, he sequence {Ei}is compu ed as ollows: Ei:= {x:∃u∈ U|Ax +Bu +Bdd∈ Ei−1,∀d∈ D Aτmax x+ τmax−1 X i=0 Aτmax−1−iBu ∈˜ Ξi−1} (10) No ice ha he u he equi emen in (10) imposes ha he consecu i e applica ion o he command inpu u o τmax ime ins an s d i es he egula ed s a e ajec o y wi hin L [ i=0 Ξi.Then, he ollowing esul s holds ue. P oposi ion 1: Le x(0) ∈ L [ i=0 Ξibe an ini ial s a e con- di ion o he egula ed sys em (3) unde he ac ion o he Nominal MPC. I a a ce ain ime ins an ˆ > 0an a ack is de ec ed, he con olle swi ching Nominal MPC → Resilien MPC is always iable. P oo - Omi ed o space easons. 2 Finally, du ing he on-line phase he Nominal MPC and Resilien MPC command inpu s a e ob ained as ollows. Gi en a gene ic s a e x∈ Ei( ), he Nominal MPC com- pu es he admissible con ol ac ion by sol ing he ollowing op imiza ion p oblem: uc( ) := a g min Fj( )(x, uc)s. . (11) Ax +Buc∈ Ei( )−1(12) whe e Fj( )(x, uc)∈F:= {Fh(x, uc)} h=1 a se o penaliz- ing unc ions ha a e andomly chosen a each ime ins an by he eal- alued unc ion j( ) : ZZ+→ {1, . . . , }. Con e sely, by assuming ha x∈Ξip ec ⊆ΞL he Resilien MPC sequence o Ncon ol mo es is ob ained by sol ing he ollowing con ex op imiza ion p oblem: min {˜uk}max ˜xk N−1 X k=0 k˜xkk2 Rx+k˜ukk2 Ru(13) ˜xk+1 =A˜xk+B˜uk;(14) ˜x0=x; ˜xN∈Ξ0;(15) ˜uk∈ U,˜xk∈ΞL;k= 0, . . . , N −1(16) whe e ˜xkis he k−s a e ahead dis u bance- ee p edic ion, Rx=RT x≥0and Ru=RT u>0s a e and inpu shaping ma ices, espec i ely. A. Anomaly de ec o This uni has he aim o check he admissibili y o ans- mi ed da a z( ), zΣ( ).and z+ Σ( ).Recall ha Σdeno es he win model o he plan P, z( ) he ecei ed s a e measu e- men esul ing om he applica ion o he cu en easible command u eas( )while zΣ( )accoun s o he ou pu o he applica ion o he inpu u( ) o Σ.Finally z−1is he a ack- ee measu emen s o ed in he Con olle Bu e which is possibly upda ed a each ime ins an . By using simila a gumen s as in [6], he ollowing wo-s eps de ec o comes ou . S a ing om he so-called easible measu emen z( ),one has ha on he con olle side he ollowing logical se - membe ship condi ions mus be e i ied: z∈Ξi⇒∃uc∈U such ha ∀d∈ D, Az+Buc∈˜ Ξi−1 he e o e i z−1∈Ξiand z( )∈Ξj, j > i, (17) hen an a ack is unde way. The FDI occu ence can be e ealed by exploi ing he concep o expec ed one-s ep p edic ion se Z+.Speci ically, Z+(z−1,u( −1)):= {z+∈IRn:z+=Az−1+Bu( −1)+Bdd, ∀d∈D} ⊂ Ξi−1 (18) whe e z−1∈Ξiis he a ailable in o ma ion a he p e ious ime ins an and u( −1) is he easible, hough no op imal, command. Then, he ollowing logics comes ou : D+(z( )) := a ack,i z( )/∈Z+(z−1, u( −1)) no a ack, o he wise (19) As he measu emen zΣ( )is conce ned, he FDI a ack de ec ion exac ly ollows he same lines abo e desc ibed, i.e. zΣ( )in place o z( )in (17). Con e sely, he de ec ion o FDI occu ences needs a u he condi ion in addi ion o (19). Since zΣ( ) ep esen s he one-s ep s a e p edic ion unde he ac ion o u( )and i is equi ed o check i u( )is admissible o he plan Pa he nex ime ins an + 1, hen one has o e i y i s se -membe ship o Z++(z−1,{u( −1), u( −1)}) := AZ+(z−1,u( −1))+Bu( −1)+BdD ⊂ Ξi−2(20) and, as a consequence, he de ec ion logics is D++(zΣ( )) := a ack, i zΣ( )/∈Z++(z−1,{u( −1), u( −1)}) no a ack, o he wise (21) Con e sely once he a ack is e ealed, he se -membe ship es s (17) and (19) will be e alua ed by conside ing he con ol mo es o uMP C ( )in place o uc( ),i.e., u( )←uMP C ( )k wi h zΣ( ) = Ax( ) + BuMP C ( )k+Bdd( )(22) Then, acco ding o he abo e analysis, he ollowing esul summa izes ha FDIs canno emain inde ini ely s eal hy. P oposi ion 2: Gi en he Cb-NCS a chi ec u e o Fig. 2, a se o penalizing unc ions Fand a eal- alued unc ion j( ).Le uc( )be he nominal command inpu solu ion o he op imiza ion (11)-(12) and uMP C ( ) he esilien sequence compu ed by sol ing he SDP (13)-(16). Then, he se - membe ship logics (17), (19) and (21) always de ec in a ini e ime FDI a acks. P oo - Omi ed o space easons. As he eco e y om he a ack is conce ned, simila a - gumen s can be exploi ed downline o he applica ion o uMP C ( ).Le de ec be he de ec ion ime ins an , one has ha i x(¯ )∈ E ,¯ > de ec , < ide ec ,(23) wi h ide ec he se -le el complying wi h (17), hen he plan Pis no longe unde a ack. Then, he ollowing esul holds ue. Co olla y 1: I he e exis s a ime ins an ¯ > de ec sa is ying (23) such ha z(¯ )∈Z+(z−1, u( −1)) wi h z(¯ ) as in (22), hen he NCS o Fig. 2 is a ack- ee. P oo - I s aigh o wa dly ollows by cons uc ion and collec ing he abo e de elopmen s. 2 B. On-line ope a ions and coun e measu es The de elopmen s o he p e ious sec ions a e he e col- lec ed o desc ibe he modus ope andi o he esilien a chi- ec u e o Fig. 2. Wi hou loss o gene ali y, i is assumed ha ini ially he plan Pis a ack- ee. Du ing his phase, he ea e deno ed as he nominal mode, he plan is egula ed by exploi ing uc( ),i.e., he admissibili y o he ecei ed command u( ) is i s checked by (19) and (21), hen i is applied a he nex ime ins an + 1.This easoning applies un il one o he logics (17), (19), (21) e eals an a ack occu ence, namely a de ec >0.As a consequence, he Nominal-MPC canno be longe upda ed and he ollowing coun e measu e akes place acco ding o an open-loop ashion. Fi s , he command u( −1) is consecu i ely e-used o he τmax −1 ime ins an s so ha he esul ing s a e condi ion x( de ec + τmax −1) ∈SiΞi,as p esc ibed in (10). Hence, he Resilien -MPC can be ac i a ed and he enc yp ed sequence [[uMP C ( de ec +τmax )]] compu ed on he basis o he las s o ed measu emen [[z−1]].This swi ching is iable in i ue o he ollowing esul . P oposi ion 3: Le z=Aτmax x+ τmax−1 X i=0 Aτmax−1−iBuc+ Bdd( ),wi h d( )∈ D,be he τmax− h s a e ahead e olu ion on he plan side o Fig. 2. Then, he dis u bance- ee e olu ion ˜zcompu ed on he con olle side ˜z:= Aτmax z−1+ τmax−1 X i=0 Aτmax−1−iBuc(24) is an admissible ini ial condi ion o he Resilien -MPC scheme. P oo - Omi ed o space easons. C. S abili y issue The nex p oposi ion s a es he s abili y p ope y o he p oposed esilien scheme. Theo em 1: Le {Ei}L i=0 and {Ξi}L i=0 be non-emp y one- s ep s a e ahead con ollable se amilies and x(0) ∈ EL. Then, he Cb-NCS a chi ec u e o Fig. 2 ensu es cons ain s sa is ac ion and UUB o any admissible a ack occu ence and dis u bance ealiza ion. P oo - Omi ed o space easons. V. ENCRYPTED CLOUD COMPUTING This sec ion is de o ed o desc ibe he enc yp ed op- e a ions wi hin he Pallie HE c yp osys em o sol e he op imiza ion (13)-(16). To his end, he i s s ep consis s in ew i ing (13)-(16) as he ollowing Quad a ic P og amming (QP) p oblem: q?(x) := a g min q∈Q(x) (x, q)(25) whe e q:= {˜uk}N−1 k=0 ∈IRNm Q(x) :=    q∈IRNm |∃{˜xk}N−1 k=0 ∈IR(N+1)n s. .(q,{˜xk}N−1 k=0 ∈IRNn) sa is ies (14) −(16) wi h ˜x0=x    and (x, q) := 1 2qTHq+xTFTq.In wha ollows, he me hod in oduced in [4], which exploi s a p ojec ed g adien scheme (PGS) belonging o he class o p oximal algo i hms [15], will be used o add essing he op imiza ion (25). In pa icula , a PGS is ca ied ou by he ollowing ecu sions qj+1 =p ojQ(x)(qj−α∆q (x, qj)) p ojQ(x)(qj−α(Hqj+Fx)) (26) wi h p ojQ(·)being he p ojec ion ope a o in o a se Q. Since he Pallie c yp osys em is no o de -p ese ing, he p ojec ion canno be pe o med in he enc yp ion domain as well as he op imal solu ion o (13)-(16). This nume ical d awback can be o e come by conside ing a single i e a ion o (26): ˜ q1=q0−α(Hq0+Fx) = (I−αH)q0−αFx (27) wi h ˜ q1compu ed on he cloud as [[˜ q1]] = (Ecd∆(I−αH)[[q0]])⊕(Ecd∆(αF)[[x]]) (28) Finally, he esilien MPC ac ion, compu ed on he ac ua o side, is: uMP C ( ) = p ojQ(x)(Dcd∆(˜ q1) := a g min q∈Q(x)kq−˜ q1k (29) VI. SIMULATIONS Conside a pla oon o wo ehicles wi h pi( ), i( )and ai( )deno ing posi ion, eloci y and accele a ion o ehicle Vi( ), espec i ely, and d2( )being hei in e -dis ance. The con ol objec i e is o ensu e ha he pla oon dynamics con e ge o an asymp o ically s able equilib ium: all in e - ehicle dis ance e o s ei( ) := d2, −d2( )con e ge o ze o wi h d2, = 2[m] he a ge dis ance be ween he ehicles. To his end he ollowing con ol-o ien ed model has been de i ed ˙x( )=       0 1 0 0 0 0−1/τ 0 0 0 100−1 0 0 0 0 0 −1 0 0 0 0 −1/τ       x( )+       0 0 1/τ 0 0 0 0 0 0 1/τ       u( ) whe e τ= 0.18s, x( ) = [ 1( ), a1( ), d2( ), 2( ), a2( )]T and u( ) = [u1( ), u2( )],wi h ui∈ U := [−2,2]m s2, he command inpu in cha ge o ac ua e he engine. In u n, his model can be ecas as (3) by a o wa d Eule disc e iza ion unde he sampling ime Ts= 0.1s. In he sequel, he ollowing ope a ing scena io is conside ed: S a ing om he ini ial condi ions x(0) = [5.93,14.60,3.20,9.47,1.43]Ti is equi ed ha he egula ed s a e ajec o y is d i en o he a ge x = [10,0,2,10,0]Twhile keeping he p esc ibed cons ain s. The Nominal-MPC is ini ialized ollowing guidelines om Sec ions III-IV, gene a ing a sequence o 150 con ollable se s cen e ed a x wi h a maximum du a ion o 14 s eps. Fo he cloud in as uc u e, a Resilien -MPC scheme is imple- men ed based on Sec ions IV and VI, wi h a con ol ho izon o 46 s eps using PGS wi h s ep size α= 0.01. Enc yp ed con olle ope a ions a e execu ed using he Py hon lib a y eclib wi h key sizes o 1024 bi s. Simula ion esul s a e p esen ed in Figu es 3-6. In an a ack- ee scena io, Figu e 3 demons a es he con e gence o he pla oon unde Nominal- MPC, eaching he e minal se Ξ0a = 13.5sas expec ed. In a co e a ack scena io anging om = 4 s o = 9s, he esilien a chi ec u e p omp ly de ec s he a ack a = 4.4swhen z(4.4) ∈Ξ96 using he D++ de ec o uni (Figu e 4). Ini ially, he sa e command sequence om he ac ua o bu e is applied o he nex 1.4s, while he cloud compu es a new Resilien -MPC sequence o be con eyed in an enc yp ed o m. Once a ailable, he enc yp ed sequence is applied o he successi e 4.6s. Despi e a sligh pe o mance loss, wi h he a ge se Ξ0 eached a = 15 s, he a acke ’s ac ions a e ende ed ine ec i e due o he use o enc yp ed da a. A = 9.2s, when z(9.2) ∈Z+(z(4.4), uMP C (7.6)16), he eco e y phase is igge ed acco ding o P oposi ion 3, swi ching he De ec o ’s s a us o ”no a ack” and e- ac i a ing he Nominal-MPC. 0 50 100 150 i( ) 0 5 10 15 20 Time [s] Fig. 3. No a ack scena io: se -membe ship le el o he quan i y x( )−x . D+ D++ 0 5 10 15 20 Time [s] no a ack a ack no a ack a ack Fig. 4. De ec o s signals -3 -2 -1 0 1 ua (1) ( ) 0 5 10 15 20 Time [s] -3 -2 -1 0 ua (2) ( ) Fig. 5. Applied inpu : esilien con olle VII. CONCLUSIONS In his pape , a no el model p edic i e con ol s a egy o cons ained cybe -physical sys ems subjec o di e en classes o a acks has been concei ed. In o de o ake ad an age o enc yp ion/dec yp ion ea u es and cloud com- pu ing, he p oposed con olle is de eloped on wo laye s: a 0 50 100 150 i( ) 0 5 10 15 20 Time [s] con eyed ac ual 3.5 4 4.5 88 90 92 94 96 Fig. 6. 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