STOCHASTIC MODEL PREDICTIVE CONTROL

Transcription

1 hesis 2015/11/21 9:18 page 1 #1 POLITECNICO DI MILANO DIPARTIMENTO DI ELETTRONICA, INFORMAZIONE E BIOINGEGNERIA DOCTORAL PROGRAMME IN INFORMATION TECHNOLOGY STOCHASTIC MODEL PREDICTIVE CONTROL WITH APPLICATION TO DISTRIBUTED CONTROL SYSTEMS Docoral Disseraion of: Luca Giulioni Supervisor: Prof. Riccardo Scaolini Co-supervisor: Prof. Marcello Farina 2015 XXVIII Cycle

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3 hesis 2015/11/21 9:18 page 1 #3 A chi crede in me da sempre.

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5 hesis 2015/11/21 9:18 page I #5 Absrac The goal of his Thesis is wofold. Firsly, we deal wih he analysis and he developmen of Sochasic Model Predicive Conrol algorihms (SMPC) for linear discree-ime sysems wih addiive sochasic disurbances and probabilisic consrains on he saes and he inpus. Secondly, we consider he developmen of disribued Model Predicive Conrol algorihms for uncerain linear discree-ime sysems and we exend he echniques described in he firs par o he disribued framework. SMPC echniques are based on he idea of aking advanage of he informaion available on he probabilisic characerizaion of he uncerainy affecing he sysem o relax he problem consrains wih respec o classical wors-case approaches. In his seup, a novel SMPC algorihm, named probabilisic-smpc or p-smpc, is proposed boh in he sae-feedback and oupu-feedback framework and is applicaion is discussed in several examples. The main advanages of p-smpc rely on he guaraneed recursive feasibiliy and convergence properies, even in he case of disurbances wih possibly unbounded suppor, and he reduced compuaional load, similar o he one required by sandard nominal MPC echniques. The idea behind disribued MPC algorihms is based on he assumpion ha he coupling erms among he subsysems can be inerpreed as disurbances o be rejeced. Iniially, a regulaion problem for dynamically coupled subsysems wih local probabilisic consrains is considered and a novel algorihm, based on p-smpc, exends he nice recursive feasibiliy and convergence properies of he cenralized approach o he disribued case. Secondly, he problem of racking a reference oupu signal is discussed and a mulilevel scheme is proposed, ha relies on robus MPC echniques o handle he couplings beween subsysems. Paricular aenion is devoed o he applicaion of he proposed approach o a real mobile robo coordinaion problem. Finally a sochasic disribued MPC algorihm for racking reference signals is presened for dynamically decoupled subsysems subjec o local and collecive probabilisic consrains. The mobile robo coordinaion problem is exended also o his framework o show he viabiliy of he approach. I

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7 hesis 2015/11/21 9:18 page III #7 Conens 1 Inroducion 1 2 Sae of he ar of Sochasic MPC Sochasic sysems and consrains Sochasic MPC, formulaions and properies Cos funcion Conrol sraegies Reformulaion of sae consrains Consrains on inpu variables Feasibiliy and convergence properies Summarizing ables Numerical examples Comparison beween sochasic and robus ighenings Choice of he conrol sraegy and ighening approaches Join chance consrain approximaion Commens Sae-feedback probabilisic SMPC Problem saemen Regulaor srucure Reformulaion of he probabilisic consrains MPC algorihm: formulaion and properies Cos funcion Terminal consrains III

8 hesis 2015/11/21 9:18 page IV #8 Conens Iniial condiions for he mean and he covariance Saemen of he sae feedback p-smpc problem Implemenaion issues Proof of he main Theorem Proof of recursive feasibiliy Proof of convergence Simulaion example Commens Applicaion examples A quick overview on some paradigmaic SMPC algorihms Sochasic ube MPC (-SMPC) Sochasic MPC for conrolling he average number of consrain violaions (av-smpc) Scenario MPC (s-smpc) The proposed approach: probabilisic MPC (p-smpc) Simulaion examples Academic benchmark example Muli-room emperaure conrol Temperaure conrol on a realisic building Building model Disurbance model Sandard nominal MPC Robus MPC sraegy Sochasic MPC sraegies Commens Oupu-feedback probabilisic SMPC Problem saemen Regulaor srucure Reformulaion of he probabilisic consrains MPC algorihm: formulaion and properies Cos funcion Terminal consrains Saemen of he oupu feedback p-smpc problem Implemenaion issues Approximaion of he oupu-feedback p-smpc for allowing a soluion wih LMIs Approximaion of p-smpc wih consan gains Boundedness of he inpu variables IV

9 hesis 2015/11/21 9:18 page V #9 Conens 5.5 Proof of he main Theorem The p-smpcl problem Proof of Lemma LMI reformulaion of he consrains Simulaion example Commens Disribued Predicive Conrol for regulaion: a sochasic approach Problem saemen Regulaor srucure Reformulaion of he probabilisic consrains The disribued SMPC algorihm: formulaion and properies Cos funcion Terminal consrains Iniial condiions Saemen of he local MPC problems and main resul Implemenaion issues Proofs Proof of Lemma Proof of Theorem Simulaion example Commens Disribued Predicive Conrol for racking reference signals: a robus approach Ineracing subsysems Conrol sysem archiecure The disribued predicive conrol algorihm Proof of he main heorem Conrol of unicycle robos The model of unicycle robos The experimenal seup Conrol of dynamically coupled subsysems Commens Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains Problem saemen Local conroller srucure Reformulaion of he probabilisic consrains Applicaion of local and coupling consrains V

10 hesis 2015/11/21 9:18 page VI #10 Conens 8.2 SDPC for racking: formulaion and properies Cos funcion Terminal consrains Choice of he iniial condiions The i-h MPC conroller Sochasic Disribued Conrol of a flee of unicycle robos Model of unicycle robos Obsacle and collision avoidance Fixed obsacle avoidance Iner-robo collision avoidance Terminal Consrains Saemen of he problem and skech of he algorihm Simulaion resuls Single robo probabilisic obsacle avoidance Muli-robo collision avoidance Applicaion o a real coordinaion problem Commens Conclusions 185 Bibliography 189 VI

11 hesis 2015/11/21 9:18 page 1 #11 CHAPTER1 Inroducion Model Predicive Conrol (MPC) is nowadays a sandard in many indusrial conexs, see e.g. [131], due o is abiliy o cope wih complex conrol problems and o he availabiliy of heoreical resuls guaraneeing feasibiliy and sabiliy properies, see [100]. These reasons have moivaed he many effors devoed o develop MPC algorihms robus wih respec o unknown, bu bounded, disurbances or model uncerainies, see for example [94,102]. The problem of designing robus deerminisic MPC schemes has many soluions, see for example [83, 139], however, hese algorihms can suffer from some issues. Firsly, feasibiliy, convergence, and sabiliy properies are usually achieved by resoring, implicily or explicily, o a wors-case analysis, which may urn ou o be very conservaive or even impossible in he case of unbounded uncerainies and i may require he soluion o difficul on-line min-max opimizaion problems, ha are compuaionally very demanding, see [94]. Secondly, he uning of robus algorihms, for insance he off-line procedures required o compue robus posiive invarian ses, can be compuaionally demanding or even impracicable for sysems of medium-high order, see [101]. Thirdly, and perhaps mos imporanly, hey do no consider he possible a-priori knowledge of he saisical properies of he disurbances, i.e. heir disribuion funcion, 1

12 hesis 2015/11/21 9:18 page 2 #12 Chaper 1. Inroducion which can be assumed o be available in many problems. In his seup, if he uncerainies or he sae and conrol disurbances are characerized as sochasic processes, consrains should be reformulaed in a probabilisic framework [68, 156]. Saring from he pioneering papers [82, 145], hese reasons have moivaed he developmen of MPC algorihms for sysems affeced by sochasic noise and subjec o probabilisic sae and/or inpu consrains. Sochasic MPC (SMPC) has already been considered in several applicaion fields, such as emperaure and HVAC conrol in buildings [39, 86, 89 91, 113, 114, 123, 159, 162], process conrol [152], power producion, managemen, and dispach in sysems wih renewable energy sources [3, 67, 98, 109, 119, 122, 124, 135, 141, 165], cellular neworks managemen [155], driver seering, scheduling, and energy managemen in vehicles [13, 37, 49, 53, 57, 85, 132, 133, 140], pah planning and formaion conrol [14, 17, 48, 158], air raffic conrol [80], invenory conrol and supply chain managemen [69, 72, 166], resource allocaion [30], porfolio opimizaion and finance [38, 61, 62, 111, 129, 146]. In spie of his large number of applicaions of SMPC and he already available heoreical resuls, many ough challenges emerge in his seup, relaed o he developmen of mehods wih guaraneed sabiliy and feasibiliy properies. Indeed, while for bounded disurbances recursive feasibiliy and convergence can be esablished boh in a deerminisic or a sochasic seup, he more general case of unbounded noise poses more difficulies and some specific soluions and reformulaions of hese properies have been adoped. For example in [28] he concep of invariance wih probabiliy p is used, while in [117] he definiion of probabilisic resolvabiliy is inroduced. Due o he large variey of available SMPC echniques, he lieraure on he opic is quie vas and someimes no very consisen. Thus, he firs par of his Thesis will be devoed o discuss he main key poins and he many possible formulaions of SMPC algorihms wih he aim of clarifying he mos imporan aspec abou he opic. Moreover, an algorihm, denoed as probabilisic-smpc or p-smpc, will be proposed and is properies and applicaions horoughly discussed. In recen years, anoher very significan opic is represened by he developmen of disribued conrol algorihms, ha allow o handle large-scale sysems whose complexiy is relaed o he he presence of a high number of small or medium-scale subsysems ineracing via inpus, saes, oupus or consrains. As discussed before, due o is flexibiliy, robusness, and he vas lieraure on he opic, among all he possible soluions, paricularly ineresing appear o be hose based on Model Predicive Conrol. The 2

13 hesis 2015/11/21 9:18 page 3 #13 mos imporan aspec, when conrolling large-scale sysems in a disribued framework, is ha using local MPC conrollers, he prediced rajecories of inpus, saes and oupus are direcly available and can be used as informaion o be ransmied o oher local conrollers o coordinae heir acions (see for example [93]). This daa exchange can grealy simplify he design of a disribued conrol sysem and can allow one o obain performances close o hose of a cenralized conroller. The goal of he second par of his Thesis is o exend he SMPC echnique proposed in he firs par o he disribued framework. In paricular, he firs problem o be addressed will be he conrol of dynamically coupled linear sysems wih addiive disurbances and subjec o local probabilisic consrains. Then, he problem of designing a disribued conroller for racking reference signals will be discussed and a soluion based on he well-esablished DPC algorihm, see e.g. [43, 47, 93], is presened. The main idea of DPC is ha, a every sampling ime, each subsysem ransmis o is neighbors he reference rajecories of is inpus and saes over he predicion horizon and guaranees ha he real values of heir inpus and saes lie in a specified invarian neighbor of he corresponding reference rajecories. In his way, each subsysem has o solve an MPC problem where he reference rajecories received from he oher conrollers represen a disurbance known over all he predicion horizon, while he differences beween he reference and he real values of is neighbors inpus and saes can be reaed as unknown bounded disurbances o be rejeced. To his end, a robus ube-based MPC formulaion is adoped and implemened using he heory of polyopic invarian ses. Finally he problem of racking a reference signals in presence of probabilisic consrains will be discussed for dynamically decoupled subsysems and he presened algorihm applied o a mobile robo coordinaion problem. Srucure of he Thesis The Thesis is organized as follows. In Chaper 2 he sochasic consrained conrol problem is inroduced and several possible modeling assumpions on he consrains, conrol laws and cos funcions are discussed wih he goal of presening he main ingrediens of a sochasic MPC (SMPC) problem. Afer his quick overview of he opic, a deailed analysis of he sae of he ar of sochasic MPC approaches already available in he lieraure is presened and resuls are summarized. In Chaper 3 he analyic p-smpc approach is presened for linear discreeime sysems wih addiive possibly unbounded disurbances, measurable sae and subjec o one or more individual chance consrains on he sae and he inpu. The consrains are reformulaed ino deerminisic ones and 3

14 hesis 2015/11/21 9:18 page 4 #14 Chaper 1. Inroducion he resuling algorihm has similar compuaional complexiy o sandard MPC approaches. Moreover, an efficien LMI formulaion of he problem o be solved is presened. Finally, resoring o proper erminal consrains and a swiching iniializaion sraegy, recursive feasibiliy and convergence are guaraneed even in he case of unbounded uncerainies. In Chaper 4 he sae-feedback p-smpc approach described in he Chaper 3 is applied o some academic and realisic benchmark examples, ogeher wih hree oher noable approaches seleced from he lieraure, namely he ube-based SMPC [77], he average SMPC [75] and a scenario MPC [144]. These algorihms are firsly presened and some hins on heir implemenaion are given wih he aim of showing he difficulies ha may arise in enforcing feasibiliy and convergence properies o he MPC scheme. Afer his quick overview, he firs example analyzes he behavior of he conrolled sysem in erms of acual violaion of one single chance consrain a a single ime insan, wih he goal of showing he effeciveness of he various approaches in reaching he desired level of relaxaion. The second example is a emperaure conrol problem for a small four-room aparmen and he goal is o show he behavior of he seleced SMPC conrollers in erms of violaion of he consrains over ime. In boh he examples he algorihms are compared in erms of conservaism (acual violaion frequency) and compuaion ime. Finally he hird example is he more complex problem of conrolling he emperaure inside a realisic model of a building. In paricular, he model of a real building is exploied ogeher wih real measuremens of he disurbance (he exernal emperaure) colleced from a meeorological company. In his example ogeher wih a classical nominal MPC algorihm and a robus wors-case approach, only he p-smpc approach proposed in Chaper 3 and he av- SMPC in [75] are seleced o be implemened hanks o heir efficiency and ease of implemenaion. Resuls are presened wih he main goal of showing he advanages of such sochasic MPC approaches. In Chaper 5 he sae-feedback p-smpc approach is exended o he case of discree-ime linear sysems wih addiive disurbances, boh on he sae and he oupu, and non measurable saes and he oupu-feedback p-smpc algorihm is presened. In paricular a sae observer is inroduced in he sochasic conrol scheme whose gain is seleced, ogeher wih he feedback conroller gain, by solving a proper MPC problem. Individual chance consrains inside such problem are handled by means of a second order descripion of he sysem variables and hrough he use of he Canelli- Chebyshev s inequaliy in case of disurbances wih unknown disribuion. The choice of proper erminal consrains for boh he expeced value and 4

15 hesis 2015/11/21 9:18 page 5 #15 he variance of he sae of he sysem and an inializaion sraegy similar o he one adoped in Chaper 3 allow o guaranee recursive feasibiliy and convergence properies even in he case of unbounded uncerainies. Moreover, he main implemenaion issues are discussed and a way of reformulaing consrains as LMI is proposed. Finally wo simulaion examples are shown o prove he efficacy of he algorihm. In Chaper 6 he problem of designing a disribued sochasic MPC conroller for dynamically coupled sysems and local probabilisic consrains is discussed. Wih respec o classical robus disribued predicive conrol echniques he aim is o cope wih disurbances wih possibly unbounded suppor. To his end, he p-smpc approach inroduced in Chaper 3 is adaped o allow for a disribued implemenaion and, in paricular, he recursive feasibiliy and convergence guaranees are exended by means of similar condiions on he iniializaion of he algorihm. A simple example shows he efficacy of he proposed approach. In Chaper 7 a disribued conrol scheme for racking reference signals is presened ha is based on well-known robus MPC echniques o ensure he coordinaion of he agens. The conrol scheme is composed by hree differen levels, namely an oupu feedback rajecory generaion level, a sae and inpu rajecory generaion level and a racking MPC level ha ake advanage of he exchange of informaion beween subsysems o achieve local goals while saisfying collecive consrains and compensaing for he coupled dynamics. Feasibiliy and convergence properies are proven by means of wors-case consideraions. In he end, a simulaion example and a real experimen are presened o show he efficacy and he flexibiliy of he approach. In paricular, he real example is a mobile robo coordinaion problem in which he single robos are required o reach heir own final goals while avoiding fixed obsacle and collisions. In Chaper 8 he disribued racking problem is exended o he sochasic case wih he aim of using i o solve he mobile robos coordinaion problem inroduced in Chaper 7. To his end, he analysis is limied only o dynamically decoupled subsysems wih local and coupling probabilisic consrains. Firsly, an approach inspired from [127] is quickly described for solving he general problem. Secondly, all he previous resuls are formulaed for he mobile agens coordinaion problem and he collision consrains are handled by exending he approach described in [2] o he probabilisic case. Some simulaion resuls are shown o analyze he sochasic behavior of he conrolled sysem in erms of number of deeced collisions. Finally some resuls from a real coordinaion example are shown o prove he efficacy of he proposed approach. 5

16 hesis 2015/11/21 9:18 page 6 #16 Chaper 1. Inroducion In Chaper 9 some conclusion on he overall work presened in his Thesis are drawn and possible fuure direcions are discussed. 6

17 hesis 2015/11/21 9:18 page 7 #17 CHAPTER2 Sae of he ar of Sochasic MPC The problem of designing robus deerminisic MPC schemes has nowadays many soluions, see for example [83, 139]. However, he available approaches are in general compuaionally very demanding, since hey eiher require he soluion o difficul on-line min-max opimizaion problems, see [94], or he off-line compuaion of polyopic robus posiive invarian ses, see [101]. In addiion hey are conservaive, mainly because hey implicily or explicily rely on wors-case approaches. Indeed, even if he uncerainies are characerized as sochasic processes, wors-case deerminisic mehods do no ake advanage of he available knowledge on he characerisics of he process noise, such as heir probabiliy densiy funcion, and canno even guaranee recursive feasibiliy in case of possibly unbounded disurbances. To overcome hese limiaions, an emerging field of research concerns he design of innovaive Sochasic Model Predicive Conrol (SMPC) algorihms, aimed a exploiing he sochasic naure of he uncerainy and, when available, is saisical descripion. In his framework, hard consrains on he sysem variables have o be reformulaed as sochasic ones, allowing he conrolled sysem o violae hem in prescribed probabilisic erms. In his scenario, i is possible o consider also unbounded disurbances and/or uncerainies, for example in case hey 7

18 hesis 2015/11/21 9:18 page 8 #18 Chaper 2. Sae of he ar of Sochasic MPC are characerized by a Gaussian disribuion. In spie of he large number of applicaions of SMPC (see Chaper 1), he classificaion of he many available SMPC algorihms can be quie difficul due o he large variey of problem formulaions and soluions. For example, design mehods have been developed for linear or nonlinear, discree-ime or coninuous-ime sysems, wih addiive, muliplicaive or parameric uncerainies, finie or infinie horizon cos funcions, polyopic, quadraic or more complex probabilisic consrains. Also, linear sysems wih known sae have generally been considered, wih he noable excepions of [24, 65, 157], where oupu feedback mehods have been proposed. For hese reasons, he goal of his chaper is o presen he mos widely used problem formulaions, wih paricular emphasis on he definiion of sae and conrol consrains in probabilisic erms, on he cos funcion o be minimized, and on he srucure of he adoped conrol law. Then, we propose a classificaion of he available mehods based on he sysem s assumpions, he adoped MPC formulaion, and he feasibiliy and convergence properies. A final secion of conclusions closes he analysis and gives he moivaion o he algorihms proposed in he nex chapers. Before coninuing, a noe on he erminology is due. Here we denoe wih he expression hard consrain he ideal consrain ha one wans o impose in a deerminisic framework while we use he erm sochasic consrain o denoe in general is relaxed version obained allowing a parial violaion, in a sense ha will be clarified laer, of he original requiremen. However, he work will be focused on a subse of hese sochasic consrains denoed as probabilisic, or chance, consrains wih he meaning ha he relaxaion is expressed in erms of maximum allowed probabiliy of violaing he requiremen. 2.1 Sochasic sysems and consrains In he mos general case assume ha he sysem under conrol is described by he following discree-ime, nonlinear model x +1 = f(x, u, w ) (2.1) In (2.1) x R n, u R m, w W R nw are he sae, inpu, and sochasic noise vecors, respecively, which mus saisfy, a leas ideally, a se of consrains described in very general erms by he inequaliies g(x, u, w ) 0 (2.2) 8

19 hesis 2015/11/21 9:18 page 9 # Sochasic sysems and consrains where he funcion g( ) : R n m nw R r, can ake differen forms, as specified in he following. In paricular, depending on he noise characerisics, hese consrains can be saisfied deerminisically, or, in order o consider ha heir deerminisic fulfillmen can be oo igh or even impossible due o he presence of he sochasic noise, a parial violaion is allowed. Many SMPC algorihms have been developed wih specific reference o linear sysems wih addiive or muliplicaive uncerainy. In he case of addiive uncerainy, he adoped model is x +1 = Ax + B u u + B w w (2.3) where he noise erm has he role of a real disurbance acing on he sysem or can be used o represen an unmodeled dynamics. Sysems wih muliplicaive uncerainy are described by he model q x +1 = Ax + B u u + [A j x + B j u ]w j (2.4) and heir use is widely popular in specific applicaion fields, such as in financial applicaions, see e.g., [38,129], where sock prices and he porfolio wealh dynamics are represened as in (2.4). Before going ino deails, i is easy o see ha, in view of he sochasic naure of he noise w, he inclusion of hard consrains (2.2) in he problem formulaion can lead o infeasibiliy. For insance, when he uncerainy acing on he sysem has an unbounded suppor, i.e. he se W is unbounded as in he case of a Gaussian noise, here is no way o ensure hard consrains on he sae variable. Moreover, even if he uncerainy is bounded, he worscase scenario ha one has o consider o ensure he saisfacion of he hard consrains migh be oo conservaive, and performances of he obained soluion could be increased by resoring o a sochasic reformulaion of (2.2). In paricular, hree ypes of sochasic consrains are usually encounered in he lieraure: j=1 Expecaion consrains E[g(x, u, w)] 0 Probabilisic or chance consrains P {g(x, u, w) 0} 1 p Inegraed chance consrains P {g(x, u, w) s} ds p 0 9

20 hesis 2015/11/21 9:18 page 10 #20 Chaper 2. Sae of he ar of Sochasic MPC where P {ϕ} is he probabiliy of ϕ and p is a design parameer o be uned in order o obain a rade-off beween performances and consrain violaion. The use of expecaion consrains a he place of deerminisic ones represens he simples soluion and amouns o ensure ha he consrains are saisfied on average for he considered problem. A noable example is [130], where i is required ha he expeced value of quadraic funcions of sae and inpu variables respecs given bounds. In his way, however, he number of occurred violaions or he amoun of he single violaion are no conrolled direcly. To his end, one can hink abou exending he formulaion considering a uning parameer, for example requiring ha E[g(x, u, w)] α for some parameer α whose choice is however difficul and somehow arificial. On he oher hand, probabilisic consrains are commonly adoped in view of he fac ha many problems can be naurally formulaed using his framework, for example every ime he violaion of cerain consrains up o a specified frequency is allowed. In his formulaion, however, he consrain is no always convex and furher approximaions are needed o use i wihin an MPC scheme, ha will be discussed in deail in he sequel. Wih respec o he previous soluion, noice ha he probabilisic consrains can be inerpreed as he expeced value of he indicaor funcion of he even consrains saisfied. Finally, inegraed chance consrains [58] [59], are a useful ool o express he idea of consrain violaion in a more quaniaive way. Roughly speaking, in his formulaion he consrain violaion is allowed wih high probabiliy if he amoun by which i is violaed is small enough. For a more deailed explanaion of he consrain models and a clear analysis of heir effecs on he problem, he reader is referred o, e.g., [15, 33]. To simplify he seup, in his Thesis focus will be placed on he second class of consrains, he so-called probabilisic, or chance, consrains. However, based on he form of g( ), anoher disincion is due, ha is widely used in he lieraure. In paricular, when g(x, u, w) is a vecor, for example when he goal is o express he probabiliy ha he sae and/or he conrol are inside a cerain se, he consrain is called join chance consrain. On he oher hand, when g(x, u, w) is a scalar funcion, he consrain is addressed 10

21 hesis 2015/11/21 9:18 page 11 # Sochasic sysems and consrains as individual chance consrain. Even if he join represenaion seems o be more naural, is exac racable represenaion usually does no exis, despie he convexiy of he consrain iself. This is because he mere evaluaion of he consrain requires he compuaion of a mulivariae inegral, which is known o become prohibiive in high dimensions. For his reason, from a pracical poin of view, join chance consrains need o be approximaed o obain a racable expression. A clear overview of he problem can be found for example in [15], [110] and in he references herein. Besides he use of confidence ellipsoids or sampling echniques ha will no be considered in his work, he simples way o work wih a join chance consrain is o approximae i by spliing he overall se ino a sequence of individual chance consrains, whose single probabiliies sum up o he original one, as described in [14]. In a more formal way, if g(x, u, w) = [g 1 (x, u, w),..., g r (x, u, w)] T, he join chance consrain can be rewrien as follows { r } P {g(x, u, w) 0} = P g i (x, u, w) 0 1 p i=1 or, in oher words, he probabiliy ha a leas one of he single violaion occurs needs o be less han p, i.e., { r } P g i (x, u, w) > 0 p i=1 Applying he Boole s inequaliy o he las expression we obain he following bound { r } r P g i (x, u, w) > 0 P {g i (x, u, w) > 0} i=1 and hus one can choose a se of r parameers p i such ha i=1 P {g i (x, u, w) 0} 1 p i i = 1... r, r p i = p Due o he fac ha r i=1 p i = p, he above equaions clearly give a conservaive approximaion of he original consrain. Based on ha, an easy choice can be o equally subdivide he overall risk p seing p i = p/r, i = 11 i=1

22 hesis 2015/11/21 9:18 page 12 #22 Chaper 2. Sae of he ar of Sochasic MPC 1,..., r or o choose he single risks offline based on same heurisics. However, if a less conservaive soluion is required, he approximaion can be reduced including he values p i as free variables in he opimizaion problem. This ieraive risk allocaion echnique is discussed in [120] and [15] for he case of Gaussian uncerainy. In he res of his work we will assume ha consrains are already given in he form of muliple individual chance consrains o ease he seup. Before coninuing in he analysis, a remark is due. Noe ha, once he model for he consrains has been defined, for example chosen from he ones presened above, we are sill dealing wih a sochasic programming problem ha is generally difficul o solve in is original version and hus proper reformulaions and/or approximaions based on he available descripion of he uncerainy are needed o make i easier o implemen. Togeher wih his, we need o sress he fac ha, when dealing wih MPC schemes, he simple applicaion of he sochasic consrains o he fuure predicions of he sysem variables is in general no sufficien o guaranee he recursive feasibiliy of he algorihm. Boh hese facs will be clarified in he sequel and an overview of he echniques used in he lieraure o enforce feasibiliy will be discussed in a simple case. 2.2 Sochasic MPC, formulaions and properies. Once he model used o represen he sochasic sysem has been chosen and he sae and conrol consrains have been properly reformulaed as discussed in he previous secion, he MPC opimizaion problem can be saed by defining a suiable cos funcion ogeher wih addiional consrains which can be added o achieve recursive feasibiliy and sabiliy properies. Then, specific algorihms can be developed according o differen approaches. I is possible o roughly cluser he differen sochasic MPC mehods nowadays available in wo main classes: he firs one, i.e. he so-called analyic mehods (referred in [163] as probabilisic approximaion mehods), is based on he reformulaion of probabilisic-ype consrains and of he cos funcion in erms of variables whose behavior can be characerized in deerminisic erms (e.g., mean values and variances), o be included in he MPC formulaion. The second class of approaches relies on he randomized, or scenario-based mehods, i.e., on he on-line random generaion of a sufficien number of noise realizaions, and on he soluion o a suiable 12

23 hesis 2015/11/21 9:18 page 13 # Sochasic MPC, formulaions and properies. consrained opimizaion problem based on hese scenarios. The main feaures of hese mehods will be now discussed Cos funcion In a deerminisic framework, he MPC cos funcion J N is usually seleced o weigh, over a finie number of seps defined by he predicion horizon N, a sage cos l(x, u), plus a cos relaed o he sae a he end of he horizon, l f (x). Since in a sochasic framework, he sae and possibly he conrol variables are random processes, J N is iself a random variable ha depends on he uncerainy affecing he sysem. This makes he derivaion of he cos funcion o be minimized in a probabilisic seup arbirary, o some exen. Some of he possible choices aken in he lieraure are lised below. The mos commonly used cos funcion in a sochasic framework is he following +N 1 J = E[ l(x i, u i ) + l f (x +N )] (2.5) i= where he expecaion is aken over he disribuion of he disurbance. In he analyic framework, e.g., [96, 130], he cos funcion above can be reformulaed as a funcion of he mean value and he variance of he sysem variables by selecing l(x, u) = x 2 Q + u 2 R and l f (x) = x 2 P. Then, defining E[x i] = x i, E[u i ] = ū i, X i =var(x i ), and U i =var(u i ), one can wrie E[l(x i, u i )] = E[ x i 2 Q + u i 2 R] = x i 2 Q + ū i 2 R + r(qx i + RU i ) (2.6a) E[l f (x +N )] = E[ x +N 2 P ] = x +N 2 P + r(p X +N ) (2.6b) An alernaive o (2.5) can be found by resoring o he cerainy equivalence principle (see, e.g., [5, 143, 144, 160]). More specifically, we can define he cos funcion as he deerminisic one Ĵ = +N 1 i= l(ˆx i, u i ) + l f (ˆx +N ) (2.7) where he nominal sysem rajecory ˆx i i =,..., + N is obained using he updae equaion ˆx i+1 = f(ˆx i, u i, ŵ i ) wih iniial condiion 13

24 hesis 2015/11/21 9:18 page 14 #24 Chaper 2. Sae of he ar of Sochasic MPC ˆx = x and where ŵ i, i =,..., + N 1, is a nominal disurbance rajecory, e.g., defined as he expeced value or he opimal predicor of w i. In a scenario-based framework (see, e.g. [4, 5, 128, 144, 160]), a sampled average over N s noise realizaions can be considered a he place of (2.5), i.e., J 1 N s N s k=1 J [k] (2.8) where +N 1 J [k] = l(x [k] i, u i ) + l f (x [k] +N ) (2.9) i= and where, denoing by w [k] i i =,..., + N 1 he k-h noise realizaion (wih k = 1,..., N s ), he rajecory x [k] i i = + 1,..., + N is compued using he updae equaion x [k] i+1 = f(x[k] i, u i, w [k] i ) wih iniial condiion x [k] = x, i.e., he measured sae. In he conex of scenario-based SMPC, a wors-case opimizaion procedure can be also employed. For example, in [20, 144] he following cos funcion is minimized J max = max (J [k] ) (2.10) k=1,...,n s in which he wors-case cos is aken from all he exraced realizaion Conrol sraegies In he following, o simplify he seup as much as possible, focus will be placed on linear sysems of he ype (2.3) wih addiive zero-mean whie noise wih bounded suppor. In view of he superposiion principle, i is always possible, a ime, o wrie he fuure evoluion of he sae variable as he sum of wo componens, x +i = x +i + e +i, wih x = x, e = 0, and where x +i evolves independenly of he noise w +i, while e +i depends (linearly) jus on he evoluion of exogenous variable w +i. Defining he sequence of possible disurbances along he horizon, w = [ w T... w T +N 1] T W N, i is 14

25 hesis 2015/11/21 9:18 page 15 # Sochasic MPC, formulaions and properies. possible o wrie e +i = E i w (2.11) where E i is a suiable marix represening he effec of he noise on he uncerainy of he evoluion of he sae variable, and in urn on he reliabiliy of he predicion given by x +i. In he following we will highligh ha E i can assume differen values depending on he adoped conrol sraegy. More specifically, we will describe how E i depends on he chosen conrol law and is degrees of freedom, similarly o he discussion given in [107]. Open loop conrol Some approaches (e.g., [17, 60, 120]) require ha, a ime, he candidae conrol sequence u,..., u +N 1 o be applied o he sysem (2.3) is compued direcly as a resul of he opimizaion problem. This means ha u,..., u +N 1 are independen of w and are deerminisically defined as a funcion of he curren sae x or, in oher words, we can wrie u +i = ū +i. Therefore, he inpu sequence is defined as u = [ ū T... ū T +N 1 ] T. Wih his choice, he evoluion of he deerminisic sae x +i is described by x k+1 = A x k + B u ū k (2.12) for k =,..., + N 1, while he open loop evoluion of perurbed componen of he sae variable is e k+1 = Ae k + B w w k for k =,..., + N 1. In his case, i follows ha he marix E i in (2.11) corresponds o he i-h row-block of he marix B w AB w B w A N 1 B w A N 2 B w... B w I is clear ha, in his case, he variance of e +i (i.e., he uncerainy on he evoluion of he sae variable) evolves in an unconrolled fashion. Especially in case he sysem is unsable, his approach has significan drawbacks, since i may induce serious feasibiliy problems. 15

26 hesis 2015/11/21 9:18 page 16 #26 Chaper 2. Sae of he ar of Sochasic MPC Disurbance feedback conrol The disurbance-feedback approach is employed in differen works, see, e.g., [36, 76, 128, 160]. In his case, exploiing he fac ha our predicion are made in a closed-loop fashion, he inpu sequence u is defined as a funcion of he disurbance sequence w. In paricular, informaion on he sae along he horizon are capured indirecly by he disurbance since, once he inpu is known, i is always possible o recover w +i from x +i+1 and x +i. The mos common choice corresponds o he affine feedback case, where i is se u = c + Θ w (2.13) Here boh c and Θ are opimal values, compued a ime, of he degrees of freedom c = [ c T... c T +N 1] T and θ +1, Θ = θ +2, θ +2, θ +N 1, θ +N 1,+1 θ +N 1, Noe ha, for causaliy reasons, Θ is se o be a lower-block riangular marix wih zero diagonal blocks. Moreover, since he number of free variables grows quadraically wih he horizon N, i is common o see reduced paramerizaions for Θ. Also in his case, he evoluion of he deerminisic sae x +i is described by (2.12) for k =,..., + N 1, where he deerminisic inpu is equivalen o ū k = H k+1 c and H i R n nn is he marix selecing he i-h vecor elemen from c. The evoluion of he perurbed componen of he sae variable is e k+1 = Ae k + B u H i Θ w + B w w k for k =,..., + N 1. In his case, i follows ha he marix E i corresponds o he i-h block-row of he marix 16

27 hesis 2015/11/21 9:18 page 17 # Sochasic MPC, formulaions and properies. B w B u AB w B w AB u B u Θ A N 1 B w A N 2 B w... B w A N 1 B u A N 2 B u... B u From he laer i is clear ha, a he opimizaion level, he choice of Θ can grealy reduce he effec of he noise sequence w on e +i, and evenually is variance. Sae feedback conrol Sae feedback approaches include, e.g., [25,27 29,77]. In his case, he inpu variable u +i is defined as a funcion of x +i. Slighly differen versions of sae feedback conrol laws have been proposed in he lieraure. More specifically in [96, 130] i is se u +i = ū +i + K +i (x +i x +i ) (2.14) where x k evolves according o (2.12) for k =,..., + N 1. In his case, he resuls of he opimizaion problem are he sequence of gains K +i, and he sequence of open-loop erms ū +i, for i = 0,..., N 1. The evoluion of perurbed componen of he sae variable is e k+1 = (A + B u K k )e k + B w w k for k =,..., + N 1. Defining Φ k = A + B u K k, i follows ha he marix E i corresponds o he i-h block-row of he marix B w Φ +1 B w B w Φ +N... Φ +1 B w Φ +N... Φ +2 B w... B w (2.15) As i is discussed in [55], i is possible o obain an equivalen disurbance feedback formulaion, wih fewer degrees of freedom wih respec o he ones in marix Θ k. I is worh menioning, however, ha boh formulaions resul in convex problems, see e.g., [130]. 17

28 hesis 2015/11/21 9:18 page 18 #28 Chaper 2. Sae of he ar of Sochasic MPC in, e.g., [24, 27, 28, 77], he conrol law is u +i = ū +i + Kx +i (2.16) Here he resul of he opimizaion problem is only he sequence of erms ū +i, for i = 0,..., N 1, while he gain K is fixed offline. The evoluion of he deerminisic sae x +i is described by x k+1 = Φ x k + B u ū k (2.17) for k =,..., + N 1, where Φ = A + B u K and he marix E i corresponds o he i-h block-row of he marix (2.15), where we se Φ k = Φ for all k. In boh cases, i is clear ha a proper choice of he conrol gain (which can be an opimizaion variable or a design parameer as in [27]) can reduce he effec of he noise sequence w on e +i, and evenually is variance. This resuls in a larger feasibiliy region wih respec o he case of open loop soluions (especially when he sysem is unsable) Reformulaion of sae consrains In his secion, o simplify he seup as much as possible, individual linear chance consrains on he sae will be considered, while he possible presence of deerminisic or sochasic consrains on he conrol variable u will be horoughly discussed in he sequel. Assume ha, a ime, he goal is o impose he following sae consrain in he nex i 1 predicion seps P { g T x +i h } 1 p (2.18) where g R n, h R are fixed and x = x. As discussed in Secion 2.2.2, in view of (2.11) we can wrie (2.18) as P { g T ( x +i + E i w ) h } 1 p (2.19) and wo main approaches are available for enforcing (2.19). Firs we discuss how his issue is approached by analyic mehods, hen he scenariobased ones will be considered. 18

29 hesis 2015/11/21 9:18 page 19 # Sochasic MPC, formulaions and properies. Analyic reformulaion of sae consrains In general, in an analyic framework, we can guaranee (2.19) by verifying he following consrain g T x +i h q i (1 p) (2.20) where he consrain ighening level, q i (1 p), can be characerized as discussed below, based on he available informaion regarding he noise sequence w, is bounds, and is properies. In paricular, consider he following cases. Requiring no consrain violaion is equivalen o impose (2.19) wih p = 0. This corresponds o he wors-case ighening adoped in he deerminisic framework, and amouns o seing q i (1) = max w W N g T E i w (2.21) In his case, i is easy o undersand ha (2.20) may admi a soluion only under he assumpion ha W is bounded. In case a non-zero probabiliy of violaion is allowed, i is possible o use he knowledge on he disribuion of he noise, if available. Following, for example, he approach proposed in [77, 87], he erm q i (1 p) can be compued as q i (1 p) = arg min q q, s.. P { g T E i w q } = 1 p (2.22) In general, he previous expression canno be compued analyically since i involves he evaluaion of a mulivariae convoluion inegral. I is hen necessary o approximae i numerically, for example by discreizing he disribuions of w or using sample-based approaches. However, if he value of p and he shape of he consrains are fixed in he problem, his compuaion can be performed off-line only once and wih an arbirary precision. Under he assumpion ha he noise is unbounded, Gaussian wih expeced value w = 0 and covariance marix W, also he variable e +i is Gaussian and he exac consrain ighening can be compued analyically (e.g., in [17, 120]). Specifically, q i (1 p) = g T E i WEi T g N 1 (1 p) (2.23) where W =diag(w,..., W ) and N is he cumulaive probabiliy funcion of a Gaussian variable wih zero mean and uniary variance. 19

30 hesis 2015/11/21 9:18 page 20 #30 Chaper 2. Sae of he ar of Sochasic MPC Finally, if he disribuion of he noise is no specified, bu is expeced value w and covariance marix W are known, i is possible o resor o he Canelli-Chebyshev inequaliy [97] ha is repored in he following lemma Lemma 1. (Canelli s inequaliy) Le y be a (scalar) random variable wih mean ȳ and variance Y. Then for every R α 0 i holds ha P(y y + α) Y Y + α 2 (2.24) Recalling ha (wihou loss of generaliy) we are now considering w = 0, i is possible o use he approach proposed in [96]. In paricular, consider he consrain in (2.18) and assume ha he following expression holds for some δ 0 g T x +i h δ (2.25) Thus, by he use of (2.25) and (2.24) we have ha P{g T x +i h} P{g T x +i g T x +i + δ} gt E i WE T i g g T E i WE T i g + δ2 where, again, W = diag(w,..., W ). Therefore, he original inequaliy (2.18) is saisfied by imposing ha in urn can be rearranged as g T E i WEi T g p (2.26) g T E i WEi T g + δ2 g T E i WE T i g( 1 p p ) δ2 (2.27) Now, combining (2.25) and (2.27), he probabilisic consrain (2.18) is approximaed by (2.20) where a bound for q i (1 p) is provided as 1 p q i (1 p) g T E i WEi T g (2.28) p Noe ha in (2.23) and (2.28) he erm q i (1 p) has he same shape if we define a funcion f(1 p) ha is f(1 p) = N 1 (1 p) 20

31 f(1! p) hesis 2015/11/21 9:18 page 21 # Sochasic MPC, formulaions and properies. whenever he uncerainy is Gaussian and is bounded by 1 p f(1 p) = p when he disribuion of he uncerainy is no known. Of course, he choice (2.28) is in general conservaive. For example, in Figure 2.1 we show he comparison beween he values of f(1 p) in (2.23) and (2.28) p 1 p p Figure 2.1: Comparison beween he values of f(1 p) = f(1 p) = N 1 (1 p) (doed line). (solid line) and From his discussion is clear ha he choice of he marix E i (which in urn depends on he adoped conrol sraegy, see Secion 2.2.2) grealy affecs he size of q i (1 p), which can be minimized o evenually enhance he feasibiliy properies of he SMPC-based conrol scheme. Reformulaion of sae consrains in a scenario framework In scenario-based sochasic MPC mehods, consrains are verified for a finie number of sampled deerminisic predicion sequences, compued on he basis of proper exracions of he uncerainy. The number of he exraced samples is carefully seleced o ensure a prescribed level of consrain violaion. 21

32 hesis 2015/11/21 9:18 page 22 #32 Chaper 2. Sae of he ar of Sochasic MPC Specifically, recalling (2.11), he probabilisic consrain (2.18) is guaraneed by enforcing, a any ime insan i along he predicion horizon, he following collecion of deerminisic consrains g T ( x +i + E i w [i,k i] ) h (2.29) for all k i = 1,..., N s,i, where w [i,k i] is he k i -h noise realizaion relaive o he i-h consrain, and where i is assumed ha w [i,k] is independen of w [j,h] for all i j or h k. The number N s,i of noise realizaions mus be carefully chosen in order o have precise sochasic guaranees on he soluion. More in deail, denoe by d i he number of variables involved in he opimizaion problem referred o he i-h predicion. The probabiliy ha g T x +i > h is smaller han p wih a given confidence level β is (see e.g. [20, 144]) d i 1 k=1 ( Ns,i The value of N s,i can hus be chosen so as o saisfy k ) p k (1 p) N s,i k β (2.30) N s,i d i ln(1/β) + 2(d i + 1) ln(1/β) p (2.31) Noe ha his number is mainly affeced by he dimension of he problem (d i ) and by he choice of he parameer p while i is only marginally increased by he confidence level β, due o he logarihmic dependence, ha hus can be chosen o be very small. Moreover, i may happen, especially when he required number of samples, N s,i is low, ha he soluion o he problem is oo conservaive due o unlucky exracions of he uncerainy used o reformulae he consrain. To avoid his siuaion, i is possible o implemen ieraive sample-removal algorihms, which are able o deec he more sringen consrains o be ignored. In his case, he expression in (2.30) is no valid anymore and a differen procedure, described for example in [23,144] is required o compue boh he number of samples N s,i and he number of consrains N r,i o be removed given he probabiliy p and he confidence β. Of course, due o he removed samples, he number of exracions ha are needed o approximae he chance consrain correcly is increased. An advanage of using his removal echnique is ha i is easier o reach he level of relaxaion required in he problem, while he main drawback is relaed o he increase of he compuaional complexiy due o boh he increased number of samples and o he removal procedure iself. 22

33 hesis 2015/11/21 9:18 page 23 # Sochasic MPC, formulaions and properies. Finally noe ha he equaion (2.29) applies when open loop predicions are performed a ime. In his case, in principle, o enforce (2.19) for each considered fuure insan i = 1,..., N, one should exrac a number N s,i of independen realizaions, resuling in a a oal number of realizaions required a ime of N i=1 N s,i. Recall ha he number of independen opimizaion variables d i (and hence N s,i ) for he single consrain increases wih i. This, however, resuls unnecessary in a closed-loop receding-horizon conrol seing, as horoughly discussed and proved in [144], for more deails see Secion Consrains on inpu variables Conrarily o he case of sae consrains, here is no general consensus in he SMPC lieraure on inpu consrains and sauraions. More specifically, he issue is weher o reformulae inpu consrains as probabilisic ones (as i is generally done in he case of sae consrains) or no. In a pracical conex, indeed, i is desirable ha inpu variables lie in specified ranges; however, his requiremen is no always compaible wih he adoped conrol sraegy and wih recursive feasibiliy requiremens. Considering open loop conrol sraegies, since he inpu sequence ū +i is deerminisically defined a ime, hard bounds on i could be enforced regardless of he eniy of he disurbance (boh in he bounded and in he unbounded case). On he oher hand, when sae feedback and disurbance feedback policies are adoped, he values aken by u +1,..., u +N 1 depend on w,..., w +N 2, and herefore sauraion consrains can be imposed a ime a mos only on u in case he noise is unbounded. Of course, if he noise suppor is bounded, hard consrains can be enforced on he whole inpu rajecory a he price of applying robus wors-case argumens. Consisenly wih his discussion, hard consrains are assumed on he inpu variables in wo cases: (i) when he noise is bounded, e.g., [27, 76]; (ii) when recursive feasibiliy requiremens are relaxed, e.g., [128, 144]. A noable excepion is discussed in [65], where a nonlinear (oupu) feedback policy is adoped; in his case, he use of bounded nonlinear funcions of y +i ŷ +i (where y is he sysem oupu and ŷ is he sysem oupu nominal predicion), ogeher wih he assumpions ha he sysem is sable and ha sae consrains are absen, allows hard bounds on inpus o be enforced a any ime insan and o guaranee recursive feasibiliy. In he sraegy presened here, consrains on inpu variables mus be formulaed in general as probabilisic ones. Some excepions will be laer discussed, e.g., when he sysem is sable and he degrees of freedom on 23

34 hesis 2015/11/21 9:18 page 24 #34 Chaper 2. Sae of he ar of Sochasic MPC K,..., K +N 1 are reduced Feasibiliy and convergence properies In he conex of sochasic MPC he problem of guaraneeing recursive feasibiliy is sill largely open. Some noable soluions are available (for boh sae feedback and disurbance feedback approaches, see e.g., [25, 27, 74, 76]), under he assumpion ha he disurbances are bounded. They are based on imposing suiable mixed probabilisic/wors-case consrain ighening o equaion (2.20) (or, in oher words, by amplifying he erm q i (1 p)) for i > 1 by accouning for bounded ses where he sae can evolve due o he bounded noise affecing he sae equaion. The more general case of unbounded noise poses more difficulies and some specific soluions and reformulaions of hese properies have been adoped; for example in [28,29] he concep of invariance wih probabiliy p is used, while in [117] he definiion of probabilisic resolvabiliy is inroduced. In some noable works (e.g., [20,60,144]), including he scenario-based mehods, he feasibiliy of he opimizaion problem is assumed a each ime sep, and possibly enforced, e.g., by reformulaing he problem consrains in a sof fashion. Concerning sabiliy and convergence resuls, while in case of bounded deerminisic disurbances pracical sabiliy (i.e., convergence in a neighborhood of he origin) can be esablished, in he sochasic framework mean square sabiliy resuls are generally addressed, wih he noable excepion of [60], where convergence of he mean value of x(k) o zero is proven. Indeed, in [24,27,65,87], and in general in case of addiive noise, i can be proven ha lim E[ x k 2 ] = lim ( E[x k ] 2 + var(x k )) cons (2.32) k k This means ha he sae of he sysem is driven o a neighborhood of he seady sae condiion (whose dimension depends on he ampliude of he inpu noise and on he adoped conrol policy). On he oher hand, when modeling uncerainies are described (e.g., for linear sysems wih muliplicaive uncerainies [28, 41, 130]), poin-wise convergence can be obained, e.g., ha lim k E[ x k 2 ] = 0. Noe ha, for simpliciy in he following summary he propery (2.32) will be denoed mean square convergence and he consan value will be possibly equal o zero. 24

35 hesis 2015/11/21 9:18 page 25 # Sochasic MPC, formulaions and properies. LINEAR NONLINEAR [85], [56] ADDITIVE NOISE MULTIPLICATIVE NOISE UNCERTAINTY BOUNDED UNBOUNDED [24], [87], [76], [7], [115], [143], [75], [24], [107], [74], [27], [78], [79], [26], [96], [112] [28], [41], [9], [25], [29], [8] [16], [104] [116], [42], [16], [154], [18] [159], [161], [113], [163], [161], [60], [45], [36], [148], [128], [117], [99], [90], [65], [164], [33], [31], [147], [118], [64], [63], [66], [15], [120], [10], [14], [157], [152], [6], [151], [150], [82], [145] [126] [81], [70], [130], [28], [34] PARAMETRIC UNCERTAINTY [20], [21] [144], [22], [108] [125] Table 2.1: Classificaion of available SMPC echniques based on sysem assumpions Summarizing ables In his secion we give a schemaic overview of he many differen approaches proposed in he recen lieraure on SMPC. Firs, in Table 2.1 we focus on he sysem assumpions required by he differen algorihms. Secondly, in Table 2.2 we sress, for he available mehods, how consrains are handled, and he informaion available/assumed on he disurbance acing on he sysem. Finally, in Table 2.3 we review he main heoreical properies guaraneed in he differen algorihms. In addiion o he informaion summarized in Tables 2.1, 2.2, 2.3, he following commens are due o highligh he main characerisics of he approaches proposed so far. The main advanage of scenario approaches is ha hey are applicable o wide classes of sysems (linear, nonlinear) affeced by general disurbances (addiive, muliplicaive, parameric, bounded or unbounded) wih consrains of general ype on he inpus, saes, and oupus, provided ha he problem is convex in he opimizaion variables. Their drawbacks come from he compuaional complexiy ha arises from he high number of samples usually required o reach he desired probabilisic guaranees and evenually o he need of sampleremoval sraegies. In addiion, sabiliy and recursive feasibiliy properies are no available a presen, as i can be deduced from a join analysis of Tables 2.2, 2.3. In he wide class of analyical mehods, algorihms wih recursive 25

36 hesis 2015/11/21 9:18 page 26 #36 Chaper 2. Sae of he ar of Sochasic MPC ANALYTICAL KNOWN PDF EXPECTED VALUE STATE CONSTRAINTS TYPE JOINT CC ALL [42], [64], [130] GAUSSIAN [81], [126] [117], [116], [118], [15], [120], [150], [82] SECOND ORDER [163], [164] RANDOMIZED (KNOWN PDF) [159], [161], [144], [115], [20], [36], [143], [22], [128], [99], [41], [16] INDEPENDENT CC [104], [125] [87], [76], [75], [24], [107], [74], [27], [147], [78], [79], [26], [25], [29], [10] [113], [148], [14], [157], [152], [151], [145] [46], [85], [45], [60], [96] Table 2.2: Classificaion of available SMPC echniques based on algorihms srucure. FEASIBILITY IN PROBABILITY EXPECTED VALUE YES [20], [22], [34] [9] NO PROBABILISTIC CONVERGENCE MEAN SQUARE [87], [45], [41], [24], [65], [107], [27], [31], [78], [79], [26], [63], [96], [25], [29] [81], [42], [130], [126] NO [85], [76], [36], [75], [74] [159], [161], [144], [113], [104], [125] [163], [115], [148], [143], [128], [99], [164], [147], [118], [64], [16], [15], [120], [10], [14], [157], [152], [151], [150], [82], [145] [60] [28] [117], [116] Table 2.3: Classificaion of available SMPC echniques based on feasibiliy and convergence properies. 26

37 hesis 2015/11/21 9:18 page 27 # Numerical examples feasibiliy and convergence are already available boh for bounded and unbounded noise. However, in order o rigorously reformulae coss and consrains in an analyical fashion, he mos popular assumpion is ha he model is linear and affeced by addiive or muliplicaive whie noise. A noable excepion is represened by he approaches ha rely on polynomial chaos expansions, e.g., [104, 125, 126]. Thanks o he possibiliy of approximaing general analyic funcions using series of basis funcions, hese mehods handle sysems affeced by parameric uncerainies wih known disribuion; for example, coninuous-ime and discree-ime linear sysems are considered in [125] and [126], respecively, while nonlinear discree-ime sysems are addressed in [104]. Very few algorihms, see [24,65], have been exended o deal wih he oupu feedback case, which sill represens a largely open issue. 2.3 Numerical examples In his secion we show, hrough some simple examples, he effec of he modeling choices discussed in his chaper. In paricular, he firs example compares a robus and a probabilisic ighening procedure for differen disribuions of he noise. To his end, he probabilisic bounds are obained based on he Chebyshev inequaliy as described in Secion The second example, similarly o he firs one, describes he effec of he several conrol policies discussed in Secion Finally, he hird example shows he approximaion we inroduce by spliing a join chance consrain ino several individual chance consrains using risk allocaion procedures Comparison beween sochasic and robus ighenings Consider he following scalar sysem x +1 = ax + u + w (2.33) where 0 < a < 1, w [ w max, w max ], w max > 0, and he measurable sae is consrained as follows x x max (2.34) The limiaions imposed by a deerminisic robus approach (for example in [102]) and a probabilisic mehod, are now compared. For boh he 27

38 hesis 2015/11/21 9:18 page 28 #38 Chaper 2. Sae of he ar of Sochasic MPC algorihms an open-loop conrol law of he form u = ū is considered, where, similarly o equaion (2.12), ū is he inpu of he deerminisic sysem x +1 = a x + bū. In a probabilisic framework, we allow he consrain (2.34) o be violaed wih probabiliy p, i.e., P {x x max } p (2.35) To verify (2.34) and (2.35) he ighened consrain x k x max x mus be fulfilled in boh he approaches where, in case of he deerminisic robus approach [102] x = x RP I = + i=0 a i w max = 1 1 a w max while, having defined w as a sochasic process wih zero mean and variance W, in he probabilisic framework x = x S (p) = X(1 p)/p and X is he seady sae variance saisfying he algebraic equaion X = a 2 X + W, i.e. X = W/(1 a 2 ). Noably, W akes differen values depending upon he noise disribuion. I resuls ha he deerminisic ighened consrains are more conservaive provided ha x S (p) < x RP I, i.e. p > (1 a) 2 b(1 a 2 ) + (1 a) 2 (2.36) Consider now he hree disribuions depiced in Figure 2.2 seing W = w 2 max/b, wih b = 3 for uniform disribuion (A) b = 18 for riangular disribuion (B) b = 25 for runcaed Gaussian disribuion (C) Seing, for example, a = 0.9, condiion (2.36) is verified for p > in case (A), p > in case (B), and p > in case (C). Noe ha, alhough formally runcaed, he disribuion in case (C) can be well approximaed wih a non-runcaed Gaussian disribuion: if his informaion were available, one could use x S (p) = X N 1 (1 p) for 28

39 hesis 2015/11/21 9:18 page 29 # Numerical examples 0 Figure 2.2: Disribuions: uniform (case A, solid line), riangular (case B, dashed line), runcaed Gaussian (case C, doed line). consrain ighening, and in his case x S (p) < x RP I would be verified wih ( ) (1 a 2 )b p > 1 N 0 (1 a) Choice of he conrol sraegy and ighening approaches In his simple example we show he effec of he choice of he conrol sraegies discussed in Secion and of he analyic ighening approaches presened in Secion In paricular, consider he following discree-ime linear scalar sysem x +1 = ax + u + w where a = 0.6 and he disurbance w is a zero mean whie noise wih variance W = 10 2 and such ha w 0.5. Again, o simplify as much as possible he seup, we do no consider inpu consrains while he focus is on a single sae chance-consrain P {gx h} 1 p, 0 wih g = 1, h = 3 and p = 0.2. Predicions are compued wihin an horizon 29

40 ighened consrain hesis 2015/11/21 9:18 page 30 #40 Chaper 2. Sae of he ar of Sochasic MPC of lengh N = 10. Resuls are shown in Figure. 2.3 and 2.4 for hree differen ighening approaches, namely he wors-case in (2.21), he direc quanile esimaion in (2.22) and he use of he Chebyshev inequaliy in (2.28) and applying respecively an open-loop policy, a fixed gain sae-feedback policy and a fixed gain affine disurbance-feedback policy wors-case quanile esimaion chebyshev fuure sep Figure 2.3: Consrain ighening on a scalar example over N = 10 predicion seps. Compued using an open-loop conrol sraegy. I is easy o show ha, besides he choice of he conrol sraegy, he use of a sochasic consrain in place of he deerminisic wors-case one allows one o reduce he ighening level boh in he case of known (quanile) and unknown (Chebyshev) disribuion. This grealy moivaes he use of such approaches inside a Model Predicive Conrol scheme. Moreover, i is clear ha he use of a closed-loop conrol sraegy such as he fixed gain sae-feedback or he fixed gain disurbance-feedback easily overcomes he open-loop soluion, and his happens wihou adding any furher degree of freedom o he policy a he price of possibly relaxing any inpu consrain along he horizon. A more general comparison of he differen conrol policies, obained using (2.22) o compue he ighenings, is shown in Figure Obviously, he ime-varying sae-feedback and disurbance- 30

41 ighened consrain hesis 2015/11/21 9:18 page 31 # Numerical examples wors-case quanile esimaion chebyshev fuure sep Figure 2.4: Consrain ighening on a scalar example over N = 10 predicion seps. Compued using a sae-feedback or an affine disurbance-feedback conrol sraegy ( in his case he same resul is obained). feedback sraegies are boh able o achieve he lowes possible ighening level, due o he fac ha we are explicily compensaing for he effec of he disurbance. However, in his simple example, he difference beween such ime-varying policies and heir simplified version wih fixed gains, is no so crucial and hus one can prefer he consan soluions ha reduces he number of degrees of freedom. This fac will be discussed in he sequel wih respec o he proposed algorihm Join chance consrain approximaion In his example we consider he problem of approximaing a single join chance consrain ino muliple individual chance consrains as described in Secion 2.1. In paricular, consider he following wo independen Gaussian sochasic variables x N ( x, 1) y N (ȳ, 1) The problem is o choose heir expeced values, respecively x and ȳ, so ha he following consrain is saisfied {[ ] [ ]} x 0 P p y 0 31

42 ighened consrain hesis 2015/11/21 9:18 page 32 #42 Chaper 2. Sae of he ar of Sochasic MPC open-loop sae-feedback disurbance-feedback opimized disurbance-feedback opimized sae-feedback fuure sep Figure 2.5: Consrain ighening on a scalar example over N = 10 predicion seps. Comparison beween differen conrol sraegies. wih p = 0.5 (of course, due o he unbounded suppor, we canno have p = 1, i.e., an hard consrain). Following he procedure described in Secion 2.1 he join chance consrain can be approximaed wih wo individual consrains as follows { P {x 0} px P {x 0 y 0} p, p x + p y = 0.5 P {y 0} p y where he erms p x and p y are now design parameers o be chosen such ha p x + p y = p. To his end, we can selec offline wo possible values or we can consider he wo parameers as exra degrees of freedom o be seleced hrough a proper algorihm (risk allocaion). Focusing on he feasible regions of boh he original join chance consrained problem and he approximaed one, resuls are shown in Figure 2.6. In paricular, i is easy o show ha he region corresponding o a single choice of p x and p u has he shape of a recangle and i resuls o be much smaller han he feasible region of he original problem. Moreover, even using a dynamic risk allocaion procedure, i.e. keeping he parameers p x and p y as free, we canno reach he same feasibiliy region of he join chance 32

43 hesis 2015/11/21 9:18 page 33 # Commens consrain, as shown by he doed line. Figure 2.6: Feasible region for he join chance consrain and for he pair of individual chance consrains corresponding o wo differen choices of p x and p y. The poins represen he envelope ha can be obained using risk allocaion. 2.4 Commens In his chaper we discussed he imporance of considering a sochasic Model Predicive Conrol framework o be able o deal wih sochasic, possibly unbounded, disurbances and o reduce he conservaiveness wih respec o classical robus approaches in he bounded case. This is paricularly significan when informaion on he probabilisic disribuion of he disurbances acing on he sysem are available and he problem consrains can be relaxed in a sochasic way, meaning ha a parial violaion can be acceped. A very simple example showed he effec of his change in perspecive. Afer an overview on he commonly used models for uncerain discreeime sysems we focused on linear sysems wih addiive disurbances. Sim- 33

44 hesis 2015/11/21 9:18 page 34 #44 Chaper 2. Sae of he ar of Sochasic MPC ilarly, among he possible ways o express sochasic consrains, we chose o adop he so-called individual chance consrains, whose formulaion appears o be more naural in several conexs like for example he emperaure and energy consumpion conrol in buildings or he power producion planning in presence of renewable energy sources, bu can be applied also o differen problems like robos pah planning and obsacle avoidance. Wih respec o his seup, he main ingrediens of an MPC conrol problem have been analyzed in a sochasic framework. The mos common ways adoped o handle sochasic consrains have been presened and heir applicaion has been discussed under several possible conrol sraegies. Moreover, we have shown how o specify cos funcions and we have summarized he main resuls available in he lieraure for convergence and feasibiliy guaranees. Based on he previous discussion, i is apparen ha here is a need for sochasic MPC algorihms ha are able o provide convergence and recursive feasibiliy guaranees wihou exploiing wors-case argumens. In paricular, his would allow handling disurbances wih a-priori unbounded suppor direcly inside an MPC scheme, wihou specifying any addiional recovery sraegy (of course a he price of some relaxaion). Togeher wih his, here is also need for oupu-feedback conrol sraegies ha share he same guaranees in erms of convergence and feasibiliy. Moivaed by his analysis, in he sequel an analyic SMPC algorihm (denoed as p-smpc) will be presened, boh for he sae-feedback and he oupu-feedback case, ha is able o give he desired guaranees a he price of relaxing hard inpu bounds ino probabilisic requiremens. Wih respec o sample-based soluions, he proposed algorihm is much less flexible, in he sense ha i can be used only in he case of linear sysems wih addiive disurbance and individual chance consrains, bu he compuaional load is similar o he one of a sandard deerminisic algorihm. 34

45 hesis 2015/11/21 9:18 page 35 #45 CHAPTER3 Sae-feedback probabilisic SMPC In his chaper a novel sochasic MPC echnique is presened for discreeime linear sysems wih addiive disurbances characerized by a possibly unbounded suppor. As horoughly discussed in Chaper 2, he knowledge of he sochasic disribuion of he uncerainy is exploied o reduce he conservaiveness wih respec o well known robus approaches and o consider also possibly unbounded disurbances. The consrains acing on he sysem are muliple individual linear chance consrains boh on he sae and he inpu, bu exensions o join chance consrains can be obained using, for example, risk allocaion echniques. The perfec knowledge of he sysem sae is firsly assumed and a sae-feedback conrol algorihm is derived and compared o oher paradigmaic examples from he lieraure. Resuls are given in erms of mean square convergence and recursive feasibiliy. For linear sysems wih addiive noise, a simple, ye effecive, way o handle probabilisic consrains and o reformulae a probabilisic MPC problem in erms of a deerminisic one has been proposed in [96]. In order o handle uncerainies wih unknown disribuion, his algorihm is based on he use of he Canelli-Chebyshev s inequaliy, as discussed in Secion The 35

46 hesis 2015/11/21 9:18 page 36 #46 Chaper 3. Sae-feedback probabilisic SMPC main weakness of he mehod in [96] is due o he assumpion ha he noise is bounded and o he inclusion in he MPC problem of some quie arificial consrains o guaranee ISS convergence. In his chaper, he algorihm described in [96] is deeply revisied and exended o derive a compuaionally efficien MPC mehod for sysems subjec o possibly unbounded disurbances. Specifically, he algorihm here proposed is characerized by a) A compuaional burden only slighly heavier ha he one required by sabilizing MPC mehods for undisurbed linear sysems b) The possibiliy o consider unbounded noises c) Guaraneed recursive feasibiliy and convergence under mild condiions. These properies are obained by considering a any ime insan he curren expeced value and he covariance of he sae as opimizaion variables, o be properly seleced according o wo alernaive sraegies, and by imposing some consrains o heir value a he end of he predicion horizon. 3.1 Problem saemen Consider, as discussed in he previous chaper, he following discree-ime linear sysem x +1 = Ax + B u u + B w w 0 (3.1) where x R n is he sae, u R m is he inpu and w R nw is a zeromean whie noise wih variance W and a-priori unbounded suppor. A his sage no exra assumpions on he noise disribuion are made. Perfec sae informaion is assumed, ogeher wih he reachabiliy of he pairs (A, B u ) and (A, B w ), where B w BT w = B w. In line wih he discussion in Chaper 2 and due o he possibly unbounded suppor of he disurbance, consrains on he sae and inpu variables of sysem (3.1) are imposed in a probabilisic sense, i.e., a ime he following muliple individual chance consrains are considered P{b T r x +k x max r } 1 p x r, k > 0, r = 1,..., n r (3.2) P{c T s u +k u max s } 1 p u s, k 0, s = 1,..., n s (3.3) 36

47 hesis 2015/11/21 9:18 page 37 # Problem saemen where P(φ) denoes he probabiliy of φ, b r, c s are consan vecors, x max r, u max s are bounds for he sae and conrol variables and p x s, p u s are design parameers ha can be chosen independenly for each consrain. I is also assumed ha he se of relaions b T r x x max r, r = 1,..., n r (c T s u u max s, s = 1,..., n s ), defines a compac se X (U) conaining he origin in is inerior Regulaor srucure For sysem (3.1), he goal is o design wih MPC a feedback conrol law of he form u = ū + K (x x ) (3.4) where he open-loop erm ū and he ime-varying gain K are free parameers ha mus be compued by solving a any ime insan a suiable opimizaion problem. Moreover, in (3.4), x denoes he (possibly condiional) expeced value of he sae x a ime, commonly denoed as x = E{x }. Remark 1. Noe ha, due o he choice of he sae-feedback conrol law (3.4), we are forced o use probabilisic consrains also on he inpus of he sysem, as discussed in Secion 2.2.2, bu we can sill choose he parameers p u s o be sufficienly small o avoid frequen violaions. In order o handle he chance consrains (3.2) and (3.3), he sysem variables, and laer heir fuure predicions, are firsly described in erms of heir expeced value and variance. In paricular, we define he nominal model ha evolves as follows x +1 = A x + B u ū k 0 (3.5) where he erm ū is he same as in (3.4) and is again a degree of freedom of our opimizaion problem. Of course, since E{w } = 0 for all, i is easy o verify ha x +1 = E[x +1 ] and hus he nominal model represens he dynamics of he expeced value of he sae. Saring from his, define he error variable δx = x x (3.6) ha has he meaning of he difference beween he real sae and is expeced value. Moreover, according o (3.1)-(3.5), he predicion of he sae error evolves as δx +1 = (A + B u K )δx + B w w (3.7) 37

48 hesis 2015/11/21 9:18 page 38 #48 Chaper 3. Sae-feedback probabilisic SMPC where again he gains K are degrees of freedom of he opimizaion problem. Due o he fac his error is zero mean, if we denoe he prediced covariance marix of he sae as X, is evoluion can be compued as X +1 = E{δx +1 δx T +1} = (A + B u K )X (A + B u K ) T + B w W B T w (3.8) Noe ha, in general, expression (3.8) is nonlinear when he gains are considered as free variables, however, resoring o sandard manipulaions in marix compuaion, i is possible o rewrie i as an LMI. This fac will be discussed in he sequel Reformulaion of he probabilisic consrains In order o sae an MPC problem efficienly solvable on-line, he probabilisic consrains (3.2) and (3.3) are now ransformed in deerminisic, alhough igher, ones using he expressions in (3.5) and (3.8) and resoring o he resuls presened in Paragraph In paricular, consrain (3.2) is saisfied if he following inequaliy holds b T r x +k x max r b T r X +k b r f(1 p x r), k 0, r = 1,..., n r (3.9) where he variance of he sae, X +k plays he role of he variance of he uncerainy, namely e +k in (2.11) ha appears in (2.28). Moreover, similarly o he discussion in Paragraph 2.2.3, he consan erm f(1 p x r) can be compued as N 1 (1 p x r), where N ( ) is he cumulaive densiy funcion of a normal disribuion, in case he uncerainy (and due o lineariy also he sae and he inpu) is Gaussian, while i can always be bounded by he erm (1 p x r)/p x r, resoring o he Canelli-Chebyshev s lemma [97], in case of unknown disribuions. For furher deails on he procedure or a comparison wih he oher available mehods he reader is referred o Chaper 2. Inequaliy (3.9), ogeher wih (3.5) and (3.8), clearly shows he differen effecs of he erms appearing in conrol law (3.4) and moivaes is srucure. In fac, he open-loop erm ū influences he evoluion of he mean x (recall (3.5)), while he gain K can be chosen o limi he variance X (recall (3.8)) which evoluion is oherwise fixed by he iniial condiion, as horoughly discussed in Secion However, when K is considered as a degree of freedom, he pracical applicaion of he deerminisic consrain (3.9) is sill hampered by is nonlinear dependence on X, so ha a furher 38

49 hesis 2015/11/21 9:18 page 39 # Problem saemen linearizaion sep is useful o finally derive a linear consrain o be included in he MPC opimizaion problem. To his end, leing δ = εx max, where ε [0, 1] is an addiional design parameer, and a he price of an addiional sligh ighening of he consrain, a sandard linearizaion procedure allows one o reformulae (3.9) as follows b T r x +k (1 0.5ε)x max r f(px r) 2 b T 2εx max r X +k b r, k 0, r = 1,..., n r r (3.10) The same procedure used for he sae consrains can be used o reformulae he inpu consrains (3.3). In paricular, denoing by U +k = E{(ū +k + K +k δx +k )(ū +k + K +k δx +k ) T } = K +k X +k K T +k (3.11) he covariance marix of he inpu variable, he corresponding deerminisic consrain becomes c T s ū +k u max s c T s U +k c s f(p u s), k 0, s = 1,..., n s (3.12) and is linear (igher) counerpar is c T s ū +k (1 0.5ε)u max s f(pu s) 2 c T 2εu max s U +k c s, k 0, s = 1,..., n s s (3.13) While he use of Canelli s inequaliy allows one o reformulae he problem wihou making any assumpion on he disribuion of he disurbance w, we sress again he fac ha he values of f(1 p) in case of unknown disribuion are greaer han in he Gaussian case (e.g., abou an order of magniude in he range (0.1, 0.4)) as horoughly discussed in he previous chaper. Remark 2. Noe ha he linearizaion procedure adoped in (5.31a) and (5.31b) is no necessary if he gain ha appears in (3.4) is fixed and no considered as a degree of freedom of he MPC algorihm. From one hand, his choice simplifies he online usage of he algorihm and reduces he conservaism of he consrain approximaion, while on he oher hand i doesn allow o explicily conrol he variances of he sae and he inpu, ha depend only on heir iniial values, hus represening a sor of rade-off in he design phase. 39

50 hesis 2015/11/21 9:18 page 40 #50 Chaper 3. Sae-feedback probabilisic SMPC 3.2 MPC algorihm: formulaion and properies To formally sae he MPC algorihm for he compuaion of he regulaor parameers ū, and K, he following noaion will be adoped: given a variable z or a marix Z, a any ime sep we will denoe by z +k and Z +k, k 0, heir generic values in he fuure, while z +k and Z +k will represen heir specific values compued based on he knowledge (e.g., measuremens) available a ime. According o he sandard procedure of MPC, a any ime insan a fuure predicion horizon of lengh N is considered and a suiable opimizaion problem is solved. The main ingrediens of he opimizaion problem are now inroduced Cos funcion Assume o be a ime and denoe by ū,...,+n 1 = {ū,..., ū +N 1 } he nominal inpu sequence over a fuure predicion horizon of lengh N. Moreover, define by K,...,+N 1 = {K,..., K +N 1 } he sequence of he fuure conrol gains, and recall ha he covariance X +k = E { } δx +k δx T +k evolves, saring from X, according o (3.8). In line wih he discussion in Secion 2.2.1, he cos funcion we consider here is he following +N 1 J = E{ ( x i 2 Q + u i 2 R) + x +N 2 S} (3.14) i= where Q and R are posiive definie, symmeric marices of appropriae size, S is he soluion of he algebraic Lyapunov equaion (A + B u K) T S(A + B u K) S = Q KT R K (3.15) and K can be any sabilizing gain for he error sysem (3.7). Wihou loss of generaliy we choose here o se K as he soluion of an LQ conrol problem, wih sae and conrol weighs Q, R, for he nominal model (3.5). To exploi he srucure of he cos funcion, define, in general, he expeced value along he horizon, x +k = E{x +k }, which are predicions compued according o x +k+1 =A x +k + B u ū +k (3.16a) 40

51 hesis 2015/11/21 9:18 page 41 # MPC algorihm: formulaion and properies Also, le u +k = ū +k +K k (x +k x +k ), and X +k = E{(x +k x +k )(x +k x T +k )}, which evolves according o X +k+1 =(A + B u K +k )X +k (A + B u K +k ) T + B w W B T w (3.16b) Accordingly, as described in Secion 2.2.1, he cos funcion (3.14) can be represened as a sum of wo componens accouning for he expeced value and he variance of he sae variable, respecively where J m = J = J m ( x, ū...+n 1 ) + J v (X, K...+N 1 ) (3.17) J v = +N 1 i= +N 1 i= ( x i 2 Q + ū i 2 R) + x +N 2 S (3.18) r{(q + K T i RK i )X i } + r{sx +N } (3.19) Noe ha J m depends on he nominal inpus ū...+n 1 = {ū,..., ū +N 1 } and on he mean value iniial condiion x. On he oher hand, J v depends on he gains K...+N 1 = {K,..., K +N 1 } and on he iniial covariance X. Therefore, in he conrol law (3.4) ū can be used o drive he mean value of he sae, while K can be seleced o reduce is variance as much as possible. Furhermore, he pair ( x, X ) will be also accouned for as an argumen of he MPC opimizaion, as laer discussed Terminal consrains As usual in MPC wih guaraneed sabiliy, see e.g. [100], also in he algorihm proposed here some consrains mus be imposed a he end of he predicion horizon for boh he mean value x +N and he variance X +N. Specifically, he erminal consrains we consider are he following x +N X f (3.20) X +N X (3.21) In (3.20), he se X f is a classic posiively invarian se (see [73]) for he sysem under he auxiliary LQ conrol law, saisfying (A + B u K) x Xf x X f (3.22) As for he erminal condiion on he variance, X, we can obain somehing wih he same invarian propery by deriving he seady sae soluion of he 41

52 hesis 2015/11/21 9:18 page 42 #52 Chaper 3. Sae-feedback probabilisic SMPC Lyapunov equaion (3.8) compued by considering a noise variance W W and assuming he consan gain K, i.e. X = (A + B u K) X(A + Bu K) T + B w W B T w (3.23) Indeed, recalling equaion (3.8) and using he consan gain K, from (3.21) we can wrie X +N+1 = (A + B u K)X+N (A + B u K) T + B w W B T w (3.24) (A + B u K) X(A + Bu K) T + B w W B T w X so ha X plays he role of an invarian se for he variance. Togeher wih he previous definiions, we need o ensure ha inside he erminal se he probabilisic requiremens on he sae and inpu are me and hus he following condiions, involving boh X f and X a he same ime, mus hold b T x (1 0.5ε)x max f(px ) 2 2εx max bt Xb c T K x (1 0.5ε)u max f(pu ) 2εu max ct Ūc (3.25a) (3.25b) for all x X f, where we denoed he seady sae inpu covariance marix as Ū = K X K T. Remark 3. In he compuaion of he erminal condiions hrough (3.25a) and (3.25b), we used he linearized version of he deerminisic consrains obained from he original probabilisic ones. Noe ha if we are able o use he nonlinear form ha appears in (3.9) and (3.12) he previous expression can be slighly modified o obain a less igh approximaion of he original consrain. This is useful when he gain sequence K +k is fixed and no considered as a degree of freedom as already discussed in Remark Iniial condiions for he mean and he covariance The proposed sochasic MPC problem aims a minimizing he cos funcion (3.14) also wih respec o he iniial condiions ( x, X ). Specifically, he seleced opimal values of x and X are x and X, respecively. In order o use he mos recen informaion available on he sae, a any ime 42

53 hesis 2015/11/21 9:18 page 43 # MPC algorihm: formulaion and properies insan i would be naural o se he curren value of he nominal sae x o x (i.e., leing x be he opimal condiional expecaion value, using he a poseriori daa x ), corresponding o seing X o zero. However, since we do no exclude he possibiliy of unbounded disurbances, his choice would lead in such cases o an infeasible opimizaion problem, and he fundamenal propery of recursive feasibiliy would be los. On he oher hand, and in view of he erminal consrains (3.20), (3.21), i is quie simple o see ha recursive feasibiliy is guaraneed provided ha he considered mean is updaed according o he predicion equaion (3.5), which corresponds o a variance updae given by updae (3.8). These consideraions lead o define he following wo alernaive sraegies. Sraegy 1 - Rese of he iniial sae: in he MPC opimizaion problem se x = x, X = 0. This corresponds o using all he informaion available a ime from he measures. Sraegy 2 - Predicion: in he MPC opimizaion problem se x = x 1, X = X 1,i.e., use he nominal predicion from he pas opimal soluion and x 1 = A x Bū 1 1 X 1 = (A + B u K 1 1 )X 1 1 (A + B u K 1 1 ) T + B w W B T w In conclusion, also he iniial condiions ( x, X ) are opimizaion parameers which mus be chosen according o eiher Sraegy 1 or Sraegy 2, i.e. ( x, X ) {(x, 0), ( x 1, X 1 )} (3.26) Saemen of he sae feedback p-smpc problem The probabilisic sae-feedback SMPC (p-smpc) problem can now be saed as follows. p-smpc problem: a any ime insan minimize, wih respec o he sequences ū...+n 1, K...+N 1, and o he pair ( x, X ), he performance index (3.17) subjec o he dynamics (3.16a) and (3.16b), o he linear deerminisic reformulaions (3.10), (3.13) of he probabilisic consrains (3.2), (3.3) for all k = 0,..., N 1, o he iniial consrain (3.26), and o he 43

54 hesis 2015/11/21 9:18 page 44 #54 Chaper 3. Sae-feedback probabilisic SMPC erminal consrains (3.20), (3.21). In paricular, he problem we solve a ime is he following min ū...+n 1, K...+N 1 ( x, X ) s.. ( x +k 2 Q + u +k 2 R) + x +N 2 S] N 1 E[ k=0 x +k+1 = A x +k + Bū +k X +k+1 = (A + BK +k )X +k + (A + BK +k ) T + B w W B T w b T r x +k (1 0.5ε)x max f(px ) 2 2εx max bt r X +k b r, r, k c T s ū +k (1 0.5ε)u max f(pu ) 2 2εu max ct s K +k X +k K+kc T s, s, k x +N X f X +N X ( x, X ) {(x, 0), ( x 1, X 1 )} (3.27) Denoing by {ū,..., ū +N 1 }, {K,..., K +N 1 }, and ( x, X ) he opimal soluion of he p-smpc problem, and according o he receding horizon principle, he feedback conrol law acually used is hen given by (3.4) as u = ū + K (x x ) (3.28) where obviously, when he rese sraegy is used, he feedback erm vanishes. In order o sae he properies of he proposed algorihm, we define he p-smpc problem feasibiliy se as he se { Ξ := ( x 0, X 0 ) : ū } 0,...,N 1, K 0,...,N 1 such ha (3.10), (3.13) hold for all k = 0,..., N 1 and (3.20), (3.21) are verified Noe ha, in view of he compacness of X, see (3.2), he se Ξ resuls o be compac. The recursive feasibiliy and convergence properies of he resuling conrol sysem are discussed in deail in he sequel. However, some 44

55 hesis 2015/11/21 9:18 page 45 # MPC algorihm: formulaion and properies preliminary commens are in order. A he iniial ime = 0, he algorihm mus be iniialized by seing x 0 0 = x 0 and X 0 0 = 0. In view of his, feasibiliy a ime = 0 amouns o (x 0, 0) Ξ. According o he problem saemen, feasibiliy of p-smpc a ime > 0 amouns o {(x, 0), ( x 1, X 1 )} Ξ. The binary choice beween Sraegies 1 and 2 for he iniializaion of x, X, see consrain (3.26), requires o solve a any ime insan wo opimizaion problems. However, noice ha in Sraegy 1 (rese of he iniial nominal sae and covariance), he on-line minimizaion of he erm J v in he cos funcion J is no needed. In fac, i is possible o compue off-line he opimal sequence of conrol gains {K 0 0,..., K N 1 0 } minimizing J v wih X 0 = 0 and o use hem in on-line operaions. I follows ha he corresponding MPC problem is a prey sandard one, where only J m is o be minimized wih respec o he sequence {ū,..., ū +N 1 }. A sequenial procedure can be adoped o reduce he average overall compuaional burden. The opimizaion problem corresponding o Sraegy 1 is firs solved, hen, if i is infeasible, Sraegy 2 mus be used. On he conrary, if i is feasible, i is possible o compare he resuling value of he opimal cos funcion wih he value of he cos ha corresponds o he use of he sequences {ū 1,..., ū +N 2 1, K x +N 1 }, {K 1,..., K +N 2 1, K}. If he opimal cos wih Sraegy 1 is lower, Sraegy 1 can be used wihou solving he MPC problem for Sraegy 2. This does no guaranee opimaliy, bu he convergence properies of he mehod saed in he resul below are recovered and he compuaional effor is reduced. Now we are in he posiion o sae he main resul concerning he convergence properies of he algorihm. Theorem 1. Assume ha here exiss ρ (0, 1) such ha he noise variance W verifies (N + λmax(s) ) λ min (Q) r(sb w W B T λ min (Q) w) < min(ρ σ 2, ρλ min ( X)) (3.29) 45

56 hesis 2015/11/21 9:18 page 46 #56 Chaper 3. Sae-feedback probabilisic SMPC where σ is he maximum radius of a ball, cenered a he origin, included in X F. Then, if a ime = 0 he S-MPC problem admis a soluion, he opimizaion problem is recursively feasible and, as +, E{ x 2 } 1 ρλ min (Q) (N + λ max(s) λ min (Q) ) r(sb ww Bw) T while saisfying he sae and inpu probabilisic consrains (3.3) and (3.2) for all Implemenaion issues In his secion he proposed algorihm is reformulaed ino a SDP problem o obain an efficien soluion procedure, based on specific solvers, ha guaranees a low compuaional cos. To his end, noe ha boh he expeced value and he variance of he sae and inpu along he predicion horizon,..., + N depend on he opimizaion variables ū...+n 1 and K...+N 1. In paricular, he expeced value is relaed only o he nominal inpu ū and his dependence is linear and governed by equaion (3.5). On he oher hand, he variance of he sae evolves as in equaion (3.8) and herefore shows a nonlinear dependence wih respec o he gain K. To ransform equaion (3.8) ino a linear one i is possible o resor o Schur complemens, following, for example, he ideas in [96, 130]. Recall firs Schur s lemma for marices (see for example Appendix 5.5 of [19]) Lemma 2. Schur complemen. Suppose ha D is a non singular marix and A, B, C are marices of proper dimensions, han he following wo expressions are equivalen: [ A ] B C D 0, A BD 1 C 0 in which 0 sands for he posiive semidefinieness of he marix equaion. Now consider he evoluion of he sae variance (3.8) and define a new se of opimizaion variables, G k, ha are compued a each ime sep as 46

57 hesis 2015/11/21 9:18 page 47 # Implemenaion issues G k = K k X k, k =,..., + N 1. Then we can wrie X k+1 = (A + B u K k )X k (A + B u K k ) T + B w W Bw T = (AX k + B u G k )X 1 k (AX k + B u G k ) T + B w W W 1 (B w W ) T which can be reformulaed as X k+1 [ (AX k + B u G k ) B w W ] [ ] [ X k 0 (AXk + B u G k ) T ] 0 W (B w W ) T = 0 Finally, recalling ha we are minimizing he variance hrough he erm J v, i is possible o relax he equaliy ino an inequaliy and use he Schur s lemma o obain he LMI reformulaion of he consrain X i+1 (AX i + B u G i ) (B w W ) (AX i + B u G i ) T X i 0 0 (3.30) (B w W ) T 0 W ha can ake he place of (3.8) in he opimizaion problem. Noice ha in (3.30) boh X k+1 and X k are considered as opimizaion variables a each sep of he predicion horizon while he gain K k is replaced by G k. Wih he same procedure we need o rewrie he covariance marix U k of he inpu variable o replace he gain K k. In paricular we obain U k = Kk T X k K k = G T k X 1 k G k and again he previous expression can be wrien as he following LMI [ Uk G T ] k 0 (3.31) G k X k in which he sequence of U k, k =,..., + N 1 mus be considered as exra opimizaion variables for he problem. The cos funcion can be reaed as follows. Saring direcly from (3.14), i 47

58 hesis 2015/11/21 9:18 page 48 #58 Chaper 3. Sae-feedback probabilisic SMPC can be wrien as +N 1 J( ) = E{ ( x i 2 Q + u i 2 R) + x +N 2 S} i= +N 1 = E{ = = i= +N 1 i= +N 1 i= E{ [ xi [ xi u i u i r{m E{ ] ] T M 2 M + x +N 2 S} [ xi u i [ xi u i ] } + E{ x +N 2 S} ] [ ] T xi }} + r{se{x +Nx T u i +N}} where M = diag(q, R). Now defining a new opimizaion variable P i R n+m n+m as [ ] [ ] T xi xi P i = E{ } (3.32) and a pariion Pi x becomes u i u i of he firs n rows and n columns, he cos funcion J( ) = +N 1 i= r{mp i } + r{sp x +N} (3.33) ha is linear in he sequence of new opimizaion variables P,...,+N. Taking he expression in (3.32) i is possible o compue he following expression [ ] [ ] T xi + δx i xi + δx i P (i) = E{ } ū i + K i δx i ū i + K i δx i ] [ ] ] [ ] [ xi I [ xi I = E{( + δx i )( + δx i ) T } = E{ = ū i K i ] ] T [ xi [ xi } + ū i ū i ] ] T [ xi [ xi + ū i ū i [ Xi G i [ Xi ] G i ] X 1 i ū i K i X 1 i E{δx i δx T i }X 1 i [ ] T Xi G i [ ] T Xi G i 48

59 hesis 2015/11/21 9:18 page 49 # Proof of he main Theorem rearranging he above erms he definiion of P i can be imposed hrough he following LMI [ ] ] Xi [ xi P i G i ū i [ ] Xi G T i X i 0 0 (3.34) ] T [ x i ū T i 0 1 ha mus be added o he problem o make use of he linear cos funcion (3.33). Noe ha his consrain holds along he predicion horizon, for i = + 1,..., + N 1, while paricular aenion mus be given o he iniial sep, i =, and o he las sep, i = + N. If a ime we have X = 0, since we are using all he available informaion on he sae, (3.34) becomes ] [ xi P i ū i ] 0 (3.35) T [ x i ū T i 1 while if one makes use of he prediced sae of he sysem, as described by Sraegy II, and hence X 0, i is possible o apply (3.34) also for i =. For = N in boh he cases i resuls [ ] P+N X +N x +N x T 0 (3.36) +N Proof of he main Theorem Proof of recursive feasibiliy Assume ha, a ime insan, a feasible soluion of p-smpc is available, i.e., ( x, X ) Ξ wih opimal sequences {ū,..., ū +N 1 } and {K,..., K +N 1 }. We prove ha, a ime + 1, a leas a feasible soluion o p-smpc exiss, i.e., ( x +1, X +1 ) Ξ wih feasible, possibly subopimal, sequences {ū +1,..., ū +N 1, K x +N } and {K +1,..., K +N 1, K}. Consrain (3.10) is verified for all pairs ( x +1+k, X +1+k ), k = 0,..., N 2, in view of he feasibiliy of he p-smpc a ime. Furhermore, in view of (3.20), (3.21), and he condiion (3.25a), we have ha 49

60 hesis 2015/11/21 9:18 page 50 #60 Chaper 3. Sae-feedback probabilisic SMPC b T r x +N (1 0.5ε)x max r f(px r) 2 2εx max bt r (1 0.5ε)x max r f(px r) 2 2εx max r Xb r b T r X +N b r for all r = 1,..., n r, i.e., consrain (3.10) is verified. Analogously, consrain (3.13) is verified for all pairs (ū +1+k, U +1+k ), k = 0,..., N 2, in view of he feasibiliy of S-MPC a ime. Furhermore, in view of (3.20), (3.21), and he condiion (3.25b), we have ha c T K s x +N (1 0.5ε)u max s f(pu s) 2 2εu max s (1 0.5ε)u max s f(pu s) 2εu max s c T s Ūc s c T s U +N c s for all s = 1,..., n s i.e., consrain (3.13) is verified. In view of (3.20) and of he invariance propery (3.22) i follows ha x +N+1 = (A + B K) x +N X f and, in view of (3.21) X +N+1 = (A + B u K)X+N (A + B u K) T + B w W B T w (A + B u K) X(A + Bu K) T + B w W B T w = X hence verifying boh (3.20) and (3.21) a ime Proof of convergence In view of he feasibiliy, a ime + 1 of he possibly subopimal soluion given by {ū +1,..., ū +N 1, K x +N }, {K +1,..., K +N 1, K}, and ( x +1, X +1 ), we have ha he opimal cos funcion compued a ime + 1 is J ( + 1) = J m( + 1) + J v ( + 1) 1. In view of he opimaliy of J ( + 1) we have J ( + 1) J m ( x +1, u +1,..., u +N 1, K x +N ) + J v (X +1, K +1,..., K +N 1, K) (3.37) 1 For breviy, we denoe J (x, x 1, X 1 ) wih J (), Jm (x, x 1, X 1 ) wih Jm (), and Jv (x, x 1, X 1 ) wih Jv () 50

61 hesis 2015/11/21 9:18 page 51 # Proof of he main Theorem Noe ha, for he expeced value componen J m ( x +1, u +1,..., u +N 1, K x +N ) = J m ( x, u,..., u +N 1 ) x 2 Q ū 2 R+ x +N 2 Q + K x +N 2 R x +N 2 S+ (A + B u K) x+n 2 S and in view of he definiion of S, given by (3.15), we have also Furhermore, noe ha and hus i is possible o wrie x +N 2 Q + K x +N 2 R x +N 2 S+ (A + B u K) x+n 2 S = 0 (3.38) J m ( x, u,..., u +N 1 ) = J m() (3.39) J m ( x +1, u +1,..., u +N 1, K x +N ) J m() x 2 Q ū 2 R (3.40) Consider now he variance componen, J v. We compue ha J v (X +1, K +1,..., K +N 1, K) = J v (X, K,..., K +N 1 ) r{(q + K T RK )X } + r{(q + K T R K)X +N + (3.41) S(A + B u K)X+N (A + B u K) T + SB w W B T w SX +N } Recall he following properies of he race: r(a + B) = r(a) + r(b), r(ab) = r(ba), being A and B marices of suiable dimensions. In view of his, and recalling (3.15) r{(q + K T R K)X +N + S(A + B u K)X+N (A + B u K) T } = r{(q + K T R K + (A + B u K) T S(A + B u K))X+N } = r{sx +N } (3.42) and herefore we have ha J v ( + 1) J v () r{(q + K T RK )X } + r(b w W B T w) (3.43) From (3.37)-(3.42) we obain J ( + 1) J () E{ x 2 Q + u 2 R } + r(sb ww B T w) J () λ min (Q)E{ x 2 } + r(sb w W B T w) 51 (3.44)

62 hesis 2015/11/21 9:18 page 52 #62 Chaper 3. Sae-feedback probabilisic SMPC where λ min (Q) denoes he smalles eigenvalue of he marix Q. Furhermore, from he definiion of J () we also have ha J () E{ x 2 Q + u 2 R } λ min (Q)E{ x 2 } (3.45) Define now he erminal se Ω F = {( x, X) : x X F, X X}. Assuming ha ( x, X ) Ω F we have ha J () Jm aux () + Jv aux (), where Jm aux () = +N 1 k= (A + B u K) k x 2 Q + K(A + B u K) k x 2 R + (A + B K) u N x 2 S since { K x,..., K(A+B u K) N 1 x } is a feasible inpu sequence. Therefore, recalling (3.15), Jm aux () = x 2 S (3.46) Similarly, and recalling he properies of he race and (3.15), we obain ha J aux v () is equal o N 1 k=0 r{(q + K T R K)[(A + B u K) k X (A + B u K) T (k) + k 1 i=0 (A + B u K) i B w W B T w(a + B u K) T (i) ]} +r{s[(a + B u K) N X (A + B u K) T (N) + N 1 i=0 (A + B K) u i B w W Bw(A T T + B u K) (i) ]} = r{[ N 1 k=0 (A + B K) u T (k) (Q + K T R K)(A k + B u K) T +(A + B u K) (N) S(A + B u K) N ]X } Therefore +r{[ N 1 k=1 k 1 i=0 (A + B u K) T (i) (Q + K T R K)(A + B u K) i + N 1 i=0 (A + B u K) T (i) S(A + B u K) i ]B w W B T w} = r{sx } + r{[s + N 1 k=1 ((A + B u K) T (k) S(A + B u K) k ) + N 1 i=1 (A + B u K) T (i) (Q + K T R K)(A + B u K) i ]B w W B T w} = r{sx } + r{[s + N 1 i=1 S]B ww B T w} J aux v () = r{sx } + Nr{SB w W B T w} (3.47) Combining (3.46) and (3.47) we obain ha J () E{ x 2 S } + N r{sb ww Bw} T λ max (S)E{ x 2 } + N r{sb w W Bw} T (3.48) 52

63 hesis 2015/11/21 9:18 page 53 # Proof of he main Theorem Remark ha assumpions (3.44), (3.45), and (3.48) are similar o he ones needed in he framework of inpu-o-sae sabiliy. From (3.44), (3.45) and (3.48) i is possible o derive robus sabiliy-relaed resuls. If ( x, X ) Ω F hen, in view of (3.48), (3.44) J ( + 1) J ()(1 λ min(q) λ max (S) ) + ( λ min(q) λ max (S) N + 1)r(SB ww B T w) (3.49) Denoe b = 1 λmax(s) (N + ). If J ρ λ min () b r(sb (Q) w W Bw), T and provided ha inequaliies (3.29) are verified, hen one can prove ha ( x, X ) IΩ F, where IΩ F denoes he inerior of Ω F. In fac, J () b r(sb w W Bw) T implies ha, in view of (3.45) E{ x 2 } = x 2 + r(x ) This, considering (3.29), implies ha b λ min (Q) r(sb ww B T w) x 2 < σ 2 r(x ) < λ min ( X) (3.50a) (3.50b) In view of (3.50a), hen x X F. Furhermore, (3.50b) implies ha λ max (X ) < λ min ( X), which in urn implies ha X X. Therefore, recalling (3.49), if J () b r(sb w W B T w), hen J (+1) b r(sb w W B T w). and he posiive invariance of he se D = {( x, X) : J () b r(sb w W B T w)} (3.51) is guaraneed. From his poin on, he proof follows similarly o [95, 136]. For ( x, X ) Ω F \D, i holds ha which, in view of (3.48), implies ha E{ x 2 } > J () > b r(sb w W B T w) (3.52) 1 λ min (Q) r{sb ww B T w} 53

64 hesis 2015/11/21 9:18 page 54 #64 Chaper 3. Sae-feedback probabilisic SMPC Therefore, considering (3.44) J ( + 1) J () < 0 (3.53) On he oher hand, here exiss consan c > 0 such ha, for all x wih ( x, X ) Ξ\Ω F here exiss x Ω wih ( x Ω, X Ω ) Ω F \D such ha λ min (Q)E{ x 2 } λ min (Q)E{ x Ω 2 } c This, in view of (3.44) and (3.53), implies ha, for all x wih ( x, X ) Ξ\Ω F J ( + 1) J () < c (3.54) and, in urn, his implies ha here exiss a ime insan T 1 > 0 such ha x +T1 is such ha ( x +T1 +T 1, X +T1 +T 1 ) Ω F. If, on he one hand ( x +T1 +T 1, X +T1 +T 1 ) D, in view of he posiive invariance of D, ( x +k +k, X +k +k ) D for all k T 1. If, on he oher hand, ( x +T1 +T 1, X +T1 +T 1 ) Ω F \D, recalling (3.49), (3.52), and (3.45) J ( + T 1 + 1) J ( + T 1 ) (1 ρ) λ min(q) λ max (S) J ( + T 1 ) (3.55) (1 ρ) λ min(q) 2 λ max (S) E{ x +T 1 2 } In view of (3.54)-(3.55) here exiss T 2 T 1 such ha, for all ε > 0 J( + k) ε + b r(sb w W B T w) for all k T 2, which proves ha E{ x 2 } 1 λ min (Q) b r(sb ww B T w) asympoically, as Simulaion example In his secion, a very simple simulaion example is proposed o show he efficacy of he p-smpc scheme presened in his chaper. In paricular, he example we consider is inspired by he one ha appears in [117] where he sysem is a double inegraor represening a poin-mass moving in a wo dimensional space subjec o uncerainy in he posiion. The conrol goal is o drive he sae of he sysem o he origin while remaining wih a desired 54

65 hesis 2015/11/21 9:18 page 55 # Simulaion example probabiliy inside a cerain area. The sysem can be described as in (3.1) wih marices / A = B = 0 1/2 1 0 F = and he disurbance w k is a whie noise wih zero-mean, covariance marix W = I 2 and unknown disribuion. The consrained area is a recangle wih verices ( 5, 3.5), (5, 3.5), (5, 3.5) and ( 5, 3.5), while he conrol acion is subjec o he consrain u k 1. In he cos funcion (3.14), he weighing marices have been chosen as Q = 10 3 I 4, R = I 2. The gain K has been obained as he soluion of he corresponding LQ conrol problem and has been used in (3.23) o compue he seady sae covariance marix X wih W = W. Seing ε = 0.2 and p u = 0.2 he resuls in Figure 3.1 have been obained for differen values of he violaion p x applied o each of he consrains. Figure 3.1: Simulaion resuls of he proposed S-MPC wih differen violaion probabiliies. From hese resuls, i is apparen ha he algorihm successfully brings he uncerain sae o a neighborhood of he origin wihou violaing he consrains on he posiion for each seleced value of p x. As expeced, higher 55

66 hesis 2015/11/21 9:18 page 56 #66 Chaper 3. Sae-feedback probabilisic SMPC probabiliy of violaion corresponds o a less conservaive algorihm and o higher values of he cos funcion associaed wih he real sae. Focusing on a single simulaion wih p x = 0.1, p u = 0.2, ε = 0.2, i is possible o see in Figure3.2 how he rese sraegy works. Clearly, Sraegy 2 (Predicion) is more frequenly used when he sae approaches he boundary region Real posiion Nominal posiion wih rese Nominal posiion wihou rese x x 1 Figure 3.2: Simulaion resuls wih p x = 0.1 and p u = 0.2. Seing p u = 0.2 and ε = 0.2 and for differen values of p x, T = 100 simulaions are made o verify he heoreical probabiliy of consrain violaion presened and he resuls are summarized in Table.3.1. p x = 0.01 p x = 0.2 p x = 0.5 Measured violaion Table 3.1: Consrain violaion in 100 simulaions. Due o he use of Canelli s inequaliy o ighen he probabilisic consrains and he linearizaion procedure applied o heir deerminisic version, he algorihm resuls o be very conservaive in erms of acual violaion. More deailed examples will be presened laer o compare some of he more ineresing echniques in he SMPC lieraure and paricular aenion will be devoed o he resuls in erms of violaion probabiliy of he consrains. 56

67 hesis 2015/11/21 9:18 page 57 # Commens 3.6 Commens In his chaper we proposed a novel algorihm, denoed as p-smpc, for sochasic Model Predicive Conrol of discree-ime linear sysem wih addiive uncerainy under a se of individual chance consrains. In line wih he analysis presened in Chaper 2, he p-smpc approach is classified as an analyic mehod, in which he probabilisic consrains are firsly reformulaed as deerminisic, alhough igher, ones and hen implemened ino a quie sandard MPC seup. Wih respec o similar algorihms already discussed in he lieraure, in he p-smpc echnique proper erminal consrains, ogeher wih a sraegy for choosing he iniial condiion of he opimizaion problem, allow o obain recursive feasibiliy and convergence properies even in he case of unbounded uncerainies. The main advanages and disadvanages of he proposed algorihm, are now summarized. In paricular, he p-smpc algorihm is very useful due o 1) is abiliy o deal wih possibly unbounded disurbances acing on he sysem, 2) he guaraneed recursive feasibiliy and convergence properies boh in he case of bounded and unbounded uncerainies, 3) a reduced online compuaional load, similar o he one of sandard nominal MPC algorihms, 4) a low design complexiy. However, some drawbacks are presen, mainly relaed o 1) he need for probabilisic consrains on he inpu (due o he choice of a feedback conrol law in presence of unbounded disurbances, we are no able o enforce hem in a wors-case fashion), 2) he abiliy o cope only wih addiive disurbances 3) he quie conservaive Canelli-Chebyschev bound in he case of unknown uncerainies disribuion 4) he choice of individual chance consrains (in case he original problem is specified wih join chance consrains we need o reformulae hem by he use, for example, of offline risk allocaion echniques). In he nex chaper, various examples of he applicaion of he proposed approach in comparison wih oher noable sochasic MPC echniques are 57

68 hesis 2015/11/21 9:18 page 58 #68 Chaper 3. Sae-feedback probabilisic SMPC presened. The aim is o furher prove he efficacy of he p-smpc algorihm and o show he main difficulies ha arise in he implemenaion of such SMPC schemes. 58

69 hesis 2015/11/21 9:18 page 59 #69 CHAPTER4 Applicaion examples In his chaper we presen some applicaion examples of four sochasic MPC algorihms for linear sysems wih addiive disurbances and measurable sae. Three of hese echniques are seleced from he lieraure for he ineresing idea on which hey are based and he las one is he probabilisic SMPC (p-smpc) presened in he previous chaper. Afer a quick overview on he seleced algorihms, ha has he only goal of showing he main raionale behind hem and giving a skech on he implemenaion issues, several examples are presened o show he efficacy of hese echniques and o make comparisons in erms of probabilisic guaranees, design complexiy and online compuaional load. In paricular he following examples are used. Firsly a oy sysem is presened, in which he effec of he probabilisic consrain a a fixed ime sep is analyzed in erms of he acual violaion over muliple simulaions. Then a simple muli-room model is used o show he efficacy of he seleced echniques in erms of reaching he desired consrain violaion level over ime. In he end he problem of conrolling he emperaure inside a realisic building is considered. In paricular resuls of he applicaion of a nominal sandard MPC algorihm, a robus wors-case algorihm and wo of he four seleced sochasic echniques are compared wih he aim of showing he real benefi 59

70 hesis 2015/11/21 9:18 page 60 #70 Chaper 4. Applicaion examples of probabilisic approaches. 4.1 A quick overview on some paradigmaic SMPC algorihms Before proceeding wih he examples, we summarize here he main guidelines of hree SMPC algorihms aken from he lieraure which will be esed in simulaion, ogeher wih he algorihm proposed here (denoed in he sequel as p-smpc). The algorihms have been seleced o represen ineresing classes of mehods; specifically and wih reference o he classificaion in Chaper 2, we consider wo analyic mehods for dealing wih disurbances wih bounded suppor, labeled as -SMPC and av-spmc, and a general scenario-based approach, denoed by s-spmc. The aim is o highligh, in a simple seup, he main difficulies ha arise when rying o ensure properies like sabiliy and recursive feasibiliy while reducing as much as possible he level of conservaism and o furher prove he efficacy of he p-smpc developed in his hesis. The seleced mehods are used o conrol linear models wih addiive uncerainy described, adoping he same noaion presened in Chaper 2, as in equaion (2.3). In paricular he sysem is described by he following expression x +1 = Ax + B u u + B w w (4.1) where he sae is assumed o be measurable. In he following descripion, a single probabilisic sae consrain of he ype (2.18) is considered, while he inpu is lef unconsrained for he sake of simpliciy Sochasic ube MPC (-SMPC) The Sochasic ube MPC mehod (-SMPC) is an analyic scheme, proposed for linear sysems affeced by addiive and/or muliplicaive uncerainies, see e.g., [24,25,27 29,77]. In [28,29] no boundedness assumpion is made on he disurbance affecing he sysem and recursive feasibiliy resuls are esablished in probabiliy. In laer works (e.g., [24, 25, 27, 77]), under he assumpion of bounded disurbances, recursive feasibiliy resuls are given. Here we consider a simplified version of he approach presened in [77]. In his work he auhors define an offline consrain ighening procedure ha, hanks o he assumpion ha w W := {w : w α} wih 60

71 hesis 2015/11/21 9:18 page 61 # A quick overview on some paradigmaic SMPC algorihms α = [α 1... α nw ] T, ensures he recursive feasibiliy of he algorihm while aking advanage of he probabilisic naure of he consrain. The resuling online compuaional load is comparable wih he one of a deerminisic MPC algorihm, a he price of reducing he flexibiliy wih respec o online ighening procedures. The conrol scheme is based on a sae feedback sraegy of he ype (2.16), i.e., we have u +i = ū +i + Kx +i (4.2) where K is a consan gain ha sabilizes he marix A + B u K and ū is compued as a soluion of a proper MPC problem. In principle, ransien probabilisic consrains are guaraneed as in (2.20), where q i (1 p) can be compued, as in (2.22), using he expression q i (1 p) = arg min q q, s.. P { g T E i w q } = 1 p (4.3) or obained, in a conservaive way, using he Chebyshev inequaliy (2.28). To ensure recursive feasibiliy, he soluion presened in [77] akes advanage of he known bounds on he disurbance o implemen a mixed sochasic/wors-case ighening procedure, which could include some conservaiveness. More specifically, (2.20) is replaced by g T x +i h β i, i = 1, 2,... (4.4) where β 1 = q 1 (1 p) and β i q i (1 p) for all i 2. For deails on he compuaion of he erms β i see [77]. The algorihm is implemened using he common dual mode predicion paradigm, where he above consrains are accouned for explicily along he horizon defined by N and implicily by means of a proper erminal consrain x +N S ˆN { g T Φ l x N h β N+l, l = 1... ˆN } S ˆN = x N : g T Φ l x N h β, l > ˆN (4.5) where i is assumed ha ū +i = 0 for all i N and ˆN defines an addiional predicion horizon. This allows o define an infinie-ime expecaion cos funcion of he ype J = E[x T +iqx +i + u T +iru +i ] (4.6) i=0 which is minimized a each ime sep. Noe ha, once all he erms required up o a desired numerical precision are compued offline, he remaining problem is no more complex han a classical deerminisic MPC, 61

72 hesis 2015/11/21 9:18 page 62 #72 Chaper 4. Applicaion examples and he algorihm shares he same feasibiliy properies on he closed-loop operaions ha we have in he robus case (for he deailed proof see [77]) Sochasic MPC for conrolling he average number of consrain violaions (av-smpc) The analyic approach presened in his secion has been firs proposed in [74], and has been laer developed in [75,76]. I applies o linear sysems wih bounded uncerainies, i.e., where w W, wih W bounded. Recursive feasibiliy properies can be esablished and hard consrains on he inpu variable u U are allowed. The approach proposed by he auhors relies on a disurbance feedback conrol sraegy wih average cos funcion of ype (2.5), alhough in [76] i is shown ha more general cases can be encompassed. The general approach is similar o he one discussed in Secion 4.1.1; however, he mixed probabilisic/wors-case consrain ighenings are relaxed hanks o he idea ha he sampled average of consrain violaions mus be limied, raher han is probabilisic counerpar. In his way, he approach resuls o be in general less conservaive a he price of a sligh increase in he compuaional burden. In his secion we describe only he simplified version of he conrol scheme, inspired by [75] where, for simpliciy, a single probabilisic consrain of he ype (2.19) is considered and inpu consrains are negleced. In his approach, denoing by v he number of consrain violaions occurred up o ime (wih v ), he consrain (2.20) for x +1 is replaced by he following one g T x +1 h q 1 (1 β ) (4.7) where β = max(p( + 1) v, p). Noe ha β > p is equivalen o v / < p, so ha, if he average rae of violaions v / occurred up o ime is smaller han he prescribed limi p, we can allow for a greaer probabiliy of consrain violaion a he nex ime sep. Also, a ime + 1, he sae mus be included in a se S r, defined in such a way ha, if x +1 S r hen, for all possible realizaions of he bounded noise sequence w, x +r S, where he se S is he so-called sochasic conrol invarian se. S is defined in such a way ha if x S hen here exiss u such ha Ax+B u u+b w w S for all w W and g T (Ax + B u u) h q 1 (1 p). Inuiively, he number r is compued as he number of seps ahead in which a consrain violaion would allow o saisfy v +i /( + i) < p for all i = 1,..., r. As discussed in [76], in our simplified seing x +1 S r is ensured if, for all 62

73 hesis 2015/11/21 9:18 page 63 # A quick overview on some paradigmaic SMPC algorihms i = 1,..., N r g T x +i + max w W N E i e +i h q 1 (1 p) I is worh noing ha in [76] a more general case is considered, where a forgeing facor is allowed in he compuaion of v, as well as he use of a penaly funcion o quanify he disance from consrain violaion (e.g., raher han jus defining v as he number of violaions, v can represen also how far he sae has been from violaing he bounds). Also, as already discussed, hard consrains on inpus are allowed Scenario MPC (s-smpc) As previously discussed, he main idea of all he scenario-based echniques, see for example [20], [144] and [36] and he references herein, is o ake advanage of he possibiliy o draw samples of he uncerainy, or o use is pas records, o formulae a sample-based version of he conrol problem. In he following simulaions we have used a general scenario-based algorihm inspired by [144]. The adoped conrol policy is he disurbance-feedback one, i.e., u +i is compued following (2.13) as i 1 u +i = c +i + θ ij w +j (4.8) where boh he open loop erm c and he parameers θ ij are free variables of a suiable MPC problem whose goal is o minimize he expeced cos funcion (2.5). The laer can be done, as discussed in Secion 2.2.1, by minimizing he sampled cos (4.9), ha is J 1 N s j=0 N s k=1 J [k] (4.9) where J [k] is he sandard average cos funcion (2.5) compued correspondingly o he k-h exracion of he disurbance sequence. As highlighed in Secion 2.2.1, oher possible cos funcions can be used in his framework. The probabilisic consrains on sae variables, similar o (2.29), are derived along he lines of he general ideas presened in Secion However, as horoughly discussed in [144], in a receding-horizon conrol framework, i is sufficien o exrac a oal number N s of realizaions, appearing 63

74 hesis 2015/11/21 9:18 page 64 #74 Chaper 4. Applicaion examples in all fuure consrains a he same ime g T ( x +i + E i w [k] ) h (4.10) for all k = 1,..., N s. In paricular, N s is he number of realizaions required for enforcing he probabilisic consrain (2.18) for i = 1. For example, N s = N s,1 as in (2.31) in case no consrain removal is performed. In addiion, for furher reducing he number of required realizaions, in [144] i is shown ha, in order o guaranee ha he average (insead of he poinwise) probabiliy of consrain violaion is equal o p, N s,1 in (2.31) can be replaced by N s d 1 p p Also in his case, he removal sraegy can be adoped for reducing he possible conservaiviy of he resuls a he price of an high increase of he online complexiy of he algorihm. For deails, see again [144] The proposed approach: probabilisic MPC (p-smpc) The approach proposed in he previous chaper falls in he caegory of analyic mehods, and has been developed specifically for linear sysems of ype (2.3) wih addiive and possibly unbounded uncerainy w, assumed o be a zero mean whie noise wih variance W. I encompasses he case where sae consrains are in he form (2.19). In view of he unboundedness of he noise affecing he sysem, inpu consrains mus also be formulaed, in general, as probabilisic ones. For a deailed explanaion he reader is referred o Chaper 3, however a quick summary is given below wih he aim of sress he differences wih he oher approaches. Here, for he sake of exposiion, inpu consrains are negleced. The approach lies on a sae feedback policy of ype (2.14) and, o reduce he compuaional complexiy bu a he price of obaining subopimal soluions, we discard he conrol gains as opimizaion variables and se K k = K for all k 0. Differenly from he case considered in Secion 2.2.2, here i is no necessarily se x = x, bu i is required, more in general, ha x = E[x ] (4.11) Due o he choice of fixing he gains, he probabilisic consrain (2.19) is formulaed as in (2.20) o be enforced for all he predicion horizon, i.e., for i =,..., + N 1 where, in line wih (2.28) q i (1 p) = 64

75 hesis 2015/11/21 9:18 page 65 # Simulaion examples gt X +i g (1 p)/p In case w is a (non-runcaed) Gaussian variable, similarly o (2.23), q i (1 p) = g T X +i gn 1 (1 p). The cos funcion o be minimized is he average cos (2.5). Recursive feasibiliy is esablished in view of wo ingrediens: Terminal consrains, boh in he mean value and in he variance, i.e. x +N X f (4.12) X +N X (4.13) Iniializaion. The iniial condiion for he pair ( x, X ) is free o ake wo differen values, i.e., ( x, X ) = (x, 0) or ( x, X ) = ( x, X 1 ), where ( x, X 1 ) is compued saring from he opimal soluion obained by he p-smpc problem solved a ime 1. For furher deails see Chaper Simulaion examples In his secion we apply he mehods described in Secion 4.1 o differen case sudies. The firs wo examples make reference o linear models wih addiive noise (4.1), wih measurable sae and only one probabilisic sae consrain of ype (2.18), while he inpu is unconsrained. The las example, inspired by a realisic problem, is more complex and several linear individual chance consrains on boh he saes and he inpus are considered. In his case, however, only he av-smpc and p-smpc algorihms are implemened. For beer comparisons, in he uning of all analyic mehods (i.e., -SMPC, av-smpc, and p-smpc), he funcions q i are compued offline as in (2.22) (wih paricular reference o av-smpc, please see deails in [76]) Academic benchmark example In his secion we consider he problem presened, e.g., in [27]. In paricular, he focus is on obaining he desired consrain violaion rae a he specific ime insan = 1 s. The sysem wih addiive uncerainy, see (4.1), is characerized by he marices A = [ ] , B u = [ ] 1, B w = 1 [ ]

76 hesis 2015/11/21 9:18 page 66 #76 Chaper 4. Applicaion examples The disurbance w is a runcaed Gaussian noise wih zero mean, variance σ 2 = 1/144 and bounds w The sysem is subjec o a single probabilisic sae consrain of he ype (2.19), i.e. P { g T x h } 1 p, 0 (4.14) where g T = [ ], h = 0.5, and he violaion level is se o p = 0.2. The feasible area is shown in Figure 4.1, ogeher wih he iniial condiion x 0 = [ 5 60 ] T. Noe ha, in view of he fac ha x0 does no lie in he feasible area, probabilisic consrains are enforced, for all mehods, from insan = 1s on. As in [27] an LQ conroller has been firs used o show he unconsrained soluion and compare i wih he consrained case. In paricular choosing he weighs Q = I 2 and R = I 2, he LQ opimal gain is K = [ ] T, and he ransien of he closed-loop sysem wih LQ conrol has he behavior shown in Figure 4.1, where all he simulaion correspond o a violaion of he consrain a ime = 1. The four SMPC algorihms previously described have been implemened by using quadraic cos funcions wih he same weighing marices Q and R considered in he design of he LQ conroller and wih a predicion horizon N = 5. The resuls achieved wih he -SMPC conroller described in secion are shown in Figure 4.2. The algorihm is able o guaranee an acual violaion rae of 0.1, more conservaive han he required value 0.2, a ime = 1 s, wih he compuaional ime of a single ieraion of solve = s. The av-smpc sraegy described in Secion produced he resuls shown in Figure 4.3. Thanks o he online ighening procedure, he overall violaion rae a he ime insan = 1 s goes up o 0.19, close o he desired level 0.2, even if he required average compuaional ime for a single ieraion increases o solve = 0.12 s. The p-smpc conroller described in Secion has been implemened by seing x 0 = E[x 0 ] = x 0, X 0 = 0. Noe ha, since he iniial sae does no lie inside he feasibiliy region, a ime = 0 he consrains (4.14) have been negleced. The obained resuls are repored in Figure 4.4. The consrain violaion rae a = 1 s is 0.18, close o he desired one. The compuaional ime for a single ieraion is approximaely solve = s. 66

77 hesis 2015/11/21 9:18 page 67 # Simulaion examples x2 x 1 Figure 4.1: Feasible region (pink area) and sysem rajecories for 100 runs wih he LQ conroller. Finally, he s-smpc scheme described in Secion 4.1.3, has been formulaed in such a way ha poin-wise probabilisic consrain saisfacion is required wih confidence level β = This resuls in he number of required samples and he number of removable consrains N s = 498 and N r = 59, respecively. The resuls are shown in Figure 4.5. The consrain violaion rae a = 1 s is 0.13 wih an average compuaional ime of solve = 1.8 s. The overall resuls, in erms of consrain violaion frequency a = 1 s, and compuaional ime are summarized in Table 4.1. This able highlighs a relaive conservaiveness of -SMPC and s-smpc, besides he high compuaional burden of s-smpc. However all he presened algorihms are able o solve he problem obaining increased performances wih respec o deerminisic sraegies wihou affecing oo much he compuaion ime. 67

78 hesis 2015/11/21 9:18 page 68 #78 Chaper 4. Applicaion examples x2 x 1 x2 x 1 Figure 4.2: Simulaions using -SMPC. On he righ he overall sysem rajecories for he 100 runs. On he lef he violaed consrain a ime = 1 s. 68

79 hesis 2015/11/21 9:18 page 69 # Simulaion examples x2 x 1 x2 x 1 Figure 4.3: Simulaions using av-smpc conroller. On he righ he overall sysem rajecories for he 100 runs. On he lef he violaed consrain a ime = 1 s Muli-room emperaure conrol In his secion we consider an example aken from [12]. The problem consiss in conrolling he emperaures, T i, i = 1,... 4 inside a simple building 69

80 hesis 2015/11/21 9:18 page 70 #80 Chaper 4. Applicaion examples x 1 x2 x2 x 1 Figure 4.4: Simulaions using p-smpc. On he righ he overall sysem rajecories for he 100 runs. On he lef he violaed consrain a ime = 1 s. composed by four rooms, each one equipped wih a radiaor supplying hea 70

81 hesis 2015/11/21 9:18 page 71 # Simulaion examples x2 x 1 x2 x 1 Figure 4.5: Simulaions using s-smpc. On he righ he overall sysem rajecories for he 100 runs. On he lef he violaed consrain a ime = 1 s. q i, i = A schemaic represenaion of he building is given in Figure 71

82 hesis 2015/11/21 9:18 page 72 #82 Chaper 4. Applicaion examples Approach Violaion frequency Compuaion ime -SMPC sec av-smpc sec p-smpc sec s-smpc sec Table 4.1: Sochasic Model Predicive Conrol approaches comparison - resuls for he academic benchmark example Figure 4.6: Schemaic represenaion of a building wih wo aparmens. The nominal exernal emperaure is fixed as T E = 0 C and, for he sake of simpliciy, solar radiaion is no considered. The model of he sysem is linear, where he four sae variables represen he emperaure variaions around he equilibrium poin T = 20 o C, for more deails see [12]. The discree-ime sysem is obained by ZOH discreizaion wih sampling ime 10 s, i.e., we have 72

83 hesis 2015/11/21 9:18 page 73 # Simulaion examples A = B u = and B w = I 4. The sae and conrol consrains are x,i [ 5, 5], u,i [ 0.038, 0.030], i = 1,..., 4 (4.15) while he iniial condiions are x 0 = [ 3.2, 2.58, 1.12, 3.55] T. The sae consrains are relaxed allowing a p = 0.2 rae of violaion for he emperaure of each room (independen chance consrain). The uncerainy acing on he sysem is a runcaed Gaussian addiive noise wih variance W = 0.05I 4 and bounds w 2. Resuls of he applicaion of he described algorihms are evaluaed over M = 100 differen runs. Differenly from he previous example, he goal here is o esimae he level of consrain violaion of he soluion on he long run. For all he differen conrol schemes, he cos funcion is seleced wih Q = I and R = 100I, while he predicion horizon N = 5 has always been used. The -SMPC algorihm, described in 4.1.1, has been uned by compuing he feedback gain and he erminal weighing marix wih LQ. Resuls, referred o he emperaure of room A, are shown in Figure 4.7. The average consrain violaion is 0.04, while he average compuaional ime for each ieraion is solve = 1.56s. The av-smpc mehod described in Secion lead o he resuls repored in Figure 4.8, again referred o he emperaure of room A. The conrolled sysem violaes he consrain wih an average frequency of 0.19, and he average compuaion ime is solve = 2.2s. The p-smpc conroller described in Secion has been uned by seing p u = 0.1 and compuing he inpu gain and he erminal weighing 73

84 hesis 2015/11/21 9:18 page 74 #84 Chaper 4. Applicaion examples Temperaure Sae rajecory Consrain Non violaing samples Violaing samples consrain violaion Measured violaion Desired violaion run Figure 4.7: Simulaions using -SMPC. On he lef a sample rajecory T 1. On he righ he violaion frequency over 100 runs. Temperaure Sae rajecory Consrain Non violaing samples Violaing samples consrain violaion Measured violaion Desired violaion run Figure 4.8: Simulaions using av-mpc conroller. On he lef a sample rajecory T 1. On he righ he violaion frequency over 100 runs. marix wih LQ. The behavior of he emperaure of room A is shown in Figure 4.9. The consrain is violaed wih an average rae of 0.19, close o he desired value p = 0.2. The average compuaional ime is solve = 1.79s. Finally, he s-smpc algorihm described in Secion has been formulaed in order o guaranee ime-average probabilisic consrain saisfacion. Considering p 1 = 4, he number of sample exracions is N s = 465 wih N r = 55 o be removed. The simulaion resuls are shown in Figure The violaion frequency of he consrain is 0.17 and he required average compuaional ime is solve = 17.6s. The resuls achieved wih he four mehods are summarized in Table 4.2. Even in his case, all he algorihms can provide beer performances (clearly 74

85 hesis 2015/11/21 9:18 page 75 # Temperaure conrol on a realisic building Measured violaion Desired violaion Temperaure Sae rajecory Consrain Non violaing samples Violaing samples consrain violaion run Figure 4.9: Simulaions using p-smpc. On he lef a sample rajecory T 1. On he righ he violaion frequency over 100 runs Measured violaion Desired violaion Temperaure Sae rajecory Consrain Non violaing samples Violaing samples consrain violaion run Figure 4.10: Simulaions using s-smpc. On he lef a sample rajecory T 1. On he righ he violaion frequency over 100 runs. in erms of power consumpion) han a robus deerminisic version, hanks o he probabilisic relaxaion. However, he -SMPC seems o be he mos conservaive one due o he offline ighening procedure coupled wih he wors-case consideraions along he horizon. On he oher hand, he s- SMPC, even if i is very flexible in is usage, requires an high compuaion ime. The av-smpc and he p-smpc are good compromises and hus are he ones ha will be esed in he nex example. 4.3 Temperaure conrol on a realisic building In his example, he av-smpc and he p-smpc algorihms previously discussed, are used o conrol he emperaure inside a realisic medium-size commercial building wih he aim of minimizing he power consumpion 75

86 hesis 2015/11/21 9:18 page 76 #86 Chaper 4. Applicaion examples Approach Violaion frequency Compuaion ime -SMPC sec av-smpc sec p-smpc sec s-smpc sec Table 4.2: Sochasic Model Predicive Conrol approaches comparison - resuls for example 3 while mainaining some comfor consrains in presence of uncerain weaher condiions and building occupaion. Noe ha, in his framework, he use of sochasic conrol echniques can be crucial, due o he explici way of handling he risk associaed o an uncerain environmen and sochasic Model Predicive Conrol approaches are very useful due o he possibiliy of exploiing he knowledge of weaher forecass or scheduled modificaions of he comfor consrains (for example desired emperaures can change during he nigh) anicipaing he reacion of he sysem. In he sequel, a realisic 3 zones office building is considered, ogeher wih real daa and forecass for he ciy of Lausanne (Swizerland) during he firs week of December 2013, colleced from he web service hp://www. wunderground.com. For people occupaion a realisic profile ha mach he dimension of he building is considered. The model used in he design of he conrol algorihms is a descripion of he hermal dynamics of he building obained as discussed in [54,134]. The enire modeling process is handled in MATLAB hanks o he use of he oolbox OpenBuild (hp://la.epfl.ch/openbuild) ha inegraes in an easy o use package sandard ools for energy efficiency analysis like for example he well-knownenergy Plus Building model Here he model of he proposed 3-zones building, resuling from he oolbox OpenBuild (see [54]), is presened. In paricular, in line wih equaion (4.1), we consider he following discree-ime linear ime-invarian model wih addiive disurbances x +1 = Ax + B u u + B d d y = Cx (4.16) 76

87 hesis 2015/11/21 9:18 page 77 # Temperaure conrol on a realisic building where he inpu vecor u represens he power used by he building (in he case considered here he heaing power), he oupu vecor y represens he emperaures inside he hree zones of he building and he vecor d collecs he disurbances coming respecively from he ouside emperaure, he weaher condiion, and he number of people inside he building. The marix B d describes he effec of he hree sources of uncerainy on he saes of he sysem. The ime sep used for compuing he model is T s = 20 minues. For he building under sudy he dimension of he sae is n x = 71, he dimension of he inpu is n u = 3 and he dimension of he oupu is n y = 3. If needed i is possible o derive a reduced order model exploiing he srucure of he building and approximaing some of he relaionships beween is surfaces. In his case he sae vecor boils down o n x = 10 elemens, a he price of a slighly increased modeling error and a he price of loosing he physical meaning of each of is componens. The reduced model will be used in he sequel. As for he disurbance vecor d, we assume o know in advance he weaher siuaion (solar radiaion profile) and he flow of people inside he building, so ha he aenion is focused only on he uncerainy coming from he unknown ouside emperaure. In his case, o predic is profile, we have access o real ime forecass, made by a weaher company, ha las for he following en days and are updaed every hour. Defining he vecor of known disurbances as d, he dynamics of he sysem become x +1 = Ax + B u u + B d d + B w w (4.17) where he marix B w is B w = [ ] T Bd due o he fac ha he forecas error for he ouside emperaure, w, is jus a scalar. The goal of he emperaure conroller we wan o design is o mainain he emperaure of each zone of he building, namely he oupus of he sysem, inside a given comfor region, while minimizing he power required. The bounds on he emperaure can be ime-varying, for example relaxing he requiremens during he nigh, bu are supposed o be known in advance, accordingly o a cerain schedule. The consrains acing on he oupu are hus he following y min while he bounds on he inpu are y y max 0 (4.18) u min u u max 0 (4.19) 77

88 hesis 2015/11/21 9:18 page 78 #88 Chaper 4. Applicaion examples where, wihou loss of generaliy, we can consider u min = 0. Due o he availabiliy of he forecass for he following hours and he explici presence of consrains on he emperaures and he inpu powers, he use of an MPC conroller can be very effecive. Furhermore, in pracice, violaions of he proposed bounds on he emperaure are usually allowed in erms of a maximum accumulaed violaion during he year, hus moivaing he use of a sochasic approach o explicily accoun for his relaxaion Disurbance model Consider now he descripion of he forecas error for he ouside emperaure, denoed wih w, in (4.17). In paricular a every ime sep we have access o a new forecas ha we can use in he MPC conroller o guess he fuure evoluion of he emperaures along he predicion horizon, and improve our decision. Pas forecass (one hour ahead) and real measuremens, can be analyzed o derive a sochasic descripion of he disurbance acing on he sysem and o improve he forecass made by he weaher company. An example of he forecas error for he period 01/12/2013 o 15/12/2013 is shown in Figure Figure 4.11: Forecas error on he ouside emperaure example. Pas daa are used o idenify he following model e +1 = e w 1 + w w N (0, 1.2) (4.20) Thanks o he esimaed model, he residual disurbance acing on he sysem has zero mean and a nearly Gaussian disribuion. Noe ha, in general, 78

89 hesis 2015/11/21 9:18 page 79 # Temperaure conrol on a realisic building he suppor of he disurbance is naurally unbounded, so ha, o apply robus policies, we have o pu some arificial bounds and allow he possibiliy o violae he consrains Sandard nominal MPC Wih he aim of highligh he advanages of probabilisic approaches, firsly we consider he applicaion of a sandard nominal MPC scheme o he problem described so far. In his case we consider he forecass we have as correc, jus neglecing any furher uncerainy. In paricular, we consider he following dynamics for he sysem o be conrolled x +1 = A x + B u ū + B d d ȳ = C x (4.21) In line wih he discussion in Chaper 2, he chosen cos funcion is he sandard quadraic cos on he oupu and he inpu J = N 1 k= x k 2 Q + ū k 2 R + x +N 2 S (4.22) For he simulaion we considered y min = 22 C and y max = 26 C, wih nighly relaxaions of 4 C, u min = 0 kw for all he acuaors and respecively u max = [10, 3, 10] kw, c i = 1. The sae and inpu weighing marices are chosen as Q = I, R = 10 I and he predicion horizon is se o N = 72 seps. As a resul, we obain an unconrolled frequency of violaion Robus MPC sraegy Consider now he applicaion of a robus MPC sraegy o avoid consrain violaions due o he parially unknown ouside emperaure. Noe ha, o compue he opimal soluion wih respec o he wors-case disurbance sequence along he predicion horizon, we need o pu arificial bounds on he disurbance ha will ac as a compromise beween he achieved robusness and he conservaiviy. In his case we ake he wors case-forecas error analyzing he pas daa and define he se W such ha w W. The error is jus a scalar, so ha he se W is an inerval W = {w : w w max } = { w : 79 [ ] } 1 w w max 1 (4.23)

90 hesis 2015/11/21 9:18 page 80 #90 Chaper 4. Applicaion examples Beween he many available conrol sraegies for formulaing a robus MPC we choose o adop an affine disurbance feedback policy as described in The simulaions are run considering he same seup used in he nominal case and bounding he disurbance o w 2 C. As a resul, we have he behavior shown in Figure 4.12 for he zone emperaures. In his case he effec of he wors case conroller is o keep he sysem far from he consrains so ha he resul is quie conservaive Zone emperaures Zone 1 Zone 2 Zone 3 21 Temperaure [?C] Time Figure 4.12: Zone emperaures and desired consrains wih robus MPC Sochasic MPC sraegies To overcome he limiaions shown in he previous secions, and in paricular o avoid he unconrolled number of consrains violaions of he nominal conroller, or he definiion of oo conservaive bounds of he robus case, we consider here wo of he sochasic MPC echniques described in Secion 4.1, namely he av-smpc and he p-smpc. In he av-smpc he consrain ighening is done accordingly o he closedloop behavior of he sysem iself, i.e., aking ino accoun he hisory of he occurred violaions. As in he robus case, a bound on he disurbance is required for feasibiliy purposes and hus we assume ha w W. In he 80

91 hesis 2015/11/21 9:18 page 81 # Temperaure conrol on a realisic building p-smpc he consrain ighening is performed in an open-loop perspecive bu he assumpion on bounded disurbance is relaxed, a he price of inroducing sochasic consrains also on he conrol inpus (for furher deails see Chaper 3). In boh he algorihms he cos funcion is he average of he classical quadraic cos in (4.24), i.e., we consider J = E { +N 1 i= x T i Qx i + u T i Ru i + x T +NSx +N } (4.24) where Q = C T C, R = 100 I 3, and S is compued as in he nominal and robus case. The disurbance is approximaed wih a Gaussian disribuion as described in Secion and is covariance (W in he p-smpc) is esimaed from he covariance of he available samples. The sae consrains are relaxed allowing for a probabiliy of violaing he consrains of p x = 0.2. For he av-smpc he parameers are chosen as ξ = 0.4, ξ = 0.2, α = 0.05, γ = 0.85 and n s = 3. As for he p-smpc he conroller gain, in order o simplify he algorihm and o use he non-linear version of he consrains (3.9), is fixed o K = K LQ and he inpu violaion probabiliy is se o p u = 0.1. The emperaure inside he zones is shown in Figure 4.13 for he av-smpc and in Figure 4.14 for he p-smpc case. As expeced in boh he cases we are able o carry he sysem o he desired posiion wih respec o he consrain and o obain an acual violaion frequency of 0.16 and 0.10, respecively. These values are much lower han he desired violaion frequency, probably due o he no exac modeling of he disurbance, however he resuling behavior is sill beer han he one obained in he robus case, hus moivaing he use of he proposed approaches. Noe ha, in general, he av-smpc is more flexible since a proper choice of he so-called loss funcion(i.e., he way he violaions are measured [76]) allows o model differen kind of sochasic consrain relaxaions (see Chaper 2. For example choosing such funcion as L(y) = max{y, 0}, insead of couning he number of violaions, allows o model expeced value consrains, exploiing specificaion on he maximum possible error averaged over ime common in he field of emperaure conrol. 81

92 i i hesis 2015/11/21 9:18 page 82 #92 i i Chaper 4. Applicaion examples Figure 4.13: Zone emperaures and desired consrains wih av-smpc. Figure 4.14: Zone emperaures and desired consrains wih p-smpc. 82 i i i i

93 hesis 2015/11/21 9:18 page 83 # Commens 4.4 Commens In his chaper we briefly presened four sochasic MPC algorihms for linear discree-ime sysems wih addiive bounded disurbances and linear individual chance consrains. Following he erminology adoped in Chaper 2, hree of hem are analyic mehods and he las one is a sample-based (scenario) mehod. These algorihms have been applied firsly o wo differen simulaion examples wih he aim of showing he behavior of he conrolled sysem over muliple simulaions wih respec o, respecively, he number of violaions of a single consrain poin-wise in ime and he long-run number of violaions of a cerain consrain in a receding horizon fashion. Comparisons are made in erm of conservaiveness, i.e. he measured frequency of violaion wih respec o he desired level, and online complexiy. Of course, due o he simpliciy of he examples adoped, we are no able o compleely show he possibiliies of each algorihm, however some general commens can be drawn. In paricular we noiced ha he -SMPC algorihm has proven o be very fas and easy o implemen, i gives beer resuls han sandard deerminisic approaches bu i is sill oo conservaive due o he offline use of mixed probabilisic wors-case ighenings. Thanks o he assumpion of bounded disurbances i guaranees feasibiliy and convergence. he av-smpc, wih he idea of adjusing he consrains based on acual violaion measuremens, represens a good compromise beween ube based sraegy, in which he ighenings are compleely offline and sample-based approaches (scenario) ha are compleely online. The main limiaion is due o he bounded disurbance assumpion ha is used o guaranee recursive feasibiliy of he scheme. he p-smpc has proven o be effecive and easy o design. Is main limiaion is relaed o he need for probabilisic inpu consrain ha is compensaed, however, by he guaraneed convergence and feasibiliy properies even in he case of unbounded disurbances. he s-smpc is really flexible and capable o solve a wide range of problems, ha he oher seleced approaches are no able o manage, wih a reduced offline complexiy. The only assumpion is basically he convexiy of he opimizaion problem once a disurbance sample is exraced. Even join chance consrains can be considered direcly wihou resoring o risk allocaion procedures. However, he algorihm can be very demanding especially if used ogeher wih sample 83

94 hesis 2015/11/21 9:18 page 84 #94 Chaper 4. Applicaion examples removal echniques. In general i no suied for small linear sysems, like he ones esed here. The hird example is he problem of conrolling he emperaure inside a realisic building based on real uncerainy (exernal emperaure) measures. The goal was o show he difference beween nominal, robus and sochasic MPC sraegies and o his end he p-smpc and he av-smpc algorihms have been implemened due o heir good compromise beween complexiy and conservaism. From he resuls i is easy o undersand he imporance of such sochasic Model Predicive Conrol echniques o overcome he limiaions of sandard robus MPC algorihms. 84

95 hesis 2015/11/21 9:18 page 85 #95 CHAPTER5 Oupu-feedback probabilisic SMPC In his chaper we propose, wih he aim of exending he p-smpc echnique presened in Chaper 3, an oupu-feedback sochasic Model Predicive Conrol algorihm for linear discree-ime sysems affeced by a possibly unbounded addiive noise wih no measurable sae and subjec o probabilisic consrains. Similarly o he p-smpc case, if he noise disribuion is unknown, he probabilisic consrains on he sae and inpu variables are reformulaed by means of he Canelli-Chebyshev s inequaliy. Again he recursive feasibiliy is guaraneed and he convergence of he sae o a suiable neighborhood of he origin is proved under mild assumpions. In addiion, implemenaion issues are horoughly addressed and wo examples are discussed in deails, wih he aim of providing an insigh ino he performance achievable by he proposed conrol scheme. As in he sae feedback case, he algorihm compuaional load can be made similar o he one of a sandard sabilizing MPC algorihm wih a proper choice of he design parameers (e.g., convering all consrains o linear marix inequaliies, LMIs), so ha he applicaion of he proposed approach o medium/large-scale problems is allowed. 85

96 hesis 2015/11/21 9:18 page 86 #96 Chaper 5. Oupu-feedback probabilisic SMPC 5.1 Problem saemen In his chaper we consider he following discree-ime linear sysem { x+1 = Ax + B u u + B w w 0 y = Cx + v (5.1) where x R n is he sae, u R m is he inpu, y R p is he measured oupu and w R nw, v R p are wo independen, zero-mean, whie noise processes wih covariance marices W 0 and V 0, respecively, and a-priori unbounded suppor. Wih respec o he previous chaper, here we remove he assumpion ha he sae is fully accessible and we focus on he use of he uncerain measures y ino he conrol scheme. To his end, he pair (A, C) is assumed o be observable, ogeher wih he reachabiliy of he pairs (A, B u ) and (A, Bw ), where marix B w saisfies T B wbw = Bw W Bw. T As in he sae-feedback case, polyopic consrains on he sae and inpu variables of sysem (5.1) are imposed as muliple independen chance consrains, i.e., i is required ha, for all 0 P{b T r x x max r } 1 p x r, r = 1,..., n r (5.2) P{c T s u u max s } 1 p u s, s = 1,..., n s (5.3) where p x r, p u s are again design parameers and i is assumed ha he se of relaions b T r x x max r, r = 1,..., n r (respecively, c T s u u max s, s = 1,..., n s ), define a convex se X (respecively, U) conaining he origin in is inerior Regulaor srucure For sysem (5.1), we wan o design a sandard regulaion scheme composed by he sae observer coupled wih he feedback conrol law ˆx +1 = Aˆx + B u u + L (y C ˆx ) (5.4) u = ū + K (ˆx x ) (5.5) where, as in Secion 3.1, x is he sae of he nominal model x +1 = A x + B u ū (5.6) 86

97 hesis 2015/11/21 9:18 page 87 # Problem saemen In (5.4), (5.5), he feedforward erm ū and he ime-varying gains L, K are design parameers o be seleced hrough a proper MPC scheme o guaranee convergence properies and he fulfillmen of he probabilisic consrains (5.2), (5.3). Wih respec o he previous case, he disance of he sae from is expeced value, namely δx, and hus is variance, depends now on he presence of boh he conroller and he observer. To model his siuaion, define he wo errors e = x ˆx ε = ˆx x (5.7a) (5.7b) Saring from (5.7) one can wrie he difference beween he sae and is expeced value as he sum of he wo errors e and ε δx = x x = e + ε (5.8) In [ order ] o ease he noaion, we can collec hem ino he vecor σ = e T ε T T whose dynamics, according o (5.1)-(5.7), is described by where σ +1 = [ ] w Φ σ + Ψ v (5.9) [ ] [ ] A L C 0 Bw L Φ =, Ψ = L C A + B u K 0 L Noe ha boh hese marices depend on he free variables L and K. In he following, similarly o Chaper 4 i is assumed ha, by a proper iniializaion, i.e. E {σ 0 } = 0, and hus recalling ha he processes v and w are zero mean, he enlarged sae σ of sysem (5.9) is zero-mean, so ha x = E{x }. Due o his assumpion and denoing by Σ = E { } σ σ T and by Ω = diag(w, V ) he covariance marices of σ and [w T v T ] T respecively, he evoluion of Σ is governed by Σ +1 = Φ Σ Φ T + Ψ ΩΨ T (5.10) 87

98 hesis 2015/11/21 9:18 page 88 #98 Chaper 5. Oupu-feedback probabilisic SMPC As in he sae-feedback algorihm, we are ineresed in using he sae and inpu covariance marices o handle probabilisic consrains, however we are forced o consider in he compuaions he overall error vecor, and is covariance Σ, due o he presence of he observer. In paricular, by definiion he variable δx defined by (5.8) is zero mean and is covariance marix X can be derived from Σ as X = E { δx δx T } = [ I I ] Σ [ I I ] T (5.11) Finally, leing also δu = u ū = K (ˆx x ), one has E {δu } = 0 and also he covariance marix U = E { } δu δu T can be obained from Σ as U = E { K ε ε T K T } = [ 0 K ] Σ [ 0 K ] T (5.12) This second order descripion of he sysem variables can now be used, as in he sae-feedback case, o reformulae he consrains Reformulaion of he probabilisic consrains To incorporae he probabilisic consrains (5.2) and (5.3) ino he MPC problem, we rely on he probabilisic approximaion mehod described in deail in Chaper 3, involving he sochasic variables second-order descripion, a he price of suiable ighening. This, save for he case of Gaussian disurbances, induces some conservaiveness. As in he sae-feedback case, we denoe here wih f(p) a funcion ha is f(p) = (1 p)/p in he case he disribuion of he sochasic variable is unknown or f(p) = N 1 (1 p), where N is he cumulaive probabiliy funcion of a Gaussian variable wih zero mean and uniary variance, in he case he variables are Gaussian. The probabilisic consrains (5.2)-(5.3), a ime i, are verified provided ha he following (deerminisic) inequaliies are saisfied b T r x i x max r b T r X i b r f(p x r) (5.13a) c T s ū i u max s c T s U i c s f(p u s) (5.13b) where he erms X i and U i depend now on he overall covariance marix Σ as in (5.11) and (5.12). Noe again ha, if he suppor of he noise erms w k and v k is unbounded, he definiion of he sae and conrol consrains in probabilisic erms is he 88

99 hesis 2015/11/21 9:18 page 89 # MPC algorihm: formulaion and properies only way o sae feasible conrol problems. In case of bounded noise he comparison, in erms of conservaiveness, beween he probabilisic framework and he deerminisic one has been discussed in he example of Secion MPC algorihm: formulaion and properies To formally sae he MPC algorihm for he compuaion of he regulaor parameers ū, L, K, he following noaion will be adoped: given a variable z or a marix Z, a any ime sep we will denoe by z +k and Z +k, k 0, heir generic values in he fuure, while z +k and Z +k will represen heir specific values compued based on he knowledge (e.g., measuremens) available a ime. The main ingrediens of he opimizaion problem are now inroduced Cos funcion Assume o be a ime and denoe by ū,...,+n 1 = {ū,..., ū +N 1 } he nominal inpu sequence over a fuure predicion horizon of lengh N. Moreover, define by K,...,+N 1 = {K,..., K +N 1 }, and L,...,+N 1 = {L,..., L +N 1 } he sequences respecively of he fuure conrol and observer gains, and recall ha he covariance Σ +k = E { σ +k σ T +k} evolves, saring from Σ, according o (5.10). In line wih he discussion in Secion and wih he choice in Chaper 3, he cos funcion o be minimized is he sum of wo componens. The firs one (J m ) accouns for he expeced values of he fuure nominal inpus and saes, while he second one (J v ) is relaed o he variances of he fuure errors e, ε, and of he fuure inpus. Specifically, he overall performance index is where J = J m ( x, ū,...,+n 1 ) + J v (Σ, K,...,+N 1, L,...,+N 1 ) (5.14) J m = J v = E E +N 1 { +N 1 i= i= { +N 1 x i 2 Q + ū i 2 R + x +N 2 S (5.15) i= x i ˆx i 2 Q L + x +N ˆx +N 2 S L } + ˆx i x i 2 Q + u i ū i 2 R + ˆx +N x +N 2 S } (5.16) 89

100 hesis 2015/11/21 9:18 page 90 #100 Chaper 5. Oupu-feedback probabilisic SMPC where he posiive definie and symmeric weighs Q, Q L, S, and S L mus saisfy he following inequaliy wih Q T S T + Φ T S T Φ 0 (5.17) [ ] A LC 0 Φ =, LC A + B u K Q T = diag(q L, Q + K T R K), S T = diag(s L, S), and he fixed gains K, L chosen so as o guaranee ha Φ,i.e., he error sysem in (5.9), is asympoically sable. By means of sandard compuaions, i is possible o wrie he cos (5.16) as follows J v = +N 1 i= r(q T,i Σ i ) + r(s T Σ +N ) (5.18) where Q T,i = diag(q L, Q+ K T i R K i ). Similarly o he sae-feedback case, from (5.14)-(5.16), i is apparen ha he goal is wofold: o drive he mean x o zero by acing on he nominal inpu componen ū,...,+n 1 and o minimize he variance of σ by acing on he gains K,...,+N 1 and L,...,+N 1. In addiion, also he pair ( x, Σ ) mus be considered as an addiional argumen of he MPC opimizaion, as laer discussed, o guaranee recursive feasibiliy Terminal consrains As usual in sabilizing MPC, see e.g. [100], some erminal consrains mus be considered. Based on he same raionale used in he Chaper 3, also he erminal condiion is applied hrough he second order descripion of he sae variables. In he new seup, he mean x +N and he variance Σ +N a he end of he predicion horizon mus saisfy x +N X F (5.19) Σ +N Σ (5.20) where X F is again he posiively invarian se such ha (A + B u K) x XF, x X F (5.21) 90

101 hesis 2015/11/21 9:18 page 91 # Saemen of he oupu feedback p-smpc problem while Σ is obained as he seady-sae soluion of he Lyapunov equaion (5.10), i.e., Σ = Φ ΣΦ T + Ψ ΩΨ T (5.22) [ ] Bw L where he marix Φ has been already defined, Ψ = and he collecive variance Ω = diag( W, V ) is buil by considering (arbirary) noise 0 L variances W W and V V. In addiion, and consisenly wih (5.13), he following coupling condiions mus be verified. b T r x x max r c T s K x u max s for all x X F, where b T Xb r r f(p x r), r = 1,..., n r (5.23a) c T s Ūc sf(p u s), s = 1,..., n s (5.23b) X = [ I I ] Σ [ I I ] T, Ū = [ 0 K] Σ [ 0 K] T (5.24) As i will be shown (see Theorem 4), (5.19) and (5.20) allow for recursive feasibiliy of he conrol scheme and enforce mean square convergence properies. Noe ha in (5.22), he choice of Ω is subjec o a radeoff. In fac, large variances W and V resul in large Σ and, in view of (5.24), large X and Ū. This enlarges he erminal consrain (5.20) bu, on he oher hand, reduces he size of he erminal se X F compaible wih (5.23). 5.3 Saemen of he oupu feedback p-smpc problem The formulaion of he main oupu-feedback p-smpc problem requires a preliminary discussion concerning he iniializaion, similar o he one in Chaper 3 bu wih he main difference ha here we have o deal wih observed daa. In principle, and in order o use he mos recen informaion available on he sae, a each ime insan i would be naural o se he curren value of he nominal sae x o he esimae ˆx and he covariance marix Σ o diag(σ 11, 1, 0), where Σ 11, 1 is he covariance of sae predicion error e obained using he observer (5.4) and canno be rese o zero. However, as already discussed, since we do no exclude he possibiliy of unbounded 91

102 hesis 2015/11/21 9:18 page 92 #102 Chaper 5. Oupu-feedback probabilisic SMPC disurbances, in some cases his choice could lead o infeasible opimizaion problems. On he oher hand, and in view of he erminal consrains (5.19), (5.20), i is quie easy o see ha recursive feasibiliy is guaraneed provided ha x is updaed according o he predicion equaion (5.6), which corresponds o he variance updae given by (5.10). These consideraions moivae he choice of accouning for he iniial condiions ( x, Σ ) as free variables, which will be seleced by he conrol algorihm (based on feasibiliy and opimaliy of he MPC opimizaion problem defined below) according o he following alernaive sraegies Sraegy 1 - Rese of he iniial sae: in he MPC opimizaion problem se x = ˆx, Σ = diag(σ 11, 1, 0). This corresponds o using all he informaion available hrough he observer a ime. Sraegy 2 - Predicion: in he MPC opimizaion problem se x = x 1, Σ = Σ 1. This corresponds o using he opimal predicion compued from he pas opimal soluion for boh he expeced value and he variance of he sae. The oupu-feedback p-smpc problem can now be saed. S-MPC problem: a any ime insan solve min x, Σ, ū,...,+n 1, J K,...,+N 1, L,...,+N 1 where J is defined in (5.14), (5.15), (5.16), subjec o - he dynamics (5.6) and (5.10); - he consrains (5.13) for all i =,..., + N 1, r = 1,..., n r, s = 1,..., n s ; - he iniializaion consrain (corresponding o he choice beween sraegies S1 and S2) ( x, Σ ) {(ˆx, diag(σ 11, 1, 0)), ( x 1, Σ 1 )} (5.25) - he erminal consrains (5.19), (5.20). Denoing by ū,...,+n 1 = {ū,..., ū +N 1 }, K,...,+N 1 = {K,..., K +N 1 }, L,...,+N 1 = {L,..., L +N 1 }, and ( x, Σ ) he opimal 92

103 hesis 2015/11/21 9:18 page 93 # Saemen of he oupu feedback p-smpc problem soluion o he p-smpc problem, he feedback conrol law acually used is hen given by (5.5) wih ū = ū, K = K, and he sae observaion evolves as in (5.4) wih L = L. Now, define he p-smpc problem feasibiliy se as Ξ := ( x 0, Σ 0 ) : ū 0,...,N 1, K 0,...,N 1, L 0,...,N 1 such ha (5.6), (5.10) and (5.13) hold for all k = 0,..., N 1 and (5.19), (5.20) are verified. Some commens are in order. 1) A he iniial ime = 0, he algorihm mus be iniialized by seing x 0 0 = ˆx 0 and Σ 0 0 = diag(σ 11,0, 0). In view of his, feasibiliy a ime = 0 amouns o (ˆx 0, Σ 0 0 ) Ξ. 2) The binary choice beween sraegies S1 and S2 requires o solve a any ime insan wo opimizaion problems. However, similarly o he saefeedback case, he following sequenial procedure can be adoped o reduce he average overall compuaional burden: he opimizaion problem corresponding o sraegy S1 is firs solved and, if i is infeasible, sraegy S2 mus be solved and adoped. On he conrary, if he problem wih sraegy S1 is feasible, i is possible o compare he resuling value of he opimal cos funcion wih he value of he cos ha corresponds o he feasible sequences {ū 1,..., ū +N 2 1, K x +N 1 }, {K 1,..., K +N 2 1, K} and {L 1,..., L +N 2 1, L}. If he opimal cos wih sraegy S1 is lower, sraegy S1 can be used wihou solving he MPC problem for sraegy S2. This does no guaranee opimaliy, bu he convergence properies of he mehod, saed in he resul below, are recovered while he compuaional effor is reduced. Now we are in he posiion o sae he main resul concerning he convergence properies of he algorihm. Theorem 2. If, a = 0, he p-smpc problem admis a soluion, i is recursively feasible and he sae and he inpu probabilisic consrains (5.2) and (5.3) are saisfied for all 0. Furhermore, if here exiss ρ (0, 1) such ha he noise variance Ω verifies (N + β ) α r(s T ΨΩΨ T ) < min(ρ σ 2, ρλ min ( Σ)) (5.26) α where σ is he maximum radius of a ball, cenered a he origin, included 93

104 hesis 2015/11/21 9:18 page 94 #104 Chaper 5. Oupu-feedback probabilisic SMPC in X F, and α = min{λ min (Q), r{q 1 + Q 1 β = max{λ max (S), r{s T }} L } 1 } (5.27a) (5.27b) hen, as + dis( x 2 + r{σ }, [0, 1 α (N + β α ) r(s T ΨΩΨ T )]) 0 (5.28) Noe ha, as expeced, for smaller and smaller values of Ω, also x and r{σ } end o zero. 5.4 Implemenaion issues In his secion wo issues are addressed. Firs, he non linear consrains (5.13) and he non linear dependence of he covariance evoluion, see (5.10), on K,...,+N 1, L,...,+N 1 make he numerical soluion of S-MPC impracical. In secions and wo possible soluions, inspired by he one adoped in Chaper 3, are described, allowing o cas he p-smpc problem as a quadraic one, wih linear consrains. The second issue concerns he fac ha, in our framework, deerminisic consrains on inpu variable (e.g., sauraions) are no accouned for. In Secion we propose some possible soluions o his problem Approximaion of he oupu-feedback p-smpc for allowing a soluion wih LMIs A soluion, based on an approximaion of S-MPC characerized by linear consrains solely, is now presened. Firs define A D = 2A, B D u = 2Bu, C D = 2C, and V D = 2V and le he auxiliary gain marices K and L be seleced according o he following assumpion. Assumpion 1. The gains K and L are compued as he seady-sae gains of he LQG regulaor for he sysem (A D, B D, C D ), wih sae and conrol weighs Q and R, and noise covariances W W and V V D. Noe ha, if a gain marix K (respecively L) is sabilizing for (A D + Bu D K) = 2(A + B u K) (respecively (A D LC D ) = 2(A LC)), i is also sabilizing for (A + B u K) (respecively (A LC)), i.e., for he original sysem. The following preliminary resul can be saed. 94

105 hesis 2015/11/21 9:18 page 95 # Implemenaion issues Lemma 3. Define A D L = A D L C D, A D K = A D + Bu D K, he block diagonal marix Σ D = diag(σ D 11,, Σ D 22,), Σ D 11, R n n, Σ D 22, R n n and he updae equaions Σ D 11,+1 =A D L Σ D 11,(A D L ) T + B w W Bw T + L V D L T (5.29a) Σ D 22,+1 =A D K Σ D 22,(A D K ) T + L C D Σ D 11,C D T L T + L V D L T (5.29b) Then I) Σ D Σ implies ha Σ D +1 = diag(σ D 11,+1, Σ D 22,+1) Σ +1. II) We can rewrie as LMIs he following inequaliies Σ D 11,+1 A D L Σ D 11,(A D L ) T + B w W Bw T + L V D L T (5.30a) Σ D 22,+1 A D K Σ D 22,(A D K ) T + L C D Σ D 11,C D T L T + L V D L T (5.30b) Based on Lemma 3-II, we can reformulae he original problem so ha he covariance marix Σ D is used insead of Σ. Accordingly, he updae equaion (5.10) is replaced by (5.29) and he oupu feedback p-smpc problem is recas as an LMI one (see Appendix 5.6), grealy reducing he required compuaional effor a he price of a sligh exra ighening. As for he nonlinear dependence of he inequaliies (5.13) on he covariance marices X and U, i is again possible o prove ha hey are saisfied if b T r x i (1 0.5α x )x max r c T s ū i (1 0.5α u )u max s bt r X i b r f(p x 2α x x r) 2 max r ct s U i c s f(p u 2α u u s) 2 max s (5.31a) (5.31b) where α x (0, 1] and α u (0, 1] are free design parameers (for a more deailed explanaion see Chaper 3). Also, noe ha X [ I I ] [ ] Σ D T I I = Σ D 11, + Σ D 22, and ha U [ ] [ ] 0 K Σ D T 0 K = K Σ D 22,K T so ha, defining X D = Σ D 11, + Σ D 22, and U D = K Σ D 22,K T, (5.31) can be wrien as follows b T r x i (1 0.5α x )x max r bt r Xi D b r f(p x 2α x x r) 2 max c T s ū i (1 0.5α u )u max s ct s Ui D c s f(p u 2α u u s) 2 max r s (5.32a) (5.32b) 95

106 hesis 2015/11/21 9:18 page 96 #106 Chaper 5. Oupu-feedback probabilisic SMPC Noe ha he reformulaion of (5.13) ino (5.32) has been performed a he price of addiional consrain ighening. For example, on he righ hand side of (5.32a), x max r is replaced by (1 0.5α x )x max r, which significanly reduces he size of he consrain se. Parameer α x canno be reduced a will, since i also appears a he denominaor in he second addiive erm. In view of Assumpion 1 and resoring o he separaion principle, i is possible o show [52] ha he soluion Σ D o he seady-sae equaion Σ D = Φ D ΣD (Φ D ) T + Ψ ΩΨ T (5.33) is block-diagonal, i.e., Σ D = diag( Σ D 11, Σ D 22), where [ A D Φ D = LC D ] 0 LC D A D + Bu D K Based on he same raional, he erminal consrain (5.20) mus be ransformed ino Σ D +N Σ D which corresponds o seing Σ D 11,+N Σ D 11, Σ D 22,+N Σ D 22 (5.34) aking advanage of he block-diagonal form. Defining X D = Σ D 11 + Σ D 22 and Ū D = K Σ D 22 K T, he erminal se condiion (5.23) mus now be reformulaed as b T r x (1 0.5α x )x max r bt X r D b r f(p x 2α x x r) 2 max c T s K x (1 0.5α u )u max s for all x X F. Also J v mus be reformulaed. Indeed J v J D v = +N 1 i= r ct s Ū D c s f(p u 2α u u s) 2 max r { Q L Σ D 11,i + QΣ D 22,i + RK i Σ D 22,iK T i + r { S L Σ D 11,+N + SΣ D 22,+N s } } (5.35a) (5.35b) (5.36) where he erminal weighs S and S L mus now saisfy he following Lyapunovlike inequaliies (ĀD K )T SĀD K S + Q + K T R K 0 (ĀD L )T S L Ā D L S L + Q L + (C D ) T LT S LC D 0 (5.37) 96

107 hesis 2015/11/21 9:18 page 97 # Implemenaion issues where ĀD K = AD + B D u K and ĀD L = AD LC D. I is now possible o formally sae wha we call he p-smpcl problem (o denoe ha here he gains are considered as free variables in opposiion o wha we sae in he nex secion). S-MPCl problem: a any ime insan solve min x, Σ D 11,, Σ D 22,, ū,...,+n 1, K,...,+N 1, L,...,+N 1 where J is defined in (5.14), (5.15), (5.36), subjec o - he dynamics (5.6) and (5.29); - he linear consrains (5.32) for all i =,..., +N 1, r = 1,..., n r, s = 1,..., n s ; - he iniializaion consrain, corresponding o he choice beween sraegies S1 and S2, i.e., ( x, Σ D 11,, Σ D 22,) {(ˆx, Σ D 11, 1, 0), ( x 1, Σ D 11, 1, ΣD 22, 1 )} - he erminal consrains (5.19), (5.34). J The following corollary follows from Theorem 4. Corollary 1. If, a ime = 0, he p-smpcl problem admis a soluion, i is recursively feasible and he sae and inpu probabilisic consrains (5.2) and (5.3) are saisfied for all 0. Furhermore, if here exiss ρ (0, 1) such ha he noise variance Ω D = diag(w, V D ) verifies hen, as + (N + β ) α r(s T ΨΩ D Ψ T ) < min(ρ σ 2, ρλ min ( Σ D )) (5.38) α dis( x 2 + r{σ D }, [0, 1 α (N + β α ) r(s T ΨΩ D Ψ T )]) 0 The proof of Corollary 1 readily follows from he proof of Theorem 4. For deails please see [46]. 97

108 hesis 2015/11/21 9:18 page 98 #108 Chaper 5. Oupu-feedback probabilisic SMPC Approximaion of p-smpc wih consan gains The soluion presened in he following is characerized by a grea simpliciy and consiss in seing L = L and K = K for all 0. In his case, he value of Σ +k (and herefore of X +k and U +k ) can be direcly compued for any k > 0 by means of (5.10) as soon as Σ is given. As a byproduc, he nonlineariy in he consrains (5.13) does no carry abou implemenaion problems. Therefore, his soluion has a wofold advanage: firs, i is simple and requires an exremely lighweigh implemenaion; secondly, i allows for he use of nonlinear less conservaive consrain formulaions. In his simplified framework, he following problem, denoed in he res as p-smpcc in conras o he p-smpcl, can be saed. p-smpcc problem: a any ime insan solve min J x,σ,ū,...,+n 1 where J is defined in (5.14), (5.15), (5.16), subjec o - he dynamics (5.6), wih K = K, and Σ +1 = ΦΣ Φ T + ΨΩΨ T (5.39) - he consrains (5.13) for all i =,..., + N 1, r = 1,..., n r, s = 1,..., n s ; - he iniializaion consrain (5.25); - he erminal consrains (5.19), (5.20). An addiional remark is due. The erm J v in (5.18) does no depend only on he conrol and observer gain sequences K,...,+N 1, L,...,+N 1, bu also on he iniial condiion Σ. Therefore, i is no possible o discard i in his simplified formulaion. The following corollary can be derived from Theorem 4. Corollary 2. If, a ime = 0, he p-smpcc problem admis a soluion, i is recursively feasible and he sae and inpu probabilisic consrains (5.2) and (5.3) are saisfied for all 0. Furhermore, if here exiss ρ (0, 1) such ha he noise variance Ω verifies (5.26), hen, as +, (5.28) holds. The proof of Corollary 2 readily follows from he proof of Theorem 4. For deails please see [46]. 98

109 hesis 2015/11/21 9:18 page 99 # Proof of he main Theorem Boundedness of he inpu variables Boh he sae-feedback p-smpc presened in Chaper 3 and he oupufeedback p-smpc scheme described in he previous secions canno handle hard consrains on he inpu variables. However, in general inpu variables are bounded in pracice, and may be subjec o Hu 1 (5.40) where H R n H m is a design marix and 1 is a vecor of dimension n H whose enries are equal o 1. Three possible approaches are proposed o accoun for his case. Inequaliies (5.40) can be saed as addiive probabilisic consrains (5.3) wih small violaion probabiliies p u s. This soluion, alhough no guaraneeing saisfacion of (5.40) wih probabiliy 1, is simple and easy. In he p-smpcc scheme, define he gain marix K in such a way ha A + B u K is asympoically sable and, a he same ime, H K = 0. From (5.5), i follows ha Hu = Hū +H K(ˆx x ) = Hū. Therefore, o verify (5.40) i is sufficien o include in he problem formulaion he deerminisic consrain Hū 1. In he p-smpcc scheme, if probabilisic consrains on u are absen, replace (5.5) wih u = ū and se Hū 1 in he p-smpc opimizaion problem o verify (5.40). If we also define û = ū + K(ˆx x ) as he inpu o equaion (5.4), he dynamics of variable σ is given by (5.9) wih [ ] A LC Bu K Φ = LC A + B u K and he argumens follow similarly o hose proposed in he paper. I is worh menioning, however, ha marix Φ mus be asympoically sable, which requires asympoic sabiliy of A. 5.5 Proof of he main Theorem Recursive feasibiliy is firs proved. Assume ha, a ime insan, a feasible soluion of S-MPC is available, i.e., ( x, Σ ) Ξ wih opimal 99

110 hesis 2015/11/21 9:18 page 100 #110 Chaper 5. Oupu-feedback probabilisic SMPC sequences ū,...,+n 1, K,...,+N 1, and L,...,+N 1. We prove ha a ime + 1 a feasible soluion exiss, i.e., in view of he iniializaion sraegy S2, ( x +1, Σ +1 ) Ξ wih admissible sequences ū f +1,...,+N = {ū +1,..., ū +N 1, K x +N }, K f +1,...,+N = {K +1,..., K +N 1, K}, and L f +1,...,+N = {L +1,..., L +N 1, L}. Consrain (5.13a) is verified for all pairs ( x +1+k, X +1+k ), k = 0,..., N 2, in view of he feasibiliy of p-smpc a ime. Furhermore, in view of (5.19), (5.20), (5.24), and he condiion (5.23a),we have ha b T x +N x max b T X +N bf(p x r), i.e., consrain (5.13a) is verified. Analogously, consrain (5.13b) is verified for all pairs (ū +1+k, U +1+k ), k = 0,..., N 2, in view of he feasibiliy of p-smpc a ime. Furhermore, in view of (5.19), (5.20), (5.24), and he condiion (5.23b), we have ha c T K x+n u max c T U +N cf(p u s), i.e., consrain (5.13b) is verified. In view of (5.19) and of he invariance propery (5.21) i follows ha x +N+1 = (A+B u K) x+n X F and, in view of (5.20), (5.22) Σ +N+1 Φ ΣΦ T + Ψ ΩΨ T = Σ, hence verifying boh (5.19) and (5.20) a ime + 1. The proof of convergence is parially inspired by [136]. In view of he feasibiliy, a ime + 1 of he possibly subopimal soluion ū f +1,...,+N, K f +1,...,+N, Lf +1,...,+N, and ( x +1, Σ +1 ), we have ha he opimal cos funcion compued a ime + 1 is J ( + 1) = J m( + 1) + J v ( + 1) 1. In view of he opimaliy of J ( + 1) J ( + 1) J m ( x +1, ū f +1,...,+N ) (5.41) Noe ha, in view of (5.17) + J v (Σ +1, K f +1,...,+N, Lf +1,...,+N ) Furhermore J m ( x +1, ū f +1,...,+N ) J m ( x, ū,...,+n 1 ) x 2 Q ū 2 R (5.42) J m ( x, ū,...,+n 1 ) = J m() (5.43) 1 For breviy, we denoe J (x, x 1, Σ 1 ) wih J (), Jm (x, x 1, Σ 1 ) wih Jm (), and Jv (x, x 1, Σ 1 ) wih Jv () 100

111 hesis 2015/11/21 9:18 page 101 # Proof of he main Theorem Now consider J v in (5.18) and noe ha, in view of he properies of he race and (5.17) J v (X +1, K f +1,...,+N, Lf +1,...,+N ) J v (X, K,...,+N 1, L,...,+N 1 ) [ ] QL 0 r{ 0 Q + K T RK Σ } + r(s T ΨΩΨ T ) (5.44) From (6.32)-(5.44) we obain J ( + 1) J () ( x 2 Q + ū 2 R [ ] ) QL 0 r{ 0 Q + K T RK Σ } + r(s T ΨΩΨ T ) (5.45) Furhermore, from he definiion of J () J () x 2 Q + ū 2 R {[ ] } QL 0 + r 0 Q + K T RK Σ (5.46) Now, denoe Ω F = {( x, Σ) : x X F, Σ Σ}. Assuming ha ( x, Σ ) Ω F we have ha J () Jm aux () + Jv aux (), where J aux m () = N 1 k=0 (A + B u K) k x 2 Q + K(A + B u K)k x 2 R + (A + B u K) N x 2 S since { K x,..., K(A+B u K) N 1 x } is a feasible inpu sequence. Therefore, from (5.17), Similarly, recalling (5.17), we obain ha J aux m () x 2 S (5.47) J aux v () r{s T Σ } + Nr{S T ΨΩΨ T } (5.48) Combining (5.47) and (5.48) we obain ha, for all ( x, Σ ) Ω F J () x 2 S + r{s T Σ } + N r{s T ΨΩΨ T } (5.49) 101

112 hesis 2015/11/21 9:18 page 102 #112 Chaper 5. Oupu-feedback probabilisic SMPC From (5.45), (5.46) and (5.49) i is possible o derive robus sabiliy-relaed resuls. Before o proceed, recall ha r{s T Σ } = r{s 1 2 T T Σ S 1 2 T } where S 1 2 T is a marix ha verifies S 1 2 T T S 1 2 T = S T. Therefore r{s T Σ } = r{s 1 2 T T Σ S 1 2 T } = Σ 1 2 S 1 2 T 2 F On he oher hand, denoing Q T = diag(q L, Q + K T RK ), i follows ha r{q T Σ } = Σ 1 2 Q 1 2 T 2 F. Moreover, r{s T Σ } Σ F S 1 2 T 2 F = r{s T }r{σ } (where F is he Frobenius norm) and, in view of he marix inversion Lemma, r{q T Σ } ( Q 1 2 T 2 F ) 1 Σ F r{(diag(q L, Q)) 1 } 1 r{σ } = r{q 1 + Q 1 L } 1 r{σ }. Define V ( x, Σ ) = x 2 + r{σ } and ω = r{s T ΨΩΨ T }. In view of his, we can reformulae (5.45), (5.46) and (5.49) as follows. J ( + 1) J () αv ( x, Σ ) + ω J () αv ( x, Σ ) J () βv ( x, Σ ) + Nω (5.50a) (5.50b) (5.50c) If ( x, Σ ) Ω F hen, in view of (5.50c), (5.50a) J ( + 1) J ()(1 α β ) + (α N + 1)ω (5.51) β Le η (ρ, 1) and denoe b = 1(N + β ). In view of (5.50b), if J () b ω η α hen V ( x, Σ ) b ω. This, considering (5.26), implies ha α x 2 ρ η σ2, r(σ ) ρ η λ min( Σ) (5.52) 102

113 hesis 2015/11/21 9:18 page 103 # The p-smpcl problem In view of (5.52), hen x X F and λ max (Σ ) < λ min ( Σ), which in urn implies ha Σ < Σ. Therefore, recalling (5.51), if J () b ω, hen J ( + 1) b ω and he posiive invariance of he se D = {( x, Σ) : J () b ω} is guaraneed. If ( x, Σ ) Ω F \D, i holds ha J () > b ω which, in view of (5.50c), implies ha V ( x, Σ ) > 1 α ω (5.53) Since ( x, Σ ) Ω F \D, recalling (5.51), (5.53), and (5.50b), here exiss c 1 > 0 (funcion of η) such ha J ( + 1) J () (1 η) α2 β V ( x, Σ ) c 1 (5.54) On he oher hand, for all x wih ( x, Σ ) Ξ\Ω F, here exiss consan c 2 > 0 such ha here exiss x Ω wih ( x Ω, Σ Ω ) Ω F \D such ha αv ( x, Σ ) αv ( x Ω, Σ Ω ) c 2. This, in view of (5.50a) and (5.54), implies ha J ( + 1) J () < c 2 (5.55) In view of (5.54)-(5.55), for all x wih ( x, Σ ) Ξ\D here exiss c (funcion of η) J ( + 1) J () < c (5.56) This implies ha, for each η (ρ, 1), here exiss T > 0 such ha x +T is such ha ( x +T +T, Σ +T +T ) D, i.e., ha αv ( x +k +k, Σ +k +k ) bω for all k T. This, for η 1, implies (5.28). 5.6 The p-smpcl problem Proof of Lemma 3 For he proof of Par I., he following resul is used. Lemma 4. Given a posiive semi-definie, symmeric marix M, hen [ ] [ ] M11 M 12 2M11 0 M = M12 T M M

114 hesis 2015/11/21 9:18 page 104 #114 Chaper 5. Oupu-feedback probabilisic SMPC Proof of Lemma 4 Since M 0, hen [ ] [ ] x1 x T 1 x T 2 M = x T 1 M 11 x 1 +x T 2 M 22 x 2 x T 1 M 12 x 2 x T 2 M x 12x T for all x 1, x 2 such ha [ x T 1 x T 2 ] 0. From his, we obain ha [ ] [ ] x1 x T 1 x T 2 M = x T 1 M 11 x 1 + x T 2 M 22 x 2 + x T 1 M 12 x 2 + x T 2 M x 12x T 1 2 2x T 1 M 11 x 1 + 2x T 2 M 22 x 2 = [ ] [ ] [ ] 2M x T 1 x T 11 0 x M 22 for all x 1, x 2 such ha [ x T 1 x T 2 ] 0. This concludes he proof of Lemma 4. Consider now marix Σ and is block decomposiion [ ] Σ11, Σ 12, Σ = Σ T 12, Σ 22, where Σ ij, R n n for all i, j = 1, 2. A bound for he ime evoluion of he covariance marix Σ is compued, ieraively, considering ha Σ +1 Φ Σ D Φ T + Ψ ΩΨ T (5.57) If we define Σ D +1 = diag(σ D 11,+1, Σ D 22,+1), where Σ D 11,+1 = 2(A L C)Σ D 11,(A L C) T + B w W B T w + 2L V L T (5.58) Σ D 22,+1 = 2(A + B u K )Σ D 22,(A + B u K ) T + 2L CΣ D 11,C T L T + 2L V L T (5.59) hen we obain ha Σ D +1 Σ +1, in view of Lemma 4. The laer corresponds wih (5.29). Par IIa. LMI reformulaion of he updae of Σ D 11,k. We define Γ k = diag(σ D 11,k, W 1, (V D ) 1 ), Θk = [ A D L k F W L k V D] and we rewrie consrain (5.30a) as Σ D 11,k+1 Θ k Γ k ΘT k 0. Resoring o he Schur complemen i is possible o derive he equivalen form Γ 1 k 104 x 2

115 hesis 2015/11/21 9:18 page 105 # The p-smpcl problem Θ T k (ΣD 11,k+1 ) 1 Θk 0. To obain a linear inequaliy from he previous expression we define Z k = (Σ D 11,k+1) 1 L k (5.60) and Σ D 11,i = (Σ D 11,i) 1, i.e., diag( Σ D 11,k, W, V D ) Φ T k ( Σ D 11,k+1 ) 1 Φ k 0, where Φ k = [( Σ D 11,k+1 AD Z k C D ), Σ D 11,k+1 F W, Z kv D ]. The laer expression can be wrien as a compac LMI as follows Σ D 11,k W V D Φ T k 0 (5.61) Φ k ΣD 11,k+1 Noice ha, however, in he consrains (5.32) and in he cos funcion (5.36), he erm Σ D 11,i appears, raher han is inverse Σ D 11,i. To solve his issue, we define marix k as an upper bound o Σ D 11,k (i.e., k Σ D 11,k ), which can be recovered from Σ D 11,k+1 hrough he following linear inequaliy [ ] k I 0 (5.62) I ΣD 11,k Then, one should replace Σ D 11,k wih k in (5.32) and (5.36). Par IIb. Reformulaion of he updae of Σ D 22,k. Consider now he inequaliy (5.30b), i.e., Σ D 22,k+1 (A D + B D u K k )Σ D 22,k(A D + B D u K k ) T L k (C D Σ D 11,kC D T + V D )L T k 0 (5.63) Recalling (5.60), (5.63) can be rewrien as Σ D 22,k+1 (AD +Bu D K k )Σ D 22,k (AD + Bu D K k ) T Σ D 11,k+1 M kσ D 11,k+1 0, where M k = Z k (C D Σ D 11,kC D T + V D )Z T k (5.64) By defining Ξ k = K k Σ D 22,k, and using he marix k+1 in place of Σ D 11,k+1, he inequaliy (5.63) can be recas as a suiable LMI. In fac, in view of he Schur complemen Lemma and leing M k = M 1, we obain k [ Σ D 22,k+1 (A D Σ D 22,k + ] BD u Ξ k ) k+1 [ (A D Σ D 22,k + BD u Ξ k ) T ] [ ] Σ D 22,k 0 k+1 0 Mk (5.65)

116 hesis 2015/11/21 9:18 page 106 #116 Chaper 5. Oupu-feedback probabilisic SMPC The equaion (5.64) can be recas as he inequaliy M k Z k (C D Σ D 11,k CD T + V D )Zk T, which can be reformulaed as [ M k Zk V D Z k C D] [ (Zk V D ) T ] [ ] V D 0 0 (5.66) (Z k C D ) T 0 ΣD 11,k Finally, concerning he equaliy M k = M 1 k, i can be solved using he approach proposed in [40]. Indeed, we solve he following LMI [ ] Mk I 0 (5.67) I Mk and, a he same ime, we minimize he addiional cos funcion r{m k Mk } (5.68) The problem (5.67)-(5.68) can be managed using he recursive cone complemenariy linearizaion algorihm proposed in [40] wih a suiable iniializaion LMI reformulaion of he consrains While he consrain (5.31a) is a linear inequaliy (and herefore i does no need o be furher reformulaed), he inequaliy (5.31b) needs special aenion. As already remarked, in (5.31b), U k mus be replaced by Ūk. In urn, he equaliy Ūk = K k Σ D 22,k KT k = Ξ k(σ D 22,k ) 1 Ξ T k mus be recas as an LMI as follows: ] [Ūk Ξ k Ξ T k Σ D 0 (5.69) 22,k 5.7 Simulaion example In his secion he efficiency of he proposed approach is shown by means of a simple example inspired by [102]. In paricular we consider he sysem in (5.1) where we se A = [ ] 1 1, B = 0 1 [ ], F = I 2, C = I 2 and we assume ha noise is Gaussian, wih W = 0.01I 2 and V = 10 4 I 2. The probabilisic consrains are P{x 2 2} 0.1, P{u 1} 0.1, and 106

117 hesis 2015/11/21 9:18 page 107 # Simulaion example P{ u 1} 0.1. In (5.14), (5.15), and (5.18) we se Q L = Q = I 2, R = 0.01, and N = 9. In Figure 5.1 we compare he feasible ses obained wih he mehods presened in Secion 5.4, and under differen assumpions concerning he noise (namely p-smpcc (1), p-smpcc (2), p-smpcl (1), p-smpcl (2), where (1) denoes he case of Gaussian disribuion and (2) denoes he case when he disribuion is unknown). Apparenly, in view of he linearizaion of he consrains (see he discussion afer (5.32)), he p-smpcl algorihm is more conservaive han p-smpcc. On he oher hand, concerning he dimension of he obained feasibiliy se, in his case he use of he Chebyshev - Canelli inequaliy does no carry abou a dramaic performance degradaion in erms of conservaiveness x S MPCc (1) S MPCc (2) S MPCl (1) S MPCl (2) x 1 Figure 5.1: Plos of he feasibiliy ses for S-MPCc (1), S-MPCc (2), S-MPCl (1), S-MPCl (2) In Figure 5.2 we show he evoluion of he sae variables x 1 and x 2, respecively, using he differen conrol approaches, for 200 Monecarlo runs, saring from iniial condiion (5, 1.5). Also, in Figure 5.3 we show he corresponding inpus. Apparenly, he fac ha he conrol and esimaion 107

118 hesis 2015/11/21 9:18 page 108 #118 Chaper 5. Oupu-feedback probabilisic SMPC gains are free variables makes he ransien behaviour of he sae responses in case of p-smpcl more performing, wih respec o he case when he p-smpcc is used. For a more deailed analysis, please see Table 5.1, where i is winessed ha he overshoo and he variance of he dynamic sae response are reduced in case of p-smpcl, a he price of a more reacive inpu response. The fac ha he conrol and esimaion gains are free variables makes he ransien behaviour of he sae responses in case of p-smpcl more performing and reduces he variance of he dynamic sae response (a he price of a more reacive inpu response), wih respec o he case when p-smpcc is used. For example, he maximum variance of x 1 (k) (resp. of x 1 (k)) is abou 0.33 (resp ) in case of p-smpcc (1) and (2), while i resuls abou 0.25 (resp ) in case of p-smpcl (1) and (2). On he oher hand, he maximum variance of u(k) is abou in case of p-smpcc, while i is in case of p-smpcl. S-MPCc (1) S-MPCc (2) S-MPCl (1) S-MPCl (2) Overshoo on E{x 1 (k)} Overshoo on E{x 2 (k)} Overshoo on E{u(k)} Max variance of x 1 (k) Max variance of x 2 (k) Max variance of u(k) Table 5.1: Comparison of he dynamic behaviour of he rajecories using he differen approaches. 5.8 Commens Wih he aim of filling a gap in he lieraure of sochasic Model Predicive Conrol, in his chaper an exension o he oupu-feedback case of he so-called p-smpc approach presened in Chaper 3 has been proposed. Wih respec o he sae-feedback case he algorihm is complicaed by he need for a sae observer whose gain is considered, similarly o he feedback conroller gain, as a free parameer of he opimizaion problem. Feasibiliy and convergence properies are sill guaraneed in he case of unbounded disurbances. The main feaures of he proposed probabilisic MPC algorihm lie in is simpliciy and in is ligh-weigh compuaional load, boh in he off-line design phase and in he online implemenaion. This allows for he applicaion of he p-smpc scheme o medium/large-scale problems, 108

119 hesis 2015/11/21 9:18 page 109 # Commens for sysems affeced by general disurbances. Similarly o he sae feedback p-smpc, he main limiaions wih respec o exising approaches are due o he difficuly o cope wih sauraions in he inpu variables and in he subopimaliy of he soluion when he Canelli-Chebyshev s inequaliy is used insead of he full noise characerizaion. 109

120 hesis 2015/11/21 9:18 page 110 #120 Chaper 5. Oupu-feedback probabilisic SMPC S MPCc Gaussian noise S MPCc Unknown disribuion 4 4 x 1 (k) 2 x 1 (k) S MPCl Gaussian noise S MPCl Unknown disribuion 4 4 x 1 (k) 2 x 1 (k) S MPCc Gaussian noise S MPCc Unknown disribuion x 2 (k) 1 x 2 (k) sep k S MPCl Gaussian noise sep k S MPCl Unknown disribuion x 2 (k) 1 x 2 (k) sep k sep k Figure 5.2: Trajecories x 1 (lef) and x 2 (k) (righ) for 200 runs (ligh grey lines), sampled mean value (black solid line line), mean value ± sampled sandard deviaion (doed dark grey lines), maximum and minimum values (dashed dark grey lines). 110

121 hesis 2015/11/21 9:18 page 111 # Commens S MPCc Gaussian noise S MPCc Unknown disribuion u(k) 0 u(k) sep k S MPCl Gaussian noise sep k S MPCl Unknown disribuion u(k) 0 u(k) sep k sep k Figure 5.3: Trajecories u(k) for 200 runs (ligh grey lines), sampled mean value (black solid line line), mean value ± sampled sandard deviaion (doed dark grey lines), maximum and minimum values (dashed dark grey lines). 111

122 hesis 2015/11/21 9:18 page 112 #122

123 hesis 2015/11/21 9:18 page 113 #123 CHAPTER6 Disribued Predicive Conrol for regulaion: a sochasic approach In his chaper, aking advanage of he sochasic perspecive discussed in he firs par of he Thesis, we propose a novel Disribued Predicive Conrol approach for linear dinamically inerconneced subsysems affeced by addiive, possibly unbounded, sochasic noise and subjec o probabilisic consrains on he sae and he inpu. The aim is o obain an algorihm ha is able o deal wih disurbances wih possibly unbounded suppor. In paricular, he p-smpc algorihm described in Chaper 3 is adoped, wih some modificaions, inside a disribued conrol scheme and recursive feasibiliy and convergence properies are proven. Even if he resuls can be apparenly very conservaive, his mehod allows o move he firs seps owards he developmen of efficien sochasic disribued predicive conrol soluions ha can overcome he limiaions of sandard robus approaches. In his chaper, in order o simplify he conrol problem, we assume ha he subsysems are coupled only hrough heir saes while each of hem is subjec only o local probabilisic consrains on he sae and he inpu. Exensions o he case of coupling probabilisic consrains are possible using, for example, he approach proposed in [127]. 113

124 hesis 2015/11/21 9:18 page 114 #124 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach 6.1 Problem saemen Consider a se of M dynamically coupled subsysems, each one described by he following linear dynamics x [i] +1 = A ii x [i] + B u,i u [i] + A ij x [j] + B w,i w [i], i = 1,..., M (6.1) j i where x [i] R n i, u [i] R m i are he sae and he inpu of he i-h subsys- R p i is a zero-mean independen whie noise wih known variance W i, possibly unbounded suppor and such ha w [i] and w [j] are uncorrelaed for all i j. Perfec sae informaion is available em, respecively, while w [i] and all he pairs (A ii, B u,i ) are assumed o be sabilizable. The i-h subsysem is denoed in he sequel wih S i, i = 1,..., M, and any subsysem S j is said o be a neighbor of subsysem S i if and only if A ij 0, i.e., if and only if he saes x [j] of S j influence he dynamics of S i. The se of neighbors of S i is denoed here by N i. In addiion, in line wih he discussion in he previous chapers, we assume ha each subsysem S i is subjec o a se of local probabilisic consrains on he sae and he inpu of he form (2.18), i.e., for each subsysem S i we consider he following se of individual chance consrains { P P b T r,ix [i] { c T s,iu [i] x max r,i u max s,i } 1 p x r,i, r = 1,..., n ri, 0 } 1 p u s,i, s = 1,..., n si, 0 (6.2a) (6.2b) where, as discussed in deail in Chaper 2, P {φ} denoes he probabiliy of φ and he values p x r,i and p u s,i are considered as design parameers, wih he meaning ha we are allowing he single consrain o be violaed up o he specified probabiliy poinwise in ime. Coupling consrains are no considered here o simplify he seup, however exensions o his case are possible following he approach in [127]. Concerning he consrain se for he sae and for he inpu X [i] : {x b T r,ix x max r,i, r = 1... n ri } U [i] : {u c T s,iu u max s,i, s = 1... n si } we assume ha hey are nonempy and conaining he origin in heir inerior. 114

125 hesis 2015/11/21 9:18 page 115 # Problem saemen For he developmen of he disribued conroller, i is worh defining he dynamics of he collecive sysem. To his end, denoing by v = (v [1],..., v [s] ) he shor-hand of a vecor wih v [1],..., v [s] componens, le he collecive sae, inpu and disurbance vecor be, respecively, x = (x [1] R n, n = M i=1 n i, u = (u [1],..., u [M] ) R m, m = M w = (w [1],..., w [M] he collecive sysem marices, i.e. A 11 A A 1M A 21 A A =, A M A MM Thus, he overall dynamical model is,..., x [M] ) i=1 m i, and ) R p, p = M i=1 p i, and define by A, B and B w B = diag(b u,1,..., B u,m ) B w = diag(b w,1,..., B w,m ) x +1 = Ax + Bu + B w w (6.3) We also define W = var(w ) = diag(w 1,..., W M ). Wih reference o he collecive sysem in (6.3), some srucural requiremens are inroduced by he following sanding assumpion Assumpion 2. (i) The pair (A, G w ) is sabilizable, where G w is such ha G w G T w = B w WB w T. (ii) There exiss a decenralized gain K = diag(k 1,..., K M ) such ha he marices (A + BK) and (A ii + B i K i ), i = 1,..., M are Schur sable Regulaor srucure Inspired by he p-smpc approach presened in Chaper 3, we define he conrol law for he i-h subsysem, i = 1,..., M, as u [i] = û [i] + K i (x [i] ˆx [i] ) (6.4) where he erm û [i] is obained as a soluion of a proper local MPC problem, while he parameer K i is seleced as a consan gain whose compuaion will be discussed in he sequel. Moreover, he variables ˆx [i] and û [i] are he sae and he inpu, respecively, of he i-h nominal sysem, denoed as Ŝi, and given for all i = 1,..., M by ˆx [i] +1 = A iiˆx [i] + B u,i û [i] (6.5) 115

126 hesis 2015/11/21 9:18 page 116 #126 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach Noe ha (6.5) is obained neglecing boh he effec of he uncerainy and he presence of he couplings wih he neighboring subsysems, and hus he M nominal sysems are compleely deerminisic and decenralized. In parallel o he real and he nominal decenralized models, (6.1) and (6.5), we need o accoun for he behavior of he expeced values of he sae and inpu of each agen, namely x [i] = E[x [i] ] and ū [i] = E[u [i] ], ha will be used, in line wih he approach proposed in Chaper 3, o reformulae he chance consrains (6.2). In paricular, saring from (6.1) and recalling ha E[w [i] ], 0, he evoluion of he expeced value of he sae of he i-h subsysem can be described as follows for all i = 1,..., M, i.e, x [i] +1 = A ii x [i] + B u,i ū [i] + A ij x [j] (6.6) j N i where he coupling erms, x [j], j N i, are he expeced values of he sae of all he neighboring subsysems of S i. Noe ha, conrarily o he nominal subsysems in (6.5), he dynamics of he expeced values in (6.6) is no decoupled and hus canno be used direcly inside each local conroller. Concerning he expeced value of he i-h inpu, ū [i], recall ha ˆx [i] and are deerminisic variables, hus, for all i = 1,..., M, we have from û [i] equaion (6.4) ha ū [i] = û [i] + K i ( x [i] ˆx [i] ) (6.7) Denoing by x = ( x [1],..., x [M] ) R n and by ū = (ū [1],..., ū [M] ) R m he collecive expeced values of he saes and of he inpus, respecively, he collecive average dynamics of he sysem is given, in view of equaion 6.6, by x +1 = A x + Bū (6.8) Similarly o he procedure adoped in Chaper 3, we define for he i-h subsysem he sae error variable, δx [i] = x [i] x [i] and, from (6.1) and (6.6), we obain he dynamics δx [i] +1 = (A ii + B u,i K i )δx [i] + A ij δx [j] + B w,i w [i] (6.9) j N i In (6.9) he erms δx [j], j N i ha appear in he sum are he deviaions of he saes of he neighboring agens from heir own expeced values and, because of ha, heir expeced value is zero. In paricular, assuming ha each subsysem is properly iniialized so ha E[δx [i] ] = 0, i = 116

127 hesis 2015/11/21 9:18 page 117 # Problem saemen ) R n be he collecive error, from (6.9) he dy- 1,..., M, i follows ha E[δx [i] +k ] = 0, k 0. δx = (δx [1],..., δx [M] namics of he overall error is Furhermore, leing δx +1 = (A + BK)δx + B w w (6.10) wih E[δx ] = 0, 0. Now define he covariance marix of he overall sysem in (6.3) as X = var(x ) = E[δx δx T ]. In view of equaion (6.10) is evoluion is described by X +1 = (A + BK)X (A + BK) T + B w WB w T (6.11) where W = var(w ) is he block diagonal variance of he overall disurbance due o he independence assumpion. Finally, noe he variance of variable δx [i] is X [i], i.e., he i-h diagonal block of he possibly full marix X. A block-diagonal upper bound for he variance Noe ha, in general, even if we assume ha, a a given ime, he marix X is block-diagonal, is evoluion X +k, k > 0 is full due o he couplings beween he subsysems, which represens a ough boleneck in he design of a disribued conroller and for he reformulaion of probabilisic consrains in deerminisic erms. In order o overcome his aspec, we define a symmeric and posiive semi-definie block-diagonal marix, X, in such a way ha X [1] X = X[2].... X (6.12) X[M] and whose evoluion X +1 = diag( following se of M updae equaions X [1] +1,..., X [i] +1 = (Ãii + B [i] u,i K i ) X (Ãii + B u,i K i ) T + à X[j] ij à T ij + B w,i W i Bw,i T j N i X [1] +1) is governed by he (6.13) where Ãii = ν i A ii, à ij = ν i A ij and B u,i = ν i B u,i, and ν i is he cardinaliy of Ñ i = {j A ji 0, j = 1,..., M}. The following lemma holds 117

128 hesis 2015/11/21 9:18 page 118 #128 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach Lemma 5. Assume ha, a ime, X X. Then if X +1 is defined according o equaion (6.13) we have ha X +1 X +1 Moreover, since X X, hen also X [i] X [i], 0. The proof of Lemma 5 is posponed o Secion Finally, in order o ighen he probabilisic consrains as described in Chaper 3, we are ineresed in characerizing he evoluion of he covariance marix of he i-h subsysem X [i]. However, due o he coupled srucure of he sysem, and in paricular o he off-diagonal erms of X, hose consrains can only be solved in a cenralized fashion. On he oher hand, based on he resul in Lemma 5 and he definiion of X and X, he new marices X, [i] i = 1,..., M can be used in place of he original ones o consruc he deerminisic, alhough igher, consrains. Noe, however, ha in view of [i] equaion (6.13) he marices X, i = 1,..., M are sill coupled, and hus he evoluion (6.13) needs o be compued in a disribued fashion. This fac will be discussed in he sequel. Concerning he covariance of he inpu, from he srucure of he conroller in (6.4) and (6.7) we can define, for all i = 1,..., M or, collecively, define U as U [i] = K i X [i] K T i (6.14) U = KX K T Moreover, using he bounding marices, i = 1,..., M, we define a new marix Ũ [i] as Ũ [i] = K X[i] i Ki T which, by consrucion, is such ha Ũ [i] U [i], 0 and hus Ũ [i], i = 1,..., M, can be used o ighen he consrain in place of he real variance of he inpu. X [i] Reformulaion of he probabilisic consrains Consider he se of local individual chance consrains ha appears in equaion (6.2). The approach described in Chaper 3 is now used o reformulae hem in deerminisic erms, based on he expeced value and variance of he sae and he inpu of he i-h subsysem. Moreover, in order o 118

129 hesis 2015/11/21 9:18 page 119 # Problem saemen mainain a disribued srucure of he conroller, he bounds and Ũ [i], i = 1,..., M in place of he original sae and conrol variances, are considered inside he definiion of he consrains. This, of course, inroduces a source of conservaism bu grealy simplifies he seup, allowing for a disribued implemenaion. In paricular, saring from he sae and inpu consrains ses, X [i] and U [i], [i] define M ighened ses, X and Ū[i], for all i = 1,..., M, 0 as { } X [i] = x b T r,ix x max r,i b T [i] r,i X b r,i f(p x i,r) r = 1... n ri (6.15) X [i] and Ū [i] = { } u c T s,iu u max s,i c T s,iũ [i] c s,i f(p u s,i) s = 1... n si (6.16) As in he previous chapers, he value of he funcion f(p) depends on he sochasic naure of he disurbance and in paricular, if he disribuion of he uncerainy is unknown, i is bounded by f(p) = (1 p)/p, while if he uncerainy is Gaussian (and his, due o lineariy, is rue also for he sae and he inpu) he exac value is compued as f(p) = N 1 (1 p). For furher deails see Chaper 2. Finally, as discussed in Chaper 3, he original consrains in (6.2) are saisfied provided ha x [i] X [i], 0 (6.17a) ū [i] Ū[i], 0 (6.17b) However, noe ha, he consrains in (6.17) are sill no suiable for he implemenaion of a disribued MPC algorihm, where a any ime he evoluion of he expeced value and of he variance of he saes and inpus of each subsysem mus be compued in a disribued way from o + N, N being he lengh of he adoped predicion horizon. In fac, he following wo main problems arise P-A in view of (6.6), also he expeced values x [i] +k, k = 0,..., N, of he saes of he subsysems S i, i = 1,..., M, depend on he expeced values x [j] +k of he saes of he neighbors S j, j N i ; 119

130 hesis 2015/11/21 9:18 page 120 #130 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach P-B in view of (6.13), he ses X [i] +k and Ū[i] +k, k = 0,..., N, are affeced [j] X +k of he neighboring subsysems S j, by all he covariance bounds j N i, so ha a disribued compuaion is required. In view of his and in order o derive he desired conroller, we need o furher reformulae he consrains. Problem P-A Firsly, in order o characerize he evoluion of he expeced value in a decenralized way, we recall ha he nominal sysems Ŝi defined in (6.5) are decenralized and we resor o an approach based on he robus ube-based soluion" developed in [102] for robus cenralized MPC. Specifically, for he i-h subsysem we define he difference z [i] +k = x[i] +k ˆx[i] +k ha, in view of (6.6), (6.5), (6.7), he dynamics of z [i] +k 1, is given by and noe, for k = 0,..., N z [i] +k+1 = (A ii + B u,i K i )z [i] +k + j N i A ij x [j] +k (6.18) = (A ii + B u,i K i )z [i] +k + j N i A ij z [j] +k + j N i A ij ˆx [j] +k Correspondingly, he collecive vecor z +k = (z [1] +k,..., z[m] +k ) evolves according o he following dynamics z +k+1 = (A + BK)z +k + d +k (6.19) where d +k = (d [1] +k,..., d[m] +k ), wih d[i] +k = j N i A ij ˆx [j] +k, can be inerpreed as a vecor of disurbances o be rejeced. Now assume ha in he formulaion of he M local MPC problems we can enforce he consrain ˆx [i] [i] [i] +k ˆX 0, i = 1,..., M, where ˆX 0, i = 1,..., M are closed and compac ses conaining he origin in heir inerior. Then, we can also guaranee ha, for all i = 1,..., M d [i] +k = j N i A ij ˆx [j] +k D[i] +k = j N i A ij ˆX[i] o (6.20) The following sanding assumpion is now in order Assumpion 3. There exiss a recangular robus posiive invarian (RPI) se for sysem (6.19), i.e., Z = Z [1] Z [2]... Z [M] 120

131 hesis 2015/11/21 9:18 page 121 # Problem saemen such ha z Z implies z +k Z, d +k D [1] +k... D[M] +k, k 0. Similarly, he ses Z [i], i = 1,..., M are RPI for he sysem in (6.18), meaning ha z [i] Z [i], for all i = 1,..., M implies z [i] +k Z[i], d [i] +k D[i] +k, z [j] +k Z[j], j N i and k 0. Problem P-B Wih reference o he problem P-B, we assume ha, a ime insan, he [i] sequence X k, k =,..., + N 1 of fuure upper bounds of he sae variances is available o subsysem i. As i will be clarified in he following, [i] we will guaranee ha X k is, a all ime insans k > 0, an upper bound of he sae variance by properly defining he corresponding MPC problems. [i] Noe ha he recursive updae of X k can be made in a disribued fashion. For example, hanks o a lighweigh neighbor-o-neighbor communicaion [j] nework, he erms X k, j N i can be made available o subsysem i, [i] and marix X k+1 can be compued using (6.13) and made available for he definiion of he MPC problem when needed. Concerning his, he following sanding assumpion is required o guaranee ha such sequence does no grow unbounded. Assumpion 4. For all i = 1,..., M here exis marices X i, which are soluions o he se of algebraic equaions (Ãii + B u,i K i ) X i (Ãii + B u,i K i ) T = X i j Ni à ij Xj à T ij B w,i W i B T w,i (6.21) I is imporan o remark ha he exisence of marices X i, i = 1,..., M, [i] guaranee ha, if X 1 X [i] i for all i = 1,..., M, hen X k X i and for all k > 0. This can be proved by inducion compuing ha if, a ime k, X[i] k X [i] i, hen X k+1 = (Ãii + B [i] u,i K i ) X k (Ãii + B u,i K i ) T + à X[j] ij k ÃT ij + B w,i W i B T [i] w,i saisfies X k+1 (Ãii + B u,i K i ) X i (Ãii + j N i B u,i K i ) T + à ij Xj à T ij + B w,i W i Bw,i T = X i. j N i Now, in order o reformulae he consrains (6.17), i is possible o define [i] wo convex ses ˆX and Û[i], saisfying ˆX [i] = X [i] Z [i], Û[i] 121 = Ū[i] K i Z [i] (6.22)

132 hesis 2015/11/21 9:18 page 122 #132 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach respecively. The following implicaions hold { ˆx [i] z [i] ˆX [i] = x [i] Z [i] = ˆx [i] + z [i] X [i] (6.23a) { û[i] z [i] Û[i] = ū [i] Z [i] = û [i] + K i z [i] Ū[i] (6.23b) so ha he sae and conrol consrains can be enforced on he nominal decenralized subsysems. Noe however ha, from (6.23a) and (6.23b), we need o correcly iniialize he problem in order o guaranee z [i] Z [i], for all i = 1,..., M. This problem will be addressed in he sequel. [i] Finally we define he ses X and Ū[i] as he ighened consrain ses in (6.15) and (6.16) compued in correspondence of he marices X i, Ū i = K i Xi Ki T, i.e., { } X [i] = x b T r,ix x max r,i b T X i,r i b r,i f(p x i,r) (6.24a) { } Ū [i] = u c T s,iu u max s,i c T s,iūic s,i f(p u s,i) (6.24b) The following sanding assumpion allows us o correcly define he ses used in he proposed conrol scheme. [i] Assumpion 5. Wih reference o he definiion in (6.24) he ses X Z [i] and Ū[i] K i Z [i], i = 1,..., M exis and conain he origin in heir inerior. The previous assumpion ensure ha all he ses 1,..., M and 0, exis and conain he origin in heir inerior. ˆX [i] and Û[i], for i = 6.2 The disribued SMPC algorihm: formulaion and properies In his secion he main feaures of he proposed algorihm are described and he i-h local MPC problem, o be solved by each subsysem a each ime sep, is saed. Similarly o he p-smpc approach, in order o give feasibiliy guaranees, he choice of he iniial condiions for he nominal sae (expeced value) and he variance is considered as an exra degree of freedom of he problem. The cos funcion, he erminal consrains, and he iniializaions of he M local MPC problems are now specified o guaranee feasibiliy and convergence of he conrol scheme. 122

133 hesis 2015/11/21 9:18 page 123 # The disribued SMPC algorihm: formulaion and properies Cos funcion The local cos funcion o be minimized, J [i], i = 1,..., M, depends on he nominal values of he sae and inpu, i.e. J [i] = +N 1 k= ( ˆx [i] k 2 Q i + û [i] k 2 R i ) + ˆx [i] +N 2 S i (6.25) where, for all i = 1,..., M, Q i > 0 and R i > 0 are uning knobs and he erminal weighs S i are defined such ha Q i +K T i R i K i + (Ãii + B u,i K i ) T S i (Ãii + B u,i K i ) S i 0 (6.26) Terminal consrains As sandard in MPC wih sabiliy guaraneed, erminal consrains mus be included ino he problem formulaion, i.e., ˆx [i] +N X[i] F (6.27) where he erminal se X [i] F is a posiive invarian se for he nominal subsysem (6.5) under he auxiliary conrol law û [i] = K iˆx [i] where he original probabilisic sae and conrol consrains (6.2a) and (6.2b) are verified. In our framework, his means ha also he ighened consrains (6.15) and (6.16) mus be saisfied inside X [i] F, i.e., X [i] F x[i] Iniial condiions x [i] K i x [i] x [i] X [i] Z [i] Ū[i] K i Z [i] ˆX [i] o (6.28) The iniial condiions of each local opimizaion problem are free variables seleced o guaranee recursive feasibiliy properies. The basic idea is o ake advanage of he informaion available a ime whenever possible and, oherwise, o use he opimal nominal soluion compued a ime 1 ha guaranees feasibiliy. Specifically, wo possible iniializaion sraegies for he choice of ˆx [i] are defined Sraegy S1 - nominal evoluion: a ime choose ˆx [i] = ˆx[i] 1. In his way 123

134 hesis 2015/11/21 9:18 page 124 #134 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach we are using he opimal soluion compued a ime 1 in order o counerac he effec of possibly unbounded disurbances ha affeced he sae. Moreover, x [i] is defined according o (6.6) wih (6.7), which implies, from Lemma 5, ha var(x [i] ()) = X [i] [i] X. Also, z [i] Z [i] by consrucion. Sraegy S2 - robus iniializaion: a ime se x [i] Z [i]. We im- = 0 and plicily assume ha x [i] z [i] Z [i]. = x [i] ˆx [i] and hus var(x [i] ()) = X [i] X [i] Saemen of he local MPC problems and main resul We are now ready o formulae he i-h local opimizaion problem. i-mpc problem: a each ime sep measure he sae x [i] and solve he opimizaion problem min ˆx [i],û[i]...+n 1 J [i] (ˆx [i], û [i],...,+n 1 ) (6.29a) subjec o ˆx [i] ˆx [i] {ˆx [i] 1 } (x[i] Z [i] ) (6.29b) k+1 = A iiˆx [i] k [i] ˆX ˆx [i] k û [i] k 0 + B u,iû [i] k ˆX[i] k, k =... + N 1 Û[i] k, k =... + N 1 ˆx [i] +N X[i] F (6.29c) (6.29d) (6.29e) (6.29f) As a soluion we obain he opimal sequence û [i]...+n 1 and he opimal value ˆx [i]. Thus, consisenly wih (6.4), a ime sep we apply he following inpu u [i] = û [i] + K[i] i (x [i] ˆx [i] ) (6.30) Moreover we compue he opimal sequence a ime, i.e., ˆx [i]...+n 1, hanks o (6.5). We are now in he posiion for sae he main resul. Theorem 3. If, a ime = 0, all i-mpc problems, i = 1,..., M, admi a soluion, hen he opimizaion problem is recursively feasible and 124

135 hesis 2015/11/21 9:18 page 125 # Implemenaion issues E{ x 2 } r{x ss } as +, where X ss is he unique posiive semidefinie soluion (recall Assumpion 2) o he Lyapunov equaion X ss (A + BK)X ss (A + BK) T = B w WB w T Furhermore, he sae and inpu probabilisic consrains (6.2) are verified for all Implemenaion issues The independen and local MPC problems previously defined require he a-priori off-line compuaion of a number of parameers and ses which mus be performed in a cenralized way. In paricular, he gains K i and K saisfying Assumpion 2 can be compued wih he procedure based on he soluion of LMI s repored in [12]. [i] Addiional parameers of he algorihm are he ses ˆX 0, see (6.20), and he recangular ses Z [i] saisfying Assumpion 3. Their choice is fundamenal, since direcly limis he feasibiliy se of he local MPC problems and hus may represen he main source of conservaism. However one can noice [i] ha, in general, he bigger are he ses ˆX o he bigger will be he ses Z [i], meaning ha he wo erms ˆx [i] and x [i] are allowed o be far from each oher. In pracice, his will affec he dimension of he ighened original [i] [i] consrain ses ˆX = X Z [i] and Û[i] = Ū[i] K i Z [i], 0 and again reduce he overall feasibiliy se for he i-h MPC problem. Thus, he choice [i] of ˆX 0 and Z [i] represens a sor of radeoff ha needs o be accouned for in he design. However, i is always possible o choose a sufficienly small ˆX [i] o such ha Z [i] is no empy. Procedures for he pracical compuaion of hese ses are also described in [12]. Finally, noe ha condiions (e.g., LMIs or based on small-gain condiions) for guaraneeing he exisence of soluions o (6.21) can be sudied along he lines raced in [44]. 6.4 Proofs Proof of Lemma5 In his secion we prove Lemma 5. In paricular, consider he overall covariance marix X and a symmeric posiive semi-definie marix X = diag(,..., ) such ha X X. From equaion (6.11) we have X [1] X [M] 125

136 hesis 2015/11/21 9:18 page 126 #136 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach ha X +1 = (A + BK)X (A + BK) T + B w WB w T (A + BK) X (A + BK) T + B w WB w T (6.31) Focus now on he firs erm of equaion (6.31), i.e., Ψ = (A + BK)X (A + BK) T and, in order o simplify he noaion, define F = (A + BK). Since X is posiive semidefinie by assumpion, hen also Ψ is posiive semidefinie and hus for every vecor of proper dimension η = [η1 T,..., ηm T ]T we have ha η T Ψη = [η1 T... ηm] T F 11 F F 1M F 21 F F M F MM X F 11 F F 1M F 21 F F M F MM Then, by means of simple compuaions, he laer expression can be rewrien as follows η T Ψη = [ ηi T F i1 i ηi T F i2... i ] ηi T F im i X [1] X[2] X[M] T η 1 Fi1η T i i Fi2η T i i. FiM T η i i. η M from which we have η T Ψη = ηi T F X[j] ij j i Fhjη T h = j h i,h ηi T F X[j] ij F hj η h Now we apply he following resul, ha can be derived from Lemma 4 in Secion 5.6, o he previous expression. If P 0 hen for every v i and v j of appropriae dimensions we have vi T P v i + vj T P v j 2vi T P v j and hus vi T P v j 1/2( v i 2 P + v j 2 [j] P ). In paricular, given ha he erms X, for all i = 1,..., M are posiive semidefinie by assumpion, we have ηi T F X[j] ij F hj η h j j i,h 1 2 (ηt i F X[j] ij i,h 126 F T ij η i + ηh T F X[j] hj Fhjη T h )

137 hesis 2015/11/21 9:18 page 127 # Proofs now he wo erms in he brackes are compleely equivalen and moreover we can define as ν j he number of elemens F ij 0 so ha we have ηi T F X[j] ij F hj η h j j i,h ν j i ηi T F X[j] ij Fij T η i = i η i 2 j ν j F T ij [j] X F ij and finally, defining F ij = ν j F ij, his corresponds o [ η T 1... ηm] T j F 1j X[j] F T 1j... j F Mj X[j] F T Mj η 1. η M Summarizing his proves ha j F 1j X[j] F T 1j... j F Mj X[j] F T Mj F X F T and finally from his, and recalling ha B w WB w T is block diagonal, he proof is compleed Proof of Theorem3 Assume ha, a ime insan and for each subsysem i = 1,..., M, a feasible soluion o i-mpc is available, i.e., ˆx [i] wih opimal inpu sequence. We prove ha, a ime + 1, a feasible soluion o each i-mpc û [i]...+n 1 problem exiss, i.e., ˆx [i] +1 (i.e., wih sraegy S1 in consrain (6.29b)), wih inpu sequence û [i],f N = {û[i] +1,..., û[i] +N 1, K iˆx [i] +N }. Since (6.29d) and (6.29e) hold for he i-mpc problem solved a ime, i.e., for all k =,..., + N 1, i holds ha (6.29d) and (6.29e) are verified also for he candidae feasible soluion o he i-mpc problem a ime + 1 for k = + 1,..., + N 1. Also, in view of he fac ha X[i] +N X i, of (6.28), and (6.29f), hen ˆx [i],f +N = ˆx [i] [i] [i] +N ˆX 0 ˆX +N, as required. Finally, since ˆx[i],f +N = X [i] F and since X[i] F is a posiively invarian se for he nominal ˆx [i] +N 127

138 hesis 2015/11/21 9:18 page 128 #138 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach subsysem (6.5) under he auxiliary conrol law, hen ˆx [i],f +N+1 = (A ii + B u,i K i )ˆx [i]. This concludes he proof of recursive feasibiliy. +N X[i] F In view of he feasibiliy, a ime + 1 of he possibly subopimal soluion û [i],f N, x[i] +1, and denoing wih J [i], ( + 1) he opimal cos funcion compued a ime + 1 by subsysem i, hen J [i], ( + 1) J [i],f ( + 1 ) (6.32) where, by sandard argumens, J [i] (+1 ) = J [i], () ˆx [i] 2 Q i û [i] 2 R i + ˆx [i] +N 2 Q i + K iˆx [i] +N 2 R i ˆx [i] +N 2 S i + (A ii + B u,i K i )ˆx [i] +N 2 S i. In view of (6.26) and (6.32), hen J [i], (+1) J [i], () ( ˆx [i] 2 Q i + û [i] 2 R i ). Using sandard argumens in MPC, his proves ha ˆx [i] 2 Q i + û [i] 2 R i 0 as + which, since Q i > 0 and R i > 0, implies ha ˆx [i] 0 and û [i] 0 as +. From (6.3) and (6.30), i follows ha x +1 = (A + BK)x + B(û Kˆx ) + B w w where (A + BK) is sable in view of Assumpion 2. Now define he saionary process described by +1 = (A + BK) + B w w wih zero mean and variance X ss. Resoring o he separaion principle, we can define x = x, which evolves according o x +1 = (A + BK) x + B(û Kˆx ) On he oher hand, B(û Kˆx ) is asympoically vanishing, and herefore x 0 as 0. Therefore E{ x 2 } = E{ x 2 } + E{ 2 } E{ 2 } = r(x ss ). Finally since, for all > 0, ˆx [i] [i] ˆX, x [i] ˆx [i] Z i hen x [i] [i] X. [i] [i] Recall ha ses ˆX k and Û[i] k are compued according o (6.22), where X k and Ū[i] k are defined in (6.15) and (6.16). In urn, hey depend on he values [i] [i] of X k and Ũ k = K [i] i X k KT i, respecively. From Lemma 5 and he Canelli inequaliy, hen (6.2a) is verified. Also, since û [i] Û[i], hen ū [i] Ū[i]. In urn, (6.2b) is also proved. 128

139 hesis 2015/11/21 9:18 page 129 # Simulaion example 6.5 Simulaion example In his secion we consider as an example he sysem described in [12] o show he effeciveness of he proposed disribued approach. In paricular, he goal is o conrol he levels, h 1, h 2, h 3 and h 4, of he four ank sysem depiced in Figure.6.1 acing on he volages of he wo pumps, v 1 and v 2. The coninuous-ime dynamics of he sysem is given by dh 1 d = a 1 2gh1 + a 4 2gh4 + γ 1k 1 v 1 A 1 A 4 A 1 dh 2 d = a 2 2gh2 + (1 γ 1)k 1 v 1 A 2 A 2 dh 3 d = a 3 2gh3 + a 2 2gh2 + γ 2k 2 v 2 A 3 A 2 A 3 dh 4 d = a 4 2gh4 + (1 γ 2)k 2 v 2 A 4 A 4 (6.33) where A i and a i are he cross-secion of Tank i and he cross secion of he oule hole of Tank i, respecively, and he parameers γ 1 (0, 1) and γ 2 (0, 1) represen he fracion (fixed) of waer ha flows inside he lower anks. The coefficiens k 1 and k 2 represen he conversion parameers from he volage applied o he pump o he flux of waer. The values are chosen as in [12]. The sysem is linearized around he equilibrium poin v 1 = v 2 = 3V, h 1 = cm, h 2 = 1.409cm, h 3 = cm and h 4 = 1.634cm and hen discreized using a sampling ime of 1 s. The resuling dynamics is pariioned ino wo differen subsysems collecing anks 1-2 and anks 3-4, respecively. Figure 6.1: Four ank sysem. 129

140 hesis 2015/11/21 9:18 page 130 #140 Chaper 6. Disribued Predicive Conrol for regulaion: a sochasic approach Wih his choice, he sysem marices ha appear in (6.1) are [ ] [ ] [ ] A 11 =, A 12 =, B u,1 =, B w,1 = I [ ] [ ] [ ] A 21 =, A 22 =, B u,2 =, B w,2 = I The disurbances are assumed o be wo addiive independen Gaussian noises wih zero means and variances W 1 = W 2 = 1. The iniial condiion are chosen as x [1] 0 = [0.274, 0.067] T and x [2] 0 = [0.203, 0.254] T. Moreover, he consrains on he inpus and saes are defined by x [1] min = [ , ] T, x [1] max = [ 40, 40 ] T [1] + x min x [2] min = [ , ] T, x [2] max = [ 40, 40 ] T [2] + x min u [1] min = u[2] min = 3, u[1] max = u [2] max = 3 and, in line wih he proposed approach, hey are imposed as individual chance consrains following (6.2). The single violaion probabiliies are fixed o a common value for he sae consrains, p x = 0.2 and o p u = 0.1 for he inpu. I is se Q 1 = Q 2 = I 2 amd R 1 = R 2 = 0.1. Also, marices K i and S i are defined o saisfy Assumpion2 and (6.26). A se of 100 simulaions is run, o show he effeciveness of he approach. Resuls are depiced in Figure 6.2 where i is possible o verify ha he proposed echnique correcly drives he sae of he sysems o a neighborhood of he origin. 6.6 Commens In his chaper, a novel Sochasic Disribued Predicive Conrol echnique has been presened for linear discree-ime sysems wih addiive, possibly unbounded, disurbances and subjec o individual chance-consrains on he sae and he inpu. In paricular, he p-smpc approach described in Chaper 3 has been exended o he disribued framework and he guaraneed feasibiliy and convergence properies have been recovered by means of a proper choice of he iniial condiions for he problem and is erminal consrains. An example showed he viabiliy of he approach. Despie he presence of sochasic consrains, he usage of robus se inclusion echniques renders he proposed algorihm quie conservaive, and increases he design complexiy. However, his Chaper has o be inended 130

141 Waer level hesis 2015/11/21 9:18 page 131 # Commens Time Figure 6.2: Example of applicaion of he disribued p-smpc approach. The colored lines represen he rajecories of he sysem saes (he levels) over ime in 100 differen simulaions. as a firs sep in he direcion of applying SMPC mehods ino disribued conrol problems, wih he aim of coping wih disurbances wih possibily unbounded suppor and probabilisic consrains on he sysem variables. 131

142 hesis 2015/11/21 9:18 page 132 #142

143 hesis 2015/11/21 9:18 page 133 #143 CHAPTER7 Disribued Predicive Conrol for racking reference signals: a robus approach. Many decenralized and disribued MPC algorihms have been recenly developed, see for example he book [92] and he review papers [32, 142]. Mos of hese mehods consider he so-called regulaion problem: given a large-scale dynamical sysem made by a number of (inerconneced or independen) subsysems, he problem is o asympoically seer o zero he sae of all he subsysems by coordinaing he local conrol acions wih a minimum amoun of ransmied informaion. However, in his seing, he soluion of he racking problem wih disribued MPC is much more difficul. In fac, he decenralizaion consrains do no allow o follow he sandard approach, based on he reformulaion of he racking problem as a regulaion one by compuing, a any se-poin change of he oupu, he corresponding sae and conrol arge values. For his reason, and o he bes of our knowledge, only he cooperaive disribued MPC algorihm described in [50] is nowadays available, while a differen approach based on a disribued reference governor has been proposed in [149]. In his chaper, a new disribued MPC mehod for he soluion of he rack- 133

144 hesis 2015/11/21 9:18 page 134 #144 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. ing problem is proposed for sysems made by he collecion of M subsysems. The algorihm is developed according o he hierarchical srucure depiced in Figure 7.1 and in paricular a) a he higher layer, given he required oupu reference signals y [i] se poin, i = 1,..., M, feasible rajecories x [i], ũ [i], and ỹ [i] of he local sae, inpu and oupu variables are compued for each subsysem S i. Noably, hese rajecories are compued according o he prescribed disribued informaion paern, also adoped a he lower layer b) a he lower layer, M regulaors are designed wih he disribued algorihm developed in [47] for he soluion of he local regulaion problems. The overall algorihm guaranees ha he conrolled oupus reach he prescribed reference values whenever possible, or heir neares feasible value when feasibiliy problems arise due o he consrains. A preliminary version of his algorihm is presened in [43] where, however, a less general conrol problem is considered and more resricive condiions are required. In paricular, DPC assumes ha he fuure sae and conrol reference rajecories are ransmied by each subsysem o is neighbors, and he differences beween hese reference rajecories and he rue ones are inerpreed as disurbances o be rejeced. As such, he robus MPC approach inroduced in [102] is used for he developmen of he disribued conrol laws. In addiion, he proposed algorihm also allows one o consider he presence of possible unfeasible reference signals, i.e. of se-poins ha canno be reached due o sae and/or conrol consrains. This problem is solved by resoring o he ideas described in [84] where, in case of unfeasible references, i is suggesed o seer he oupu variables o he neares (in some sense) value compaible wih he process consrains. Noes on he noaion adoped in his chaper In order o be consisen wih he well-esablished lieraure on disribued predicive conrol, e.g. [88], he noaion adoped in his chaper is slighly differen from he one used in he res of he Thesis. In he sequel a quick descripion is given, however, whenever he meaning is no clarified by he conex, more deails will be found locally. A marix is Schur if all is eigenvalues lie in he inerior of he uni circle. The shor-hand v = (v 1,..., v s ) denoes a column vecor wih s (no necessarily scalar) componens v 1,..., v s. The symbols and denoe 134

145 hesis 2015/11/21 9:18 page 135 # Ineracing subsysems [1] y se poin [M] y se poin Disribued reference generaor [1] [1] [1] x u y [M] [M] [M] x u y C M C 1 S M S 1 Figure 7.1: Overall conrol archiecure. he Minkowski sum and Ponryagin difference, respecively, [137], while M i=1 A i = A 1 A M. A generic q-norm ball cenered a he origin in he R dim space is defined as follows B q,ε (dim) (0) := {x R dim : x q ε}. For a discree-ime signal s and a, b N, a b, (s a, s a+1,..., s b ) is denoed wih s [a:b]. 7.1 Ineracing subsysems Consider M dynamically ineracing subsysems which, according o he noaion used in [88], are described by x [i] +1 = A ii x [i] + B ii u [i] + E i s [i] y [i] z [i] = C i x [i] = C zi x [i] + D zi u [i] (7.1a) (7.1b) (7.1c) where x [i] R n i and u [i] R m i are he saes and inpus, respecively, of he i-h subsysem, while y [i] R p i is is oupu, wih p i m i. In line wih he ineracion-oriened models inroduced in [88], he coupling inpu and oupu vecors s [i] and z [i], respecively, are defined o characerize he 135

146 hesis 2015/11/21 9:18 page 136 #146 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. inerconnecions among he subsysems, i.e., M s [i] = L ij z [j] (7.2) j=1 We say ha subsysem j is a dynamic neighbor of subsysem i if and only if L ij 0, and we denoe as N i he se of dynamic neighbors of subsysem i (which excludes i). The inpu and sae variables are subjec o he local" consrains u [i] U i R m i and x [i] X i R n i, respecively, where he ses U i and X i are convex. Furhermore, we allow for n c linear consrains involving he oupu variables of more han one subsysem: he h-h consrain inequaliy is M H [j] h y[j] l h (7.3) j=1 where H [j] h R1 p j are row vecors, which can be possibly equal o zero, for some values of j. Wihou loss of generaliy, he oupus involved in coupling consrains are accouned for as coupling oupus, i.e., for all h = 1,..., n c and for all j = 1,..., M, here exiss a marix H [j]z h such ha H [j] h y[j] = H [j]z h z[j] (7.4) under (7.1b) and (7.1c). We say ha he h-h inequaliy is a consrain on subsysem i if H [i] h 0, and we denoe he se of consrains on subsysem i as C i = {h {1,..., n c }: he h-h inequaliy is a consrain on i}, and wih n [i] c he number of elemens of C i. Subsysem j i is a consrain neighbor of subsysem i if here exiss h C i such ha H [j] h 0. For all h = 1,..., n c, we finally denoe wih S h he se of subsysems for which he h-h inequaliy is a consrain, i.e., S h = {i : H [i] h 0} and wih n h = S h he cardinaliy of S h. Collecing he subsysems (7.1) for all i = 1,..., M, we obain he collecive dynamical model x +1 = A x + Bu y = C x (7.5a) (7.5b) where x = (x [1],..., x [M] ) R n, n = M i=1 n i, u = (u [1],..., u [M] ) R m, m = M i=1 m i, and y = (y [1],..., y [M] ) R p, p = M i=1 p i, are he 136

147 hesis 2015/11/21 9:18 page 137 # Conrol sysem archiecure collecive sae, inpu, and oupu vecors, respecively. The sae ransiion marices A 11 R n 1 n 1,..., A MM R n M n M of he M subsysems are he diagonal blocks of A, whereas he dynamic coupling erms beween subsysems correspond o he non-diagonal blocks of A, i.e., A ij = E i L ij C zj, wih j i. Correspondingly, B ii, i = 1,..., M, are he diagonal blocks of B, whereas he influence of he inpu of a subsysem upon he sae of differen subsysems is represened by he off-diagonal erms of B, i.e., B ij = E i L ij D zj, wih j i. The collecive oupu marix is defined as C =diag(c 11,..., C MM ). Concerning sysem (7.5a) and is pariion, he following main assumpion on decenralized sabilizabiliy is inroduced: Assumpion 6. There exiss a block-diagonal marix K =diag(k 1,..., K M ), wih K i R m i n i, i = 1,..., M such ha: (i) F = A + BK is Schur, (ii) F ii = (A ii + B ii K i ) is Schur, i = 1,..., M. Remark 4. The design of he sabilizing marix K can be performed according o he procedure proposed in [47] or by resoring o an LMI formulaion, see [11], based on well known resuls in decenralized conrol, see e.g. [153]. The following sandard assumpion is made. Assumpion 7.. ([ ]) In A B rank = n + p C Conrol sysem archiecure The higher layer of he hierarchical scheme of Figure 7.1 is iself made by wo sub-layers, as shown in Figure 7.2: a reference oupu rajecory layer compues in a disribued way he oupu reference rajecories ỹ [i] given he ideal" se-poins y [i] se poin, while a reference sae and inpu rajecory layer deermines he corresponding sae and conrol rajecories x [i] and ũ [i]. A he lower layer of he srucure of Figure 7.1, a disribued robus MPC layer is designed o drive he real sae and inpu rajecories x [i] and u [i] of he subsysems as close as possible o x [i], ũ [i], while saisfying he consrains. Noably, a each level, informaion is required o be ransmied only among neighboring subsysems. 137

148 hesis 2015/11/21 9:18 page 138 #148 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. Figure 7.2: Disribued archiecure for racking reference signals. The reference oupu rajecory layer A any ime, he reference rajecories ỹ [i] +k, k = 1,..., N 1, are regarded as an argumen of an opimizaion problem iself (see he following Paraghaph IV.b) raher han a fixed parameer, similarly o he approach in [84]. However, in he considered disribued conex, oo rapid changes of he oupu reference rajecory of a given subsysem could grealy affec he performance and he behavior of he oher subsysems. Therefore, in order o limi he rae of variaion, i is required ha, for all i = 1,..., M, for all 0 ỹ [i] +1 ỹ [i] B (p i) q,ε (0) (7.6) The reference oupu rajecory managemen layer is also commied o defining suiable updae laws for ỹ [i] in such a way ha, for all ime seps, (7.3) is verified for all h = 1,..., n c. The reference sae and inpu rajecory layer Given, a any ime sep, he fuure reference rajecories ỹ [i] k, k =,..., + N 1, in order o define he sae and conrol reference rajecories ( x [i], ũ [i] ), 138

149 hesis 2015/11/21 9:18 page 139 # Conrol sysem archiecure consider he sysems where, similarly o (7.1c) and (7.2) x [i] +1 = A ii x [i] + B ii ũ [i] + E i s [i] ẽ [i] +1 = ẽ [i] + ỹ [i] +1 C i x [i] z [i] s [i] = C zi x [i] = j N i L ij z [j] + D zi ũ [i] (7.7a) (7.7b) (7.7c) (7.7d) Define χ [i] A ij = = ( x [i], ẽ [i] ), [ ] Aii 0 C i I pi [ ] Aij and consider he conrol law K e i where K i = [ Ki x herefore defined by if j = i if j i ũ [i], B ij = = K i χ [i] [ Bij 0 ] [ ] 0, G i = I pi (7.7e) (7.7f) ]. Leing Fij = A ij + B ij K j, he dynamics of χ [i] χ [i] +1 = F ii χ [i] + F ij χ [j] + G i ỹ [i] +1 (7.8) j N i The gain marix K i is o be deermined as follows: denoing by A and B he marices whose block elemens are A ij and B ij, respecively, and K =diag(k 1,..., K M ), he following assumpion mus be fulfilled Assumpion 8. The marix F = A + BK is Schur sable. The synhesis of he K i s can be performed provided ha Assumpion 7 is verified and according o he procedures proposed in Remark 4. Moreover, when p i > m i for some i, he available degrees of freedom can be used o selec K i s fulfilling some addiional opimizaion crieria. Define, for all i = 1,..., M and for all 0, χ [i]ss = (x [i]ss is, e [i]ss ), as he seady-sae condiion for (7.8) corresponding o he reference oupus ỹ [i] assumed consan, i.e., ỹ [i] +1 = ỹ [i], and saisfying for all i = 1,..., M χ [i]ss = F ii χ [i]ss + j N i F ij χ [j]ss + G i ỹ [i] (7.9) 139

150 hesis 2015/11/21 9:18 page 140 #150 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. In view of (7.7) and Assumpion 8, C i x [i]ss Then a soluion o he sysem (7.9) exiss and is unique. Collecively define = (χ [1]ss,..., χ [M]ss ), χ = (χ [1] From (7.7e)-(7.9) we can collecively wrie χ ss = ỹ [i] +1 and F is Schur sable.,..., χ [M] ), and ỹ = (ỹ [1],..., ỹ [M] ). χ ss +1 χ ss = (I n+p F) 1 G(ỹ +1 ỹ ) (7.10) where G =diag(g 1,..., G M ). Therefore χ +1 χ ss +1 =F(χ χ ss ) (I n+p F) 1 FG(ỹ +1 ỹ ) (7.11) which can be rewrien as χ +1 χ ss +1 = F(χ χ ss ) + w (7.12) where w can be seen as a bounded disurbance. In fac, in view of (7.6), w W = (I n+p F) 1 FG M i=1 B(p i) q,ε (0). Under Assumpion 8, for he sysem (7.12) here exiss a possibly non-recangular Robus Posiive Invarian (RPI) se χ such ha, if χ χ ss χ, hen i is guaraneed ha χ +k χ ss +k χ for all k 0. This, in urn, implies ha he convex ses χ i exis and can be defined in such a way ha χ M i=1 χ i, so ha, for any iniial condiion χ 0 χ ss 0 χ, for all 0 i holds ha χ [i] The robus disribued MPC layer χ [i]ss χ i (7.13) The DPC algorihm described in [47] is used o drive he real sae and inpu rajecories x [i] and u [i] as close as possible o heir references x [i], ũ [i]. Specifically, by adding suiable consrains o he disribued MPC problem formulaion, for each subsysem and for all 0 i is possible o guaranee ha he acual coupling oupu rajecories lie in specified imeinvarian neighborhoods of heir reference rajecories. More formally, if z [i] z [i] Z i, where Z i is compac, convex and 0 Z i, in view of (7.7d) i is guaraneed ha s [i] s [i] S i, where S i = j N i L ij Z j. In his way, (7.1a) can be wrien as where E i (s [i] x [i] +1 = A ii x [i] + B ii u [i] + E i s [i] + E i (s [i] s [i] ) (7.14) s [i] ) can be seen as a bounded disurbance, while E i s [i] +k can be inerpreed as an inpu, known in advance over he predicion horizon k = 0,..., N

151 hesis 2015/11/21 9:18 page 141 # Conrol sysem archiecure For he saemen of he individual MPC sub-problems, henceforh called i-dpc problems, define he i-h subsysem nominal model associaed o equaion (7.14) and le ˆx [i] +1 = A ii ˆx [i] + B ii û [i] + E i s [i] (7.15) ẑ [i] = C zi ˆx [i] + D zi û [i] (7.16) The conrol law for he i-h subsysem (7.14), for all 0, is assumed o be given by u [i] = û [i] + K i (x [i] where K i saisfies Assumpion 6. Leing ε [i] (7.15) and (7.17) i follows ha where ˆx [i] ) (7.17) = x [i] ˆx [i] from (7.14), ε [i] +1 = F ii ε [i] + w [i] (7.18) w [i] is a bounded disurbance since s [i] = E i (s [i] s [i] s [i] ) (7.19) S i. I follows ha w [i] W i = E i S i (7.20) Since w [i] is bounded and F ii is Schur, here exiss an RPI E i for (7.18) such ha, for all ε [i] E i, hen ε [i] +1 E i. Therefore a ime +1, in view of (7.1c) and (7.16), i holds ha z +1 ẑ [i] [i] +1 = (C zi +D zi K i )ε [i] +1 (C zi +D zi K i )E i. In order o guaranee ha, a ime +1, z [i] +1 z [i] +1 Z i can be sill verified by adding suiable consrains o he opimizaion problems, he following assumpion mus be fulfilled. Assumpion 9. For all i = 1,..., M, here exiss a posiive scalar ρ i such ha (C zi + D zi K i )E i B q,ρi (0) Z i (7.21) If Assumpion 9 is fulfilled define, for all i = 1,..., M, he convex neighborhood of he origin z i saisfying z i Z i (C zi + D zi K i )E i (7.22) 141

152 hesis 2015/11/21 9:18 page 142 #152 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. and consider he consrain ẑ [i] +1 z [i] +1 z i, in such a way ha z [i] +1 z [i] +1 = z [i] +1 ẑ [i] +1+ẑ [i] +1 z [i] +1 (C zi +D zi K i )E i z i Z i (7.23) as required a all ime seps The disribued predicive conrol algorihm The overall design problem is composed by a preliminary cenralized offline design and an on-line soluion of he M i-dpc problems, as now deailed. Off-line design The off-line design consiss of he following procedure: 1) compue he marices K and K saisfying Assumpions 6 and 8 (see Remark 4); (for he compuaion of RPIs see [137]); 3) compue he RPI ses E i for he subsysems (7.18) and he ses z i saisfying (7.22) and (7.23); 4) compue ˆX i X i E i, Ûi U i K i E i, he posiively invarian se Σ i for he equaion 2) define B (p i) q,ε (0), compue χ (an RPI for (7.12)) and χ i such ha δx [i] +1 = F ii δx [i] (7.24) (C zi + D zi K i )Σ i z i (7.25a) and he convex ses Y i such ha [ Ini 0 K x i K e i ] ( Γ i (I n+p F) 1 G ) M j=1 Y j χ i [ Ini K i ] Σ i ˆX i Ûi (7.25b) where Γ i is he marix, of suiable dimensions, ha selecs he subvecor χ [i] ou of χ. Specifically, Y i is he se associaed o ỹ [i] such ha he corresponding seady-sae sae and inpu saisfy he conrol and sae consrains defined by ˆX i and Ûi. Concerning he se-heoreical condiions guaraneeing he design of he 142

153 hesis 2015/11/21 9:18 page 143 # The disribued predicive conrol algorihm conrol scheme, i is worh menioning ha, a he price of a more conservaive scheme and slower seling imes (e.g., small parameer ε), equaions (7.25) can always be verified. On he oher hand, i is no always possible o selec ses Z i such ha (7.21) is verified; as invesigaed in [47] i consiss in a nework-wide small gain condiion. On-line design The on-line design is based on he soluion of he following disribued and independen opimizaion problems. I is worh remarking ha he reference oupu layer and he MPC problem consis in wo independen opimizaion problems. This enhances he reliabiliy of he approach and reduces is compuaional load, a he price of limiing he rae of variaion of he oupu reference o he value ha guaranees consrain saisfacion in all possible condiions, conrarily o [84]. 1) Compuaion of he reference oupus. The oupu reference rajecories ỹ [i] +N are compued o minimize he disance from he ideal se-poins y [i] se poin and o fulfill he consrains, including he coupling ones (7.3). Concerning he laer, define { H [j] [ h Cj 0 ] } χ + max H [j]z h z (7.26) z Z j lh = l h j Sh and, for all 0 max χ χ j k h,+n 1 = j S h H [j] h ỹ[j] +N 1 (7.27) Then, i is possible o show (see he Appendix) ha H [i] h ỹ[i] +N l h j S h \{i} H [j] h ỹ[j] +N 1 (n h 1) n h ( l h k h,+n 1 ) (7.28) guaranees ha (7.3) holds and recursive feasibiliy. Therefore, (7.28) can be used in place of (7.3) in he following opimizaion problem associaed wih he reference oupu rajecory layer. min ȳ [i] +N V y i (ȳ[i] +N, ) (7.29) 143

154 hesis 2015/11/21 9:18 page 144 #154 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. subjec o and (7.28), where ȳ [i] +N ỹ[i] +N 1 B(p i) q,ε (0) (7.30) ȳ [i] +N Y i (7.31) V y i (ȳ[i] +N ) = γ ȳ[i] +N ỹ[i] +N ȳ [i] +N y[i] se poin 2 T i The weigh T i mus verify he inequaliy T i > γi pi (7.32) while γ is an arbirarily small posiive consan. 2) Compuaion of he conrol variables. The i-dpc problem solved by he i-h robus MPC layer uni is defined as follows: where Vi N (ˆx [i], û [i] min ˆx [i],û[i] [:+N 1] +N 1 [:+N 1] ) = k= Vi N (ˆx [i], û [i] [:+N 1] ) (7.33) ˆx [i] k subjec o (7.15) and, for k =,..., + N 1, and o he erminal consrain x[i] k 2 Q i + û [i] k ũ[i] k 2 R i + ˆx [i] +N x[i] +N 2 P i (7.34) x [i] ẑ [i] k ˆx [i] E i (7.35a) z[i] k z i (7.35b) ˆx [i] k ˆX i (7.35c) û [i] k Ûi (7.35d) ˆx [i] +N x[i] +N Σ i (7.36) The weighs Q i and R i in (7.34) mus be aken as posiive definie marices while, in order o prove he convergence properies of he proposed approach, selec he marices P i as he soluions of he (fully independen) Lyapunov equaions F T ii P i F ii P i = (Q i + K T i R i K i ) (7.37) 144

155 hesis 2015/11/21 9:18 page 145 # The disribued predicive conrol algorihm A ime, he (ˆx [i], û[i] [:+N 1], ȳ[i] +N ) is he soluion o he i-dpc problem and û [i] is he inpu o he nominal sysem (7.15). Remark 5. Noe ha he problems (7.29) and (7.33) are independen of each oher. In fac, (7.29) does no depend on ˆx [i] and û [i] [:+N 1]. Moreover, boh he cos funcion Vi N and he consrains (7.35) are independen of ȳ [i] +N. According o (7.17), he inpu o he sysem (7.1a) is u [i] = û [i] + K i(x [i] ˆx [i] ) (7.38) Moreover, se ỹ [i] +N = ȳ[i] +N and compue he references ẽ[i] +N and x[i] +N+1 wih (7.7b) and (7.7a), respecively. Finally se ũ [i] +N = Kx i x [i] +N + Ke i ẽ [i] +N [:+N 1] +N from (7.7f). Denoing by ˆx [i] k he sae rajecory of sysem (7.15) semming from ˆx[i] and û [i], a ime i is also possible o compue ˆx[i]. The properies of he proposed disribued MPC algorihm for racking can now be summarized in he following resul. Theorem 4. Le Assumpions 6-9 be verified and he uning parameers be seleced as previously described. If a ime = 0 a feasible soluion o he consrained problems (7.29), (7.33) exiss and k h,n 1 l h (see (7.26), (7.27)) for all h = 1,..., n c hen, for all i = 1,..., M I) Feasible soluions o (7.29), (7.33) exis for all 0, i.e., consrains (7.28), (7.30), (7.31) and (7.35), (7.36), respecively, are verified. Furhermore, he consrains (x [i], u [i] ) X i U i and for all i = 1,..., M, and (7.3) for all h = 1,..., n h, are fulfilled for all 0. II) If coupling consrains (7.3) are absen, hen he resuling MPC conroller asympoically seers he i-h sysem o he admissible se-poin, where y[i] is he soluion o y [i] feas.se poin feas.se poin y [i] feas.se poin =argmin y [i] y [i] se poin 2 T i (7.39) y [i] Y i When coupling saic consrains are presen, he convergence o he neares feasible soluion o he prescribed se-poin may be prevened for some iniial condiions. These siuaions are denoed deadlock soluions in [149]. Fuure work will be specifically devoed o his issue. 145

156 hesis 2015/11/21 9:18 page 146 #156 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. 7.4 Proof of he main heorem Proof of recursive feasibiliy of problem (7.28)-(7.31) Assume ha, a sep, a soluion ȳ [i] +N o (7.29) exiss for all i = 1,..., M and ha k h,+n 1 l h for all h = 1..., n c. Firs noe ha, since k h,+n 1 lh, j S h H [j] h ỹ[j] +N 1 l h. Recall ha, in view of he relaionship beween he marices H [i], and C i used in (7.23), for all i = 1,..., M h, H[i]z h H [i] h y[i] +N 1 = H[i]z h z[i] +N 1 H[i]z h z[i] +N 1 H[i]z h Z i Since, in view of (7.4), H [i]z h z[i] +N 1 = H[i] h C i x [i] +N 1, we obain H [i] h y[i] +N 1 H[i] h C i x [i] +N 1 H[i]z h Z i Recall ha C i x [i] +N 1 = [ C i 0 ] χ [i] +N 1 [ Ci 0 ] χ [i] +N 1 [ C i 0 ] (χ [i]ss +N 1 χ i ). Since [ C i i follows ha H [i] h y[i] +N 1 H [i] h ỹ[i] +N 1 H[i] h and ha, from (7.13), we have 0 ] χ [i]ss +N 1 = ỹ[i] +N 1, [ Ci 0 ] χ i H[i]z h Z i From he definiion of l h, i is easy o see ha he ighened consrain k h,+n 1 l h implies (7.3). Furhermore, he fulfillmen of (7.28) a ime implies ha k h,+n = i S h H [i] h ỹ[i] +N (n h 1)k h,+n 1 + n h lh (n h 1)( l h k h,+n 1 ) l h (7.40) which, in urn, implies ha (7.3) will be verified also a ime + N. Finally we prove ha a soluion o (7.29) exiss a sep + 1 for all i = 1,..., M. In fac, aking ȳ [i] +N+1 = ȳ[i] +N = ỹ [i] +N one has ȳ[i] +N ỹ[i] +N = 0 B (p i) q,ε(0) and ȳ[i] +N Y i, hence verifying (7.30) and (7.31), respecively. Furhermore H [i] h ỹ[i] +N+1 = H [i] h ỹ[i] +N l h j S h \{i} H[j] h ỹ[j] +N (n h 1) n h ( l h k h,+n ) In fac k h,+n l h (n h 1) n h ( l h k h,+n ) in view of he fac ha k h,+n l h, as i is proved in (7.40). 146

157 hesis 2015/11/21 9:18 page 147 # Proof of he main heorem Proof of convergence for he reference managemen layer In absence of coupling consrains (7.3), since a ime +1, ȳ [i] +N+1 = ỹ[i] +N is a feasible soluion, in view of he opimaliy of he soluion ȳ [i] +N+1 +1 V y i (ȳ[i] +N+1 +1, + 1) V y i (ȳ[i] +N, + 1) ȳ[i] +N y[i] se poin 2 T i (7.41) In view of he fac ha ȳ [i] +N+1 +1 = ỹ[i] +N+1 for all, we can wrie he erm V y i (ȳ[i] +N+1 +1, + 1) = γ ỹ[i] +N+1 ỹ[i] +N 2 + ỹ [i] +N+1 y[i] and rewrie (7.41) as ỹ [i] +N+1 y[i] se poin 2 T i se poin 2 T i, ỹ [i] +N y[i] se poin 2 T i γ ỹ [i] +N+1 ỹ[i] +N 2. From his we infer ha, as, ỹ [i] +N+1 ỹ[i] +N 0 and ỹ [i] +N y[i] se poin 2 T i cons (7.42) Assume, by conradicion, ha ỹ [i] +N y[i] se poin 2 T i c i, wih c i > c o i, where c o i = y [i] feas.se poin y[i] se poin 2 T i (7.43) Noe ha his implies ha ỹ [i] +N y[i] feas.se poin. Assume ha, given, for all he opimal soluion o (7.29) is ȳ [i] ȳ [i] where ȳ [i] y [i] se poin 2 T i +N = = c i. I resuls ha V y i (ȳ[i] +N, ) = c i. On he oher hand, an alernaive soluion is given by ȳ [i] +N, where ȳ[i] +N = λ i ȳ [i] + (1 λ i )y [i] feas.se poin, wih λ i [0, 1). This soluion is feasible provided ha (I) ȳ [i] +N ȳ[i] B p i is sufficienly small, (II) ȳ [i] +N q,ε(0) which can be verified if (1 λ i ) Y i which is also saisfied if (1 λ i ) is sufficienly small (since Y i is convex and ȳ [i] y [i] feas.se poin ). According o his alernaive soluion V y i (ȳ[i] +N, ) = γ ȳ[i] +N ȳ[i] 2 + ȳ [i] +N y[i] se poin 2 T i Now if (7.32) is verified, hen V y i (ȳ[i] +N, ) < V y i (ȳ[i], ). This conradics he assumpion ha ỹ [i] +N y[i] se poin 2 T i c i, wih c i > c o i. Therefore, he only asympoic soluion compaible wih (7.29), is ha corresponding wih ỹ [i] +N = y[i] feas.se poin. 147

158 hesis 2015/11/21 9:18 page 148 #158 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. I is now proved ha ỹ [i] y [i] feas.se poin for. In view of As- y [i] feas.se poin for all i = sumpion 7, his implies ha C i χ [i] 1,..., M. = C i x [i] Proof of recursive feasibiliy of he i-dpc problem Assume ha, a sep, a soluion o (7.33) exiss for all i = 1,..., M, i.e., (ˆx [i], û[i] [:+N 1] ). Nex we prove ha, a sep +1, a soluion o (7.33) exiss for all i = 1,..., M. To do so, we prove ha he uple (ˆx [i] +1, û[i] [+1:+N] ) saisfies he consrains (7.15), (7.35a)-(7.36) and is herefore a feasible (possibly subopimal) soluion o (7.33). Here û [i] [+1:+N] is obained wih û [i] +N = ũ[i] +N + K i(ˆx [i] +N x[i] +N ) (7.44) Firs, noe ha, in view of he robus posiive invariance of ses E i wih respec o equaion (7.18), i = 1,..., M, x [i] +1 ˆx [i] +1 E i, and herefore (7.35a) is verified. Furhermore, in view of he feasibiliy of (7.35b)-(7.35d) a sep, i follows ha (7.35b)-(7.35d) are saisfied a sep + 1 for k = + 1,..., + N 1 and, from (7.36) and (7.25a), ẑ [i] +N z[i] +N = (C zi + D zi K i )(ˆx [i] +N x[i] +N ) (C zi + D zi K i )Σ i z i Hence consrain (7.35b) is verified for k = + N. Furhermore ] [i] [ˆx +N û [i] +N ] [i] [ x +N ũ [i] +N [ Ini K i ] Σ i where, from (7.13) ] [i] [ ] [ x +N Ini 0 ( ) ũ [i] K x +N i Ki e χ [i]ss +N χ i (7.45) In urn, in view of (7.9) and similarly o (7.10), χ [i]ss +N Γ i(i n+p F) 1 G 148 M Y j (7.46) i=1

159 hesis 2015/11/21 9:18 page 149 # Conrol of unicycle robos This evenually implies ha, in view of (7.25b) ] [i] [ [ˆx +N Ini 0 û [i] K x +N i Ki e ] ( Γ i (I n+p F) 1 G ) M i=1 Y j χ i [ Ini K i ] Σ i ˆX i Ûi which verifies consrains (7.35c) and (7.35d) for k = + N. Noe ha, in view of (7.7a), (7.15), and (7.44) ˆx [i] +N+1 x[i] +N+1 = F ii(ˆx [i] +N x[i] +N ) (7.47) and herefore ˆx [i] +N+1 x[i] +N+1 Σ i in view of he definiion of Σ i as a posiively invarian se for (7.24), hence verifying (7.36). Therefore also he consrain (7.36) is verified a sep + 1. Proof of convergence for he robus MPC layer A ime he pair (ˆx [i], û[i] [:+N 1] ) is a soluion o (7.33), leading o he opimal cos Vi N (). Since (ˆx [i] +1, û[i] [+1:+N] ) is a feasible soluion o (7.33) a ime + 1, by opimaliy V N i ( + 1) V N i (ˆx [i] +1, û[i] [+1:+N] ) and, by applying sandard argumens in MPC [138] Vi N (+1) Vi N () ( ˆx [i] x[i] 2 Q i + û [i] ũ[i] 2 R i ), so ha, for all i = 1,..., M, ˆx [i] x[i] and û [i] as. Now, consider he model (7.5a) and, collecively, ũ[i] he model (7.7a) and equaion (7.17). We have ha x +1 = A x + B ( û + K(x ˆx ) ) x +1 = A x + Bũ + BK( x x ) (7.48) for all 0. Denoe x = x x, ˆx = ˆx x and û = û ũ. From (7.48), x +1 = F x +B ( û K ˆx ). Since B ( û K ˆx ) 0 as, in view of Assumpion 6 i holds ha x 0 as, which implies ha asympoically C i x [i] 7.5 Conrol of unicycle robos C i x [i]. In his secion he proposed algorihm is applied o he problem of posiioning a number of mobile robos in specified posiions, while guaraneeing collision avoidance. In principle, wo main limiaions have been 149

160 hesis 2015/11/21 9:18 page 150 #160 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. deeced for he applicaion of he discussed conrol echnique o he considered problem. Firs, as i is widely known, he model of unicycle robos is nonlinear and nonholonomic consrains affec is dynamics, so ha i is no possible o conrol hese vehicles using linear models obained hrough linearizaion. However, by resoring o a feedback linearizaion procedure, previously discussed in [121], a linear model of he robos will be used o describe he sysem s dynamics. Furhermore, collision avoidance consrains (on he surface) should be in principle non-convex and described using nonlinear inequaliies. This would preven he applicaion of our algorihm, since he allowed coupling consrains correspond o linear inequaliies of he form (7.3). To circumven his problem, suiable linear consrains are defined o replace nonconvex ones. Experimenal resuls will be shown o winess he viabiliy of our approach The model of unicycle robos The coninuous-ime dynamics of a single robo is described by a modified version of he firs-order kinemaic model [121] ẋ = v cos φ ẏ = v sin φ φ = ω v = a (7.49a) (7.49b) (7.49c) (7.49d) where he pair (x, y) represen he caresian posiion of he robo on is working area, φ denoes is orienaion angle, and v is is linear velociy. The linear acceleraion a and he angular velociy ω are considered as conrol inpus o be seleced hrough he proposed algorihm. By resoring o a feedback linearizaion procedure (see [121]) a linear model of he robos can be derived o describe he sysem s dynamics. Namely, define η 1 = x, η 2 = ẋ, η 3 = y, η 4 = ẏ, and he dynamics resuling from (7.49) is η 1 = η 2 η 2 = a cos φ vω sin φ η 3 = η 4 η 4 = a sin φ + vω cos φ (7.50a) (7.50b) (7.50c) (7.50d) 150

161 hesis 2015/11/21 9:18 page 151 # Conrol of unicycle robos Now define wo new ficiious inpu variables a x = a cos φ vω sin φ and a y = a sin φ + vω cos φ ha represen he acceleraion along he wo caresian axes. In his way, from (7.50) he model (7.49) is ransformed in a se of wo decoupled double inegraors wih inpus a x and a y. To recover he real inpus (ω, a) from (a x, a y ) i is sufficien o compue [ ] ω = 1 [ ] [ ] sin φ cos φ ax (7.51) a v v cos φ v sin φ Noe ha, for obaining (7.51), i is assumed ha v 0. This singulariy poin mus be accouned for when designing conrol laws on he equivalen linear model [121]. The linear model from (7.50) is hen discreized wih sampling ime τ, obaining he following marices for he dynamics τ 1 τ A ii = A = τ, B τ 0 ii = B = (7.52) τ τ The oupu variables are he caresian coordinaes, x and y, i.e., η 1 and η 3 in (7.50) and hus [ ] C i = Noe ha his case sudy is characerized by (i) no dynamically coupling erms, i.e., E i = 0 and L ij = 0 for all i, j = 1,..., M; (ii) saic coupling consrains on he posiion variables guaraneeing collision avoidance. Therefore, se C zi = C i and D zi = 0. a y The experimenal seup The experimenal se-up consiss of hree e-puck mobile robos [106]. To simplify he applicaion of he algorihm, he conrol law is designed on a porable compuer communicaing wih he e-puck robos hrough wireless connecion. The measuremen sysem consiss of a camera, insalled on he op of he cm 2 working area. Posiion and orienaion of each robo are deeced using wo colored circles, placed on he op of each agen, see Figure 7.3. The linear velociy is reconsruced from measured daa using 151

162 hesis 2015/11/21 9:18 page 152 #162 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. Figure 7.3: Skech of he experimenal se-up. a filering procedure. Noe ha he fac ha no coupling erms are presen (i.e., E i = 0 and F ij = 0 for all i, j) grealy simplifies he design phase. More specifically, wih reference o he off-line design seps oulined in Secion 7.3 1) Assumpion 6 and Assumpion 8 correspond o solve cenralized smallscale (e.g., eigenvalue assignmen) problems. 2) Similarly o he previous sep, χ i can be compued as he RPI se for χ [i] +1 χ [i]ss +1 = F ii (χ [i] χ [i]ss ) + w [i] wih w [i] = (I 6 F i ) 1 F i G i B (2) q,ε and ε = 5 cm. 3) Since E i = 0, w [i] = 0 in (7.18), E i is an arbirarily small posiively invarian se, and z i can be chosen arbirarily. 4) Since, in (7.25b), we have ha M Γ i (I n+p F) 1 G Y j = (I 6 F i ) 1 G i Y i i=1 152

163 hesis 2015/11/21 9:18 page 153 # Conrol of dynamically coupled subsysems his sep can be verified in a decenralized fashion. In his experimen, only he collision avoidance wih respec o oher agens is considered. However, noe ha he fixed obsacle case, generically denoed wih he erm obsacle avoidance, can be easily derived wih similar consideraions. To handle collisions in he problem each agen is considered as a circular objec characerized by he ime-varying posiion of is cener and a fixed radius R i. In general, collision avoidance consrains are non-convex and described using nonlinear inequaliies and hus paricular aenion is needed o handle hem inside he proposed framework. Assume for example a 2D circular obsacle cenered in z wih radius d, he non collision requiremens corresponds o he saisfacion of a minimum disance consrain, and hus he feasibiliy region O C is O C = {z z z 2 > d} (7.53) ha is clearly non-convex. To circumven his problem, suiable linear consrains are defined o replace non-convex ones. In paricular, each of hem is obained by racing a line semming from he cener of he i-h robo and angen o a circumference whose cener corresponds o he posiion of he neighboring robo j, and whose radius is d = R i + R j. An example of he approximaion is shown in Figure 7.4. Beween he wo compued consrains (a T 1 z b 1 and a T 2 z b 2 in his simple example) only one mus be seleced and in paricular we choose he one ha allows he agen o say closer o is final goal. Of course his procedure inroduces some conservaism, since he new feasibiliy region is much smaller han he original one, however i allows o efficienly solve he opimizaion problem. A more efficien soluion, inspired from he work in [2], will be presened in he nex chaper for he sochasic framework. For all i = 1, 2, 3, in he cos funcions V y i and V N i we se γ = 1, T i = 4I 2 and Q i = I 4, R i = 0.01I 2, respecively. In he repored real experimen he hree robos are iniially placed (a ime = 1) a posiions (28, 52), (39, 16), and (90, 39) - all coordinaes are in cm. Figure 7.5 shows he evoluion of heir moion in reaching he goal posiions - i.e., (86, 13), (77, 55), and (20, 39) - a ime = 45 s while fulfilling collision avoidance consrains. 7.6 Conrol of dynamically coupled subsysems Consider now he four-ank sysem (see Figure 7.6) described in [71] and used as a benchmark o es several conrol algorihms, boh cenralized and 153

164 hesis 2015/11/21 9:18 page 154 #164 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. Figure 7.4: Linear approximaion of he avoidance consrain. The obsacle (red ball) is approximaed choosing one of he wo angen lines semming from he curren posiion of he i-h agen (blue poin). disribued, e.g., in [1,35,51,103]. The goal is o conrol he waer levels h 1 and h 3 of anks 1 and 3 using he pump command volages v 1 and v 2. The variables x [i], i = 1,..., 4 and u [j], j = 1, 2 are he variaions of he levels h i and of he volages v j wih respec o he corresponding nominal working poins. The obained linearized and discreized sysem wih sampling ime τ = 0.5 s has he form (7.5) wih n = 4, p = m = 2 where A = B = , C = [ ] The sysem is decomposed ino wo subsysems: he firs one is composed by Tank 1 and Tank 2, and he second one by Tank 3 and Tank 4. We se x [1] = (x 1, x 2 ), u [1] = u 1, y [1] = y 1, x [2] = (x 3, x 4 ), u [2] = u 2, and y [2] = 154

165 hesis 2015/11/21 9:18 page 155 # Conrol of dynamically coupled subsysems Figure 7.5: Plos of he robo rajecories. Robo 1: ; robo 2: ; robo 3:. Symbols wih whie surface denoe he posiion of he robos, while symbols wih black surface denoe he goal posiions. Large circles wih grey dashed line denoe he area occupied by he robos. Figure 7.6: Schemaic represenaion of he four-anks sysem 155

166 hesis 2015/11/21 9:18 page 156 #166 Chaper 7. Disribued Predicive Conrol for racking reference signals: a robus approach. y 2. The consrains on he sysem s variables are x [1] min = [ ] T, x [1] max = [ ] T [2], x min = [ ] T [2], x max = [ ] T, u [1] min = 3, u[1] max = 3, u [2] min = 3, u[2] max = 3. The marices K i and K i fulfilling Assumpion 6 and Assumpion 7 have been compued as described in Remark 4. The weighs used are Q 1 = Q 2 = I 2, R 1 = R 2 = 1, T 1 = T 2 = 1, γ = 10 6, N = 3. In he simulaions, he reference rajecories y [i] se poin, i = 1, 2 are piece-wise consan, see Figure 7.7. The resuls achieved are depiced in Figure 7.7. Noably, he se-poin y [1] se poin = 2.5 resuls infeasible o our algorihm, and hence he sysem oupu y [1] neares feasible value. converges o he Waer level of Tank 1 (cm) Waer level of Tank 2 (cm) Time (s) Figure 7.7: Trajecories of he oupu variables y [1] (above) and y [2] (below) (black solid lines) and reference oupus ỹ [1] (above) and ỹ [2] (below) (black dashed lines). Grey dash-doed lines: desired se-poins y [1,2] se poin. 7.7 Commens In his chaper a novel disribued scheme for racking reference signals has been proposed. As in all exising noncooperaive schemes, a degree of conservaiviy is brough abou when addressing couplings among subsysems and for obaining simple problems a he reference generaor level. On he oher hand, he main advanages are: scalabiliy of he online implemenaion, limied ransmission and compuaional load (also in view of he fac ha he reference generaor layer is independen from he robus MPC layer, and hence compuaions can be performed in a parallelized fashion), 156

167 hesis 2015/11/21 9:18 page 157 # Commens and simpliciy of implemenaion. The algorihm has also proven o be very flexible, i.e., in he unicycle robo applicaion, where i has already been successfully used for he soluion of leader-following and/or formaion conrol problems. In he nex chaper an exension of he proposed algorihm for racking reference signals under probabilisic consrains is considered for he simpler case of dynamically decoupled sysems and wih he aim o use i in he unicycle coordinaion problem described here. 157

168 hesis 2015/11/21 9:18 page 158 #168

169 hesis 2015/11/21 9:18 page 159 #169 CHAPTER8 Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains In he previous chaper we presened a disribued predicive conrol approach for racking reference signals ha is based on he use of robus/worscase echniques o handle couplings and uncerainies and saisfy local and coupling deerminisic consrains. Wih he aim of obaining similar resuls, we consider in his chaper a disribued predicive conrol approach for racking reference signals in a sochasic framework, i.e. in presence of probabilisic consrains, as discussed in Chaper 2. In paricular, inspired by he work in [127] and some resuls presened in Chaper 6, an algorihm is presened for he case of dynamically independen sysems wih coupling chance consrains and hen reformulaed specifically in he case of he unicycle coordinaion problem inroduced in Secion 7.5. In order o show he viabiliy of he approach in a real es case, some resuls from [105] are presened and discussed. 159

170 hesis 2015/11/21 9:18 page 160 #170 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains 8.1 Problem saemen Similarly o he previous chapers, consider a se of M sysems, denoed by S i, i = 1,..., M, each of which is described by he independen dynamics x [i] +1 = A ii x [i] + B ii u [i] + B w,i w [i] (8.1) where, for he i-h subsysem, x [i] R n i is he sae, u [i] R m i is he inpu and w [i] R p i is a zero-mean whie noise wih variance W i and possibly unbounded suppor. In addiion, we assume ha he uncerainies w [i] and are uncorrelaed for all j i. In order o derive a racking conroller, w [j] i is useful o define he oupu of subsysem S i as z [i] = C i x [i] (8.2) where z [i] R m i for all i = 1,..., M. In line wih he probabilisic MPC algorihm presened in Chaper 3 we assume perfec sae informaion and he sabilizabiliy of he pairs (A ii, B ii ) and (A ii, B w,i ) wih B w,i W i BT w,i = B w,i. Noe also ha, due o he assumpion of independen dynamics, he decenralized sabilizabiliy required in he general disribued framework, amouns here o he sabilizabiliy of he single subsysems. Finally, as in Assumpion 7 of he previous chaper, we define he marix [ ] I Aii B ii S i = C i 0 and we assume ha rank(s i ) = n i + m i for all i = 1,..., M. The purpose is o develop a sochasic disribued conrol scheme guaraneeing ha boh local and collecive consrains are verified while achieving racking properies. However, being in a probabilisic framework, some remarks are in order Concerning he racking propery, if z [i] G is he desired goal for he oupu variable z [i], we aim o guaranee ha E[z [i] ] z [i] G as. As for he consrains, we mus allow for a probabiliy of consrain violaion. Indeed, based on he definiions given in Chaper 2 and similarly o he soluion adoped in Chaper 3, for each subsysem S i he local consrains are defined as a se of individual chance consrains { P P b T r,ix [i] { c T s,iu [i] } x max r,i 1 p x r,i, r = 1,..., n r,i } u max s,i 1 p u s,i, s = 1,..., n s,i (8.3a) (8.3b) 160

171 hesis 2015/11/21 9:18 page 161 # Problem saemen In addiion, we define a se of collecive probabilisic consrains ha involve he saes of possibly all he subsysems a he same ime, i.e., { M } P d l,i x [i] 1 1 p c l, l = 1,..., n c (8.3c) i=1 In line wih he noaion adoped in he previous chapers, in (8.3) he erms b r,i, c s,i and d l,i are consan vecors, while he maximal allowed probabiliies of consrain violaion p x r,i, p u s,i and p c l are design parameers. Concerning (8.3a), he ses and X i = {x b T r,ix [i] U i = {u c T s,iu [i] x max r,i, r = 1,..., n r,i }, i = 1,..., M u max s,i, s = 1,..., n s,i }, i = 1,..., M are assumed o be bounded and conaining he origin in heir inerior. Finally, based on he srucure in (8.3c) and using he erminology inroduced in Chaper 7, we say ha sysems S i and S j are consrain neighbors if here exiss l [1,..., n c ] such ha boh d l,i 0 and d l,j 0. The se of neighbors of subsysem S i is denoed again by N i Local conroller srucure According o he p-smpc approach presened in Chaper 3, he local conrol law for he i-h subsysem is composed by an open-loop and a feedback erm and akes he form u [i] = ū [i] + K [i] (x [i] x [i] ) (8.4) where ū [i] and x [i] are he inpu and he sae, respecively, of he nominal sysem S i, defined as x [i] +1 = A ii x [i] + B ii ū [i] z [i] = C i x [i] (8.5a) (8.5b) and z [i] represens he nominal oupu. Noe ha, also in his case, by properly iniializing he nominal sysem and given ha he uncerainy in (8.1) has zero mean, i is possible o prove ha x [i] and ū [i] represens he expeced values of he sae and he inpu, respecively. In (8.4) boh he gain K [i] and he open-loop erm ū [i] need o be compued as he soluions of a 161

172 hesis 2015/11/21 9:18 page 162 #172 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains suiable local MPC opimizaion problem. Conrarily o he case discussed in Chaper 6, hanks o he assumpion of independen dynamics, he expeced values of he saes of each subsysem are now accessible in a decenralized fashion and, hus, hey can be direcly used in our seup Reformulaion of he probabilisic consrains Besides minimizing local cos funcions and saisfying local consrains (8.3a) and (8.3b) we require ha collecive goals are aained, i.e., ha he coupling consrains (8.3c) are verified a all ime insans. As in he previous chaper, in order o ensure he required global properies, we assume ha informaion are ransmied beween agens a each ime insan, and in paricular ha subsysem S i is able o communicae wih is neighbors S j, j N i. More deails on he ype and he amoun of informaion o be ransmied will be given laer. As in he res of he Thesis, he following noaion will be adoped o denoe he opimal predicions along he horizon. Given a generic variable z and a generic marix Z, a any ime sep we will denoe by z +k and Z +k, k 0, he specific values of z +k and Z +k compued based on he knowledge (e.g. measuremens) available a ime. According o he discussion in Chaper 3, i is possible o consider x [i] (respecively, u [i] ) as a sochasic variable wih mean x [i] (respecively, ū [i] ) and variance X [i] (respecively, U [i] ) and use he second order descripion of he sysem variables o enforce probabilisic consrains. In paricular, consrains (8.3) are verified if he following inequaliies can be guaraneed for all 0 ν i=1 b T r,i x [i] c T s,iū [i] d T z,i x [i] x max r,i u max s,i 1 ν b T r,i X[i] b r,i f(p x i ) r = 1,..., n ri (8.6a) c T s,i U [i] c s,i f(p u i ) s = 1,..., n si (8.6b) i=1 d T z,i X[i] d z,i f(p c ) l = 1,..., n c (8.6c) As described in Secion he funcion f(p) depends on he paricular disribuion of he uncerainy and is defined as f(p) = (1 p)/p if he disribuion is no known or f(p) = N 1 (1 p) if he uncerainy is Gaussian (and hus, due o lineariy, also he sae and inpu variables are 162

173 hesis 2015/11/21 9:18 page 163 # Problem saemen Gaussian). Noe ha he wo local consrains, (8.6a) and (8.6b), are derived exacly as described in Chaper 3, while he collecive consrain (8.6c) is obained in view of he uncorrelaion of he noises and he fac ha he subsysems are independen from each oher by assumpion Applicaion of local and coupling consrains As in he p-smpc approach, local consrains (8.6a) and (8.6b) can be direcly enforced in he MPC opimizaion problem by each local conroller. On he oher hand, consrain (8.6c) should be solved in a cenralized way due o he fac ha i includes, a he same ime, he decision variables of more han one subsysem. An alernaive is o handle (8.6c) in a disribued fashion, as proposed for example in [127]. Adoping an approach similar o he one described in Chaper 7, his is achieved by means of wo elemens. Firsly, a ime, he couplings are handled by using he opimal sae rajecories (in erms of expeced value and variance in his case) compued a he previous ime insan, 1, and received by he neighboring agens. Secondly, i is ensured ha hese rajecories a ime mainain similar properies wih respec o he ones ha have been previously ransmied by means of addiional consrains o be applied boh on he expeced value and he variance of he sae. Such addiional consrains on he predicions, ha need o be verified for all k = 0,..., N 1, are obained by considering he opimal disance from a cerain consrain a he previous ime insan, spliing i beween he agens involved in he consrain and, finally, allowing each of hem o deviae from heir opimal prediced rajecory of a mos he resuling value. In paricular, and for all l = 1,..., n c and i = 1,..., M d T l,i( x [i] +k x[i] +k 1 ) 1 ν l δ l ( + k 1) X [i] +k X[i] +k 1 (8.7a) (8.7b) where ν l is he number of subsysems such ha d l,i 0 (i.e., he number of subsysems consrained by he l-h coupling consrain). Consisenly wih he proposed probabilisic approach, for consrains (8.6c), he value δ l ( + k 1) is compued as follows δ l ( + k 1) = 1 M i=1 d T l,i x [i] +k M i=1 d T l,i X[i] +k 1 dt l,i f(pc l ) (8.8)

174 hesis 2015/11/21 9:18 page 164 #174 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains and represens he disance, wih respec o he opimal rajecories obained a ime 1 by all he agens from he violaion of he l-h consrain a ime + k. A his poin some commens are in order. In paricular, noe ha he values δ l ( + k 1), k = 0,..., N 1 and l = 1,..., n c are compleely deermined a ime by he opimal expeced value and variance rajecories of he neighboring agens compued a ime 1. These informaion are hus he ones we need o ransmi beween neighboring agens in order o o solve he coupling in a disribued way. Of course, similarly o he approach discussed in Chaper 7 his choice inroduces a source of conservaism bu allows o compleely decouple he collecive consrains and o parallelize he soluion of he problem. 8.2 SDPC for racking: formulaion and properies As discussed in Chaper 7, in order o guaranee he racking propery, a each ime sep we need o add o he problem an exra degree of freedom, consising of he virual oupu, z [i], ha is acually racked by he local (probabilisic) MPC conroller while moving owards is final goal z [i] G. In line wih he muli-layer sraegy, saring from he oupu reference, ha is kep consan along he predicion horizon, we need o define he consan reference rajecories for he sae and he inpu o be racked by he sae and he inpu of he nominal model, x [i] +k and ū[i] +k, respecively, for all k = 0,..., N. In Chaper 7 his is done by he so-called reference sae/inpu rajecory layer hrough he definiion of a new sysem ha acs as an observer fed by he desired oupu reference o rack. However, in he case considered here, we aim o simplify he formulaion and hus we resor o he following equivalence [ x [i] ũ [i] ] = S 1 i where he marix S i is always full-rank by assumpion. [ ] 0 z [i] = M i z [i] (8.9) 1 Consider now he problem of generaing he virual oupu reference signal, z [i], and he problem of racking he generaed sae and inpu references x [i] and ũ [i]. Differenly from he previous chaper, in which he conrol scheme is composed by several levels, he goal here is o solve he reference generaion and reference racking problems ogeher in one single opimizaion. Consisenly wih he sochasic framework, inroduced in he previous secion, he local MPC conroller we wan o design o perform his ask is 164

175 hesis 2015/11/21 9:18 page 165 # SDPC for racking: formulaion and properies similar o he p-smpc described in his Thesis. In paricular, he main ingrediens are described below Cos funcion As for he local cos funcion, we consider here, similarly o Chaper 3, he following expression for all i = 1,..., M, i.e., +N 1 J() [i] = E[ k= + z [i] x [i] k z [i] G 2 T i x[i] 2 Q i + u [i] k ũ[i] 2 R i + x [i] +N x[i] 2 P i ] = J m [i] ( x [i], ū [i]...+n 1, z[i] ) + J v [i] (X [i], K [i]...+n 1 ) (8.10) where he wo componens ha accoun respecively for he expeced value and he variance of he sysem variables are defined as +N 1 J m [i] = k= + z [i] x [i] k x[i] 2 Q i + ū [i] k ũ[i] 2 R i + x [i] +N x[i] 2 P i z [i] G 2 T i (8.11a) +N 1 J v [i] = r{(q i + K [i]t k R i K [i] k )X[i] k } + r{p ix [i] +N } k= (8.11b) The main difference wih respec o he approach proposed in Chaper 3 is ha J m [i] is now formulaed for racking and ha here is an exra erm ha accouns for he evoluion of he virual reference o he final goal z [i] G. Consisenly wih he p-smpc case, for all i = 1,..., M we compue he erminal weigh P i as (A ii + B ii Ki ) T P i (A ii + B ii Ki ) P i = Q i K T i R i Ki (8.12) where he consan gain K i can be any sabilizing gain for he i-h nominal model, S i in (8.5a) Terminal consrains As in sandard MPC schemes, erminal consrains are enforced o guaranee recursive feasibiliy properies. Accordingly o he discussion in Chaper 3 and 5, a simple way o handle erminal consrains in he probabilisic 165

176 hesis 2015/11/21 9:18 page 166 #176 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains framework is o involve he expeced value and he variance of he sae variables. Similarly o equaion (3.8), he covariance marix of he sae of he i-h subsysem, X [i], under he conrol law (8.4) evolves as X [i] +1 = (A ii + B ii K [i] )X [i] (A ii + B ii K [i] ) T + B w,i W i B T w,i (8.13) Based on equaion (8.13) and similarly o he resul in(3.23), we define, for each i = 1,..., M he seady sae covariance marix X i as he soluion of he Lyapunov equaion (A ii + B ii Ki ) T Xi (A ii + B ii Ki ) X i = B w,i Wi B T w,i (8.14) where he covariance of he disurbance is chosen as W i W i. Similarly o equaion (3.25), using he seady sae covariance X i, we define for each subsysem he ighened convex se of local consrains X i as { } X i = x [i] b T r,i x [i] x max r,i b Tr,i X i b r,i f(p xi ), r = 1,..., n ri (8.15) Moreover, he seady sae covariance marix of he conrol inpu is obained as Ūi = K i Xi KT i and, based on his, we define he local probabilisic se Ū i as follows { } Ū i = ū [i] c T s,iū [i] u max s,i c Ts,iŪic s,i f(p ui ), s = 1,..., n si (8.16) However, wih respec o he definiion of he erminal consrains in Chaper 3, he ses (8.15) and (8.16) are no sufficien, in he racking framework, o compue a erminal se ha guaranees he desired properies and furher consideraions are required. In paricular, we adop here a zero erminal consrain sraegy for he choice of he expeced value of he sae, i.e., we require ha x [i] +N = [ I 0 ] M i z [i] (8.17) Furhermore, wih his choice, he erm z [i] mus be seleced in order o saisfy he local ighened consrains in (8.15) and in (8.16) and hus we require ha M i z [i] λ ( Xi ) Ūi (8.18) where λ (0, 1). Finally, z [i], i = 1,..., M mus verify also he collecive consrains in (8.6c) and, o his end, we resor again o a disribued soluion. In paricular, we assume ha, a ime, each agen S i, i = 1,..., M 166

177 hesis 2015/11/21 9:18 page 167 # SDPC for racking: formulaion and properies can use he opimal soluions obained a ime 1 by is neighbors, namely z [j] 1 1, j N i, o compue δ [z] l ( 1) = 1 M i=1 d T z,i [ I 0 ] Mi z [i] 1 1 and hen o define he erminal consrain d T i,z [ I 0 ] Mi ( z [i] M d T X z,i i d z,i f(p c l )1 i=1 (8.19) z [i] 1 1 ) 1 ν l δ [z] l ( 1) (8.20) Concerning he variance, similarly o Chaper 3 we require for all i = 1,..., M ha he following inequaliy holds X [i] +N Choice of he iniial condiions X [i] i (8.21) In order o apply a sae-feedback p-smpc like conroller, as described in Chaper 3, we need o consider he iniial condiions for he expeced value and he variance of he sae, namely x [i] and X[i], as exra degrees of freedom of he opimizaion problem and selec hem according o he following sraegy Sraegy 1 - Rese of he iniial sae: in order o use all he informaion available a ime from measures, se x [i] = x[i], X [i] = 0. Sraegy 2 - Predicion: in order o mainain feasibiliy of he problem when he firs sraegy is no applicable (due o he possibly unbounded disurbance ha can drive he sae ouside he feasibiliy region), se x [i] = x[i] 1, X [i] = X[i] 1, i.e., use he pas opimal predicions. For furher deails on he raionale behind his choice and on he implemenaion issues please see Chaper The i-h MPC conroller Based on he ingrediens discussed so far, he local deerminisic opimizaion problem for he i-h subsysem S i (denoed by i-psmpc) can now be saed as follows 167

178 hesis 2015/11/21 9:18 page 168 #178 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains i-psmpc problem: a any ime insan solve he minimizaion problem min ū [i]...+n 1, K[i]...+N 1 z [i], ( x [i], X [i] ) J m [i] ( x [i], ū [i]...+n 1, z[i] ) + J v [i] (X [i], K [i]...+n 1 ) (8.22) subjec o he dynamics (8.5a) and (8.13), o he local consrains (8.6a), (8.6b) and he collecive consrains (8.7) for all k = 0,..., N 1, o he erminal consrains (8.17), (8.18), (8.20) and (8.21) and o he iniializaion sraegy ( x [i], X[i] ) {(x[i], 0), ( x [i] 1, X[i] 1 )} As a soluion o he i-psmpc problem we obain he opimal sequences of open-loop erms and gains, respecively ū [i]...+n 1 and K[i]...+N 1, and he opimal values x [i],x[i] and z[i]. Thus, consisenly wih (8.4), a ime sep we apply he following inpu u [i] = ū [i] + K[i] (x[i] x [i] ) (8.23) Moreover, in view of (8.5a), (8.13) and (8.5b), we can compue he opimal sequences a ime, i.e., x [i]...+n 1, X[i]...+N 1 and z[i]...+n 1, o be ransmied o he oher agens accordingly wih he neighboring scheme. Noe ha, in line wih he discussion in Chaper 3, he i-psmpc opimizaion problem can be simplified by choosing offline a fixed gain, i.e., seing K [i] = K i, 0, i = 1,..., M. In view of his, he compuaional load associaed o each i-psmpc problem would be he same of a small-scale nominal MPC problem and, moreover, he nonlineariy in he consrains does no carry abou implemenaion problems (for furher deails on he implemenaion in he complee case, see Chaper 3). However, even in his case he erm J v in (8.10) canno be discarded in he opimizaion because, alhough i does no depend anymore on he conrol gain sequence K [i]...+n 1, i sill depends upon he iniial condiion X[i]. 8.3 Sochasic Disribued Conrol of a flee of unicycle robos In his secion we discuss he applicaion of he algorihm described in his chaper o he problem of conrolling he flee of mobile robos presened in Secion 7.5. Wih respec o he robus seup, he wors-case consideraions are relaxed ino he probabilisic framework by allowing a cerain probabiliy of violaing he obsacle/collision avoidance consrains. Besides he 168

179 hesis 2015/11/21 9:18 page 169 # Sochasic Disribued Conrol of a flee of unicycle robos he apparen naivey of he applicaion, he seup allows o show, hrough a simple disribued conrol problem (due o he independen dynamics), he effeciveness of such echniques and once again, sress he fac ha probabilisic approaches represen he only way we have o handle possibly unbounded disurbances. In view of he special srucure of he mobile robos coordinaion problem, where he iner-robo collision avoidance represens he only coupling erm, several deails mus be considered and he opimizaion problem needs o be re-defined Model of unicycle robos The model of he i-h subsysem is he unicycle model described in Secion 7.5, i.e., ẋ = v cos φ ẏ = v sin φ φ = ω v = a (8.24a) (8.24b) (8.24c) (8.24d) The sysem dynamics are linearized hrough a feedback-linearizaion loop (see e.g. [121]) and discreized wih sample ime τ so ha he model of he subsysem S i is he one in (8.1) wih τ 1 τ A ii = τ, B τ 0 ii = τ 0 2, C i = τ [ ] , B w,i = I where he uncerainy is used o consider unmodeled dynamics, acuaion disurbances, ec. Furher deails on he idenificaion of he uncerainy disribuions will be given laer in he example secion Obsacle and collision avoidance In Secion 7.5 he collision (and obsacle) avoidance propery has been quickly discussed and a soluion o replace nonconvex disance consrains wih linear, alhough conservaive, consrains has been proposed. However, in order o inroduce he concep of avoidance in he probabilisic framework, here a differen seup, aken from [2], is presened. More 169

180 hesis 2015/11/21 9:18 page 170 #180 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains specifically, our goal is o define an algorihm such ha, for any fixed obsacle h wih known posiion or any oher moving agen j = 1,..., M, j i, he collision wih he i-h robo is avoided wih a cerain probabiliy. To his end, and in order o simplify he seup, each obsacle (robo) is assumed o be circular, cenered in zh o (z[i] ) and wih radius Rh o (R i). Moreover, for each robo i we define he se of n o i proximal obsacles as O i {1,..., n o i }, i.e., he se of obsacles ha are o be accouned for by robo i and he se of proximal neighbors as C i N \{i}, i.e., he se of oher agens wih which collision avoidance mus be enforced. For consisency we assume ha j C i if and only if i C j. The obsacle and collision avoidance properies will be saisfied by simply incorporaing suiable consrains ino he local opimizaion during he compuaion of he reference rajecories o be pursued by he robos. Noe ha, in view of he non-convexiy of boh ypes of consrains, hey canno be simply expressed in he forms (8.3c), and hus furher discussions are needed. In order o obain linear bounds ha approximae he obsacle and collision avoidance consrains, firsly consider he generic minimum disance consrain z z 2 d (8.25) where z and d are given. As proposed in [2], define as an ouer approximaion of he circle a symmeric polyope P, ha is defined using r linear inequaliies, i.e., P = {z h T k (z z) d} (8.26) The idea is now o noe ha he violaion of any of he inequaliies h T k (z z) d, i.e., he fac ha here exiss an index k {1,..., r} such ha h T k (z z) > d, implies ha (8.25) is verified and hus ha we are in a collision siuaion. Wih his in mind wo sraegies for handling collisions wih fixed obsacles and oher moving agens, respecively, are presened. Boh of hem are inspired from he resuls in [2] and exended o he adoped probabilisic framework using he approach in [127] Fixed obsacle avoidance Consider now he problem of avoiding he collisions wih he h-h fixed obsacle, cenered in z o h and wih radius Ro h. Denoing wih R i he radius of he i-h unicycle robo posiioned a ime in z [i], o preven collision wih 170

181 hesis 2015/11/21 9:18 page 171 # Sochasic Disribued Conrol of a flee of unicycle robos a mos probabiliy p o hi, we ideally require ha { } P z [i] zh o 2 d o hi (8.27) where he overall radius is defined as d o hi = R i + Rh o. However, due o he lack of convexiy of (8.27), we need o use he sraegy described above o derive a linear approximaion. In paricular, define he ouer approximaing polyope P o hi, described by a se of ro hi linear inequaliies P o hi = {(h [o,hi] k ) T (z [i] zh) o d o hi, k = 1,..., rhi} o (8.28) Now, similarly o (8.3a), he probabilisic obsacle avoidance consrain in (8.27) is reformulaed { as he exisence of a } leas a value l [1,..., rhi o ] such ha P (h [0,hi] ) l T (z [i] zh o) do hi. This, similarly o (8.6a) and based on he resuls discussed in Chaper 3, is verified if (h [o,hi] l ) T ( z [i] zh) o d o hi + (h [o,hi] l ) T Z [i] (h [o,hi] l ) f(p o hi) (8.29) where he expeced value of he posiion of he robo is z [i] variance is defined as Z [i] = CX [i] C T. = C x [i] and is Inspired by he approach in [2] we need o define along he predicion horizon, i.e., for all k =,..., + N 1, he funcion and he index δhi,l(k o 1) = (h [o,hi] l ) T ( z [i] k 1 zo h) d o hi ) T Z [i] (h [o,hi] l k 1 h[o,hi] l f(p o hi) (8.30) lo hi (k 1) = argmax δhi,l(k o 1) (8.31) l [1,...,rhi o ] Noe ha boh (8.30) and (8.31) are compleely defined by he knowledge of he opimal rajecory of robo i compued a he previous ime insan 1. Based on he compued values, a ime sep, he following linear consrain, o be guaraneed by robo i, may be used o replace (8.27) (h [o,hi] lo hi (k 1) )T ( z [i] k z[i] k 1 ) δo hi, l hi o (k 1)(k 1) (8.32) Iner-robo collision avoidance The iner-robo collision avoidance issue is more challenging han he plain obsacle avoidance one, since pairs of moving objecs mus coordinae heir 171

182 hesis 2015/11/21 9:18 page 172 #182 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains acions in a disribued compuaion scheme. In oher words, while he obsacle avoidance requiremen generaes only local consrains o be applied on he i-h agen, he iner-robo collision avoidance requires he use of collecive consrains. However, similar consideraions can be used o handle he non convex disance consrains and he sraegy described in [127] applied o handle collecive consrains. To preven collision beween robos i and j, cenered respecively in z [i], wih a mos probabiliy p ij, a ime, we ideally require ha z [j] P and { } z [i] z [j] 2 d ij p ij (8.33) where d ij = R i + R j. Again, we sar defining he ouer approximaing polyope, P ij, circumscribing he circle cenered a z [j] wih radius d ij, ha is represened by a se of r ij linear inequaliies, i.e., P ij = {(h [ij] k )T (z [i] z [j] ) d ij k = 1,..., r ij } (8.34) In his way, as in he fixed obsacle case, iner-robo collision avoidance can be saed { as he exisence of a leas a value } of he index l [1,..., r ij ] such ha P ) T (z [i] z [j] ) d ij p ij. This is verified if (h [ij] l (h [hi] l ) T ( z [i] z [j] ) d ij + (h [ij] l ) T (Z [i] k 1 + Z[j] k 1 )h[ij] l f(p ij ) (8.35) In order o apply he consrain o he problem we need o define along he horizon, i.e., for all k =,..., + N 1 he following funcion δ ij,l (k 1) = (h [ij] l ) T ( z [i] ) T (Z [i] (h [ij] l k 1 z[j] k 1 ) d ij k 1 + Z[j] k 1 )h[ij] l f(p ij ) (8.36) and in urn we need o compue he index corresponding o he maximum value of δ ij,l (k 1) as lij (k 1) = argmax l [1,...,r ij ] δ ij,l (k 1) (8.37) Noe ha he values (8.36) and (8.37) depend from he opimal rajecories of boh he agens i and j bu also ha hese values are compued a ime 1 and hus are compleely known a ime under he assumpion ha he neighboring agens are able o communicae heir fuure seps. According o he procedure described in Secion 8.1, in order o apply he avoidance 172

183 hesis 2015/11/21 9:18 page 173 # Sochasic Disribued Conrol of a flee of unicycle robos consrain (8.33) a ime sep, he following linear consrain, o be guaraneed by robo i and j, may be used along he predicion horizon, i.e. for all k = 0,..., N 1 (h [ij] lij (k 1) )T ( z [i] k z[i] k 1 ) 1 2 δ ij, l ij (k 1)(k 1) (h [ij] lij (k 1) )T ( z [j] k 1 z[j] k ) 1 2 δ ij, l ij (k 1)(k 1) (8.38a) (8.38b) The main difference wih (8.32) is ha in he iner-robo collision avoidance case, he single avoidance consrain compued for robo i affecs also he se of consrains o be applied o he robo j. However, applying he proposed sraegy, he resuling opimizaion problem for agen i is compleely independen from he degrees of freedom of he neighboring agens, hus allowing for a disribued implemenaion a he price of a sligh conservaism due o he limiaion imposed o he evoluion of he oupu variable beween wo consecuive seps Terminal Consrains As discussed in Secion 8.2.2, also in he framework of disribued conrol for mobile robos i is useful o adop a zero erminal consrain-ype sraegy. In paricular, for he sae a he end of he horizon, x [i] +N, i.e., we require ha x [i] +N = [ I 0 ] M i z [i] (8.39) Concerning z [i] i mus, naurally, firs be seleced in such a way ha [ I 0 ] Mi z [i] λ X i (8.40) where he se X i represens he local consrains o be saisfied and is defined as { X i = x [i] b T r,i x [i] 1 b Tr,i X [i] b r,i f(p xr,i), } r = 1,..., n r (8.41) Secondly, i mus be seleced in order o preven collisions. This is done by considering he properies of he soluion o he problem obained a he previous ime sep, 1, i.e., z [i] 1 1. Concerning obsacle avoidance beween he i-h robo and he h-h obsacle, his requires he definiion of δ hi,l( o 1) = (h [o,hi] l ) T ( z [i] 1 1 zo h) d o hi ) T Z [i] h [o,hi] (h [o,hi] l 173 l f(p o hi) (8.42)

184 hesis 2015/11/21 9:18 page 174 #184 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains where Z [i] is he seady sae value of he covariance marix of he oupu and, in urn, is compued saring from he seady sae covariance marix of he sae from (8.14) as Z [i] = C X [i] C T. In addiion, he index corresponding o he maximum value of δ hi,l o ( 1) is compued as lo hi ( 1) = argmax δo hi,( 1) ( 1) (8.43) l [1,...,rhi o ] and a ime sep, he following linear consrain, o be guaraneed by robo i, mus be fulfilled (h [o,hi] lo hi ( 1) )T ( z [i] z [i] 1 1 ) δ o hi, l hi o ( 1)( 1) (8.44) in order o safely choose z [i] a he end of he horizon. Similarly, concerning iner-robo collision avoidance, consider he seady sae variances of robo i and j, namely Z [i] and Z [j], and define he funcion δ ij,l ( 1) = (h [ij] l ) T ( z [i] 1 1 z[j] 1 1 ) d ij (h [ij] l ) T ( Z [i] + Z [j] )h [ij] l f(p ij ) and he corresponding index (8.45) lij ( 1) = argmax l [1,...,r ij ] δ ij,l ( 1) (8.46) A ime sep, he following linear consrain, o be guaraneed by boh he robo i and j, mus be fulfilled (h [ij] lij ( 1) )T ( z [i] z [i] 1 1 ) 1 2 δ ij, li j( 1) ( 1) (8.47a) (h [ij] lij ( 1) )T ( z [j] 1 1 z[j] ) 1 2 δ ij, lij ( 1)( 1) (8.47b) Saemen of he problem and skech of he algorihm We are now in he posiion o derive he main opimizaion problem, solved by he local conroller embedded in he i-h robo. Wih respec o he algorihm presened in he previous secion and wih he aim of reducing he compuaional load and simplifying he applicaion we fix he value of he conroller gain, i.e. we se K [i] = K i, where K i is a generic sabilizing gain for he nominal sysem. 174

185 hesis 2015/11/21 9:18 page 175 # Simulaion resuls i-psmpc problem: a any ime insan solve min ū [i]...+n 1, x[i],x[i], z[i] J m [i] ( x [i], ū [i]...+n 1, z[i] ) + J v [i] (X [i] ) (8.48) subjec o he dynamics (8.5a) and (8.13), consrains (8.6a), obsacle avoidance consrains (8.32) for all h O i, iner-robo collision avoidance consrains (8.38) for all j C j and for all k = 0,..., N 1 he iniial consrain ( x [i], X[i] ) {(x[i], 0), ( x [i] 1, X[i] 1 )}, he erminal consrains (8.39), (8.40), (8.44) and (8.47). Similarly o he general case presened in he previous secion, as a soluion o he i-psmpc problem we obain he opimal sequences for he open-loop erm, ū [i]...+n 1, and he opimal values x[i],x[i] and z[i]. Thus, consisenly wih (8.4), a ime sep we apply he following inpu u [i] = ū [i] + K i (x [i] x [i] ) (8.49) Moreover, he opimal sequences, in view of (8.5a), (8.13) and (8.5b), allow o define he opimal predicions sequences a ime o be ransmied o he oher agens accordingly wih he neighboring scheme, i.e., x [i] X [i]...+n 1 and z[i]...+n 1, respecively....+n 1, 8.4 Simulaion resuls In his secion wo simulaion examples are presened o show he efficacy of he adoped approach. In paricular, he firs example shows he behavior of a single robo while avoiding a fixed obsacle o reach a desired goal. In he second example wo differen agens are considered and he resuls of he iner-robo collision avoidance sraegy are presened and discussed. Wih respec o he algorihm described in he previous secion, a summary of he implemenaion seps is given below for he general muli-agen case. Coordinaion algorihm - Iniializaion A ime = 0 and for all i = 1,..., M I) Se z [i] 0 = z [i] 0, where z [i] 0 is he iniial posiion of robo i 175

186 hesis 2015/11/21 9:18 page 176 #186 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains II) Se z [i] k 0 = z[i] 0, k = 1,..., N III) Se X [i] 0 0 fixed gain K [i] k = 0 and compue X[i] k 0, k = 1,..., N using (8.13), wih a = K i. Concerning his, check ha he uning knobs N and W i are compaible wih he fulfillmen of he erminal consrain (8.21), i.e., X [i] N 0 X i IV) Compue Z [i] k 0 = CX[i] k 0 CT, k = 1,..., N Online implemenaion A each ime sep > 0 Ia) Updae he se of proximal obsacles O i Ib) Updae he se of proximal neighbors C i Ic) Receive, for all j C i, he quaniies z [j] 1 1 and R j, and he sequences z [j]...+n 1 1,Z[j]...+N 1 1 II) Solve he i-rsmpc problem and compue he sequences z [i]...+n 1, Z [i]...+n 1 III) Apply inpu u [i] compued as in (8.4) Single robo probabilisic obsacle avoidance The firs simulaion is run considering a single unicycle robo and a single fixed obsacle wih he aim o analyze he effec of he probabilisic avoidance consrains in erms of number of collisions deeced on a bunch of differen simulaions (noe ha in his case he i-psmpc conroller is reduced o a sandard p-smpc conroller for racking). The obsacle is cenered in z1 o = (5, 5) wih radius R1 o = 2, he goal is in z1 G = (9, 9) and, wihou loss of generaliy, he robo is supposed o be a poin wih iniial condiions x [1] = (0, 0, 0, 0) and X [1] = 0. The disurbance acing on he robo is assumed o be a zero mean Gaussian whie noise wih variance W 1 = 0.01 I 4. The parameers of he i-psmpc conroller are chosen as Q = I 4, R = I 2, T = 100I 2, he gain K 1 is fixed and compued wih LQ, 176

187 y hesis 2015/11/21 9:18 page 177 # Simulaion resuls ogeher wih he erminal weigh P 1. The avoidance consrain is allowed o be violaed up o p o 1 = 0.2, he sae is lef unconsrained and he inpu consrains are seleced o ensure ha u [1] 1 is violaed wih probabiliy a mos p u 1 = 0.1. In Figure.8.1 he compued rajecories corresponding o 100 differen simulaions are shown. The measured frequency of collision is 0.17, close o he desired value p o x Figure 8.1: Example of probabilisic obsacle avoidance. Trajecories of he robo over 100 differen simulaions saring from he green square and poining o he yellow riangle. The measured collision frequency is Muli-robo collision avoidance The second simulaion is a collision avoidance es beween wo differen agens each running he i-psmpc algorihm. Also in his case, he aim is o show he effeciveness of he approach and o discuss he probabilisic properies of he soluion in erms of deeced collisions over differen simulaions. To his end, he seup is consruced so ha he wo robos mee in he cener of he working area and hus need o sar an avoidance maneuver as shown in Figure 8.2. The red dashed lines denoe he opimal rajecories prediced by he conroller. 177

188 y y y y hesis 2015/11/21 9:18 page 178 #188 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains 60 = = x = x = x x Figure 8.2: Example of collision avoidance beween wo unicycles. The grey circles represen he wo robos, he red riangles are he wo goals, respecively z [1] G = (90, 50) and z [2] G = (10, 51) and he red dashed lines are he opimal prediced rajecories a he curren insan. The unicycles are assumed o be circular wih radius R 1 = R 2 = 1.5. The wo goals are seleced as z [1] G = (90, 50) and z[2] G = (10, 51) and he iniial condiions for he wo robos se o x [1] = (0, 0, 50, 0), x [2] = (100, 0, 50, 0) and X [1] = X [2] = 0. The disurbances acing on he wo agens are independen zero-mean Gaussian whie noises wih variances W 1 = W 2 = 0.1I 4. The parameers of he wo i-psmpc conrollers are Q = I 4, R = I 2, T = 100 I 2. As in he previous example, he conrol gains for boh he agens, K1 and K 2 are fixed and compued wih LQ, ogeher wih he erminal weighs P 1 and P 2. The collision is allowed wih a probabiliy up o p 12 = 0.3, and as in he previous example he sae is lef unconsrained while he inpu consrains are seleced o ensure ha u [1] 1 is violaed wih probabiliy a mos p u 1 = 0.1. In Figure 8.3 he evoluion over ime of he relaive disance beween he wo robos is shown for 250 differen simulaions. The measured frequency of deeced collisions (ha happen around he ime insan = 40, as apparen from Figure 8.2) is 0.29, hus confirming he efficacy of he approach. 178

189 iner-robo disance hesis 2015/11/21 9:18 page 179 # Applicaion o a real coordinaion problem ime Figure 8.3: Example of probabilisic collision avoidance beween wo unicycles. Blue lines represen he relaive disance over ime for 250 differen exracions of he uncerainy. The red doed line is he minimum disance o avoid collisions se o 3 cm The measured collision frequency is Applicaion o a real coordinaion problem In his secion he approach discussed in his chaper is applied o a real coordinaion problem run on he experimenal seup described in Secion 7.5. Firsly, in order o idenify he model disurbance acing on he i-h robo, several open-loop ess are made obaining he measuremens in Figure 8.4. In paricular, heir mean resuls o be zero and he disribuion nearly Gaussian as depiced in Figure 8.5, hus moivaing he choice made in he previous secion. The esimaed covariance marix is block diagonal W Using he idenified disurbance model, wo differen ess are made. Obviously, in boh he cases, he goal is no o check he probabilisic behavior 179

190 hesis 2015/11/21 9:18 page 180 #190 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains Figure 8.4: Measured disurbance samples in one of he ess. The wo figures on he op represen he disurbance on he posiion while he boom figures represen he disurbance on he speed. of he sysem bu jus o show he efficacy of he approach ha allows o solve he problem even in he case of unbounded disurbances for which a wors-case echnique like he one used in Chaper 7 is no applicable. In pracice, he collision probabiliy will be se o a sufficienly small value. The firs es is similar o he second simulaion example in which wo unicycles are required o swich heir posiion hus generaing a possible collision siuaion close o he cener of he working area. The resul is shown in Figure 8.6 by means of a series of picures of he working plane aken a differen ime insans. The second example is a complee coordinaion ask in which hree agens need o reach heir own parking spos while avoiding collisions beween each oher and wih fixed obsacles. The resuling behavior is described in Figure 8.7 where he red circles represen virual obsacles along he pah. 180

191 hesis 2015/11/21 9:18 page 181 # Commens Figure 8.5: Example of disribuion of he measured disurbance samples. The firs figure shows he disurbance on he posiion along he x axis and he second figure he disurbance on he velociy along he x axis. 8.6 Commens In his chaper we described a disribued sochasic MPC algorihm for racking reference signals in he case of dynamically decoupled subsysems and probabilisic local and coupling consrains, based on he preliminary work in [127]. The algorihm is hen specialized o he mobile robos coordinaion problem where he previous resuls are used, ogeher wih he procedure suggesed in [2], for avoiding collisions wih fixed obsacle and wihin he agens. Two simulaion experimens has shown he efficacy of he proposed approach in erms of probabilisic characerizaion of he conrolled sysem rajecories, i.e. he number of deeced collisions over muliple simulaions, boh in he case of fixed obsacles avoidance and in he case of an iner-robo coordinaion ask. A hird example originally presened in [105] is repored in order o show he effeciveness of he proposed approach in a real unicycles coordinaion problem. 181

192 hesis 2015/11/21 9:18 page 182 #192 Chaper 8. Sochasic Disribued Predicive Conrol for racking of independen sysems wih coupling consrains Figure 8.6: Coordinaion es wih wo unicycles (red and blue). For boh of hem he aim is o reach heir own local goals while avoiding collision in he cenral zone wih high probabiliy. 182

193 hesis 2015/11/21 9:18 page 183 # Commens Figure 8.7: Example of collision free navigaion in presence of several agens (blue, yellow and red) and fixed (virual) obsacles. The robos, equipped wih he i-psmpc conrollers, are able o reach heir goal while avoiding collisions and resolving conflics. 183