Safety control with performance guarantees of cooperative systems using compositional abstractions

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1 Safety ontrol with performane garantees of ooperative systems sing ompositional abstrations Pierre-Jean Meyer, Antoine Girard, Emmanel Witrant To ite this version: Pierre-Jean Meyer, Antoine Girard, Emmanel Witrant. Safety ontrol with performane garantees of ooperative systems sing ompositional abstrations. 5th IFAC Conferene on Analysis and Design of Hybrid Systems, Ot 2015, Atlanta, United States. Elsevier, 48 27, pp , 2015, IFAC- PapersOnLine. < /j.ifaol >. <hal > HAL Id: hal Sbmitted on 29 Jl 2015 HAL is a mlti-disiplinary open aess arhive for the deposit and dissemination of sientifi researh doments, whether they are pblished or not. The doments may ome from teahing and researh instittions in Frane or abroad, or from pbli or private researh enters. L arhive overte plridisiplinaire HAL, est destinée a dépôt et à la diffsion de doments sientifiqes de nivea reherhe, pbliés o non, émanant des établissements d enseignement et de reherhe français o étrangers, des laboratoires pblis o privés.

2 Safety ontrol with performane garantees of ooperative systems sing ompositional abstrations Pierre-Jean Meyer, Antoine Girard Emmanel Witrant Univ. Grenoble Alpes, CNRS, LJK, F Grenoble, Frane {Pierre-Jean.Meyer ; Antoine.Girard}@imag.fr Univ. Grenoble Alpes, CNRS, GIPSA-lab, F Grenoble, Frane Emmanel.Witrant@jf-grenoble.fr Abstrat: In this paper, the monotoniity property is exploited to obtain symboli abstrations, in the sense of alternating simlation, of a lass of nonlinear ontrol systems sbjet to distrbanes. Both a entralized and a ompositional approahes are presented to obtain sh abstrations, from whih ontrollers are synthesized to satisfy safety speifiations and optimize a performane riterion sing a reeding horizon approah. Performane garantees on the trajetories of the ontrolled system an be obtained with both approahes. The ontroller synthesis and performane garantees are illstrated and ompared on the temperatre reglation in a bilding. Keywords: Symboli ontrol; Compositional synthesis; Monotone systems. 1. INTRODUCTION The problem of ontrol synthesis for omplex dynamial systems has been approahed by varios methods, sh as robst H ontrol Skogestad and Postlethwaite, 2005 or model preditive ontrol Rawlings and Mayne, The method onsidered in this paper onsists in reating an abstration of the model with some behavioral relationship ensring that the ontrol applied to the abstration also ontrols the original model. These abstrations an be obtained in several ways, sing a bisimlation algorithm Alr et al., 2000, ompting reahable sets Reißig, 2009 or by state qantization Pola et al., All these approahes provide an abstration where the states are symbols representing sets of states in the original model, with the nmber of symbols ating as a trade-off between the preision on the state information and the simpliity of the abstration model. The abstration model may differ depending on the ontrol objetives, the desired simplifiations on the original model, the available information and to what extend we an ignore part of this information. When the abstration is a finite model, as it is the ase in this paper, we an enfore safety speifiations by sing a fixed point algorithm and optimize a performane riterion sing a reeding horizon ontroller Ding et al., 2014 similarly to a model preditive ontrol strategy Rawlings and Mayne, The advantage of the abstration in this ase is that all the omptations an be done offline and the reslting ontrol implementation ths only orresponds to a look-p table. To overome the exponential omplexity of reating sh a finite model, we propose a ompositional approah taking an additional tradeoff between preision and sim- This work was partly spported by a PhD sholarship and the researh projet COHYBA fnded by Région Rhône-Alpes. pliity of the abstration by deomposing the system into sbsystems with partial information on the state. Similarly to an assme-garantee reasoning Alr and Henzinger, 1999, the ontroller synthesis for eah sbsystem assmes that the safety speifiations are met for the others. We onsider a lass of monotone nonlinear systems more preisely, ooperative systems, implying that their trajetories preserve some partial order on the state e.g. see Smith 1995 for atonomos systems and Angeli and Sontag 2003 for ontrolled systems. This monotoniity property signifiantly failitates reating an abstration and proving the reqired behavioral relationship. For example in Moor and Raish 2002, over-approximations of reahable sets are diretly obtained sing the monotoniity. This method an be extended to a lass of non-monotone systems satisfying the mixed-monotoniity property, where the dynamis are deomposed into inreasing and dereasing omponents Coogan and Arak, Monotone systems appear in nmeros fields, sh as molelar biology Sontag, 2007, hemial networks Belgaem and Gozé, 2013 or thermal dynamis in bildings, whih is the appliation onsidered in this paper. The strtre of this paper is as follows. The lass of systems onsidered and some preliminary definitions are presented in Setion 2, followed by the problem formlation in Setion 3. In Setion 4 we present a entralized method to synthesize a ontroller based on a symboli abstration of the system. A ompositional approah is then given in Setion 5 where the previos method is sed on sbsystems with a partially observable state. For both methods, we provide performane garantees on the ontrolled trajetories of the original system. Finally, in Setion 6, or methodologial reslts are illstrated and ompared on the temperatre ontrol in a bilding.

3 2.1 Cooperative systems 2. PRELIMINARIES We onsider a lass of nonlinear systems given by: ẋ = fx,, w, 1 where x R n, [, ] R p and w [w, w] R q denote the state, the ontrol inpt and the distrbane inpt, respetively. The trajetories of 1 are denoted Φ, x 0,, w where Φt, x 0,, w is the state reahed at time t R + 0 from initial state x 0 R n, nder pieewise ontinos ontrol and distrbane inpts : R + 0 [, ] and w : R + 0 [w, w], respetively. Let and > denote the omponentwise ineqalities on the appropriate spae R m, m {n, p, q}. We also extend the definition of these ineqalities to fntions of time z, z : R + 0 Rm with z z t 0, zt z t. In Angeli and Sontag 2003, a dynamial system with inpts is said to be monotone when its trajetories preserve a partial ordering on the states. In this paper, we fos on ooperative systems where the partial orderings are. Definition 1. Cooperative system. System 1 is ooperative if for all x x,, w w it holds for all t 0, Φt, x,, w Φt, x,, w. Definition 1 is assmed to be satisfied for all the reslts of this paper. Charaterization of sh systems based on the vetor field f an be fond in Angeli and Sontag Alternating simlation In Tabada 2009, a system is defined as a qadrple S = X, X 0, U, onsisting of: a set of states X; a set of initial states X 0 X; a set of inpts U; a transition relation X U X. A transition x,, x is eqivalently written x x or x P ostx,. Ux denotes the set of inpts sh that P ostx,. A trajetory of S is an infinite seqene x 0, 0, x 1, 1,... sh that x 0 X 0 and for all i N, i Ux i and x i+1 P ostx i, i. Complex dynamial systems may motivate reating an abstration of their model. Ideally, finding a ontrol strategy for the abstration wold be simpler than for the original model. However, to ontrol the original model with the ontroller of the abstration, the systems mst satisfy a formal behavioral relationship sh as simlation or bisimlation. In the ase of ontrol systems with distrbanes, we are interested in alternating simlation relations, defined in Tabada Definition 2. Alternating simlation. Consider two systems S a and S b. A map H : X b X a is an alternating simlation relation from S a to S b if it holds: x a0 X a0, x b0 X b0 x a0 = Hx b0 ; x a = Hx b, a U a x a, b U b x b sh that x b P ost bx b, b, Hx b P ost ax a, a. We say that S b alternatingly simlates the abstration S a and denote it as S a AS S b. The seond ondition means that all the inpts of abstration S a have an eqivalent in S b sh that all transitions in S b are mathed by a transition in the abstration. 3. PROBLEM FORMULATION We onsider the system S = X, X 0, U, orresponding to a sampled version with a onstant sampling period τ R + of 1 where X = R n, X 0 = [x, x is a halflosed interval x [x, x x > x x, U = [, ] R p and x x if w : [0, τ] [w, w] x = Φτ, x,, w. Or ontrol objetive is to meet the safety speifiation of maintaining the state of S in the interval [x, x. In addition to the safety speifiation that may allow several vales of the ontrol inpt, we want to optimize a performane riterion given for a trajetory x 0, 0, x 1, 1,... of the ontrolled system S by i=0 λi gx i, i, where gx, is the ost of hoosing inpt when the state of S is x and λ 0, 1 is a disont fator to rede the inflene of the steps frther in the ftre. In what follows, we present two approahes based on abstrations to obtain sh ontrollers of S. In Setion 4, we se a entralized approah where we reate an abstration of the whole system S, while a ompositional approah is onsidered in Setion ABSTRACTION BASED CONTROL SYNTHESIS 4.1 Symboli abstration We want to reate a finite abstration S a of the sampled system S. To take advantage of the ooperativeness of or system Definition 1, the set of initial states is hosen as a niform partition X a0 = P 0 of [x, x into smaller idential half-losed intervals. For an element s P 0, we denote as s and s its lower and pper bonds, respetively: s = [s, s R n. If we want α N intervals per dimension in the partition, P 0 an be expressed as follows: P 0 = {[ s, s + x x α s x + x x α Zn } [x, x, where denotes the omponentwise mltipliation of vetors. The set of states of S a is taken as X a = P 0 {Ot} with Ot = R n \[x, x sh that X a is a partition of R n. S a is alled a symboli abstration bease eah element s P an be seen as a symbol for all the states x s of the original model S. To obtain a finite-transition system, we first need a finite inpt set. Similarly to the lower bonds s in P 0, we disretize U = [, ] reglarly into β 2 vales per dimension, inlding the lower and pper bonds: U a = + β 1 Zp U, 2 Then, we se the fat that 1 is ooperative to ompte an over-approximation of the reahable set P ost[s, s, : for all x s = [s, s, w : [0, τ] [w, w], Definition 1 gives Φτ, x,, w [Φτ, s,, w, Φτ, s,, w]. 3 Hene, the sessors of a symbol s P 0 are those interseting this over-approximation interval: s s a if s [Φτ, s,, w, Φτ, s,, w]. This method was presented in Moor and Raish 2002 for the ase of systems withot distrbanes. For simpliity and to ensre the alternating simlation, we onsider that all possible transitions exist from symbol Ot: U a, s P, Ot s. This hoie has no onseqene in what a

4 follows sine we are interested in the invariane of the interval [x, x and these transitions will soon be disarded. Proposition 3. The symboli model S a is alternatingly simlated by the original system S: S a AS S Proof. Consider s = Hx x s as the andidate alternating simlation relation. The first ondition of Definition 2 is immediately satisfied. For the seond ondition, let s = [s, s P 0, x s, U a U and x P ostx,. From the definition of the transitions of S, P ostx, [Φτ, s,, w, Φτ, s,, w] whih means that the symbol s = Hx is sh that s P ost a s, and x s. Lastly, U a, P ost a Ot, = P, therefore any transition in S from x Ot an be mathed by a transition in S a. We an note that the symboli model S a desribed above is a finite-state and finite-transition abstration of the initial system S. In addition, for a pair s,, heking the existing otgoing transitions s s only reqires a to ompte two sessors in S the bonds of s and interset the obtained over-approximation interval with the finite partition P. This symboli model an ths be bilt with a finite nmber of operations. 4.2 Reeding horizon ontrol Safety We now aim to synthesize ontrollers of S that meet the speifiation of invariane of the interval [x, x. The orresponding safety speifiation for the abstration S a is the set P 0. The safety game on S a an be solved by introding the operator F P 0 : 2 P 2 P sh that: F P 0Z = {s Z P 0 U a, P ost a s, Z}. The set F P 0Z ontains all symbols s Z P 0 whose sessors stay in Z for some U a. Note that P ost a s, sine for all s, U a s = U a. Then the maximal fixedpoint Z a = lim i FP i P 0 of F 0 P 0 is omptable in a finite nmber of steps and allows the definition of a nondeterministi ontroller C a : Z a 2 Ua solving the safety game for S a if Z a Tabada, 2009: C a s = { U a P ost a s, Z a }. 4 Performane optimization Then, we se a dynami programming algorithm Bertsekas, 1995 to minimize the ost of system S a ontrolled with C a over a finite horizon of N sampling periods. For a known initial state s 0, this ost J 0 s 0 is ompted iteratively following the priniple of optimality Bellman, 1957 for all k from N to 0: J k s = min g a s, + λ max J k+1s, 5 C as s P ost as, where J N+1 s = 0 for all s P 0 and the ost fntion g a is defined sing the ost g on S from Setion 3: g a s, = max gx,. 6 x s At eah iteration of 5, we minimize over safe inpts C a s the sm of the ost of the rrent step and the worst ase additive ost of all the following steps. We an then apply a reeding horizon ontrol sheme Cas = arg min g a s, + λ max J 1s 7 C as s P ost as, where at eah sampling period we measre the rrent symbol s and only apply the first element of the ontrol poliy provided by 5. This approah is the basis of model preditive ontrol Rawlings and Mayne, 2009, with the differene that all the omptations of 5 and 7 an be done offline for or finite transition system S a. With the alternating simlation in Proposition 3, we an obtain a ontroller C X a : [x, x U of the sampled system S: s Z a, x s, C X a x = C as C a s Safety and performane garantee In this setion, we show that the trajetories of S ontrolled with the reeding horizon ontroller 8 satisfy the safety speifiation and we provide an expliit bond on their osts. We onsider the following intermediate reslt. Lemma 4. Let M = max J N s = max min g as,. s Z a s Z a C as Then J 0 s J 1 s + λ N M for all s Z a. Proof. This is proved by indtion. For the initial step, we onsider 5 with k = N 1, the inpt C a s satisfying J N s = g a s, and the definition of M: J N 1 s J N s + λ max J Ns J N s + λm. s P ost as, Assme now that J k s J k+1 s + λ N k M, then: J k 1 s= min g a s, + λ max J ks C as s P ost as, min g a s, + λ max J k+1s C as s P ost as, +λ N k+1 M J k s + λ N k+1 M. With k = 1, we obtain the reslt from Lemma 4. An pper bond on the performane riterion of S is then obtained when sing the reeding horizon ontroller Ca X. Theorem 5. Let x 0, 0, x 1, 1,... be a trajetory of S ontrolled with Ca X in 8, then k N, x k [x, x. Moreover, let s 0, s 1, P 0 sh that for all k N, x k s k. Then, for all k N, + λ j gx k+j, k+j J 0 s k + λn+1 1 λ M. j=0 Proof. We know from 4 that C a renders S a invariant in the fixed point Z a. With the alternating simlation from Proposition 3 and the definition of Ca X in 8, if x 0 is initialized in a symbol s 0 Z a, then the state of S ontrolled with Ca X remains in {x R n s Z a, x s} [x, x. For the seond part of the proposition, let Js = J 0 s + λn+1 1 λ M. We start from the definition of J 0 s k in 5 with k = Cas k as in 7: Js k =g a s k, k + λ max s P ost as k, k J 1 s + λn+1 1 λ M g a s k, k + λj 1 s k+1 + λn+1 1 λ M g a s k, k + λ J 0 s k+1 λ N M + λn 1 λ M gx k, k + λjs k+1 The first ineqality is obtained for a partilar vale s = s k+1 of the possible sessors, the seond omes

5 from Lemma 4 and the third from the definition 6 of g a. Ths, if the ineqality obtained above is applied to all the following states of the trajetory, we have for any k: Js k gx k, k + λjs k+1 gx k, k + λgx k+1, k+1 + λ 2 Js k+2... Expanding these ineqalities to all states of the trajetory leads to the reslts from Theorem 5. Note that the onstant part λn+1 1 λ M of the pper bond in Theorem 5 goes to zero when the size N of the horizon sed in the dynami programming grows. 5. COMPOSITIONAL APPROACH It is well known that salability is one of the main limitations of the symboli method presented in Setion 4 de to the exponential omplexity in the dimension n of the system. The idea presented in this setion is to rede the omptational brden at the ost of lower preision: we deompose the system S into sbsystems by onsidering only a sbset of the state and ontrol inpt omponents. 5.1 Sbsystems Let m be the nmber of sbsystems, I 1,..., I m a partition of {1,..., n} and J 1,..., J m a partition of {1,..., p}. For sbsystem i, x Ii and Ji are the vetors of observable states and ontrollable inpts, respetively. The remaining state and inpt omponents, denoted as x Ki and Li with K i = {1,..., n}\i i and L i = {1,..., p}\j i, are onsidered as distrbanes. For a set of indies I and a fntion, set or variable V, V I represents the projetion of V on the dimensions of indies in I. Similarly to an assme-garantee reasoning Alr and Henzinger, 1999, we assme that the other sbsystems ensre the safety speifiation for the nobservable states: x Ki [x Ki, x Ki. The symboli abstration orresponding to sbsystem i is the finite atomaton S i = X i, X i0, U i, i where X i0 = PI 0 i and U i = U a,ji are the projetions of P 0 and U a on the dimensions I i of R n or J i of R p. Let Ot i be sh that X i = PI 0 i {Ot i } is a partition of X Ii projetion of the ontinos state spae X on the dimensions in I i. Using the simplified notations ϕ i s Ii, Ji = Φ Ii τ, s Ii, x Ki, Ji, Li, w and ϕ i s Ii, Ji = Φ Ii τ, s Ii, x Ki, Ji, Li, w, the transition relation is defined as follows for all s Ii X i0, Ji U i and s I i X i : s Ii Ji i s I i s I i [ϕ i s Ii, Ji, ϕ i s Ii, Ji ] ; Ot i J i i s I i. Sine Φ Ii denotes the projetion of Φ on the dimensions in I i, we an learly see that we obtain a less preise over-approximation of the reahable set de to the loss of observability of x Ki and Li. The seond part of the transition definition is similar to the one of S a. Using a ost fntion g i : X i U i R +, we an apply the ontroller synthesis approah presented in Setion 4.2 to sbsystem S i and obtain the maximal fixed point Z i X i0 = PI 0 i and the following three ontrollers for all s Z i : C i s = { U i P ost i s, Z i }, 9a Ci s = arg min g i s, + λ max J 1s i, 9b C is s P ost is, Ci X x = Ci s C i s, x s, 9 where C i is the safety ontroller assoiated to the fixed point Z i, Ci is the reeding horizon ontroller for S i obtained from 5 and Ci X the orresponding reeding horizon ontroller on the ontinos state spae X Ii. 5.2 Composition Let S = P, P 0, U a, orrespond to the omposition of all sbsystems S i, with a transition relation defined by, s, s P 0, U a : s s Ot s if i {1,..., m}, s Ii Ji i s I i, Ot if i {1,..., m} s Ii Ji i Ot i, s and Ot Ot. We an then prove the alternating simlation between S and S a and, by transitivity of AS, between S and S. Proposition 6. S AS S a and S AS S. Proof. Consider the identity as the andidate alternating simlation relation. The first ondition of Definition 2 is immediately satisfied. Let s P, U a and s P ost a s,. If s, s P 0, then for all i {1,..., m}, s I i [Φ Ii τ, s,, w, Φ Ii τ, s,, w]. Definition 1 gives: { ΦIi τ, s Ii, x Ki, Ji, Li, w Φ Ii τ, s,, w 10 Φ Ii τ, s Ii, x Ki, Ji, Li, w Φ Ii τ, s,, w whih implies that s I i P ost i s Ii, Ji for all i and then s P ost s,. If s = Ot, Ot P ost a s, means that there exists j {1,..., n} sh that Φ j τ, s,, w < x j or Φ j τ, s,, w x j. Let i {1,..., m} sh that j I i, then 10 gives Ot i P ost i s Ii, Ji whih implies Ot P ost s,. If s = Ot then s P ost Ot, = P. The seond reslt is obtained by transitivity of the alternating simlation Tabada, 2009 sing Proposition 3. Sine the sbsystems partition the sets of state and inpt indies, the fixed points or ontrollers an simply be omposed with a Cartesian prodt: let Z = Z 1 Z m and for all s Z and x s, C s = C 1 s I1 C m s Im 11a C s = C1 s I1,..., Cms Im 11b C X x = C1 X x I1,..., Cmx X Im 11 To obtain a reslt similar to Theorem 5, we need to introde the following assmption. Assmption 7. g a s, m i=1 gi s Ii, Ji s Z, U a and M m i=1 M i with M i = max min gi s i, i. i C is i s i Z i Theorem 8. Let x 0, 0, x 1, 1,... be a trajetory of S ontrolled with C X from 11, then k N, x k [x, x. Moreover, let s 0, s 1, P 0 sh that for all k N, x k s k. Then, nder Assmption 7, for all k N,

6 + j=0 λ j gx k+j, k+j m i=1 J i 0s k I i + λn+1 1 λ m M i. i=1 Proof. Let s Z and x s. By onstrtion, P ost i s Ii, Ci Xx I i Z i. Then by definition of S, Z and C X, we have P ost s, C X x Z. Let x P ostx, C X x and s P sh that x s. The alternating simlation from Proposition 6 gives s P ost s, C X x Z whih implies that x {x R n s Z, x s} [x, x. For the seond reslt, for eah sbsystem S i, Assmption 7 gives J0s i Ii J1s i Ii +λ N M i for all s Ii Z i similarly to Lemma 4. Then, as in the proof of Theorem 5 we obtain + j=0 λ j g i s k+j I i, k+j J i J0s i k I i + λn+1 1 λ M i. 12 Taking the sm of 12 over all sbsystems, we obtain the ineqality in Theorem 8 sing 6 and Assmption 7: for all x s, gx, g a s, m i=1 gi s Ii, Ji. From the first part of Theorem 8, we an dede the following reslt. Corollary 9. Z Z a. Proof. Z is an invariant set for S and with the alternating simlation from Proposition 6, it is also an invariant set for S a. We also know that Z a in Setion 4.2 is the maximal invariant set of S a for safety speifiations in P 0. We an also show that the ost obtained in 5 for S a is smaller than the sm of those for the sbsystems S i. Proposition 10. Under Assmption 7, for all s Z we have J 0 s m i=1 J 0s i Ii. Proof. Let the ost fntion g s, = m i=1 gi s Ii, Ji. Sine two fntions g i and g j with j i are independent, the min and max operators on g an be deomposed into looking for its extrema for eah i {1,..., m}. The dynami programming 5 applied to S sing g and the safety ontroller C from 11a ths has for soltion the fntions Jk s = m i=1 J k is I i for all s Z. Define G k and k as G ks, = g s, + λ max J k+1s, s P ost s, k = arg ming ks,, C s sh that Jk s = min C s G k s, = G k s, k and assme we have similar notations G a k s, and a k for S a. Sine Z Z a and J k is not defined on Z a \Z, we only fos on s Z and prove the ineqality by indtion. The initial ineqality J N s JN s is a onseqene of the first part of Assmption 7. Next, assme that for all s Z, we have J k+1 s Jk+1 s. Then we have: J k s=g a ks, a k G a ks, k g s, k + λ max J k+1s s P ost as, k g s, k + λ max J k+1s s P ost s, k g s, k + λ max J k+1s = Jks. s P ost s, k The for ineqalities are obtained sing, in this order, the definition of a k, Assmption 7, the alternating simlation in Proposition 6 P ost a s, P ost s, and the indtion hypothesis. Combining Proposition 10 with Assmption 7, we an see that, as expeted, the performane garantees from Theorem 5 are smaller than those from Theorem TEMPERATURE REGULATION IN BUILDINGS System In this setion, we illstrate the reslts of this paper on the temperatre reglation in a flat eqipped with UnderFloor Air Distribtion, an alternative soltion to traditional eiling ventilation in bildings Baman and Daly, 2003 where the air is ooled in an nderfloor plenm before being sent into eah room. The temperatre in a room is assmed to be niform and the model of its variations is derived from the energy and mass onservation eqations in this room. For a n-room bilding, the nonlinear dynamis of the system an ths be desribed by T = ft,, w, δ, 13 where the n-dimensional vetor field f depends on the temperatre T R n, the ontrol inpt [0, 1] n orresponds to the ventilation in eah room, the exogenos inpts w R 3 ontains nontrolled temperatres nderfloor, eiling and otside and the binary distrbane δ {0, 1} q represents the disrete state of heat sores in the rooms and the opening of doors. A detailed desription of the system and the hypotheses to obtain 13 is given in Witrant et al and Meyer et al. 2013, where we also prove that the model is ooperative as in Definition 1. Abstration The methods in Setion 4 have been validated in Meyer et al on an experimental 4-room small-sale bilding. To failitate visalization, in this paper we onsider the model 13 for a 2-room flat. We define the symboli model S a as in Setion 4.1 with a partition P 0 of the state interval [T, T R 2 into symbols half-losed intervals and a ontrol set U a obtained as in 2 from a disretization of U = [0, 1] 2 with β = 5 vales per dimension. The sampling period τ has to be hosen as a tradeoff between large vales to avoid self loops for the optimization and small vales to avoid large variations that might prevent finding a non-empty fixed-point. Safety An illstration of the ontroller synthesis proposed in Setion 4.2 on the symboli abstration S a is depited in Figre 1. In all three examples of this figre, we represent the partition P 0 of the target interval as the red grid and the symbols olored in yellow are those in the fixed-point Z a solving the safety speifiation. In the first graph of Figre 1, we want to keep the temperatre of both rooms in an interval [22, 24] and we an see that Z a = P 0, whih means that for eah symbol in the interval there exists a vale of the ventilation keeping both temperatres in their intervals in any ondition of the distrbanes. This ase an be linked to the notion of robst ontrolled invariant interval defined in Meyer et al The seond graph of Figre 1 orresponds to a similar partition bt with the target interval for room 2 shifted to [20, 22]. In these onditions, we an see that the largest fixed-point Z a for S a does not over the whole interval bt is mh larger than the maximal robst ontrolled invariant sb-interval for the ontinos system 1. This is de to the fat that

7 Criterion k C a X C X + j=0 λj gx k+j, k+j k = 0 Garanteed pper bondx k j=0 λj gx k+j, k+j mean Garanteed pper bondx k k N Table 1. Costs on the trajetories of Figre 2. Fig. 1. Fixed-points Z a yellow for S a and largest robst ontrolled invariant sb-interval blak. the symboli approah allows non-retangle invariants. In the same onditions, the third graph of Figre 1 represents the fixed-point Z a when the symboli model S a is reated with a finer partition symbols of the interval. If we keep inreasing the preision of the partition and rede the sampling time aordingly, Z a onverges to the maximal robst ontrolled invariant sbset of the interval and we an see that we already obtain a good approximation with symbols. With the ompositional approah from Setion 5, the fixed-points Z obtained in the three onditions of Figre 1 are Z = Z a = P 0 in the first ase and Z = in the others, whih onfirms that Z Z a. Complexity On the other hand, the ompositional approah ompensates the loss of preision in its sbsystems by a signifiant inrease in omptation effiieny. If we onsider the model 13 for a n-room bilding with α symbols and β ontrol vales per dimension as in Setion 4.1, the symboli abstration S a an be obtained by ompting 2αβ n sessors of the sampled system S, while the omposed system S only needs n 2αβ. Ths we go from an exponential omplexity in n and a polynomial one in α and β for S a to a linear omplexity in n, α and β for S. In the onfigration presented below for the 2-room bilding, S a and C were obtained after 26s while S and C only reqired 0.13s on a 3GHz CPU. In a 4-room bilding, the entralized method S a, C for the experimental implementation presented in Meyer et al took a ople of days while the ompositional method only needs 0.4s in the same onditions. Performanes The dynami programming algorithm is rn over a finite time window of size N = 5 and with a disont fator λ = 0.5. These vales are hosen sh that the onstant part of the pper bond in Theorems 5 and 8 is small enogh: λ N+1 /1 λ 3%. The ost fntion g at step k is defined as the ombination of three riteria to be minimized: the norm k of the ontrol inpt, its variations k k 1 reqiring to introde an extended state z k = T k, k 1 Bertsekas, 1995 and the distane T k T between the state T k and the enter T of the interval. To have omparable inflenes, the three riteria are normalized and assoiated with a 1/3 weight. Using the sqare of the 2-norm, the first part of Assmption 7 is satisfied with an eqality. The seond part of Assmption 7 is verified nmerially: M 1 = M 2 = M/2 = To ompare the reslts with both the entralized and the ompositional approahes, we onsider the onditions of the first graph of Figre 1 where both Z a and Z are nonempty. We an first hek that Proposition 10 is satisfied: over all symbols s Z a, J0 1 s 1 + J0 2 s 2 J 0 s varies between 0 and 0.92 with a mean vale of 0.016, while the maximal vale of either side of the ineqality is In Figre 2, we represent the temperatre and ventilation of system 13 ontrolled with Ca X from the entralized method in Setion 4 red dashed rve or pward triangles and C X from the ompositional approah in Setion 5 ble plain rve or downward triangles. The simlations are rn with sinsoidal exogenos inpts and the disrete distrbanes are sh that all possible ombinations appear: the heat sore in room 1 is on from 13 to 36 and from 61 to 84 mintes, in room 2 from 25 to 72 mintes and the door is open after 49 mintes ntil the end. First, we an see that both ontrollers orretly maintain the state of the system in the presribed bonds. For the seond part of Theorems 5 and 8, we ompte the ost + j=0 λj gx k+j, k+j of the trajetory from any sampling time k disarding the last 10 having too short trajetories and verify that it is smaller than the garanteed pper bond provided in Theorems 5 and 8 orresponding to the same sampling time k. The vale of both sides of these ineqalities for the initial state x 0 of the trajetories of S respetively ontrolled with Ca X and C X are reported in the top of Table 1. We only look at the initial state sine it is the only step where we know that the trajetories are in the same state. We an first see two expeted reslts: the ost of the trajetories are smaller than their respetive garanteed pper bonds as in Theorems 5 and 8 and the garanteed pper bond for the ompositional method is larger than the entralized one as in Proposition 10. There is however a srprising reslt that the atal performane from the initial state are better for the ompositional method. This is onfirmed when ompting the average vale of the performane riterion from any point of the trajetories, while the average on the orresponding garanteed pper bonds are omparable for both methods bottom of Table 1. This old be explained by the fat that Ca X and C X are obtained with worst-ase onsiderations on the distrbanes. Sine more distrbanes are involved in the ompositional method nobservable states and inpts, C X natrally is more onservative, whih gave better performanes in this partilar simlation.

8 Fig. 2. 2D system 13 ontrolled with the ontrollers from the entralized red and ompositional ble methods. 7. CONCLUSION We presented two methods to obtain a symboli abstration, in the sense of alternating simlation, of a lass of non-linear systems sbjet to distrbanes and satisfying the ooperativeness property. The first one is a entralized approah where an abstration of the whole system is reated, while a ompositional approah is onsidered in the seond method to rede the omplexity of the problem at the ost of the preision of the abstration. These symboli abstrations are then sed to synthesize a ontroller ensring a safety speifiation on the original system and minimizing some ost riterion. For both approahes, we provide performane garantees in the form of an pper bond for the total ost of the ontrolled original system on any infinite time horizon. We also show that the garanteed performanes for the entralized method are, as expeted, better than those for the ompositional approah. Finally, these theoretial reslts are illstrated on the temperatre reglation in a 2-room bilding. In partilar, we an observe on these simlations the large redtion in omptation time with the ompositional method for a relatively small or no loss of performane ompared to the entralized approah. This ompositional approah ths provides the opportnity to apply symboli methods in a real-time implementation and to systems of larger dimensions that old not be handled otherwise. REFERENCES Alr, R. and Henzinger, T.A Reative modles. Formal Methods in System Design, 151, Alr, R., Henzinger, T.A., Lafferriere, G., and Pappas, G.J Disrete abstrations of hybrid systems. Proeedings of the IEEE, 887, Angeli, D. and Sontag, E.D Monotone ontrol systems. IEEE Transations on Atomati Control, 4810, Baman, F.S. and Daly, A Underfloor air distribtion UFAD design gide. ASHRAE. Belgaem, I. and Gozé, J.L Global stability of enzymati hains of fll reversible Mihaelis-Menten reations. Ata biotheoretia, 613, Bellman, R Dynami Programming. Prineton University Press. Bertsekas, D.P Dynami programming and optimal ontrol, volme 1. Athena Sientifi Belmont, MA. Coogan, S. and Arak, M Salable finite abstration of mixed monotone systems. In Hybrid Systems: Comptation and Control, Ding, X., Lazar, M., and Belta, C LTL reeding horizon ontrol for finite deterministi systems. Atomatia, 502, Meyer, P.J., Girard, A., and Witrant, E Controllability and invariane of monotone systems for robst ventilation atomation in bildings. In Proeedings of the 52 nd IEEE Conferene on Deision and Control, Meyer, P.J., Girard, A., and Witrant, E Symboli ontrol of monotone systems, appliation to ventilation reglation in bildings. In Hybrid Systems: Comptation and Control, Moor, T. and Raish, J Abstration based spervisory ontroller synthesis for high order monotone ontinos systems. In Modelling, Analysis, and Design of Hybrid Systems, Springer. Pola, G., Girard, A., and Tabada, P Approximately bisimilar symboli models for nonlinear ontrol systems. Atomatia, 4410, Rawlings, J.B. and Mayne, D.Q Model preditive ontrol: Theory and design. Nob Hill Pb. Reißig, G Comptation of disrete abstrations of arbitrary memory span for nonlinear sampled systems. In Hybrid Systems: Comptation and Control, Skogestad, S. and Postlethwaite, I Mltivariable feedbak ontrol: analysis and design. Wiley, 2 nd edition. Smith, H.L Monotone dynamial systems: an introdtion to the theory of ompetitive and ooperative systems, volme 41. Amerian Mathematial So. Sontag, E.D Monotone and near-monotone biohemial networks. Systems and Syntheti Biology, 12, Tabada, P Verifiation and ontrol of hybrid systems: a symboli approah. Springer. Witrant, E., Di Maro, P., Park, P., and Briat, C Limitations and performanes of robst ontrol over WSN: UFAD ontrol in intelligent bildings. IMA Jornal of Mathematial Control and Information, 274,