Event-Triggered Strategies for Decentralized Model Predictive Controllers

Transcription

1 Preprints o the 18th IFAC World Congress Event-Triggered Strategies or Decentralized Model Predictive Controllers Alina Eqtami Dimos V. Dimarogonas Kostas J. Kyriakopoulos Control Systems Lab, Department o Mechanical Engineering, National Technical University o Athens, 9 Heroon Polytechniou Street, Zograou 15780, Greece ( alina,kkyria@mail.ntua.gr). School o Electrical Engineering, Royal Institute o Technology (KTH), SE , Stockholm, Sweden ( dimos@ee.kth.se). Abstract: In this paper, novel event-triggered strategies or the design o model predictive (MPC) controllers are presented. The MPC ramework consists in inding the solution to a constraint optimal-control problem at every time-step. The case o triggering the optimization o the MPC only when is needed, is investigated. The centralized case is treated irst and the results are then extended to a decentralized ormulation. We consider a system composed by a number o interconnected subsystems, each one o them controlled by a robust MPC algorithm. Using the Input-to-State (ISS) property o the decentralized MPC controller we reach to an event-triggering rule, or each o the subsystems. 1. INTRODUCTION Decentralized control is an eicient ormulation or controlling large scale systems, while model predictive control ormulations have the capability to deal with nonlinearities and state and control constraints. Thereore, decentralized approaches to MPC have gained much interest in recent years. Related results on decentralized MPC can be ound in Dunbar et al. [2006], Franco et al. [2008], Keviczky et al. [2006], Raimondo et al. [2007], Raimondo et al. [2009] and in the review paper Scattolini [2009] and the number o papers quoted therein. One o the aspects that should be taken in consideration when implementing decentralized control laws, is the communication and controller schemes. The scheduling o the actuation updates can be done in a time-driven or in an event-driven ashion. The irst one might be proven a conservative choice, while when we have limited resources, the second choice may be more avorable. In recent years the ramework o event-driven eedback and sampling has been developed. Related works can be ound in Heemels et al. [2007] and Tabuada [2007] or general nonlinear systems, while similar approaches or decentralized rameworks can be ound in Dimarogonas et al. [2009], Mazo et al. [2010], and Wang et al. [2009]. The main assumption used or these event-triggered policies, is the Input-to-State property o the plant. There has been a lot o research on ISS properties o the MPC ramework or general discrete-time systems. Recent results can be ound in Lazar et al. [2009] and Pin et al. [2008]. In this work, in order to ind a triggering condition or sampling in the centralized case, we appropriately modiy the ormulation presented in Marruedo et al. [2002]. The ISS property o the system under decentralized MPC controllers is discussed in Franco et al. [2008], Raimondo et al. [2009] and Raimondo et al. [2007]. Here we employ the ISS property o the decentralized MPC in Raimondo et al. [2007] in order to reach to a triggering condition or each o the subsystems o the ramework. The contribution o this paper relies in inding a triggering rule or sampling in the case o nonlinear systems, under a nonlinear MPC strategy. In the aorementioned eventtriggering techniques, the control value is held constant between the actuator updates. Nevertheless, in this paper the act that predictive controllers provide a control sequence at each time-step is used. In particular, the control sequence that is provided by the MPC controller is applied in an open-loop ashion between the actuator updates. First, this event-triggering scheme is applied to a centralized MPC controller. Then, we consider a system composed by a number o interconnected subsystems, each one o them controlled by a robust MPC algorithm. Using this decentralized MPC controller we derive an eventtriggering rule or each o the subsystems. Although the event-triggering setup or MPC controllers is quite new, some results have already been presented, Varutti et al. [2009], Iino et al. [2009], Grune et al. [2009] and Sijs et al. [2010]. Related to the above reerences, we provide here a dierent event triggering strategy to compute the event times. The remainder o the paper is organized as ollows. In Section 2 and in Section 3, the problem statement and the triggering condition or the centralized case is presented respectively. The decentralized counterpart is presented in the ollowing two Sections; the problem statement or the decentralized ramework is provided in Section 4, while in Section 5 the triggering rule is given. Section 6 summarizes the results o this paper and indicates urther research eorts. Copyright by the International Federation o Automatic Control (IFAC) 10068

2 Preprints o the 18th IFAC World Congress 2. PROBLEM STATEMENT FOR THE CENTRALIZED NMPC CASE In order to ind a triggering condition or the discretetime nonlinear system under an NMPC control law, a modiication o the analysis proposed in Marruedo et al. [2002], will be used in the ollowing, in order to ind a triggering condition. Consider the nonlinear discrete-time dynamic system x k+1 = (x k,u k ) + w k (1) where x k R n denotes the system s state, u k R m the control variables and w k W R n is the additive disturbance. Assume that W is a compact set, containing the origin, and that the admissible set o uncertainties are bounded, thus w k W, w k γ (2) The state and control variables are subjected to the ollowing constraints x k X, u k U, k Z 0 (3) For control design purposes, we assume the noal model o the system is o the orm x k+1 = (x k,u k ) (4) Assume that (0,0) = 0 and that (x,u) is locally Lipschitz in x in the domain X U, with Lipschitz constant L. Given the system (4), the ollowing double subscript notation will be used hereater ˆx(k + j + 1 k) = (ˆx(k + j k),u k+j ) where ˆx( ) is the predicted state and ˆx(k + j + 1 k) is the predicted state at time k + j + 1, based on the measured state o the real system at time k, u k+j is a control sequence or time k until k + j, and x k = ˆx(k k) is the measured state at time step k. The uncertainty o the system can cause discrepancies between the actual state trajectory o the system, given rom (1), subject to a speciic sequence o inputs and the predicted state given rom (4), or the same sequence o inputs. In order to account or this mismatch the error e, is introduced in the analysis. The error e is deined as the norm o the dierence between the predicted and the real evolution o the state. In the sequel, the double subscript notation will be reserved or the error. The error e(k+j k) will thus be given by e(k + j k) = x k+j ˆx(k + j k) (5) Note, that the sequence o states o the actual system is thought to be measurable. It can be proven that the error e(k + j k) is bounded, and that it depends on the error at time k + 1, i.e. or a given sequence o inputs the error e(k + j k), at time step k + j, will satisy e(k + j k) L j NMPC or Discrete-Time Systems e(k + 1 k) + Lj 1 L γ (6) NMPC involves solving on-line a inite-horizon, openloop optimal control problem (abbr. OCP), based on the measurement provided by the plant. The OCP in the discrete-time case, consists in imizing, with respect to a control sequence u F (k) [u(k k),u(k+1 k),...,u(k+n 1 k)], a cost unction J N (x k,u F (k)). The positive integer N Z 0, denotes the prediction horizon. Thus, the OCP or the noal system (4), is given by u F (k) J N(x k,u F (k)) = i=n 1 u F (k) L( x(k + i k),u(k + i k)) + V ( x(k + N k)) (7a) s.t. x(k + j k) X j j = 1,...,N (7b) u(k + j k) U j = 0,...,N (7c) x(k + N k) X (7d) where X denotes the teral constraint set and denotes the controller internal variables and x(k k) = x k. Following a similar procedure as in Marruedo et al. [2002], the ollowing assumptions or the stage cost L( ) and the teral cost V ( ) rom (7a), are stated Assumption 1. i) The stage cost L(x,u) is Lipschitz continuous in X U, with a Lipschitz constant L c. Assume that L(0,0) = 0 and that there are positive integers α > 0 and ω 1, such that L(x,u) α (x,u) ω. ii) Let the teral region X rom (7d) be a subset o an admissible positively invariant set Φ o the noal system. Assume that there is a local stabilizing controller h(x k ) or the teral state X. The associated Lyapunov unction V ( ) has the ollowing properties V ((x k,h(x k )) V (x k ) L(x k,h(x k )), x k Φ, and is Lipschitz in Φ, with Lipschitz constant L V. The set Φ is given by Φ = {x R n : V (x) α Φ } such that Φ X h = {x X N 1 : h(x) U}. The set X = {x R n : V (x) α ν } is such that or all x Φ, (x,h(x)) X. The state constraint set X o the standard MPC ormulation, is being replaced by a restricted constraint set X j rom (7b). This state constraints tightening or the noal system with additive disturbance, guarantees that the evolution o the real system will be admissible or all time. Using the supremum o (6), there is X j = X B j where B j = {x R n : x Lj 1 L 1 γ}. The set operator denotes the Pontryagin dierence TRIGGERING CONDITION FOR THE CENTRALIZED NMPC CASE In this section the triggering condition or the discretetime system under the NMPC control will be provided. Beore tackling this problem, some general concepts about using the ISS property o a system in order to ind a triggering condition, are going to be presented irst. A continuous unction V : R n R 0 is an ISS-Lyapunov unction or a system o the orm x k+1 = (x k,e k ), i there exist K unctions α 1,α 2, such that α 1 ( x k ) V (x k ) α 2 ( x k ) or all x X and or some class K unction, α 3, and some class K, α 4, unction V (x k ) also satisies V ((x k,e k )) V (x k ) α 3 ( x k ) + α 4 ( e k ) (8) In this case the system is ISS with respect to the measurement error e k. System remains stable, i e k satisies α 4 ( e k ) σα 3 ( x k ) (9) 10069

3 Preprints o the 18th IFAC World Congress with 0 < σ < 1, because invoking this rule into equation (8), V is still guaranteed to be decreasing. A triggering condition is thus given by (9) as stated in Eqtami et al. [2010]. Once the error is large enough so that (9) is violated, the control law must be updated and the error is reset to zero. Hence, V remains strictly decreasing under the triggering rule. 3.1 Feasibility and ISS Stability In the classic discrete-time NMPC strategy the control law is updated at each time-step k. The control law is given by u k = u (k k), where u (k k) is the irst term o the optimal solution provided by the discrete-time OCP (7a)- (7d). In the event-triggered setup the rest o the optimal sequence might be used provided that some conditions are ulilled. In this case the optimal control sequence is recalculated at the discrete time instants {t 0,t 1,t 2,...} {k 0,k 1,k 2,...}. Assume that or every triggering instant t i a new OCP is triggered too, and that t i coincides with k j. During the time span [t i,t i+1 ), where t i+1 is the next triggering instant when the new OCP is triggered, the control law provided at t i k j is implemented in openloop ashion. The triggering condition is going to be stated later in the text. In the NMPC case considered here, consider control trajectories ū F (k + m), or time steps m = 0,...,N, based on the optimal solution in k, u F (k ), i.e. or m = 0,...,N ū(k + j k + m) = { u = (k + j k ) orj = m,...,n 2 (10) h(ˆx(k + N k + m)) orj = m + N From easibility o u F (k ) it ollows that or m = 0,...,N there is ū(k + j k + m) U, and according to Marruedo et al. [2002] ˆx(k + N k + m) X, or all m = 0,...,N, provided that the uncertainties are bounded by γ (α Φ α v )/(L V L N 1 ). The optimal cost at time step k is JN (k ) and the cost o the easible sequence at a time step j [0,N 1] is indicated by J N (k + j). Then the dierence o these costs is J j = J N (k + j) JN(k ) (11) The next theorem can now be stated Theorem 1. Consider the system (1) subject to (3) and assume that the previously presented Assumption 1 holds. Then, using the control law rom (10), the dierence between the cost o a easible sequence at time step k + j and the optimal cost o at time step k is bounded by j J j L Zj e(k + j k ) α x k i+j ω (12) where L Zj is given by L Zj = L V L (N 1) j + L C L(N 1) j L Proo. First, the dierence (11) is calculated or j = 0. A similar procedure as in Marruedo et al. [2002] is going to be used. Then the calculation will be repeated or j = 1, and inally the general rule or random j will be stated. For j = 0 the dierence (11) is J 0 = J N (k) J N(k ) N 1 = {L( x(k + i k),ū(k + i k)) L(ˆx(k + i k ),u (k + i k ))} + V ( x(k + N k)) V (ˆx(k + N k )) N 2 = {L( x(k + i k),ū(k + i k)) L(ˆx(k + i k ),u (k + i k ))} + L( x(k + N k),h( x(k + N k)) L(x k 1,u k 1 ) + V ( x(k + N k)) V (ˆx(k + N k )) (13) From deinition o (10) we have ū(k+i k) = u (k+i k 1). Imposing the control law (10) or m = 0 to the system, by induction we get ˆx(k + j k) ˆx(k + j k ) L j e(k k ) (14) where the error e(k k ), using (5), is given by e(k k ) = x k ˆx(k k ) (15) The dierence between the running costs, with the help o (14) and (15), is L( x(k + j k),ū(k + j k)) (16) L(ˆx(k + j k ),u (k + j k )) L c L j e(k k ) Analogously, V ( x(k + N k)) V (ˆx(k + N k )) L V L N 1 e(k k ) (17) Consider also, that rom Assumption 1i) we have L(x,u) α (x,u) ω α x ω. Substituting these expressions to (13), the ollowing is derived J 0 L Z0 e(k k ) α x k 1 ω (18) with L Z0 = L V L N 1 + L c LN 1 1 L 1. For j = 1 the dierence (11) becomes J 1 = J N (k + 1) J N(k ) N 1 = {L( x(k + i + 1 k + 1),ū(k + i + 1 k + 1)) L(ˆx(k + i k ),u (k + i k ))} + V ( x(k + N + 1 k + 1)) V (ˆx(k + N k )) N 3 = {L( x(k + i + 1 k + 1),ū(k + i + 1 k + 1)) L(ˆx(k + i + 1 k ),u (k + i + 1 k ))} + L( x(k + N k + 1),h( x(k + N k + 1)) L(x k 1,u k 1 ) + L( x(k + N k + 1),h( x(k + N k + 1)) L(x k,u k ) + V ( x(k + N + 1 k + 1)) V (ˆx(k + N k )) + V ( x(k + N k + 1) V ( x(k + N k + 1) (19) From Assumption 1ii), we have 10070

4 Preprints o the 18th IFAC World Congress L( x(k + N k + 1),h( x(k + N k + 1))) + V ( x(k + N + 1 k + 1)) V ( x(k + N k + 1)) 0 (20) Moreover, V ( x(k + N k + 1)) V (ˆx(k + N k )) L V L N 2 e(k + 1 k ) (21) Substituting these expressions to (19), it can be concluded that the dierence J 1 is bounded by J 1 L Z1 e(k + 1 k ) α x k 1 ω α x k ω (22) with L Z1 = L V L N 2 +L c LN 2 L 1. From the above it can be concluded using the same calculation, that or random j [0,N ] the dierence J j = J N (k + j) JN (k ), is given rom (12), and hence the proo is completed. 3.2 Triggering Condition System (1), subject to (3), which satisies the Assumption 1, is ISS stable with respect to measurement errors, under an NMPC strategy. This can be concluded by the optimality o the solution that results to J N(k) J N(k ) J 0 L Z0 e(k k ) α x k 1 ω Using this ISS property o the system, and based on the general rule given by (9), the triggering rule or discretetime systems (1) is written in this case as L Z0 e(k k ) σ α x k 1 ω (23) The next OCP is thus triggered i condition (23) is violated, otherwise the control law rom (10) is used or m = 0. The triggering rule (23), is valid in the irst step. In order to ensure that the system stays stable, using control law (10), or m 0, there are ew more things to consider. According to Marruedo et al. [2002] and the proo o Theorem 1, optimality o the solution is not necessary to guarantee convergence o the closed-loop system. Thus, in order to maintain stability we must ensure that J j is strictly decreasing or all m 0. The system can use the control law (10), as long as J j+1 J j (24) In this case the convergence o the closed-loop system is guaranteed. Consequently, the triggering rule can be stated as j L Zj e(k + j k ) σ α x k i+j ω and L Zj e(k + j k ) σ α 1 j x k i+j ω j 1 L Zj 1 e(k + j k ) σ α x k i+j ω (25a) (25b) The next OCP is triggered whenever condition (25a) or (25b) is violated. Note, that the state vector x k is assumed to be available through measurements and that it provides the current plant inormation. The previous analysis guarantees that the closed loop system will have the same convergence properties as in Marruedo et al. [2002]. However, the OCP in the case o this paper is not calculated at each time instant, but only when the triggering condition is violated. Thus the convergence to a compact set and ultimate boundedness properties o Marruedo et al. [2002] are preserved in the event-triggered ormulation: Theorem 2. Consider the system (1), subject to (3) under an NMPC strategy and assume that the previously presented Assumption 1 holds. Then the NMPC control law given by (7a)-(7d) along with the triggering rule (25a)- (25b), drives the closed loop system towards a compact set where it is ultimately bounded. 4. PROBLEM STATEMENT FOR THE DECENTRALIZED NMPC CASE In the ollowing, the proposed ramework or inding eventtriggering condition or sampling is extended to a general discrete-time system which is composed by the interconnection o M local subsystems, controlled by a decentralized NMPC control law. The ramework is considered to be ully decentralized, i.e., there is no inormation exchange between the subsystems. A stabilizing decentralized MPC algorithm o nonlinear systems composed by a number o subsystems has been presented in Raimondo et al. [2007]. Each subsystem is locally controlled by an NMPC controller with guaranteed ISS property. Considering the eect o interconnections as perturbation terms it is shown that the overall system is also ISS. In accordance with the previous centralized approach, a modiication o the analysis in Raimondo et al. [2007] is going to be equipped in the sequel, in order to derive the event-triggering condition o the system. Consider the nonlinear perturbed discrete-time dynamic subsystem x s (k + 1) = s (x s (k),u s (k)) + g s (x(k)) + ψ s (k) (26) with k Z 0 and s = 1,...,M. The state o the s- th subsystem is x s (k) R ns while u s (k) R ms is the control variable and inally ψ s (k) R ns is the additive uncertainty. Assume also that s (0,0) = 0. The overall state is given as x(k) [x 1 (k),x 2 (k),...,x M (k)] R n (27) with n = M s=1 n s. Notice that g s depends on the overall state and describes the inluence o M subsystems on the s-subsystem and is such that g s (0) = 0. The state and the disturbance are required to ulil the ollowing constraints x s X s u s (k) U s ψ s Ψ s (28) where X s and Ψ s are compact sets o R ns and U s is a compact set o R ms all o them containing the origin as an interior point. Deining (x,u) [ 1 (x 1,u 1 ),..., M (x M,u M )], g(x) [g 1 (x),...,g M (x)] and ψ [ψ 1,...,ψ M ], the whole system can be written as x(k + 1) = (x(k),u(k)) + g(x(k)) + ψ(k) (29) with k Z 0. Assumption 2: i) There exists positive Lipschitz constant L s such that s (x 1,u s ) s (x 2,u s ) L s x 1 x 2, or all x 1, x 2 X s and or all u s U s and all s = 1,...,M. ii) There exist M positive constants L gs,j with s = 1,...,M, such that g s (x) M j=1 L g s,j x j

5 Preprints o the 18th IFAC World Congress iii) For each subsystem, the sum o interaction term and the disturbance term is restricted to ulil the ollowing constraint w s {g s (x) + ψ s } W s, or all x X and or all ψ s Ψ s, where W s is a compact set o R ns, containing the origin as an interior point, while X X 1 X M. 4.1 NMPC or Decentralized Discrete-Time Systems The Optimal Control Problem (abbr. OCP) or each subsystem is solved or the noal state dynamics o the system (26), i.e., x s (k + 1) = s (x s (k),u s (k)) (30) As in the centralized case, the OCP or the noal system (30), is obtained by locally imizing at time instant k with respect to a control sequence u sf (k) [u s (k k),...,u s (k + N s k)], the ollowing perormance index u sf (k) J s(x s,u sf (k)) = N s 1 u sf (k) L s ( x s (k + i k),u s (k + i k)) + V s ( x s (k + N s k)) (31a) s.t. x s (k + j k) X js j = 1,...,N s (31b) u s (k + j k) U s j = 0,...,N s (31c) x s (k + N s k) X s (31d) where X s denotes the teral constraint set. As in the centralized case, the state constraint set is being replaced with a restricted set X js. Following a similar procedure as in Raimondo et al. [2007], we make the ollowing assumptions or the stage cost L s ( ) and the teral cost V s ( ) rom (31a): Assumption 3: i) The stage cost L s (x s,u s ) is Lipschitz continuous in X s U s, with a Lipschitz constant L cs. Let L s (0,0) = 0, and assume that L s (x s,u s ) α s ( x s ) where α s is a class K -unction. ii) Let the teral region X s rom (31d) be a subset o an admissible positively invariant set X s o the noal system. Assume that there is a local controller h s or the teral state X s. The associated teral penalty V s ( ) has the ollowing property α Vs ( x s ) V s (x s ) β Vs ( x s ) or all x s X s, where α Vs and β Vs are class K -unctions. We also assume that, V s ( s (x s,h s ))) V s (x s ) L s (x s,h s (x s )) x s X s and that V s is Lipschitz in X s, with Lipschitz constant L Vs. 5. TRIGGERING CONDITION FOR THE DECENTRALIZED NMPC CASE The triggering condition or each o the subsystem o the orm (26) under a decentralized NMPC control law, can be reached ollowing a similar procedure as in the previously presented, centralized case. In the ollowing, a similar notation as in the previous sections will be used. Since there are mismatches between the real subsystem (26) and the noal subsystem (30), the predicted evolution using the noal model might dier rom the real evolution o each o the subsystem. Next, a bound o the dierence between the real and the predicted evolution is given. This bound depends on the error e s (k + j k) and on the interaction and disturbance term that aects each subsystem. The error e s (k + j k) is e s (k + j k) = x s,k+j ˆx s (k + j k) (32) where x s,k+j is the state o the subsystem s, measured at time step k + j, and ˆx s (k + j k) is the predicted state o the same subsystem at the same time step, measured rom time step k. Lemma 1. Consider subsystems (26) and (30), subject to the constraints (28) and assume that the previously presented Assumption 2, holds. Then, or a given sequence o inputs, the error e s (k + j k), at time step k + j, will satisy e s (k + j k) L j 1 s e s (k + 1 k)+ + Lj 1 j 1 s L s ( j 1 ψ s (k + i) + Proo. We have so that and i=1 i=1 e s (k k) = x s,k ˆx s (k k) = 0 M L gs,t x t,k+i ) e s (k + 1 k) = x s,k+1 ˆx s (k + 1 k) e s (k + 2 k) = x s,k+2 ˆx s (k + 2 k) (33) L s x s,k+1 ˆx s (k + 1 k) + ψ s (k + 1) + g s (x k+1 ) M L s e s (k + 1 k) + ψ s (k + 1) + L gs,t x t,k+1 e s (k + 3 k) = x s,k+3 ˆx s (k + 3 k) M L s x s,k+2 ˆx s (k+2 k) + ψ s (k+2) + L gs,t x t,k+2 L s (L s e s (k +1 k)+ ψ s (k +1) + + ψ s (k + 2) + M L gs,t x t,k+1 )+ M L gs,t x t,k+2. Using similar calculations, we can derive + Lj 1 s e s (k + j k) L j 1 s j 1 L s ( i=1 j 1 ψ s (k + i) + e s (k + 1 k)+ i=1 M L gs,t x t,k+i ) Finding the triggering condition or each o the subsystems (26), under a decentralized NMPC control law o the orm (31a)-(31d), can be treated as an extension o the centralized case. Following the procedure o the centralized case, previously presented, and with the help o Raimondo et al. [2007], it can be shown that in the same manner as in the centralized case, the triggering rule or each o the subsystems s, can be stated as 10072

6 Preprints o the 18th IFAC World Congress and L Zs,j e s (k + j k ) σ L Zs,j e s (k + j k ) σ j α s ( x s,k i+j ) j α s ( x s,k i+j ) (34a) j 1 L Zs,j 1 e s (k + j k ) σ α s ( x s,k i+j ) with L Zs,j = L Vs L (Ns 1) j s + L cs L(Ns 1) j 1 s L s 1. (34b) The next OCP is triggered whenever condition (34a) or (34b) is violated. Note, that the state vector x k is assumed to be available through measurements and that it provides the current plant inormation. Theorem 3. Consider the system (26), subject to (28) under a decentralized NMPC strategy and assume that the previously presented Assumption 2 and Assumption 3, holds. Then the NMPC control law given by (31a)- (31d) along with the triggering rule (34a)-(34b), drives the closed loop system to a compact set where it is ultimately bounded. 6. CONCLUSION We provided an event-triggered ormulation o model predictive control based systems. The main idea is to trigger the solution o the optimal control problem only when it is needed, and not at every time-step as in the case o classic discrete time MPC. The event-based strategy is possible to alleviate the computational burden o a MPC ramework. The centralized case is treated irst, where we reached to an explicit event-triggered condition. Then the results has been extended to the decentralized ormulation, where an event-triggered condition is given or each o the subsystems. Future work involves inding the triggering condition in a cooperative control problem o a system o distributed agents which operate in a common environment. REFERENCES D.V. Dimarogonas, K.H. Johansson. Event-triggered control or multi-agent systems. 48th IEEE Con. Decision and Control, pages , W.B. Dunbar, R.M. Murray. Distributed receding horizon control with application to multi-vehicle ormation stabilization. Automatica, volume 42, pages , A. Eqtami, D.V. Dimarogonas, K.J. Kyriakopoulos. Event- triggered control or discrete-time systems. American Control Conerence, pages , E. Franco, L. Magni,T. Parisini, M. Polycarpou, D. Raimondo. Cooperative constraint control o distributed agents with nonlinear dynamics and delayed inormation excange: a stabilizing receding-horizon approach. IEEE Trans. Autom. Control, volume 53, pages 1: , L. Grune, F. Muller. An algorithm or event-based optimal eedback control. 48th IEEE Con. Decision and Control, pages , W.P.M.H. Heemels, J.H. Sandee, P.P.J. Van Den Bosch. Analysis o event-driven controllers or linear systems. International Journal o Control, volume 81, pages 4: , Y. Iino, T. Hatanaka, M. Fujita. Event-predictive control or energy saving o wireless networked control system. American Control Conerence, pages , T. Keviczky, F. Borrelli, G.J. Balas. Decentralized receding horizon control or large scale dynamically decoupled systems. Automatica, volume 42, pages 13: , M. Lazar, W.P.M.H. Heemels. Predictive control o hybrid systems: stability results or suboptimal solutions. Automatica, volume 42, pages 1: , D. Limon Marruedo, T. Alamo, E.F. Camacho. Input-to- State stable MPC or constrained discrete-time nonlinear systems with bounded additive uncertainties. 41st IEEE Con. Decision and Control, pages , M. Mazo Jr., and P. Tabuada. Decentralized eventtriggered control over wireless sensor/actuator networks. Submitted, G.Pin, L. Magni, T. Parisini, D.M. Raimondo. Robust receding-horizon control o nonlinear systems with state dependent uncertainties: an input-to-state approach. American Control Conerence, pages , D.M. Raimondo, P. Hokayem, J. Lygeros, M. Morari. An iterative decentralized MPC algorithm or largescale nonlinear systems. Proceedings o the 1st IFAC Workshop on Estimation and Control o Networked Systems, pages , D.M. Raimondo, L. Magni, R. Scattolini. Decentralized MPC o nonlinear systems: an input-to-state stability approach. International Journal o Robust and Nonlinear Control, volume 17, pages 17: , R. Scattolini. Architectures or distributed and hierarchical model predictive control - a review. Journal o Process Control, volume 19, pages , J. Sijs, M. Lazar, W.P.M.H. Heemels. On integration o event-based estimation and robust MPC in a eedback loop. HSCC 10:Proc. o the 13th ACM International Conerence on Hybrid Systems: Computation and Control, pages 31 40, P. Tabuada. Event-triggered real-time scheduling o stabilizing control tasks. IEEE Transactions on Automatic Control, volume 52, pages 9: , P. Varutti, R. Findeisen. On the synchronization problem or the stabilization o networked control systems over nondeteristic networks. American Control Conerence, pages , X. Wang, M.D. Lemmon. Event-triggering in distributed networked systems with data dropouts and delays. HSCC 09:Proc. o the 12th International Conerence on Hybrid Systems: Computation and Control, volume 52, pages ,