Economic and Distributed Model Predictive Control: Recent Developments in Optimization-Based Control

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1 SICE Journal of Control, Measurement, and System Integration, Vol. 10, No. 2, pp , March 2017 Economic and Distributed Model Predictive Control: Recent Developments in Optimization-Based Control Matthias A. MÜLLER and Frank ALLGÖWER Abstract : The goal of this paper is to give an overview of some recent developments in the field of model predictive control. After a brief introduction to the basic concepts and available stability results, we in particular set our focus on the areas of distributed and economic model predictive control, where more general control objectives than setpoint stabilization are typically of interest. Key Words : model predictive control, distributed MPC, economic MPC, nonlinear systems. 1. Introduction Model predictive control (MPC), also called receding horizon control, is a very successful modern control technology. Its basic idea is as follows: at each sampling instant, the future behavior of the system is predicted over some finite time horizon using some prediction model, and an open-loop optimal control problem is solved to determine the optimal input trajectory over this horizon. Then, the first part of this optimal input is applied to the system until the next sampling instant, at which the horizon is shifted and the whole procedure is repeated again. The main advantages of MPC and the reasons for its widespread success include (i) guarantees for closed-loop satisfaction of hard input and state constraints, (ii) the possibility to directly include the optimization of some performance criterion in the controller design, and (iii) its applicability to nonlinear systems with possibly multiple inputs. In the last decades, a very rich body of literature has emerged discussing both various theoretical issues such as stability and robustness as well as successful applications in many industrial fields, see, e.g., the books and survey articles [1] [9] and the references therein. Besides a brief introduction to the basic concepts and available stability results in MPC (see Section 2), the main focus of this review article are the areas of distributed and economic model predictive control, which have both received a significant amount of attention in recent years. One unifying feature of the underlying control tasks in both these areas is that typically more general objectives than the classical goal of setpoint stabilization are considered. For example, in a typical distributed cooperative control task, multiple dynamical systems have to achieve some common cooperative goal such as multivehicle platooning and formation stabilization, or distributed power generation. In other situations, some general objective is considered which is possibly related to the economics of the system, such as maximization of a certain product in the process industry, minimization of energy consumption, or maxi- Institute for Systems Theory and Automatic Control, University of Stuttgart, Stuttgart, Germany matthias.mueller@ist.uni-stuttgart.de, frank.allgower@ist.uni-stuttgart.de (Received November 28, 2016) (Revised December 22, 2016) mization of the achievable profit in general. While the former setting can be handled within a suitable distributed MPC framework, the latter is the distinctive feature of so-called economic MPC schemes. When considering large-scale systems, the development of a single (centralized) model predictive controller for the overall system is in general not possible or not desirable, since, e.g., such a central control entity having enough computational resources is not available, such an approach is not scalable, or the information/communication flow between the systems is restricted to a certain (given) communication topology. Hence, in distributed MPC, the main challenge is how the repeated online optimization can be performed in a distributed fashion such that for the overall closed-loop system, desired guarantees such as constraint satisfaction, stability, or achievement of the cooperative control objective can still be established. To this end, various solution approaches suitable for different system classes and interconnection structures have been proposed; Section 3 provides an overview of some of these approaches. As motivated above, in economic MPC some general cost function is used for controller design, which is not necessarily positive definite with respect to some setpoint to be stabilized. This results in the fact that the closed-loop system might not converge to some steady-state, since the optimal behavior of the system can be more complex (e.g., periodic), depending on the system dynamics, the cost function and the constraints. Important questions in such a setting are in particular the classification of the optimal operating behavior as well as establishing desired closed-loop performance and convergence guarantees. These issues will be discussed in Section 4. We close this section by noting that the goal of this article is not to give an exhaustive overview of all the available results in the above areas, which by the immense number of existing publications is almost impossible. Rather, we attempt to summarize the main ideas in a coherent fashion such as to provide the reader a concise introduction to the field of model predictive control in general, and in particular to the areas of distributed and economic MPC. The exposition will be done for discretetime systems, as is the case in most of the available results in the literature. Nevertheless, the same (or similar) results are typically also available for continuous-time systems. JCMSI 0002/17/ c 2016 SICE

2 40 SICE JCMSI, Vol. 10, No. 2, March Notation Let I [a,b] denote the set of integers in the interval [a, b] R, and I a the set of integers greater than or equal to a. Boldface symbols denote finite sequences v : I [0,N] R n for N I 0, or infinite sequences v : I 0 R n, i.e., v = {v(0),...,v(n)} or v = {v(0), v(1),...}, respectively. 2. MPC - Basic Principle and Stability In this section, we give a brief introduction to the basic principle of MPC and available stability results. In the following, the system dynamics are given by a nonlinear difference equation of the form x(t + 1) = f (x(t), u(t)), x(0) = x 0, (1) where f : R n R m R n, x(t) R n and u(t) R m are the system state and the control input, respectively, at time t I 0, and x 0 R n is the initial condition. The system has to satisfy pointwise-in-time state and input constraints x(t) X, u(t) U (2) for all t I 0,whereX R n and U R m are the state and input constraint sets, respectively 1. In order to define the model predictive control law, at each time t I 0 with measured state x(t), for a given prediction horizon N I 1 the following optimization problem is solved: Problem 1 minimize u(t) subject to J N (x(t), u(t)) x(k + 1 t) = f (x(k t), u(k t)), k I [0,N 1], (3a) x(0 t) = x(t), (3b) x(k t) X, u(k t) U, k I [0,N 1], (3c) x(n t) X f, (3d) where N 1 J N (x(t), u(t)) := l(x(k t), u(k t)) + V f (x(n t)). (4) k=0 In this problem, u(t) := {u(0 t),...,u(n 1 t)} and x(t) := {x(0 t),...,x(n t)} are input and corresponding state sequences predicted at time t over the prediction horizon N. The constraints (3) ensure that predicted states and inputs have to obey the system model (1) and satisfy the given state and input constraints, and that the final predicted state is contained in the terminal region X f X. Furthermore, the stage cost function l : X U R specifies the considered performance criterion, and V f : X f R is the terminal cost function. The terminal constraint (3d) and the terminal cost V f are used in various schemes in the literature in order to guarantee closedloop stability (cf. Subsection 2.1); other approaches omit these additional ingredients (cf. Subsection 2.2), as is also typically 1 Instead of decoupled state and input constraints, also more general (possibly coupled) constraints of the form (x(t), u(t)) Z R n R m for all t I 0 can be considered. done in industrial practice. Under some mild technical conditions (such as continuity of the involved functions and compactness/closedness of the involved sets), Problem 1 is either infeasible or a (possibly non-unique) minimizer exists, which is denoted by u 0 (t) := {u 0 (0 t),...,u 0 (N 1 t)} and in which case x 0 (t) := {x 0 (0 t),...,x 0 (N t)} and J 0 N (x(t)) := J N(x(t), u 0 (t)) denote the corresponding optimal state sequence and optimal value function, respectively. A standard model predictive controller is now defined by the following algorithm. Algorithm 1 Consider system (1). At each time t I 0, measure the state x(t), solve Problem 1 and apply the control input u MPC (x(t)) := u 0 (0 t). Algorithm 1 results in the closed-loop system x(t + 1) = f (x(t), u MPC (x(t))), x(0) = x 0. (5) In the following, we discuss how desired guarantees for this closed-loop system can be derived. The classical control objective, which is considered in most of the available MPC literature, is to asymptotically stabilize a given equilibrium setpoint of system (1), which without loss of generality is assumed to be the origin. In this case, a positive definite stage cost function l needs to be chosen, i.e., l(0, 0) = 0andl(x, u) > 0forall (x, u) 0. However, in general the closed loop (5) is not necessarily asymptotically stable if Problem 1 is not designed properly. In fact, it is rather easy to construct examples where this is the case; for an experimental study of this fact, see, e.g., [10]. In the literature, two main approaches can be found how stability in MPC can be established. One uses a suitable choice of the terminal region and cost, while the other establishes stability without these ingredients by a suitable (long enough) choice of the prediction horizon N. 2.1 Stability in MPC with Additional (Terminal) Constraints When using a framework with a terminal constraint (3d) and a terminal cost function, the main assumptions to ensure stability are as follows. Assumption 1 There exists a local auxiliary control law u = κ f (x) such that the following is satisfied inside the terminal region, i.e., for all x X f : i) κ f (x) U, ii) f (x,κ f (x)) X f, iii) V f ( f (x,κ f (x))) V f (x) l(x,κ f (x)). Assumption 1 means that the local auxiliary controller satisfies the input constraints (when applied inside the terminal region), that the terminal region is invariant under application of the local auxiliary controller, and that the terminal cost serves as a control Lyapunov function inside the terminal region. Note that the auxiliary local controller is never applied to the system, but is only needed in order to properly design the terminal region X f and the terminal cost V f. If Assumption 1 (and some additional, nonrestrictive technical assumptions) are satisfied, one can show that (i) Problem 1 is feasible for all t I 0 if it is initially feasible (i.e., for t = 0) and (ii) the origin is an asymptotically stable equilibrium for the closed-loop system (5). A

3 SICE JCMSI, Vol. 10, No. 2, March proof of this result was first obtained in [11] for continuoustime systems (compare also [3],[12]); for the discrete-time setting, see, e.g., [2],[4],[5]. The survey paper [2] also gives a good overview on the historical development of stability results in MPC. The main idea in order to prove the above result on recursive feasibility and asymptotic stability is to construct a feasible candidate solution to Problem 1 at time t + 1 from the optimal solution at time t. In particular, such a feasible candidate solution ũ(t + 1) is given by appending the tail of the previously optimal solution by the local control law, i.e., ũ(t + 1) = {u 0 (1 t),...,u 0 (N 1 t),κ f (x 0 (N t))}. Feasibility of this candidate solution follows from feasibility of the optimal solution u 0 (t) at time t and properties i)andii) of Assumption 1. Given this feasible candidate solution and the assumption of initial feasibility, recursive feasibility can easily be established by induction. Furthermore, one can use the optimal value function JN 0 as a Lyapunov function for the closed-loop system (5). In particular, the above defined feasible (but in general suboptimal) candidate solution ũ results in a cost decrease given by JN 0 (x(t +1)) J N(x(t +1), ũ(t +1)) JN 0 (x(t)) l(x(t), umpc (t)), from which asymptotic stability can be concluded. A constructive method how to compute X f, V f and κ f such that Assumption 1 is satisfied was first proposed in [11] (for continuous-time systems). This procedure is based on the assumption that the linearization of system (1) at the origin is stabilizable and considers the case where the stage cost function l is quadratic. The local auxiliary controller is chosen as a linear control law κ f (x) = Kxwhich stabilizes the linearization. Then, a quadratic terminal cost V f is computed such that condition iii) of Assumption 1 is satisfied for the linearized system by solving a Lyapunov equation, and the terminal region X f is chosen as a sublevel set of V f. Since the linearization error consists of quadratic (and higher order) terms, one can show that condition iii) of Assumption 1 is also satisfied for the original nonlinear system if a small enough sublevel set is chosen as terminal region. Furthermore, by assuming that 0 is contained in the input constraint set U, it follows that condition i) is also satisfied for X f small enough. Finally, condition ii) follows from satisfaction of iii) and the fact that X f is a sublevel set of V f. A discrete-time version of this procedure can, e.g., be found in [4]; also, various different approaches and generalizations to obtain possibly larger terminal sets are available, see, e.g., [12] [14]. 2.2 Stability in MPC without Terminal Constraints The second main approach to ensure stability omits the use of a terminal constraint (3d) and a terminal cost in (4). Instead, closed-loop stability is typically shown by choosing the prediction horizon N large enough, given that some controllability condition is satisfied. While early results showed that under certain conditions, such a large enough prediction horizon exists (see, e.g., [15],[16]), more recent publications also give constructive (although possibly conservative) estimates for the required prediction horizon length using a suitable controllability condition [5],[17] [21]. This controllability condition is typically formulated as the existence of a suitable upper bound on the optimal value function: Assumption 2 There exist L > 0 and numbers 0 L N L, such that for all x X and all N I 0, the following inequality holds: JN 0 (x) L N inf l(x, u). (6) u U Assumption 2 can, e.g., be verified using certain asymptotic or exponential controllability conditions, although this can be a difficult task in general. Using Assumption 2, one can show that there exists a number α(n) such that the relaxed dynamic programming inequality JN 0 (x(t + 1)) J0 N (x(t)) α(n)l(x(t), u MPC (t)) holds. In case that α(n) > 0, again asymptotic stability follows. Furthermore, α(n) provides a suboptimality estimate for the resulting MPC closed-loop cost compared to the infinite horizon optimal cost. Namely, let J MPC (x 0 ) := t=0 l(x(t), u MPC (x(t))) denote the closed-loop cost resulting from application of Algorithm 1 (with prediction horizon N), and J (x 0 0 ) be the infinite horizon optimal cost. Then from the relaxed dynamic programming inequality it follows that α(n)j MPC (x 0 ) J (x 0 0 )forallx 0 X. In [5],[20], a detailed analysis can be found how α(n) can be obtained using Assumption 2 (see also the continuous-time version [21]). In particular, one can show that α(n) 1forN. This implies that (i) closed-loop stability is guaranteed for a large enough prediction horizon N (since for such N, α(n) > 0) and (ii) for N,the MPC closed loop recovers infinite horizon optimality. 2.3 Discussion and Further Results We now briefly discuss some of the advantages and disadvantages of the above described methods to establish stability in MPC. The main benefits of the framework of Subsection 2.1 with terminal constraints and a terminal cost are that a shorter prediction horizon than in a setting without terminal constraints might be sufficient to ensure closed-loop stability, and that a systematic procedure how to satisfy the crucial Assumption 1 is available for a large class of systems (in particular, in case that the linearization is stabilizable as described above). Furthermore, this assumption allows to establish recursive feasibility of the repeatedly solved optimization problem as well as closed-loop stability in a straightforward manner. On the other hand, adding a terminal constraint might be restrictive and the feasible set (i.e., the set of states for which a solution to Problem 1 exists) might be unnecessarily small, depending on the size of the terminal region. Also, the computational complexity is increased through the additional terminal constraint. The approach of Subsection 2.2 does not exhibit these drawbacks through omitting the use of a terminal constraint, resulting in a simpler optimization problem and a possibly larger feasible region. On the other hand, recursive feasibility is not as straightforward, but requires additional assumptions and/or arguments, see, e.g., [5],[15],[22]. Furthermore, the required controllability condition (Assumption 2) can be difficult to verify, and for general nonlinear systems, no systematic procedure exists to this end. An additional advantage of MPC without terminal constraints is that closed-loop performance estimates can be given via the suboptimality index α(n). When using the setting of Subsection 2.1, the establishment of such closed-loop performance estimates is not as straightforward and in general depends on properties of the terminal cost V f. Certain suboptimality estimates can be established if the terminal cost is a good approximation of the infinite horizon optimal cost, see, e.g., [5].

4 42 SICE JCMSI, Vol. 10, No. 2, March 2017 In conclusion, each of the two presented schemes has its advantages and disadvantages, and the choice of a suitable stabilizing MPC scheme depends on the considered application. An alternative to the two stabilizing MPC approaches described above are Lyapunov-based MPC methods (see, e.g., [23]). Here, a (global) control Lyapunov function is assumed to be known, and instead of terminal constraints, a constraint requiring a certain decrease of the control Lyapunov function is included into Problem 1. Furthermore, the basic stabilizing MPC approaches discussed in this section have been extended in many different ways. For example, output feedback MPC schemes, the stabilization of more general sets and trajectory tracking problems have been studied, as well as MPC schemes for different system classes such as hybrid, periodic, or timedelay systems. Also, important issues concerning robustness in the presence of uncertainties and disturbances have been investigated both in a deterministic and a stochastic framework. All these areas are currently still very active fields of research, and it is well beyond the scope of this article to discuss all these approaches in detail; a recent survey summarizing some of the existing results (with a particular focus on MPC for uncertain systems) can be found in [9]. 3. Distributed MPC When considering large-scale systems, the development of a single (centralized) model predictive controller, i.e., repeatedly solving Problem 1 for the overall system, is in general not possible or not desirable. This can be due to several reasons. For example, such a central entity having enough computational power might not be available. In other situations, the communication infrastructure is such that information can only be shared locally between subsystems, but not with some central coordinating unit. Furthermore, in a competitive setting, systems might not want to share all available information with everyone due to privacy issues. In such cases, distributed MPC schemes are of interest. Here, the main challenge is how to design suitable optimization problems (similar to Problem 1) and how to develop suitable algorithms for a distributed solution, such that guarantees for the overall closed-loop system can be given. In such a distributed setting, coupling/interactions between subsystems can be of different nature. First, physical couplings can exist, i.e., the dynamics of the individual subsystems are coupled. This means that the dynamics of the individual subsystems depend on other subsystems states and/or inputs, i.e., the dynamics of each subsystem i is given by x i (t + 1) = f i (x(t), u(t)), (7) where x denotes the state of the overall system x = [x T 1...xT P ]T and P is the number of subsystems, and similarly u denotes the input of the overall system. Such couplings appear in many applications such as, e.g., large chemical plants consisting of multiple interconnected reactors, in electrical grids, or in coupled mechanical systems. In other settings, the overall system is composed of a team of dynamically independent subsystems whose dynamics only depend on the own system state and input, i.e., x i (t + 1) = f i (x i (t), u i (t)). (8) This is typically the case in multi-agent settings such as multirobot coordination or formation flight. Second, couplings between the subsystems can stem from coupling constraints. Typical examples include collision avoidance or connectivity maintenance constraints. Third, a common objective is another source of coupling between subsystems, resulting in couplings in the cost functions J i,n of the individual subsystems. In the above mentioned examples, this could be a system-wide performance optimization of a chemical plant or a certain cooperative task such as formation flight. Finally, the available communication topology, i.e., which systems can exchange information with each other, is crucial for the selection of a suitable distributed MPC scheme. In general, the graph representing this communication topology need not be the same as the one resulting from the three sources of couplings between the subsystems discussed above. However, in many of the available results, it is assumed that each system can (at least) communicate with every other system to which it is coupled. A distinctive feature of distributed MPC compared to other distributed control methods is that the information, which is exchanged between subsystems, typically consists of predicted state and/or input trajectories (compared to, e.g., only the current state), which allows satisfaction of constraints despite couplings and allows for improved performance. The existing types of coupling between the subsystems as well as the available communication infrastructure are crucial for designing suitable optimization problems (similar to Problem 1) for each of the subsystems and for developing suitable algorithms for a distributed solution. In recent years, a large body of literature with hundreds of papers has been developed for various different settings including different types of system dynamics, couplings, and control objectives and using different distributed solution methods, see, e.g., the survey articles [24],[25] and the edited volume [26], which presents 35 distributed MPC approaches using a unified notation. In the following, we do not attempt to provide an exhaustive survey of all available distributed MPC results, but rather give a brief overview of how the different approaches can be classified in terms of the employed solution strategies (see Subsection 3.1). Furthermore, in contrast to the above mentioned existing surveys, we put a particular focus on distributed MPC schemes which have been developed for more general cooperative control tasks than setpoint stabilization (see Subsection 3.2). 3.1 Solution Approaches in Distributed MPC One possibility to classify distributed MPC schemes is according to the employed solution strategy, i.e., how the distributed MPC problem is formulated and solved, and what the main distinctive feature is regarding the establishment of closed-loop guarantees. In the following, we discuss four different classes, to which most of the existing distributed MPC schemes can be allocated. First, we consider iterative schemes, where some sort of distributed optimization algorithm is used. After that, in non-iterative schemes, we distinguish between sequential approaches, schemes based on robustness considerations, and methods using additional consistency constraints. Finally, we discuss various advantages and disadvantages of the different schemes. Iterative methods. Many distributed MPC schemes use iterative methods. Here, in each time step the optimization prob-

5 SICE JCMSI, Vol. 10, No. 2, March lem for the overall system, i.e., Problem 1, is solved by some sort of distributed optimization algorithm. This means that at each time t, all subsystems iteratively solve an optimization problem (in parallel) and exchange information multiple times such that the solutions converge to the solution of the centralized, overall optimization problem. Distributed MPC schemes belonging to this category can, e.g., be found in [27] [29] using a Jacobi method or in [30],[31] based on dual decomposition. The alternating direction method of multipliers (ADMM) was employed in [32], whereas in [33] and [34] the authors use a sensitivity-based coordination scheme and a block-coordinate descend method, respectively. In [35], the authors present a framework for the distributed synthesis of suitable terminal regions and terminal cost functions, which can then be used for distributed MPC using suitable distributed optimization algorithms. Finally, Lyapunov-based techniques are used in [36],[37]. Most of the existing iterative distributed MPC schemes are designed for linear systems and the obtained results build on convexity of the problem (which is in particular needed for convergence of the distributed optimization algorithms); there are few exceptions which also treat nonlinear systems such as [28],[36]. Some of the available schemes need the requirement that the distributed optimization algorithm converges at each time t, which might be impractical since a large number of iterations might be necessary. Other schemes provide feasibility and stability already for a finite number of iterations, some even for the case of only one iteration. Ensuring satisfaction of coupling constraints is typically difficult, at least in case of a finite number of iterations; a notable exception can be found in [31]. Sequential methods. Adifferent approach to distributed MPC are sequential methods. Here, the distinctive feature is that the subsystems do not solve their optimization problems in parallel but sequentially, i.e., only one subsystem optimizes while all subsystems keep their predicted trajectories fixed. Using such a strategy allows to establish feasibility and fulfillment of coupling constraints in a rather straightforward fashion. A first sequential distributed MPC scheme of this type appeared in [38], where only couplings due to constraints were considered. These ideas were subsequently extended to more general settings, including other cooperative control tasks than stabilization and uncertain system dynamics, see, e.g., [39] [43]. A benefit of such sequential schemes is that they easily carry over to nonlinear systems and also coupling constraints can be taken into account in a straightforward fashion. On the other hand, such approaches are typically not well scalable if all subsystems solve their optimization problem at each time t in a sequential order, or the performance might be compromised if only one (or few) subsystems are allowed to optimize at each time t, while the other subsystems follow their previously computed trajectories. Depending on the interconnection structure, this drawback can be overcome since subsystems which are not coupled to each other can optimize in parallel without compromising the theoretical feasibility and stability guarantees. The above schemes were developed for settings where no couplings in the dynamics are present. If this is the case, further negotiation between the systems [44] or a special system structure [36],[45] are necessary in order to establish closed-loop guarantees. Approaches based on robustness considerations. Another class of distributed MPC schemes is based on robustness considerations. Here, the influences of neighboring subsystems are (partially) treated as unknown disturbances, and certain concepts for the analysis and controller design of uncertain systems are employed either within the analysis of the resulting closed-loop system, or already in the design of the optimization problems in the distributed MPC scheme. For example, in [46] the authors show for nonlinear subsystems coupled by a common objective that each subsystem is input-to-state stable (ISS) with respect to the disturbance being composed of the influences of neighboring subsystems, and by a small gain argument that in turn also the overall closed-loop system is asymptotically stable. In [47], a tube-based robust MPC approach is utilized. Each subsystem assumes that trajectories of neighboring subsystems are contained within a tube around a communicated nominal trajectory, and ensures that its own trajectory is contained in such a tube. This design can be used in order to show asymptotic stability of the resulting overall closed-loop system. In the distributed MPC scheme presented in [48], subsystems exchange a sequence of sets (called contracts ), in which predicted trajectories are guaranteed to be contained. For the resulting closed-loop system, ISS with respect to the contract size can be established. While in the above references, predicted trajectories (and/or tubes containing them) are exchanged between the subsystems, the idea of treating influences of neighboring subsystems as disturbances can also be employed in an even simpler, yet more conservative, setting, where no communication between the subsystems takes place. Such schemes are typically referred to as decentralized MPC schemes, see, e.g., [49],[50], where min-max or again tube-based MPC techniques, respectively, are employed. Methods employing consistency constraints. Finally, some distributed MPC approaches impose additional constraints on predicted state and/or input sequences in order to establish desired closed-loop properties. One typical form of such constraints is to require predicted trajectories to be close enough to previously communicated ones, see, e.g., [51] [53]. This ensures that at each time t, the new optimal trajectory computed by each subsystem i is close enough to what the other subsystems assumed for this subsystem, which is why such constraints are sometimes called consistency constraints or compatibility constraints in the literature. A similar approach was taken in [54], where overall closed-loop stability was shown if a certain compatibility condition is satisfied. A different approach is taken in [55] and [56] for systems only coupled via a common objective, where a constraint tightening on predicted inputs and states, respectively, is employed in order to obtain desired closed-loop guarantees. Discussion. We now briefly compare the above distinguished classes of distributed MPC schemes. In general, the best system-wide performance will be obtained by iterative schemes. As indicated above, for enough iterations the performance will converge to that of a centralized MPC approach; this is typically not the case in the other distributed MPC frameworks. Hence if both enough communication and computational capacity are available for possibly multiple iterations per time instant, iterative schemes can be well suited. On the other hand, they might not be applicable when considering nonlinear systems or if coupling constraints are present. Sequen-

6 44 SICE JCMSI, Vol. 10, No. 2, March 2017 tial schemes have the advantage that possibly less communication (typically only once per time instant) is needed, and also coupling constraints as well as nonlinear system dynamics can be taken into account in a fairly straightforward fashion. On the other hand, couplings in the dynamics might be difficult to consider, and there is also a potential scalability problem as discussed above. Approaches based on robustness consideration also need less communication than iterative schemes, and compared to sequential schemes, they might be better scalable (depending on the system structure) and also couplings in the system dynamics can be considered. However, such schemes can be quite conservative and hence might only work if the coupling strength is rather weak. Similar considerations also hold for methods based on additional (consistency) constraints. Namely, while they potentially share similar advantages as robustness-based approaches compared to iterative and sequential schemes, the degrees of freedom within the optimization might be quite small, depending on the exact formulation of the consistency constraints, resulting in a quite conservative solution with diminished performance. In summary, each of the different classes of distributed MPC schemes has its own advantages and disadvantages, and the choice of a suitable distributed MPC algorithm heavily depends on the properties of the considered application, i.e., on the considered system class, the types of couplings, and the available computational resources and communication infrastructure. 3.2 Distributed MPC for General Cooperative Control Tasks The vast majority of existing distributed MPC schemes, including most of those referenced above, consider the classical control objective of stabilizing the overall system at some (a priori given) setpoint. However, when considering typical cooperative control problems, often different, more general, control objectives are of interest. This is in particular the case in settings where a team of dynamically independent subsystems has to pursue a common, cooperative task such as vehicle platooning or formation flight, corresponding to couplings in constraints and cost functions. Cooperative control tasks of particular importance, which have received significant attention in recent years, are consensus and synchronization problems, where the systems have to agree on an a priori unknown state (or output) value or trajectory. These type of problems appear in many applications such as multi-vehicle platooning and formation stabilization, distributed sensor fusion, the synchronization of coupled oscillators and others (see, e.g., [57]). When considering such general cooperative control objectives, typically different methods have to be used for the MPC design and/or closed-loop analysis compared to the case where a setpoint (or trajectory) to be stabilized is known a priori. For example, the design of suitable terminal cost functions and terminal regions has to be suitably adapted to such a setting. Furthermore, while some approaches still use the optimal value function as a Lyapunov function for convergence analysis, others rather exploit certain geometric properties of the resulting closed-loop dynamics. In the following, we briefly review some distributed MPC schemes which were specifically developed for such more general control objectives than setpoint stabilization. In [58], consensus for single and double integrators subject to input constraints is considered, exploiting certain geometric properties of the system trajectories resulting from the rather simple dynamics. Reaching consensus via distributed MPC is also considered in [59] for single integrators, for double integrators [60], for scalar linear systems [61], and for general linear systems [62]. Here, the closed-loop analysis is based on the explicit solution to the respective optimization problems, which is characterized for the case that no constraints are present. A similar approach is taken in [63], where consensus for doubleintegrators is shown in the presence of input constraints which are active only a finite number of times, and in [64], where flocking of double-integrators is considered. In [65], for general linear systems, the authors calculate an optimal consensus point at each time step by iteratively solving a centralized optimization problem, where this point is used as setpoint in the MPC formulation. A theoretical analysis of this scheme was done in [66], where convergence to a common consensus point corresponding to an equilibrium of the system is established, both for the case where the negotiations about the optimal consensus point have converged in each time step, as well as when this negotiation is terminated prior to convergence. In [67], a consensus problem is solved for linear systems using standard methods, and then the resulting solution is used as a reference signal in a tracking MPC framework. Furthermore, in [68] the authors propose a robust distributed MPC scheme for linear time-varying uncertain systems to achieve practical consensus. The sequential distributed MPC scheme in [40] is designed for general cooperative control tasks which can be formulated as the stabilization of some (possibly non-compact) set, which includes consensus/synchronization as a special case. Convergence to the desired set, i.e., fulfillment of the cooperative control task, is shown for nonlinear systems including coupling constraints under suitable assumptions on the terminal regions and terminal cost functions. Furthermore, it is shown how these conditions can be satisfied for the special cooperative control task of consensus/synchronization. Distributed MPC for vehicle platooning is considered in [69]. Stability and string stability for the resulting closed-loop system is achieved by penalizing deviations from previously communicated trajectories in the cost function. In [70], the authors consider distributed MPC for two single-integrator systems with a possibly competitive objective, such as different desired distances to each other. Finally, also the framework of economic MPC (compare Section 4), where some general performance criterion is considered, can be exploited for certain cooperative control problems. To this end, recently some first distributed economic MPC schemes have been proposed in the literature, which will be discussed at the end of Section Economic MPC Economic MPC is a variant of MPC where compared to the classical stabilizing MPC setting as in Section 2, some more general performance criterion is considered. The motivation (and also the naming) for this more general MPC formulation came from the process industry. There, the classical approach is to apply a two-layer control structure, where at the upper layer the so-called real-time optimization (RTO) determines economically optimal setpoints or trajectories for the system, which are then tracked by a controller on the lower level, see, e.g., [71] [74] and the references therein. However, it was noticed that the performance can possibly be significantly im-

7 SICE JCMSI, Vol. 10, No. 2, March proved if the economic cost function is directly incorporated into the lower level MPC controller [73],[75],[76]. On a technical level, the only difference of economic MPC compared to standard stabilizing MPC is that the stage cost function l in Problem 1 is not chosen to be positive definite with respect to an a priori given setpoint as in Section 2, but can be some general function (possibly related to the economics of the considered system). This results in the fact that the closed-loop system might not converge to some equilibrium setpoint, since other trajectories (e.g., periodic) exist resulting in a better performance. Important questions in this context are (i) how to classify the optimal operating behavior, (ii) whether the closedloop system finds the optimal operating behavior (i.e., converges there), and (iii) what statements about closed-loop performance can be made. These issues will be discussed in the following subsections. 4.1 Classifying Optimal Operating Behaviors In general, the optimal operating behavior depends on the given system dynamics f, the stage cost function l, and the input and state constraint sets U and X. The most simple case is when steady-state operation is optimal, i.e., the optimal operating behavior is to stay at the best steady-state. This is the best studied case in the literature, which we also first discuss in the following. We then mention possible extensions to more general cases (such as optimal periodic operation). Let Z 0 denote the largest forward invariant set contained in X U, i.e., the set which contains all elements in X U which are part of a feasible state/input sequence pair: Z 0 := { (x, u) X U : v : I 0 U s.t. (z(0), v(0)) = (x, u), z(t + 1) = f (z(t), v(t)), (z(t), v(t)) X U t I 0, } X U. (9) Furthermore, let S be defined as the set of all feasible state/input equilibrium pairs of system (1), i.e., S := {(x, u) :x X, u U, x = f (x, u)} Z 0, (10) which is assumed to be non-empty. Under some mild technical conditions (such as, e.g., continuity of f and l and compactness of X and U), existence of a (possibly non-unique) optimal state/input equilibrium pair (x, u ) is guaranteed, i.e., (x, u ) satisfies l(x, u ) = inf l(x, u). (11) (x,u) S Optimal steady-state operation can now been defined as follows [75],[77]. Definition 3 The system (1) is optimally operated at steadystate, if for each solution satisfying (x(t), u(t)) X U for all t I 0 the following holds: Tt=0 l(x(t), u(t)) lim inf l(x, u ). (12) T T + 1 The definition of optimal operation at steady-state is such that no feasible solution to system (1) leads to an average performance (measured in terms of the stage cost l) which is better than the optimal steady-state cost l(x, u ), which is the average cost when operating the system at the optimal steady-state x. In order to study optimal steady-state operation of a system, a certain dissipativity condition has turned out to play a crucial role. The concept of dissipativity dates back to Willems [78] (see also [79] for a discrete time version) and is as follows. Definition 4 The system (1) is dissipative with respect to the supply rate s : R n R m R if there exists a storage function λ : R n R 0 such that the following inequality is satisfied for all (x, u) R n R m : λ( f (x, u)) λ(x) s(x, u). (13) If there exists ρ K such that for all (x, u) R n R m λ( f (x, u)) λ(x) ρ( x x ) + s(x, u), (14) then system (1) is strictly dissipative. If the dissipation inequality (13) is only satisfied on some set W R n R m, we say that system (1) is dissipative on W. Given the above notation and definitions, it was shown in [75] (see also [80] for a continuous-time version) that if system (1) is dissipative on Z 0 with respect to the supply rate s(x, u) = l(x, u) l(x, u ), then it is optimally operated at steady-state 2. Furthermore, if system (1) is strictly dissipative with respect to this supply rate, a slightly stronger property than optimal steady-state operation is satisfied, which was termed (uniform) suboptimal operation off steady-state [75],[81]. In the most general case, the converse result is not true, which was shown by two counterexamples in [82]. However, under suitable controllability conditions, necessity of dissipativity for optimal steady-state operation can indeed be established. Namely, if system (1) is optimally operated at steady-state, it is dissipative with respect to the supply rate s(x, u) = l(x, u) l(x, u )onthe intersection of the M-step controllable and M-step reachable set, for arbitrary M I 1 (cf. [83] for a more precise statement of this result). Furthermore, if system (1) is uniformly suboptimally operated off steady-state and locally controllable at x, then it is strictly dissipative on Z 0 [81]. In summary, dissipativity with respect to the supply rate s(x, u) = l(x, u) l(x, u ) serves as an (almost) equivalent characterization of the property of optimal steady-state operation. The more general case where the system is optimally operated at a periodic orbit can be treated in a similar fashion. Namely, one can define optimal operation of system (1) at a P- periodic orbit Π analogously to Definition 3, but with l(x, u ) on the right hand side of (12) replaced by the average cost of the P-periodic orbit Π. Then, a suitable cyclic dissipativity condition or a dissipativity condition for a suitably defined P-step system can be used as a sufficient and necessary condition (the latter again under a suitable controllability assumption) for optimal P-periodic operation [81],[84] [86]. Furthermore, a (partial) generalization of the above results to the case of time-varying systems and cost functions is available [87], where again a suitable dissipativity condition is used as a sufficient condition to establish optimal operation of the system at some optimal time-varying trajectory. 2 We remark that in the stabilizing MPC setting of Section 2, i.e., with positive definite stage cost l, strict dissipativity with respect to the supply rate s(x, u) = l(x, u) l(x, u ) trivially follows with λ 0.

8 46 SICE JCMSI, Vol. 10, No. 2, March Closed-Loop Convergence Having established a systems-theoretic characterization of optimal operating behaviors, we now discuss whether the closed-loop system (5) resulting from application of an economic MPC scheme (Algorithm 1) converges to the optimal operating behavior. Again, the exposition follows the basic case where steady-state operation is optimal, and we subsequently discuss available extensions to more general cases. In case that steady-state operation is optimal, it is desirable that the closed-loop system converges to the optimal steadystate x.again,different economic MPC schemes are available in the literature, in particular again with and without an additional terminal constraint and/or terminal cost in Problem 1. The main difficulty in both approaches is the following: since the stage cost l is in general not positive definite with respect to x, the optimal value function JN 0 might increase along closed-loop trajectories and can therefore not be used as a Lyapunov function, as was done in Section 2. Convergence to the optimal steady-state was first established in [88] for linear systems and strictly convex cost functions, by using convexity arguments. A major step forward in the general nonlinear case was done in [89] under a strong duality condition, which was later generalized in [75] to a strict dissipativity condition, both using a terminal equality constraint (i.e., X f = {x }). Namely, in these references the rotated cost function L(x, u) := l(x, u) l(x, u ) + λ(x) λ( f (x, u)) (15) is considered. Under the assumption that system (1) is strictly dissipative with respect to the supply rate s(x, u) = l(x, u) l(x, u ), from (14) it directly follows that the rotated stage cost L is positive definite with respect to x. Furthermore, it can be shown that when replacing the cost function l in (4) by the rotated cost L, for this modified optimization problem one still obtains the same solution as for the original optimization problem. This is due to the fact that the constraints are the same, and one can show that thanks to the terminal equality constraint, the cost functions only differ by a constant term. This implies that for the convergence analysis, the modified optimization problem using the rotated stage cost function L can be used. But since L is positive definite as discussed above, results from standard stabilizing MPC can be used in order to establish closed-loop asymptotic convergence to the optimal steady-state x. In case that a (nontrivial) terminal region instead of a terminal equality constraint shall be used, again Assumption 1 (with l(x,κ f (x)) on the right hand side of iii) replaced by l(x,κ f (x)) + l(x, u )) has to be satisfied. Again, since l is in general not positive definite with respect to x,a different procedure than the one described in Subsection 2.1 has to be used to calculate a suitable terminal region and terminal cost satisfying Assumption 1 as shown in [90] (see also [87] for a continuous-time version). In economic MPC schemes without terminal constraints, the main insight employed in order to establish closed-loop convergence to x is that the optimal solutions to Problem 1 exhibit a certain turnpike behavior. Here, turnpike behavior [91] [93] means that the open-loop optimal trajectories x 0 resulting from application of optimal solutions u 0 to Problem 1 spend most of the time in a neighborhood E of x, where (an upper bound on) the number of time instants outside this neighborhood depends on the size of E, but is independent of the prediction horizon N. In [94],[95], it was shown that strict dissipativity with respect to the supply rate s(x, u) = l(x, u) l(x, u )together with suitable controllability assumptions implies such a turnpike behavior (see also [80] for a continuous-time version); converse statements were recently studied in [96]. Using this turnpike property, (practical) asymptotic stability of the optimal steady-state x for the closed-loop system (5) is shown in [97], using again the optimal value function of a suitably modified optimization problem as a (practical) Lyapunov function (see also [98] for a continuous-time version). Here, the size of the neighborhood into which the closed loop converges depends on the prediction horizon N, and goes to zero as N. Interestingly, a candidate solution ũ(t + 1) for the next time instant t +1 is constructed by appending an additional control value not necessarily at the end as described in Subsection 2.1 (and as is also the case in economic MPC with terminal constraints), but at some point where the optimal open-loop trajectory predicted at time t, x 0 (t), is close to x. Existence of such a time point is guaranteed by the turnpike property. Similar to Subsection 4.1, generalizations of the above results are available to cases where different operating behaviors than steady-state operation are optimal. In case of optimal periodic operation, closed-loop convergence to the optimal periodic orbit was established in [85],[99] [102] using suitable (periodic) terminal constraints and in [86] without such additional constraints exploiting a certain periodic turnpike property. Furthermore, in the case of time-varying systems and cost functions, closed-loop convergence to some optimal time-varying optimal trajectory is established in [87] using again a suitable time-varying dissipativity condition and suitable time-varying terminal constraints. 4.3 Closed-Loop Performance As discussed above, the primary objective in economic MPC is not the stabilization of an a priori given setpoint (or trajectory), but rather the optimization of some general performance criterion. Hence closed-loop performance guarantees are of particular interest. The first such result was obtained for a setting with terminal (equality) constraint [75]. Namely, there it was shown that when using the optimal steady-state x as a terminal equality constraint, the asymptotic average performance of the resulting closed-loop system (5) is better than or equal to the optimal steady-state cost, i.e., Tt=0 l(x(t), u MPC (x(t))) lim sup l(x, u ). (16) T T + 1 This result was extended in [90] to a setting using a terminal region instead of a terminal equality constraint. Furthermore, in [75] it was also shown that when using an arbitrary periodic orbit as a periodic terminal constraint, again (16) is satisfied with l(x, u ) on the right hand side replaced by the average cost along this periodic orbit. The proof of (16) uses the same candidate input sequence ũ(t+1) as defined in Subsection 2.1 to show that JN 0 (x(t + 1)) J0 N (x(t)) l(x(t), umpc (t)) + l(x, u ), from which under some mild technical assumptions (16) follows. In case that steady-state operation is optimal (compare Subsection 4.1), (12) together with (16) implies that the closedloop exhibits optimal asymptotic average performance. Similar results are also available for economic MPC schemes without terminal constraints. Namely, for both the cases of opti-

9 SICE JCMSI, Vol. 10, No. 2, March mal steady-state [94] and optimal periodic operation [86], the same conditions which allow to conclude practical convergence to the optimal operating behavior (see Subsection 4.2) also allow to conclude that the closed-loop asymptotic average performance is approximately optimal (with vanishing error term as N ). While asymptotic average performance results allow for statements about the long-term behavior of the closed-loop system, they do not allow for statements about transient (finitetime) performance. Transient performance estimates were obtained for the case where steady-state operation is optimal for both settings with [103] and without [97] terminal constraints. Namely, there it was shown that among all trajectories which steer the system into a certain neighborhood of the optimal steady-state in a certain number of steps K, the closed-loop (5) yields the best performance over this time interval up to an error term depending on K and N (see [97],[103] for a more precise statement). This means that the closed-loop also exhibits an approximately optimal transient performance. The generalization of this result to the case where steady-state operation is not optimal is still open. A different finite-time performance estimate was obtained in [104]. There, the authors establish that the finite-time performance of a certain shrinking-horizon economic MPC scheme is at least as good as the one of a certain stabilizing MPC controller. 4.4 Discussion The results of the preceding subsections show that dissipativity plays a crucial role in the context of economic MPC. Namely, certain (strict) dissipativity conditions can both be used to classify different optimal operating behaviors as well as to show that the closed loop converges there as desired. In [89],[95], it is shown that a linear system with a strictly convex stage cost l subject to convex constraints (with nonempty interior) is strictly dissipative with respect to the supply rate s(x, u) = l(x, u) l(x, u ) (using a linear storage function λ) and hence optimally operated at steady-state. For general nonlinear systems, computing a suitable storage function λ in order to verify the above dissipativity conditions can be a difficult task, and systematic approaches are only available for certain special system classes such as, e.g., sum of squares techniques for polynomial systems [105]. On the other hand, explicit knowledge of a storage function λ is not necessary for the implementation of economic MPC, and the results of the previous subsections also allow (at least partially) to conclude a desired closed-loop behavior without having to verify the strict dissipation inequality (14). Namely, if the optimal operating behavior for a system is steady-state or periodic operation (in its strict form), then the corresponding strict dissipativity condition is satisfied (cf. Subsection 4.1), which in turn can be used to conclude that the closed-loop system converges to the optimal steady-state / periodic orbit (cf. Subsection 4.2). Loosely speaking, this means that the closed-loop system does the right thing, i.e., it finds the optimal operating behavior. However, in a setting with terminal constraints, these conclusions are limited to the case where the optimal steady-state / periodic orbit is known a priori, since it has to be used as a terminal constraint 3. This is not the case when using an economic MPC scheme without terminal constraints; here, however, the closedloop convergence results of Subsection 4.2 crucially depend on the assumption that an optimal solution to Problem 1 can be found, which might not be the case in the general nonlinear (nonconvex) case. 4.5 Further Results We now briefly discuss some further available results in the field of economic MPC. Lyapunov-based economic MPC schemes are, e.g., developed in [104],[106],[107]. Here, constraints based on a known control Lyapunov function are added to the repeatedly solved optimization problem to ensure that the system stays within a certain stability region or converges there. Economic MPC using a generalized terminal constraint was, e.g., considered in [101],[108] [111]. Here, the predicted terminal state is not required to be equal to the optimal steadystate or periodic orbit, but only to some (arbitrary) steadystate [108] [110] or periodic orbit [101],[111]. Advantages compared to standard terminal constraints are that the optimal steady-state/periodic orbit need not be known a priori, and the feasible set can be larger. Closed-loop performance and convergence guarantees can again be established under suitable conditions. Generalizations of some of the above results to time-varying cost functions, systems, and/or constraints were, e.g., developed in [87],[107],[109],[112],[113]. Besides standard pointwise-in-time constraints (2), also (less restrictive) average constraints can be of interest in economic MPC. In [75],[114],[115] it was shown how Problem 1 can be modified such that the resulting closed-loop system satisfies these constraints. The articles [116],[117] consider combined economic and tracking cost functions and establish closed-loop convergence or ultimate boundedness depending on how the economic and tracking part are weighted. For the case where steady-state operation is optimal, a tracking MPC problem is formulated in [118],[119] which is shown to be (locally) equivalent to economic MPC. The above results have been developed for nominal systems, i.e., without uncertainties or disturbances. As was discussed in [120], just transferring robust MPC approaches from a stabilizing to an economic context might not result in an optimal performance, and hence novel approaches have to be developed. Some robust and stochastic economic MPC schemes can, e.g., be found in [111],[120] [126]. Some of these references focus on applications, while others establish theoretical properties such as closed-loop performance and convergence guarantees in the presence of disturbances. Concerning applications, economic MPC schemes have successfully been applied in many different fields, and it is well beyond the scope of this article to mention all of these. Examples include, among others, the process industry [106],[127], building climate control [128], the control of wind turbines [129], supply chain management [130], refrigeration systems [131], power systems [132], and the development of sustainable climate policies [133]. Finally, concerning distributed economic MPC, there are only few theoretical results available in the literature. In [134],[135], the approach of [27] was extended to a setting with 3 If a wrong terminal constraint is used (e.g., the optimal steady-state x is used as terminal constraint even though steady-state operation is not optimal), closed-loop convergence to the optimal operating behavior cannot be guaranteed. Nevertheless, the asymptotic average performance result (16) holds.

10 48 SICE JCMSI, Vol. 10, No. 2, March 2017 economic objective function. A distributed economic MPC scheme for output stabilization in a network of competing systems was proposed in [136]. Furthermore, the distributed implementation of a Lyapunov-based economic MPC scheme is considered in [137], and a hierarchical structure is proposed in [138], where a centralized economic MPC problem is solved on the upper level on a slow time scale, while a distributed controller on the lower level is implemented on a faster time scale to react on disturbances. Finally, in [139],[140] a distributed economic MPC scheme is considered for a setting of self-interested systems which have to satisfy some cooperative task, where the overall optimal steady-state is not known a priori, but negotiated online while the systems already take control actions. 5. Conclusions This paper is intended to give an overview of recent developments in the field of model predictive control. A particular focus was set on the areas of distributed and economic MPC, where typically more general control objectives than setpoint stabilization are of interest. In conclusion, both areas have seen significant progress within the last years and various important results have been obtained regarding the design and the analysis of suitable MPC schemes for such settings. Furthermore, an ever-growing number of successful implementations of both distributed and economic MPC schemes in a multitude of different application areas has been reported in the literature. Both fields of distributed and economic MPC are currently still very active fields of research, and many interesting issues deserve further attention in the future. For example, further research is needed in both distributed and economic MPC in the context of uncertain systems or in a time-varying and/or adaptive setting. 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12 50 SICE JCMSI, Vol. 10, No. 2, March 2017 IEEE Transactions on Cybernetics, Vol. 46, No. 9, pp , [62] H. Li and W. Yan: Receding horizon control based consensus scheme in general linear multi-agent systems, Automatica, Vol. 56, pp , [63] Z. Cheng, H.T. Zhang, M.C. Fan, and G. Chen: Distributed consensus of multi-agent systems with input constraints: A model predictive control approach, IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 62, No. 3, pp , [64] L. Zhou and S. Li: Distributed model predictive control for multi-agent flocking via neighbor screening optimization, International Journal of Robust and Nonlinear Control, RNC3606, [65] B.J. Johansson, A. Speranzon, M. Johansson, and K.H. Johansson: Distributed model predictive consensus, Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, pp , [66] T. Keviczky and K.H. 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13 SICE JCMSI, Vol. 10, No. 2, March Journal of Process Control, Vol. 24, No. 8, pp , [98] T. Faulwasser and D. Bonvin: On the design of economic NMPC based on approximate turnpike properties, Proceedings of the 54th IEEE Conference on Decision and Control, pp , [99] R. Huang, E. Harinath, and L.T. Biegler: Lyapunov stability of economically oriented NMPC for cyclic processes, Journal of Process Control, Vol. 21, No. 4, pp , [100] M. Zanon, S. Gros, and M. Diehl: A Lyapunov function for periodic economic optimizing model predictive control, Proceedings of the 52nd IEEE Conference on Decision and Control, pp , [101] D. Limon, M. Pereira, D. Muñoz de la Peña, T. Alamo, and J.M. Grosso: Single-layer economic model predictive control for periodic operation, Journal of Process Control, Vol. 24, No. 8, pp , [102] B. Houska: Enforcing asymptotic orbital stability of economic model predictive control, Automatica, Vol. 57, pp , [103] L. Grüne and A. 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Gros, and M. Diehl: A tracking MPC formulation that is locally equivalent to economic MPC, Journal of Process Control, Vol. 45, pp , [120] F.A. Bayer, M.A. Müller, and F. Allgöwer: Tube-based robust economic model predictive control, Journal of Process Control, Vol. 24, No. 8, pp , [121] F.A. Bayer, M. Lorenzen, M.A. Müller, and F. Allgöwer: Robust economic model predictive control using stochastic information, Automatica, Vol. 74, pp , [122] T.G. Hovgaard, L.F.S. Larsen, and J.B. Jørgensen: Robust economic MPC for a power management scenario with uncertainties, Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pp , [123] R. Huang, L.T. Biegler, and E. Harinath: Robust stability of economically oriented infinite horizon NMPC that include cyclic processes, Journal of Process Control, Vol. 22, No. 1, pp , [124] L.E. Sokoler, B. Dammann, H. Madsen, and J.B. 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14 52 SICE JCMSI, Vol. 10, No. 2, March 2017 [134] J. Lee and D. Angeli: Cooperative distributed model predictive control for linear plants subject to convex economic objectives, Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pp , [135] J. Lee and D. Angeli: Distributed cooperative nonlinear economic MPC, Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems, [136] P.A.A. Driessen, R.M. Hermans, and P.P.J. van den Bosch: Distributed economic model predictive control of networks in competitive environments, Proceedings of the 51st IEEE Conference on Decision and Control, pp , [137] X. Chen, M. Heidarinejad, J. Liu, and P.D. Christofides: Distributed economic MPC: Application to a nonlinear chemical process network, Journal of Process Control, Vol. 22, No. 4, pp , [138] I.J. Wolf, H. Scheu, and W. Marquardt: A hierarchical distributed economic NMPC architecture based on neighboringextremal updates, Proceedings of the American Control Conference, pp , [139] M.A. Müller and F. Allgöwer: Distributed economic MPC: A framework for cooperative control problems, Proceedings of the 19th IFAC World Congress, pp , [140] P.N. Köhler, M.A. Müller, and F. Allgöwer: A distributed economic MPC scheme for coordination of self-interested systems, Proceedings of the American Control Conference, pp , Matthias A. MÜLLER He received a Diploma degree in Engineering Cybernetics from the University of Stuttgart, Germany, and an M.S. in Electrical and Computer Engineering from the University of Illinois at Urbana-Champaign, US, both in In 2014, he obtained a Ph.D. in Mechanical Engineering, also from the University of Stuttgart, Germany, for which he received the 2015 European Ph.D. award on control for complex and heterogeneous systems. He is currently working as a lecturer and postdoctoral researcher (Akademischer Rat) at the Institute for Systems Theory and Automatic Control at the University of Stuttgart. In 2012, he was a visiting researcher at the Imperial College, London, UK. He is a member of the Eliteprogram for PostDocs of the Baden-Württemberg Foundation and was a semi-plenary speaker at the 5th IFAC Conference on Nonlinear Model Predictive Control His research interests include nonlinear control and estimation, model predictive control, distributed control and switched systems. Frank ALLGÖWER He studied Engineering Cybernetics and Applied Mathematics in Stuttgart and at the University of California, Los Angeles (UCLA), respectively, and received his Ph.D. degree from the University of Stuttgart in Germany. He is the Director of the Institute for Systems Theory and Automatic Control and Executive Director of the Stuttgart Research Centre Systems Biology at the University of Stuttgart. His research interests include cooperative control, predictive control, and nonlinear control with application to a wide range of fields including systems biology. At present he is President Elect of the International Federation of Automatic Control IFAC and serves as Vice President of the German Research Foundation DFG.

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