A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> This paper presents a novel Distributed Predictive Control (DPC) algorithm for linear discrete-time systems. This method enjoys the following properties: (i) state and input constraints can be considered; (ii) under mild assumptions, convergence of the closed loop control system is proved; (iii) it is not necessary for each subsystem to know the dynamical models of the other subsystems; (iv) the transmission of information is limited, in that each subsystem only needs the reference trajectories of the state variables of its neighbors. A simulation example is reported to illustrate the main characteristics and performance of the algorithm. <s> BIB001 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> In this paper we consider a linear system structured into physically coupled subsystems and propose an innovative distributed control scheme capable of guaranteeing asymptotic stability and satisfaction of constraints on system inputs and states. Our method hinges on the availability of a decentralized stabilizing regulator for the unconstrained system and provides a two-layer controller for each subsystem. Upper controllers receive planned state trajectories from neighboring subsystems and exploit the notion of tubes (Langson, Chryssochoos, Rakovic, & Mayne, 2004) for achieving robustness of stability with respect to coupling. Lower controllers generate planned trajectories using Model Predictive Control (MPC) independently of the other subsystems. The proposed control scheme is arguably easier to design and apply than existing distributed controllers with similar features. A comparison of the proposed approach with existing centralized and distributed MPC regulators is conducted using an illustrative example. <s> BIB002 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> Dual decomposition is a powerful technique for deriving decomposition schemes for convex optimization problems with separable structure. Although the Augmented Lagrangian is computationally more stable than the ordinary Lagrangian, the prox-term destroys the separability of the given problem. In this paper we use another approach to obtain a smooth Lagrangian, based on a smoothing technique developed by Nesterov, which preserves separability of the problem. With this approach we derive a new decomposition method, called proximal center algorithm, which from the viewpoint of efficiency estimates improves the bounds on the number of iterations of the classical dual gradient scheme by an order of magnitude. This can be achieved with the new decomposition algorithm since the resulting dual function has good smoothness properties and since we make use of the particular structure of the given problem. <s> BIB003 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> We propose a distributed optimization algorithm for mixed L"1/L"2-norm optimization based on accelerated gradient methods using dual decomposition. The algorithm achieves convergence rate O(1k^2), where k is the iteration number, which significantly improves the convergence rates of existing duality-based distributed optimization algorithms that achieve O(1k). The performance of the developed algorithm is evaluated on randomly generated optimization problems arising in distributed model predictive control (DMPC). The evaluation shows that, when the problem data is sparse and large-scale, our algorithm can outperform current state-of-the-art optimization software CPLEX and MOSEK. <s> BIB004 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> In this paper, a novel distributed Kalman filter (KF) algorithm along with a distributed model predictive control (MPC) scheme for large-scale multi-rate systems is proposed. The decomposed multi-rate system consists of smaller subsystems with linear dynamics that are coupled via states. These subsystems are multi-rate systems in the sense that either output measurements or input updates are not available at certain sampling times. Such systems can arise, e.g., when the number of sensors is smaller than the number of variables to be controlled, or when measurements of outputs cannot be completed simultaneously because of practical limitations. The multi-rate nature gives rise to lack of information, which will cause uncertainty in the system's performance. To circumvent this problem, we propose a distributed KF-based MPC scheme, in which multiple control and estimation agents each determine actions for their own parts of the system. Via communication, the agents can in a cooperative way take one another's actions into account. The main task of the proposed distributed KF is to compensate for the information loss due to the multi-rate nature of the systems by providing optimal estimation of the missing information. A demanding two-area power network example is used to demonstrate the effectiveness of the proposed method. <s> BIB005 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> In distributed model predictive control (DMPC), where a centralized optimization problem is solved in distributed fashion using dual decomposition, it is important to keep the number of iterations in the solution algorithm, i.e. the amount of communication between subsystems, as small as possible. At the same time, the number of iterations must be enough to give a feasible solution to the optimization problem and to guarantee stability of the closed loop system. In this paper, a stopping condition to the distributed optimization algorithm that guarantees these properties, is presented. The stopping condition is based on two theoretical contributions. First, since the optimization problem is solved using dual decomposition, standard techniques to prove stability in model predictive control (MPC), i.e. with a terminal cost and a terminal constraint set that involve all state variables, do not apply. For the case without a terminal cost or a terminal constraint set, we present a new method to quantify the control horizon needed to ensure stability and a prespecified performance. Second, the stopping condition is based on a novel adaptive constraint tightening approach. Using this adaptive constraint tightening approach, we guarantee that a primal feasible solution to the optimization problem is found and that closed loop stability and performance is obtained. Numerical examples show that the number of iterations needed to guarantee feasibility of the optimization problem, stability and a prespecified performance of the closed-loop system can be reduced significantly using the proposed stopping condition. <s> BIB006 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> Due to the ubiquitous existence of external disturbances, the design of distributed control algorithms with robustness is an urgent demand for multi-agent system applications. This technical note investigates the robust distributed model predictive control (MPC) problem for a group of nonlinear agents (subsystems) subject to control input constraints and external disturbances. A robustness constraint is proposed to handle the external disturbances. Based on this, a novel robust distributed MPC scheme is designed for the overall agent system. Furthermore, the feasibility of the robust distributed MPC scheme and the robust stability of the overall agent system are analyzed, respectively. The conditions under which the proposed MPC is feasible and the overall agent system is robustly stabilized are established. Finally, the application of the robust distributed MPC to a group of cart-damper-spring systems verifies the theoretical results. <s> BIB007 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> Abstract We address the problemof controlling interconnected, possibly large-scale systems of systems using distributed model predictive control. We consider the case in which the nonlinear subsystems can be coupled physically and can have shared constraints. The presented approach is based on the transmission of contracts between neighboring subsystems. Contracts are guaranteed sequences of possible future trajectories or trajectory sets of the coupling variables of the subsystems. We derive for the approach sufficient conditions for guaranteeing recursive feasibility and Input-to-State stability. Furthermore, we discuss the case of the so-called Plug & Play operations in which a subsystem in the network is replaced by a new, possibly different, one and when a subsystem is removed or added to the network. <s> BIB008 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> In this letter, a distributed model predictive control strategy for battery energy storage systems is proposed to regulate voltage in distribution network with high-renewable penetration. Control actions are calculated based on communication between interconnected neighboring subsystems and a multistep receding optimization, also considering system and battery constraints. The proposed approach is shown to be highly effective through a simulation case study, indicating high potential for applications. <s> BIB009 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> This technical note develops a new form of distributed stochastic model predictive control (DSMPC) algorithm for a group of linear stochastic subsystems subject to additive uncertainty and coupled probabilistic constraints. We provide an appropriate way to design the DSMPC algorithm by extending a centralized SMPC (CSMPC) scheme. To achieve the satisfaction of coupled probabilistic constraints in a distributed manner, only one subsystem is permitted to optimize at each time step. In addition, by making explicit use of the probabilistic distribution of the uncertainties, probabilistic constraints are converted into a set of deterministic constraints for the predictions of nominal models. The distributed controller can achieve recursive feasibility and ensure closed-loop stability for any choice of update sequence. Numerical examples illustrate the efficacy of the algorithm. <s> BIB010 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> Abstract In this paper, a distributed model predictive control scheme is proposed for linear, time-invariant dynamically coupled systems. Uniquely, controllers optimize state and input constraint sets, and exchange information about these–rather than planned state and control trajectories–in order to coordinate actions and reduce the effects of the mutual disturbances induced via dynamic coupling. Mutual disturbance rejection is by means of the tube-based model predictive control approach, with tubes optimized and terminal sets reconfigured on-line in response to the changing disturbance sets. Feasibility and exponential stability are guaranteed under provided sufficient conditions on non-increase of the constraint set parameters. <s> BIB011 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> In this paper, we consider the problem of coordinating self-interested interacting dynamical systems by means of a distributed economic MPC framework. The self-interest of the systems is reflected by an individual local objective function each agent is trying to minimize, while cooperation is required with respect to coupling constraints and an asymptotic cooperative goal, which is represented by a particular steady state of the overall system. Our basic premise is that this steady state, which fulfills the cooperative goal, is not known a priori but has to be negotiated online, while already taking control actions. For the purpose of determining this steady state in a distributed way, arbitrary distributed computation algorithms can be incorporated into the proposed framework. We show that satisfaction of coupling constraints and convergence to the desired overall steady state can be established. Examples for an asymptotic cooperative goal include synchronization under conflicting objectives or sensor coverage, which are both studied in the work at hand and are also illustrated by numerical simulations. <s> BIB012 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> A distributed model predictive control strategy is proposed for subsystems sharing a limited resource. Self-organized Time Division Multiple Access is used to coordinate subsystem controllers in a sequence such that no two re-optimize simultaneously. This new approach requires no central coordination or pre-organized optimizing sequence. The scheme guarantees satisfaction of coupled constraints despite dynamic entry and exit of subsystems. <s> BIB013 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> This paper proposes a distributed model predictive control (MPC) approach for a family of discrete-time linear systems with local (uncoupled) and global (coupled) constraints. The proposed approach is based on the dual problem of an overall MPC optimization problem involving all systems, which is then solved distributively using a modified distributed Nesterov-accelerated-gradient algorithm. To further reduce the computational requirement, this approach allows for early termination of the distributed gradient algorithm. This is made possible via a consensus algorithm that determines the satisfaction of the termination condition and by appropriate tightening of the coupled constraints. Under reasonable assumptions, the approach is able to produce a suboptimal solution as long as the network of the systems is connected while ensuring recursive feasibility and exponential stability of the closed-loop system. The performance of the proposed approach is demonstrated by a numerical example. <s> BIB014 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> A two-layer control scheme based on model predictive control (MPC) operating at two different timescales is proposed for the energy management of a grid-connected microgrid (MG), including a battery, a microturbine, a photovoltaic (PV) system, a partially non predictable load, and the input from the electrical network. The high-level optimizer runs at a slow timescale, relies on a simplified model of the system, and is in charge of computing the nominal operating conditions for each MG component over a long time horizon, typically one day, with sampling period of 15 min, so as to optimize an economic performance index on the basis of available predictions for the PV generation and load request. The low-level controller runs at higher frequency, typically 1 min, relies on a stochastic MPC (sMPC) algorithm, and adjusts the MG operation to minimize the difference, over each interval of 15 min, between the planned energy exchange and the real one, so avoiding penalties. The sMPC method is implemented according to a shrinking horizon strategy and ensures probabilistic constraints satisfaction. Detailed models and simulations of the overall control system are presented. <s> BIB015 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> This note proposes a distributed model predictive control (DMPC) scheme with switched cost functions for a class of spatially interconnected systems with communication constraints. Non-iterative and parallel communication strategy is considered to ensure that all distributed controllers complete input updates at each single information exchange step. The proposed DMPC scheme switches the optimization index on a switching surface generated by control invariant sets. With the index-switching strategy, stability of the origin is ensured by a terminal control law. Convergence conditions of the optimal cost to zero are established taking into account the causal link between the presumed trajectory and the optimized trajectory of the previous step. The compatibility constraints preserve the quadratic program property that is desired in practical applications. It is also observed that the proposed DMPC scheme has benefits dealing with systems that need to take into account safety-related spatial constraints. <s> BIB016 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> 1) Case With Coupled Constraints: <s> In many of today's control systems, the control loop is closed over a communication network with limited resources that requires explicit consideration of the network utilization in the controller design. At first, this paper deals with the decentralized control of discrete-time linear time-invariant systems consisting of multiple dynamically coupled subsystems subject to state and input constraints where the sensors send the measurements over a network to the controllers. A novel design for an event trigger mechanism for reducing the communication effort over the network in combination with an output-based model predictive controller is proposed. The approach is enhanced with another event trigger for a configuration where additionally the controllers send the inputs to the actuators over the network. Furthermore, it is discussed how the scheme can be extended for the control of sampled-data systems and how the network imperfections bounded time-varying delays, and sampling intervals can be considered in the design. Recursive feasibility of the control law and exponential stability of the origin are guaranteed for all considered setups. Illustrative examples demonstrate the clear reduction of the communication effort compared to a time-triggered output-based model predictive controller. <s> BIB017
In many practical engineering, lots of distributed systems, such as power systems and cooperative robot systems, are governed by a group of uncoupled system dynamics but a possibly coupled constraint on global system states and control inputs BIB012 . This global constraint can be described by where m is the number of subsystems (see BIB013 for more details). Recently, it is disclosed that the optimality of the overall system is dependent on a dual problem of the Lagrangian function. Under this dual problem, dual variables associated with the abovementioned constraint can be regarded as consensus variables and therefore the solution of DMPC can be transformed into a distributed consensus optimization problem, which can be dealt with decomposition methods BIB003 , alternating direction multiplier methods BIB014 , dual accelerated gradient methods BIB004 , and other optimization-based approaches. Among others, serial DMPC schemes BIB013 , in which one subsystem is optimized at a time slot while holding all others constant. In other words, local optimization problems are solved sequentially in each subsystem, where neighboring interaction and/or solutions have to be taken into consideration. Unfortunately, the optimality applying this kind of methods is unclear although it offers greater flexibility in communication and computation. DMPC with probabilistic coupled constraints also becomes a concern BIB010 , BIB015 , and the corresponding form is P where b is a known threshold and p is the permitted probability. In contrast to deterministic forms, addressed issues allow constraint violations with the occurring probability below a prespecified threshold. For this topic, the stochastic tube approach embedded in serial DMPC schemes is utilized to ensure the probabilistic constraint while satisfying the recursive feasibility and closed-loop stability (see BIB010 and the references therein). 2) Case With Coupled Dynamics: As is well known, the resultant feature of interconnection leads to significant difficulties for the application of model predictive control (MPC) in largescale systems with coupled dynamics. The impact from neighboring states and/or control is generally handled by regarding them as a bounded mutual disturbance when performing the prediction of system behavior by resorting to both a nominal model of each subsystem and local interactions. Specifically, each subsystem transmits first its future reference trajectory to its neighbors while guaranteeing that the actual trajectory lies within a certain bound BIB001 . Some standard MPC tools are then employed to solve the addressed local optimization problem with various communication constraints. This route gives rise to some interesting tube DMPC approaches (see BIB002 for more details). It is worth pointing out that most existing approaches can satisfy the primary concern on feasibility and stability, but expose two critical drawbacks: the conservatism induced by the boundedness of the mutual disturbance and the deficiency in global performance due to the local optimization. In order to overcome the conservatism drawback, some improved DMPC schemes are developed in recent years, such as DMPC with sharing contract sets BIB008 , DMPC with terminal sets reconfigured online BIB011 , and DMPC with switched cost functions BIB016 . At the same time, some prediction schemes are proposed to compensate the missing variables caused by event-triggered protocols and multirate mechanism in BIB017 and BIB005 . Furthermore, to improve the global performance, some iterative coordination-based algorithms are proposed to solve a centralized optimization problem in a distributed fashion BIB006 . 3) Case With Coupled Cost Functions: The essential difference from above-mentioned two cases is that the connection among subsystems is realized by means of coupled cost functions BIB014 , BIB009  For instance, a dual-mode MPC strategy, which satisfies the feasibility and the stability for overall system, is developed in BIB007 to increase the robustness against the external disturbances. Note that the above result is dependent on a necessary assumption that a state-feedback control law of uncoupling subsystems is predetermined such that there is a constraint admissible invariant set, which plays an important role to guarantee the stabilization of closed-loop systems and the feasibility of developed DMPC schemes BIB009 .