Distributed Model Predictive Control Based on Dynamics Games

Abstract

Model predictive control (MPC) is widely recognized as a high performance, yet practical,control technology. This model-based control strategy solves at each sample a discrete-timeoptimal control problem over a finite horizon, producing a control input sequence. Anattractive attribute of MPC technology is its ability to systematically account for systemconstraints. The theory of MPC for linear systems is well developed; all aspects suchas stability, robustness,feasibility and optimality have been extensively discussed in theliterature (see, e.g., (Bemporad & Morari, 1999; Kouvaritakis & Cannon, 2001; Maciejowski, 2002; Mayne et al., 2000)). The effectiveness of MPC depends on model accuracy and the availability of fast computational resources. These requirements limit the application base for MPC. Even though, applications abound in process industries (Camacho & Bordons, 2004), manufacturing (Braun et al., 2003), supply chains (Perea-Lopez et al., 2003), among others, are becoming more widespread.Two common paradigms for solving system-wide MPC calculations are centralised anddecentralised strategies. Centralised strategies may arise from the desire to operate thesystem in an optimal fashion, whereas decentralised MPC control structures can result fromthe incremental roll-out of the system development. An effective centralised MPC can bedifficult, if not impossible to implement in large-scale systems (Kumar & Daoutidis, 2002;Lu, 2003). In decentralised strategies, the system-wide MPC problem is decomposed intosubproblems by taking advantage of the system structure, and then, these subproblemsare solved independently. In general, decentralised schemes approximate the interactionsbetween subsystems and treat inputs in other subsystems as external disturbances. Thisassumption leads to a poor systemperformance (Sandell Jr et al., 1978; ?iljak, 1996). Therefore, there is a need for a cross-functional integration between the decentralised controllers, in which a coordination level performs steady-state target calculation for decentralised controller (Aguilera & Marchetti, 1998; Aske et al., 2008; Cheng et al., 2007; 2008; Zhu & Henson, 2002).Several distributed MPC formulations are available in the literature. A distributed MPCframework was proposed by Dumbar and Murray (Dunbar & Murray, 2006) for the classof systems that have independent subsystem dynamic but link through their cost functionsand constraints. Then, Dumbar (Dunbar, 2007) proposed an extension of this framework thathandles systemswith weakly interacting dynamics. Stability is guaranteed through the use ofa consistency constraint that forces the predicted and assumed input trajectories to be close toeach other. The resulting performance is different from centralised implementations in mostof cases. Distributed MPC algorithms for unconstrained and LTI systems were proposed in(Camponogara et al., 2002; Jia & Krogh, 2001; Vaccarini et al., 2009; Zhang & Li, 2007). In (Jia & Krogh, 2001) and (Camponogara et al., 2002) the evolution of the states of each subsystem is assumed to be only influenced by the states of interacting subsystems and local inputs, while these restrictions were removed in (Jia & Krogh, 2002; Vaccarini et al., 2009; Zhang & Li, 2007). This choice of modelling restricts the system where the algorithm can be applied, because inmany cases the evolution of states is also influenced by the inputs of interconnected subsystems. More critically for these frameworks is the fact that subsystems-based MPCs only know the cost functions and constraints of their subsystem. However, stability and optimality as well as the effect of communication failures has not been established.The distributed model predictive control problem from a game theory perspective for LTIsystems with general dynamical couplings, and the presence of convex coupled constraintsis addressed. The original centralised optimisation problem is transformed in a dynamicgame of a number of local optimisation problems, which are solved using the relevantdecision variables of each subsystem and exchanging information in order to coordinatetheir decisions. The relevance of proposed distributed control scheme is to reduce thecomputational burden and avoid the organizational obstacles associated with centralisedimplementations, while retains its properties (stability, optimality, feasibility). In this context,the type of coordination that can be achieved is determined by the connectivity and capacity of the communication network as well as the information available of system?s cost function and constraints. In this work we will assume that the connectivity of the communication network is sufficient for the subsystems to obtain information of all variables that appear in their local problems. We will show that when system?s cost function and constraints are known by all distributed controllers, the solution of the iterative process converge to the centralised MPC solution. This means that properties (stability, optimality, feasibility) of the solution obtained using the distributed implementation are the same ones of the solution obtained using the centralised implementation. Finally, the effects of communication failures on the system?s properties (convergence, stability and performance) are studied. We will show the effect of the system partition and communication on convergence and stability, and we will find a upper bound of the system performance.