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-time optimal control problem over a finite horizon, producing a control input sequence. An attractive attribute of MPC technology is its ability to systematically account for system constraints. The theory of MPC for linear systems is well developed; all aspects such as stability, robustness,feasibility and optimality have been extensively discussed in the literature. 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, manufacturing, supply chains, among others, are becoming more widespread. Two common paradigms for solving system-wide MPC calculations are centralised and decentralised strategies. Centralised strategies may arise from the desire to operate the system in an optimal fashion, whereas decentralised MPC control structures can result from the incremental roll-out of the system development. An effective centralised MPC can be difficult, if not impossible to implement in large-scale systems. In decentralised strategies, the system-wide MPC problem is decomposed into subproblems by taking advantage of the system structure, and then, these subproblems are solved independently. In general, decentralised schemes approximate the interactions between subsystems and treat inputs in other subsystems as external disturbances. This assumption leads to a poor systemperformance. 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. Several distributed MPC formulations are available in the literature. A distributed MPC framework was proposed by Dumbar and Murray for the class of systems that have independent subsystem dynamic but link through their cost functions and constraints. Then, Dumbar proposed an extension of this framework that handles systemswith weakly interacting dynamics. Stability is guaranteed through the use of a consistency constraint that forces the predicted and assumed input trajectories to be close to each other. The resulting performance is different from centralised implementations in most of cases. Distributed MPC algorithms for unconstrained and LTI systems were proposed. 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 then removed. 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 LTI systems with general dynamical couplings, and the presence of convex coupled constraints is addressed. The original centralised optimisation problem is transformed in a dynamic game of a number of local optimisation problems, which are solved using the relevant decision variables of each subsystem and exchanging information in order to coordinate their decisions. The relevance of proposed distributed control scheme is to reduce the computational burden and avoid the organizational obstacles associated with centralised implementations, 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.