Alejandro Cesar Limache

Distributed Model Predictive Control Based on Dynamic Games

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 (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 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 (Kumar & Daoutidis, 2002; Lu, 2003). 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 (Sandell Jr et al., 1978; Siljak, 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 MPC framework was proposed by Dumbar and Murray (Dunbar & Murray, 2006) for the class 4

On the issue that Finite Element discretizations violate, nodally, Clausius’s postulate of the second law of thermodynamics

Discretization processes leading to numerical schemes sometimes produce undesirable effects. One potentially serious problem is that a discretization may produce the loss of validity of some of the physical principles or mathematical properties originally present in the continuous equation. Such loss may lead to uncertain results such as numerical instabilities or unexpected non-physical solutions. As a consequence, the compatibility of a discrete formulation with respect to intrinsic physical principles might be essential for the success of a numerical scheme. This paper addresses such type of issue. Its main objective is to demonstrate that standard Finite Element discretizations of the heat conduction equation violate Clausius’s postulate of the second law of thermodynamics, at nodal level. The problem occurs because non-physical, reversed nodal heat-fluxes arise in such discretizations. Conditions for compatibility of discrete nodal heat-fluxes with respect to Clausius’s postulate are derived here and named discrete thermodynamic compatibility conditions (DTCC). Simple numerical examples are presented to show the undesirable consequences of such failure. It must be pointed out that such DTCCs have previously appeared in the context of the study of the conditions that make discrete solutions to satisfy the discrete maximum principle (DMP). However, the present article does not put attention on such mathematical principle but on the satisfaction of a fundamental physical one: the second law of thermodynamics. Of course, from the presented point of view, it is clear that the violation of such fundamental law will cause, among different problems, the violation of the DMP.