Model predictive PDE control for energy efficient building operation:Economic model predictive control and time varying systems

供暖、通风和空调（HVAC）设施构成了一类具有巨大节能潜力的控制系统。为了实现这些节省，我们建议使用模型预测控制（MPC）作为基于优化的控制技术。为了获得MPC所需的精确模型，我们打算明确考虑状态变量的时空分布，即，使用基于偏微分方程（PDE）的动态模型。该提案将与托马斯默勒（Thomas Meurer）和斯特凡·斯维温（Stefan Schwein）的合作伙伴提案密切合作，旨在分析和设计空间分布和时变模型的MPC方案，实现相应的算法，并将其应用于联合基准问题。根据预期的能源效率，经济MPC公式将成为提案的重点。这里，“经济MPC”代表一类MPC算法，其中控制目标不是平衡的稳定或时变参考轨迹的跟踪。相反，目标是遵循能量最优路径，该路径不是先验给定的，而是由MPC优化本身的目标隐含地定义的。经济预测控制的关键问题是如何设计目标和约束条件，使得移动时域上的迭代优化在长的、可能无限的时域上产生近似最优的闭环轨迹。为此，在第一个工作包的经济MPC偏微分方程将进行详细研究。关键的困难在于系统动力学是在无限维的状态空间中演化的。我们的目标是开发新的方法，利用HVAC控制问题的特殊结构。在第二个工作包中，考虑了时变系统动态或依赖于时变数据的问题。在这里，特别是，一个适当的推广的概念，最佳平衡必须找到和适当的终端约束，在时变设置应制定。第三个工作包涉及执行。除了将与合作伙伴项目密切合作进行的新开发例程的编码之外，用于稳定MPC的后验性能测量应扩展到经济MPC，作为测量在优化中使用降阶模型引入的误差的装置。所有工作包的结果将被应用，特别是集中在一个基准问题的暖通空调控制。这些活动在第四个工作包中有详细说明，将与其他工作包同时进行。

Heating, Ventilation and Air Conditioning (HVAC) facilities form a class of control systems which have a huge potential for energy savings. In order to realize these savings, we propose to use Model Predictive Control (MPC) as an optimization based control technique. In order to obtain the accurate models needed for MPC, we intend to explicitly take into account the spatio-temporal distribution of the state veriable, i.e., to use dynamic models based on Partial Differential Equations (PDEs). This proposal, which shall be carried out in close cooperation with the partner proposals by Thomas Meurer and Stefan Volkwein, aims at analyzing and designing MPC schemes for spatially distributed and time varying models, at implementing the respective algorithms, and at applying it to a joint benchmark problem. In light of the intended energy efficiency, economic MPC formulations will be in the focus of the proposal. Here "economic MPC" stands for a class of MPC algorithms in which the control objective is not the stabilization of an equilibrium or the tracking of a time varying reference trajectory. Instead, the goal is to follow an energy optimal path which is not given a priori but implicitly defined by the objective of the MPC optimization, itself. The key question in economic MPC is how to design the objective and constraints such that the iterative optimization on moving horizons yields an approximately optimal closed loop trajectory on a long, possibly infinite time horizon. To this end, in the first work package economic MPC for PDEs will be investigated in detail. Key difficult is the fact that the system dynamics evolve in an infinite dimensional state space. Our goal is to develop new methods exploiting the special stuctures of HVAC control problems. In the second work package, problems with time varying system dynamics or depending on time varying data are considered. Here, in particular, an appropriate generalization of the concept of an optimal equilibrium must be found and appropriate terminal constraints in the time varying setting shall be developed. The third work package concerns the implementation. Beyond the coding of the newly developed routines, which will be carried out in close collaboration with the partner projects, the a posteriori performance measure for stabilizing MPC shall be extended to economic MPC, as a device to measure the errors introduced by using reduced order models in the optimization. The results from all work packages will be applied and particularly focused to a benchmark problem for HVAC control. These activities are detailed in the fourth work package and will be carried out in parallel with the other work packages.