Flatness-based MPC and observer design for PDE systems

建筑部门被认为是最大的能源消耗部门之一，在建筑生命周期中，80%以上的能源用于建筑运营。有鉴于此，自动化、控制和优化可以被确定为实现建筑节能运行的关键要素。由于建筑物本质上是沿多个时间和空间尺度演化的，数学建模通常会导致以偏微分方程组(PDE)的形式表示系统。在此分布参数系统描述的基础上，本项目旨在开发基于平坦度的偏微分方程优化和模型预测控制(MPC)设计方法。这包括设计状态观测器以实现MPC方案，以及使用适当的降阶模型进行有效的数值实施。利用分布参数系统的平坦性，将具有偏微分方程、状态约束和输入约束的动态最优控制问题归结为平坦输出轨迹及其导数的静态约束最优控制问题，从而推导出最优控制方法。在此基础上，一方面希望解决PDE系统的最优轨迹规划和前馈控制问题。另一方面，这个参数化的静态优化问题构成了发展偏微分方程系统的稳定化预测控制方案的基础。除了将所确定的最优前馈控制与适当的反馈控制策略相结合来稳定分布参数误差系统外，还可以通过引入适当的优化范围来连续地确定平坦输出轨迹中引入的自由度。这导致了一种基于平坦度的滚动域算法，其闭环系统的稳定性将使用Lyapunov技术来解决。这些分析设计技术与近似方法和模型降阶方法相结合，形成了计算效率高的半数值设计方法。这里，将特别强调从PDE系统到ODE表示保持平坦性的方案。观测器的设计是基于BackStep方法和滚动水平估计方法在高维空间域的半线性偏微分方程组上的扩展。这将包括严格的泛函分析公式和结合近似方法来解决实时能力。所开发的方法将以节能建筑运行为例进行评估。为此，在模拟场景中将考虑不同复杂性的模型，从受热控(通风口)影响的单个房间的表示开始，到具有对流的质量和热能交换的相互关联的房间。

The building sector is considered as one of the largest energy consumers with more that 80% of the provided energy spent for building operation during the building life cycle. In view of this, automation, control and optimization can be identified as key ingredients to achieve energy efficient building operation. Since buildings inherently evolve along multiple time and spatial scales, mathematical modeling typically leads to a system representation in terms of partial differential equations (PDEs). Based on this distributed-parameter system description, this project aims at the development of flatness-based optimal and model predictive control (MPC) design methods for PDEs. This includes the design of state observers to enable the realization of the MPC schemes as well as the efficient numerical implementation using appropriate reduced-order models. Here, the flatness property of the distributed-parameter system is exploited to deduce methods for optimal control by reducing the dynamic optimal control problem with PDE, state and input constraints to a constrained static optimal control problem in the flat output trajectory and its derivatives. With this, it on the one hand desired to address optimal trajectory planning and feedforward control for PDE systems. On the other hand, this parametrized static optimization problem constitutes the basis to develop stabilizing MPC schemes for PDE systems. In addition to the combination of the determined optimal feedforward control with suitable feedback control strategies to stabilize the distributed-parameter error system, the degrees-of-freedom introduced in the flat output trajectory can be determined consecutively by introducing an appropriate optimization horizon. This leads to a flatness-based receding horizon algorithm, whose closed-loop stability properties will be addressed using Lyapunov techniques. These analytic design techniques are merged with methods of approximation and model order reduction, which leads to computationally efficient semi-numeric design approaches. Here, special emphasis will be given to schemes that preserve the flatness property from the PDE system to the ODE representation. Observer design is based on the extension of backstepping-based approaches and moving horizon estimation to semilinear PDEs with higher-dimensional spatial domain. This will include a rigorous functional analytic formulation and the incorporation of approximation methods to address real-time capabilities. The developed methods will be evaluated for the example of energy efficient building operation. For this, models of different complexity will be considered in simulation scenarios starting from the representation of a single room influenced by thermal control (vents) to interconnected rooms with convective exchange of mass and thermal energy.