Flatness-based MPC and observer design for PDE systems

建筑行业被认为是最大的能源消费者之一，在建筑生命周期中，超过80%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节能建筑运营的关键要素。由于建筑物本质上是沿着多个时间和空间尺度沿着发展的，因此数学建模通常会导致用偏微分方程（PDE）表示的系统表示。基于此分布参数系统描述，本项目旨在开发基于平坦度的偏微分方程最优和模型预测控制（MPC）设计方法。这包括状态观测器的设计，使MPC计划的实现，以及使用适当的降阶模型的有效的数值实现。在这里，利用分布参数系统的平坦性，推导出最优控制方法，通过减少PDE，状态和输入约束的动态最优控制问题，在平坦的输出轨迹及其衍生物的约束静态最优控制问题。因此，一方面需要解决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基础，以发展稳定的MPC计划PDE系统。除了将确定的最优前馈控制与适当的反馈控制策略相结合以稳定分布参数误差系统之外，还可以通过引入适当的优化时域来连续地确定在平坦输出轨迹中引入的自由度。这导致了一个平坦的滚动时域算法，其闭环稳定性将使用李雅普诺夫技术解决。这些分析设计技术与近似和模型降阶方法相结合，从而产生计算效率高的半数值设计方法。在这里，将特别强调的计划，保持平坦性从偏微分方程系统的常微分方程表示。观测器的设计是基于扩展的backstepping为基础的方法和滚动时域估计的半线性偏微分方程与高维空间域。这将包括一个严格的功能分析公式，并纳入近似方法，以解决实时能力。所开发的方法将进行评估的例子，节能建筑运营。为此，将在模拟场景中考虑不同复杂性的模型，从受热控制（通风口）影响的单个房间到具有质量和热能对流交换的互连房间。

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.