Efficient calculation of explicit model predictive control laws using topological equivalence classes of critical points

建立了模型预测控制方法。一种称为显式预测控制(EMPC)的较新变体与其他预测控制方法的不同之处在于，不需要在控制器运行时进行数值优化。由于在运行时不需要迭代算法，EMPC适用于具有非常高的采样频率和严格的实时要求的系统。然而，EMPC只能应用于约束较少、预测范围较短的低阶线性系统。无论是EMPC控制律的计算，还是这些控制律的快速评估，目前都是世界各地的研究人员正在研究的问题。目前，在快速计算方法取得进展后，显式控制律的实际计算是瓶颈。本项目的目标是设计一种新的方法(简称顶点法)来计算具有线性输入、状态和输出约束的线性系统的显式控制律。这一目标的三个方面可以具体如下：(1)开发一种计算复杂显式控制律的健壮方法：顶点法是计算量大的。它与现有的方法具有相同的特点。然而，与现有的方法相比，该方法所需的计算更简单，而且可以更可靠地进行。假设所提出的方案能够实施，顶点方法将是计算量较大的，但由于其简单和可靠，可以应用于比现有方法更苛刻的MPC问题。此外，顶点法以一种特别适合于最近提出的快速估计EMPC律的方法的形式提供了显式控制律。(2)显式律通用化描述的基础：虽然项目的第一阶段只涉及线性系统，但具有范式和等价类的顶点的描述是长期感兴趣的，因为它原则上适合于描述非线性MPC问题的显式控制律的结构。顶点法是基于分叉奇点理论的中心思想，该理论是一种起源于非线性的理论。(3)在线预测控制与显式预测控制相结合的在线预测控制加速：项目的一部分研究了一种利用在线预测控制中显式控制律结构的新方法。在控制器的运行时不使用实际的显式控制律，但其结构用于简化在线预测控制问题。核心思想不是存储关于哪些约束对于每个x(T)是活动的信息，而是存储关于哪些约束是不活动的信息。后一种信息可以与多面体一起存储。这些多面体一般不是凸多面体，但这些多面体的数量只随着控制问题的大小线性增长，这与EMPC中计算工作量和存储的指数增长形成了对比。

Model predictive control (MPC) methods are established. A more recent variant known as explicit MPC (EMPC) differs from other MPC approaches in that no numerical optimization is required at runtime of the controller. Because no iterative algorithms are required at runtime, EMPC is suitable for systems with very high sampling frequencies and strict realtime requirements. EMPC can only be applied, however, to low order linear systems with few constraints and short prediction horizons. Both, the calculation of EMPC control laws, and the fast evaluation of these laws, are currently investigated by researchers worldwide. Today the first step, the actual calculation of the explicit laws, is the bottleneck, after methods for the fast evaluation have recently made progress.It is the goal of the project to devise a new method (the vertex-method for short) for the calculation of explicit control laws for linear systems with linear input, state and output constraints. Three aspects of this goal can be detailed as follows: (1) Development of a robust method for the calculation of complex explicit control laws: The vertex-method is computational demanding. It shares this trait with existing approaches. The calculations required in the proposed method are, however, simpler and can be carried out more reliably than for the existing methods. Assuming the proposed project can be carried out, the vertex-method will be computational demanding but can be applied to more demanding MPC problems than current methods due to its simplicity and reliability. Moreover, the vertex-method provides the explicit control law in a form which is particularly suitable for the recently proposed approaches to the fast evaluation of EMPC laws. (2) Foundations of a generalizable description of explicit laws: While the first period of the project exclusively deals with linear systems, the description of vertices with normal forms and equivalence classes is of long-term interest, because it is in principle suitable to describe the structure of explicit control laws of nonlinear MPC problems. The vertex-method is based on central ideas of the singularity theory of bifurcations, which is a nonlinear theory by origin. (3) Acceleration of online MPC by combining explicit and online MPC: A part of the project investigates a new approach to exploiting the structure of an explicit control law in online MPC. The actual explicit control law is not used at the runtime of the controller, but its structure is used to simplify the online MPC problem. The central ideas is not to store information about which constraints are active for every x(t), but about which constraints are inactive. The latter information can be stored with polytopes. These polytopes are not convex in general, but the number of these polytopes grows only linearly in the size of the control problem, in contrast to the exponential growth of computational effort and storage in EMPC.