Altmann R, Heiland J
1Institut für Mathematik MA4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.
2Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany.
Multibody Syst Dyn. 2017;40(1):75-98. doi: 10.1007/s11044-016-9558-z. Epub 2016 Nov 10.
We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques. By means of the well-known -car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.
我们考虑动力学部分受限于规定轨迹的机械系统。这类系统的一个例子是带有负载的建筑起重机，且要求负载沿特定路径移动。直接以伺服约束的形式强制执行此条件会导致任意高阶指数的微分代数方程（DAE）。通常，模型方程的指数为5，这已经带来了很高的正则性条件。如果我们放宽伺服约束并从最优控制的角度考虑该系统，强正则性条件就会消失，并且可以通过标准技术获得解。借助著名的“ - 车”示例和桥式起重机，说明了直接DAE和替代建模方法在理论上和预期的数值困难。我们展示了在最优控制背景下问题的表述方式，并讨论了最优控制系统的可解性。我们讨论了有问题的DAE行为在最优控制系统中仍然是固有的，并展示了其表现如何依赖于优化的正则化参数。
