Theory and Methods of Control and Optimization of Dynamical Systems for Engineering Applications

这个德俄合作项目的目标是开发分析和数值方法，用于俄罗斯合作伙伴提出和研究的某些动态过程的优化和最优控制，这些动态过程显示出对工程应用重要的复杂特性。在没有外力的阻性环境中，机器人仅通过移动内部质量或改变构型来进行运动，会导致具有状态相关不连续性的机械控制问题，特别是由于库仑摩擦以及大量的控制和状态约束。欠驱动多刚体系统具有粘弹性、状态部分可测、系统参数不确定和外部摄动等特点，需要快速反馈控制律来实现高精度定位。高速(超音速)客机的节油飞行规划需要建立复杂的长程混合整数控制模型，其中包含大量的状态约束和控制约束，考虑了技术约束和乘客的舒适性和安全性。为了满足这些特殊的要求，将开发新的解析和数值最优控制方法。Heidelberg团队将考虑一类相对一般的非线性最优控制问题(OCP)，该问题具有整数值控制，由约束边界条件以及状态和控制约束的常微分方程或DAE控制。动力学的右侧可能在依赖于状态的开关函数的零点处不连续，特别是由库仑摩擦引起的。原则上，这类最优控制问题可以通过扩展庞特里亚金最大值原理(PMP)来解决。然而，在上述情况下，间接PMP导致了带有开关函数的多点边值问题和伴随变量的额外跳跃，这是非常困难的。因此，拟议的项目将建立在海德堡小组开发的直接多次激发方法的基础上，以数字方式解决这些问题。将开发两种处理库仑摩擦的方法，一种是“正向积分”方法，另一种是“析取编程”方法。对于具有不确定性的系统，最优反馈控制将通过“非线性模型预测控制”和“移动水平估计”相结合的快速多层迭代来计算，并将其推广到上述一般的OCP类。我们将开发一种新的解析方法来推导快速、局部有效的分段仿射反馈控制律，它将进一步显著减少采样次数，将其与PMP方法进行比较，以计算“相邻反馈”控制，并开发可能的交叉技术。专门的新数值方法的开发将与具有挑战性的工程问题的分析和解决齐头并进，并将在莫斯科和海德堡团队的密切合作下进行。

This German-Russian collaboration project aims at the development of both analytical and numerical methods for optimization and optimal control of certain classes of dynamical processes, proposed and studied by the Russian partners, which exhibit complex characteristic properties important for engineering applications. Robotic locomotion in resistive environment without external forces only by moving internal masses or configuration changes leads to mechanical control problems with state-dependent discontinuities, especially due to Coulomb friction, and numerous control and state constraints. Underactuated multibody systems with visco-elastic components, only partially measurable states, uncertain system parameters and external perturbations call for fast feedback control laws to achieve high-precision positioning. Fuel efficient flight planning of high-speed (supersonic) passenger aircraft leads to complex models of longhorizon mixed-integer control problems with numerous state and control constraints accounting for technical restrictions and passenger comfort and safety. In order to meet these particular requirements new analytical and numerical optimal control methods will be developed. The Heidelberg team will consider the relatively general class of nonlinear optimal control problems (OCP) with integer-valued controls and governed by ODE or DAE subject to boundary conditions and state and control constraints. The right hand sides of the dynamics may be discontinuous at zeros of state-dependent switching functions, arising particularly from Coulomb friction. In principle, such optimal control problems can be solved by extensions of Pontryagin’s Maximum Principle (PMP). However, in the above case the indirect PMP results in multipoint boundary value problems with switching functions and additional jumps in the adjoint variables, which are extremely difficult to solve. The proposed project will therefore build on the direct multiple shooting approach developed by the Heidelberg group to solve such problems numerically. Two ways to handle Coulomb friction will be developed, a “forward integration” approach and a novel “disjunctive programming” method. For systems with uncertainties, optimal feedback controls will be computed by fast multilevel iterations for “Nonlinear Model Predictive Control” combined with “Moving Horizon Estimation”, which will be generalized to the general OCP class above. We will develop a new analytical approach to derive fast, locally valid, piecewise affine feedback control laws, which will significantly reduce sampling times further, compare it to the PMP approach to compute “neighbouring feedback” controls, and develop possible cross-over techniques. The development of the dedicated new numerical methods will go hand in hand with analysis and solution of the challenging engineering problems and will be conducted in close cooperation of the Moscow and the Heidelberg team.