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

这个德国-俄罗斯合作项目旨在发展分析和数值方法，以优化和最优控制某些类型的动态过程，由俄罗斯合作伙伴提出和研究，这些过程具有复杂的特性，对工程应用很重要。机器人在无外力的阻力环境中，仅通过内部质量的移动或结构的改变来进行运动，会导致具有状态依赖不连续的机械控制问题，特别是由于库仑摩擦，以及众多的控制和状态约束。欠驱动多体系统具有粘弹性部件、部分状态可测、系统参数不确定和外部扰动等特点，需要快速反馈控制律来实现高精度定位。高速（超声速）客机的节油飞行规划涉及复杂的长视界混合整数控制问题模型，该模型包含大量的状态约束和控制约束，考虑到技术限制和乘客的舒适性和安全性。为了满足这些特殊的要求，新的解析和数值最优控制方法将得到发展。海德堡团队将考虑相对一般的非线性最优控制问题（OCP），具有整数值控制，由边界条件、状态和控制约束的ODE或DAE控制。在状态相关开关函数的零点处，动力学的右侧可能是不连续的，特别是由库仑摩擦引起的。原则上，这类最优控制问题可由庞特里亚金极大值原理（PMP）的扩展来解决。然而，在上述情况下，间接PMP导致具有切换函数和伴随变量附加跳变的多点边值问题，这是极难解决的。因此，拟议的项目将建立在海德堡集团开发的直接多次射击方法的基础上，以数字方式解决这些问题。本文将提出两种处理库仑摩擦的方法，即“前向积分”方法和一种新的“析取规划”方法。对于具有不确定性的系统，将“非线性模型预测控制”与“移动地平线估计”相结合，通过快速多层迭代计算出最优反馈控制，并将其推广到上述一般OCP类。我们将开发一种新的分析方法来推导快速，局部有效，分段仿射反馈控制律，这将大大减少采样时间，将其与PMP方法进行比较，以计算“邻近反馈”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方案齐头并进，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.