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

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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.