Time-optimal nonlinear model predictive control

Abstract

This thesis deals with the development and analysis of novel time-optimal model predictive control concepts for nonlinear systems. Common realizations of model predictive controllers apply direct transcription methods to first discretize and then optimize the subordinate optimal control problems. The key idea of the proposed concepts is to introduce discretization grids in which the underlying discretization is explicitly treated as temporally variable during optimization. A single optimization parameter for all grid intervals leads to the global uniform grid, while the definition of an individual parameter for each interval results in the local uniform and quasi-uniform grid representations. The proposed grids are well-suited for established direct transcription methods such as multiple shooting and collocation. In addition, a proposed non-uniform grid with extended multiple shooting is highly beneficial for bang-singular-bang control systems with simple constraint sets. Minimizing the local time information in all grids leads to the overall time-optimal transition. Integration with state feedback does not immediately guarantee asymptotic stability and recursive feasibility. To this end, the thesis provides a grid adaptation scheme capable of ensuring practical stability and, under more restricted conditions, also asymptotic stability while maintaining feasibility. The practical stability results facilitate the systematic dual-mode control design that restores asymptotic stability and establishes smooth stabilization. The secondary objective of this thesis is the computationally efficient realization of time-optimal model predictive control by exploiting the inherent sparse structures in the optimal control problems. In particular, the efficient computation of first- and second-order derivatives required for iterative optimization is facilitated by a so-called hypergraph. The hypergraph captures the structure of the transcribed optimal control problems and leads to an almost linear relation between computation time and grid size. In addition, the hypergraph shows negligible computation times for each reconfiguration that is essential for grid adaptation. Numerous examples in simulation and with a real experimental system demonstrate the capabilities and potentials of the proposed concepts. Extensive benchmarks in C++ compare the proposed methods with each other and the current state of the art. The methods based on variable discretization outperform the current time-optimal model predictive control methods in the literature, especially with regard to computation time.