Optimization-based Solutions to Constrained Trajectory-tracking and Path-following Problems

Systems and control theory is of great importance for the analysis, the design, and the operation of complex dynamical systems. The prototypical problem in control theory is the stabilization of a set point. This is the problem of designing a feedback such that the closed-loop solutions stay in a neighborhood of the set point and converge to it. When, instead of a set point, a time-varying reference needs to be stabilized-i.e., the closed-loop solutions shall converge to a time-varying reference trajectory-then the problem is called trajectory tracking. Typical examples of trajectory-tracking problems are set point changes along precomputed references, synchronization tasks or startup of processes. While stabilization and trajectory tracking are well-understood for a wide range of systems, not all control tasks arising in practise belong to these categories.

For example, consider the case of steering a car automatically along a road. Usually, the driving velocity is not predetermined. The only requirements are to keep the car on the road while driving sufficiently fast. Formally, we can reformulate this task as follows: the system state or output should converge to a known geometric curve and follow it along in a specific direction. These kinds of control tasks are known as path-following problems. The major difference to trajectory tracking is that the speed along the path along is not fixed a priori. Instead it is an additional degree of freedom which can be tuned to achieve fast convergence, and small deviations of the system state or output to the path. Path-following problems typically arise in robotics and vehicle-like applications, such as unmanned aerial vehicles, robots, or milling machines. Moreover, some process control problems can be formulated as path following, for example, the task of steering a batch reactor along a temperature profile while maximizing the process productivity.

This thesis deals with the design of optimization-based control schemes for trajectorytracking and path-following problems in the presence of constraints. We present novel results on the design of nonlinear model predictive control (NMPC) schemes for both problem classes.

Sufficient conditions for the stability of NMPC for trajectory tracking based on timevarying terminal regions are derived. The key feature of our approach is the introduction of positive invariant time-varying level sets of Lyapunov functions as terminal regions. These sets allow enlarging the region of attraction of the proposed control schemes and thereby improve the control performance.

Furthermore, we discuss path-following problems for constrained nonlinear systems. We propose using a nonlinear normal form of an augmented system to analyze these problems. We investigate sufficient conditions for exact path followability of unconstrained and constrained nonlinear systems. Based on these results we present tailored predictive control schemes for path following. We derive sufficient conditions which guarantee the convergence of the system output to the path. In contrast to previous works-for example, in the field of robotics-our schemes can handle constraints on states and inputs as well as situations where the system does not start on the path. In other words, using the proposed control schemes makes it possible to stabilize the motion of a system in the presence of constraints with respect to a path defined in an output space. Examples from robotics and chemical engineering are drawn upon to support our results.

The main intention of this thesis is twofold: Firstly, we show that nonlinear model predictive control is very well applicable to problems beyond set point stabilization. Secondly, we demonstrate that path-following concepts provide a suitable framework for many challenging control problems ranging from robotics to chemical engineering.