Optimal trajectory tracking

TL;DR

This thesis develops a framework for optimal trajectory tracking in nonlinear systems with affine controls, introducing exactly realizable trajectories and leveraging linearization and perturbation methods for analytical solutions.

Contribution

It introduces the concept of exactly realizable trajectories, characterizes them via a constraint equation, and derives analytical solutions for nonlinear control problems using linearization and perturbation techniques.

Findings

Exact solutions for nonlinear trajectory tracking are derived.

Linear structures underlying nonlinear control problems are identified.

The approach applies to mechanical systems and neural models like FitzHugh-Nagumo.

Abstract

This thesis investigates optimal trajectory tracking of nonlinear dynamical systems with affine controls. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. The concept of so-called exactly realizable trajectories is proposed. For exactly realizable desired trajectories exists a control signal which enforces the state to exactly follow the desired trajectory. For a given affine control system, these trajectories are characterized by the so-called constraint equation. This approach does not only yield an explicit expression for the control signal in terms of the desired trajectory, but also identifies a particularly simple class of nonlinear control systems. Based on that insight, the regularization parameter is used as the small parameter for a perturbation expansion. This results in a reinterpretation of affine optimal control problems with small regularization term as singularly perturbed differential equations. The small parameter originates from the formulation of the control problem and does not involve simplifying assumptions about the system dynamics. Combining this approach with the linearizing assumption, approximate and partly linear equations for the optimal trajectory tracking of arbitrary desired trajectories are derived. For vanishing regularization parameter, the state trajectory becomes discontinuous and the control signal diverges. On the other hand, the analytical treatment becomes exact and the solutions are exclusively governed by linear differential equations. Thus, the possibility of linear structures underlying nonlinear optimal control is revealed. This fact enables the derivation of exact analytical solutions to an entire class of nonlinear trajectory tracking problems with affine controls. This class comprises mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model.