[Paper Review] Time-optimal control with direct collocation and variable discretization

Christoph Rösmann, Artemi Makarow+1 more|arXiv (Cornell University)|May 25, 2020

Advanced Control Systems Optimization20 references4 citations

TL;DR

This paper proposes a time-optimal control framework using direct collocation with variable discretization and Hermite-Simpson collocation, where time intervals are optimized alongside control trajectories. It demonstrates that linear or constant control parameterizations outperform quadratic splines by avoiding constraint violations and oscillations, enabling robust, accurate solutions for stiff systems with improved real-time applicability.

ABSTRACT

This paper deals with time-optimal control of nonlinear continuous-time systems based on direct collocation. The underlying discretization grid is variable in time, as the time intervals are subject to optimization. This technique differs from approaches that are usually based on a time transformation. Hermite-Simpson collocation is selected as common representative in the field of optimal control and trajectory optimization. Hereby, quadratic splines approximate the system dynamics. Several splines of different order are suitable for the control parameterization. A comparative analysis reveals that increasing the degrees of freedom in control, e.g. quadratic splines, is not suitable for time-optimal control problems due to constraint violation and inherent oscillations. However, choosing constant or linear control splines points out to be very effective. A major advantage is that the implicit solution of the system dynamics is suited for stiff systems and often requires smaller grid sizes in practice.

Motivation & Objective

• To develop a direct collocation approach for time-optimal control of nonlinear continuous-time systems with variable time discretization.
• To investigate the impact of different control parameterizations (linear, quadratic, constant) on constraint satisfaction and solution accuracy.
• To enable real-time and predictive control applications by ensuring numerical stability and convergence for stiff systems.
• To compare collocation methods (Hermite-Simpson, finite differences) and multiple shooting in terms of dynamics accuracy and constraint handling.
• To validate the method on benchmark systems, including the Van der Pol oscillator, under realistic control and state constraints.

Proposed method

• Employs direct collocation with Hermite-Simpson quadrature rules to approximate state and control trajectories using piecewise polynomial splines.
• Introduces variable time discretization by optimizing time intervals between collocation points, enabling final time to be a decision variable.
• Uses quadratic splines for control parameterization in Hermite-Simpson collocation to achieve high accuracy in trajectory approximation.
• Applies sparse finite difference Jacobian and Hessian computation via hypergraph representation to exploit sparsity and improve solver efficiency.
• Implements a grid adaptation scheme in model predictive control to reduce horizon length iteratively, ensuring forward invariance and convergence.
• Solves the resulting nonlinear programs using IPOPT with HSL-MA57 as the linear solver, leveraging sparsity for computational efficiency.

Experimental results

Research questions

• RQ1 How does the choice of control parameterization (linear, quadratic, constant) affect constraint satisfaction and solution quality in time-optimal control problems?
• RQ2 Can variable discretization via optimized time intervals improve convergence and accuracy compared to fixed-grid or time-transformation methods?
• RQ3 What is the trade-off between solution accuracy and constraint violation when using higher-order control splines in direct collocation?
• RQ4 How do different collocation schemes (Hermite-Simpson, finite differences) compare in terms of dynamics accuracy and robustness for stiff systems?
• RQ5 To what extent can this approach support real-time or predictive control applications with guaranteed stability and constraint compliance?

Key findings

• Quadratic control splines in Hermite-Simpson collocation achieve the highest dynamics accuracy but frequently violate control bounds due to interpolation at collocation points.
• Linear and constant control parameterizations yield more reliable solutions with fewer constraint violations and reduced oscillations, despite lower approximation order.
• The method successfully handles stiff systems due to implicit integration of dynamics, requiring smaller grid sizes in practice compared to explicit methods.
• Forward difference collocation exhibits the largest dynamics error, confirming its first-order accuracy and instability in high-precision tasks.
• The grid adaptation scheme ensures forward invariance and convergence toward the target set, enabling stable model predictive control with reduced horizon length.
• Hermite-Simpson collocation with linear control splines provides the best balance of accuracy, constraint compliance, and computational efficiency for real-time applications.