Overview
This research investigates the problem of determining the best approximation control for a double integrator system characterized by infeasibility. The infeasibility arises from control function constraints, specifically upper and lower bounds that are too restrictive. The objective is to minimize the squared $\mathcal{L}^2$-norm of a gap function, which quantifies the separation between two constraint sets, thereby identifying the optimal approximate control solution.
Research Context
The study builds upon existing results concerning problems involving general linear control systems. These prior findings provide a foundational understanding for addressing control systems where constraints may lead to infeasibility. The focus here narrows to the specific case of an infeasible double integrator, examining both analytical and numerical methodologies for control approximation.
Approach
The methodological approach proceeds in several stages:
- Review of Existing Literature: Initially, the study reviews established findings related to optimal control problems within general linear control systems. This provides a baseline and context for the subsequent specific analysis.
- Analytical Solution for Double Integrator: For the particular case of the infeasible double integrator problem, an analytical solution is derived. This solution pertains to a bang-bang control strategy, specifically one exhibiting at most a single switching event. The infinite-dimensional optimization problem inherent in this control strategy is consequently transformed.
- Problem Reduction: The aforementioned infinite-dimensional optimization problem is reduced to a more tractable form: solving two algebraic equations involving two variables. These variables are specifically computed to determine the switching time of the bang-bang control and the value of the gap function.
- Numerical Approaches for Algebraic Equations: The study discusses various numerical methods applicable to solving the system of two algebraic equations identified.
- Douglas-Rachford Algorithm Implementation: A relaxed version of the Douglas-Rachford algorithm is described for its application to the double integrator problem.
- Numerical Experimentation: Numerical experiments are conducted to demonstrate the practical implementation of the Douglas-Rachford algorithm and to evaluate its performance characteristics.
Findings
- The infeasibility in the double integrator problem is characterized by a gap function, whose squared $\mathcal{L}^2$-norm is minimized to find the best approximation control.
- An analytical solution was developed for the bang-bang control in the infeasible double integrator problem, limited to at most one switching.
- This analytical solution reduces the infinite-dimensional optimization problem to a system of two algebraic equations in two variables. These variables define the switching time and the gap function.
- Numerical approaches for solving this system of algebraic equations were discussed.
- The (relaxed) Douglas-Rachford algorithm was implemented for the double integrator problem.
- Numerical experiments illustrated the implementation and performance of the Douglas-Rachford algorithm.