Accurate Solutions to Optimal Control Problems via a Flexible Mesh and Integrated Residual Transcription

TL;DR

The results show that the proposed method can be more than two times more accurate than conventional fixed mesh collocation for the same computational time and more than three times more accurate for the same problem size.

Abstract

We propose joining a flexible mesh design with an integrated residual transcription in order to improve the accuracy of numerical solutions to optimal control problems. This approach is particularly useful when state or input trajectories are non-smooth, but it may also be beneficial when dynamics constraints are stiff. Additionally, we implement an initial phase that will ensure a feasible solution is found and can be implemented immediately in real-time controllers. Subsequent iterations with warm-starting will improve the solution until optimality is achieved. Optimizing over the mesh node locations allows for discontinuities to be captured exactly, while integrated residuals account for the approximation error in-between the nodal points. First, we numerically show the improved convergence order for the flexible mesh. We then present the feasibility-driven approach to solve control problems and show how flexible meshing and integrated residual methods can be used in practice. The presented numerical examples demonstrate for the first time the numerical implementation of a flexible mesh for an integrated residual transcription. The results show that our proposed method can be more than two times more accurate than conventional fixed mesh collocation for the same computational time and more than three times more accurate for the same problem size.

Figures (12)

• Figure 1: Meshing for state and input approximations $\tilde{x}$ and $\tilde{u}$ used in constructing numerical solutions for \ref{['eq:equation2']}. Interpolation mesh is denoted by $\tau_i^j$ and $\mu_i^j$, quadrature mesh is $\rho_i^k$ and decision variables are $s_i^j$ and $c_i^j$.
• Figure 2: Order of convergence plot for approximating $u(t)=|\cos(\pi t)|$ for fixed vs flexible mesh as polynomial degree $P$ is increased for various mesh size $N$ between $10$ and $100$. The residual function is $\epsilon_r=\int_0^2 (\tilde{u}(t)-u(t))^2 dt$.
• Figure 3: Numerical solution and absolute error plots of $\dot{x}(t)=-x(t)\cdot\text{sgn}(t-1)$ with initial condition $x(0)=1$ using $N=7$ intervals and polynomial degree $a=2$.
• Figure 4: Convergence plot for a stiff Van der Pol system ($\nu=500$) where the integrated residual was defined as $\epsilon_{r}:=\int_{t_0}^{t_f}{\epsilon(t)}\mathop{}\!\mathrm{d} t$ with the residual function $\epsilon(t):=\left\lVert F(\dot{\tilde{x}}(t),\tilde{x}(t),\tilde{u}(t),t)\right\rVert^2_2$
• Figure 5: Feasibility driven solution approach: A schematic overview of the solution method. Note quadrature error was checked before feasibility and optimality in both stages, but it was omitted from the figure for enhanced readability.