Numerical methods for closed-loop systems with non-autonomous data

TL;DR

This work addresses numerical challenges in closed-loop control of non-autonomous systems by integrating an LQR-based feedback design with nonlinear simulations. It develops non-autonomous, low-rank DRE solvers using both non-autonomous BDF and splitting schemes to obtain time-varying gains $K(t)$, and couples this with a fractional-step-theta time stepping framework that adapts to rapidly changing controls. The authors demonstrate, across moving-boundary problems such as a steel profile cooling scenario and a two-phase Stefan problem, that time-adaptive fractional-step-theta methods provide reliable, efficient simulation of strongly varying controls, while non-autonomous BDF methods handle truly time-dependent DREs better than splitting schemes in large-scale settings. The results offer practical pathways for robust, scalable closed-loop simulations in complex systems with moving interfaces and non-autonomous data, with potential extensions to alternative ARE solvers and adaptive indicators.

Abstract

By computing a feedback control via the linear quadratic regulator (LQR) approach and simulating a non-linear non-autonomous closed-loop system using this feedback, we combine two numerically challenging tasks. For the first task, the computation of the feedback control, we use the non-autonomous generalized differential Riccati equation (DRE), whose solution determines the time-varying feedback gain matrix. Regarding the second task, we want to be able to simulate non-linear closed-loop systems for which it is known that the regulator is only valid for sufficiently small perturbations. Thus, one easily runs into numerical issues in the integrators when the closed-loop control varies greatly. For these systems, e.g., the A-stable implicit Euler methods fails.\newline On the one hand, we implement non-autonomous versions of splitting schemes and BDF methods for the solution of our non-autonomous DREs. These are well-established DRE solvers in the autonomous case. On the other hand, to tackle the numerical issues in the simulation of the non-linear closed-loop system, we apply a fractional-step-theta scheme with time-adaptivity tuned specifically to this kind of challenge. That is, we additionally base the time-adaptivity on the activity of the control. We compare this approach to the more classical error-based time-adaptivity.\newline We describe techniques to make these two tasks computable in a reasonable amount of time and are able to simulate closed-loop systems with strongly varying controls, while avoiding numerical issues. Our time-adaptivity approach requires fewer time steps than the error-based alternative and is more reliable.

Figures (12)

• Figure 1: Convergence orders of BDF with different $n_{\wp}$, small scale example.
• Figure 2: Convergence of splitting and BDF and the theoretical convergence orders $\wp$ (dashed), small scale example.
• Figure 3: Errors of splitting and BDF for $n_t = 512$ and $n_{\wp} = 10$, small scale example.
• Figure 4: Runtime of splitting and BDF, steel profile example. All the methods use the same temporal grid.
• Figure 5: Runtime of splitting and BDF, steel profile example. All the methods use the same temporal grid.