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.

翻译：通过线性二次型调节器（LQR）方法计算反馈控制，并利用该反馈模拟非线性非自治闭环系统，我们结合了两个数值上具有挑战性的任务。对于第一个任务（反馈控制的计算），我们采用非自治广义微分Riccati方程（DRE），其解决定了时变反馈增益矩阵。针对第二个任务，我们期望能够模拟非线性闭环系统，而此类系统的调节器仅对足够小的扰动有效。因此，当闭环控制变化剧烈时，积分器极易出现数值问题——例如，A稳定的隐式Euler方法会失效。 一方面，我们实现了求解非自治DRE的分裂格式和BDF方法的非自治版本，这些方法在自治情形下已是成熟的DRE求解器。另一方面，为应对非线性闭环系统模拟中的数值难题，我们采用了一种专门针对此类挑战设计的分数步theta格式，并引入时间自适应策略——即，额外基于控制活动的强度来调整时间步长。我们将此方法与更经典的基于误差的时间自适应方法进行了比较。 我们描述了使这两个任务在合理时间内可计算的技术，并成功模拟了控制量剧烈变化的闭环系统，同时避免了数值问题。与基于误差的替代方法相比，我们的时间自适应策略所需的步数更少，且结果更可靠。