非自治数据下闭环系统的数值方法
Numerical methods for closed-loop systems with non-autonomous data
AI总结：
本文针对非自治闭环系统，结合LQR反馈控制与非自治DRE求解，提出基于控制活动的时间自适应分数步theta格式，以更少步数和更高可靠性完成强变化控制下的闭环仿真。
摘要：
通过线性二次调节器（LQR）方法计算反馈控制，并利用该反馈对非线性非自治闭环系统进行仿真，我们将两个具有数值挑战性的任务结合在一起。对于第一个任务，即反馈控制的计算，我们使用非自治广义微分Riccati方程（DRE），其解确定了时变反馈增益矩阵。对于第二个任务，我们希望能够在已知调节器仅对足够小的扰动有效的情况下，对非线性闭环系统进行仿真。因此，当闭环控制变化很大时，积分器中很容易出现数值问题。对于这些系统，例如，A-稳定的隐式Euler方法会失效。一方面，我们实现了分裂格式和BDF方法的非自治版本，用于求解非自治DRE。这些方法在自治情形下是成熟的DRE求解器。另一方面，为了解决非线性闭环系统仿真中的数值问题，我们采用了一种分数步theta格式，并专门针对这类挑战调整了时间自适应策略。也就是说，我们额外将时间自适应建立在控制活动的基础上。我们将该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，并能够在避免数值问题的同时，对具有剧烈变化控制的闭环系统进行仿真。我们的时间自适应方法所需的步数少于基于误差的替代方案，并且更加可靠。
英文摘要：
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.
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