液体降膜的线性二次调节控制

IF 1.9 4区 数学 Q1 MATHEMATICS, APPLIED
Oscar A. Holroyd, Radu Cimpeanu, Susana N. Gomes
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引用次数: 0

摘要

SIAM 应用数学杂志》第 84 卷第 3 期第 940-960 页,2024 年 6 月。摘要我们提出并分析了一种基于线性二次调节(LQR)的新方法,用于通过底部的吹气和吸力稳定下降的液膜。LQR 方法通过预先计算增益矩阵实现快速响应反馈控制,但只适用于线性常微分方程(ODE)系统。相比之下,描述液膜在斜面上流动动态的纳维-斯托克斯方程过于复杂,无法用标准的控制理论技术来稳定。为了弥补这一差距,我们使用了通过渐近分析获得的降阶模型--本尼方程和加权残余积分边界层模型--来推导多级控制框架。该框架包括一个 LQR 反馈控制,它是为近似降阶系统的线性化和离散化 ODE 系统设计的,然后将其应用于完整的 Navier-Stokes 系统。该控制方案通过直接数值模拟(DNS)进行测试,并与线性稳定性阈值和最小所需执行器数量的分析预测进行比较。通过比较两个降阶模型之间的策略,我们发现在两种情况下,我们都能在各自的有效参数范围内成功稳定地获得均匀的平膜,而更精确的加权残差模型优于 Benney 衍生的控制方案。此外,我们还发现加权残差控制的效果远远超出了预期的适用范围。所提出的方法提高了将稳健控制技术应用于现实世界系统的可行性,并可推广到其他形式的驱动系统。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Linear Quadratic Regulation Control for Falling Liquid Films
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 940-960, June 2024.
Abstract. We propose and analyze a new methodology based on linear-quadratic regulation (LQR) for stabilizing falling liquid films via blowing and suction at the base. LQR methods enable rapidly responding feedback control by precomputing a gain matrix, but they are only suitable for systems of linear ordinary differential equations (ODEs). By contrast, the Navier–Stokes equations that describe the dynamics of a thin liquid film flowing down an inclined plane are too complex to stabilize with standard control-theoretical techniques. To bridge this gap, we use reduced-order models—the Benney equation and a weighted-residual integral boundary layer model—obtained via asymptotic analysis to derive a multilevel control framework. This framework consists of an LQR feedback control designed for a linearized and discretized system of ODEs approximating the reduced-order system, which is then applied to the full Navier–Stokes system. The control scheme is tested via direct numerical simulation (DNS) and compared to analytical predictions of linear stability thresholds and minimum required actuator numbers. Comparing the strategy between the two reduced-order models, we show that in both cases we can successfully stabilize towards a uniform flat film across their respective ranges of valid parameters, with the more accurate weighted-residual model outperforming the Benney-derived controls. The weighted-residual controls are also found to work successfully far beyond their anticipated range of applicability. The proposed methodology increases the feasibility of transferring robust control techniques towards real-world systems and is also generalizable to other forms of actuation.
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来源期刊
CiteScore
3.60
自引率
0.00%
发文量
79
审稿时长
12 months
期刊介绍: SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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