Linear quadratic control of plane Poiseuille flow–the transient behaviour
Linear quadratic control of plane Poiseuille flow–the transient behaviour
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平面泊肃叶流的线性二次控制——瞬态行为
DOI：
10.1080/00207170701477764
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发表时间：
2007
影响因子：
2.1
通讯作者：
G. Papadakis
中科院分区：
文献类型：
--
作者：
J. McKernan;J. Whidborne;G. Papadakis
本文描述了平面Poietille流中单波数对周期性2-D扰动的最优线性二次型控制器的设计，以及随后使用有限体积全Navier-Stokes求解器在线性和非线性初始条件下的验证，所选择的初始条件产生最大的线性瞬态能量增长。对于线性量级的初始条件，开环和闭环有限体积求解器的结果与线性模拟吻合良好。瞬态能量增长是流体流动问题中的一个重要性能指标。控制器降低了瞬态能量增长，并且通常可以看到非线性效应将能量水平保持在缩放的线性值以下，尽管它们在一个模拟中确实引起不稳定性。需要相对大量的局部蒸发流体。负责的瞬态能量增长的模式被确定。模式被示出不变得更加正交的应用程序的控制。状态估计器的合成需要更高的离散化水平比状态反馈控制器的合成。一个简单的调整估计器的权重，提出了改进的收敛均匀权重从零初始估计。名义希腊文符号α流向（x）波数，每2 π距离的周期数β展向（z）波数，每2 π距离的周期数ε（t）时间t的同步瞬态能量约束时间t的同步误差能量约束同步瞬态能量约束本征值系统η（x，y，z，t）波数对α处的壁面法向涡度扰动η傅立叶系数，β θ历时瞬态能量界θ误差历时误差能量界θ E st估计能量界ith特征值对角特征值矩阵μ动粘度ρ流体密度ith A的奇异值（A）谱范数或A模态振幅向量的最大奇异值，χ 0初始χ，在时间t = 0处，右特征向量的矩阵，i i i在时间t E误差（t）误差能量，在时间t E 0 E的最坏开环扰动，在t = 0（2.26 × 10 − 9）E对，边界 模对能量增长的上界h通道壁分离单位矩阵j 状态反馈增益矩阵估计器增益矩阵N所用的最高切比雪夫多项式次数，最终配置点指数P压力P B稳定基流压力p压力扰动状态变量加权（能量）矩阵控制加权矩阵R雷诺数r控制权重乘数s测量噪声权重乘数用于状态变量之间转换的可逆矩阵，不包括壁旁速度和壁涡t时间x，y，z流向、壁法向和展向坐标流速矢量稳定基流速度U cl U B在中心线速度扰动矢量u，v，w波数对α，β控制矢量测量噪声功率谱密度过程噪声功率谱密度状态变量矢量状态估计矢量估计误差矢量， 从而产生θ 从而产生θ误差 转换为配置点处的值，状态变量转换为，因此测量向量y n y在第n个Chebyshev-Gauss-Lobatto配置点处
This paper describes the design of optimal linear quadratic controllers for single wavenumber-pair periodic 2-D disturbances in plane Poiseuille flow, and subsequent verification using a finite-volume full Navier–Stokes solver, at both linear and non-linear levels of initial conditions selected to produce the largest linear transient energy growth. For linear magnitude initial conditions, open and closed-loop finite-volume solver results agree well with a linear simulation. Transient energy growth is an important performance measure in fluid flow problems. The controllers reduce the transient energy growth, and the non-linear effects are generally seen to keep energy levels below the scaled linear values, although they do cause instability in one simulation. Comparatively large local quantities of transpiration fluid are required. The modes responsible for the transient energy growth are identified. Modes are shown not to become significantly more orthogonal by the application of control. The synthesis of state estimators is shown to require higher levels of discretiation than the synthesis of state-feedback controllers. A simple tuning of the estimator weights is presented with improved convergence over uniform weights from zero initial estimates. Nomenclature Greek symbols α streamwise (x) wave number, cycles per 2π distance β spanwise (z) wave number, cycles per 2π distance ε(t) synchronic transient energy bound at time t synchronic error energy bound at time t ζ eigenvalue in synchronic transient energy bound eigensystem η(x,y,z,t) wall-normal vorticity perturbation η Fourier coefficient at wavenumber pair α,β θ diachronic transient energy bound θ Error diachronic error energy bound θ Est estimated energy bound ith eigenvalue diagonal eigenvalue matrix μ dynamic viscosity ρ fluid density ith singular value of A (A) spectral norm or largest singular value of A modal amplitude vector, χ0 initial χ, at time t = 0 Ψ matrix of right eigenvectors ψ i ith right eigenvector frequency Roman symbols system matrix input matrix output matrix multiplying co-efficient for nth Chebyshev polynomial c i amplitude of mode i E(t) transient energy, , at time t E Est(t) estimated transient energy, , at time t E Error (t) error energy , at time t E 0 E of worst open-loop perturbation of , at t = 0 (2.26 × 10− 9) E pair,bound upper bound on mode pair energy growth h channel wall separation identity matrix j state feedback gain matrix estimator gain matrix N highest Chebyshev polynomial degree used, final collocation point index P pressure P b steady base flow pressure p pressure perturbation state variable weighting (energy) matrix control weighting matrix R Reynolds number r control weight multiplier s measurement noise weight multiplier invertible matrix for conversion between state variables and , excludes next-to-wall velocities and wall vorticities t time x,y,z streamwise, wall-normal and spanwise co-ordinates flow velocity vector steady base flow velocity U cl Ub at centreline velocity perturbation vector u,v,w Fourier coefficients at wavenumber pair α,β control vector measurement noise power spectral density process noise power spectral density state variable vector state estimates vector estimate error vector, which generates θ which generates θ Error transformed to values at collocation points state variables transformed to , thus measurement vector y n y at nth Chebyshev–Gauss–Lobatto collocation point