Optimal Control of Plane Poiseuille Flow Workshop on Flow Control, - PowerPoint PPT Presentation

� � ✁ � ✁ Optimal Control of Plane Poiseuille Flow Workshop on Flow Control, Poitiers 11-14th Oct 04 J. Mckernan , J.F .Whidborne , G.Papadakis Cranfield University, U.K. King’s College, London, U.K. Optimal Control of Plane Poiseuille Flow – p. 1/22

� � � ✁ Content Background Schmid and Henningson’s spectral model (2001) Linear state-space model (modified Bewley 1998, Hogberg 2003) State feedback estimator and controller (Joshi 1995, ) Initial conditions (Butler and Farrell 1992) Controller implementation in full FV Navier-Stokes solver Results and Conclusions Optimal Control of Plane Poiseuille Flow – p. 2/22

✆✝ ✄ ✝ ✞ ✁ � ✆ ☎ ✄ � ✂ ✁ � ✝ ✝ ✝ ✝ ✝ Background Theoretical, rather than experimental approach Laminar Poiseuille (closed channel) flow Simple base flow (constant), geometry, boundary conditions Linearly unstable , Transition Here , 2D (streamwise, wall-normal) Synthesize linear optimal controllers ‘Minimise’ Transient Energy Growth (Time integral) Test controllers in full Navier-Stokes Solver Optimal Control of Plane Poiseuille Flow – p. 3/22

Control of Poiseuille Flow Upper Wall wall−normal, y streamwise, x Plane Poiseuille Flow + Flow Disturbance Lower Wall Actuation Controller Sensing Optimal Control of Plane Poiseuille Flow – p. 4/22

� ✝ ☎ ✆ ✞ ✆ ✂ ✞ Spectral Model Schmid and Henningson 2001 Linearised Navier-Stokes equations Velocity-vorticity formulation Spectral discretisation Chebyshev in wall-normal direction Fourier in streamwise and spanwise directions ✁✄✂ One wave number pair (here ) Homogeneous wall boundary conditions Optimal Control of Plane Poiseuille Flow – p. 5/22

✡ ✝ ✝ ✂ ✟ ✝ ✂ ✂ ✝ ✏ ✂ ☎ ✂ ✎ ✂ � ✂ ✂ ☎ ✟ ✁ ✞ ✂ ✞ � ✆ State-Space Model Form;- Input : rate of change of wall-normal velocities at walls �✄✂✆☎✞✝ ( now inhomogeneous ) ☎✠✟ ✂☛✡ Output : Wall-shear stress measurements States : Coeffs of Novel Chebyshev Recombinations ☞✍✌ Wall-normal velocities at walls Optimal Control of Plane Poiseuille Flow – p. 6/22

✕ ✭ ✑ ☛ ✔ ✁ ✄ ☛ ✡ ✁ ✄ ✍ ✧ ✜ ✆ ✥ ✬ � ✜ � ✥ ✍ � ✪ ✆ ✥ ✜ ✓ ✍ ✧ ☛ ✧ ✆ ✥ ✡ ☛ ✁ ✄ ✆ ☞ ✣ ✡ ✌ ✑ ✜ ✆ ☞ ✁ ✄ ✡ ☛ ✌ ✮ ✍ ✎ ✏ ✣ Optimal Control of Plane Poiseuille Flow – p. 7/22 ✩✗✩ ✜✤✣ ✘✛✚ ✖✗✖ ✁✞✝✠✟ State-Space Model ✜✢✭ ✁☎✄ ☛✂✒ ✁☎✄ ✜✤✣✫✧ ✜✤✣ ✩✗✩ Navier-Stokes equations;- Linear state-Space form;- ✜✤✣ ✘✛✚ ✁✞✝✠✟ ✖✗✖ �✂✁☎✄ ✩✗✩ ✣★✧ �✙✘✛✚ �✢✜✤✣✦✥ ✖✗✖

✞ ✏ ✆ ✡ ✠ ✞ ✝ ✟ ✞ ✆ ✄ ✟ ✝ ✏ ✠ ✆ ✡ ✆ ✏ ✆ ✆ ✏ ✞ ✂ ✁ � ✝ ✁ ✞ ✂ ✞ � ✞ Optimal State Feedback Given the real system;- Feedback control signal to minimize;- ✎ ☎✄ ✎ ☎✄ ✎ ☎✄ ✎ ☎✄ Given by where from ARE Optimal Control of Plane Poiseuille Flow – p. 8/22

✞ ☎ ✝ ✝ � ☎ ✝ ✞ ✂ ✁ � ✞ ☎ ✄ ✝ ✆ ✟ ✁✂ ✆ ✞ ✆ ✞ Weighting Matrices Choose to form energy density (Bewley 1998);- ✄✆☎ Curtis-Clenshaw quadrature for integration ✞✠✟ Choose Max energy vs plotted from linear simulations Choose Optimal Control of Plane Poiseuille Flow – p. 9/22

✆ ✄ ✆ ✁ ☎ ✟ � ✁ ✂ ✆ ✡ ✞ ✁ ✁ ✠ ✡ ✠ ✞ ✆ ✝ ✆ ✆ ✂ � ✁ ✎ � � ✁ ✂ ✞ � ✁ ✂ ✄ ✟ ✟ ✁ � ✁ ✂ ✏ ✝ ✄ ✞ ☎ ✟ Optimal State Estimation Given the real estimator;- The optimal to minimize;- Is given by where from ARE Optimal Control of Plane Poiseuille Flow – p. 10/22

✞ ✝ ✞ ✆ ✝ ✆ ✆ ✂ ✟ ✆ � ✁ ☎ ☎ ✞ ✂ � ✞ ✏ ✞ ✏ ☎ � ✞ ✎ ✟ ✆ ✎ ✞ ✂ ✆✝ ✝ Weighting Matrices represents covariance of process noise Choose is physical distance between states, represents covariance of measurement noise Choose Fastest pole vs plotted from linear simulations Choose , for fastest estimator pole plant Optimal Control of Plane Poiseuille Flow – p. 11/22

☎ ✆ ✏ � ✎ ✝ ✏ ✎ ✂ ✂ � � ✎ ✝ ✂ ✞ � ✁ ✝ � ✟ � ✡ ✞ ✁ � ✏ ✝ ✎ ☎ ✄ ✠ ✏ ✞ � ✁ ☎ ✂ � ✞ ✞ ✏ ✆ ✏ ✏ ✝ � Initial Conditions ‘Worst’ not unstable eigenvector Non-orthogonal system matrix (Trefethen 1993) Variational method used (Butler and Farrell 1992, Bewley 1998) ✂ ✄✂ ✎ ☎✄ ✎ ☎✄ ✎ ☎✄ States , Energy Transient Energy Growth from ✆✞✝ Hence (TS waves) Optimal Control of Plane Poiseuille Flow – p. 12/22

✄ ✁ ✆ ☎ � ✂ ✁ ✁ ✂ Non-Linear Simulations Finite-volume full Navier-Stokes solver Second order in space (CDS), implicit first order in time PISO algorithm, Collocated grid ( ) BCs: Streamwise - cyclic Walls - transpiration Spanwise - symmetric ✄✆☎ Code modified to solve for the perturbation about base flow Optimal Control of Plane Poiseuille Flow – p. 13/22

✞ ☎ ✁ ✂ ✏ ✂ ✝ ✂ � ✂ ☎ ✂ � ✂ ✟ ✁ ✝ ✆ ✟ ✞ ✟ ✝ ✂ ✞ ✟ � ✁ ✂ � ✟ ✂ ✡ ✞ ✎ ✂ ✡ ✝ ✂ ✂ ✡ ✟ ✏ ✆ ✟ ✝ ✎ ✂ ✡ ✟ ☎ � � ✁ ✂ ✞ � ✁ ✂ ✄ ☎ Implementation of Controller FFT to compute measurements from States estimated using;- Control signals computed using;- ☎✍✝ (LQR) (LQG) ✂✆☎ Integration for and Inverse FFT for Optimal Control of Plane Poiseuille Flow – p. 14/22

� ✟ ✎ ✝ ✏ �✁ ✂ ✞ ✝ ☎ ✝ ✝ ✝ ✆ ✄ Open Loop Results −6 8 x 10 Non−linear Simulation Non−linear Simulation Estimate 7 Linear Simulation Linear Simulation Estimate Transient Energy Density/ ρ U cl 2 6 5 4 3 2 1 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow – p. 15/22

☎ ✎ ✝ ✏ �✁ ✂ ✞ ✝ � ✝ ✆ ✄ ✟ Open Loop Results 0.08 Non−linear Simulation Non−linear Simulation Estimate 0.07 Linear Simulation Linear Simulation Estimate Transient Energy Density/ ρ U cl 2 0.06 0.05 0.04 0.03 0.02 0.01 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow – p. 16/22

✡ ✂ ✁ ✡ ☎ ✄ ✁ ✂ ✁ ✏ ✝ ✎ ✞ ✟ ✄ ✏ ✝ ✎ Closed Loop Results - LQR State Feedback, ✟ ✁� ✞✠✟ ✁ ✝✆ 0.012 Non−linear Simulation Linear Simulation 0.01 Transient Energy Density/ ρ U cl 2 0.008 0.006 0.004 0.002 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow – p. 17/22

✡ ✂ ✁ ✡ ☎ ✄ ✁ ✂ ✁ ✏ ✝ ✎ ✞ ✟ ✄ ✏ ✝ ✎ Closed Loop Results - LQG Output Feedback, ✟ ✁� ✞✠✟ ✁ ✝✆ 0.025 Non−linear Simulation Non−linear Simulation Estimate Linear Simulation Linear Simulation Estimate 0.02 Transient Energy Density/ ρ U cl 2 0.015 0.01 0.005 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow – p. 18/22

✡ ✏ ✄ ✁ ✂ ✁ ✡ ✂ ✝ ✁ ✎ ✞ ✟ ✄ ✏ ✝ ✎ ☎ Closed Loop Results Output Feedback, ✟ ✁� ✞✠✟ ✁ ✝✆ 0.025 Non−linear Simulation Linear Simulation 0.02 0.015 Upper wall transpiration vel/U cl 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow – p. 19/22

Conclusions Optimal controllers for 2D Poiseuille Flow synthesized. Controller implemented in FV CFD code. Small Perturbations Spectral Linear and FV CFD results identical. Large Perturbations Open loop - CFD simulation saturates, but still unstable. Closed Loop - simulations stabilised, CFD needs longer than linear. Optimal Control of Plane Poiseuille Flow – p. 20/22

✎ ✁ ✏ ✎ � ✂ ☎ ✏ � ✂ Future Targets Investigate control of linearly stable but ‘worst’ perturbation CFD simulations - Streamwise Vortices rather than TS waves Actuation by wall-normal and tangential transpiration More control degrees of freedom LMI Controllers Minimise upper bound on max transient energy growth Optimal Control of Plane Poiseuille Flow – p. 21/22

The End Thank you. Optimal Control of Plane Poiseuille Flow – p. 22/22