1 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Low dimensional model for control of the Blasius boundary layer by balanced truncation Espen Åkervik in collaboration with Shervin Bagheri, Luca Brandt and Dan S. Henningson

2 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Outline of talk Flow instability, examining the initial value problem Adding inputs and outputs: The lifting procedure for wall actuation Reduced order models preserving input-output characteristics, balanced truncation LQG feedback control results based on reduced order model Conclusions

3 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Flow set up and state space form Linearized Navier-Stokes equations Evolution operator central to both stability investigation and control design Stability deals with full system

4 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Matrix-free methods for stability Asymptotic stability (modal) Short time stability (non-modal) Both are eigenvalue problems to be solved by the Arnoldi method through Krylov sequences Snapshots from DNS are used

5 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Non-modal growth Asymptotically stable; single eigenmodes are not observed Optimal growth yields wavepacket propagation Later stage governed by cooperation of TS modes

6 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Adding inputs and outputs to the NS equations Forced NS stokes equations with outputs Input-output behaviour For control design it is sufficient to capture the I-O behaviour controller performance

7 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Obtaining a state space formulation for wall actuation Boundary controlled system not on state space form Lifting procedure to obtain a volume forced system (Högberg et al 2003) Steady state solution Form an augmented system steady state solution from DNS

8 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Reduced order model for control Full system too big for optimization problem and online time integration Approximate by reduced system Such that the IO is preserved One systematic approach is balanced truncation (Moore 81)

9 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Controllability and observability How do the inputs affect the outputs? Generated states from input (controllability) Generated signal from states (observability) Hankel operator Gramians observable states controllable states

10 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Balanced modes Find states that maximizes the effect from inputs to outputs Eigenvalue problem for balanced modes Balanced modes mutually diagonalize Gramians Computed using snapshot method (Rowley 2005):

11 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Balanced modes Direct and adjoint balanced modes Use these for oblique Galerkin projection to create reduced order model

12 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Preserving input-output behaviour Impulse response from disturbance to objective function Frequency response from all inputs to all outputs

13 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Control of the propagating wavepacket energy Optimal controller computed with LQG on small system Small estimator running online Control signal fed into actuator in DNS

14 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Conclusions Complex stability/control problems solved using Krylov/Arnoldi methods based on snapshots of forward and adjoint Navier-Stokes solutions Balanced modes give low order models preserving input- output relationship between sensors and actuators Feedback control of Blasius flow –Reduced order models with balanced modes used in LQG control –Controller based on small number of modes works well in DNS Framework enables LQG control for many complex flows –DNS/ADNS is all that is needed

15 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Extra slides

16 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Modal stability Least stable eigenmodes equivalent using time-stepper and matrix solver Least stable branch is a global representation of Tollmien- Schlichting (TS) modes No single eigenmode will be observed

17 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Non-modal growth Convectively unstable flow, asymptotically stable Optimals come in pairs with similar eigenvalues Many structures have potential for growth

18 Espen Åkervik 7 th ERCOFTAC SIG33 WORKSHOP Wall-shear stress sensors Approximate the sensors by Gaussians Adjoint sensor given by inner product relation Integration by parts