Review Article

Adaptive and Model-Based Control Theory Applied to Convectively Unstable Flows

[+] Author and Article Information
Nicolò Fabbiane, Shervin Bagheri, Dan S. Henningson

Department of Mechanical Engineering,
Linnè FLOW Centre,
Royal Institute of Technology (KTH),
Stockholm S-10044, Sweden

Onofrio Semeraro

Laboratoire d'Hydrodynamique (LadHyX),
CNRS-Ecole Polytechnique,
Palaiseau 91128, France

1http://www.mech.kth.se/~nicolo/ (Nicolò Fabbiane)

Manuscript received December 20, 2013; final manuscript received April 11, 2014; published online June 17, 2014. Assoc. Editor: James J. Riley.

Appl. Mech. Rev 66(6), 060801 (Jun 17, 2014) (20 pages) Paper No: AMR-13-1104; doi: 10.1115/1.4027483 History: Received December 20, 2013; Revised April 11, 2014

Research on active control for the delay of laminar–turbulent transition in boundary layers has made a significant progress in the last two decades, but the employed strategies have been many and dispersed. Using one framework, we review model-based techniques, such as linear-quadratic regulators, and model-free adaptive methods, such as least-mean square filters. The former are supported by an elegant and powerful theoretical basis, whereas the latter may provide a more practical approach in the presence of complex disturbance environments that are difficult to model. We compare the methods with a particular focus on efficiency, practicability and robustness to uncertainties. Each step is exemplified on the one-dimensional linearized Kuramoto–Sivashinsky equation, which shows many similarities with the initial linear stages of the transition process of the flow over a flat plate. Also, the source code for the examples is provided.

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Bushnell, D. M., and Moore, K. J., 1991, “Drag Reduction in Nature,” Annu. Rev. Fluid Mech., 23, pp. 65–79. [CrossRef]
Kim, J., and Bewley, T. R., 2007, “A Linear Systems Approach to Flow Control,” Annu. Rev. Fluid Mech., 39, pp. 39–383. [CrossRef]
Schlichting, H., and Gersten, K., 2000, Boundary-Layer Theory, Springer, Heidelberg, NY.
Saric, W. S., Reed, H. L., and Kerschen, E. J., 2002, “Boundary-Layer Receptivity to Freestream Disturbances,” Annu. Rev. Fluid Mech., 34(1), pp. 291–319. [CrossRef]
Schmid, P. J., Henningson, D. S., 2001, Stability and Transition in Shear Flows (Applied Mathematical Sciences), Vol. 142, Springer, New York.
Jovanovic, M. R., and Bamieh, B., 2005, “Componentwise Energy Amplification in Channel Flows,” J. Fluid Mech., 534, pp. 145–183. [CrossRef]
Schmid, P. J., 2007, “Nonmodal Stability Theory,” Annu. Rev. Fluid Mech., 39, pp. 129–62. [CrossRef]
Glad, T., and Ljung, L., 2000, Control Theory, Taylor & Francis, London.
Huerre, P., and Monkewitz, P. A., 1990, “Local and Global Instabilities in Spatially Developing Flows,” Annu. Rev. Fluid Mech., 22, pp. 473–537. [CrossRef]
Joshi, S. S., Speyer, J. L., and Kim, J., 1997, “A Systems Theory Approach to the Feedback Stabilization of Infinitesimal and Finite-Amplitude Disturbances in Plane Poiseuille Flow,” J. Fluid Mech., 332, pp. 157–184.
Bewley, T. R., and Liu, S., 1998, “Optimal and Robust Control and Estimation of Linear Paths to Transition,” J. Fluid Mech., 365, pp. 305–349. [CrossRef]
Cortelezzi, L., Speyer, J. L., Lee, K. H., and Kim, J., 1998, “Robust Reduced-Order Control of Turbulent Channel Flows via Distributed Sensors and Actuators,” IEEE 37th Conference on Decision and Control, Tampa, FL, Dec. 16–18, pp. 1906–1911.
Högberg, M., Bewley, T. R., and Henningson, D. S., 2003, “Linear Feedback Control and Estimation of Transition in Plane Channel Flow,” J. Fluid Mech., 481, pp. 149–175. [CrossRef]
Chevalier, M., Hœpffner, J., Akervik, E., and Henningson, D. S., 2007, “Linear Feedback Control and Estimation Applied to Instabilities in Spatially Developing Boundary Layers,” J. Fluid Mech., 588, pp. 163–187. [CrossRef]
Monokrousos, A., Brandt, L., Schlatter, P., and Henningson, D. S., 2008, “DNS and LES of Estimation and Control of Transition in Boundary Layers Subject to Free-Stream Turbulence,” Intl J. Heat Fluid Flow, 29(3), pp. 841–855. [CrossRef]
Lee, K. H., Cortelezzi, L., Kim, J., and Speyer, J., 2001, “Application of Reduced-Order Controller to Turbulent Flow for Drag Reduction,” Phys. Fluids, 13, pp. 1321–1330. [CrossRef]
Högberg, M., Bewley, T. R., and Henningson, D. S., 2003, “Relaminarization of Reτ = 100 Turbulence Using Gain Scheduling and Linear State-Feedback Control Flow,” Phys. Fluids, 15, pp. 3572–3575. [CrossRef]
Chevalier, M., Hœpffner, J., Bewley, T. R., and Henningson, D. S., 2006, “State Estimation in Wall-Bounded Flow Systems. Part 2: Turbulent Flows,” J. Fluid Mech., 552, pp. 167–187. [CrossRef]
Ljung, L., 1999, System Identification, Wiley, New York.
Elliott, S., and Nelson, P., 1993, “Active Noise Control,” IEEE Signal Process. Mag., 10(4), pp. 12–35. [CrossRef]
Milling, R. W., 1981, “Tollmien–Schlichting Wave Cancellation,” Phys. Fluids, 24, pp. 979–981. [CrossRef]
Jacobson, S. A., and Reynolds, W. C., 1998, “Active Control of Streamwise Vortices and Streaks in Boundary Layers,” J. Fluid Mech., 360, pp. 179–211. [CrossRef]
Sturzebecher, D., and Nitsche, W., 2003, “Active Cancellation of Tollmien–Schlichting Instabilities on a Wing Using Multi-Channel Sensor Actuator Systems,” Intl J. Heat Fluid Flow, 24, pp. 572–583. [CrossRef]
Rathnasingham, R., and Breuer, K. S., 2003, “Active Control of Turbulent Boundary Layers,” J. Fluid Mech., 495, pp. 209–233. [CrossRef]
Lundell, F., 2007, “Reactive Control of Transition Induced by Free-Stream Turbulence: An Experimental Demonstration,” J. Fluid Mech., 585, pp. 41–71. [CrossRef]
McKeon, B. J., Sharma, A. S., and Jacobi, I., 2013, “Experimental Manipulation of Wall Turbulence: A Systems Approach” Phys. Fluids, 25(3), p. 031301. [CrossRef]
Goldin, N., King, R., Pätzold, A., Nitsche, W., Haller, D., and Woias, P., 2013, “Laminar Flow Control With Distributed Surface Actuation: Damping Tollmien–Schlichting Waves With Active Surface Displacement,” Exp. Fluids, 54(3), pp. 1–11. [CrossRef]
Sipp, D., Marquet, O., Meliga, P., and Barbagallo, A., 2010, “Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach,” Appl. Mech. Rev., 63(3), p. 030801. [CrossRef]
Bagheri, S., and Henningson, D. S., 2011, “Transition Delay Using Control Theory,” Philos. Trans. R. Soc., 369, pp. 1365–1381. [CrossRef]
Bagheri, S., Hœpffner, J., Schmid, P. J., and Henningson, D. S., 2009, “Input-Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows,” Appl. Mech. Rev., 62, p. 020803. [CrossRef]
Sipp, D., and Schmid, P. J., 2013, “Closed-Loop Control of Fluid Flow: A Review of Linear Approaches and Tools for the Stabilization of Transitional Flows,” AerospaceLab J., 6.
el-Hak, M. G., 1996, “Modern Developments in Flow Control,” Appl. Mech. Rev., 49, pp. 365–379. [CrossRef]
Bewley, T. R., 2001, “Flow Control: New Challenges for a New Renaissance,” Prog. Aerospace. Sci., 37, pp. 21–58. [CrossRef]
Collis, S. S., Joslin, R. D., Seifert, A., and Theofilis, V., 2004, “Issues in Active Flow Control: Theory, Control, Simulation, and Experiment,” Prog. Aerosp. Sci., 40(4), pp. 237–289. [CrossRef]
Bagheri, S., Brandt, L., and Henningson, D. S., 2009, “Input–Output Analysis, Model Reduction and Control of the Flat-Plate Boundary Layer,” J. Fluid Mech., 620(1), pp. 263–298. [CrossRef]
Chevalier, M., Schlatter, P., Lundbladh, A., and Henningson, D. S., 2007, “A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows,” KTH Mechanics, Stockholm, Sweden, Technical Report No. TRITA-MEK 2007:07.
Grundmann, S., and Tropea, C., 2008, “Active Cancellation of Artificially Introduced Tollmien–Schlichting Waves Using Plasma Actuators,” Exp. Fluids, 44(5), pp. 795–806. [CrossRef]
Kuramoto, Y., and Tsuzuki, T., 1976, “Persistent Propagation of Concentration Waves in Dissipative Media Far From Thermal Equilibrium,” Prog. Theor. Phys., 55(2), pp. 356–369. [CrossRef]
Sivashinsky, G. I., 1977, “Nonlinear Analysis of Hydrodynamic Instability in Laminar Flames—I. Derivation of Basic Equations,” Acta Astronaut., 4, pp. 1177–1206. [CrossRef]
Manneville, P., 1995, Dissipative Structures and Weak Turbulence, Springer, Berlin, Germany.
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., and Vattay, G., 2012, “Turbulence?” Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen, Denmark, Chap. 4. http://ChaosBook.org/version14ChaosBook.org/version14
Charru, F., 2011, Hydrodynamic Instabilities, 1st ed., Cambridge University, Cambridge, UK.
Skogestad, S., and Postlethwaite, I., 2005, Multivariable Feedback Control, Analysis to Design, 2nd ed, Wiley, Chichester, UK.
Aström, K. J., and Wittenmark, B., 1995, Adaptive Control, 2nd ed. Addison-Wesley, Reading, MA.
Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A., 1989. “State-Space Solutions to Standard H2 and H Control Problems,” IEEE Trans. Autom. Control, 34, pp. 831–847. [CrossRef]
Zhou, K., Doyle, J. C., and Glover, K., 2002, Robust and Optimal Control. Prentice Hall, Englewood Cliffs, NJ.
el-Hak, M. G., 2007, Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University, Cambridge, UK.
Julliet, F., Schmid, P. J., and Huerre, P., 2013, “Control of Amplifier Flows Using Subspace Identification Techniques,” J. Fluid Mech., 725, pp. 522–565. [CrossRef]
Belson, B. A., Semeraro, O., Rowley, C. W., and Henningson, D. S., 2013, “Feedback Control of Instabilities in the Two-Dimensional Blasius Boundary Layer: The Role of Sensors and Actuators,” Phys. Fluids, 25, p. 054106. [CrossRef]
Lewis, F. L., and Syrmos, L. V., 1995, Optimal Control, Wiley, New York.
Bewley, T. R., Moin, P., and Temam, R., 2001, “DNS-Based Predictive Control of Turbulence: An Optimal Benchmark for Feedback Algorithms,” J. Fluid Mech., 447(1), pp. 179–225. [CrossRef]
Gunzburger, M., 2003, Perspectives in Flow Control and Optimization. SIAM, Philadelphia, PA.
Boyd, S., and Vandenberghe, L., 2004, Convex Optimization, Cambridge University, Cambridge, UK.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 2007, Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed., Cambridge University, Cambridge, UK.
Corbett, P., and Bottaro, A., 2001, “Optimal Control of Nonmodal Disturbances in Boundary Layers,” Theor. Comput. Fluid Dyn., 15(2), pp. 65–81. [CrossRef]
Arnold, W. I., and Laub, A., 1984, “Generalized Eigenproblem Algorithms and Software for Algebraic Riccati Equations,” Proc. IEEE, 72(12), pp. 1746–1754. [CrossRef]
Benner, P., Li, J., and Penzl, T., 2008, “Numerical Solution of Large-Scale Lyapunov Equations, Riccati Equations, and Linear-Quadratic Optimal Control Problems,” Numer. Linear Algebra Appl., 15, pp. 755–777. [CrossRef]
Banks, H. T., and Ito, K., 1991, “A numerical Algorithm for Optimal Feedback Gains in High Dimensional Linear Quadratic Regulator Problems,” SIAM J. Control Optim., 29(3), pp. 499–515. [CrossRef]
Benner, P., 2004, “Solving Large-Scale Control Problems,” Control Syst. IEEE, 24(1), pp. 44–59. [CrossRef]
Bamieh, B., Paganini, F., and Dahleh, M., 2002, “Distributed Control of Spatially Invariant Systems,” IEEE Trans. Autom. Control, 47(7), pp. 1091–1107. [CrossRef]
Högberg, M., and Bewley, T. R., 2000, “Spatially Localized Convolution Kernels for Feedback Control of Transitional Flows,” IEEE 39th Conference on Decision and Control, pp. 3278–3283. [CrossRef]
Akhtar, I., Borggaard, J., Stoyanov, M., and Zietsman, L., 2010, “On Commutation of Reduction and Control: Linear Feedback Control of a Von Kármán Street,” 5th Flow Control Conference, American Institute of Aeronautics and Astronautics, pp. 1–14.
Martensson, K., 2012, “Gradient Methods for Large-Scale and Distributed Linear Quadratic Control,” Ph.D. thesis, Department of Automatic Control, Lund University, Sweden.
Pralits, J. O., and Luchini, P., 2010, “Riccati-Less Optimal Control of Bluff-Body Wakes,” Seventh IUTAM Symposium on Laminar-Turbulent Transition, P.Schlatter and D. S.Henningson, eds., Springer, Dordrecht, Vol. 18, pp. 325–330.
Semeraro, O., Pralits, J. O., Rowley, C. W., and Henningson, D. S., 2013, “Riccati-Less Approach for Optimal Control and Estimation: An Application to Two-Dimensional Boundary Layers,” J. Fluid Mech., 731, pp. 394–417. [CrossRef]
Garcia, C. E., Prett, D. M., and Morari, M., 1989, “Model Predictive Control: Theory and Practice—A Survey,” Automatica, 25(3), pp. 335–348. [CrossRef]
Qin, S. J., and Badgwell, T. A., 2003, “A Survey of Industrial Model Predictive Control Technology,” Control Eng. Pract., 11(7), pp. 733–764. [CrossRef]
Noack, B. R., Morzynski, M., and Tadmor, G., 2011, Reduced-Order Modelling for Flow Control, Vol. 528, Springer, Milan, Italy.
Bryd, R. H., Hribar, M. E., and Nocedal, J., 1999, “An Interior Point Algorithm for Large-Scale Nonlinear Programming,” SIAM J. Optim., 9, pp. 877–900. [CrossRef]
Corke, T. C., Enloe, C. L., and Wilkinson, S. P., 2010, “Dielectric Barrier Discharge Plasma Actuators for Flow Control,” Annu. Rev. Fluid Mech., 42(1), pp. 505–529. [CrossRef]
Suzen, Y., Huang, P., Jacob, J., and Ashpis, D., 2005, “Numerical Simulations of Plasma Based Flow Control Applications,” AIAA Paper No. 2005-4633.
Kriegseis, J., 2011, “Performance Characterization and Quantification of Dielectric Barrier Discharge Plasma Actuators,” Ph.D. thesis, TU Darmstadt, Germany.
Coleman, T. F., and Li, Y., 1996, “A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables,” SIAM J. Optim., 6(4), pp. 1040–1058. [CrossRef]
Anderson, B., and Moore, J., 1990, Optimal Control: Linear Quadratic Methods, Prentice Hall, New York.
Penrose, R., 1955, “A Generalized Inverse for Matrices,” Math. Proc. Cambridge Philos. Soc., 51, pp. 406–413. [CrossRef]
Luenberger, D. G., 1979, Introduction to Dynamic System, Wiley, New York.
Hervé, A., Sipp, D., Schmid, P. J., and Samuelides, M., 2012, “A Physics-Based Approach to Flow Control Using System Identification,” J. Fluid Mech., 702, pp. 26–58. [CrossRef]
Haykin, S., 1986, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, NJ.
Choi, H., Moin, P., and Kim, J., 1994, “Active Turbulence Control for Drag Reduction in Wall-Bounded Flows,” J. Fluid Mech., 262, pp. 75–110. [CrossRef]
Doyle, J. C., 1978, “Guaranteed Margins for LQG Regulators,” IEEE Trans. Autom. Control, AC-23(4), pp. 756–757. [CrossRef]
Erdmann, R., Pätzold, A., Engert, M., Peltzer, I., and Nitsche, W., 2012, “On Active Control of Laminar-Turbulent Transition on Two-Dimensional Wings,” Philos. Trans. R. Soc., 369, pp. 1382–1395. [CrossRef]
Anderson, B., and Liu, Y., 1989, “Controller Reduction: Concepts and Approaches,” IEEE Trans. Autom. Control, 34, pp. 802–812. [CrossRef]
Akervik, E., Hœpffner, J., Ehrenstein, U., and Henningson, D. S., 2007, “Optimal Growth, Model Reduction and Control in a Separated Boundary-Layer Flow Using Global Eigenmodes,” J. Fluid Mech., 579, pp. 305–314. [CrossRef]
Moore, B., 1981, “Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction,” IEEE Trans. Autom. Control, 26(1), pp. 17–32. [CrossRef]
Rowley, C. W., 2005, “Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition,” Int. J. Bifurcation Chaos, 15(03), pp. 997–1013. [CrossRef]
Ilak, M., and Rowley, C. W., 2008, “Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition,” Phys. Fluids, 20, p. 034103. [CrossRef]
Barbagallo, A., Sipp, D., and Schmid, P. J., 2009, “Closed-Loop Control of an Open Cavity Flow Using Reduced Order Models,” J. Fluid Mech., 641, pp. 1–50. [CrossRef]
Semeraro, O., Bagheri, S., Brandt, L., and Henningson, D. S., 2011, “Feedback Control of Three-Dimensional Optimal Disturbances Using Reduced-Order Models,” J. Fluid Mech., 677, pp. 63–102. [CrossRef]
Noack, B. R., Afanasief, K., Morzynski, M., Tadmor, G., and Thiele, F., 2003, “A Hierarchy of Low-Dimensional Models for the Transient and Post-Transient Cylinder Wake,” J. Fluid Mech., 497, pp. 335–363. [CrossRef]
Siegel, S. G., Siegel, J., Fagley, C., Luchtenburg, D. M., Cohen, K., and McLaughlin, T., 2008, “Low Dimensional Modelling of a Transient Cylinder Wake Using Double Proper Orthogonal Decomposition,” J. Fluid Mech., 610, pp. 1–42. [CrossRef]
Ilak, M., Bagheri, S., Brandt, L., Rowley, C. W., and Henningson, D. S., 2010, “Model Reduction of the Nonlinear Complex Ginzburg—Landau Equation,” SIAM J. Appl. Dyn. Sys., 9(4), pp. 1284–1302. [CrossRef]
Huang, S., and Kim, J., 2008, “Control and System Identification of Separated Flow,” Phys. Fluids, 20, p. 101509. [CrossRef]
Ma, Z., Ahuja, S., and Rowley, C. W., 2011, “Reduced-Order Models for Control of Fluids Using the Eigensystem Realization Algorithm,” Theor. Comput. Fluid Dyn., 25(1–4), pp. 233–247. [CrossRef]
Semeraro, O., Bagheri, S., Brandt, L., and Henningson, D. S., 2013, “Transition Delay in a Boundary Layer Flow Using Active Control,” J. Fluid Mech., 731(9), pp. 288–311. [CrossRef]
Dadfar, R., Fabbiane, N., Bagheri, S., and Henningson, D. S., 2014, “Centralised Versus Decentralised Active Control of Boundary Layer Instabilities,” Flow, Turbul. Combust. to be published.
Quarteroni, A., 2009, Numerical Models for Differential Problems, Springer, Milan, Italy.


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Fig. 1

Scheme of a Blasius boundary-layer flow developing over a flat plate. A disturbance modeled by d grows exponentially while convected downstream. The actuator u is used to attenuate the disturbance before it triggers transition to turbulence; the actuation signal is computed based on the measurements provided by the sensor y. The output z, located downstream of the actuator, estimates the efficiency of the control action.

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Fig. 2

Response to a small, localized initial condition in a Blasius boundary-layer flow. A Tollmien–Schlichting wave-packet emerges and grows exponentially while propagating downstream. Contours of the streamwise component of the velocity are shown as a function of the streamwise direction (x) and time (t). The location along the normal-direction y is chosen in the vicinity of the wall.

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Fig. 3

The real frequency ωr and its imaginary part ωi are shown as a function of the spatial frequency α, in (a) and (b), respectively. The relation among the spatial and temporal frequencies is given by the dispersion relation (8). Positive values of ωi characterize unstable waves (gray region).

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Fig. 4

Response to a small, localized initial condition in a 1D KS flow (6) with ℜ = 0.25,P = 0.05, and V = 0.4. The contours are shown as a function of the streamwise direction (x) and the time (t). The initial condition triggers a growing and traveling wave-packet, similar to the 2D boundary-layer flow shown in Fig. 2. [script00.m].

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Fig. 6

Top frame (a) shows the spatiotemporal response to white noise d(t). (b) The velocity contours are shown as a function of the streamwise direction (x) and time (t). The signals y(t) and z(t) are shown for two different realizations (black and gray lines) in (c) and (d), respectively. Dashed lines indicate the standard deviation of the signals. [script01.m].

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Fig. 5

Spatial support of the inputs and outputs along the streamwise direction. All the elements are modeled as a Gaussian function (14), with σd = σu = σy = σz = 4.

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Fig. 7

Time discrete impulse response (○) between the input u to the output z; due to the presence of strong time delays in the system, a lag of t ≈ 550 is observed. The relevant part of the kernel is reconstructed via a FIR filter (◻). [script02.m].

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Fig. 8

Controllability (Gc,u) and observability (Go,y) Gramians, normalized by their trace; the absolute values are reported in logarithmic scale as a function of the streamwise direction (x). Due to the symmetry, only the upper/lower triangular part of each Gramian is shown. [script03.m].

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Fig. 9

Schematic figure showing the 5 transfer functions defining the closed-loop system (35). The transfer functions Pyd,Pzd describe the input/output behavior between the disturbance d and the outputs y and z, respectively; Pyu and Pzu relate the actuator u to the two outputs y and z, respectively, while Kuy is the compensator transfer-function. Because of the convectively unstable nature of the flow, Pyu is negligible for the chosen sensor/actuator locations; thus it does not allow any feedback.

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Fig. 10

The disturbance generated by the impulse response of the system at the actuator location u in (a) is shown as a function of the streamwise direction (x) and time (t). The wave-packet is detected only by the output z (c); due to the convective nature of the flow, the sensor placed upstream of the actuator can not detect the propagating disturbance, and the resulting signal is practically null (b). [script02.m].

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Fig. 12

MPC strategy: the controller is computed over a finite time-horizon Tc, based on the predicted time-horizon Tp. Once the solution is available, the control signal is applied on a shorter time windows Ta. In the successive step, the time-window slides forward in time and the optimization is performed again, starting from a new initial condition at t = Ta. The procedure is iterated while proceeding forward in time.

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Fig. 11

Control gain K computed using the LQR technique for wz = 1 and wu = 1, (see Sec. 3.1.1). [script04.m]

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Fig. 17

Spatiotemporal evolution of the response of the system to a disturbance d(t) (a), compared to the estimated full-order state, using a Kalman filter (b); the contours are shown as a function of the streamwise direction (x) and time (t). The error-norm between the original state and the estimated state is shown in (c). The vertical blue, dashed line indicates when the estimator is turned on. [script06.m].

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Fig. 13

Control design in presence of constraints: the gray regions indicate the limits imposed to the amplitude of the control signal u(t). The control u(t) is designed following two different strategies: LQR with a saturation function (–) and constrained MPC (–), see Sec. 3.2.2. The LQR solution (– –) is introduced as reference. The performances of the controllers are shown in terms of rms-velocity reduction in Fig. 14.

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Fig. 14

Control of the KS equation. The rms velocity as a function of the x direction is analyzed; the uncontrolled configuration (–) is compared to three different control strategies already considered in Fig. 13 (same legend).

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Fig. 15.

In (a) the LQR solution (Sec. 3.1.1) is compared to the MPC gains computed for two different times of optimization Tp without constraints, see Sec. 3.2.3. The optimization times are compared to the impulse response Pzu(t) (b). Note that for longer time Tp, covering the main dynamics of the impulse response Pzu(t), the MPC and LQR solutions are equivalent.

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Fig. 16

Kalman estimation gain L computed for Rd = 1 and Rn = 0.1, (see Sec. 4.1.1). [script06.m].

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Fig. 20

Block diagram of the closed-loop system. The compensator, consisting of a controller coupled to an estimator, computes the control signal u(t) given the measurement y(t). The minimization of the measurement z(t) is the target parameter of the controller. Note that in a feedforward controller, the output z can be used to add robustness to the compensator (for instance, in adaptive filters, Sec. 5.4).

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Fig. 18

In (a) the evolution of E˜zy(i) is calculated by an adaptive LMS filter and shown as a function of the discrete time (iΔt). The estimation starts at t = 4000, as indicated by a blue dashed line (–). As the iteration progresses, the error-norm constantly reduces (b). [script07.m].

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Fig. 19

Impulse responses (yz) of the estimator as a function of the discrete time. Red circles (○) correspond to the FIR time-discrete Kalman-filter-based kernel E˜zy(i) and the blue squares (◻) to the one identified by the LMS algorithm. [script07.m].

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Fig. 23

Robustness to uncertainties of the system: the actuator is displaced of 5 length units from its nominal position. The performance of the adaptive filter FXLMS (– – and ·-) is compared to the LQR (– –), LQG (–) and P-τ (–) compensators; as a reference, the uncontrolled case is shown (–). The rms-velocity is shown as a function of the streamwise direction (x). The adaptive filter performs reasonably well in the presence of unmodeled dynamics; the performances are enhanced by the use of a online identified P˜zu (– –). The performances of the LQG (–) and P-τ (–) compensators are significantly reduced (compare with Fig. 22). [script10.m].

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Fig. 21

Spatiotemporal response in presence of a white noise input d(t) for the closed-loop system (a) and the compensator (b); the disturbance is shown as a function of the streamwise direction (x) and time (t). The measurement y(t), feeding the compensator, is shown in (c). At t = 4000 (– –), the compensator starts its action and after a short lag the actuator is fed with the computed control signal u(t). The perturbation is canceled, as shown in the contours reported in (a) and the output z(t) minimized (t  > 5000). [script08.m].

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Fig. 24

Robustness to uncertainties of the system: FXLMS control gain K˜uy(i) (◻) is shifted along the time-discrete coordinate if compared to the static LQG gain (○) to compensate for the unmodeled shift in actuator position. [script10.m].

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Fig. 22

The rms velocity as a function of the streamwise location x is shown for the uncontrolled case (–), the LQG (–), the LQR (– –) and the opposition controller P − τ (–). [script08.m, script09.m].

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Fig. 25

Two strategies are possible to compute a reduced-order compensator, reduce-then-design and design-then-reduce. In general, the two paths do not lead at the same results.

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Fig. 26

Control configuration for a 3D flow developing over a flat plate. A possible configuration consists of localized sensors and actuators placed along the spanwise direction.



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