Skip to main content Accessibility help

Sensor and actuator placement trade-offs for a linear model of spatially developing flows

  • Stephan F. Oehler (a1) and Simon J. Illingworth (a1)


We consider feedback flow control of the linearised complex Ginzburg–Landau system. The particular focus is on any trade-offs present when placing a single sensor and a single actuator. The work is presented in three parts. First, we consider the estimation problem in which a single sensor is used to estimate the entire flow field (without any control). Second, we consider the full information control problem in which the entire flow field is known, but only a single actuator is available for control. By considering the optimal sensor placement and optimal actuator placement while varying the stability of the system, a fundamental trade-off for both problems is made clear. Third, we consider the overall feedback control problem in which only a single sensor is available for measurement; and only a single actuator is available for control. By varying the stability of the system, similar fundamental trade-offs are made clear. We discuss implications for effective feedback control with a single sensor and a single actuator and compare it to previous placement methods.


Corresponding author

Email address for correspondence:


Hide All
Åkervik, E., Hœpffner, J., Ehrenstein, U. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305314.
Akhtar, I., Borggaard, J., Burns, J. A., Imtiaz, H. & Zietsman, L. 2015 Using functional gains for effective sensor location in flow control: a reduced-order modelling approach. J. Fluid Mech. 781, 622656.
Aström, K. J. & Murray, R. M. 2010 Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press.
Bagheri, S., Henningson, D. S., Hœpffner, J. & Schmid, P. J. 2009 Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 020803.
Belson, B. A., Semeraro, O., Rowley, C. W. & Henningson, D. S. 2013 Feedback control of instabilities in the two-dimensional Blasius boundary layer: the role of sensors and actuators. Phys. Fluids 25 (5), 054106.
Bensoussan, A. 1972 Optimization of sensors’ location in a distributed filtering problem. In Stability of Stochastic Dynamical Systems, pp. 6284. Springer.
Berger, E. 1967 Suppression of vortex shedding and turbulence behind oscillating cylinders. Phys. Fluids 10 (9), S191S193.
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305349.
Bewley, T. R, Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179225.
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67 (5), 050801.
Burns, J. A. & King, B. B. 1994 Optimal sensor location for robust control of distributed parameter systems. In Proc. 33rd IEEE Conf. Decision Control, pp. 39673972. IEEE.
Chen, K. K. & Rowley, C. W. 2011 H 2 optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system. J. Fluid Mech. 681, 241260.
Chen, K. K. & Rowley, C. W. 2014 Fluid flow control applications of ℋ2 optimal actuator and sensor placement. In American Control Conference (ACC), pp. 40444049. IEEE.
Chen, K. K. & Rowley, C. W. 2015 Heuristics for effective actuator and sensor placement in feedback flow control. In Active Flow and Combustion Control 2014, pp. 115129. Springer.
Chen, W. H. & Seinfeld, J. H. 1975 Optimal location of process measurements. Intl J. Control 21 (6), 10031014.
Choi, H., Jeon, W. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113139.
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1987 Models of hydrodynamic resonances in separated shear flows. In Proc. 6th Symp. on Turbulent Shear Flows, Toulouse, pp. 321326.
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60 (1), 25.
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1990 The effect of nonlinearity and forcing on global modes. In New Trends in Nonlinear Dynamics and Pattern-Forming Phenomena, pp. 259274. Springer.
Chomaz, J. M., Huerre, P. & Redekopp, L. G. 1991 A frequency selection criterion in spatially developing flows. Stud. Appl. Maths 84 (2), 119144.
Cohen, K., Siegel, S., McLaughlin, T., Gillies, E. & Myatt, J. 2005 Closed-loop approaches to control of a wake flow modeled by the Ginzburg–Landau equation. Comput. Fluids 34 (8), 927949.
Cohen, K., Siegel, S., McLaughlin, T. & Myatt, J. 2003 Fuzzy logic control of a circular cylinder vortex shedding model. In 41st Aerospace Sciences Meeting and Exhibit, p. 1290. AIAA.
Colburn, C. H.2011 Estimation techniques for large-scale turbulent fluid systems. PhD thesis, University of California, San Diego.
Cossu, C. & Chomaz, J. M. 1997 Global measures of local convective instabilities. Phys. Rev. Lett. 78 (23), 4387.
Gad-el-Hak, M. 1996 Modern developments in flow control. Appl. Mech. Rev. 49 (7), 365379.
Gad-el-Hak, M., Pollard, A. & Bonnet, J. 2003 Flow Control: Fundamentals and Practices. Springer Science and Business Media.
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.
Gillies, E. A. 1998 Low-dimensional control of the circular cylinder wake. J. Fluid Mech. 371, 157178.
Gillies, E. A. 2001 Multiple sensor control of vortex shedding. AIAA J. 39 (4), 748750.
Hiramoto, K., Doki, H. & Obinata, G. 2000 Optimal sensor/actuator placement for active vibration control using explicit solution of algebraic Riccati equation. J. Sound Vib. 229 (5), 10571075.
Hu, W., Morris, K. & Zhang, Y. 2016 Sensor location in a controlled thermal fluid. In 55th Conf. on Decision and Control (CDC), pp. 22592264. IEEE.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Illingworth, S. J. 2014 Estimation and control of vortex shedding at low Reynolds numbers. In 19th Australasian Fluid Mechanics Conference.
Illingworth, S. J. 2015 Dynamic estimation of vortex shedding. In 9th Intl Symp. on Turbulence and Shear Flow Phenomena (TSFP9).
Illingworth, S. J. 2016 Model-based control of vortex shedding at low Reynolds numbers. Theor. Comput. Fluid Dyn. 30 (5), 429448.
Illingworth, S. J., Morgans, A. S. & Rowley, C. W. 2012 Feedback control of cavity flow oscillations using simple linear models. J. Fluid Mech. 709, 223248.
Joshi, S. S., Speyer, J. L. & 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, 157184.
Juillet, F., Schmid, P. J. & Huerre, P. 2013 Control of amplifier flows using subspace identification techniques. J. Fluid Mech. 725, 522565.
Kasinathan, D. & Morris, K. 2013 -optimal actuator location. IEEE Trans. Autom. Control. 58 (10), 25222535.
Khan, T., Morris, K. & Stastna, M. 2015 Computation of the optimal sensor location for the estimation of an 1-D linear dispersive wave equation. In American Control Conference (ACC), 2015, pp. 52705275. IEEE.
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39 (1), 383417.
Kumar, S. & Seinfeld, J. 1978 Optimal location of measurements for distributed parameter estimation. IEEE Trans. Autom. Control 23 (4), 690698.
Lauga, E. & Bewley, T. R. 2003 The decay of stabilizability with Reynolds number in a linear model of spatially developing flows. Proc. R. Soc. Lond. A 459 (2036), 20772095.
Lauga, E. & Bewley, T. R. 2004 Performance of a linear robust control strategy on a nonlinear model of spatially developing flows. J. Fluid Mech. 512, 343374.
Liepmann, H. W. & Nosenchuck, D. M. 1982 Active control of laminar-turbulent transition. J. Fluid Mech. 118, 201204.
Litrico, X. & Georges, D. 1999 Robust continuous-time and discrete-time flow control of a dam–river system. (ii) Controller design. Appl. Math. Model. 23 (11), 829846.
Monkewitz, P. 1989 Feedback control of global oscillations in fluid systems. In 2nd Shear Flow Conference, p. 991. AIAA.
Mons, V., Chassaing, J. & Sagaut, P. 2017 Optimal sensor placement for variational data assimilation of unsteady flows past a rotationally oscillating cylinder. J. Fluid Mech. 823, 230277.
Morris, K. 2011 Linear-quadratic optimal actuator location. IEEE Trans. Autom. Control 56 (1), 113124.
Natarajan, M., Freund, J. B. & Bodony, D. J. 2016 Actuator selection and placement for localized feedback flow control. J. Fluid Mech. 809, 775792.
Noack, B. R., Morzynski, M. & Tadmor, G. 2011 Reduced-order Modelling for Flow Control. Springer Science and Business Media.
Oehler, S., Ooi, A. & Illingworth, S. J. 2016 Actuator and sensor selection for feedback control of the linearised Ginzburg–Landau equation. 20th Australasian Fluid Mechanics Conference. Australasian Fluid Mechanics Society.
Park, D. S., Ladd, D. M. & Hendricks, E. W. 1993 Feedback control of a global mode in spatially developing flows. Phys. Lett. A 182 (2–3), 244248.
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54, 263288.
Roussopoulos, K. 1993 Feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 248 (1993), 267296.
Rowley, C. W. & Juttijudata, V. 2005 Model-based control and estimation of cavity flow oscillations. In 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, pp. 512517. IEEE.
Schmid, P. J. & Brandt, L. 2014 Analysis of fluid systems: stability, receptivity, sensitivity lecture notes from the FLOW-NORDITA summer school on advanced instability methods for complex flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66 (2), 024803.
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows. Springer Science and Business Media.
Skogestad, S. & Postlethwaite, I. 2007 Multivariable Feedback Control: Analysis and Design. Wiley.
Son, D., Jeon, S. & Choi, H. 2011 A proportional–integral–differential control of flow over a circular cylinder. Phil. Trans. R. Soc. Lond. A 369 (1940), 15401555.
Taira, K., Brunton, S. L., Dawson, S. T. M., Rowley, C. W., Colonius, T., McKeon, B. J., Schmidt, O. T., Gordeyev, S., Theofilis, V. & Ukeiley, L. S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55 (12), 40134041.
Trefethen, L. N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578584.
Wehrmann, O. H. 1965 Reduction of velocity fluctuations in a Kármán vortex street by a vibrating cylinder. Phys. Fluids 8 (4), 760761.
Weideman, J. A. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465519.
Williams, J. E. F. & Zhao, B. C. 1989 The active control of vortex shedding. J. Fluids Struct. 3 (2), 115122.
Yu, T. K. & Seinfeld, J. H. 1973 Observability and optimal measurement location in linear distributed parameter systems. Intl J. Control 18 (4), 785799.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed