Abstract
We present a method that seeks to combine the properties of optimal control with the robust character of closed-loop control. The method relies on the availability of a reduced model of the system to be controlled, in order to express the control problem in a low-dimensional space where the system-state-dependent optimal control law is subsequently approximated in a preprocessing stage. A polynomial expansion is used for the approximation, enabling fast update of the optimal control law each time a new observation of the system state is made available. It results in a real-time compatible, efficient (optimal-like), and robust control strategy. A compressed-sensing approach is proposed to efficiently construct the approximate control law, exploiting the compressible character of the optimal control law in the retained approximation basis. The method is demonstrated for the control of the flow around a cylinder and is shown to perform as well as the much more costly receding-horizon optimal control approach, where the exact optimal control problem is actually recomputed, even in the presence of large aleatoric perturbations. Potential and remaining issues toward application to larger dimensional reduced systems are also discussed, and some directions for improvement are suggested.
Similar content being viewed by others
References
Abramowitz M., Stegun I.: Handbook of mathematical functions. Dover, NY (1970)
Åkervik E., Hœpffner J., Ehrenstein U., Henningson D.S.: Optimal growth, model reduction and control in a separated boundary layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314 (2007)
Becker, S, Bobin, J, Candès, E.: NESTA: a fast and accurate first-order method for sparse recovery. Caltech Institute of Technology, Tech. Rep. (2009)
Benner P., Jing-Rebecca L., Pentzl T.: Numerical solution of large-scale Lyapunov equation, Riccati equations, and linear-quadratic optimal control problems. Num. Lin. Alg. Appl. 15, 755–777 (2008)
Candès E.J., Tao T.: Near-optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inform. Theory. 52, 5406–5425 (2006)
Candès E.J., Romberg J., Tao T.: Stable signal recovery from incomplete measurements. Comm. Pure Appl. Math. 59, 1207–1223 (2005)
Candès E.J., Romberg J.: Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6(2), 227–254 (2006)
Clenshaw C., Curtis A.: A method for numerical integration on a automatic computer. Numer. Math. 2, 197–205 (1960)
Donoho D.L.: Compressed sensing. IEEE Trans. Inform. Theory. 52(4), 1289–1306 (2006)
Efron B., Hastie T., Johnstone I., Tibshirani R.: Least Angle Regression. Ann. Phys. 32, 407–499 (2004)
Gerstner T., Griebel M.: Numerical integration using sparse grids. Numer. Algorithm 18, 209–232 (1998)
Gilbert J.C., Lemaréchal C.: Some numerical experiments with variable-storage quasi-Newton algorithms. Math. Program. 45, 407–435 (1989)
Iollo A., Lanteri S., Désidéri J.-A.: Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations. Theor. Comput. Fluid Dyn. 13, 377–396 (2000)
Iooss G., Joseph D.D.: Elementary stability and bifurcation theory. 2nd edn. Springer, Berlin (1990)
Kim J., Bewley T.R.: A Linear Systems Approach to Flow Control. Ann. Rev. Fluid Mech. 39, 383–417 (2007)
Laub A.J.: A Schur method for solving algebraic Riccati equations. IEEE Trans. Auto. Control. 24, 913–921 (1979)
LeMaître O., Scanlan R.H., Knio O.M.: Estimation of the flutter derivatives of an NACA airfoil by means of Navier-Stokes simulation. J. Fluid Struct. 17(1), 1–28 (2003)
Le Maître O., Mathelin L.: Equation-free model reduction for complex dynamical systems. Int. J. Numer. Meth. Fluid. 63(2), 163–184 (2009)
Lewis F., Syrmos V.: Optimal control. 2nd edn. Wiley, New York (1995)
Lumley, J.L.: The structure of inhomogeneous turbulent flows. In: Iaglom, A.M., Tatarski, V.I. (eds.) Atmospheric Turbulence and Radio Wave Propagation, 221–227 (1967)
Lumley, J.L.: Coherent structures in turbulence. In: Meyer, R.E. (ed.) Transition Turbul. 315–342 (1981)
Mathelin, L, Gallivan, K.: A compressed sensing approach for partial differential equations with random input data. Comm. in Comput. Phys., submitted (2010)
Mathelin L., Le Maître O.: Robust control of uncertain cylinder wake flows based on robust reduced order models. Comput. Fluid. 38, 1168–1182 (2009)
Morris, K, Navasca, C.: Solution of Algebraic Riccati Equations Arising in Control of Partial Differential Equations. In: Zolesio J.P., Cagnol J. (eds.) Control and Boundary Analysis. Lecture Notes in Pure Appl. Math., vol. 240. CRC Press, Boca Raton (2005)
Nobile F., Tempone R., Webster C.G.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Num. Anal. 46, 2411–2442 (2008)
Novak E., Ritter K.: High-dimensional integration of smooth functions over cubes. Numer. Math. 75, 79–97 (1996)
Novak E., Ritter K.: Simple cubature formulas with high polynomial exactness. Constr. Approx. 15, 499–522 (1999)
Petras K.: On the Smolyak cubature error for analytic functions. Adv. Comput. Math. 12, 71–93 (2000)
Petras K.: Fast calculation in the Smolyak algorithm. Numer. Algorithm. 26, 93–109 (2001)
Rugh W.J., Shamma J.S.: Research on gain scheduling. Automatica. 36, 1401–1425 (2000)
Sirovitch L.: Turbulence and the dynamics of coherent structures part i: Coherent structures. Quart. Appl. Math. 45(3), 561–571 (1987)
Smolyak S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk. SSSR 4, 240–243 (1963)
Takens F.: Detecting strange attractors in turbulence. Lect. Notes Math. 898, 366–381 (1981)
Van den Berg E., Friedlander M.P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)
Wright S.J.: Primal-Dual Interior-Point Methods. SIAM Publications, Philadelphia, PA (1997)
Zibulevsky M., Elad M.: L1-L2 optimization in signal and image processing. IEEE Signal Proc. Mag. 27(3), 76–88 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Knio.
Rights and permissions
About this article
Cite this article
Mathelin, L., Pastur, L. & Le Maître, O. A compressed-sensing approach for closed-loop optimal control of nonlinear systems. Theor. Comput. Fluid Dyn. 26, 319–337 (2012). https://doi.org/10.1007/s00162-011-0235-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-011-0235-9