On the Sample Complexity of Stabilizing Linear Dynamical Systems from Data
Tóm tắt
Từ khóa
Tài liệu tham khảo
Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems, Adv. Des. Control, vol. 6. SIAM, Philadelphia, PA (2005). https://doi.org/10.1137/1.9780898718713
Armstrong, E.: An extension of Bass’ algorithm for stabilizing linear continuous constant systems. IEEE Trans. Autom. Control 20(1), 153–154 (1975). https://doi.org/10.1109/TAC.1975.1100874
Armstrong, E., Rublein, G.: A stabilization algorithm for linear discrete constant systems. IEEE Trans. Autom. Control 21(4), 629–631 (1976). https://doi.org/10.1109/TAC.1976.1101295
Beattie, C.A., Gugercin, S.: Realization-independent $$\cal{H}_2$$-approximation. In: 51st IEEE Conference on Decision and Control (CDC), pp. 4953–4958 (2012). https://doi.org/10.1109/CDC.2012.6426344
Behr, M., Benner, P., Heiland, J.: Example setups of Navier-Stokes equations with control and observation: Spatial discretization and representation via linear-quadratic matrix coefficients. e-print arXiv:1707.08711, arXiv (2017). https://doi.org/10.48550/arXiv.1707.08711. Mathematical Software (cs.MS)
Benner, P., Castillo, M., Quintana-Ortí, E.S., Hernández, V.: Parallel partial stabilizing algorithms for large linear control systems. J. Supercomput. 15(2), 193–206 (2000). https://doi.org/10.1023/A:1008108004247
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015). https://doi.org/10.1137/130932715
Benner, P., Heiland, J., Werner, S.W.R.: Robust output-feedback stabilization for incompressible flows using low-dimensional $$\cal H\it _{\infty }$$-controllers. Comput. Optim. Appl. 82(1), 225–249 (2022). https://doi.org/10.1007/s10589-022-00359-x
Benner, P., Li, J.R., Penzl, T.: Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer. Lin. Alg. Appl. 15(9), 755–777 (2008). https://doi.org/10.1002/nla.622
Benner, P., Schilders, W., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Silveira, L.M.: Model Order Reduction. Volume 1: System- and Data-Driven Methods and Algorithms. De Gruyter, Berlin, Boston (2021). https://doi.org/10.1515/9783110498967
Benner, P., Schilders, W., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Silveira, L.M.: Model Order Reduction. Volume 2: Snapshot-Based Methods and Algorithms. De Gruyter, Berlin, Boston (2021). https://doi.org/10.1515/9783110671490
Benner, P., Werner, S.W.R.: MORLAB – Model Order Reduction LABoratory (version 5.0) (2019). https://doi.org/10.5281/zenodo.3332716. See also: https://www.mpi-magdeburg.mpg.de/projects/morlab
Benner, P., Werner, S.W.R.: MORLAB—The Model Order Reduction LABoratory. In: P. Benner, T. Breiten, H. Faßbender, M. Hinze, T. Stykel, R. Zimmermann (eds.) Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics, vol. 171, pp. 393–415. Birkhäuser, Cham (2021). https://doi.org/10.1007/978-3-030-72983-7_19
Berberich, J., Koch, A., Scherer, C.W., Allgöwer, F.: Robust data-driven state-feedback design. In: 2020 American Control Conference (ACC), pp. 1532–1538 (2020). https://doi.org/10.23919/ACC45564.2020.9147320
Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory, Studies in Applied and Numerical Mathematics, vol. 15. SIAM, Philadelphia, PA (1994). https://doi.org/10.1137/1.9781611970777
Breiten, T., Morandin, R., Schulze, P.: Error bounds for port-Hamiltonian model and controller reduction based on system balancing. Comput. Math. Appl. 116, 100–115 (2021). https://doi.org/10.1016/j.camwa.2021.07.022
Brunton, S.L., Brunton, B.W., Proctor, J.L., Kutz, J.N.: Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control. PLoS ONE 11(2), e0150171 (2016). https://doi.org/10.1371/journal.pone.0150171
Brunton, S.L., Kutz, J.N.: Data-driven science and engineering: machine learning, dynamical systems, and control. Cambridge University Press, Cambridge (2019). https://doi.org/10.1017/9781108380690
Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U. S. A. 113(15), 3932–3937 (2016). https://doi.org/10.1073/pnas.1517384113
Campi, M.C., Lecchini, A., Savaresi, S.M.: Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automatica J. IFAC 38(8), 1337–1346 (2002). https://doi.org/10.1016/S0005-1098(02)00032-8
Datta, B.N.: Numerical Methods for Linear Control Systems: Design and Analysis. Academic Press, San Diego, CA (2004). https://doi.org/10.1016/B978-0-12-203590-6.X5000-9
De Hoop, M.V., Kovachki, N.B., Nelsen, N.H., Stuart, A.M.: Convergence rates for learning linear operators from noisy data. e-print 2108.12515, arXiv (2021). https://doi.org/10.48550/arXiv.2108.12515. Statistics Theory (math.ST)
De Persis, C., Tesi, P.: Formulas for data-driven control: Stabilization, optimality, and robustness. IEEE Trans. Autom. Control 65(3), 909–924 (2020). https://doi.org/10.1109/TAC.2019.2959924
Dean, S., Mania, H., Matni, N., Recht, B., S., T.: On the sample complexity of the linear quadratic regulator. Found. Comput. Math. 20(4), 633–679 (2020). https://doi.org/10.1007/s10208-019-09426-y
Dragan, V., Halanay, A.: Stabilization of Linear Systems. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA (1999). https://doi.org/10.1007/978-1-4612-1570-7
Drmač, Z., Mezić, I., Mohr, R.: Data driven modal decompositions: Analysis and enhancements. SIAM J. Sci. Comput. 40(4), A2253–A2285 (2018). https://doi.org/10.1137/17M1144155
Fliess, M., Join, C.: Model-free control. Int. J. Control 86(12), 2228–2252 (2013). https://doi.org/10.1080/00207179.2013.810345
Gevers, M., Bazanella, A.S., Bombois, X., Miskovic, L.: Identification and the information matrix: How to get just sufficiently rich? IEEE Trans. Autom. Control 54(12), 2828–2840 (2009). https://doi.org/10.1109/TAC.2009.2034199
Golub, G.H., Van Loan, C.F.: Matrix Computations, fourth edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2013)
Gosea, I.V., Gugercin, S., Beattie, C.: Data-driven balancing of linear dynamical systems. SIAM J. Sci. Comput. 44(1), A554–A582 (2022). https://doi.org/10.1137/21M1411081
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.2. http://cvxr.com/cvx (2020)
Grant, M.C., Boyd, S.P.: Graph implementations for nonsmooth convex programs. In: V.D. Blondel, S.P. Boyd, H. Kimura (eds.) Recent Advances in Learning and Control, Lect. Notes Control Inf. Sci., vol. 371, pp. 95–110. Springer, London (2008). https://doi.org/10.1007/978-1-84800-155-8_7
Gurobi Optimization, LLC: Gurobi optimizer reference manual (2022). https://www.gurobi.com
Ho, B.L., Kalman, R.E.: Editorial: Effective construction of linear state-variable models from input/output functions. at-Automatisierungstechnik 14(1–12), 545–548 (1966). https://doi.org/10.1524/auto.1966.14.112.545
Jonckheere, E.A., Silverman, L.M.: A new set of invariants for linear systems–application to reduced order compensator design. IEEE Trans. Autom. Control 28(10), 953–964 (1983). https://doi.org/10.1109/TAC.1983.1103159
Kaiser, E., Kutz, J.N., Brunton, S.L.: Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proc. R. Soc. A: Math. Phys. Eng. Sci. 474(2219), 20180335 (2018). https://doi.org/10.1098/rspa.2018.0335
Kaiser, E., Kutz, J.N., Brunton, S.L.: Data-driven discovery of Koopman eigenfunctions for control. Mach. Learn.: Sci. Technol. 2(3), 035023 (2021). https://doi.org/10.1088/2632-2153/abf0f5
Kramer, B., Peherstorfer, B., Willcox, K.: Feedback control for systems with uncertain parameters using online-adaptive reduced models. SIAM J. Appl. Dyn. Syst. 16(3), 1563–1586 (2017). https://doi.org/10.1137/16M1088958
Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1995)
Leibfritz, F.: $$COMPl_{e}ib$$: COnstrained Matrix-optimization Problem library – a collection of test examples for nonlinear semidefinite programs, control system design and related problems. Tech.-report, University of Trier (2004). http://www.friedemann-leibfritz.de/COMPlib_Data/COMPlib_Main_Paper.pdf
Lequin, O., Gevers, M., Mossberg, M., Bosmans, E., Triest, L.: Iterative feedback tuning of PID parameters: comparison with classical tuning rules. Control Eng. Pract. 11(9), 1023–1033 (2003). https://doi.org/10.1016/S0967-0661(02)00303-9
Mayo, A.J., Antoulas, A.C.: A framework for the solution of the generalized realization problem. Linear Algebra Appl. 425(2–3), 634–662 (2007). https://doi.org/10.1016/j.laa.2007.03.008. Special issue in honor of P. A. Fuhrmann, Edited by A. C. Antoulas, U. Helmke, J. Rosenthal, V. Vinnikov, and E. Zerz
MOSEK ApS: The MOSEK optimization toolbox for MATLAB manual. Version 9.1.9 (2019). https://docs.mosek.com/9.1/toolbox/index.html
Peherstorfer, B.: Multifidelity Monte Carlo estimation with adaptive low-fidelity models. SIAM/ASA J. Uncertainty Quantification 7(2), 579–603 (2019). https://doi.org/10.1137/17M1159208
Peherstorfer, B.: Sampling low-dimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference. SIAM J. Sci. Comput. 42(5), A3489–A3515 (2020). https://doi.org/10.1137/19M1292448
Peherstorfer, B., Gugercin, S., Willcox, K.: Data-driven reduced model construction with time-domain Loewner models. SIAM J. Sci. Comput. 39(5), A2152–A2178 (2017). https://doi.org/10.1137/16M1094750
Peherstorfer, B., Willcox, K.: Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016). https://doi.org/10.1016/j.cma.2016.03.025
Quarteroni, A., Rozza, G.: Reduced Order Methods for Modeling and Computational Reduction, MS &A – Modeling, Simulation and Applications, vol. 9. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-02090-7
Rosenbrock, H.H.: State-space and Multivariable Theory, Studies in dynamical systems, vol. 3. Wiley, New York (1970)
Safonov, M.G., Tsao, T.C.: The unfalsified control concept: A direct path from experiment to controller. In: B.A. Francis, A.R. Tannenbaum (eds.) Feedback Control, Nonlinear Systems, and Complexity, Lect. Notes Control Inf. Sci., vol. 202, pp. 196–214. Springer, Berlin, Heidelberg (1995). https://doi.org/10.1007/BFb0027678
Schaeffer, H., Caflisch, R., Hauck, C.D., Osher, S.: Sparse dynamics for partial differential equations. Proc. Natl. Acad. Sci. U. S. A. 110(17), 6634–6639 (2013). https://doi.org/10.1073/pnas.1302752110
Schaeffer, H., Tran, G., Ward, R.: Extracting sparse high-dimensional dynamics from limited data. SIAM J. Appl. Math. 78(6), 3279–3295 (2018). https://doi.org/10.1137/18M116798X
Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010). https://doi.org/10.1017/S0022112010001217
Schulze, P., Unger, B.: Data-driven interpolation of dynamical systems with delay. Syst. Control Lett. 97, 125–131 (2016). https://doi.org/10.1016/j.sysconle.2016.09.007
Schulze, P., Unger, B., Beattie, C., Gugercin, S.: Data-driven structured realization. Linear Algebra Appl. 537, 250–286 (2018). https://doi.org/10.1016/j.laa.2017.09.030
Sturm, J.F.: Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999). https://doi.org/10.1080/10556789908805766
Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: Theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014). https://doi.org/10.3934/jcd.2014.1.391
Tu, S., Boczar, R., Packard, A., Recht, B.: Non-asymptotic analysis of robust control from coarse-grained identification. e-print 1707.04791, arXiv (2017). https://doi.org/10.48550/arXiv.1707.04791. Optimization and Control (math.OC)
Van Overschee, P., De Moor, B.: Subspace Identification for Linear Systems: Theory, Implementation, Applications. Springer, Boston, MA (1996). https://doi.org/10.1007/978-1-4613-0465-4
Van Waarde, H.J., Eising, J., Trentelman, H.L., Camlibel, M.K.: Data informativity: A new perspective on data-driven analysis and control. IEEE Trans. Autom. Control 65(11), 4753–4768 (2020). https://doi.org/10.1109/TAC.2020.2966717
Voigt, M.: On linear-quadratic optimal control and robustness of differential-algebraic systems. Dissertation, Otto-von-Guericke-Universität, Magdeburg, Germany (2015)
Werner, S.W.R.: Code, data and results for numerical experiments in “On the sample complexity of stabilizing linear dynamical systems from data” (version 1.0) (2022). https://doi.org/10.5281/zenodo.5902997
Willems, J.C., Rapisarda, P., Markovsky, I., De Moor, B.L.M.: A note on persistency of excitation. Syst. Control Lett. 54(4), 325–329 (2005). https://doi.org/10.1016/j.sysconle.2004.09.003
Ziegler, J., Nichols, N.: Optimum settings for automatic controllers. Trans. ASME 64, 759–768 (1942)