Computation of open-loop inputs for uniformly ensemble controllable systems

Michael Schönlein

doi:10.3934/mcrf.2021046

Article Contents

2022, Volume 12, Issue 3: 813-829. Doi: 10.3934/mcrf.2021046

This issue Previous Article A nonlinear version of Halanay's inequality for the uniform convergence to the origin Next Article Controllability to rest of the Gurtin-Pipkin model

Computation of open-loop inputs for uniformly ensemble controllable systems

Michael Schönlein

Institute for Mathematics, University of Würzburg, 97074 Würzburg, Germany

This article was recruited by Birgit Jacob

Received: September 2019

Revised: August 2020

Early access: September 2021

Published: July 2022

The author is supported by the DFG grant SCHO 1780/1-1

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This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.

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Mathematics Subject Classification: Primary: 45A05, 93B05, 93B40.

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[1]	A. Agrachev, Y. Baryshnikov and A. Sarychev, Ensemble controllability by Lie algebraic methods, ESAIM Control Optim. Calc. Var., 22 (2016), 921-938. doi: 10.1051/cocv/2016029.
[2]	B. D. O. Anderson, S. Mou, A. S. Morse and U. Helmke, Decentralized gradient algorithm for solution of a linear equation, Numer. Algebra Control Optim., 6 (2016), 319-328. doi: 10.3934/naco.2016014.
[3]	A. Becker and T. Bretl, Approximate steering of a unicycle under bounded model perturbation using ensemble control, IEEE Transactions on Robotics, 28 (2012), 580-591. doi: 10.1109/TRO.2011.2182117.
[4]	M. Belhadj, J. Salomon and G. Turinici, Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups, Eur. J. Control, 22 (2015), 23-29. doi: 10.1016/j.ejcon.2014.12.003.
[5]	J. Bolte, Continuous gradient projection method in Hilbert spaces, J. Optim. Theory Appl., 119 (2003), 235-259. doi: 10.1023/B:JOTA.0000005445.21095.02.
[6]	R. Brockett, Notes on the control of the Liouville equation, in Control of Partial Differential Equations (eds. F. Alabau-Boussouira, R. Brockett, O. Glass, J. LeRousseau and E. Zuazua), vol. 2048 of Lecture Notes in Mathematics, Springer, Heidelberg, (2012), 101–129. doi: 10.1007/978-3-642-27893-8_2.
[7]	Y. Chen, T. T. Georgiou and M. Pavon, Optimal transport over a linear dynamical system, IEEE Trans. Automat. Control, 62 (2017), 2137-2152. doi: 10.1109/TAC.2016.2602103.
[8]	F. C. Chittaro and J.-P. Gauthier, Asymptotic ensemble stabilizability of the Bloch equation, Systems Control Lett., 113 (2018), 36-44. doi: 10.1016/j.sysconle.2018.01.008.
[9]	G. Dirr, Ensemble controllability of bilinear systems, Oberwolfach Reports, 9 (2012), 674-676.
[10]	G. Dirr, U. Helmke and M. Schönlein, Controlling mean and variance in ensembles of linear systems, IFAC-PapersOnLine, 49 (2016), 1018-1023.
[11]	G. Dirr and M. Schönlein, Uniform and $L^q$-ensemble ensemble reachability of parameter-dependent linear systems, J. Differential Equations, 283 (2021), 216-262. doi: 10.1016/j.jde.2021.02.032.
[12]	H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, 1996.
[13]	A. Fleig and L. Grüne, Estimates on the minimal stabilizing horizon length in model predictive control for the Fokker-Planck equation, IFAC-PapersOnLine, 49 (2016), 260-265. doi: 10.1016/j.ifacol.2016.07.451.
[14]	P. A. Fuhrmann and U. Helmke, The Mathematics of Networks of Linear Systems, Springer International Publishing, Switzerland, 2015. doi: 10.1007/978-3-319-16646-9.
[15]	B. K. Ghosh, Some new results on the simultaneous stabilizability of a family of single input, single output systems, Systems Control Lett., 6 (1985), 39-45. doi: 10.1016/0167-6911(85)90052-0.
[16]	B. K. Ghosh, An approach to simultaneous system design. I: Semialgebraic geometric methods, SIAM J. Control Optim., 24 (1986), 480-496. doi: 10.1137/0324027.
[17]	B. K. Ghosh, An approach to simultaneous system design. II: Nonswitching gain and dynamic feedback compensation by algebraic geometric methods, SIAM J. Control Optim., 26 (1988), 919-963. doi: 10.1137/0326051.
[18]	B. K. Ghosh, Transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems, SIAM J. Control Optim., 24 (1986), 1091-1109. doi: 10.1137/0324066.
[19]	H. Gzyl and J. L. Palacios, The Weierstrass approximation theorem and large deviations, Amer. Math. Monthly, 104 (1997), 650-653. doi: 10.2307/2975059.
[20]	U. Helmke and M. Schönlein, Uniform ensemble controllability for one-parameter families of time-invariant linear systems, Systems Control Lett., 71 (2014), 69-77. doi: 10.1016/j.sysconle.2014.05.015.
[21]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
[22]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.
[23]	R. Kress, Linear Integral Equations, 3rd edition, Springer, New York, 2014 doi: 10.1007/978-1-4614-9593-2.
[24]	J.-S. Li, Ensemble control of finite-dimensional time-varying linear systems, IEEE Trans. Automat. Control, 56 (2011), 345-357. doi: 10.1109/TAC.2010.2060259.
[25]	J.-S. Li and N. Khaneja, Control of inhomogeneous quantum ensembles, Physical review A, 73 (2006), 030302. doi: 10.1103/PhysRevA.73.030302.
[26]	J.-S. Li and J. Qi, Ensemble control of time-invariant linear systems with linear parameter variation, IEEE Trans. Automat. Control, 61 (2016), 2808-2820. doi: 10.1109/TAC.2015.2503698.
[27]	S. Mou and A. S. Morse, A fixed-neighbor, distributed algorithm for solving a linear algebraic equation, 2013 European Control Conference (ECC), (2013), 2269–2273. doi: 10.23919/ECC.2013.6669741.
[28]	M. Z. Nashed and G. Wahba, Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind, Math. Comp., 28 (1974), 69-80. doi: 10.1090/S0025-5718-1974-0461895-1.
[29]	J. W. Neuberger, Sobolev Gradients and Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04041-2.
[30]	G. Pedrick, Theory of Reproducing Kernels for Hilbert Spaces of Vector Valued Functions, PhD. Thesis, University of Kansas, 1958.
[31]	M. Schönlein and U. Helmke, Controllability of ensembles of linear dynamical systems, Math. Comput. Simulation, 125 (2016), 3-14. doi: 10.1016/j.matcom.2015.10.006.
[32]	G. Shi, B. D. O. Anderson and U. Helmke, Network flows that solve linear equations, IEEE Trans. Automat. Control, 62 (2017), 2659-2674. doi: 10.1109/TAC.2016.2612819.
[33]	E. D. Sontag, Mathematical Control Theory. Deterministic Finite Dimensional Systems, 2nd Ed., Springer, New York, 1998. doi: 10.1007/978-1-4612-0577-7.
[34]	A. Tannenbaum, Invariance and System Theory: Algebraic and Geometric Aspects., Lecture Notes in Mathematics. 845. Berlin-Heidelberg-New York: Springer-Verlag, 1981.
[35]	L. Tie, W. Zhang, S. Zeng and J.-S. Li, Explicit input signal design for stable linear ensemble systems, IFAC-PapersOnLine, 50 (2017), 3051-3056.
[36]	R. Triggiani, Controllability and observability in Banach space with bounded operators, SIAM J. Control, 13 (1975), 462-491. doi: 10.1137/0313028.
[37]	G. Turinici, V. Ramakhrishna, B. Li and H. Rabitz, Optimal discrimination of multiple quantum systems: Controllability analysis, J. Phys. A, 37 (2004), 273-282. doi: 10.1088/0305-4470/37/1/019.
[38]	G. Wahba, Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind, J. Approximation Theory, 7 (1973), 167-185. doi: 10.1016/0021-9045(73)90064-6.
[39]	S. Zeng and F. Allgöwer, A moment-based approach to ensemble controllability of linear systems, Systems Control Lett., 98 (2016), 49-56. doi: 10.1016/j.sysconle.2016.09.020.
[40]	S. Zeng, H. Ishii and F. Allgöwer, Sampled observability and state estimation of discrete ensembles, IEEE Trans. Autom. Contr., 62 (2017), 2406-2418. doi: 10.1109/TAC.2016.2613478.
[41]	S. Zeng, S. Waldherr, C. Ebenbauer and F. Allgöwer, Ensemble observability of linear systems, IEEE Trans. Automat. Control, 61 (2016), 1452-1465. doi: 10.1109/TAC.2015.2463631.
[42]	S. Zeng, W. Zhang and J. Li, On the computation of control inputs for linear ensembles, 2018 Annual American Control Conference (ACC), (2018), 6101–6107. doi: 10.23919/ACC.2018.8431390.