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doi: 10.3934/mcrf.2021046
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Computation of open-loop inputs for uniformly ensemble controllable systems

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

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

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.

Citation: Michael Schönlein. Computation of open-loop inputs for uniformly ensemble controllable systems. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021046
References:
[1]

A. AgrachevY. 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. AndersonS. MouA. 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. BelhadjJ. 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. ChenT. 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. DirrU. 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. ShiB. 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. TieW. ZhangS. 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. TuriniciV. RamakhrishnaB. 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. ZengH. 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. ZengS. WaldherrC. 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.

show all references

References:
[1]

A. AgrachevY. 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. AndersonS. MouA. 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. BelhadjJ. 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. ChenT. 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. DirrU. 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. ShiB. 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. TieW. ZhangS. 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. TuriniciV. RamakhrishnaB. 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. ZengH. 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. ZengS. WaldherrC. 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.

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