# American Institute of Mathematical Sciences

September  2014, 4(3): 381-399. doi: 10.3934/mcrf.2014.4.381

## Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems

 1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Received  November 2013 Revised  February 2014 Published  April 2014

This paper is addressed to the disturbance decoupling and almost disturbance decoupling problems in infinite dimensions. We introduce a class of approximate finite dimensional systems, and show that if the systems are disturbance decoupled, so does the original infinite dimensional system. It is also shown that this approach can be employed to solve the almost disturbance decoupling problem. Finally, some illustrative examples are provided.
Citation: Xiuxiang Zhou. Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems. Mathematical Control & Related Fields, 2014, 4 (3) : 381-399. doi: 10.3934/mcrf.2014.4.381
##### References:
 [1] J. H. Bramble and V. Thomée, Discrete time Galerkin methods for a parabolic boundary value problem,, Ann. Mat. Pura Appl., 101 (1974), 115. doi: 10.1007/BF02417101. Google Scholar [2] R. F. Curtain, Disturbance decoupling by measurement feedback with stability for infinite dimensional systems,, Internat. J. Control, 43 (1986), 1723. doi: 10.1080/00207178608933569. Google Scholar [3] R. F. Curtain, Invariance concepts in infinite dimensions,, SIAM J. Control Optim., 24 (1986), 1009. doi: 10.1137/0324059. Google Scholar [4] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control and Information Sciences, (1978). Google Scholar [5] R. E. Edwards, Fourier Series, a Modern Introduction, vol.II,, $2^{nd}$ edition, (1982). Google Scholar [6] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar [7] K. A. Morris and R. Rebarber, Feedback invariance of SISO infinite dimensional systems,, Math. Control Signals Systems, 19 (2007), 311. doi: 10.1007/s00498-007-0021-9. Google Scholar [8] L. Pandolfi, Disturbance decoupling and invariant subspaces for delay systems,, Appl. Math. Optim., 14 (1986), 55. doi: 10.1007/BF01442228. Google Scholar [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [10] E. J. P. G. Schmidt and R. J. Stern, Invariance theory for infinite dimensional linear control systems,, Appl. Math. Optim., 6 (1980), 113. doi: 10.1007/BF01442887. Google Scholar [11] J. M. Schumacher, A direct approach to compensator design for distributed parameter systems,, SIAM J. Control Optim., 21 (1983), 823. doi: 10.1137/0321050. Google Scholar [12] H. L. Trentelman, Almost Invariant Subspaces and High Gain Feedback,, Ph.D thesis, (1986). Google Scholar [13] J. C. Willems, Almost invariant subspaces: An approach to high gain feedback design-part I: Almost controlled invariant subspaces,, IEEE Trans. Automat. Control, 26 (1981), 235. doi: 10.1109/TAC.1981.1102551. Google Scholar [14] J. L. Willems, Disturbance isolation in linear feedback systems,, Int. J. Syst. Sci., 6 (1975), 233. doi: 10.1080/00207727508941812. Google Scholar [15] W. M. Wonham, Linear Multivariable Control: A Geometric Approach,, $2^{nd}$ edition, (1979). Google Scholar [16] H. J. Zwart, Geometric Theory for Infinite Dimensional Systems,, Lecture Notes in Control and Information Sciences, (1989). doi: 10.1007/BFb0044353. Google Scholar [17] H. J. Zwart, Equivalence between open-loop and closed-loop invariance for infinite-dimensional systems: a frequency domain approach,, SIAM J. Control Optim., 26 (1988), 1175. doi: 10.1137/0326065. Google Scholar

show all references

##### References:
 [1] J. H. Bramble and V. Thomée, Discrete time Galerkin methods for a parabolic boundary value problem,, Ann. Mat. Pura Appl., 101 (1974), 115. doi: 10.1007/BF02417101. Google Scholar [2] R. F. Curtain, Disturbance decoupling by measurement feedback with stability for infinite dimensional systems,, Internat. J. Control, 43 (1986), 1723. doi: 10.1080/00207178608933569. Google Scholar [3] R. F. Curtain, Invariance concepts in infinite dimensions,, SIAM J. Control Optim., 24 (1986), 1009. doi: 10.1137/0324059. Google Scholar [4] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control and Information Sciences, (1978). Google Scholar [5] R. E. Edwards, Fourier Series, a Modern Introduction, vol.II,, $2^{nd}$ edition, (1982). Google Scholar [6] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar [7] K. A. Morris and R. Rebarber, Feedback invariance of SISO infinite dimensional systems,, Math. Control Signals Systems, 19 (2007), 311. doi: 10.1007/s00498-007-0021-9. Google Scholar [8] L. Pandolfi, Disturbance decoupling and invariant subspaces for delay systems,, Appl. Math. Optim., 14 (1986), 55. doi: 10.1007/BF01442228. Google Scholar [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [10] E. J. P. G. Schmidt and R. J. Stern, Invariance theory for infinite dimensional linear control systems,, Appl. Math. Optim., 6 (1980), 113. doi: 10.1007/BF01442887. Google Scholar [11] J. M. Schumacher, A direct approach to compensator design for distributed parameter systems,, SIAM J. Control Optim., 21 (1983), 823. doi: 10.1137/0321050. Google Scholar [12] H. L. Trentelman, Almost Invariant Subspaces and High Gain Feedback,, Ph.D thesis, (1986). Google Scholar [13] J. C. Willems, Almost invariant subspaces: An approach to high gain feedback design-part I: Almost controlled invariant subspaces,, IEEE Trans. Automat. Control, 26 (1981), 235. doi: 10.1109/TAC.1981.1102551. Google Scholar [14] J. L. Willems, Disturbance isolation in linear feedback systems,, Int. J. Syst. Sci., 6 (1975), 233. doi: 10.1080/00207727508941812. Google Scholar [15] W. M. Wonham, Linear Multivariable Control: A Geometric Approach,, $2^{nd}$ edition, (1979). Google Scholar [16] H. J. Zwart, Geometric Theory for Infinite Dimensional Systems,, Lecture Notes in Control and Information Sciences, (1989). doi: 10.1007/BFb0044353. Google Scholar [17] H. J. Zwart, Equivalence between open-loop and closed-loop invariance for infinite-dimensional systems: a frequency domain approach,, SIAM J. Control Optim., 26 (1988), 1175. doi: 10.1137/0326065. Google Scholar
 [1] Harald Fripertinger. The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications, 2011, 5 (2) : 407-416. doi: 10.3934/amc.2011.5.407 [2] Tania Biswas, Sheetal Dharmatti. Control problems and invariant subspaces for sabra shell model of turbulence. Evolution Equations & Control Theory, 2018, 7 (3) : 417-445. doi: 10.3934/eect.2018021 [3] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019099 [4] David Gómez-Ullate, Niky Kamran, Robert Milson. Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 85-106. doi: 10.3934/dcds.2007.18.85 [5] Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135 [6] Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64 [7] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [8] Matthias Täufer, Martin Tautenhahn. Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1719-1730. doi: 10.3934/cpaa.2017083 [9] Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 [10] Michel Duprez, Guillaume Olive. Compact perturbations of controlled systems. Mathematical Control & Related Fields, 2018, 8 (2) : 397-410. doi: 10.3934/mcrf.2018016 [11] Achilles Beros, Monique Chyba, Oleksandr Markovichenko. Controlled cellular automata. Networks & Heterogeneous Media, 2019, 14 (1) : 1-22. doi: 10.3934/nhm.2019001 [12] Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 [13] Victor Magron, Marcelo Forets, Didier Henrion. Semidefinite approximations of invariant measures for polynomial systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6745-6770. doi: 10.3934/dcdsb.2019165 [14] V. Casarino, K.-J. Engel, G. Nickel, S. Piazzera. Decoupling techniques for wave equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 761-772. doi: 10.3934/dcds.2005.12.761 [15] Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050 [16] Mary Chern, Barbara Lee Keyfitz. The unsteady transonic small disturbance equation: Data on oblique curves. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4213-4225. doi: 10.3934/dcds.2016.36.4213 [17] Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47. [18] David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321 [19] H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133 [20] Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307

2018 Impact Factor: 1.292