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

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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

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R. F. Curtain, Invariance concepts in infinite dimensions,, SIAM J. Control Optim., 24 (1986), 1009.  doi: 10.1137/0324059.  Google Scholar

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R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control and Information Sciences, (1978).   Google Scholar

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R. E. Edwards, Fourier Series, a Modern Introduction, vol.II,, $2^{nd}$ edition, (1982).   Google Scholar

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L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

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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

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L. Pandolfi, Disturbance decoupling and invariant subspaces for delay systems,, Appl. Math. Optim., 14 (1986), 55.  doi: 10.1007/BF01442228.  Google Scholar

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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

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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

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