Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems

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

Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems

Xiuxiang Zhou ^1,

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

Received November 2013 Revised February 2014 Published April 2014

Abstract
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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.

Keywords: disturbance decoupling problem, Almost controlled invariant subspaces, almost disturbance decoupling problem, controlled invariant subspaces, finite dimensional approximations..

Mathematics Subject Classification: Primary: 93C25; Secondary: 41A6.

Citation: Xiuxiang Zhou. Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems. Mathematical Control and 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), 115-152. doi: 10.1007/BF02417101.
[2]	R. F. Curtain, Disturbance decoupling by measurement feedback with stability for infinite dimensional systems, Internat. J. Control, 43 (1986), 1723-1743. doi: 10.1080/00207178608933569.
[3]	R. F. Curtain, Invariance concepts in infinite dimensions, SIAM J. Control Optim., 24 (1986), 1009-1031. doi: 10.1137/0324059.
[4]	R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8, Springer-Verlag, Berlin, 1978.
[5]	R. E. Edwards, Fourier Series, a Modern Introduction, vol.II, $2^{nd}$ edition, Springer-Verlag, New York, 1982.
[6]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 1998.
[7]	K. A. Morris and R. Rebarber, Feedback invariance of SISO infinite dimensional systems, Math. Control Signals Systems, 19 (2007), 311-335. doi: 10.1007/s00498-007-0021-9.
[8]	L. Pandolfi, Disturbance decoupling and invariant subspaces for delay systems, Appl. Math. Optim., 14 (1986), 55-72. doi: 10.1007/BF01442228.
[9]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[10]	E. J. P. G. Schmidt and R. J. Stern, Invariance theory for infinite dimensional linear control systems, Appl. Math. Optim., 6 (1980), 113-122. doi: 10.1007/BF01442887.
[11]	J. M. Schumacher, A direct approach to compensator design for distributed parameter systems, SIAM J. Control Optim., 21 (1983), 823-836. doi: 10.1137/0321050.
[12]	H. L. Trentelman, Almost Invariant Subspaces and High Gain Feedback, Ph.D thesis, Rijksuniversiteit Groningen, 1986.
[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), 235-252. doi: 10.1109/TAC.1981.1102551.
[14]	J. L. Willems, Disturbance isolation in linear feedback systems, Int. J. Syst. Sci., 6 (1975), 233-238. doi: 10.1080/00207727508941812.
[15]	W. M. Wonham, Linear Multivariable Control: A Geometric Approach, $2^{nd}$ edition, Springer-Verlag, New York, 1979.
[16]	H. J. Zwart, Geometric Theory for Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, 115, Springer-Verlag, Berlin, 1989. doi: 10.1007/BFb0044353.
[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), 1175-1199. doi: 10.1137/0326065.

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), 115-152. doi: 10.1007/BF02417101.
[2]	R. F. Curtain, Disturbance decoupling by measurement feedback with stability for infinite dimensional systems, Internat. J. Control, 43 (1986), 1723-1743. doi: 10.1080/00207178608933569.
[3]	R. F. Curtain, Invariance concepts in infinite dimensions, SIAM J. Control Optim., 24 (1986), 1009-1031. doi: 10.1137/0324059.
[4]	R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8, Springer-Verlag, Berlin, 1978.
[5]	R. E. Edwards, Fourier Series, a Modern Introduction, vol.II, $2^{nd}$ edition, Springer-Verlag, New York, 1982.
[6]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 1998.
[7]	K. A. Morris and R. Rebarber, Feedback invariance of SISO infinite dimensional systems, Math. Control Signals Systems, 19 (2007), 311-335. doi: 10.1007/s00498-007-0021-9.
[8]	L. Pandolfi, Disturbance decoupling and invariant subspaces for delay systems, Appl. Math. Optim., 14 (1986), 55-72. doi: 10.1007/BF01442228.
[9]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[10]	E. J. P. G. Schmidt and R. J. Stern, Invariance theory for infinite dimensional linear control systems, Appl. Math. Optim., 6 (1980), 113-122. doi: 10.1007/BF01442887.
[11]	J. M. Schumacher, A direct approach to compensator design for distributed parameter systems, SIAM J. Control Optim., 21 (1983), 823-836. doi: 10.1137/0321050.
[12]	H. L. Trentelman, Almost Invariant Subspaces and High Gain Feedback, Ph.D thesis, Rijksuniversiteit Groningen, 1986.
[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), 235-252. doi: 10.1109/TAC.1981.1102551.
[14]	J. L. Willems, Disturbance isolation in linear feedback systems, Int. J. Syst. Sci., 6 (1975), 233-238. doi: 10.1080/00207727508941812.
[15]	W. M. Wonham, Linear Multivariable Control: A Geometric Approach, $2^{nd}$ edition, Springer-Verlag, New York, 1979.
[16]	H. J. Zwart, Geometric Theory for Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, 115, Springer-Verlag, Berlin, 1989. doi: 10.1007/BFb0044353.
[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), 1175-1199. doi: 10.1137/0326065.

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