-
Previous Article
Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks
- NACO Home
- This Issue
-
Next Article
Examination of solving optimal control problems with delays using GPOPS-Ⅱ
Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay
Department of Applied Mathematics, ORT Braude College of Engineering, Karmiel, Israel, and, Independent Center for Studies, in Control Theory and Applications, Haifa, Israel |
A singularly perturbed linear time-dependent controlled system with a point-wise nonsmall (of order of $ 1 $) delay in the input (the control variable) is considered. Sufficient conditions of the complete Euclidean space controllability for this system, robust with respect to the parameter of singular perturbation, are derived. This derivation is based on an asymptotic analysis of the controllability matrix for the considered system and on such an analysis of the determinant of this matrix. However, this derivation does not use a slow-fast decomposition of the considered system. The theoretical result is illustrated by an example.
References:
[1] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhuser, Boston, 2007. |
[2] |
M. G. Dmitriev and G. A. Kurina,
Singular perturbations in control problems, Automat. Rem. Contr., 67 (2006), 1-43.
doi: 10.1134/S0005117906010012. |
[3] |
E. Fridman,
Robust sampled-data $H_\infty$ control of linear singularly perturbed systems, IEEE Trans. Automat. Control, 51 (2006), 470-475.
doi: 10.1109/TAC.2005.864194. |
[4] |
R. Gabasov and F. M. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker Inc., New York, 1976. |
[5] |
V. Y. Glizer,
Novel controllability conditions for a class of singularly perturbed systems with small state delays, J. Optim. Theory Appl., 137 (2008), 135-156.
doi: 10.1007/s10957-007-9324-8. |
[6] |
V. Y. Glizer,
Cheap quadratic control of linear systems with state and control delays, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 277-301.
|
[7] |
V. Y. Glizer,
Controllability conditions of linear singularly perturbed systems with small state and input delays, Math. Control Signals Systems, 28 (2016), 1-29.
doi: 10.1007/s00498-015-0152-3. |
[8] |
V. Y. Glizer,
Euclidean space output controllability of singularly perturbed systems with small state delays, J. Appl. Math. Comput., 57 (2018), 1-38.
doi: 10.1007/s12190-017-1092-5. |
[9] |
V. Y. Glizer,
Euclidean space controllability conditions for singularly perturbed linear systems with multiple state and control delays, Axioms, 8 (2019), 1-27.
doi: 10.1007/s12190-017-1092-5. |
[10] |
V. Y. Glizer, Euclidean space controllability conditions of singularly perturbed systems with multiple state and control delays, in Proceedings of the 15th IEEE International Conference on Control and Automation, Edinburgh, Scotland, (2019), 1144–1149. Google Scholar |
[11] |
V. Y. Glizer, Conditions of functional null controllability for some types of singularly perturbed nonlinear systems with delays, Axioms, 8 (2019), 1-19. Google Scholar |
[12] |
V. Y. Glizer and V. Turetsky, Robust Controllability of Linear Systems, Nova Science Publishers Inc., New York, 2012. Google Scholar |
[13] |
R. E. Kalman,
Contributions to the theory of optimal control, Bol. Soc. Mat. Mex., 5 (1960), 102-119.
|
[14] |
J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991. |
[15] |
J. Klamka,
Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Technical Sciences, 61 (2013), 335-342.
|
[16] |
P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.
![]() |
[17] |
T. B. Kopeikina,
Controllability of singularly perturbed linear systems with time-lag, Differ. Equ., 25 (1989), 1055-1064.
|
[18] |
T. B. Kopeikina,
Unified method of investigating controllability and observability problems of time-variable differential systems, Funct. Differ. Equ., 13 (2006), 463-481.
|
[19] |
C. Kuehn, Multiple Time Scale Dynamics, Springer, New York, 2015.
doi: 10.1007/978-3-319-12316-5. |
[20] |
G. A. Kurina,
Complete controllability of singularly perturbed systems with slow and fast modes, Math. Notes, 52 (1992), 1029-1033.
doi: 10.1007/BF01210436. |
[21] |
C. G. Lange and R. M. Miura,
Singular perturbation analysis of boundary-value problems for differential-difference equations. Part V: small shifts with layer behavior, SIAM J. Appl. Math., 54 (1994), 249-272.
doi: 10.1137/S0036139992228120. |
[22] |
L. Pavel, Game Theory for Control of Optical Networks, Birkhauser, Basel, Switzerland, 2012.
doi: 10.1007/978-0-8176-8322-1. |
[23] |
M. L. Pe$\stackrel{ }{ n }$a,
Asymptotic expansion for the initial value problem of the sunflower equation, J. Math. Anal. Appl., 143 (1989), 471-479.
doi: 10.1016/0022-247X(89)90053-X. |
[24] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962. |
[25] |
P. B. Reddy and and P. Sannuti, Optimal control of a coupled-core nuclear reactor by singular perturbation method, IEEE Trans. Automat. Control, 20 (1975), 766-769. Google Scholar |
[26] |
P. Sannuti,
On the controllability of singularly perturbed systems, IEEE Trans. Automat. Control, 22 (1977), 622-624.
doi: 10.1109/tac.1977.1101568. |
[27] |
P. Sannuti,
On the controllability of some singularly perturbed nonlinear systems, J. Math. Anal. Appl., 64 (1978), 579-591.
doi: 10.1016/0022-247X(78)90006-9. |
[28] |
E. Schöll, G. Hiller, P. Hövel and M. A. Dahlem,
Time-delayed feedback in neurosystems, Phil. Trans. R. Soc. A, 367 (2009), 1079-1096.
doi: 10.1098/rsta.2008.0258. |
[29] |
N. Stefanovic and L. Pavel, A Lyapunov-Krasovskii stability analysis for game-theoretic based power control in optical links, Telecommun. Syst., 47 (2011), 19-33. Google Scholar |
[30] |
O. Tsekhan, Complete controllability conditions for linear singularly perturbed time-invariant systems with multiple delays via Chang-type transformation, Axioms, 8 (2019), 1-19. Google Scholar |
[31] |
Y. Zhang, D. S. Naidu, C. Cai and Y. Zou, Singular perturbations and time scales in control theories and applications: an overview 2002–2012, Int. J. Inf. Syst. Sci. 9 (2014), 1-36. |
show all references
References:
[1] |
A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhuser, Boston, 2007. |
[2] |
M. G. Dmitriev and G. A. Kurina,
Singular perturbations in control problems, Automat. Rem. Contr., 67 (2006), 1-43.
doi: 10.1134/S0005117906010012. |
[3] |
E. Fridman,
Robust sampled-data $H_\infty$ control of linear singularly perturbed systems, IEEE Trans. Automat. Control, 51 (2006), 470-475.
doi: 10.1109/TAC.2005.864194. |
[4] |
R. Gabasov and F. M. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker Inc., New York, 1976. |
[5] |
V. Y. Glizer,
Novel controllability conditions for a class of singularly perturbed systems with small state delays, J. Optim. Theory Appl., 137 (2008), 135-156.
doi: 10.1007/s10957-007-9324-8. |
[6] |
V. Y. Glizer,
Cheap quadratic control of linear systems with state and control delays, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 277-301.
|
[7] |
V. Y. Glizer,
Controllability conditions of linear singularly perturbed systems with small state and input delays, Math. Control Signals Systems, 28 (2016), 1-29.
doi: 10.1007/s00498-015-0152-3. |
[8] |
V. Y. Glizer,
Euclidean space output controllability of singularly perturbed systems with small state delays, J. Appl. Math. Comput., 57 (2018), 1-38.
doi: 10.1007/s12190-017-1092-5. |
[9] |
V. Y. Glizer,
Euclidean space controllability conditions for singularly perturbed linear systems with multiple state and control delays, Axioms, 8 (2019), 1-27.
doi: 10.1007/s12190-017-1092-5. |
[10] |
V. Y. Glizer, Euclidean space controllability conditions of singularly perturbed systems with multiple state and control delays, in Proceedings of the 15th IEEE International Conference on Control and Automation, Edinburgh, Scotland, (2019), 1144–1149. Google Scholar |
[11] |
V. Y. Glizer, Conditions of functional null controllability for some types of singularly perturbed nonlinear systems with delays, Axioms, 8 (2019), 1-19. Google Scholar |
[12] |
V. Y. Glizer and V. Turetsky, Robust Controllability of Linear Systems, Nova Science Publishers Inc., New York, 2012. Google Scholar |
[13] |
R. E. Kalman,
Contributions to the theory of optimal control, Bol. Soc. Mat. Mex., 5 (1960), 102-119.
|
[14] |
J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991. |
[15] |
J. Klamka,
Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Technical Sciences, 61 (2013), 335-342.
|
[16] |
P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.
![]() |
[17] |
T. B. Kopeikina,
Controllability of singularly perturbed linear systems with time-lag, Differ. Equ., 25 (1989), 1055-1064.
|
[18] |
T. B. Kopeikina,
Unified method of investigating controllability and observability problems of time-variable differential systems, Funct. Differ. Equ., 13 (2006), 463-481.
|
[19] |
C. Kuehn, Multiple Time Scale Dynamics, Springer, New York, 2015.
doi: 10.1007/978-3-319-12316-5. |
[20] |
G. A. Kurina,
Complete controllability of singularly perturbed systems with slow and fast modes, Math. Notes, 52 (1992), 1029-1033.
doi: 10.1007/BF01210436. |
[21] |
C. G. Lange and R. M. Miura,
Singular perturbation analysis of boundary-value problems for differential-difference equations. Part V: small shifts with layer behavior, SIAM J. Appl. Math., 54 (1994), 249-272.
doi: 10.1137/S0036139992228120. |
[22] |
L. Pavel, Game Theory for Control of Optical Networks, Birkhauser, Basel, Switzerland, 2012.
doi: 10.1007/978-0-8176-8322-1. |
[23] |
M. L. Pe$\stackrel{ }{ n }$a,
Asymptotic expansion for the initial value problem of the sunflower equation, J. Math. Anal. Appl., 143 (1989), 471-479.
doi: 10.1016/0022-247X(89)90053-X. |
[24] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962. |
[25] |
P. B. Reddy and and P. Sannuti, Optimal control of a coupled-core nuclear reactor by singular perturbation method, IEEE Trans. Automat. Control, 20 (1975), 766-769. Google Scholar |
[26] |
P. Sannuti,
On the controllability of singularly perturbed systems, IEEE Trans. Automat. Control, 22 (1977), 622-624.
doi: 10.1109/tac.1977.1101568. |
[27] |
P. Sannuti,
On the controllability of some singularly perturbed nonlinear systems, J. Math. Anal. Appl., 64 (1978), 579-591.
doi: 10.1016/0022-247X(78)90006-9. |
[28] |
E. Schöll, G. Hiller, P. Hövel and M. A. Dahlem,
Time-delayed feedback in neurosystems, Phil. Trans. R. Soc. A, 367 (2009), 1079-1096.
doi: 10.1098/rsta.2008.0258. |
[29] |
N. Stefanovic and L. Pavel, A Lyapunov-Krasovskii stability analysis for game-theoretic based power control in optical links, Telecommun. Syst., 47 (2011), 19-33. Google Scholar |
[30] |
O. Tsekhan, Complete controllability conditions for linear singularly perturbed time-invariant systems with multiple delays via Chang-type transformation, Axioms, 8 (2019), 1-19. Google Scholar |
[31] |
Y. Zhang, D. S. Naidu, C. Cai and Y. Zou, Singular perturbations and time scales in control theories and applications: an overview 2002–2012, Int. J. Inf. Syst. Sci. 9 (2014), 1-36. |
[1] |
Pavel Krejčí, Giselle A. Monteiro. Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3051-3066. doi: 10.3934/dcdsb.2018299 |
[2] |
Ralf W. Wittenberg. Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5325-5357. doi: 10.3934/dcds.2014.34.5325 |
[3] |
Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 |
[4] |
Péter Koltai, Alexander Volf. Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times. Journal of Computational Dynamics, 2014, 1 (2) : 339-356. doi: 10.3934/jcd.2014.1.339 |
[5] |
El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control & Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013 |
[6] |
Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control & Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189 |
[7] |
Saroj P. Pradhan, Janos Turi. Parameter dependent stability/instability in a human respiratory control system model. Conference Publications, 2013, 2013 (special) : 643-652. doi: 10.3934/proc.2013.2013.643 |
[8] |
Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 |
[9] |
Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019 |
[10] |
Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743 |
[11] |
Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026 |
[12] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[13] |
Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263 |
[14] |
R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497 |
[15] |
Piotr Gwiazda, Sander C. Hille, Kamila Łyczek, Agnieszka Świerczewska-Gwiazda. Differentiability in perturbation parameter of measure solutions to perturbed transport equation. Kinetic & Related Models, 2019, 12 (5) : 1093-1108. doi: 10.3934/krm.2019041 |
[16] |
Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2561-2573. doi: 10.3934/dcdss.2020138 |
[17] |
Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020096 |
[18] |
Ross Callister, Duc-Son Pham, Mihai Lazarescu. Using distribution analysis for parameter selection in repstream. Mathematical Foundations of Computing, 2019, 2 (3) : 215-250. doi: 10.3934/mfc.2019015 |
[19] |
Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665 |
[20] |
Kasthurisamy Jothimani, Kalimuthu Kaliraj, Sumati Kumari Panda, Kotakkaran Sooppy Nisar, Chokkalingam Ravichandran. Results on controllability of non-densely characterized neutral fractional delay differential system. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020083 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]