American Institute of Mathematical Sciences

Advanced
doi: 10.3934/naco.2020027

Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay

Valery Y. Glizer

Department of Applied Mathematics, ORT Braude College of Engineering, Karmiel, Israel, and, Independent Center for Studies, in Control Theory and Applications, Haifa, Israel

Received  August 2019 Revised  March 2020 Published  May 2020

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.

Keywords: Controlled system, input delay, singular perturbation, parameter-dependent controllability matrix, asymptotic analysis, parameter-free controllability conditions.
Mathematics Subject Classification: Primary:34K26, 93B05, 93C23;Secondary:65F40.
Citation: Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020027
References:
[1]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhuser, Boston, 2007.  Google Scholar

[2]

M. G. Dmitriev and G. A. Kurina, Singular perturbations in control problems, Automat. Rem. Contr., 67 (2006), 1-43.  doi: 10.1134/S0005117906010012.  Google Scholar

[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.  Google Scholar

[4]

R. Gabasov and F. M. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker Inc., New York, 1976.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991.  Google Scholar

[15]

J. Klamka, Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Technical Sciences, 61 (2013), 335-342.   Google Scholar

[16] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.   Google Scholar
[17]

T. B. Kopeikina, Controllability of singularly perturbed linear systems with time-lag, Differ. Equ., 25 (1989), 1055-1064.   Google Scholar

[18]

T. B. Kopeikina, Unified method of investigating controllability and observability problems of time-variable differential systems, Funct. Differ. Equ., 13 (2006), 463-481.   Google Scholar

[19]

C. Kuehn, Multiple Time Scale Dynamics, Springer, New York, 2015. doi: 10.1007/978-3-319-12316-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Pavel, Game Theory for Control of Optical Networks, Birkhauser, Basel, Switzerland, 2012. doi: 10.1007/978-0-8176-8322-1.  Google Scholar

[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.  Google Scholar

[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. G. Dmitriev and G. A. Kurina, Singular perturbations in control problems, Automat. Rem. Contr., 67 (2006), 1-43.  doi: 10.1134/S0005117906010012.  Google Scholar

[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.  Google Scholar

[4]

R. Gabasov and F. M. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker Inc., New York, 1976.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991.  Google Scholar

[15]

J. Klamka, Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Technical Sciences, 61 (2013), 335-342.   Google Scholar

[16] P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.   Google Scholar
[17]

T. B. Kopeikina, Controllability of singularly perturbed linear systems with time-lag, Differ. Equ., 25 (1989), 1055-1064.   Google Scholar

[18]

T. B. Kopeikina, Unified method of investigating controllability and observability problems of time-variable differential systems, Funct. Differ. Equ., 13 (2006), 463-481.   Google Scholar

[19]

C. Kuehn, Multiple Time Scale Dynamics, Springer, New York, 2015. doi: 10.1007/978-3-319-12316-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Pavel, Game Theory for Control of Optical Networks, Birkhauser, Basel, Switzerland, 2012. doi: 10.1007/978-0-8176-8322-1.  Google Scholar

[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.  Google Scholar

[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 - A, 2014, 34 (12) : 5325-5357. doi: 10.3934/dcds.2014.34.5325
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665
[16]

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
[17]

Therese Mur, Hernan R. Henriquez. Relative controllability of linear systems of fractional order with delay. Mathematical Control & Related Fields, 2015, 5 (4) : 845-858. doi: 10.3934/mcrf.2015.5.845
[18]

Antonio Marigonda. Second order conditions for the controllability of nonlinear systems with drift. Communications on Pure & Applied Analysis, 2006, 5 (4) : 861-885. doi: 10.3934/cpaa.2006.5.861
[19]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[20]

Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611

 Impact Factor: 

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences