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June  2021, 11(2): 307-320. doi: 10.3934/naco.2020027

## 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

Received  August 2019 Revised  March 2020 Published  June 2021 Early access  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.

Citation: Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 307-320. 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. [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. [11] V. Y. Glizer, Conditions of functional null controllability for some types of singularly perturbed nonlinear systems with delays, Axioms, 8 (2019), 1-19. [12] V. Y. Glizer and V. Turetsky, Robust Controllability of Linear Systems, Nova Science Publishers Inc., New York, 2012. [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. [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. [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. [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.

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##### 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. [11] V. Y. Glizer, Conditions of functional null controllability for some types of singularly perturbed nonlinear systems with delays, Axioms, 8 (2019), 1-19. [12] V. Y. Glizer and V. Turetsky, Robust Controllability of Linear Systems, Nova Science Publishers Inc., New York, 2012. [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. [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. [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. [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.
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