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September  2020, 13(9): 2489-2508. doi: 10.3934/dcdss.2020119

## Minimum energy compensation for discrete delayed systems with disturbances

 Faculty of Sciences Ain Chock, University Hassan II, B.P.5366, Maȃrif, Casablanca, Morocco

* Corresponding author

Received  October 2018 Revised  January 2019 Published  October 2019

This work is devoted to the remediability problem for a class of discrete delayed systems. We investigate the possibility of reducing the disturbance effect with a convenient choice of the control operator. We give the main properties and characterization results of this concept, according to the delay and the observation. Then, under an appropriate hypothesis, we demonstrate how to find the optimal control which ensures the compensation of a disturbance measured through the observation (measurements, signals, ...). The discrete version of the wave equation, as well as the usual actuators and sensors, are examined. Numerical results are also presented.

Citation: Salma Souhaile, Larbi Afifi. Minimum energy compensation for discrete delayed systems with disturbances. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2489-2508. doi: 10.3934/dcdss.2020119
##### References:
 [1] L. Afifi, M. Bahadi, A. Chafiai and A. El Mizane, Asymptotic compensation in discrete distributed systems: Analysis, approximations and simulations, Applied Mathematical Sciences, 2 (2008), 99-137.   Google Scholar [2] L. Afifi and A. El Jai, Systémes Distribués Perturbés, Presses Universitaires de Perpignan (frensh), 2015. Google Scholar [3] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar [4] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [5] A. El Jaï and A. J. Pritchard, Sensors and actuators in distributed systems, International Journal of Control, 46 (1987), 1139-1153.  doi: 10.1080/00207178708933956.  Google Scholar [6] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Systems & Control: Foundations & Applications, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.  Google Scholar [7] S. Hadd and A. Idrissi, Regular linear systems governed by systems with state, input and output delays, IMA Journal of Mathematical Control and Information, 22 (2005), 423-439.  doi: 10.1093/imamci/dni035.  Google Scholar [8] S. Hadd, An evolution equation approach to nonautonomous linear systems with state, input, and output delays, SIAM Journal on Control and Optimization, 45 (2006), 246-272.  doi: 10.1137/040612178.  Google Scholar [9] S. Hadd and Q.-C. Zhong, On feedback stabilizability of linear systems with state and input delays in Banach spaces, IEEE Transactions on Automatic Control, 54 (2009), 438-451.  doi: 10.1109/TAC.2009.2012969.  Google Scholar [10] H. Shi, G. M. Xie and W. G. Luo, Controllability of linear discrete time systems with both delayed states and delayed inputs, Abstract and Applied Analysis, (2013), Art. ID 975461, 5 pp. doi: 10.1155/2013/975461.  Google Scholar [11] V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34. American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.  Google Scholar [12] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York-London-Sydney, 1972.  Google Scholar [13] J.-L. Lions, Contrȏle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, Gauthier-Villars, Paris, 1968.  Google Scholar [14] M. Naim, F. Lahmidi, A. Namir and M. Rachik, On the output controllability of positive discrete linear delay systems, Abstract and Applied Analysis, (2017), Art. ID 3651271, 12 pp. doi: 10.1155/2017/3651271.  Google Scholar [15] V. N. Phat and T. C. Dieu, Constrained controllability of linear discrete nonstationary systems in banach spaces, SIAM J. Control Optim., 30 (1992), 1311-1318.  doi: 10.1137/0330069.  Google Scholar [16] V. N. Phat, Controllability of discrete-time systems with multiple delays on controls and states, International Journal of Control, 49 (1989), 1645-1654.  doi: 10.1080/00207178908559731.  Google Scholar [17] R. Rabah and M. Malabare, Structure at infinity revisited for delay systems, IEEE-SMC-IMACS Multiconference, Symposium on Robotics and Cybernetics (CESA'96). Symposium in Modelling, Analysis and Simulation, (1996), 87–90. https://hal.archives-ouvertes.fr/hal-01466183 Google Scholar [18] R. Rabah and M. Malabare, Weak structure at infinity and row-by-row decoupling for linear delay systems, Kybernetika, 40 (2004), 181-195.   Google Scholar [19] M. Rachik, M. Lhous and A. Tridane, Controllability and Optimal Control Problem for Linear Time-varying Discrete Distributed Systems, Systems Analysis Modelling Simulation, 43 (2003), 137-164.   Google Scholar [20] J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar [21] S. Souhaile and L. Afifi, Cheap compensation in disturbed linear dynamical systems with multi-input delays, International Journal of Dynamics and Control. https://doi.org/10.1007/s40435-018-00505-6. doi: 10.1007/s40435-018-00505-6.  Google Scholar [22] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

show all references

##### References:
 [1] L. Afifi, M. Bahadi, A. Chafiai and A. El Mizane, Asymptotic compensation in discrete distributed systems: Analysis, approximations and simulations, Applied Mathematical Sciences, 2 (2008), 99-137.   Google Scholar [2] L. Afifi and A. El Jai, Systémes Distribués Perturbés, Presses Universitaires de Perpignan (frensh), 2015. Google Scholar [3] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.  Google Scholar [4] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [5] A. El Jaï and A. J. Pritchard, Sensors and actuators in distributed systems, International Journal of Control, 46 (1987), 1139-1153.  doi: 10.1080/00207178708933956.  Google Scholar [6] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Systems & Control: Foundations & Applications, Birkhäuser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.  Google Scholar [7] S. Hadd and A. Idrissi, Regular linear systems governed by systems with state, input and output delays, IMA Journal of Mathematical Control and Information, 22 (2005), 423-439.  doi: 10.1093/imamci/dni035.  Google Scholar [8] S. Hadd, An evolution equation approach to nonautonomous linear systems with state, input, and output delays, SIAM Journal on Control and Optimization, 45 (2006), 246-272.  doi: 10.1137/040612178.  Google Scholar [9] S. Hadd and Q.-C. Zhong, On feedback stabilizability of linear systems with state and input delays in Banach spaces, IEEE Transactions on Automatic Control, 54 (2009), 438-451.  doi: 10.1109/TAC.2009.2012969.  Google Scholar [10] H. Shi, G. M. Xie and W. G. Luo, Controllability of linear discrete time systems with both delayed states and delayed inputs, Abstract and Applied Analysis, (2013), Art. ID 975461, 5 pp. doi: 10.1155/2013/975461.  Google Scholar [11] V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34. American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.  Google Scholar [12] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley-Interscience, New York-London-Sydney, 1972.  Google Scholar [13] J.-L. Lions, Contrȏle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, Gauthier-Villars, Paris, 1968.  Google Scholar [14] M. Naim, F. Lahmidi, A. Namir and M. Rachik, On the output controllability of positive discrete linear delay systems, Abstract and Applied Analysis, (2017), Art. ID 3651271, 12 pp. doi: 10.1155/2017/3651271.  Google Scholar [15] V. N. Phat and T. C. Dieu, Constrained controllability of linear discrete nonstationary systems in banach spaces, SIAM J. Control Optim., 30 (1992), 1311-1318.  doi: 10.1137/0330069.  Google Scholar [16] V. N. Phat, Controllability of discrete-time systems with multiple delays on controls and states, International Journal of Control, 49 (1989), 1645-1654.  doi: 10.1080/00207178908559731.  Google Scholar [17] R. Rabah and M. Malabare, Structure at infinity revisited for delay systems, IEEE-SMC-IMACS Multiconference, Symposium on Robotics and Cybernetics (CESA'96). Symposium in Modelling, Analysis and Simulation, (1996), 87–90. https://hal.archives-ouvertes.fr/hal-01466183 Google Scholar [18] R. Rabah and M. Malabare, Weak structure at infinity and row-by-row decoupling for linear delay systems, Kybernetika, 40 (2004), 181-195.   Google Scholar [19] M. Rachik, M. Lhous and A. Tridane, Controllability and Optimal Control Problem for Linear Time-varying Discrete Distributed Systems, Systems Analysis Modelling Simulation, 43 (2003), 137-164.   Google Scholar [20] J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5.  Google Scholar [21] S. Souhaile and L. Afifi, Cheap compensation in disturbed linear dynamical systems with multi-input delays, International Journal of Dynamics and Control. https://doi.org/10.1007/s40435-018-00505-6. doi: 10.1007/s40435-018-00505-6.  Google Scholar [22] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar
Control observation for $N = 10$
Control observation for $N = 20$
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