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2015, 4(2): 131-142. doi: 10.3934/eect.2015.4.131

## On observers and compensators for infinite dimensional semilinear systems

 1 Département de Mathématiques, Université Chouaib Doukkali, Faculté des Sciences, BP 20 El Jadida 24000, Morocco

Received  May 2014 Revised  December 2014 Published  May 2015

This paper investigates two important questions for a class of partially observed infinite-dimensional semilinear systems.The first one is the design of an exponential Luenberger-like observer for this class of systems. Then, the stabilization problem around a desired equilibrium profile of the system is solved, yielding a compensator based on the Luenberger-like observer. Finally, the main result is applied to a nonisothermal chemical plug flow reactor model. The approach is illustrated by numerical simulations. The paper also gives a short overview of selected works considering the same questions for linear distributed parameter systems.
Citation: Mohammed Elarbi Achhab. On observers and compensators for infinite dimensional semilinear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 131-142. doi: 10.3934/eect.2015.4.131
##### References:
 [1] B. Abouzaid, M. E. Achhab and V. Wertz, Regulator problem for infinite-dimensional linear systems with constraints on control and its rate,, European Journal of Control, 17 (2011), 183. doi: 10.3166/ejc.17.183-190. [2] B. Abouzaid, M. E. Achhab and V. Wertz, Stabilization of a class of partially observed infinite-dimensional systems with control constraints,, IMA Journal of Mathematical Control and Information, 26 (2009), 79. doi: 10.1093/imamci/dnn014. [3] M. E. Achhab, M. Laabissi, J. Winkin and D. Dochain, State trajectory analysis of plug flow nonisothermal reactors using a nonlinear model,, Proceedings of the 38th IEEE Conference on Decision and Control. Vol. 1, (1999), 663. doi: 10.1109/CDC.1999.832862. [4] M. E. Achhab and M. Laabissi, Feedback stabilization of a class of distributed parameter systems with control constraints,, Systems & Control Letters, 45 (2002), 163. doi: 10.1016/S0167-6911(01)00171-2. [5] M. E. Achhab and V. Wertz, On stabilization of partially observed infinite-dimensional semilinear systems,, in 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations CPDE. Vol. 1, (2013), 161. doi: 10.3182/20130925-3-FR-4043.00040. [6] I. Aksikas, J. Winkin and D. Dochain, Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor,, Systems & Control Letters, 56 (2007), 122. doi: 10.1016/j.sysconle.2006.08.012. [7] R. Al-Saphory and and A. El Jai, Sensors and asymptotic $\omega$-observer for distributed diffusion systems,, Sensors, 1 (2001), 161. [8] B. Aylaj, M. E. Achhab and M. Laabissi, Asymptotic behaviour of state trajectories for a class of tubular reactor nonlinear models,, IMA Journal of Mathematical Control and Information, 24 (2007), 163. doi: 10.1093/imamci/dnl013. [9] N. Barje, M. E. Achhab and V. Wertz, Exponential observer for a class of nonlinear distributed parameter systems with application to a nonisothermal tubular reactor,, in Proceedings of the 5th International Conference on informatics in Control, (2008). [10] N. Barje, M. E. Achhab and V. Wertz, State observers for a class of semilinear infinite-dimensional systems,, International Journal of Mathematics & Statistics, 4 (2009), 69. [11] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998). [12] S. Cherkaoui and A. El jai, Consistent estimators for a class of distributed stochastic systems,, in Proceedings of the 3rd IFAC Symposium on Control of Distributed parameter Systems, (1983), 409. [13] S. Cherkaoui and M. E. Achhab, State estimation for partially observed evolution equations systems with state dependent noise,, Preprints of IFAC 9th World Congress. VI, (1984), 21. [14] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control & Information Sciences, (1978). [15] R. F. Curtain, Finite dimensional compensators for prabonic distributed systems with unbounded control and observation,, SIAM J. Control & Optimization, 22 (1984), 255. doi: 10.1137/0322018. [16] R. F. Curtain and D. Salomon, Finite dimensional compensators for infinite dimensional systems with unbounded input operators,, SIAM J. Control & Optimization, 24 (1986), 797. doi: 10.1137/0324050. [17] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4224-6. [18] A. El Jai and M. Amouroux, Sur la détermination d'un observateur de dimension finie pour une classe de systèemes linéaires à paramètres répartis,, C. R. Acad. Sc. Paris, 287 (1978). [19] A. El Jai, M. Amouroux and S. Cherkaoui, A modified Luenberger observer for parabolic systems,, in Proceedings of the 8th IFAC World Congress, (1981). [20] R. V. Gressang and G. B. Lamont, Observers for systems chracterized by semi-groups,, IEEE Trans. Autom. Control, 20 (1975), 523. [21] M. Laabissi, M. E. Achhab, J. J. Winkin and D. Dochain, Trajectory analysis of nonisothermal tubular reactor nonlinear models,, Systems & Control Letters, 42 (2001), 169. doi: 10.1016/S0167-6911(00)00088-8. [22] M. Laabissi, M. E. Achhab, J. J. Winkin and D. Dochain, Multiple equilibrium profiles for nonisothermal tubular reactor nonlinear models,, Dynamics of Continuous, 11 (2004), 339. [23] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces,, Wiley, (1976). [24] R. Nagel, ed., One-Parameter Semigroups of Positive Operators,, Lecture Notes in Mathematics, (1184). [25] A. Namir, A. Bennar and M. Laklalech, Asymptotic estimation for retarded stochastic systems,, International J. of Pure & Applied Mathematics, 41 (2007), 637. [26] C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [28] J. M. Schumacher, Dynamic Feedback in Finite and Infinite Dimensional Linear Systems,, Mathematical Center Tracts, (1981). [29] J. M. Schumacher, A direct approach to compensator design for distributed parameter systems,, SIAM J. Control & Optimization, 21 (1983), 823. doi: 10.1137/0321050. [30] I. Smets, D. Dochain and J. van Impe, Optimal temperature control of a steady-state exothermic plug flow reactor,, AIChE Journal, 48 (2002), 279. doi: 10.1002/aic.690480212. [31] M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with a priori bounded control,, Math. Control Signals Syst., 2 (1989), 265. doi: 10.1007/BF02551387. [32] J. Winkin, D. Dochain and P. Ligarius., Dynamical analysis of distributed parameter tubular reactors,, Automatica, 36 (2000), 349. doi: 10.1016/S0005-1098(99)00170-3.

show all references

##### References:
 [1] B. Abouzaid, M. E. Achhab and V. Wertz, Regulator problem for infinite-dimensional linear systems with constraints on control and its rate,, European Journal of Control, 17 (2011), 183. doi: 10.3166/ejc.17.183-190. [2] B. Abouzaid, M. E. Achhab and V. Wertz, Stabilization of a class of partially observed infinite-dimensional systems with control constraints,, IMA Journal of Mathematical Control and Information, 26 (2009), 79. doi: 10.1093/imamci/dnn014. [3] M. E. Achhab, M. Laabissi, J. Winkin and D. Dochain, State trajectory analysis of plug flow nonisothermal reactors using a nonlinear model,, Proceedings of the 38th IEEE Conference on Decision and Control. Vol. 1, (1999), 663. doi: 10.1109/CDC.1999.832862. [4] M. E. Achhab and M. Laabissi, Feedback stabilization of a class of distributed parameter systems with control constraints,, Systems & Control Letters, 45 (2002), 163. doi: 10.1016/S0167-6911(01)00171-2. [5] M. E. Achhab and V. Wertz, On stabilization of partially observed infinite-dimensional semilinear systems,, in 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations CPDE. Vol. 1, (2013), 161. doi: 10.3182/20130925-3-FR-4043.00040. [6] I. Aksikas, J. Winkin and D. Dochain, Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor,, Systems & Control Letters, 56 (2007), 122. doi: 10.1016/j.sysconle.2006.08.012. [7] R. Al-Saphory and and A. El Jai, Sensors and asymptotic $\omega$-observer for distributed diffusion systems,, Sensors, 1 (2001), 161. [8] B. Aylaj, M. E. Achhab and M. Laabissi, Asymptotic behaviour of state trajectories for a class of tubular reactor nonlinear models,, IMA Journal of Mathematical Control and Information, 24 (2007), 163. doi: 10.1093/imamci/dnl013. [9] N. Barje, M. E. Achhab and V. Wertz, Exponential observer for a class of nonlinear distributed parameter systems with application to a nonisothermal tubular reactor,, in Proceedings of the 5th International Conference on informatics in Control, (2008). [10] N. Barje, M. E. Achhab and V. Wertz, State observers for a class of semilinear infinite-dimensional systems,, International Journal of Mathematics & Statistics, 4 (2009), 69. [11] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998). [12] S. Cherkaoui and A. El jai, Consistent estimators for a class of distributed stochastic systems,, in Proceedings of the 3rd IFAC Symposium on Control of Distributed parameter Systems, (1983), 409. [13] S. Cherkaoui and M. E. Achhab, State estimation for partially observed evolution equations systems with state dependent noise,, Preprints of IFAC 9th World Congress. VI, (1984), 21. [14] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control & Information Sciences, (1978). [15] R. F. Curtain, Finite dimensional compensators for prabonic distributed systems with unbounded control and observation,, SIAM J. Control & Optimization, 22 (1984), 255. doi: 10.1137/0322018. [16] R. F. Curtain and D. Salomon, Finite dimensional compensators for infinite dimensional systems with unbounded input operators,, SIAM J. Control & Optimization, 24 (1986), 797. doi: 10.1137/0324050. [17] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4224-6. [18] A. El Jai and M. Amouroux, Sur la détermination d'un observateur de dimension finie pour une classe de systèemes linéaires à paramètres répartis,, C. R. Acad. Sc. Paris, 287 (1978). [19] A. El Jai, M. Amouroux and S. Cherkaoui, A modified Luenberger observer for parabolic systems,, in Proceedings of the 8th IFAC World Congress, (1981). [20] R. V. Gressang and G. B. Lamont, Observers for systems chracterized by semi-groups,, IEEE Trans. Autom. Control, 20 (1975), 523. [21] M. Laabissi, M. E. Achhab, J. J. Winkin and D. Dochain, Trajectory analysis of nonisothermal tubular reactor nonlinear models,, Systems & Control Letters, 42 (2001), 169. doi: 10.1016/S0167-6911(00)00088-8. [22] M. Laabissi, M. E. Achhab, J. J. Winkin and D. Dochain, Multiple equilibrium profiles for nonisothermal tubular reactor nonlinear models,, Dynamics of Continuous, 11 (2004), 339. [23] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces,, Wiley, (1976). [24] R. Nagel, ed., One-Parameter Semigroups of Positive Operators,, Lecture Notes in Mathematics, (1184). [25] A. Namir, A. Bennar and M. Laklalech, Asymptotic estimation for retarded stochastic systems,, International J. of Pure & Applied Mathematics, 41 (2007), 637. [26] C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. [28] J. M. Schumacher, Dynamic Feedback in Finite and Infinite Dimensional Linear Systems,, Mathematical Center Tracts, (1981). [29] J. M. Schumacher, A direct approach to compensator design for distributed parameter systems,, SIAM J. Control & Optimization, 21 (1983), 823. doi: 10.1137/0321050. [30] I. Smets, D. Dochain and J. van Impe, Optimal temperature control of a steady-state exothermic plug flow reactor,, AIChE Journal, 48 (2002), 279. doi: 10.1002/aic.690480212. [31] M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with a priori bounded control,, Math. Control Signals Syst., 2 (1989), 265. doi: 10.1007/BF02551387. [32] J. Winkin, D. Dochain and P. Ligarius., Dynamical analysis of distributed parameter tubular reactors,, Automatica, 36 (2000), 349. doi: 10.1016/S0005-1098(99)00170-3.
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