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

On observers and compensators for infinite dimensional semilinear systems

Mohammed Elarbi Achhab 1,

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.
Keywords: stabilization, state estimaton, compensator, chemical reactor., Semilinear infinite-dimensional systems.
Mathematics Subject Classification: Primary: 93B50, 93C25; Secondary: 47D0.
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Al-Saphory and and A. El Jai, Sensors and asymptotic $\omega$-observer for distributed diffusion systems,, Sensors, 1 (2001), 161. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[11]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control & Information Sciences, (1978). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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). Google Scholar

[20]

R. V. Gressang and G. B. Lamont, Observers for systems chracterized by semi-groups,, IEEE Trans. Autom. Control, 20 (1975), 523. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces,, Wiley, (1976). Google Scholar

[24]

R. Nagel, ed., One-Parameter Semigroups of Positive Operators,, Lecture Notes in Mathematics, (1184). Google Scholar

[25]

A. Namir, A. Bennar and M. Laklalech, Asymptotic estimation for retarded stochastic systems,, International J. of Pure & Applied Mathematics, 41 (2007), 637. Google Scholar

[26]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[28]

J. M. Schumacher, Dynamic Feedback in Finite and Infinite Dimensional Linear Systems,, Mathematical Center Tracts, (1981). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Al-Saphory and and A. El Jai, Sensors and asymptotic $\omega$-observer for distributed diffusion systems,, Sensors, 1 (2001), 161. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[11]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford University Press, (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory,, Lecture Notes in Control & Information Sciences, (1978). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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). Google Scholar

[20]

R. V. Gressang and G. B. Lamont, Observers for systems chracterized by semi-groups,, IEEE Trans. Autom. Control, 20 (1975), 523. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces,, Wiley, (1976). Google Scholar

[24]

R. Nagel, ed., One-Parameter Semigroups of Positive Operators,, Lecture Notes in Mathematics, (1184). Google Scholar

[25]

A. Namir, A. Bennar and M. Laklalech, Asymptotic estimation for retarded stochastic systems,, International J. of Pure & Applied Mathematics, 41 (2007), 637. Google Scholar

[26]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992). Google Scholar

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[28]

J. M. Schumacher, Dynamic Feedback in Finite and Infinite Dimensional Linear Systems,, Mathematical Center Tracts, (1981). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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