Volume 16 Number 4
August 2019
Article Contents
Mohamed YagoubiA Linear Quadratic Controller Design Incorporating a Parametric Sensitivity Constraint. International Journal of Automation and Computing, vol. 16, no. 4, pp. 553-563, 2019. doi: 10.1007/s11633-016-1048-5
Cite as: Mohamed YagoubiA Linear Quadratic Controller Design Incorporating a Parametric Sensitivity Constraint. International Journal of Automation and Computing, vol. 16, no. 4, pp. 553-563, 2019.

Author Biography:
• Corresponding author: Mohamed Yagoubi received M. Sc. degree in automatic control engineering from the Institut National Polytechnique de Grenoble (INPG), France in 1999, and the Ph. D. degree in automatic control from Ecole Centrale de Nantes, France in 2003. In 2004, he was a faculty member at Mines Nantes, France. Currently, he is an associate professor in the Department of Automatic Control, Production and Computer Science at Mines Nantes, France. He has published about 100 refereed journal and conference papers.
His research interests include robust control, LPV systems, descriptor systems, automotive systems, and control theory in general. He is a member of IEEE.
E-mail: mohamed.yagoubi@mines-nantes.fr (Corresponding author)
ORCID ID: 0000-0002-2586-6236
• Accepted: 2016-06-17
• Published Online: 2017-01-18
• The purpose of this paper is to propose a synthesis method of parametric sensitivity constrained linear quadratic (SCLQ) controller for an uncertain linear time invariant (LTI) system. System sensitivity to parameter variation is handled through an additional quadratic trajectory parametric sensitivity term in the standard LQ criterion to be minimized. The main purpose here is to find a suboptimal linear quadratic control taking explicitly into account the parametric uncertainties. The paper main contribution is threefold: 1) A descriptor system approach is used to show that the underlying singular linear-quadratic optimal control problem leads to a non-standard Riccati equation. 2) A solution to the proposed control problem is then given based on a connection to the so-called Lur'e matrix equations. 3) A synthesis method of multiple parametric SCLQ controllers is proposed to cover the whole parametric uncertainty while degrading as less as possible the intrinsic robustness properties of each local linear quadratic controller. Some examples are presented in order to illustrate the effectiveness of the approach.
•  [1] M. Yagoubi. A parametric sensitivity constrained linear quadratic controller. In Proceedings of the 22nd Mediterranean Conference on Control and Automation, IEEE, Palermo, Italy 2014. [2] M. Yagoubi. Synthesis of multiple sensitivity constrained controllers for parametric uncertain LTI systems. In Proceedings of the 23rd Mediterranean Conference on Control and Automation, IEEE, Torremolinos, Spain, 2015. [3] E. Kreindler. Formulation of the minimum trajectory sensitivity problem. IEEE Transactions on Automatic Control, vol. 14, no. 2, pp. 206-207, 1969.  doi: 10.1109/TAC.1969.1099130 [4] M. M. Newmann. On attempts to reduce the sensitivity of the optimal linear regulator to a parameter change. International Journal of Control, vol. 11, no. 6, pp. 1079-1084, 1970.  doi: 10.1080/00207177008905987 [5] P. J. Fleming, M. M. Newmann. Design algorithms for a sensitivity constrained suboptimal regulator. International Journal of Control, vol. 25, no. 6, pp. 965-978, 1977.  doi: 10.1080/00207177708922282 [6] D. Kyr, M. Buchner. A parametric LQ approach to multiobjective control system design. In Proceedings of the 27th IEEE Conference on Decision and Control, IEEE, Austin, USA, pp. 1278-1282, 1988. [7] A. Ansari, T. Murphey. Minimal parametric sensitivity trajectories for nonlinear systems. In Proceedings of 2013 American Control Conference, IEEE, Washington, USA, 2013. [8] G. P. Liu, R. J. Patton. Insensitive and robust control design via output feedback eigenstructure assignment. International Journal of Robust and Nonlinear Control, vol. 10, no. 9, pp. 687-697, 2000.  doi: 10.1002/(ISSN)1099-1239 [9] P. Apkarian. Nonsmooth \begin{document}$\mu$\end{document}-synthesis. International Journal of Robust and Nonlinear Control, vol. 21, no. 13, pp. 1493-1508, 2011.  doi: 10.1002/rnc.v21.13 [10] X. G. Guo, G. H. Yang. Insensitive output feedback \begin{document}$H_{\infty}$\end{document} control of delta operator systems with insensitivity to sampling time jitter. International Journal of Robust and Nonlinear Control, vol. 24, no. 4, pp. 725-743, 2014.  doi: 10.1002/rnc.v24.4 [11] M. Yagoubi, P. Chevrel. A solution to the insensitive \begin{document}$H_2$\end{document} problem and its application to automotive control design. In Proceedings of the 15th Triennial World Congress, IFAC, Barcelona, Spain, 2002. [12] M. Yagoubi, P. Chevrel. A convex method for the parametric insensitive \begin{document}$H_2$\end{document} control problem. In Proceedings of the IFAC World Congress, IFAC, Czech Republic, 2005. [13] P. Chevrel, M. Yagoubi. A parametric insensitive \begin{document}$H_2$\end{document} control design approach. International Journal of Robust and Nonlinear Control, vol. 14, no. 5, pp. 1283-1297, 2004. [14] D. J. Bender, A. J. Laub. The linear-quadratic optimal regulator for descriptor systems. IEEE Transactions on Automatic Control, vol. 32, no. 8, pp. 672-688, 1987.  doi: 10.1109/TAC.1987.1104694 [15] J. Y. Ishihara, M. H. Terra, R. M. Sales. The full information and state feedback \begin{document}$H_2$\end{document} optimal controllers for descriptor systems. Automatica, vol. 39, no. 3, pp. 391-402, 2003.  doi: 10.1016/S0005-1098(02)00243-1 [16] V. Ionescu, C. Oarǎ. Generalized continuous-time Riccati theory. Linear Algebra and its Applications, vol. 232, pp. 111-130, 1996.  doi: 10.1016/0024-3795(94)00035-2 [17] A. Ferrante, L. Ntogramatzidis. The generalized continuous algebraic Riccati equation and impulse-free continuous-time LQ optimal control. Automatica, vol. 50, no. 4, pp. 1176-1180, 2014.  doi: 10.1016/j.automatica.2014.02.014 [18] J. Choi, R. Nagamune, R. Horowitz. Synthesis of multiple robust controllers for parametric uncertain LTI systems. In Proceedings of American Control Conference, IEEE, Minneapolis, USA, 2006. [19] W. Hafez, K. Loparo, M. Hashish, E. Rasmy. Sensitivity-constrained controller design for multi-parameter systems. In Proceedings of IASTED International Conference on Applied Control and Identification, Los Angles, USA, 1986. [20] K. Takaba, T. Katayama. \begin{document}$H_2$\end{document} output feedback control for descriptor systems. Automatica, vol. 34, no. 7, pp. 841-850, 1998.  doi: 10.1016/S0005-1098(98)00025-9 [21] A. I. Lur'e. Some non-linear Problems in the Theory of Automatic Control: Nekotorye Nelineinye Zadachi Teorii Avtomaticheskogo Regulirovaniya, H. M. Stationery, 1957. [22] T. Reis. Lur'e equations and even matrix pencils. Linear Algebra and its Applications, vol. 434, no. 1, pp. 152-173, 2011. [23] J. C. Willems. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control, vol. 16, no. 6, pp. 621-634, 1971.  doi: 10.1109/TAC.1971.1099831 [24] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, Philadelphia, USA: SIAM, 1994. [25] G. Calafiore, F. Dabbene, R. Tempo. Radial and uniform distributions in vector and matrix spaces for probabilistic robustness. Topics in Control and its Applications: A Tribute to Edward J. Davison, Springer, London, UK, pp. 17-31, 1999. [26] R. C. Eberhart, J. Kennedy. A new optimizer using particle swarm theory. In Proceedings of the 6th International Symposium on Micromachine and Human Science, IEEE, Nagoya, Japan, pp. 39-43, 1995. [27] I. C. Trelea. The particle swarm optimization algorithm: Convergence analysis and parameter selection. Information Processing Letters, vol. 85, no. 6, pp. 317-325, 2003.  doi: 10.1016/S0020-0190(02)00447-7 [28] F. van den Bergh, A. P. Engelbrecht. A new locally convergent particle swarm optimizer. In Proceedings of 2002 IEEE Conference on Systems, Man and cybernetics, IEEE, Hammamet, Tunisia, 2002. [29] J. Löfberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of 2004 IEEE International Symposium on Computer Aided Control Systems Design, IEEE, Taipei, China, 2004. [30] J. F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, vol. 11, no. 1-4, pp. 625-653, 1999.  doi: 10.1080/10556789908805766 [31] F. Gay, P. De Larminat. Robust vehicle dynamics control under cornering stiffness uncertainties with insensitive \begin{document}$H_2$\end{document} theory. In Proceedings of the 4th World Multiconference on Circuits, Systems, Communications, Computers, CSCC, Vouliagmeni, Greece, 2000. [32] J. Vojtesek, P. Dostal. Optimal choice of weighting factors in adaptive linear quadratic control. International Journal of Automation and Computing, vol. 11, no. 3, pp. 241-248, 2014.  doi: 10.1007/s11633-014-0786-5
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###### Corresponding author:Mohamed Yagoubi received M. Sc. degree in automatic control engineering from the Institut National Polytechnique de Grenoble (INPG), France in 1999, and the Ph. D. degree in automatic control from Ecole Centrale de Nantes, France in 2003. In 2004, he was a faculty member at Mines Nantes, France. Currently, he is an associate professor in the Department of Automatic Control, Production and Computer Science at Mines Nantes, France. He has published about 100 refereed journal and conference papers. His research interests include robust control, LPV systems, descriptor systems, automotive systems, and control theory in general. He is a member of IEEE. E-mail: mohamed.yagoubi@mines-nantes.fr (Corresponding author) ORCID ID: 0000-0002-2586-6236

Abstract: The purpose of this paper is to propose a synthesis method of parametric sensitivity constrained linear quadratic (SCLQ) controller for an uncertain linear time invariant (LTI) system. System sensitivity to parameter variation is handled through an additional quadratic trajectory parametric sensitivity term in the standard LQ criterion to be minimized. The main purpose here is to find a suboptimal linear quadratic control taking explicitly into account the parametric uncertainties. The paper main contribution is threefold: 1) A descriptor system approach is used to show that the underlying singular linear-quadratic optimal control problem leads to a non-standard Riccati equation. 2) A solution to the proposed control problem is then given based on a connection to the so-called Lur'e matrix equations. 3) A synthesis method of multiple parametric SCLQ controllers is proposed to cover the whole parametric uncertainty while degrading as less as possible the intrinsic robustness properties of each local linear quadratic controller. Some examples are presented in order to illustrate the effectiveness of the approach.

Mohamed YagoubiA Linear Quadratic Controller Design Incorporating a Parametric Sensitivity Constraint. International Journal of Automation and Computing, vol. 16, no. 4, pp. 553-563, 2019. doi: 10.1007/s11633-016-1048-5
 Citation: Mohamed YagoubiA Linear Quadratic Controller Design Incorporating a Parametric Sensitivity Constraint. International Journal of Automation and Computing, vol. 16, no. 4, pp. 553-563, 2019.
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