Geodesic
Transformation of optimal control problems of descriptor systems into problems with state-space systems
Kybernetika, Tome 48 (2012) no. 6, pp. 1156-1179 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We show how we can transform the ${\mathscr H}_\infty$ and ${\mathscr H}_2$ control problems of descriptor systems with invariant zeros on the extended imaginary into problems with state-space systems without such zeros. Then we present necessary and sufficient conditions for existence of solutions of the original problems. Numerical algorithm for ${\mathscr H}_\infty$ control is given, based on the Nevanlinna-Pick theorem. Also, we present an explicit formula for the optimal ${\mathscr H}_2$ controller.
We show how we can transform the ${\mathscr H}_\infty$ and ${\mathscr H}_2$ control problems of descriptor systems with invariant zeros on the extended imaginary into problems with state-space systems without such zeros. Then we present necessary and sufficient conditions for existence of solutions of the original problems. Numerical algorithm for ${\mathscr H}_\infty$ control is given, based on the Nevanlinna-Pick theorem. Also, we present an explicit formula for the optimal ${\mathscr H}_2$ controller.
Classification : 49J15, 93D15
Keywords: parametrization of stabilizing controllers; inner matrices; ${\mathscr H}_\infty$ and ${\mathscr H}_2$ control 
@article{KYB_2012_48_6_a5,
     author = {Stefanovski, Jovan},
     title = {Transformation of optimal control problems of descriptor systems into problems with state-space systems},
     journal = {Kybernetika},
     pages = {1156--1179},
     year = {2012},
     volume = {48},
     number = {6},
     mrnumber = {3052879},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2012_48_6_a5/}
}
TY  - JOUR
AU  - Stefanovski, Jovan
TI  - Transformation of optimal control problems of descriptor systems into problems with state-space systems
JO  - Kybernetika
PY  - 2012
SP  - 1156
EP  - 1179
VL  - 48
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/KYB_2012_48_6_a5/
LA  - en
ID  - KYB_2012_48_6_a5
ER  - 
%0 Journal Article
%A Stefanovski, Jovan
%T Transformation of optimal control problems of descriptor systems into problems with state-space systems
%J Kybernetika
%D 2012
%P 1156-1179
%V 48
%N 6
%U http://geodesic.mathdoc.fr/item/KYB_2012_48_6_a5/
%G en
%F KYB_2012_48_6_a5
Stefanovski, Jovan. Transformation of optimal control problems of descriptor systems into problems with state-space systems. Kybernetika, Tome 48 (2012) no. 6, pp. 1156-1179. http://geodesic.mathdoc.fr/item/KYB_2012_48_6_a5/

[1] Ball, J. A., Gohberg, I., Rodman, L.: Interpolation of Rational Matrix Functions. Birkhauser Verlag, 1990. | MR | Zbl

[2] Gahinet, P., Apkarian, P.: A linear matrix inequality approach to ${{\mathscr H}_\infty}$-control. Internat. J. Robust and Nonlinear Control 4 (1990), 4, 421-448. | DOI | MR

[3] Green, M., Glover, K., Limebeer, D., Doyle, J.: A $J$-spectral factorization approach to ${{\mathscr H}_\infty}$ control. SIAM J. Control Optim. 28 (1990), 6, 1350-1371. | DOI | MR

[4] Green, M., Limebeer, D. J. N.: Linear Robust Control. Information and System Science Series, Prentice Hall, 1994.

[5] Hunt, K. J., M., M. Šebek, Kučera, V.: Polynomial solution of the standard multivariable ${{\mathscr H}_2}$-optimal control problem. IEEE Trans. Automat. Control 39 (1994), 1502-1507. | DOI | MR

[6] Ishihara, J. Y., Terra, M. H.: Impulse controllability and observability of rectangular descriptor systems. IEEE Trans. Automat. Control 46 (2001), 6, 991-994. | DOI | MR | Zbl

[7] Iwasaki, T., Skelton, R. E.: All controllers for the general control problem: LMI existence conditions and state space formulas. Automatica 30 (1994), 8, 1307-1317. | DOI | MR

[8] Kučera, V.: A comparison of approaches to solving ${{\mathscr H}_2}$ control problems. Kybernetika 44 (2008), 3, 328-335. | MR

[9] Kučera, V.: The ${{\mathscr H}_2}$ control problem: a general transfer-function solution. Internat. J. Control 80 (2007), 5, 800-815. | DOI | MR

[10] Kwakernaak, H.: Frequency domain solution of the ${{\mathscr H}_\infty}$ problem for descriptor systems. Learning, Control and Hybrid Systems, Lecture Notes in Control and Information Sciences, Springer 241 (1999), 317-336. | MR

[11] Kwakernaak, H.: ${{\mathscr H}_2}$-optimization - Theory and applications to robust control design. Annual Reviews in Control 26 (2002), 45-56. | DOI

[12] Masubuchi, I.: Dissipativity inequalities for continuous-time descriptor systems with applications to synthesis of control gains. Systems Control Lett. 55 (2006), 158-164. | DOI | MR | Zbl

[13] Masubuchi, I.: Output feedback controller synthesis for descriptor systems satisfying closed-loop dissipativity. Automatica 43 (2007), 339-345. | DOI | MR | Zbl

[14] Saberi, A., Sannuti, P., Chen, B.: ${{\mathscr H}_2}$ Optimal Control. Prentice Hall, Englewood Cliffs, NJ 1995.

[15] Stefanovski, J.: Transformation of $J$-spectral factorization of improper matrices to proper matrices. Systems Control Lett. 59 (2009), 1, 48-49. | DOI | MR | Zbl

[16] Stefanovski, J.: Simplified formula for the controller in optimal control problems. SIAM J. Control Appl. 45 (2007), 5, 2011-2034. | DOI | MR

[17] Stefanovski, J.: On general ${{\mathscr H}_2}$ control: From frequency to time domain. Internat. J. Control 83 (2010), 12, 2519-2545. | DOI | MR

[18] Stefanovski, J.: New results and application of singular control. IEEE Trans. Automat. Control 56 (2011), 3, 632-637. | DOI | MR

[19] Takaba, K., Morihira, N., Katayama, T.: ${{\mathscr H}_\infty}$ control for descriptor systems - A $J$-spectral factorization approach. In: Proc. 33rd IEEE Conf. Decision and Control Lake Buena Vista 1994, pp. 2251-2256.

[20] Takaba, K., Katayama, T.: ${{\mathscr H}_2}$ output feedback control for descriptor systems. Automatica 34 (1998), 841-850. | DOI | MR

[21] Xin, X.: Reduced-order controllers for the ${{\mathscr H}_\infty}$ control problem with unstable invariant zeros. Automatica 40 (2004), 319-326. | DOI | MR

[22] Xin, X., Anderson, B. D. O., Mita, T.: Complete solution of the $4$-block ${{\mathscr H}_\infty}$ control problem with infinite and finite j$\omega$ - axis zeros. Internat. J. Robust and Nonlinear Control 10 (2000), 59-81. | DOI | MR

[23] Xin, X., Hara, S., Kaneda, M.: Reduced-order proper ${{\mathscr H}_\infty}$ controllers for descriptor systems: Existence conditions and LMI-based design algorithms. IEEE Trans. Automat. Control 53 (2008), 5, 1253-1258. | DOI | MR

[24] Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ 1996. | Zbl