Evaluation and attainment of weighted performance measures in descriptor control systems

  • A. G. Mazko Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev
Keywords: Descriptor system, external disturbances, robust stabilization, weighted H_inf optimization

Abstract

UDC 517.925.51; 681.5.03

We consider a class of linear descriptor control systems with bounded perturbations and establish a criterion and sufficient conditions for the existence of static controllers guaranteeing that the closed-loop system is admissible and satisifies a desired estimate for the weighted  level of damping of the external and initial disturbances.  We propose new methods for the synthesis of generalized state-feedback $H_\infty$-controllers that are reduced to solving linear matrix inequalities without additional constraints. In order to illustrate the accumulated results, we present an example of descriptor system for the stabilization of an electric circuit.

Author Biography

A. G. Mazko, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev


 

References

L. Dai, Singular control systems, Springer, New York (1989), https://doi.org/10.1007/BFb0002475

R. Riaza, Differential-algebraic systems. Analytical aspects and circuit applications, World Sci., Singapore (2008), https://doi.org/10.1142/6746

Guang-Ren Duan, Analysis and design of descriptor linear systems, Springer, New York etc. (2010), https://doi.org/10.1007/978-1-4419-6397-0

Yu Feng, Mohamed Yagoubi, Robust control of linear descriptor systems, Springer Nature Singapore Pte Ltd. (2017), https://doi.org/10.1007/978-981-10-3677-4

S. Campbell, A. Ilchmann, V. Mehrmann, T. Reis (Eds.), Applications of differential-algebraic equations: examples and benchmarks, Differential-Algebraic Equations Forum, Springer Nature, Switzerland AG (2019), https://doi.org/10.1007/978-3-642-34928-7_2

A. A. Belov, A. P. Kurdyukov, Deskriptornye sistemy i zadachi upravleniya, Fizmatlit, Moskva (2015).

S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishman, Linear matrix inequalities in system and control theory, SIAM Stud. Appl. Math., 15 (1994), https://doi.org/10.1137/1.9781611970777

B. T. Polyak, P. S. Shcherbakov, Robastnaya ustojchivost' i upravlenie, Nauka, Moskva (2002).

D. V. Balandin, M. M. Kogan, Obobshchennoe $H_{∞}$-optimal'noe upravlenie kak kompromiss mezhdu $H_{∞}$-optimal'nym i $γ$-optimal'nym upravleniyami, Avtomatika i telemekhanika, № 6, 20 – 38 (2010).

Z. Feng, J. Lam, S. Xu, S. Zhou, $H_{∞}$ control with transients for singular systems, Asian Journal of Control, 18, № 3, 817 – 827 (2016), https://doi.org/10.1002/asjc.1163

A. G. Mazko, Robastnaya ustojchivost' i stabilizaciya dinamicheskih sistem. Metody matrichnyh i konusnyh neravenstv, Pr. In-tu matematiki NAN Ukraїni, 102 (2016).

P. Gahinet, P. Apkarian, A linear matrix inequality approach to $H_{∞}$ control, Intern. J. of Robust and Nonlinear Control, 4, 421 – 448 (1994), https://doi.org/10.1002/rnc.4590040403

M. Chadli, P. Shi, Z. Feng, J. Lam, New bounded real lemma formulation and $H_{∞}$ control for continuous-time descriptor systems, Asian J. Control, 20, № 1, 1 – 7 (2018), https://doi.org/10.1002/asjc.1606

I. Masubushi, Y. Kamitane, A. Ohara, N. Suda, $H_{∞}$ Control for descriptor systems: a matrix inequalities approach, Automatica, 1997, 33, № 4, 669 – 673 (1997), https://doi.org/10.1016/S0005-1098(96)00193-8

O. G. Mazko, Zvazhena ocinka i ponizhennya rivnya vplivu obmezhenih zburen' u deskriptornih sistemah keruvannya, Ukr. mat. zhurn., 72, № 11, 1510 – 1523 (2020).

O. G. Mazko, Ocinka zvazhenogo rivnya gasinnya obmezhenih zburen' u deskriptornih sistemah, Ukr. mat. zhurn., 70, № 11, 1541 – 1552 (2018).

O. G. Mazko, Weighted damping of external and initial disturbances in descriptor control systems, Ukr. mat. zhurn., 73, № 10, 1377 – 1390 (2021), https://doi.org/10.37863/umzh.v73i10.6698

D. Cobb, Robust controllability, observability and duality in singular systems, IEEE Transactions on Automatic Control, 29, 1076 – 1082 (1984), https://doi.org/10.1109/TAC.1984.1103451

K. Takaba, Robust $H^2$ control of descriptor system with time-varying uncertainty, Intern. J. of Robust and Nonlinear Control, 71, № 4, 559 – 579 (1998), https://doi.org/10.1080/002071798221678

Published
09.08.2022
How to Cite
Mazko, A. G. “Evaluation and Attainment of Weighted Performance Measures in Descriptor Control Systems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 7, Aug. 2022, pp. 980 -90, doi:10.37863/umzh.v74i7.7167.
Section
Research articles