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


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



L. Dai, Singular control systems, Springer, New York (1989), https://doi.org/10.1007/BFb0002475 DOI: 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 DOI: 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 DOI: https://doi.org/10.1007/978-1-4419-6397-0_3

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 DOI: 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 DOI: https://doi.org/10.1007/978-3-030-03718-5

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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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). DOI: https://doi.org/10.37863/umzh.v72i11.2389

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 DOI: 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 DOI: 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 DOI: https://doi.org/10.1080/002071798221678

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.
Research articles