We establish necessary and sufficient conditions for the realization of the upper bounds in the performance criteria for linear descriptor systems characterizing the weighted level of damping of the external and initial disturbances. The verification of these conditions is reduced to the solution of matrix equations and inequalities. The main statements are formulated with an aim of their subsequent application in the problems of robust stabilization and H∞-optimization of descriptor control systems.
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References
B. T. Polyak and P. S. Shcherbakov, Robust Stability and Control [in Russian], Nauka, Moscow (2002).
B. T. Polyak, M. V. Khlebnikov, and P. S. Shcherbakov, Control over Linear Systems under External Perturbations. Technique of Linear Matrix Inequalities [in Russian], Lenand, Moscow (2014).
D. V. Balandin and M. M. Kogan, Synthesis of the Regularities of Control on the Basis of Linear Matrix Inequalities [in Russian], Fizmatlit, Moscow (2007).
G. E. Dullerud and F. G. Paganini, A Course in Robust Control Theory. A Convex Approach, Springer, Berlin (2000).
P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H ∞ control,” Internat. J. Robust Nonlin. Control, 4, 421–448 (1994).
I. M. Inoue, T.Wada, M. Ikeda, and E. Uezato, “State-space H ∞ controller design for descriptor systems,” Automatica, 59, 164–170 (2015).
A. G. Mazko, Robust Stability and Stabilization of Dynamical Systems. Methods of Matrix and Conic Inequalities [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 102, Kiev (2016).
P. P. Khargonekar, K. M. Nagpal, and K. R. Poolla, “H ∞ control with transients,” SIAM J. Control Optim., 29, No. 6, 1373–1393 (1991).
D. V. Balandin andM. M. Kogan, “GeneralizedH ∞-optimal control as a compromise betweenH ∞-optimal and 𝛾-optimal controls,” Avtomat. Telemekh., No. 6, 20–38 (2010).
D. V. Balandin, M. M. Kogan, L. N. Krivdina, and A. A. Fedyukov, “Synthesis of a generalized H ∞-optimal control in discrete time on finite and infinite intervals,” Avtomat. Telemekh., No. 1, 3–22 (2014).
R. S. Biryukov, “Generalized H ∞-optimal filter for a continuous object on the basis of time-discrete observations,” Inform. Sist. Upravl., No. 4, (42), 89–101 (2014).
O. H. Mazko and S. N. Kusii, “Robust stabilization and damping of external disturbances in systems with control and observable outputs,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 13, No. 3 (2016), pp. 129–145.
A. G. Mazko and S. N. Kusii, “Stabilization with respect to output and weighted suppression of disturbances in discrete control systems,” Probl. Upravl. Inform., No. 6, 78–93 (2017).
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishman, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia (1994).
S. Xu, J. Lam, and Y. Zou, “New versions of bounded real lemmas for continuous and discrete uncertain systems,” Circuits, Systems, Signal Process, 26, 829–838 (2007).
M. Chadli, P. Shi, Z. Feng, and J. Lam, “New bounded real lemma formulation and H ∞ control for continuous-time descriptor systems,” Asian J. Control, 20, No. 1, 1–7 (2018).
F. Gao, W. Q. Liu, V. Sreeram, and K. L. Teo, “Bounded real lemma for descriptor systems and its application,” in: IFAC 14th Triennial World Congress (Beijing, China) (1999), pp. 1631–1636.
I. Masubushi, Y. Kamitane, A. Ohara, and N. Suda, “H ∞ control for descriptor systems: a matrix inequalities approach,” Automatica, 33, No. 4, 669–673 (1997).
L. Dai, Singular Control Systems, Springer, New York (1989).
R. Riaza, Differential-Algebraic Systems. Analytical Aspects and Circuit Applications, World Scientific, Singapore (2008).
A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Linear Systems of Differential Equations with Degenerations [in Ukrainian], Vyshcha Shkola, Kyiv (2000).
A. A. Boichuk, A. A. Pokutnyi, and V. F. Chistyakov, “Application of perturbation theory to the solvability analysis of differential algebraic equations,” Comput. Math. Math. Phys., 53, No. 6, 777–788 (2013).
P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, The LMI control Toolbox. For Use with Matlab. User’s Guide, The Math- Works, Inc., Natick (1995).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).
D. J. Bender and A. J. Laub, “The linear-quadratic optimal regulator for descriptor systems,” IEEE Trans. Automat. Control, AC-32, No. 8, 672–688 (1987).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 11, pp. 1541–1552, November, 2018.
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Mazko, A.G. Evaluation of the Weighted Level of Damping of Bounded Disturbances in Descriptor Systems. Ukr Math J 70, 1777–1790 (2019). https://doi.org/10.1007/s11253-019-01606-x
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DOI: https://doi.org/10.1007/s11253-019-01606-x