Abstract
A class of systems, described by algebraic-difference equations, is under consideration. Such systems are called descriptor (singular). For these systems the conditions of anisotropic norm boundedness are obtained. Anisotropic norm describes the root-mean-square gain of the system with respect to random Gaussian stationary disturbances, which are characterized by mean anisotropy. The conditions are formulated in the form of the theorem, detailed proof is given. Numerical example, illustrating anisotropic norm computation method for descriptor systems based of the proven theorem, is considered.
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References
I. G. Vladimirov, A. P. Kurdyukov, and A. V. Semenov, “Anisotropy of signals and entropy of linear stationary systems,” Dokl. Math. 51(3), 388–390.
I. G. Vladimirov, P. Diamond, and P. Kloeden, “Anisotropy-based robust performance analysis of finite horizon linear discrete timevarying systems,” Automat. Remote Control 67(8), 1265–1282.
I. G. Vladimirov, A. P. Kurdjukov, and A. V. Semyonov, “State-space solution to anisotropy-based stochastic H ∞-optimization problem,” in Proc. 13th IFAC World Congress (CA, San-Francisco, 1996), pp. 427–432.
M. M. Tchaikovsky, A. P. Kurdyukov, and V. N. Timin, “Synthesis of anisotropic suboptimal controllers by convex optimization,” Submitted to European J. Contr. Preprint is available from http://arxiv.org/abs/1108.4982, 2011.
B. L. Stevens and F. L. Lewis, Aircraft Modeling, Dynamics and Control (Wiley, N.Y., 1991).
R. W. Newcomb, “The semistate description of nonlinear time-variable circuits,” IEEE Trans. Circuits Syst. 28, 62–71 (1981).
R. W. Newcomb and B. Dziurla, “Some circuits and systems applications of semistate theory,” Circuit, Syst., Sig. Process. 8, 235–260 (1989).
H. Hemami and B. F. Wyman, “Modeling and control of constrained dynamic systems with application to biped locomotion in the frontal plane,” IEEE Trans. Automat. Control. 24, 526–535 (1979).
A. Luenberger and D. G. Arbel, “Singular dynamic Leontief systems,” Econometrica 45, 991–995 (1977).
M. Sh. Misrikhanov and V. N. Ryabchenko, “A new criteria of static stability of electrical power system with algebraic-differential mathematical model,” Vestnik Ivanovo State Power Univ., No. 2, 76–81 (2009).
M. Sh. Misrikhanov and V. N. Ryabchenko, “Iterative criteria of static stability of electrical power system in algebraic-differential form,” Stochastic Optimization Informatics 5(1–1), 209–225 (2009).
A. P. Kurdyukov, E. A. Maximov, and M. M. Tchaikovsky, “Anisotropy-based bounded real lemma,” in Proc. 19th Intern. Sympos. of Mathematical Theory of Networks and Systems (Budapest, Hungary, 2010), pp. 2391–2397.
A. A. Belov, “Anisotropy-based control synthesis for descriptor systems,” Ph. D. Thesis (M., 2011).
L. Dai, Singular Control Systems, Lecture Notes in Control and Information Sciences (Springer-Verlag, N.Y., 1989).
S. Xu and J. Lam, Robust Control and Filtering of Singular Systems. Lecture Notes in Control and Information Sciences (Springer-Verlag, Berlin, 2006).
A. V. Semyonov, I. G. Vladimirov, and A. P. Kurdjukov, “Stochastic approach to H ∞-optimization,” in Proc. 33rd IEEE Conf. Decision and Control (Florida, 1994), pp. 2249–2250.
R. M. Gray, Entropy and Information Theory (Springer-Verlag, N.Y., 1991).
U. Grenander and G. Szegö, Toeplitz Forms and Their Applications (University of California Press, Berkeley, 1958).
R. A. Horn and C. R. Johnson, Matrix Analysis (Cambrige University Press, N.Y., 2007).
P. Diamond, I. G. Vladimirov, A. P. Kurdjukov, and A. V. Semyonov, “Anisopropy-based performance analysis of linear discrete-time-invariant control systems,” Int. J. Control. 74, 28–42 (2001).
I. G. Vladimirov, A. P. Kurdjukov, and A. V. Semyonov, “On computing the anisotropic norm of linear discretetime-invariant systems,” in Proc. 13th IFAC World Congr. (CA, San-Francisco, 1996), pp. 179–184.
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Original Russian Text © O.G. Andrianova, A.A. Belov, A.P. Kurdyukov, 2015, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2015, No. 1, pp. 29–40.
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Andrianova, O.G., Belov, A.A. & Kurdyukov, A.P. Conditions of anisotropic norm boundedness for descriptor systems. J. Comput. Syst. Sci. Int. 54, 27–38 (2015). https://doi.org/10.1134/S1064230714060021
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DOI: https://doi.org/10.1134/S1064230714060021