Asymptotics of the splitting transformation for a linear stationary singularly perturbed system with delay

V. Tsekhan ; Ch.-A. Naligama

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings – XXXI". Voronezh, May 3-9, 2020, Tome 204 (2022), pp. 170-184.

Résumé

The splitting transformation is a generalization of the well-known Chang transformation for linear, stationary, singularly perturbed system with many delays in slow-state variables; it reduces the original two-speed system to two independent subsystems of smaller dimensions with different rates of change of variables. The splitting transformation leads us to Riccati and Sylvester equations for functional matrices, which can be found in the form of asymptotic series in powers of the small parameter. In this work, we prove that asymptotic approximations of any order of accuracy based on these series can be represented as finite sums in powers of $\lambda$. We compare exact solutions with approximations obtained by the method proposed.

Détail

Keywords: singularly perturbed system, delay, splitting transformation, asymptotic approximation
Mots-clés : decomposition.

Bibliographie

[1] Vasileva A. B., Dmitriev M. G., “Singulyarnye vozmuscheniya v zadachakh optimalnogo upravleniya”, Itogi nauki i tekhn. Ser. Mat. anal., 20 (1982), 3–77 | MR | Zbl

[2] Dmitriev M. G., Kurina G. A., “Singulyarnye vozmuscheniya v zadachakh upravleniya”, Avtomat. telemekh., 2006, no. 1, 3–51 | Zbl

[3] Kopeikina T. B., “Ob upravlyaemosti lineinykh singulyarno vozmuschennykh sistem s zapazdyvaniem”, Differ. uravn., 1989, 1508–1518 | Zbl

[4] Kurina G. A., “O polnoi upravlyaemosti raznotempovykh singulyarno vozmuschennykh sistem”, Mat. zametki., 52:4 (1992), 56–61 | Zbl

[5] Kheil Dzh., Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984

[6] Tsekhan O. B., “Rasscheplyayuschee preobrazovanie dlya lineinoi statsionarnoi singulyarno vozmuschennoi sistemy s zapazdyvaniem i ego primenenie k analizu i upravleniyu spektrom”, Vesn. Grozd. un-ta. Ser. 2. Mat., 7:1 (2017), 50–61

[7] Bellman R., Cooke K., Differential Difference Equations, Academic Press, New York, 1963 | MR | Zbl

[8] Chang K., “Singular perturbations of a general boundary-value problem”, SIAM J. Math. Anal., 3:3 (1972), 520–526 | DOI | MR | Zbl

[9] Chen X., Heidarinejad M., Liu J., Christofides P. D., “Composite fast-slow MPC design for nonlinear singularly perturbed systems”, AIChE J., 58:6 (2012), 1802–1811 | DOI

[10] Fridman E., “Decoupling transformation of singularly perturbed systems with small delays and its applications”, Z. Angew. Math. Mech., 76:2 (1996), 201–204 | Zbl

[11] Gajic Z., Shen X., Parallel Algorithms for Optimal Control of Large Scale Linear Systems, Springer Verlag, London, 1993 | MR | Zbl

[12] Glizer V. Ya., “$L_2$-Stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems”, Dynam. Contin. Discr. Impulsive Syst. Ser. B: Appl. Algorithms., 16 (2009), 181–213 | MR | Zbl

[13] Glizer V. Ya., “Approximate state-space controllability of linear singularly perturbed systems with two scales of state delays”, Asympt. Anal., 107:1-2 (2018.), 73–114 | MR | Zbl

[14] Johnson R., Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering, Springer-Verlag, 2005 | MR | Zbl

[15] Kokotovic P. V., Haddad A. H., “Controllability and time-optimal control of systems with slow and fast modes”, IEEE Trans. Automat. Control., 20:1 (1975), 111–113 | DOI | MR | Zbl

[16] Kokotovic P. V., Khalil H. K., O'Reilly J., Singular Perturbation Methods in Control: Analysis and Design, Academic Press, New York, 1986 | MR | Zbl

[17] Magalhaes L. T., “Exponential estimates for singularly perturbed linear functional differential equations”, J. Math. Anal. Appl., 103 (1984), 443–460 | DOI | MR | Zbl

[18] Manitius A. Z., Olbrot A. W., “Finite spectrum assignment problem for systems with delays”, IEEE Trans. Automat. Control., 24 (1979), 541–553 | DOI | MR | Zbl

[19] Michiels W., Niculescu S. I., Stability and Stabilization of Time Delay Systems: An Eigenvalue Based Approach, SIAM, Philadelphia, 2007 | MR | Zbl

[20] Niculescu S. I., Delay Effects on Stability: A Robust Control Approach, Springer, New York, 2001 | MR | Zbl

[21] Pekař L., Gao Q., “Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results”, IEEE Access., 6 (2018), 35457–35491 | DOI

[22] Prljaca N., Gajic Z., “General transformation for block diagonalization of multitime-scale singularly perturbed linear systems”, IEEE Trans. Automat. Control., 53:5 (2008), 1303–1305 | DOI | MR | Zbl

[23] Seuret A., Özbay H., Bonnet C. Mounier H. (eds.)., Low-Complexity Controllers for Time-Delay Systems, Springer, Cham, 2014 | MR | Zbl

[24] Tsekhan O., “Complete controllability conditions for linear singularly perturbed time-invariant systems with multiple delays via Chang-type transformation”, Axioms., 8:2 (2019), 71 | DOI | MR | Zbl

[25] Yang X., Zhu J. J., “A generalization of Chang transformation for linear time-varying systems”, Proc. 49 IEEE Conf. on Decision and Control (Atlanta, GA), 2010, 6863–6869

[26] Yang X., Zhu J. J., “Chang transformation for decoupling of singularly perturbed linear slowly time-varying systems”, Proc. 51 IEEE Conf. on Decision and Control (Maui, Hawaii, USA), 2012, 5755–5760 | MR

[27] Zhang Y., Naidu D. S., Cai C., et al., “Singular perturbations and time scales in control theories and applications: An overview 2002–2012”, Int. J. Inf. Syst. Sci., 9:1 (2014), 1–36 | MR