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Abstrakty
The aim of this paper is to show that a real order generalization of the dissipative concepts is a useful tool to determine the stability (in the Lyapunov and in the input-output sense) and to design control strategies not only for fractional order non-linear systems, but also for systems composed of integer and fractional order subsystems (mixed-order systems). In particular, the fractional control of integer order system (e.g. PI? control) can be formalized. The key point is that the gradations of dissipativeness, passivity and positive realness concepts are related among them. Passivating systems is used as a strategy to stabilize them, which is studied in the non-adaptive as well as in the adaptive case.
Rocznik
Tom
Strony
445--454
Opis fizyczny
Bibliogr. 33 poz.
Twórcy
autor
- Electrical Engineering, University of Chile, Av. Tupper 2007, Casilla 412-3, Santiago, Chile
- Advanced Mining Technology 2007, Casilla 412-3, Santiago, Chile
autor
- Departmento de Electricidad, Facultad de Ingeniería, Universidad Teconológica Metropolitana, Av. José Pedro Alessandri 1242, Santiago, Chile
Bibliografia
- [1] H.K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, Prentice Hall, NJ, 1996.
- [2] A. van der Schaft, L2-Gain and Passivity Techniques inNonlinear Control, Springer-Verlag, Berlin, Germany, 2000.
- [3] J.C. Willems, “Dissipative dynamical system, part I”, Arch. Rational Mech. Anal. 45 (5), 321–351 (1972).
- [4] C.I. Byrnes, A. Isidori, and J.C. Willems, “Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems”, IEEE Trans. Aut. Control 36 (11), 1228?1240 (1991).
- [5] D.J. Hill and P.J. Moylan, “Connections between finite-gain and asymptotic stability”, IEEE Trans. Automat. Contr. AC-25, 931?936 (1980).
- [6] N. Chopra and M.W. Spong, “Passivity-based control of multiagent systems”, in Advances in Robot Control, pp. 107–134, Springer-Verlag, Berlin, Germany, 2006.
- [7] N. Kottenstette, J. Hall, X. Koutsoukos, J. Sztipanovits, and P. Antsaklis, “Design of networked control systems using passivity”, IEEE Trans. Control Syst. Technol. 21 (3), 649–665 (2013).
- [8] M. Arcak. “Passivity as a design tool for group coordination”. IEEE Transactions on Automatic Control 52 (8), 1380–1390 (2007).
- [9] J.A. Gallegos and M. Duarte-Mermoud, “On the Lyapunov Theory for fractional order systems”, Appl. Math. Comput, 287, 161?170 (2016).
- [10] W. Chao-Jun, Z. Yan-Bin, and Y. Ning-Ning, “The synchronization of a fractional order hyperchaotic system based on passive control”, Chin. Phys. B 20 (6), 605051–605057 (2011).
- [11] S. Kuntanapreeda, “Adaptive control of fractional-order unified chaotic systems using a passivity-based control approach”, Nonlinear Dynamics 84 (4), 2505–2515 (2016).
- [12] K. Diethelm, The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics 2004, Springer- Verlag, Berlin, 2010.
- [13] I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
- [14] J. Sabatier and C. Farges, “Theory and applications of fractional differential equations”, 19th IFAC World Congress, Aug 2014, Cape Town, South Africa. pp. 2884–2890, 2014.
- [15] G. Montseny, “Diffusive representation of pseudo differential time operators”, Proceedings ESSAIM 5, 159–175 (1998).
- [16] R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, Fractional Order Systems: Modeling and Control Applications, World Scientific, Singapore 2010.
- [17] M. Rakhshan, V. Gupta, and B. Godwine, On Passivity of Fractional Order Systems, arXiv:1706.07551v1 [math.DS], 2017.
- [18] G. Fernández-Anaya, J.J. Flores-Godoy, and A.F. Lugo-Penaloza, “Stabilization and passification of distributed-order fractional linear systems using methods of preservation”, J. Franklin. Inst. 350 (10), 2881–2900 (2013).
- [19] A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
- [20] N. Kottenstette, M.J. McCourt, M. Xia, V. Gupta, and P.J. Antsaklis, “On relationships among passivity, positive realness, and dissipativity in linear systems”, Automatica 50 (4), 1003–1016 (2014).
- [21] F. Kerber and A. van der Schaft, “Compositional properties of passivity”, Proc. IEEE CDC-ECC, pp. 4628–4633, 2011.
- [22] S.Z. Kong and A. van der Schaft, The converse of the passivity and small-gain theorem for nonlinear input-output maps, arXiv preprint arXiv:1707.00148.
- [23] J.A. Gallegos, M.A. Duarte-Mermoud, N. Aguila-Camacho, and R. Castro-Linares, “On fractional extensions of Barbalat Lemma”, Syst. Control Lett. 84, 7–12 (2015).
- [24] J.A. Gallegos and M. Duarte-Mermoud, “Boundedness and convergence on fractional order system”, J. Comput. Appl. Math. 29 (6), 815–826 (2016).
- [25] M. Seron, D. Hill, and A.L. Fradkov, “Nonlinear adaptive control of feedback passivity systems”, Automatica 31, 1053–1060 (1995).
- [26] M.A. Duarte-Mermoud, R. Castro-Linares, and A. Castillo-Facuse, “Direct passivity of a class of MIMO non-linear systems using adaptive feedback”, Int. J. Control 75, 23–33 (2016).
- [27] J.C.Willems, “Dissipative dynamical system, part II”, Arch. Rational Mech. Anal. 45 (5), 352–393 (1972).
- [28] H. Tuan and H. Trinh, Stability of fractional-order nonlinear systems by Lyapunov direct method, arXiv:1712.02921v1 [math.CA], 2017.
- [29] S. Sastry, Nonlinear Systems: Analysis, Stability, and Control, Springer, New York, 1999.
- [30] C. Bonnet and J.R. Partington, “Coprime factorizations and stability of fractional differential systems”, Syst. Control Lett. 41, 167–174 (2000).
- [31] I. Bar-Kana, “On positive realness in multivariable stationary linear systems”, Proceedings of the Conference on Information Sciences and Systems, Baltimore MD, USA, 1989.
- [32] T.T. Hartley and C.F. Lorenzo, “Dynamics and control of initialized fractional-order systems”, Nonlinear Dynamics 29, 201–233 (2002).
- [33] J.A. Gallegos and M. Duarte-Mermoud, “Convergence of fractional adaptive systems using gradient approach”, ISA Transactions 69, 31–42 (2017).
Uwagi
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
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