Extension principle and controller design for systems with distributed time-delay

İftar, Altuğ

Kybernetika, Tome 53 (2017) no. 4, pp. 630-652

Abstract (VO)

Extension principle is defined for systems with distributed time-delay and the necessary and sufficient conditions for one system being an extension of the other are presented. Preservation of stability properties between two systems, one of which is an extension of the other is also discussed and it is shown that when the expanded system is an extension of the original system, (i) the original system is bounded-input bounded-output stable if and only if the expanded system is bounded-input bounded-output stable and (ii) the original system is exponentially stable if the expanded system is exponentially stable. Controller design using the extension principle is then considered. It is shown that, if the expanded system is an extension of the original system, then any controller designed for the expanded system can be contracted for implementation on the original system. Furthermore, if the controller designed for the expanded system stabilizes the expanded system and satisfies certain performance requirements, then the contracted controller stabilizes the original system and satisfies corresponding performance requirements for the original system. Finally, overlapping decompositions and controller design using overlapping decompositions are demonstrated. A highway traffic congestion control problem is then considered to demonstrate a possible application of the presented controller design approach.

Détail

DOI : 10.14736/kyb-2017-4-0630

Classification : 93A14, 93A15, 93C23
Keywords: large-scale systems; time-delay systems; distributed time-delay; overlapping decompositions; decentralized control; controller design

Bibliographie

[1] Ataşlar, B., İftar, A.: Decentralized routing controller design using overlapping decompositions. Int. J. Control 72 (1999), 1175-1192. | DOI | MR

[2] Atay, F. M., Hutt, A.: Neural fields with distributed transmission speeds and long-range feedback delays. SIAM J. Appl. Dynam. Systems 5 (2006), 670-698. | DOI | MR

[3] Aybar, A., İftar, A.: Overlapping decompositions and expansions of Petri nets. IEEE Trans. Automat. Control 47 (2002), 511-515. | DOI | MR

[4] Bakule, L.: Decentralized control: An overview. Ann. Rev. Control 32 (2008), 87-98. | DOI

[5] Bakule, L., Rodellar, J., Rossell, J. M.: Overlapping guaranteed cost control for uncertain continuous-time delayed systems. In: Proc. 16th IFAC World Congress, Prague 2005. | DOI

[6] Bakule, L., Rodellar, J., Rossell, J. M.: Overlapping resilient $H_\infty$ control for uncertain time-delayed systems. In: Proc. IEEE Conference on Decision and Control, Seville 2005, pp. 2290-2295. | DOI

[7] Bakule, L., Rossell, J. M.: Overlapping controllers for uncertain delay continuous-time systems. Kybernetika 44 (2008), 17-34. | MR

[8] Berezansky, L., Braverman, E.: Oscillation properties of a logistic equation with distributed delay. Nonlinear Analysis: Real World Applications 4 (2003), 1-19. | DOI | MR

[9] Cooke, K. L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86 (1082), 592-627. | DOI | MR

[10] Ding, Z.: Decentralized output regulation of large scale nonlinear systems with delay. Kybernetika 45 (2009), 33-48. | MR | Zbl

[11] Erol, H. E., İftar, A.: Stabilization of decentralized descriptor-type neutral time-delay systems by time-delay controllers. Automatica 64 (2016), 262-269. | DOI | MR

[12] Hale, J. K., Verduyn-Lunel, S. M.: Introduction to Functional Differential Equations. Springer-Verlag, New York 1993. | MR

[13] İftar, A.: Decentralized estimation and control with overlapping input, state, and output decomposition. Automatica 29 (1993), 511-516. | DOI | MR

[14] İftar, A.: Overlapping decentralized dynamic optimal control. Int. J. Control 58 (1993), 187-209. | DOI | MR

[15] İftar, A.: Decentralized robust control based on overlapping decompositions. In: Preprints 10th IFAC Symposium on Large Scale Systems, Osaka 2004, pp. 605-609. | DOI

[16] İftar, A.: Decentralized robust control of large-scale time-delay systems. In: Proc. 17th IFAC Word Congress, Seoul 2008, pp. 9332-9337. | DOI

[17] İftar, A.: Inclusion principle and overlapping decompositions of distributed-time-delay systems. In: Proc. 19th IFAC World Congress, Cape Town 2014, pp. 5796-5801. | DOI

[18] İftar, A.: Overlapping decompositions and decentralized robust control of large-scale systems with distributed time-delay. In: Proc. IEEE Conference on Decision and Control, Los Angeles 2014, pp. 463-468. | DOI

[19] İftar, A., Khorrami, F.: A comparison of multiple time-scale analysis and overlapping decomposition. IEEE Trans. Systems, Man, Cybernet. 19 (1989), 1296-1300. | DOI

[20] İftar, A., Özgüner, Ü.: Decentralized LQG/LTR controller design for interconnected systems. In: Proc. American Control Conference, Minneapolis 1987, pp. 1682-1687.

[21] İftar, A., Özgüner, Ü.: Local LQG/LTR controller design for decentralized systems. IEEE Trans. Automat. Control 32 (1987), 926-930. | DOI

[22] İftar, A., Özgüner, Ü.: Contractible controller design and optimal control with state and input inclusion. Automatica 26 (1990), 593-597. | DOI | MR

[23] İftar, A., Özgüner, Ü.: Overlapping decompositions, expansions, contractions, and stability of hybrid systems. IEEE Trans. Automat. Control 43 (1998), 1040-1055. | DOI | MR

[24] Ikeda, M., Šiljak, D. D.: Overlapping decompositions, expansions, and contractions of dynamic systems. Large Scale Systems 1 (1980), 29-38. | MR

[25] Ikeda, M., Šiljak, D. D.: Overlapping decentralized control with input, state, and output inclusion. Control Theory Advanced Technol. 2 (1986), 155-172.

[26] Ikeda, M., Šiljak, D. D., White, D. E.: Decentralized control with overlapping information sets. J. Optim. Theory Appl. 34 (1981), 279-310. | DOI | MR

[27] Ikeda, M., Šiljak, D. D., White, D. E.: An inclusion principle for dynamic systems. IEEE Trans. Automat. Control 29 (1984), 244-249. | DOI | MR

[28] Liu, S., Yu, W., Zhang, F.: Output feedback regulation for large-scale uncertain nonlinear systems with time delays. Kybernetika 51 (2015), 874-889. | DOI | MR

[29] Ma, L., Xu, M., Jia, R., Ye, H.: Exponential $H_\infty$ filter design for stochastic Markovian jump systems with both discrete and distributed time-varying delays. Kybernetika 50 (2014), 491-511. | DOI | MR

[30] Michiels, W., Morărescu, C.-I., Niculescu, S.-I.: Consensus problems with distributed delays, with application to traffic flow models. SIAM J. Control Optim. 48 (2009), 77-101. | DOI | MR

[31] Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach. Lect. Notes Control Inform. Sci. 269, Springer-Verlag, London 2001. | MR

[32] Özbay, H., Bonnet, C., Clairambault, J.: Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics. In: Proc. {IEEE} Conference on Decision and Control, Cancun 2008, pp. 2050-2055. | DOI

[33] Özer, S. M., Gülmez, G., İftar, A.: A software to design decentralized controllers for time-delay systems. In: Proc. 2016 IEEE Conference on Computer Aided Control System Design, Buenos Aires 2016, pp. 868-873. | DOI

[34] Özer, S. M., İftar, A.: Decentralized controller design for time-delay systems by optimization. In: Preprints 12th IFAC Workshop on Time-delay Systems Ann Arbor 2015, pp. 462-467. | DOI

[35] Özgüner, Ü., Hatipoğlu, C., İftar, A., Redmill, K.: Hybrid control design for a three vehicle scenario demonstration using overlapping decompositions. In: Hybrid Systems IV, Lecture Notes in Computer Science, 1273, (P. J. Antsaklis, ed.), pp. 294-328, Springer Verlag, 1997. | DOI | MR

[36] Papageorgiou, M., Blosseville, J., Hadj-Salem, H.: Macroscopic modeling of traffic flow on the Boulevard Peripherique in Paris. Transport. Res., Part B 23 (1989), 29-47. | DOI

[37] Papageorgiou, M., Blosseville, J., Hadj-Salem, H.: Modeling and real-time control of traffic flow on the southern part of Boulevard Peripherique in Paris: Part I: Modelling. Transport. Res. Part A 24 (1990), 345-359.

[38] Papageorgiou, M., Blosseville, J., Hadj-Salem, H.: Modeling and real-time control of traffic flow on the southern part of Boulevard Peripherique in Paris: Part II: Coordinated on-ramp metering. Transport. Res., Part A 24 (1990), 361-370.

[39] Penrose, R., Todd, J. A.: On best approximate solutions of linear matrix equations. Math. Proc. Cambridge Philosoph. Soc. 52 (1956), 17-19. | DOI | MR

[40] Rahman, B., Blyuss, K. B., Kyrychko, Y. N.: Dynamics of neural systems with discrete and distributed time delays. SIAM J. Applied Dynamical Systems 14 (2015), 2069-2095. | DOI | MR

[41] Rao, V. S. H., Rao, P. R. S.: Global stability in chemostat models involving time delays and wall growth. Nonlinear Analysis: Real World Appl. 5 (2004), 141-158. | DOI | MR

[42] Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39 (2003), 1667-1694. | DOI | MR

[43] Šiljak, D. D.: Decentralized Control of Complex Systems. Academic Press, Inc., San Diego 1991. | DOI | MR

[44] Sipahi, R., Atay, F. M., Niculescu, S.-I.: Stability of traffic flow behaviour with distributed delays modeling the memory effects of the drivers. SIAM J. Appl. Math. 68 (2008), 738-759. | DOI | MR

[45] Stanković, S. S., Stanojević, M. J., Šiljak, D. D.: Decentralized overlapping control of a platoon of vehicles. IEEE Trans. Control Systems Technol. 8 (2000), 816-832. | DOI

[46] Vanbiervliet, J., Verheyden, K., Michiels, W., Vandewalle, S.: A nonsmooth optimisation approach for the stabilisation of time-delay systems. ESAIM: Control, Optimisation and Calculus of Variations 14 (2008), 3, 478-493. | DOI | MR

[47] Xie, L., Fridman, E., Shaked, U.: Robust ${\cal H_\infty}$ control of distributed delay systems with application to combustion control. IEEE Trans. Automat Control 46 (2001), 1930-1935. | DOI | MR