##### Controller design for delay-independent stability of multiple time-delay systems via Déscartes's rule of signs

###### Time Delay Systems, Volume # 9 | Part# 1

**Location:**, Czech Republic

**National Organizing Committee Chair:**Zitek, Pavel, Zuna, Petr

**International Program Committee Chair:**Lafay, Jean-Francois , Havlena, Vladimmir, Vyhlidal, Tomas

**Conference Editor:**Vyhlidal, Tomas, Zitek, Pavel

Authors

Delice, Ismail Ilker; Sipahi, Rifat

Digital Object Identifier (DOI)

10.3182/20100607-3-CZ-4010.00027

Page Numbers:

144-149

Index Terms

multiple time-delay systems,delay-independent stability,controller synthesis,iterated discriminant

Abstract

A general class of multi-input linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in order to obtain a control law which stabilizes the LTI-MTDS independently of all the delays. The method commences by reformulating the infinite-dimensional analysis as a finite-dimensional algebraic one without any sacrifice of accuracy and exactness. After this step, iterated discriminant allows one to construct a single-variable polynomial, coefficients of which are the controller gains. This crucial step succinctly formulates the delay-independent stability (DIS) condition of the controlled MTDS based on the roots of the single-variable polynomial. Implementation of the Déscartes's rule of signs then reveals, without computing these roots, the sufficient conditions on the controller gains to make the LTI-MTDS delay-independent stable. Case studies are provided to demonstrate the effectiveness of the proposed methodology.

References

[1] Abhyankar, S.S. (1990).Algebraic Geometry for Scientists and Engineers, volume 35 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, R.I. [2] Baser, U. (2003). Output feedbackH∞ control problem for linear neutral systems: Delay independent case.ASME Journal of Dynamic Systems, Measurement, and Control, 125(2), 177-185. [3] Chen, J., Xu, D., and Shafai, B. (1995). On sufficient conditions for stability independent of delay.IEEE Transactions on Automatic Control, 40(9), 1675-1680. [4] Courant, R. (1988).Differential and Integral Calculus, volume 2. Interscience Publishers, New York. [5] Datko, R. (1978). A procedure for determination of the exponential stability of certain differential-difference equations.Quarterly of Applied Mathematics, 36, 279-292. [6] Delice, I.I. and Sipahi, R. (2009). Exact upper and lower bounds of crossing frequency set and delay independent stability test for multiple time delayed systems. In8th IFAC Workshop on Time-Delay Systems. Sinaia, Romania. [7] Engelborghs, K. (2000). DDE-BIFTOOL: A Matlab package for bifurcation analysis of delay differential equations. TW Report 305, Department of Computer Science, Katholieke Universiteit Leuven, Belgium. [8] Fazelinia, H., Sipahi, R., and Olgac, N. (2007). Stability analysis of multiple time delayed systems using 'Building Block' concept.IEEE Transactions on Automatic Control, 52(5), 799-810. [9] Fridman, E., Gouaisbaut, F., Dambrine, M., and Richard, J.P. (2003). Sliding mode control of systems with time-varying delays via descriptor approach.International Journal of Systems Science, 34(8), 553-559. [10] Fridman, E. and Shaked, U. (2002). A descriptor system approach toH∞ control of linear time-delay systems.IEEE Transactions on Automatic Control, 47(2), 253-270. [11] Gelfand, I.M., Kapranov, M.M., and Zelevinsky, A.V. (1994).Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston. [12] Gu, K., Kharitonov, V.L., and Chen, J. (2003).Stability of Time-Delay Systems. Birkhäuser, Boston. [13] Gu, N., Tan, M., and Yu, W. (2001). An algebra test for unconditional stability of linear delay systems. InProceedings of the 40th IEEE Conference on Decision and Control, volume 5, 4746-4747. Orlando, Florida USA. [14] Gündes, A.N., Özbay, H., and Özgüler, A. (2007). PID controller synthesis for a class of unstable MIMO plants with I/O delays.Automatica, 43(1), 135-142. [15] Hertz, D., Jury, E.I., and Zeheb, E. (1984). Stability independent and dependent of delay for delay differential systems.Journal of The Franklin Institute, 318(3), 143-150. [16] Kamen, E. (1980). On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations.IEEE Transactions on Automatic Control, 25(5), 983-984. [17] Loiseau, J.J., Michiels, W., Niculescu, S.I., and Sipahi, R. (eds.) (2009).Topics in Time Delay Systems: Analysis, Algorithms and Control, volume 388 ofLecture Notes in Control and Information Sciences. Springer-Verlag, Berlin Heidelberg. [18] Michiels, W., Engelborghs, K., Vansevenant, P., and Roose, D. (2002). Continuous pole placement for delay equations.Automatica, 38(5), 747-761. [19] Michiels, W. and Niculescu, S.I. (2007).Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach, volume 12 ofAdvances in Design and Control. SIAM, Philadelphia. [20] Moon, Y.S., Park, P., Kwon, W.H., and Lee, Y.S. (2001). Delay-dependent robust stabilization of uncertain state-delayed systems.International Journal of Control, 74(14), 1447-1455. [21] Rekasius, Z.V. (1980). A stability test for systems with delays. InProceedings Joint Automatic Control Conference, TP9-A. San Francisco, CA. [22] Silva, G.J., Datta, A., and Bhattacharyya, S.P. (2001). PI stabilization of first-order systems with time delay.Automatica, 37(12), 2025-2031. [23] Sipahi, R. and Delice, I.I. (2010). On some features of potential stability switching hypersurfaces of time-delay systems.IMA Journal of Mathematical Control and Information, submitted. [24] Sipahi, R. and Olgac, N. (2005). Complete stability robustness of third-order LTI multiple time-delay systems.Automatica, 41(8), 1413-1422. [25] Souza, F.O., de Oliveira, M.C., and Palhares, R.M. (2009). Stability independent of delay using rational functions.Automatica, 45(9), 2128-2133. [26] Stépán, G. (1989).Retarded Dynamical Systems: Stability and Characteristic Functions, volume 210 ofPitman Research Notes in Mathematics Series. Longman Scientific & Technical, co-publisher John Wiley & Sons, Inc., New York. [27] Sturmfels, B. (2002).Solving Systems of Polynomial Equations, volume 97 ofConference Board of the Mathematical Sciences regional conference series in mathematics. American Mathematical Society, Providence, Rhode Island. [28] Thowsen, A. (1982). Delay-independent asymptotic stability of linear systems.IEE Proceedings D Control Theory & Applications, 129(3), 73-75. [29] Wang, Z.H. and Hu, H.Y. (1999). Delay-independent stability of retarded dynamic systems of multiple degrees of freedom.Journal of Sound and Vibration, 226(1), 57-81. [30] Wei, P., Guan, Q., Yu, W., and Wang, L. (2008). Easily testable necessary and sufficient algebraic criteria for delay-independent stability of a class of neutral differential systems.Systems & Control Letters, 57(2), 165-174. [31] Wu, S. and Ren, G. (2004). Delay-independent stability criteria for a class of retarded dynamical systems with two delays.Journal of Sound and Vibration, 270(4-5), 625-638.