Research Papers

Eigenvalue Assignment for Control of Time-Delay Systems Via the Generalized Runge–Kutta Method

[+] Author and Article Information
JinBo Niu, LiMin Zhu, Han Ding

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China

Ye Ding

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: y.ding@sjtu.edu.cn

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 28, 2013; final manuscript received April 18, 2015; published online June 2, 2015. Assoc. Editor: Joseph Beaman.

J. Dyn. Sys., Meas., Control 137(9), 091003 (Sep 01, 2015) (7 pages) Paper No: DS-13-1367; doi: 10.1115/1.4030418 History: Received September 28, 2013; Revised April 18, 2015; Online June 02, 2015

This paper presents an eigenvalue assignment method for the time-delay systems with feedback controllers. A new form of Runge–Kutta algorithm, generalized from the classical fourth-order Runge–Kutta method, is utilized to stabilize the linear delay differential equation (DDE) with a single delay. Pole placement of the DDEs is achieved by assigning the eigenvalue with maximal modulus of the Floquet transition matrix obtained via the generalized Runge–Kutta method (GRKM). The stabilization of the DDEs with feedback controllers is studied from the viewpoint of optimization, i.e., the DDEs are controlled through optimizing the feedback gain matrices with proper optimization techniques. Several numerical cases are provided to illustrate the feasibility of the proposed method for control of linear time-invariant delayed systems as well as periodic-coefficient ones. The proposed method is verified with high computational accuracy and efficiency through comparing with other methods such as the Lambert W function and the semidiscretization method (SDM).

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Richard, J.-P., 2003, “Time-Delay Systems: An Overview of Some Recent Advances and Open Problems,” Automatica, 39(10), pp. 1667–1694. [CrossRef]
Niculescu, S.-I., 2001, Delay Effects on Stability: A Robust Control Approach, Springer, Berlin.
Kuang, Y., 1993, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, London.
Foryś, U., 2004, “Biological Delay Systems and the Mikhailov Criterion of Stability,” J. Biol. Syst., 12(1), pp. 45–60. [CrossRef]
Silva, G. J., Datta, A., and Bhattacharyya, S., “Controller Design Via Padé Approximation Can Lead to Instability,” 40th IEEE Conference on the Decision and Control, Orlando, FL, pp. 4733–4737. [CrossRef]
Manitius, A., and Olbrot, A., 1979, “Finite Spectrum Assignment Problem for Systems With Delays,” IEEE Trans. Autom. Control, 24(4), pp. 541–552. [CrossRef]
Smith, O. J., 1957, “Posicast Control of Damped Oscillatory Systems,” Proc. IRE, 45(9), pp. 1249–1255. [CrossRef]
Smith, O. J., 1959, “A Controller to Overcome Dead Time,” ISA J., 6(2), pp. 28–33.
Sánchez-Peña, R. S., Bolea, Y., and Puig, V., 2009, “MIMO Smith Predictor: Global and Structured Robust Performance Analysis,” J. Process Control, 19(1), pp. 163–177. [CrossRef]
Fliess, M., Marquez, R., and Mounier, H., 2002, “An Extension of Predictive Control, PID Regulators and Smith Predictors to Some Linear Delay Systems,” Int. J. Control, 75(10), pp. 728–743. [CrossRef]
Krasovskiĭ, N. N., 1963, Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations With Delay, Stanford University Press, Stanford, CA.
Kolmanovskiĭ, V. B., 1986, Stability of Functional Differential Equations, Academic Press, London.
Gu, K., 2001, “Discretization Schemes for Lyapunov–Krasovskii Functionals in Time-Delay Systems,” Kybernetika, 37(4), pp. 479–504.
Niu, Y., Lam, J., and Wang, X., “Sliding-Mode Control for Uncertain Neutral Delay Systems,” IEE Proceedings–Control Theory and Applications, IET, 151(1), pp. 38–44. [CrossRef]
Fattouh, A., Sename, O., and Dion, J.-M., “A LMI Approach to Robust Observer Design for Linear Time-Delay Systems,” 39th IEEE Conference on Decision and Control, Sydney, Australia, Dec. 12–15, pp. 1495–1500. [CrossRef]
Sakthivel, R., Vadivel, P., Mathiyalagan, K., and Arunkumar, A., 2014, “Fault-Distribution Dependent Reliable H∞ Control for Takagi–Sugeno Fuzzy Systems,” ASME J. Dyn. Syst., Meas., Control, 136(2), p. 021021. [CrossRef]
Sakthivel, R., Mathiyalagan, K., and Anthoni, S. M., 2012, “Robust H∞ Control for Uncertain Discrete-Time Stochastic Neural Networks With Time-Varying Delays,” IET Control Theory Appl., 6(9), pp. 1220–1228. [CrossRef]
Mathiyalagan, K., Sakthivel, R., and Anthoni, S. M., 2011, “New Stability and Stabilization Criteria for Fuzzy Neural Networks With Various Activation Functions,” Phys. Scr., 84(1), p. 015007. [CrossRef]
Vadivel, P., Sakthivel, R., Mathiyalagan, K., and Thangaraj, P., 2012, “Robust Stabilisation of Non-Linear Uncertain Takagi–Sugeno Fuzzy Systems by H∞ Control,” IET Control Theory Appl., 6(16), pp. 2556–2566. [CrossRef]
Sakthivel, R., Raja, U. K., Mathiyalagan, K., and Leelamani, A., 2012, “Design of a Robust Controller on Stabilization of Stochastic Neural Networks With Time Varying Delays,” Phys. Scr., 85(3), p. 035003. [CrossRef]
Sakthivel, R., Mathiyalagan, K., and Anthoni, S. M., 2012, “Robust Stability and Control for Uncertain Neutral Time Delay Systems,” Int. J. Control, 85(4), pp. 373–383. [CrossRef]
Zhang, J., Knopse, C. R., and Tsiotras, P., 2001, “Stability of Time-Delay Systems: Equivalence Between Lyapunov and Scaled Small-Gain Conditions,” IEEE Trans. Autom. Control, 46(3), pp. 482–486. [CrossRef]
Michiels, W., Engelborghs, K., Vansevenant, P., and Roose, D., 2002, “Continuous Pole Placement for Delay Equations,” Automatica, 38(5), pp. 747–761. [CrossRef]
Burke, J. V., Lewis, A. S., and Overton, M. L., 2005, “A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization,” SIAM J. Optim., 15(3), pp. 751–779. [CrossRef]
Cai, G., and Huang, J., 2002, “Optimal Control Method for Seismically Excited Building Structures With Time-Delay in Control,” J. Eng. Mech., 128(6), pp. 602–612. [CrossRef]
Cai, G.-P., Huang, J.-Z., and Yang, S. X., 2003, “An Optimal Control Method for Linear Systems With Time Delay,” Comput. Struct., 81(15), pp. 1539–1546. [CrossRef]
Olgac, N., and Sipahi, R., 2002, “An Exact Method for the Stability Analysis of Time-Delayed Linear Time-Invariant (LTI) Systems,” IEEE Trans. Autom. Control, 47(5), pp. 793–797. [CrossRef]
Sipahi, R., and Olgac, N., 2003, “Active Vibration Suppression With Time Delayed Feedback,” ASME J. Vib. Acoust., 125(3), pp. 384–388. [CrossRef]
Breda, D., Maset, S., and Vermiglio, R., 2005, “Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations,” SIAM J. Sci. Comput., 27(2), pp. 482–495. [CrossRef]
Stépán, G., 1989, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical Marlow, New York.
Insperger, T., and Stepan, G., 2002, “Semi-Discretization Method for Delayed Systems,” Int. J. Numer. Methods Eng., 55(5), pp. 503–518. [CrossRef]
Insperger, T., and Stépán, G., 2004, “Updated Semi-Discretization Method for Periodic Delay-Differential Equations With Discrete Delay,” Int. J. Numer. Methods Eng., 61(1), pp. 117–141. [CrossRef]
Sheng, J., Elbeyli, O., and Sun, J., 2004, “Stability and Optimal Feedback Controls for Time-Delayed Linear Periodic Systems,” AIAA J., 42(5), pp. 908–911. [CrossRef]
Sheng, J., and Sun, J., 2005, “Feedback Controls and Optimal Gain Design of Delayed Periodic Linear Systems,” J. Vib. Control, 11(2), pp. 277–294. [CrossRef]
Stépán, G., and Insperger, T., 2006, “Stability of Time-Periodic and Delayed Systems—A Route to Act-and-Wait Control,” Annu. Rev. Control, 30(2), pp. 159–168. [CrossRef]
Sun, J.-Q., and Song, B., 2009, “Control Studies of Time-Delayed Dynamical Systems With the Method of Continuous Time Approximation,” Commun. Nonlinear Sci. Numer. Simul., 14(11), pp. 3933–3944. [CrossRef]
Song, B., and Sun, J.-Q., 2010, “Supervisory Control of Dynamical Systems With Uncertain Time Delays,” ASME J. Vib. Acoust., 132(6), p. 061003. [CrossRef]
Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., and Knuth, D. E., 1996, “On the LambertW function,” Adv. Comput. Math., 5(1), pp. 329–359. [CrossRef]
Asl, F. M., and Ulsoy, A. G., 2003, “Analysis of a System of Linear Delay Differential Equations,” ASME J. Dyn. Syst., Meas., Control, 125(2), pp. 215–223. [CrossRef]
Yi, S., Nelson, P., and Ulsoy, A., 2007, “Survey on Analysis of Time Delayed Systems Via the Lambert W Function,” Adv. Dyn. Syst., 14(S2), pp. 296–301.
Yi, S., Nelson, P., and Ulsoy, A., 2010, “Eigenvalue Assignment Via the Lambert W Function for Control of Time-Delay Systems,” J. Vib. Control, 16(7–8), pp. 961–982. [CrossRef]
Wang, Z., and Hu, H., 2008, “Calculation of the Rightmost Characteristic Root of Retarded Time-Delay Systems Via Lambert W Function,” J. Sound Vib., 318(4), pp. 757–767. [CrossRef]
Yi, S., Nelson, P. W., and Ulsoy, A. G., 2010, Time-Delay Systems: Analysis and Control Using the Lambert W Function, World Scientific, Singapore. [CrossRef]
Niu, J., Ding, Y., Zhu, L., and Ding, H., 2014, “Runge–Kutta Methods for a Semi-Analytical Prediction of Milling Stability,” Nonlinear Dyn., 76(1), pp. 289–304. [CrossRef]
Berrut, J.-P., and Trefethen, L. N., 2004, “Barycentric Lagrange Interpolation,” SIAM Rev., 46(3), pp. 501–517. [CrossRef]
Schmitt, L. M., 2001, “Theory of Genetic Algorithms,” Theor. Comput. Sci., 259(1), pp. 1–61. [CrossRef]
Yang, W. Y., Cao, W., Chung, T.-S., and Morris, J., 2005, Applied Numerical Methods Using MATLAB, Wiley, Hoboken, NJ. [CrossRef]
Yi, S., Nelson, P. W., and Ulsoy, A. G., “Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations Via Eigenvalue Assignment,” American Control Conference, Seattle, WA, pp. 4928–4933.
Yi, S., Nelson, P. W., and Ulsoy, A. G., “Analysis and Control of Time Delayed Systems Via the Lambert W Function,” 17th IFAC World Congress 2008, pp. 13414–13419.
Insperger, T., and Stepan, G., 2003, “Stability of the Damped Mathieu Equation With Time Delay,” ASME J. Dyn. Syst., Meas., Control, 125(2), pp. 166–171. [CrossRef]


Grahic Jump Location
Fig. 1

Resultant rightmost poles corresponding to different computational results of ad in the complex plane with the GRKM

Grahic Jump Location
Fig. 2

Transient responses of DDEs with rightmost poles of different imaginary parts

Grahic Jump Location
Fig. 3

Contour lines of the rightmost poles of Eq. (20) in parameter space (a,ad)

Grahic Jump Location
Fig. 4

Contour lines of desired rightmost characteristic roots of the DDE in Example 3

Grahic Jump Location
Fig. 5

Convergence rate comparisons between the GRKM and the SDM




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In