Abstract

A recent work by Mazenc and Malisoff provides a trajectory-based approach for proving stability of time-varying systems with time-varying delays. Here, we provide several significant applications of their approach. In two results, we use a Lyapunov function for a corresponding undelayed system to provide a new method for proving stability of linear continuous-time time-varying systems with bounded time-varying delays. We allow uncertainties in the coefficient matrices of the systems. Our main results use upper bounds on an integral average involving the delay. The results establish input-to-state stability with respect to disturbances. We also provide a novel reduction model approach that ensures global exponential stabilization of linear systems with a time-varying pointwise delay in the input, which allows the delay to be discontinuous and uncertain. Finally, we provide an alternative to the reduction model method, based on a different dynamic extension. Our examples demonstrate the usefulness of our findings in several settings.

Keywords

  1. delay
  2. reduction approach
  3. time-varying
  4. stability

MSC codes

  1. 93Cxx
  2. 93Dxx

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Trans. Automat. Control, 27 (1982), pp. 869--879.
2.
N. Bekiaris-Liberis and M. Krstic, Nonlinear Control under Nonconstant Delays, Adv. Des. Control 25, SIAM, Philadelphia, 2013.
3.
N. Bekiaris-Liberis and M. Krstic, Nonlinear control under delays that depend on delayed states, Eur. J. Control, 19 (2013), pp. 389--398.
4.
N. Bekiaris-Liberis and M. Krstic, Compensation of state-dependent input delay for nonlinear systems, IEEE Trans. Automat. Control, 58 (2013), pp. 275--289.
5.
N. Bekiaris-Liberis and M. Krstic, Robustness of nonlinear predictor feedback laws to time- and state-dependent delay perturbations, Automatica J. IFAC, 49 (2013), pp. 1576--1590.
6.
M. Belem Saldivar Marquez, I. Boussaada, H. Mounier, and S-I. Niculescu, Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach, Adv. Ind. Control, Springer, New York, 2015.
7.
D. Bresch-Pietri, J. Chauvin, and N. Petit, Adaptive control scheme for uncertain time-delay systems, Automatica J. IFAC, 48 (2012), pp. 1536--1552.
8.
D. Bresch-Pietri, J. Chauvin, and N. Petit, Prediction-based stabilization of linear systems subject to input-dependent input delay of integral-type, IEEE Trans. Automat. Control, 59 (2014), pp. 2385--2399.
9.
D. Bresch-Pietri and N. Petit, Robust compensation of a chattering time-varying input delay, in Proceedings of the IEEE Conference on Decision and Control, Los Angeles, CA, 2014, IEEE, Piscataway, NJ, 2014, pp. 457--462.
10.
J.-Y. Dieulot and J.-P. Richard, Tracking control of a nonlinear system with input-dependent delay, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, 2001, IEEE, Piscataway, NJ, 2001, pp. 4027--4031.
11.
R. Downey, R. Kamalapurkar, N. Fischer, and W. Dixon, Compensating for fatigue-induced time-varying delayed muscle response in neuromuscular electrical stimulation control, in Recent Results on Nonlinear Delay Control Systems, Adv. Delays Dyn. 4, Springer, New York, 2015, pp. 143--161.
12.
L. Figueredo, J. Ishihara, G. Borges, and A. Bauchspiess, Robust stability criteria for uncertain systems with delay and its derivative varying within intervals, in Proceedings of the American Control Conference, San Francisco, CA, IEEE, 2011, Piscataway, NJ, 2011, pp. 4884--4889.
13.
E. Fridman and S.-I. Niculescu, On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays, Internat. J. Robust Nonlinear Control, 18 (2008), pp. 364--374.
14.
E. Fridman, A. Seuret, and J.-P. Richard, Robust sampled-data stabilization of linear systems: An input delay approach, Automatica J. IFAC, 40 (2004), pp. 1441--1446.
15.
E. Fridman and U. Shaked, Delay-dependent stability and $H_{\infty}$ control: Constant and time-varying delays, Internat. J. Control, 76 (2003), pp. 48--60.
16.
H. Gao, T. Chen, and J. Lam, A new delay system approach to network-based control, Automatica J. IFAC, 44 (2008), pp. 39--52.
17.
K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Control Eng., Birkhäuser, Boston, 2003.
18.
I. Haidar, P. Mason, S.-I. Niculescu, M. Sigalotti, and A. Chaillet, Further remarks on Markus-Yamabe instability for time-varying delay differential equations, in Proceedings of the 12th IFAC Workshop on Time Delay Systems, Ann Arbor, MI, 2015, pp. 33--38.
19.
A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic, New York, 1966.
20.
J. Hale, Ordinary Differential Equations, 2nd ed., Krieger, Malabar, FL, 1980.
21.
A. Ivanov, E. Liz, and S. Trofimchuk, Halanay inequality, Yorke $3/2$ stability criterion, and differential equations with maxima, Tohoku Math. J., 54 (2002), pp. 277--295.
22.
M. Jankovic, Recursive predictor design for linear systems with time delay, in Proceedings of the American Control Conference, Seattle, WA, 2008, IEEE, Piscataway, NJ, 2008, pp. 4904--4909.
23.
I. Karafyllis and M. Krstic, On the relation of delay equations to first-order hyperbolic partial differential equations, ESAIM Control Optim. Calc. Var., 20 (2014), pp. 894--923.
24.
I. Karafyllis, M. Malisoff, and M. Krstic, Sampled-data feedback stabilization of age-structured chemostat models, in Proceedings of the 2015 American Control Conference, Chicago, 2015, IEEE, Piscataway, NJ, 2015, pp. 4549--4554.
25.
H. Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ, 2002.
26.
V. Kharitonov and S.-I. Niculescu, On the stability of linear systems with uncertain delay, IEEE Trans. Automat. Control, 48 (2002), pp. 127--132.
27.
M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems, Systems and Control: Found. Appl., Birkhäuser, Boston, 2009.
28.
M. Krstic, Lyapunov stability of linear predictor feedback for time-varying input delay, IEEE Trans. Automat. Control, 55 (2010), pp. 554--559.
29.
W. Kwon and A. Pearson, Feedback stabilization of linear systems with delayed control, IEEE Trans. Automat. Control, 25 (1980), pp. 266--269.
30.
K. Liu, V. Suplin, and E. Fridman, Stability of linear systems with general sawtooth delay, IMA J. Math. Control Inform., 27 (2010), pp. 419--436.
31.
J. Louisell, New examples of quenching in delay differential equations having time-varying delay, in Proceedings of the 5th European Control Conference, Karlsruhe, Germany, 1999.
32.
A. Manitius and A. Olbrot, Finite spectrum assignment problem for systems with delays, IEEE Trans. Automat. Control, 24 (1979), pp. 541--552.
33.
L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J., 12 (1960), pp. 305--317.
34.
F. Mazenc and M. Malisoff, Trajectory based approach for the stability analysis of nonlinear systems with time delays, IEEE Trans. Automat. Control, 60 (2015), pp. 1716--1721.
35.
F. Mazenc and M. Malisoff, Reduction model approach for systems with a time-varying delay, in Proceedings of the IEEE 54th Conference on Decision and Control, Osaka, Japan, 2015, IEEE, Piscataway, NJ, 2015, pp. 7723--7727.
36.
F. Mazenc, M. Malisoff, and T. Dinh, Robustness of nonlinear systems with respect to delay and sampling of the controls, Automatica J. IFAC, 49 (2013), pp. 1925--1931.
37.
F. Mazenc, M. Malisoff, and Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks, Automatica J. IFAC, 44 (2008), pp. 2415--2421.
38.
F. Mazenc, M. Malisoff, and S.-I. Niculescu, Reduction model approach for linear time-varying systems with delays, IEEE Trans. Automat. Control, 59 (20147), pp. 2068--2082.
39.
F. Mazenc, M. Malisoff, and S.-I. Niculescu, Stability analysis for systems with time-varying delay: Trajectory based approach, in Proceedings of the IEEE 54th Conference on Decision and Control, Osaka, Japan, 2015, IEEE, Piscataway, NJ, 2015, pp. 1811--1816.
40.
F. Mazenc, S.-I. Niculescu, and M. Krstic, Lyapunov-Krasovskii functionals and application to input delay compensation for linear time-invariant systems, Automatica J. IFAC, 48 (2012), pp. 1317--1323.
41.
W. Michiels and S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems, Adv. Des. Control 12, SIAM, Philadelphia, 2007.
42.
S. Mondié and W. Michiels, Finite spectrum assignment of unstable time-delay system with a safe implementation, IEEE Trans. Automat. Control, 48 (2003), pp. 2207--2212.
43.
N. Petit, Y. Creff, and P. Rouchon, Motion planning for two classes of nonlinear systems with delays depending on the control, in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, 1998, IEEE, Piscataway, NJ, 1998, pp. 1007--1011.
44.
J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), pp. 1667--1694.
45.
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1976.
46.
M. Sbarciog, R. De Keyser, S. Cristea, and C. De Prada, Nonlinear predictive control of processes with variable time delay. A temperature control case study, in Proceedings of the IEEE International Conference on Control Applications, San Antonio, TX, 2008, IEEE, Piscataway, NJ, 2008, pp. 1001--1006.
47.
O. Smith, A controller to overcome dead time, ISA Journal, 6 (1959), pp. 28--33.
48.
B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer, London, 2014.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 533 - 556
ISSN (online): 1095-7138

History

Submitted: 25 June 2015
Accepted: 22 December 2016
Published online: 22 February 2017

Keywords

  1. delay
  2. reduction approach
  3. time-varying
  4. stability

MSC codes

  1. 93Cxx
  2. 93Dxx

Authors

Affiliations

Silviu-Iulian Niculescu

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.