Technical Brief

Positive Finite-Time Stabilization for Discrete-Time Linear Systems

[+] Author and Article Information
Wenping Xue

School of Electrical and Information Engineering,
Jiangsu University,
Zhenjiang 212013, China
e-mail: xwping@ujs.edu.cn

Kangji Li

School of Electrical and Information Engineering,
Jiangsu University,
Zhenjiang 212013, China
e-mail: likangji@ujs.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 March 5, 2013; final manuscript received July 28, 2014; published online August 28, 2014. Assoc. Editor: Qingze Zou.

J. Dyn. Sys., Meas., Control 137(1), 014502 (Aug 28, 2014) (5 pages) Paper No: DS-13-1100; doi: 10.1115/1.4028141 History: Received March 05, 2013; Revised July 28, 2014

In this paper, a new finite-time stability (FTS) concept, which is defined as positive FTS (PFTS), is introduced into discrete-time linear systems. Differently from previous FTS-related papers, the initial state as well as the state trajectory is required to be in the non-negative orthant of the Euclidean space. Some test criteria are established for the PFTS of the unforced system. Then, a sufficient condition is proposed for the design of a state feedback controller such that the closed-loop system is positively finite-time stable. This condition is provided in terms of a series of linear matrix inequalities (LMIs) with some equality constraints. Moreover, the requirement of non-negativity of the controller is considered. Finally, two examples are presented to illustrate the developed theory.

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


Dorato, P., 1961, “Short Time Stability in Linear Time-Varying Systems,” Proceedings of the IRE International Convention Record Part 4, New York, March 21–25, pp. 83–87.
Weiss, L., and Infante, E., 1967, “Finite Time Stability Under Perturbing Forces and on Product Spaces,” IEEE Trans. Autom. Control, 12(1), pp. 54–59. [CrossRef]
Amato, F., Ariola, M., and Dorato, P., 2001, “Finite-Time Control of Linear Systems Subject to Parametric Uncertainties and Disturbances,” Automatica, 37(9), pp. 1459–1463. [CrossRef]
Amato, F., and Ariola, M., 2005, “Finite-Time Control of Discrete-Time Linear Systems,” IEEE Trans. Autom. Control, 50(5), pp. 724–729. [CrossRef]
Garcia, G., Tarbouriech, S., and Bernussou, J., 2009, “Finite-Time Stabilization of Linear Time-Varying Continuous Systems,” IEEE Trans. Autom. Control, 54(2), pp. 364–369. [CrossRef]
Amato, F., Ariola, M., and Cosentino, C., 2010, “Finite-Time Control of Discrete-Time Linear Systems: Analysis and Design Conditions,” Automatica, 46(5), pp. 919–924. [CrossRef]
Amato, F., Ambrosino, R., Ariola, M., Cosentino, C., and Tommasi, G. D., 2014, Finite-Time Stability and Control, Springer, London, UK.
Luan, X., Liu, F., and Shi, P., 2010, “Robust Finite-Time H Control for Nonlinear Jump Systems Via Neural Networks,” Circuits Syst. Signal Process., 29(3), pp. 481–498. [CrossRef]
Xu, J., Sun, J., and Yue, D., 2012, “Stochastic Finite-Time Stability of Nonlinear Markovian Switching Systems With Impulsive Effects,” ASME J. Dyn. Syst. Meas. Control, 134(1), p. 011011. [CrossRef]
He, S., and Liu, F., 2010, “Stochastic Finite-Time Stabilization for Uncertain Jump Systems Via State Feedback,” ASME J. Dyn. Syst. Meas. Control, 132(3), p. 034504. [CrossRef]
Zuo, Z., Liu, Y., Wang, Y., and Li, H., 2012, “Finite-Time Stochastic Stability and Stabilisation of Linear Discrete-Time Markovian Jump Systems With Partly Unknown Transition Probabilities,” IET Control Theory Appl., 6(10), pp. 1522–1526. [CrossRef]
Xu, J., and Sun, J., 2010, “Finite-Time Stability of Linear Time-Varying Singular Impulsive Systems,” IET Control Theory Appl., 4(10), pp. 2239–2244. [CrossRef]
Zhang, Y., Liu, C., and Mu, X., 2012, “Robust Finite-Time Stabilization of Uncertain Singular Markovian Jump Systems,” Appl. Math. Modell., 36(10), pp. 5109–5121. [CrossRef]
Xue, W., and Mao, W., 2013, “Admissible Finite-Time Stability and Stabilization of Uncertain Discrete Singular Systems,” ASME J. Dyn. Syst. Meas. Control, 135(3), p. 031018. [CrossRef]
Stojanović, S. B., Debeljković, D. L., and Dimitrijević, N., 2012, “Finite-Time Stability of Discrete-Time Systems with Time-Varying Delay,” Chem. Ind. Chem. Eng. Q., 18(4), pp. 525–533. [CrossRef]
Zuo, Z., Li, H., and Wang, Y., 2013. “New Criterion for Finite-Time Stability of Linear Discrete-Time Systems with Time-Varying Delay,” J. Franklin Inst., 350(9), pp. 2745–2756. [CrossRef]
Lazarević, M. P., and Spasić, A. M., 2009, “Finite-Time Stability Analysis of Fractional Order Time-Delay Systems: Gronwall's Approach,” Math. Comput. Modell., 49(3–4), pp. 475–481. [CrossRef]
Xiang, Z., Sun, Y., and Mahmoud, M. S., 2012, “Robust Finite-Time H Control for a Class of Uncertain Switched Neutral Systems,” Commun. Nonlinear Sci. Numer. Simul., 17(4), pp. 1766–1778. [CrossRef]
Wang, Y., Shi, X., Zuo, Z., Chen, M. Z. Q., and Shao, Y., 2013, “On Finite-Time Stability for Nonlinear Impulsive Switched Systems,” Nonlinear Anal.: Real World Appl., 14(1), pp. 807–814. [CrossRef]
Lin, X., Du, H., Li, S., and Zou, Y., 2013, “Finite-Time Stability and Finite-Time Weighted L2-Gain Analysis for Switched Systems with Time-Varying Delay,” IET Control Theory Appl., 7(7), pp. 1058–1069. [CrossRef]
Caswell, H., 2001, Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer Associates, Sunderland, MA.
Farina, L., and Rinaldi, S., 2000, Positive Linear Systems: Theory and Applications, Wiley, New York.
Kaczorek, T., 2002, Positive 1D and 2D Systems, Springer, London, UK.
Gao, H., Lam, J., Wang, C., and Xu, S., 2005, “Control for Stability and Positivity: Equivalent Conditions and Computation,” IEEE Trans. Circuits Syst. II, 52(9), pp. 540–544. [CrossRef]
Rami, M. A., and Tadeo, F., 2007, “Controller Synthesis for Positive Linear Systems with Bounded Controls,” IEEE Trans. Circuits Syst. II, 54(2), pp. 151–155. [CrossRef]
Liu, X., 2009, “Constrained Control of Positive Systems with Delays,” IEEE Trans. Autom. Control, 54(7), pp. 1596–1600. [CrossRef]
Zhao, X., Zhang, L., Shi, P., and Liu, M., 2012, “Stability of Switched Positive Linear Systems with Average Dwell Time Switching,” Automatica, 48(6), pp. 1132–1137. [CrossRef]
Chen, G., and Yang, Y., 2014, “Finite-Time Stability of Switched Positive Linear Systems,” Int. J. Robust Nonlinear Control, 24(1), pp. 179–190. [CrossRef]
Gahinet, P., Nemirovski, A., Laub, A. J., and Chilali, M., 1995, LMI Control Toolbox, The MathWorks, Inc., Natick, MA.
El Ghaoui, L., Oustry, F., and AitRami, M., 1997, “A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems,” IEEE Trans. Autom. Control, 42(8), pp. 1171–1176. [CrossRef]
Young, S. L., Young, S. M., Wook, H. K., and Kwan, H. L., 2001, “Delay-Dependent Robust H Control for Uncertain Systems With Time-Varying State-Delay,” Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, Dec. 4–7, pp. 3208–3213.
Zhang, L., Huang, B., and Lam, J., 2003, “H Model Reduction of Markovian Jump Linear Systems,” Syst. Control Lett., 50(2), pp. 103–118. [CrossRef]
Jørgensen, N. L., 1976, “A Stability Analysis of a Dynamic Leontief Model of Exponential Growth in Consumption,” Scandinavian J. Econ., 78(4), pp. 561–570. [CrossRef]
Luenberger, D. G., and Arbel, A., 1977, “Singular Dynamic Leontief Systems,” Econometrica, 45(4), pp. 991–995. [CrossRef]
Jódar, L., and Merello, P., 2010, “Positive Solutions of Discrete Dynamic Leontief Input–Output Model with Possibly Singular Capital Matrix,” Math. Comput. Modell., 52(7–8), pp. 1081–1087. [CrossRef]


Grahic Jump Location
Fig. 1

err(c2) introduced by the sufficient condition in Theorem 2

Grahic Jump Location
Fig. 2

Closed-loop weighted state norm from four different initial conditions for Example 1

Grahic Jump Location
Fig. 3

Closed-loop system from Example 2: state responses and ‖x∧‖R (top) as well as the control input (bottom)




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