Document Type : Research Articles

Authors

Faculty of Electrical and Computer Engineering, University of Birjand, Birjand, Iran

Abstract

This article aims to derive new sufficient conditions to guarantee the stability of piecewise affine systems with time-varying delay (PWA-TVD). The set of delay-dependent linear matrix inequality (LMI) describes the novel stability criteria. This approach considers the PWA-TVD system with a time-delayed state-dependent switching signal. The newly suggested Lyapunov-Krasovskii functional (L-K-F) and improved estimation of its derivative have a crucial role in decreasing the complexity and conservatism of the proposed stability results. The suggested L-K-F belongs to the current and time-delayed states, the integral of the states over the time-varying delay, and time derivation of the states. A new inequality was used to obtain an upper bound (UB) for the time derivation of the Lyapunov functional. Then based on this UB, less conservative results are achieved. The theoretical results are applied to the numerical examples. The results confirm the effectiveness of the presented method. The conservative index is the maximum admissible UB of time delay.

Keywords

Main Subjects

[1] M. S. Branicky, "Studies in hybrid systems: Modeling,
analysis, and control," Sc.D. dissertation, Dept. of Electrical
Engineering and Computer Science, MIT, Cambridge, MA,
June 1995
.
[2] J. P. Richard, "Time-delay systems: An overview of some
recent advances and open problems," Automatica, vol. 39, no.
10, pp. 16671694, 2003.

[3] H. Ye, A. N. Michel, and L. Hou, "Stability theory for hybrid
dynamical systems," in Proc. IEEE Decision and Control
Conference, New Orleans, LA, Dec. 1995.

[4] M. Johansson and A. Rantzer, "Computation of piecewise
quadratic Lyapunov functions for hybrid systems," Dept.
Automatic Control, Lund Institute of Technology, Tech Rep.
TFRT-7549, June 1996.

[5] M. S. Branicky, "Multiple lyapunov functions and other
analysis tools for switched and hybrid systems," IEEE
Transaction on Automatic Control, vol. 43, no. 4, pp. 475
482, April 1998.

[6] A. Hassibi, and S. Boyd, "Quadratic stabilization and control
of piecewise-linear systems," Proceedings of the American
Control Conference, Philadelphia, PA, vol. 6, pp. 3659
3664, June 1998.

[7] A. Bemporad, G. Ferrari-Trecate, and M. Morari,
"Observability and controllability of piecewise affine and
hybrid systems," IEEE Transaction on Automatic Control,
vol. 45, no. 10, pp. 18641876, Oct. 2000.

[8] K. Gu, V. L. Kharitonov, and J. Chen, "Stability of Time-
Delay Systems, " Springer Science & Business Media, 2003.

[9] H. B. Zeng, Y. He, M. Wu, and S. P. Xiao, "Less conservative
results on stability for linear systems with a time-varying
delay," Optimal control, application and Method., vol. 34, no.
6, pp. 670679, 2013.

[10] W. kwon, B. Koo, and S. M. Lee, "Novel Lyapunov
Krasovskii functional with delay-dependent matrix for
stability of time-varying delay systems", Applied
Mathematics and Computation, vol. 320, pp.149-157, 2018.

[11] F. Long, L. Jiang, Y. He, and M. Wu, "Stability analysis of
systems with time-varying delay via novel augmented
LyapunovKrasovskii functionals and an improved integral
inequality", Applied Mathematics and Computation, vol. 357,
pp.325-337, 2019.

[12] T. H. Lee, and J. H. Park, "Improved stability conditions of
time-varying delay systems based on new Lyapunov
functionals", Journal of the Franklin Institute, vol. 355, no.3,
pp.1176-1191, 2018.

[13] A. Seuret and F. Gouaisbaut, "Wirtinger-based integral
inequality: Application to time-delay systems," Automatica,
vol. 49, no. 9, pp. 28602866, 2013.

[14] H. Zeng, Y. He, M. Wu, and J. She, "Free-Matrix-Based
integral inequality for stability analysis of systems with time-
varying delay," IEEE Transaction on Automatic Control, vol.
60, no. 10, pp. 27682772, 2015.

[15] R. Datta, R. Dey, B. Bhattacharya, R. Saravanakumar, and C.
K. Ahn, "New double integral inequality with application to
stability analysis for linear retarded systems", IET Control
Theory & Applications, vol. 13, no. 10, pp.1514-1524,2019.

[16] R. Zhang, D. Zeng, J. H. Park, S. Zhong, Y. Liu, and X. Zhou,
"New approaches to stability analysis for time-varying delay
systems", Journal of the Franklin Institute, vol. 356, no.7, pp.
4174-4189, 2019.


[17] V. Kulkarni, M. Jun, and J. Hespanha, "Piecewise quadratic
Lyapunov functions for piecewise affine time-delay
systems," Proceedings of the American Control Conference,
vol. 5, no.30, pp. 3885- 3889, 2004.

[18] K. Moezzi, L. Rodrigues, and A. G Aghdam, "Stability of
Uncertain Piecewise Affine Systems with Time Delay:
Delay-Dependent Lyapunov Approach," International
Journal of Control, vol. 82, no. 8, pp. 1423-1434, 2009.

[19] K. Moezzi, L. Rodrigues, and A. G. Aghdam, "Stability of
uncertain piecewise affine systems with time-delay,"
American Control Conference, pp. 2373-2378, 2009.

[20] S. Duan, J. Ni, and A. G. Ulsoy, "An improved LMI-based
approach for stability of piecewise affine time-delay systems
with uncertainty," International Journal of Control, vol. 85,
no. 9, pp. 1218-1234, 2012.

[21] S. Duan, J. Ni, and A. G. Ulsoy, "Stability criteria for
uncertain piecewise affine time-delay systems," American
Control Conference (ACC), pp. 5460-5465, 2012.

[22] C. Fiter and E. Fridman, "Stability of
piecewise affine
systems with state
-dependent delay, and application to
congestion control,"
in 52nd IEEE conference on decision
and control
, pp. 1572-1577, 2013.

[23] M. Johansson, "Piecewise Linear Control Systems,"
Springer-Verlag, 2003.

[24] T. H. Lee, H. Ju, and S. Xu, "Relaxed conditions for stability
of time-varying delay systems," Automatica, vol. 75, pp. 11-
15, 2017.

[25] R. E. Skelton, T. Iwasaki, and K. M. Grigoradis, "A unified
algebraic approach to linear control," design. New York:
Taylor and Francis, 1997.