Abstract
This paper presents the design of an observer-based stabilizing controller for linear discrete-time systems subject to interval time-varying state-delay. In this work, the problem has been formulated in convex optimization framework by constructing a new Lyapunov–Krasovskii (LK) functional to derive delay-dependent stabilization criteria. The summation inequality and the extended reciprocally convex inequality are exploited to obtain a less conservative delay upper bound in linear matrix inequality (LMI) framework. The derived stability conditions are delay-dependent and thus ensure global asymptotic stability in presence of any time-delay less than the obtained delay upper bound. Numerical examples are included to demonstrate the usefulness of the developed results.
Issue Section:
Technical Briefs
References
1.
Dey
,
R.
,
Ray
,
G.
, and
Balas
,
V. E.
, 2017
, Stability and Stabilization of Linear and Fuzzy Time-Delay Systems: A Linear Matrix Inequality Approach
, Vol.
141
,
Springer
, Berlin
.2.
Fridman
,
E.
, and
Shaked
,
U.
, 2002
, “
An Improved Stabilization Method for Linear Time-Delay Systems
,” IEEE Trans. Autom. Control
,
47
(11
), pp. 1931
–1937
.10.1109/TAC.2002.8044623.
Kwon
,
O.
,
Park
,
M.-J.
,
Park
,
J. H.
,
Lee
,
S.-M.
, and
Cha
,
E.-J.
, 2013
, “
Stability and Stabilization for Discrete-Time Systems With Time-Varying Delays Via Augmented Lyapunov–Krasovskii Functional
,” J. Franklin Inst.
,
350
(3
), pp. 521
–540
.10.1016/j.jfranklin.2012.12.0134.
Moon
,
Y. S.
,
Park
,
P.
,
Kwon
,
W. H.
, and
Lee
,
Y. S.
, 2001
, “
Delay-Dependent Robust Stabilization of Uncertain State-Delayed Systems
,” Int. J. Control
,
74
(14
), pp. 1447
–1455
.10.1080/002071701100671165.
Park
,
P.
,
Ko
,
J. W.
, and
Jeong
,
C.
, 2011
, “
Reciprocally Convex Approach to Stability of Systems With Time-Varying Delays
,” Automatica
,
47
(1
), pp. 235
–238
.10.1016/j.automatica.2010.10.0146.
Seuret
,
A.
,
Gouaisbaut
,
F.
, and
Fridman
,
E.
, 2015
, “
Stability of Discrete-Time Systems With Time-Varying Delays Via a Novel Summation Inequality
,” IEEE Trans. Autom. Control
,
60
(10
), pp. 2740
–2745
.10.1109/TAC.2015.23988857.
Zhang
,
C.-K.
,
He
,
Y.
,
Jiang
,
L.
,
Wu
,
M.
, and
Wang
,
Q.-G.
, 2017
, “
An Extended Reciprocally Convex Matrix Inequality for Stability Analysis of Systems With Time-Varying Delay
,” Automatica
,
85
, pp. 481
–485
.10.1016/j.automatica.2017.07.0568.
Nam
,
P. T.
,
Trinh
,
H.
, and
Pathirana
,
P. N.
, 2015
, “
Discrete Inequalities Based on Multiple Auxiliary Functions and Their Applications to Stability Analysis of Time-Delay Systems
,” J. Franklin Inst.
,
352
(12
), pp. 5810
–5831
.10.1016/j.jfranklin.2015.09.0189.
Chen
,
J.
,
Lu
,
J.
, and
Xu
,
S.
, 2016
, “
Summation Inequality and Its Application to Stability Analysis for Time-Delay Systems
,” IET Control Theory Appl.
,
10
(4
), pp. 391
–395
.10.1049/iet-cta.2015.057610.
Chen, J., Xu, S., Jia, X., and Zhang, B.,
2017
, “
Novel Summation Inequalities and Their Applications to Stability Analysis for Systems With Time-Varying Delay
,” IEEE Trans. Automatic Control
,
62
(5
), pp. 2470
–2475
.10.1109/TAC.2016.260690211.
Lee
,
S. Y.
,
Park
,
J.
, and
Park
,
P.
, 2019
, “
Bessel Summation Inequalities for Stability Analysis of Discrete-Time Systems With Time-Varying Delays
,” Int. J. Robust Nonlinear Control
,
29
(2
), pp. 473
–491
.10.1002/rnc.439812.
Zhao
,
T.
,
Liang
,
W.
,
Dian
,
S.
,
Xiao
,
J.
, and
Wei
,
Z.
, 2018
, “
Improved Stability and Stabilisation Criteria for Discrete Time-Delay Systems Via a Novel Double Summation Inequality
,” IET Control Theory Appl.
,
12
(3
), pp. 327
–337
.10.1049/iet-cta.2017.079113.
Venkatesh
,
M.
,
Patra
,
S.
, and
Ray
,
G.
, 2021
, “
Improved Robust Stability and Stabilization Conditions for Discrete-Time Linear Systems With Time-Varying Delay
,” Int. J. Autom. Control
(in press).14.
Maity
,
D.
,
Mamduhi
,
M. H.
,
Hirche
,
S.
,
Johansson
,
K. H.
, and
Baras
,
J. S.
, 2019
, “
Optimal LQG Control Under Delay-Dependent Costly Information
,” IEEE Control Syst. Lett.
,
3
(1
), pp. 102
–107
.10.1109/LCSYS.2018.285364815.
He
,
Y.
,
Wu
,
M.
,
Liu
,
G.-P.
, and
She
,
J.-H.
, 2008
, “
Output Feedback Stabilization for a Discrete-Time System With a Time-Varying Delay
,” IEEE Trans. Autom. Control
,
53
(10
), pp. 2372
–2377
.10.1109/TAC.2008.200752216.
Chen
,
J.
,
Lin
,
C.
,
Chen
,
B.
, and
Wang
,
Q.-G.
, 2017
, “
Improved Stability Criterion and Output Feedback Control for Discrete Time-Delay Systems
,” Appl. Math. Modell.
,
52
, pp. 82
–93
.10.1016/j.apm.2017.07.04817.
Hassan
,
L.
, 2013
, “
Observation and Control of Nonlinear Time-Delay Systems
,” Ph.D. dissertation,
Universite de Lorraine
, Nancy, France
.18.
Echi
,
N.
, 2021
, “
Observer Design and Practical Stability of Nonlinear Systems Under Unknown Time-Delay
,” Asian J. Control
,
23
(2
), pp. 685
–696
.10.1002/asjc.227119.
Zhu
,
Q.
,
Lu
,
K.
, and
Zhu
,
Y.
, 2016
, “
Observer-Based Feedback Control of Networked Control Systems With Delays and Packet Dropouts
,” ASME J. Dyn. Syst., Meas., Control
,
138
(2
), p. 021011
.10.1115/1.403213520.
Li
,
J.-N.
,
Xu
,
Y.-F.
,
Bao
,
W.-D.
,
Li
,
Z.-J.
, and
Li
,
L.-S.
, 2019
, “
Finite-Time Non-Fragile State Estimation for Discrete Neural Networks With Sensor Failures, Time-Varying Delays and Randomly Occurring Sensor Nonlinearity
,” J. Franklin Inst.
,
356
(3
), pp. 1566
–1589
.10.1016/j.jfranklin.2018.10.03221.
Guo
,
J.
,
Wang
,
Y.
, and
Bo
,
Y.
, 2021
, “
Adaptive Global Stability of Nonlinear Pure-Feedback Systems With Unknown Time-Varying Delays
,” ASME J. Dyn. Syst., Meas., Control
,
143
(3
), p. 034502
.10.1115/1.404858722.
Li
,
L.
, and
Jia
,
Y.
, 2009
, “
Non-Fragile Dynamic Output Feedback Control for Linear Systems With Time-Varying Delay
,” IET Control Theory Appl.
,
3
(8
), pp. 995
–1005
.10.1049/iet-cta.2008.000823.
Lo
,
J.-C.
, and
Lin
,
M.-L.
, 2004
, “
Observer-Based Robust H∞ Control for Fuzzy Systems Using Two-Step Procedure
,” IEEE Trans. Fuzzy Syst.
,
12
(3
), pp. 350
–359
.10.1109/TFUZZ.2004.82599224.
Xu
,
S.
,
Lam
,
J.
,
Zhang
,
B.
, and
Zou
,
Y.
, 2014
, “
A New Result on the Delay-Dependent Stability of Discrete Systems With Time-Varying Delays
,” Int. J. Robust Nonlinear Control
,
24
(16
), pp. 2512
–2521
.10.1002/rnc.3006Copyright © 2021 by ASME
You do not currently have access to this content.
