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This study considers the problem of state bounding for a class of discrete-time systems with interval time-varying delay and bounded disturbance inputs. By using an improved Lyapunov–Krasovskii functional combining with the delay-decomposition technique and the reciprocally convex approach, the authors first derive new delay-dependent conditions in terms of matrix inequalities to guarantee the existence of a ball such that, for any initial condition, the state trajectory of the system is either bounded within that ball or converges exponentially within it. On the basis of these new conditions, the authors then derive an improved ellipsoid reachable set bounding and a new result on exponential stability of discrete-time systems with interval time-varying delay. Numerical examples are presented to show the effectiveness of the obtained results and improvement over existing results.
References
-
-
1)
-
20. Seuret, A., Gouaisbaut, F.: ‘Wirtinger-based integral inequality: application to time-delay systems’, Automatica, 2013, 49, (9), pp. 2860–2866 (doi: 10.1016/j.automatica.2013.05.030).
-
2)
-
10. Wang, R., Shi, P., Wu, Z.G., Sun, Y.T.: ‘Stabilization of switched delay systems with polytopic uncertainties under asynchronous switching’, J. Franklin Inst., 2013, 350, (8), pp. 2028–2043 (doi: 10.1016/j.jfranklin.2013.05.023).
-
3)
-
31. Oucheriah, S.: ‘Robust exponential convergence of a class of linear delayed systems with bounded controllers and disturbances’, Automatica, 2006, 42, (11), pp. 1863–1867 (doi: 10.1016/j.automatica.2006.05.023).
-
4)
-
R. Sipahi ,
S.-I. Niculescu ,
C.T. Abdallah ,
W. Michiels ,
K. Gu
.
Delay effects on the stability of dynamical systems, limitations and opportunities in control.
IEEE Control Syst. Mag.
,
1 ,
38 -
65
-
5)
-
41. Xiao, N., Jia, Y., Matsuno, F.: ‘Delay distribution dependent stability criteria for discrete-time systems with interval time-varying delay’. Proc. American Control Conf., Washington, DC, 17–19 June 2013, pp. 1733–1738.
-
6)
-
8. Fernando, T.L., Phat, V.N., Trinh, H.M.: ‘Decentralized stabilization of large-scale systems with interval time-varying delays in interconnections’, Int. J. Adapt. Control Signal Process., 2012, 26, (6), pp. 541–554 (doi: 10.1002/acs.2274).
-
7)
-
C. Peng ,
Y.C. Tian
.
Improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay.
IET Control Theory Appl.
,
9 ,
752 -
761
-
8)
-
21. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: ‘Linear matrix inequalities in system and control theory’ (SIAM, Philadelphia, 1994).
-
9)
-
Z. Zuo ,
D.W.C. Ho ,
Y. Wang
.
Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov-Krasovskii functional approach.
Automatica
,
949 -
952
-
10)
-
Z. Zuo ,
D.W.C. Ho ,
Y. Wang
.
Reachable set estimation for linear systems in the presence of both discrete and distributed delays.
IET Control Theory Appl.
,
1808 -
1812
-
11)
-
32. Nam, P.T., Pathirana, P.N., Trinh, H.: ‘Exponential convergence of time-delay systems in the presence of bounded disturbances’, J. Optim. Theory Appl., 2013, 157, (3), pp. 843–852 (doi: 10.1007/s10957-012-0240-1).
-
12)
-
J.H. Kim
.
Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances.
Automatica
,
2940 -
2943
-
13)
-
38. Dong, X.Z.: ‘LMI-based H∞ control for linear singular discrete systems: a novel method’, IET Control Theory Appl., 2013, 7, (16), pp. 2028–2036 (doi: 10.1049/iet-cta.2013.0197).
-
14)
-
40. Lam, J., Zhang, B., Chen, Y., Xu, S.: ‘Reachable set estimation for discrete-time linear systems with time delays’, Int. J. Robust. Nonlinear Control, 2013, .
-
15)
-
X. Meng ,
J. Lam ,
B. Du ,
H. Gao
.
A delay-partitioning approach to the stability analysis of discrete-time systems.
Automatica
,
3 ,
610 -
614
-
16)
-
37. Hou, L., Zong, G., Wu, Y., Cao, Y.: ‘Exponential l2 − l∞ output tracking control for discrete-time switched system with time-varying delay’, Int. J. Robust. Nonlinear Control, 2012, 22, (11), pp. 1175–1194 (doi: 10.1002/rnc.1743).
-
17)
-
E. Fridman ,
U. Shaked
.
On reachable sets for linear systems with delay and bounded peak inputs.
Automatica
,
2005 -
2010
-
18)
-
3. Erneux, T.: ‘Applied delay differential equations’ (Springer, New York, 2009).
-
19)
-
Y. He ,
Q.G. Wang ,
L. Xie ,
C. Lin
.
Further improvement of free-weighting matrices technique for systems with time-varying delay.
IEEE Trans. Autom. Control
,
2 ,
293 -
299
-
20)
-
E. Fridman ,
U. Shaked
.
Stability and guaranteed cost control of uncertain discrete delay systems.
Int. J. Control
,
235 -
246
-
21)
-
30. Zuo, Z., Fu, Y., Chen, Y., Wang, Y.: ‘A new method of reachable set estimation for time delay systems with polytopic uncertainties’, Appl. Math. Comput., 2013, 221, (15), pp. 639–646 (doi: 10.1016/j.amc.2013.06.099).
-
22)
-
4. Smith, H.: ‘An introduction to delay differential equations with applications to the life sciences’ (Springer, New York, 2011).
-
23)
-
P. Park ,
J.W. Ko ,
C. Jeong
.
Reciprocally convex approach to stability of systems with time-varying delays.
Automatica
,
1 ,
235 -
238
-
24)
-
28. Zuo, Z., Fu, Y., Wang, Y.: ‘Results on reachable set estimation for linear systems with both discrete and distributed delays’, IET Control Theory Appl., 2012, 6, (14), pp. 2346–2350 (doi: 10.1049/iet-cta.2012.0491).
-
25)
-
39. That, N.D., Nam, P.T., Ha, Q.P.: ‘Reachable set bounding for linear discrete-time systems with delays and bounded disturbances’, J. Optim. Theory Appl., 2013, 157, (1), pp. 96–107 (doi: 10.1007/s10957-012-0179-2).
-
26)
-
16. Feng, Z., Lam, J., Yang, G.H.: ‘Optimal partitioning method for stability analysis of continuous/discrete delay systems’, Int. J. Robust Nonlinear Control, 2013, .
-
27)
-
L.V. Hien ,
V.N. Phat
.
Exponential stability and stabilization of a class of uncertain linear time-delay systems.
J. Franklin Inst.
,
611 -
625
-
28)
-
17. Zhu, S., Li, Z., Zhang, C.: ‘Delay decomposition approach to delay-dependent stability for singular time-delay systems’, IET Control Theory Appl., 2010, 4, (11), pp. 2613–2620 (doi: 10.1049/iet-cta.2009.0426).
-
29)
-
29. Zuo, Z., Chen, Y., Wang, Y., Ho, D.W.C., Chen, M.Z.Q., Li, H.: ‘A note on reachable set bounding for delayed systems with polytopic uncertainties’, J. Franklin Inst., 2013, 350, (7), pp. 1827–1835 (doi: 10.1016/j.jfranklin.2013.04.025).
-
30)
-
36. Li, Z., Sun, G., Gao, H.: ‘Guaranteed cost control for discrete-time Markovian jump linear system with time delay’, Int. J. Syst. Sci., 2013, 44, (7), pp. 1312–1324 (doi: 10.1080/00207721.2012.659713).
-
31)
-
35. Kwon, O.M., Park, M.J., Park, J.H., Lee, S.M., Cha, E.J.: ‘Improved robust stability criteria for uncertain discrete-time systems with interval time-varying delays via new zero equalities’, IET Control Theory Appl., 2012, 6, (16), pp. 2567–2575 (doi: 10.1049/iet-cta.2012.0257).
-
32)
-
D. Yue ,
E. Tian ,
Y. Zhang
.
A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay.
Int. J. Robust Nonlinear Control
,
3 ,
1493 -
1518
-
33)
-
7. Hien, L.V., Phat, V.N.: ‘New exponential estimate for robust stability of nonlinear neutral time-delay systems with convex polytopic uncertainties’, J. Nonlinear Conv. Anal., 2011, 12, (3), pp. 541–552.
-
34)
-
1. Komanovskii, V.B., Myshkis, A.D.: ‘Applied theory of functional differential equations’ (Kluwer, Dordrecht, 1992).
-
35)
-
P.T. Nam ,
P.N. Pathirana
.
Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays.
Automatica
,
1838 -
1841
-
36)
-
9. Fernando, T.L., Phat, V.N., Trinh, H.M.: ‘Output feedback guaranteed cost control of uncertain linear discrete systems with interval time-varying delays’, Appl. Math. Model., 2013, 37, (3), pp. 1580–1589 (doi: 10.1016/j.apm.2012.04.035).
-
37)
-
11. Chen, H., Hu, P.: ‘New result on exponential stability for singular systems with two interval time-varying delays’, IET Control Theory Appl., 2013, 7, (15), pp. 941–1949 (doi: 10.1049/iet-cta.2013.0396).
-
38)
-
5. Phat, V.N., Hien, L.V.: ‘An application of Razumikhin theorem to exponential stability for linear non-autonomous systems with time-varying delay’, Appl. Math. Lett., 2009, 22, (9), pp. 1412–1417 (doi: 10.1016/j.aml.2009.01.053).
-
39)
-
O.M. Kwon ,
S.M. Lee ,
J.H. Park
.
On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances.
Inf. Sci.
,
3735 -
3748
-
40)
-
24. Kwon, O.M., Park, M.J., Park, Ju.H., Lee, S.M., Cha, E.J.: ‘New delay-partitioning approaches to stability criteria for uncertain neutral systems with time-varying delays’, J. Franklin Inst., 2012, 349, (9), pp. 2799–2823 (doi: 10.1016/j.jfranklin.2012.08.013).
-
41)
-
2. Chiasson, J., Loiseau, J.J.: ‘Applications of time delay systems’ (Springer, Berlin, 2007).
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