Reachable Set Bounding Estimation for Distributed Delay Systems with Disturbances

The reachable set bounding estimation for distributed
delay systems with disturbances is a new problem. In this paper,we
consider this problem subject to not only time varying delay and
polytopic uncertainties but also distributed delay systems which is
not studied fully untill now. we can obtain improved non-ellipsoidal
reachable set estimation for neural networks with time-varying delay
by the maximal Lyapunov-Krasovskii fuctional which is constructed
as the pointwise maximum of a family of Lyapunov-Krasovskii
fuctionals corresponds to vertexes of uncertain polytope.On the other
hand,matrix inequalities containing only one scalar and Matlabs
LMI Toolbox is utilized to give a non-ellipsoidal description of the
reachable set.finally,numerical examples are given to illustrate the
existing results.

• Li Xu
• Shouming Zhong

• Reachable set
• Distributed delay
• Lyapunov-Krasovskii function
• Polytopic uncertainties.

[1] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnam, Linear Matrix
Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA,
1994.
[2] Boyd S,Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities
in system and control theory. Society for Industrial an Applied
Mathematics, PA:Philadelphia 1994.
[3] S. Boyd, Lecture Notes for Convex Optimization II,
2007.¡http://www.stanford.edu/class/ee364b/lectures.html¿
[4] Q.L. Han, on robust stability of neutral systems with time-varying
discrete delay and norm-bounded uncertainty, Automatica 40 (2004)
1087-1092.
[5] J.H. Kim, Improved ellipsoidal bound of reachable sets for time-delayed
linear systems with disturbances, Automatica 44 (2008) 2940-2943
[6] Z. Zuo, D.W.C. Ho, Y. Wang, Reachable set bounding for delayed systems
with polytopic uncertainties: the maximal LyapunovCKrasovskii
functional approach, Automatica 46 (2010) 949-952
[7] 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. 5 (15) (2011) 1808-1812.
[8] Z.Zuo,C.Yang,Y Wang,A new method for stability analysis of recurrent
neural networks with interval time-varying delay,IEEE Trans.Neural
Networks 21(2)(2010)339-344.
[9] D. Zhang, L. Yu, H∞ filtering for linear neutral systems with mixed
time-varying delays and nonlinear perturbations, Journal of the Franklin
Institute 347 (2010) 1374-1390.
[10] C. Shen, S. Zhong, The ellipsoidal bound of reachable sets for linear
neutral systems with disturbances, J. Frankl. Inst. 348 (2011) 2570-2585
[11] E. Fridman, U. Shaked, On reachable sets for linear systems with delay
and bounded peak inputs, Automatica 39 (2003) 2005-2010.
[12] A. Alessandri, M. Baglietto, G. Battistelli, On estimation error bounds
for receding-horizon filters using quadratic boundedness, IEEE Transactions
on Automatic Control 49 (2004) 1350-1355.
[13] O.M. Kwon, S.M. Lee, Ju H. Park, On reachable set bounding of
uncertain dynamic systems with time-varying delays and disturbances,
Inf. Sci. 181 (2011) 3735-3748.
[14] A. Alessandri, M. Baglietto, G. Battistelli, Design of state estimators
for uncertain linear systems using quadratic boundedness, Automatica
42 (2006) 497-502.
[15] Y.H.Du,S.M.Zhong,Exponential passivity of BAM neural networks
with time-varying delays.Applied Mathematics and Computation
221(2013)727-740.
[16] Y. He, Q.G. Wang, L.H. Xie, C. Lin, Further improvement of freeweighting
matrices technique for systems with time-varying delay, IEEE
Transactions on Automatic Control 52 (2) (2007) 293-299.
[17] P.T. Nam, P.N. Pathirana, Further result on reachable set bounding for
linear uncertain polytopic systems with interval time-varying delays,
Automatica 47 (2011) 1838-1841
[18] N. Ramdani, N. Meslem, Y. Candau, A hybrid bounding method for
computing an over-approximation for the reachable set of uncertain
nonlinear systems, IEEE Transactions on Automatic Control 54 (10)
(2009) 2352-2364