# State estimation of static neural networks with interval time-varying delays and sampled-data control

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## Abstract

In this paper, we consider the problem of sampled-data control for static neural networks with interval time-varying delays. As opposed to the continuous measurement, the sampled measurement is used to estimate the neuron states, and a sampled-data estimator is constructed. By converting the sampling period into a bounded time-varying delays, the error dynamics of the considered neural network is derived in terms of a dynamic system with two different time-delays. By constructing a suitable Lyapunov–Krasovskii functional with double and triple integral terms and using Jensen inequality, delay-dependent criteria are presented, so that the error system is asymptotically stable. Delay-dependent asymptotically stability condition is established in terms of linear matrix inequality (LMI) framework, which can be readily solved using the LMI toolbox. Finally, three examples are given to show the effectiveness of the theoretical results.

## Keywords

Lyapunov method Linear matrix inequality Static neural networks Sample-data control Time-varying delay## Mathematics Subject Classification

93CXX 93DXX 68TXX 65KXX 65LXX## References

- Boyd B, Ghoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaCrossRefGoogle Scholar
- Dongsheng Y, Liu X, Xu Y, Wang Y, Liu Z (2013) State estimation of recurrent neural networks with interval time-varying delay: an improved delay-dependent approach. Neural Comput Appl 23:1149–1158CrossRefGoogle Scholar
- Du B, Lam J (2009) Stability analysis of static recurrent neural networks using delay-partitioning and projection. Neural Netw 22:343–347CrossRefGoogle Scholar
- Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338CrossRefGoogle Scholar
- Feng W, Yang SX, Wu H (2012) Improved asymptotical stability criteria for static recurrent neural networks. Int J Comput Math 89:597–605MathSciNetCrossRefGoogle Scholar
- Gao H, Wu J (2009) Robust sampled-data \(H_{\infty }\) control with stochastic sampling. Automatica 45:1729–1736MathSciNetCrossRefGoogle Scholar
- He Y, Wang QG, Lin C, Wu M (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica 43:371–376MathSciNetCrossRefGoogle Scholar
- Hu J, Li N, Liu X, Zhang G (2013) Sampled-data state estimation for delayed neural networks with Markovian jumping parameters. Nonlinear Dyn 73:275–284MathSciNetCrossRefGoogle Scholar
- Huang H, Feng G, Cao J (2011) An LMI approach to delay-dependent state estimation for delayed neural networks. Neurocomputing 74:792–796CrossRefGoogle Scholar
- Huang H, Feng G, Cao J (2011) Guaranteed performance state estimation of static neural networks with time-varying delay. Neurocomputing 74:606–616CrossRefGoogle Scholar
- Kim SH, Park PG, Jeong C (2010) Robust /spl alpha/ stabilisation of networked control systems with packet analyser. IET Control Theory Appl 4:1828–1837CrossRefGoogle Scholar
- Lee TH, Park JH (2017) Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach. IEEE Trans Autom Control. doi: 10.1109/TAC.2017.2670786
- Lee TH, Park JH (2017) Improved criteria for sampled-data synchronization of chaotic Lur’e systems using two new approaches. Nonlinear Anal Hybrid Syst 24:132–145MathSciNetCrossRefGoogle Scholar
- Li P, Lam J, Shu Z (2010) On the transient and steady-state estimates of interval genetic regulatory networks. IEEE Trans Syst Man Cybern Part B Cybern 40:336–349CrossRefGoogle Scholar
- Li P, Cao J (2006) Stability in static delayed neural networks: a nonlinear measure approach. Neurocomputing 69:13–15Google Scholar
- Liu PL (2013) Improved delay-dependent robust stability criteria for recurrent neural networks with time-varying delays. ISA Trans 52:30–35CrossRefGoogle Scholar
- Lu CY, Su TJ, Huang SC (2008) Delay- dependent stability analysis for recurrent neural networks with time-varying delay. IET Control Theory Appl 8:736–742MathSciNetCrossRefGoogle Scholar
- Mahmoud MS (2009) New exponentially convergent state estimation method for delayed neural networks. Neurocomputing 72:3935–3942CrossRefGoogle Scholar
- Park PG, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47:235–238MathSciNetCrossRefGoogle Scholar
- Rakkiyappan R, Sakthivel N, Park JH, Kwon OM (2013) Sampled-data state estimation for Markovian jumping fuzzy cellular neural networks with mode-dependent probabilistic time-varying delays. Appl Math Comput 221:741–769MathSciNetzbMATHGoogle Scholar
- Ren J, Zhu H, Zhong S, Ding Y, Shi K (2015) State estimation for neural networks with multiple time delays. Neurocomputing 151:501–510CrossRefGoogle Scholar
- Sakthivel R, Arunkumar A, Mathiyalagan K (2015) Robust sampled-data \(H_\infty \) control for mechanical systems. Complexity 20:19–29MathSciNetCrossRefGoogle Scholar
- Saravanakumar R, Syed Ali M, Cao J, Huang H (2016) \(H_\infty \) state estimation of generalised neural networks with interval time-varying delays. Int J Syst Sci 1 – 12Google Scholar
- Shen H, Zhu Y, Zhang L, Park JH (2017) Extended dissipative state estimation for markov jump neural networks with unreliable links. IEEE Trans Neural Netw Learn Syst 28:346–358MathSciNetCrossRefGoogle Scholar
- Song X, Park JuH (2017) Linear optimal estimation for discrete-time measurement delay systems with multichannel multiplicative noise. IEEE Trans Circuits Syst II Exp Briefs 64:156–160CrossRefGoogle Scholar
- Sun J, Liu GP, Chen J, Rees D (2010) Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica 46:466–470MathSciNetCrossRefGoogle Scholar
- Sun J, Chen J (2013) Stability analysis of static recurrent neural networks with interval time-varying delay. Appl Math Comput 221:111–120MathSciNetzbMATHGoogle Scholar
- Syed Ali M, Saravanakumar R, Arik S (2015) Novel \(H\infty \) state estimation of static neural networks with interval time-varying delays via augmented Lyapunov-Krasovskii functional. Neurocomputing. doi: 10.1016/j.neucom.2015.07.038 CrossRefGoogle Scholar
- Syed Ali M (2015) Stability of Markovian jumping recurrent neural networks with discrete and distributed time-varying delays. Neurocomputing 149:1280–1285CrossRefGoogle Scholar
- Syed Ali M, Gunasekaran N, Kwon OM (2016) Delay-dependent \(H_\infty \) performance state estimation of static delayed neural networks using sampled-data control. Neural Comput Appl. doi: 10.1007/s00521-016-2671-3
- Syed Ali M, Gunasekaran N, Zhu Q (2017) State estimation of T–S fuzzy delayed neural networks with Markovian jumping parameters using sampled-data control. Fuzzy Set Syst. doi: 10.1016/j.fss.2016.03.012
- Wang H, Song Q (2010) State estimation for neural networks with mixed interval time-varying delays. Neurocomputing 73:1281–1288CrossRefGoogle Scholar
- Wu SL, Li KL, Huang TZ (2011) Exponential stability of static neural networks with time delay and impulses. IET Control Theory Appl 5:943–951MathSciNetCrossRefGoogle Scholar
- Wu Z, Lam J, Su H, Chu J (2012) Stability and dissipativity analysis of static neural networks with time delay. IEEE Trans Neural Netw Learn Syst 23:199–210CrossRefGoogle Scholar
- Wu YY, Wu YQ (2009) Stability analysis for recurrent neural networks with time-varying delay. Int J Autom Comput 6:223–227CrossRefGoogle Scholar
- Xu Z, Qiao H, Peng J, Zhang B (2004) A comparative study of two modeling approaches in neural networks. Neural Netw 17:73–85CrossRefGoogle Scholar
- Zhang W, Yu L (2010) Stabilization of sampled-Data control systems with control inputs missing. IEEE Trans Autom Control 55:447–452MathSciNetCrossRefGoogle Scholar
- Zheng CD, Zhang H, Wang Z (2009) Delay-dependent globally exponential stability criteria for static neural networks: an LMI approach. IEEE Trans Circuits Syst II Exp Briefs 56:605–609CrossRefGoogle Scholar
- Zhu XL, Wang Y (2011) Stabilization for sampled-data neural-network-based control systems. IEEE Trans Syst Man Cybern B Cybern 41:210–221CrossRefGoogle Scholar
- Zuo Z, Yang C, Wang Y (2010) A new method for stability analysis of recurrent neural networks with interval time-varying delay. IEEE Trans Neural Netw 21:339–344CrossRefGoogle Scholar