# American Institute of Mathematical Sciences

April  2021, 14(4): 1415-1428. doi: 10.3934/dcdss.2020358

## A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances

 Department of Computer Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, 34320 Avcilar, Istanbul, Turkey

* Corresponding author: Sabri Arik

Received  September 2019 Revised  November 2019 Published  April 2021 Early access  May 2020

It is well-known that the global asymptotic stability analysis of neutral systems is an important concept in designing the appropriate controllers or filters for this class of systems. This paper carries out a delay-independent stability analysis of neutral systems possessing discrete time delays in the states and discrete neutral delays in the time derivative of the states in the presence of nonlinear disturbances. Some new global asymptotic stability criteria are proposed by introducing a novel Lyapunov functional. The obtained delay-independent stability criteria establish some simple and easily verifiable mathematical expressions involving the elements of the system matrices and the disturbance parameters of the neutral system. Different from the most of the previously reported stability results for neutral systems, the conditions obtained in this paper are not expressed in terms the Linear Matrix Inequalities (LMIs). Therefore, the criteria presented in this paper can be considered as the alternative results to previously published stability results stated in the LMI forms. A comparison between the results of this paper and some of previously published corresponding stability results is made to substantiate the significant improvement of the proposed results. A constructive numerical example is also presented to show applicability and the effectiveness of the proposed stability condition.

Citation: Sibel Senan, Eylem Yucel, Zeynep Orman, Ruya Samli, Sabri Arik. A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1415-1428. doi: 10.3934/dcdss.2020358
##### References:
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Wang, J. Qiu and H. Gao, Adaptive neural control of stochastic nonlinear time-delay systems with multiple constraints, IEEE Trans. Syst. Man Cybernetics Syst., 47 (2017), 1875-1883.  doi: 10.1109/TSMC.2016.2562511. [7] Y. Wei, J. Qiu and S. Fu, Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay, Nonlinear Anal. Hybrid Syst., 16 (2015), 52-71.  doi: 10.1016/j.nahs.2014.11.003. [8] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5. [9] S. I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269, Springer-Verlag, London, 2001. doi: 10.1007/1-84628-553-4. [10] V. B. Kolmanovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Stability of Functional-Differential Equations, Mathematics in Science and Engineering, 180, Academic Press, Inc., London, 1986. [11] J. K. Hale, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] S. Long, Y. Wu, S. Zhong and D. Zhang, Stability analysis for a class of neutral type singular systems with time-varying delay, Appl. Math. Comput., 339 (2018), 113-131.  doi: 10.1016/j.amc.2018.06.058. [13] H. B. Chen, New criteria on stability analysis for uncertain neutral system with mixed time-varying delays, Optimal Control Appl. Methods, 34 (2013), 487-501.  doi: 10.1002/oca.2036. [14] J. Sun, G. P. Liu, J. Chen and D. Rees, Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica J. IFAC, 46 (2010), 466-470.  doi: 10.1016/j.automatica.2009.11.002. [15] H.-B. Zeng, Y. He, M. Wu and J. She, New results on stability analysis for systems with discrete distributed delay, Automatica J. IFAC, 60 (2015), 189-192.  doi: 10.1016/j.automatica.2015.07.017. [16] M. Park, O. M. Kwon, J. H. Park, S. Lee and E. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica J. IFAC, 55 (2015), 204-208.  doi: 10.1016/j.automatica.2015.03.010. [17] L. Ding, Y. He, M. Wu and C. Ning, Improved mixed-delay-dependent asymptotic stability criteria for neutral systems, IET Control Theory Appl., 9 (2015), 2180-2187.  doi: 10.1049/iet-cta.2015.0022. [18] Y. G. Chen, W. Qian and S. M. Fei, Improved robust stability conditions for uncertain neutral systems with discrete and distributed delays, J. Franklin Inst., 352 (2015), 2634-2645.  doi: 10.1016/j.jfranklin.2015.03.040. [19] P. L. Liu, Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays, ISA Trans., 60 (2016), 53-66.  doi: 10.1016/j.isatra.2015.11.004. [20] L. L. Xiong, S. M. Zhong and D. Y. Li, Novel delay-dependent asymptotical stability of neutral systems with nonlinear perturbations, J. Comput. Appl. Math., 232 (2009), 505-513.  doi: 10.1016/j.cam.2009.06.026. [21] Y. J. Liu, W. B. Ma, M.S. Mahmoud and S. M. Lee, Improved delay-dependent exponential stability criteria for neutral-delay systems with nonlinear uncertainties, Appl. Math. Model., 39 (2015), 3164-3174.  doi: 10.1016/j.apm.2014.11.036. [22] T. Wang, T. Li, G. Zhang and S. Fei, Further triple integral approach to mixed-delay-dependent stability of time-delay neutral systems, ISA Trans., 70 (2017), 116-124.  doi: 10.1016/j.isatra.2017.05.010. [23] T. Wu, L. Xiong, J. Cao, Z. Liu and H. Zhang, New stability and stabilization conditions for stochastic neural networks of neutral type with Markovian jumping parameters, J. Franklin Inst., 355 (2018), 8462-8483.  doi: 10.1016/j.jfranklin.2018.09.032. [24] Z.-Y. Li, J. Lam and Y. Wang, Stability analysis of linear stochastic neutral-type time-delay systems with two delays, Automatica J. IFAC, 91 (2018), 179-189.  doi: 10.1016/j.automatica.2018.01.014. [25] N. Zhao, X. Zhang, Y. Xue and P. Shi, Necessary conditions for exponential stability of linear neutral type systems with multiple time delays, J. Franklin Inst., 355 (2018), 458-473.  doi: 10.1016/j.jfranklin.2017.11.016. [26] R. Mohajerpoor, L. Shanmugam, H. Abdi, R. Rakkiyappan, S. Nahavandi and J. H. Park, Improved delay-dependent stability criteria for neutral systems with mixed interval time-varying delays and nonlinear disturbances, J. Franklin Inst., 354 (2017), 1169-1194.  doi: 10.1016/j.jfranklin.2016.11.015. [27] Y. Wang, X. Zhang and X. Zhang, Neutral-delay-range-dependent absolute stability criteria for neutral-type Lur'e systems with time-varying delays, J. Franklin Inst., 353 (2016), 5025-5039.  doi: 10.1016/j.jfranklin.2016.09.014. [28] Y. Wang, Y. Xue and X. Zhang, Less conservative robust absolute stability criteria for uncertain neutral-type Lur'e systems with time-varying delays, J. Franklin Inst., 353 (2016), 816-833.  doi: 10.1016/j.jfranklin.2016.01.001. [29] W. Duan, B. Du, Z. Liu and Y. Zou, Improved stability criteria for uncertain neutral-type Lur'e systems with time-varying delays, J. Franklin Inst., 351 (2014), 4538-4554.  doi: 10.1016/j.jfranklin.2014.06.008. [30] S. S. Alaviani, A necessary and sufficient condition for delay-independent stability of linear time-varying neutral delay systems, J. Franklin Inst., 351 (2014), 2574-2581.  doi: 10.1016/j.jfranklin.2013.12.003. [31] Y. Liu, S. M. Lee, O. M. Kwon and J. H. Park, Delay-dependent exponential stability criteria for neutral systems with interval time-varying delays and nonlinear perturbations, J. Franklin Inst., 350 (2013), 3313-3327.  doi: 10.1016/j.jfranklin.2013.07.010. [32] F. Deng, W. Mao and A. Wan, A novel result on stability analysis for uncertain neutral stochastic time-varying delay systems, Appl. Math. Comput., 221 (2013), 132-143.  doi: 10.1016/j.amc.2013.05.071. [33] C. Gao, Z. Liu and R. Xu, On exponential stabilization for a class of neutral-type systems with parameter uncertainties: An integral sliding mode approach, Appl. Math. Comput., 219 (2013), 11044-11055.  doi: 10.1016/j.amc.2013.04.038. [34] H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 1996. [35] S. Arik, A modified Lyapunov functional with application to stability of neutral-type neural networks with time delays, J. Franklin Inst., 356 (2019), 276-291.  doi: 10.1016/j.jfranklin.2018.11.002. [36] Y. Sun, On simple stability criteria for nonlinear neutral systems with multiple time delays, Appl. Math. Lett., 25 (2012), 1911-1915.  doi: 10.1016/j.aml.2012.02.066.

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##### References:
 [1] E. Fridman, Lyapunov-Based Stability Analysis: Introduction to Time-Delay Systems, Springer International Publishing, Switzerland, 2014. [2] X. Liu, H. J. Marquez, K. D. Kumar and Y. Lin, Sampled-data control of networked nonlinear systems with variable delays and drops, Internat. J. Robust Nonlinear Control, 25 (2015), 72-87.  doi: 10.1002/rnc.3074. [3] A. Seuret, A novel stability analysis of linear systems under asynchronous samplings, Automatica J. IFAC, 48 (2012), 177-182.  doi: 10.1016/j.automatica.2011.09.033. [4] T. Wang, H. Gao and J. Qiu, A combined fault-tolerant and predictive control for network-based industrial processes, IEEE Trans. Industrial Electronics, 63 (2016), 2529-2536.  doi: 10.1109/TIE.2016.2515073. [5] T. Wang, H. Gao and J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 416-425.  doi: 10.1109/TNNLS.2015.2411671. [6] T. Wang, J. Qiu and H. Gao, Adaptive neural control of stochastic nonlinear time-delay systems with multiple constraints, IEEE Trans. Syst. Man Cybernetics Syst., 47 (2017), 1875-1883.  doi: 10.1109/TSMC.2016.2562511. [7] Y. Wei, J. Qiu and S. Fu, Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay, Nonlinear Anal. Hybrid Syst., 16 (2015), 52-71.  doi: 10.1016/j.nahs.2014.11.003. [8] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.  doi: 10.1016/S0005-1098(03)00167-5. [9] S. I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269, Springer-Verlag, London, 2001. doi: 10.1007/1-84628-553-4. [10] V. B. Kolmanovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Stability of Functional-Differential Equations, Mathematics in Science and Engineering, 180, Academic Press, Inc., London, 1986. [11] J. K. Hale, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] S. Long, Y. Wu, S. Zhong and D. Zhang, Stability analysis for a class of neutral type singular systems with time-varying delay, Appl. Math. Comput., 339 (2018), 113-131.  doi: 10.1016/j.amc.2018.06.058. [13] H. B. Chen, New criteria on stability analysis for uncertain neutral system with mixed time-varying delays, Optimal Control Appl. Methods, 34 (2013), 487-501.  doi: 10.1002/oca.2036. [14] J. Sun, G. P. Liu, J. Chen and D. Rees, Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica J. IFAC, 46 (2010), 466-470.  doi: 10.1016/j.automatica.2009.11.002. [15] H.-B. Zeng, Y. He, M. Wu and J. She, New results on stability analysis for systems with discrete distributed delay, Automatica J. IFAC, 60 (2015), 189-192.  doi: 10.1016/j.automatica.2015.07.017. [16] M. Park, O. M. Kwon, J. H. Park, S. Lee and E. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica J. IFAC, 55 (2015), 204-208.  doi: 10.1016/j.automatica.2015.03.010. [17] L. Ding, Y. He, M. Wu and C. Ning, Improved mixed-delay-dependent asymptotic stability criteria for neutral systems, IET Control Theory Appl., 9 (2015), 2180-2187.  doi: 10.1049/iet-cta.2015.0022. [18] Y. G. Chen, W. Qian and S. M. Fei, Improved robust stability conditions for uncertain neutral systems with discrete and distributed delays, J. Franklin Inst., 352 (2015), 2634-2645.  doi: 10.1016/j.jfranklin.2015.03.040. [19] P. L. Liu, Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays, ISA Trans., 60 (2016), 53-66.  doi: 10.1016/j.isatra.2015.11.004. [20] L. L. Xiong, S. M. Zhong and D. Y. Li, Novel delay-dependent asymptotical stability of neutral systems with nonlinear perturbations, J. Comput. Appl. Math., 232 (2009), 505-513.  doi: 10.1016/j.cam.2009.06.026. [21] Y. J. Liu, W. B. Ma, M.S. Mahmoud and S. M. Lee, Improved delay-dependent exponential stability criteria for neutral-delay systems with nonlinear uncertainties, Appl. Math. Model., 39 (2015), 3164-3174.  doi: 10.1016/j.apm.2014.11.036. [22] T. Wang, T. Li, G. Zhang and S. Fei, Further triple integral approach to mixed-delay-dependent stability of time-delay neutral systems, ISA Trans., 70 (2017), 116-124.  doi: 10.1016/j.isatra.2017.05.010. [23] T. Wu, L. Xiong, J. Cao, Z. Liu and H. Zhang, New stability and stabilization conditions for stochastic neural networks of neutral type with Markovian jumping parameters, J. Franklin Inst., 355 (2018), 8462-8483.  doi: 10.1016/j.jfranklin.2018.09.032. [24] Z.-Y. Li, J. Lam and Y. Wang, Stability analysis of linear stochastic neutral-type time-delay systems with two delays, Automatica J. IFAC, 91 (2018), 179-189.  doi: 10.1016/j.automatica.2018.01.014. [25] N. Zhao, X. Zhang, Y. Xue and P. Shi, Necessary conditions for exponential stability of linear neutral type systems with multiple time delays, J. Franklin Inst., 355 (2018), 458-473.  doi: 10.1016/j.jfranklin.2017.11.016. [26] R. Mohajerpoor, L. Shanmugam, H. Abdi, R. Rakkiyappan, S. Nahavandi and J. H. Park, Improved delay-dependent stability criteria for neutral systems with mixed interval time-varying delays and nonlinear disturbances, J. Franklin Inst., 354 (2017), 1169-1194.  doi: 10.1016/j.jfranklin.2016.11.015. [27] Y. Wang, X. Zhang and X. Zhang, Neutral-delay-range-dependent absolute stability criteria for neutral-type Lur'e systems with time-varying delays, J. Franklin Inst., 353 (2016), 5025-5039.  doi: 10.1016/j.jfranklin.2016.09.014. [28] Y. Wang, Y. Xue and X. Zhang, Less conservative robust absolute stability criteria for uncertain neutral-type Lur'e systems with time-varying delays, J. Franklin Inst., 353 (2016), 816-833.  doi: 10.1016/j.jfranklin.2016.01.001. [29] W. Duan, B. Du, Z. Liu and Y. Zou, Improved stability criteria for uncertain neutral-type Lur'e systems with time-varying delays, J. Franklin Inst., 351 (2014), 4538-4554.  doi: 10.1016/j.jfranklin.2014.06.008. [30] S. S. Alaviani, A necessary and sufficient condition for delay-independent stability of linear time-varying neutral delay systems, J. Franklin Inst., 351 (2014), 2574-2581.  doi: 10.1016/j.jfranklin.2013.12.003. [31] Y. Liu, S. M. Lee, O. M. Kwon and J. H. Park, Delay-dependent exponential stability criteria for neutral systems with interval time-varying delays and nonlinear perturbations, J. Franklin Inst., 350 (2013), 3313-3327.  doi: 10.1016/j.jfranklin.2013.07.010. [32] F. Deng, W. Mao and A. Wan, A novel result on stability analysis for uncertain neutral stochastic time-varying delay systems, Appl. Math. Comput., 221 (2013), 132-143.  doi: 10.1016/j.amc.2013.05.071. [33] C. Gao, Z. Liu and R. Xu, On exponential stabilization for a class of neutral-type systems with parameter uncertainties: An integral sliding mode approach, Appl. Math. Comput., 219 (2013), 11044-11055.  doi: 10.1016/j.amc.2013.04.038. [34] H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 1996. [35] S. Arik, A modified Lyapunov functional with application to stability of neutral-type neural networks with time delays, J. Franklin Inst., 356 (2019), 276-291.  doi: 10.1016/j.jfranklin.2018.11.002. [36] Y. Sun, On simple stability criteria for nonlinear neutral systems with multiple time delays, Appl. Math. Lett., 25 (2012), 1911-1915.  doi: 10.1016/j.aml.2012.02.066.
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