# American Institute of Mathematical Sciences

doi: 10.3934/naco.2019050

## Stabilization on input time-varying delay for linear switched systems with truncated predictor control

 Department of Mathematics, Amrita School of Engineering, Coimbatore-641 112, Amrita Vishwa Vidyapeetham, India

* Corresponding author: A. Vinodkumar

Received  May 2019 Revised  July 2019 Published  September 2019

Fund Project: This work was supported by Science & Engineering Research Board (DST-SERB) project file number: ECR/ 2015/000301 in India

This study is concerned with the stabilization problem for input time-varying delay switched system under the truncated predictor control scheme. The delay in the prediction feedback, is subjected by predicting the future trajectory of the states by system equations and initial conditions, which is known as truncated prediction feedback (TPF). The TPF is used to construct the state feedback law for stabilizing the linear switched system. By constructing Lyapunov-Krasovskii functions and, the stability condition is derived to ensure the globally asymptotically stable of the state feedback stabilization at the origin. When switching system is unstable, truncated predictor control method and Hurwitz convex combination makes the system stable. Finally, a numerical example and their simulation results are given to show the effectiveness of the proposed approach.

Citation: K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2019050
##### References:
 [1] B. Aguirre-Hernández, F. R. Garcća-Sosa, C. A. Loredo-Villalobos, R. Villafuerte-Segura and E. Campos-Cantón, Open problems related to the hurwitz stability of polynomials segments, Mathematical Problems in Engineering, 9-10 (2018), 1-8.  doi: 10.1155/2018/2075903.  Google Scholar [2] Y. Ariba and F. Gouaisbaut, Construction of lyapunov-krasovskii functional for time-varying delay systems, in 2008 47th IEEE Conference on Decision and Control, IEEE, (2008), 3995–4000. Google Scholar [3] S. Białas, A sufficient condition for hurwitz stability of the convex combination of two matrices, Control and Cybernetics, 33 (2004), 109-112.   Google Scholar [4] J. D. Boskovic and R. K. Mehra, A multiple model-based reconfigurable flight control system design, in Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No. 98CH36171), vol. 4, IEEE, (1998), 4503–4508. Google Scholar [5] Z. Ding and Z. Lin, Truncated state prediction for control of lipschitz nonlinear systems with input delay, in 53rd IEEE Conference on Decision and Control, IEEE, (2014), 1966–1971. Google Scholar [6] A. Emadi, A. Khaligh, C. H. Rivetta and G. A. Williamson, Constant power loads and negative impedance instability in automotive systems: definition, modeling, stability, and control of power electronic converters and motor drives, IEEE Transactions on Vehicular Technology, 55 (2006), 1112-1125.   Google Scholar [7] K. Engelborghs, M. Dambrine and D. Roose, Limitations of a class of stabilization methods for delay systems, IEEE Transactions on Automatic Control, 46 (2001), 336-339.  doi: 10.1109/9.905705.  Google Scholar [8] T. Erneux, J. Javaloyes, M. Wolfrum and S. Yanchuk, Introduction to focus issue: Time-delay dynamics, Chao, 27 (2017), 114201. doi: 10.1063/1.5011354.  Google Scholar [9] R. Francisco, F. Mazenc and S. Mondié, Global asymptotic stabilization of a pvtol aircraft model with delay in the input, in Applications of Time Delay Systems, Springer, (2007), 343–356. doi: 10.1007/978-3-540-49556-7_21.  Google Scholar [10] Z. Gong, K. L. Teo, C. Liu and E. Feng, Horizontal well's path planning: An optimal switching control approach, Applied Mathematical Modelling, 39 (2015), 4022-4032.  doi: 10.1016/j.apm.2014.12.014.  Google Scholar [11] K. Gu, An integral inequality in the stability problem of time-delay systems, in Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No. 00CH37187), vol. 3, IEEE, (2000), 2805–2810. Google Scholar [12] K. Gu, J. Chen and V. L. Kharitonov, Stability of Time-Delay Systems, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4612-0039-0.  Google Scholar [13] K. Gu and S.-I. Niculescu, Survey on recent results in the stability and control of time-delay systems, Journal of Dynamic Systems, Measurement, and Control, 125 (2003), 158-165.   Google Scholar [14] M. H. H. Kani, M. J. Yazdanpanah and A. H. Markazi, Stability analysis of a class of uncertain switched time-delay systems with sliding modes, International Journal of Robust and Nonlinear Control, 29 (2019), 19-42.  doi: 10.1002/rnc.4369.  Google Scholar [15] Z. Lin and H. Fang, On asymptotic stabilizability of linear systems with delayed input, IEEE Transactions on Automatic Control, 52 (2007), 998-1013.  doi: 10.1109/TAC.2007.899007.  Google Scholar [16] L.-L. Liu, J.-G. Peng and B.-W. Wu, On parameterized lyapunov–krasovskii functional techniques for investigating singular time-delay systems, Applied Mathematics Letters, 24 (2011), 703-708.  doi: 10.1016/j.aml.2010.12.010.  Google Scholar [17] D. Ma, W.-H. Ki and C.-Y. 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George, Adaptive control using multiple models, switching and tuning, International Journal of Adaptive Control and Signal Processing, 17 (2003), 87-102.   Google Scholar [22] A. R. Oliveira, S. B. Gonçalves, M. de Carvalho and M. T. Silva, Development of a musculotendon model within the framework of multibody systems dynamics, in Multibody Dynamics, Springer, (2016), 213–237. Google Scholar [23] V. Pinon, F. Hasbani, A. Giry, D. Pache and C. Garnier, A single-chip wcdma envelope reconstruction ldmos pa with 130mhz switched-mode power supply, in 2008 IEEE International Solid-State Circuits Conference-Digest of Technical Papers, IEEE, (2008), 564–636. Google Scholar [24] L. Rodrćguez-Guerrero, A. Ramirez and C. Cuvas, Predictive control and truncated predictor: A comparative study on numerical benchmark problems, in 2014 11th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), IEEE, (2014), 1–6. Google Scholar [25] E. Savku and G.-W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, Journal of Optimization Theory and Applications, 179 (2018), 696-721. doi: 10.1007/s10957-017-1159-3.  Google Scholar [26] X.-M. Sun, W. Wang, G.-P. Liu and J. Zhao, Stability analysis for linear switched systems with time-varying delay, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 528-533.   Google Scholar [27] V. Van Assche, M. Dambrine, J.-F. Lafay and J.-P. Richard, Some problems arising in the implementation of distributed-delay control laws, in Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No. 99CH36304), vol. 5, IEEE, (1999), 4668–4672. Google Scholar [28] R. A. van Santen, Role of time delay in chemical reaction rates, The Journal of Chemical Physics, 57 (1972), 5418-5426.   Google Scholar [29] D. Wang, P. Shi, W. Wang and H. R. Karimi, Non-fragile $h_\infty$ control for switched stochastic delay systems with application to water quality process, International Journal of Robust and Nonlinear Control, 24 (2014), 1677-1693.  doi: 10.1002/rnc.2956.  Google Scholar [30] Y. Wei and Z. Lin, On the delay bounds of linear systems under delay independent truncated predictor feedback: the state feedback case, in 2015 54th IEEE Conference on Decision and Control (CDC), IEEE, (2015), 4642–4647. Google Scholar [31] C. Wu, K. L. Teo, R. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067.  doi: 10.1016/j.aml.2005.11.018.  Google Scholar [32] L. Wu, R. Yang, P. Shi and X. Su, Stability analysis and stabilization of 2-d switched systems under arbitrary and restricted switchings, Automatica, 59 (2015), 206-215.  doi: 10.1016/j.automatica.2015.06.008.  Google Scholar [33] H. Xu and K. L. Teo, Robust stabilization of uncertain impulsive switched systems with delayed control, Computers & Mathematics with Applications, 56 (2008), 63-70.  doi: 10.1016/j.camwa.2007.11.032.  Google Scholar [34] H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 1419-1422.   Google Scholar [35] S. Y. Yoon and Z. Lin, Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay, Systems & Control Letters, 62 (2013), 837-844.  doi: 10.1016/j.sysconle.2013.05.013.  Google Scholar [36] B. Zhou, Z. Lin and G.-R. Duan, Global and semi-global stabilization of linear systems with multiple delays and saturations in the input, SIAM Journal on Control and Optimization, 48 (2010), 5294-5332.  doi: 10.1137/090771673.  Google Scholar [37] B. Zhou, Z. Lin and G.-R. Duan, Truncated predictor feedback for linear systems with long time-varying input delays, Automatica, 48 (2012), 2387-2399.  doi: 10.1016/j.automatica.2012.06.032.  Google Scholar [38] B. Zhou, Z. Lin and G. Duan, Stabilization of linear systems with input delay and saturation parametric lyapunov equation approach, International Journal of Robust and Nonlinear Control, 20 (2010), 1502-1519.  doi: 10.1002/rnc.1525.  Google Scholar [39] Z. Zuo, Z. Lin and Z. Ding, Truncated predictor control of lipschitz nonlinear systems with time-varying input delay, IEEE Transactions on Automatic Control, 62 (2017), 5324-5330.  doi: 10.1109/TAC.2016.2635021.  Google Scholar

show all references

##### References:
 [1] B. Aguirre-Hernández, F. R. Garcća-Sosa, C. A. Loredo-Villalobos, R. Villafuerte-Segura and E. Campos-Cantón, Open problems related to the hurwitz stability of polynomials segments, Mathematical Problems in Engineering, 9-10 (2018), 1-8.  doi: 10.1155/2018/2075903.  Google Scholar [2] Y. Ariba and F. Gouaisbaut, Construction of lyapunov-krasovskii functional for time-varying delay systems, in 2008 47th IEEE Conference on Decision and Control, IEEE, (2008), 3995–4000. Google Scholar [3] S. Białas, A sufficient condition for hurwitz stability of the convex combination of two matrices, Control and Cybernetics, 33 (2004), 109-112.   Google Scholar [4] J. D. Boskovic and R. K. Mehra, A multiple model-based reconfigurable flight control system design, in Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No. 98CH36171), vol. 4, IEEE, (1998), 4503–4508. Google Scholar [5] Z. Ding and Z. Lin, Truncated state prediction for control of lipschitz nonlinear systems with input delay, in 53rd IEEE Conference on Decision and Control, IEEE, (2014), 1966–1971. Google Scholar [6] A. Emadi, A. Khaligh, C. H. Rivetta and G. A. Williamson, Constant power loads and negative impedance instability in automotive systems: definition, modeling, stability, and control of power electronic converters and motor drives, IEEE Transactions on Vehicular Technology, 55 (2006), 1112-1125.   Google Scholar [7] K. Engelborghs, M. Dambrine and D. Roose, Limitations of a class of stabilization methods for delay systems, IEEE Transactions on Automatic Control, 46 (2001), 336-339.  doi: 10.1109/9.905705.  Google Scholar [8] T. Erneux, J. Javaloyes, M. Wolfrum and S. Yanchuk, Introduction to focus issue: Time-delay dynamics, Chao, 27 (2017), 114201. doi: 10.1063/1.5011354.  Google Scholar [9] R. Francisco, F. Mazenc and S. Mondié, Global asymptotic stabilization of a pvtol aircraft model with delay in the input, in Applications of Time Delay Systems, Springer, (2007), 343–356. doi: 10.1007/978-3-540-49556-7_21.  Google Scholar [10] Z. Gong, K. L. Teo, C. Liu and E. Feng, Horizontal well's path planning: An optimal switching control approach, Applied Mathematical Modelling, 39 (2015), 4022-4032.  doi: 10.1016/j.apm.2014.12.014.  Google Scholar [11] K. Gu, An integral inequality in the stability problem of time-delay systems, in Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No. 00CH37187), vol. 3, IEEE, (2000), 2805–2810. Google Scholar [12] K. Gu, J. Chen and V. L. Kharitonov, Stability of Time-Delay Systems, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4612-0039-0.  Google Scholar [13] K. Gu and S.-I. Niculescu, Survey on recent results in the stability and control of time-delay systems, Journal of Dynamic Systems, Measurement, and Control, 125 (2003), 158-165.   Google Scholar [14] M. H. H. Kani, M. J. Yazdanpanah and A. H. Markazi, Stability analysis of a class of uncertain switched time-delay systems with sliding modes, International Journal of Robust and Nonlinear Control, 29 (2019), 19-42.  doi: 10.1002/rnc.4369.  Google Scholar [15] Z. Lin and H. Fang, On asymptotic stabilizability of linear systems with delayed input, IEEE Transactions on Automatic Control, 52 (2007), 998-1013.  doi: 10.1109/TAC.2007.899007.  Google Scholar [16] L.-L. Liu, J.-G. Peng and B.-W. Wu, On parameterized lyapunov–krasovskii functional techniques for investigating singular time-delay systems, Applied Mathematics Letters, 24 (2011), 703-708.  doi: 10.1016/j.aml.2010.12.010.  Google Scholar [17] D. Ma, W.-H. Ki and C.-Y. Tsui, A pseudo-ccm/dcm simo switching converter with freewheel switching, IEEE Journal of Solid-State Circuits, 38 (2003), 1007-1014.   Google Scholar [18] A. Manitius and A. Olbrot, Finite spectrum assignment problem for systems with delays, IEEE Transactions on Automatic Control, 24 (1979), 541-552.  doi: 10.1109/TAC.1979.1102124.  Google Scholar [19] F. Mazenc and M. Malisoff, Extensions of razumikhin theorem and lyapunov–krasovskii functional constructions for time-varying systems with delay, Automatica, 78 (2017), 1-13.  doi: 10.1016/j.automatica.2016.12.005.  Google Scholar [20] F. Mazenc, S.-I. Niculescu and M. Krstic, Lyapunov–krasovskii functionals and application to input delay compensation for linear time-invariant systems, Automatica, 48 (2012), 1317-1323.  doi: 10.1016/j.automatica.2012.04.002.  Google Scholar [21] K. S. Narendra, O. A. Driollet, M. Feiler and K. George, Adaptive control using multiple models, switching and tuning, International Journal of Adaptive Control and Signal Processing, 17 (2003), 87-102.   Google Scholar [22] A. R. Oliveira, S. B. Gonçalves, M. de Carvalho and M. T. Silva, Development of a musculotendon model within the framework of multibody systems dynamics, in Multibody Dynamics, Springer, (2016), 213–237. Google Scholar [23] V. Pinon, F. Hasbani, A. Giry, D. Pache and C. Garnier, A single-chip wcdma envelope reconstruction ldmos pa with 130mhz switched-mode power supply, in 2008 IEEE International Solid-State Circuits Conference-Digest of Technical Papers, IEEE, (2008), 564–636. Google Scholar [24] L. Rodrćguez-Guerrero, A. Ramirez and C. Cuvas, Predictive control and truncated predictor: A comparative study on numerical benchmark problems, in 2014 11th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), IEEE, (2014), 1–6. Google Scholar [25] E. Savku and G.-W. Weber, A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance, Journal of Optimization Theory and Applications, 179 (2018), 696-721. doi: 10.1007/s10957-017-1159-3.  Google Scholar [26] X.-M. Sun, W. Wang, G.-P. Liu and J. Zhao, Stability analysis for linear switched systems with time-varying delay, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 528-533.   Google Scholar [27] V. Van Assche, M. Dambrine, J.-F. Lafay and J.-P. Richard, Some problems arising in the implementation of distributed-delay control laws, in Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No. 99CH36304), vol. 5, IEEE, (1999), 4668–4672. Google Scholar [28] R. A. van Santen, Role of time delay in chemical reaction rates, The Journal of Chemical Physics, 57 (1972), 5418-5426.   Google Scholar [29] D. Wang, P. Shi, W. Wang and H. R. Karimi, Non-fragile $h_\infty$ control for switched stochastic delay systems with application to water quality process, International Journal of Robust and Nonlinear Control, 24 (2014), 1677-1693.  doi: 10.1002/rnc.2956.  Google Scholar [30] Y. Wei and Z. Lin, On the delay bounds of linear systems under delay independent truncated predictor feedback: the state feedback case, in 2015 54th IEEE Conference on Decision and Control (CDC), IEEE, (2015), 4642–4647. Google Scholar [31] C. Wu, K. L. Teo, R. Li and Y. Zhao, Optimal control of switched systems with time delay, Applied Mathematics Letters, 19 (2006), 1062-1067.  doi: 10.1016/j.aml.2005.11.018.  Google Scholar [32] L. Wu, R. Yang, P. Shi and X. Su, Stability analysis and stabilization of 2-d switched systems under arbitrary and restricted switchings, Automatica, 59 (2015), 206-215.  doi: 10.1016/j.automatica.2015.06.008.  Google Scholar [33] H. Xu and K. L. Teo, Robust stabilization of uncertain impulsive switched systems with delayed control, Computers & Mathematics with Applications, 56 (2008), 63-70.  doi: 10.1016/j.camwa.2007.11.032.  Google Scholar [34] H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 1419-1422.   Google Scholar [35] S. Y. Yoon and Z. Lin, Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay, Systems & Control Letters, 62 (2013), 837-844.  doi: 10.1016/j.sysconle.2013.05.013.  Google Scholar [36] B. Zhou, Z. Lin and G.-R. Duan, Global and semi-global stabilization of linear systems with multiple delays and saturations in the input, SIAM Journal on Control and Optimization, 48 (2010), 5294-5332.  doi: 10.1137/090771673.  Google Scholar [37] B. Zhou, Z. Lin and G.-R. Duan, Truncated predictor feedback for linear systems with long time-varying input delays, Automatica, 48 (2012), 2387-2399.  doi: 10.1016/j.automatica.2012.06.032.  Google Scholar [38] B. Zhou, Z. Lin and G. Duan, Stabilization of linear systems with input delay and saturation parametric lyapunov equation approach, International Journal of Robust and Nonlinear Control, 20 (2010), 1502-1519.  doi: 10.1002/rnc.1525.  Google Scholar [39] Z. Zuo, Z. Lin and Z. Ding, Truncated predictor control of lipschitz nonlinear systems with time-varying input delay, IEEE Transactions on Automatic Control, 62 (2017), 5324-5330.  doi: 10.1109/TAC.2016.2635021.  Google Scholar
Unstable behavior mode:1 of the linear system $(16)$
Unstable behavior mode:2 of the linear system $(16)$
Phase plot mode:1 of the systems $(16)$ with initial condition $x(t) = [0.3, 0.1]^{T}$
Phase plot mode:2 of the systems $(16)$ with initial condition $x(t) = [0.3, 0.1]^{T}$
Stability for the mode:1 of system $(16)$ with initial condition $x(t) = [0.3, 0.1]^{T}$
Stability for the mode:2 of system $(16)$ with initial condition $x(t) = [0.3, 0.1]^{T}$
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