# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2017061

## LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback

 1 Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet Road, Hanoi, Vietnam 2 Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi, Vietnam

Received  October 2016 Revised  January 2017 Published  June 2017

This paper deals with the exponential stabilization problem by means of memory state feedback controller for linear singular positive systems with delay. By using system decomposition approach, singular systems theory and Lyapunov function method, we obtain new delay-dependent sufficient conditions for designing such controllers. The conditions are given in terms of standard linear programming (LP) problems, which can be solved by LP optimal toolbox. A numerical example is given to illustrate the effectiveness of the proposed method.

Citation: Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2017061
##### References:
 [1] H. Arneson and C. Langbort, A linear programming approach to routing control in networks of constrained linear positive systems, Automatica, 48 (2012), 800-807. [2] E. K. Boukas and Y. Xia, Descriptor discrete-time systems with random abrupt changes: Stability and stabilisation, International Journal of Control, 81 (2008), 1311-1318. doi: 10.1080/00207170701769822. [3] R. Bru and S. Romero-Vivo, Positive Systems, Lecture Notes in Control and Information Sciences, vol. 389, Berlin: Springer, 2009. [4] S. L. V. Campbell, Singular Systems of Differential Equations, Boston, Mass. -London, 1980. [5] L. Dai, Singular Control Systems, Berlin: Springer, 1989. [6] Y. Ebihara, D. Peaucelle and D. Arzelier, LMI approach to linear positive system analysis and synthesis, Systems & Control Letters, 63 (2014), 50-56. doi: 10.1016/j.sysconle.2013.11.001. [7] D. Efimov, A. Polyakov and J. P. Richard, Interval observer design for estimation and control of time-delay descriptor systems, European Journal of Control, 23 (2015), 26-35. doi: 10.1016/j.ejcon.2015.01.004. [8] H. Fan, J.-E. Feng and M. Meng, Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays, Journal of Industrial and Management Optimization, 12 (2016), 1535-1556. doi: 10.3934/jimo.2016.12.1535. [9] L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, New York: Wiley-Interscience, 2000. [10] A. Ilchmann and P. H. A. Ngoc, On positivity and stability of linear time-varying Volterra equations, Positivity, 13 (2009), 671-681. [11] T. Kaczorek, Positive 1-D and 2-D Systems, Berlin: Springer, 2002. [12] J. Lam and S. Xu, Robust Control and Filtering of Singular Systems, Berlin: Springer, 2006. [13] X. Liu, Constrained control of positive systems with delays, IEEE Transactions on Automatic Control, 54 (2009), 1596-1600. doi: 10.1109/TAC.2009.2017961. [14] I. Malloci and J. Daafouz, Stabilisation of polytopic singularly perturbed linear systems, International Journal of Control, 85 (2012), 135-142. doi: 10.1080/00207179.2011.641128. [15] Y. S. Moon, P. Park and W. H. Kwon, Robust stabilization of uncertain input-delayed systems using reduction method, Automatica, 37 (2001), 307-312. doi: 10.1016/S0005-1098(00)00145-X. [16] V. N. Phat and N. H. Sau, On exponential stability of linear singular positive delayed systems, Applied Mathematics Letters, 38 (2014), 67-72. doi: 10.1016/j.aml.2014.07.003. [17] M. A. Rami, Solvability of static output-feedback stabilization for LTI positive systems, Systms & Control Letters, 60 (2011), 704-708. doi: 10.1016/j.sysconle.2011.05.007. [18] M. A. Rami, F. Tadeo and U. Helmke, Positive observers for linear positive systems, and their implications, International Journal of Control, 84 (2011), 716-725. [19] L. F. Shampine and P. Gahinet, Delay differential-algebraic equations in control theory, Applied Numerical Mathematics, 56 (2006), 574-588. [20] Z. Shu and J. Lam, Exponential estimates and stabilization of uncertain singular systems with discrete and distributed delays, International Journal of Control,, 81 (2008), 865-882. doi: 10.1080/00207170701261986. [21] R. J. Vanderbei, Linear Programming: Foundations and Extensions, International Series in Operations Research & Management Science, vol. 37,2001. [22] S. Xu, P. Dooren, R. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control, 47 (2002), 1122-1128. doi: 10.1109/TAC.2002.800651. [23] L. Zhang, J. Lam and S. Xu, On positive realness of descriptor systems, IEEE Transactions on Circuits Systems I, Fundamental Theory & Applications, 49 (2002), 401-407. [24] Y. Zhang, Q. Zhang, T. Tanaka and X. G. Yan, Positivity of continuous-time descriptor systems with time delays, IEEE Trans Auto. Contr., 59 (2014), 3093-3097. [25] Y. Zhao, R. Wang and C. Yin, Optimal dividends and capital injections for a spectrally positive Lévy process, Journal of Industrial and Management Optimization, 13 (2017), 1-21. doi: 10.3934/jimo.2016001. [26] B. Zhou, J. Hu and G. Duan, Strict linear matrix inequality characterization of positive realness for linear discrete-time descriptor systems, IET Control Theory & Applications, 7 (2010), 1277-1281. [27] S. Zhu, Z. Li and C. Zhang, Exponential stability analysis for positive systems with delays, IET Control Theory & Applications, 6 (2012), 761-767.

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##### References:
 [1] H. Arneson and C. Langbort, A linear programming approach to routing control in networks of constrained linear positive systems, Automatica, 48 (2012), 800-807. [2] E. K. Boukas and Y. Xia, Descriptor discrete-time systems with random abrupt changes: Stability and stabilisation, International Journal of Control, 81 (2008), 1311-1318. doi: 10.1080/00207170701769822. [3] R. Bru and S. Romero-Vivo, Positive Systems, Lecture Notes in Control and Information Sciences, vol. 389, Berlin: Springer, 2009. [4] S. L. V. Campbell, Singular Systems of Differential Equations, Boston, Mass. -London, 1980. [5] L. Dai, Singular Control Systems, Berlin: Springer, 1989. [6] Y. Ebihara, D. Peaucelle and D. Arzelier, LMI approach to linear positive system analysis and synthesis, Systems & Control Letters, 63 (2014), 50-56. doi: 10.1016/j.sysconle.2013.11.001. [7] D. Efimov, A. Polyakov and J. P. Richard, Interval observer design for estimation and control of time-delay descriptor systems, European Journal of Control, 23 (2015), 26-35. doi: 10.1016/j.ejcon.2015.01.004. [8] H. Fan, J.-E. Feng and M. Meng, Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays, Journal of Industrial and Management Optimization, 12 (2016), 1535-1556. doi: 10.3934/jimo.2016.12.1535. [9] L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, New York: Wiley-Interscience, 2000. [10] A. Ilchmann and P. H. A. Ngoc, On positivity and stability of linear time-varying Volterra equations, Positivity, 13 (2009), 671-681. [11] T. Kaczorek, Positive 1-D and 2-D Systems, Berlin: Springer, 2002. [12] J. Lam and S. Xu, Robust Control and Filtering of Singular Systems, Berlin: Springer, 2006. [13] X. Liu, Constrained control of positive systems with delays, IEEE Transactions on Automatic Control, 54 (2009), 1596-1600. doi: 10.1109/TAC.2009.2017961. [14] I. Malloci and J. Daafouz, Stabilisation of polytopic singularly perturbed linear systems, International Journal of Control, 85 (2012), 135-142. doi: 10.1080/00207179.2011.641128. [15] Y. S. Moon, P. Park and W. H. Kwon, Robust stabilization of uncertain input-delayed systems using reduction method, Automatica, 37 (2001), 307-312. doi: 10.1016/S0005-1098(00)00145-X. [16] V. N. Phat and N. H. Sau, On exponential stability of linear singular positive delayed systems, Applied Mathematics Letters, 38 (2014), 67-72. doi: 10.1016/j.aml.2014.07.003. [17] M. A. Rami, Solvability of static output-feedback stabilization for LTI positive systems, Systms & Control Letters, 60 (2011), 704-708. doi: 10.1016/j.sysconle.2011.05.007. [18] M. A. Rami, F. Tadeo and U. Helmke, Positive observers for linear positive systems, and their implications, International Journal of Control, 84 (2011), 716-725. [19] L. F. Shampine and P. Gahinet, Delay differential-algebraic equations in control theory, Applied Numerical Mathematics, 56 (2006), 574-588. [20] Z. Shu and J. Lam, Exponential estimates and stabilization of uncertain singular systems with discrete and distributed delays, International Journal of Control,, 81 (2008), 865-882. doi: 10.1080/00207170701261986. [21] R. J. Vanderbei, Linear Programming: Foundations and Extensions, International Series in Operations Research & Management Science, vol. 37,2001. [22] S. Xu, P. Dooren, R. Stefan and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control, 47 (2002), 1122-1128. doi: 10.1109/TAC.2002.800651. [23] L. Zhang, J. Lam and S. Xu, On positive realness of descriptor systems, IEEE Transactions on Circuits Systems I, Fundamental Theory & Applications, 49 (2002), 401-407. [24] Y. Zhang, Q. Zhang, T. Tanaka and X. G. Yan, Positivity of continuous-time descriptor systems with time delays, IEEE Trans Auto. Contr., 59 (2014), 3093-3097. [25] Y. Zhao, R. Wang and C. Yin, Optimal dividends and capital injections for a spectrally positive Lévy process, Journal of Industrial and Management Optimization, 13 (2017), 1-21. doi: 10.3934/jimo.2016001. [26] B. Zhou, J. Hu and G. Duan, Strict linear matrix inequality characterization of positive realness for linear discrete-time descriptor systems, IET Control Theory & Applications, 7 (2010), 1277-1281. [27] S. Zhu, Z. Li and C. Zhang, Exponential stability analysis for positive systems with delays, IET Control Theory & Applications, 6 (2012), 761-767.
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