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April  2018, 14(2): 583-596. doi: 10.3934/jimo.2017061

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

Nguyen H. Sau 1, and  Vu N. Phat 2,

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  April 2018 Early access  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.

Keywords: Stabilization, state feedback controller, singular systems, time-delay, linear programming.
Mathematics Subject Classification: Primary: 93D20, 34D20; Secondary: 37C75.
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, 2018, 14 (2) : 583-596. 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.   Google Scholar

[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.  Google Scholar

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R. Bru and S. Romero-Vivo, Positive Systems, Lecture Notes in Control and Information Sciences, vol. 389, Berlin: Springer, 2009. doi: 10.1007/978-3-642-02894-6.  Google Scholar

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S. L. V. Campbell, Singular Systems of Differential Equations, Boston, Mass. -London, 1980.  Google Scholar

[5]

L. Dai, Singular Control Systems, Berlin: Springer, 1989. doi: 10.1007/BFb0002475.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, New York: Wiley-Interscience, 2000. doi: 10.1002/9781118033029.  Google Scholar

[10]

A. Ilchmann and P. H. A. Ngoc, On positivity and stability of linear time-varying Volterra equations, Positivity, 13 (2009), 671-681.   Google Scholar

[11]

T. Kaczorek, Positive 1-D and 2-D Systems, Berlin: Springer, 2002. Google Scholar

[12]

J. Lam and S. Xu, Robust Control and Filtering of Singular Systems, Berlin: Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

L. F. Shampine and P. Gahinet, Delay differential-algebraic equations in control theory, Applied Numerical Mathematics, 56 (2006), 574-588.   Google Scholar

[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.  Google Scholar

[21]

R. J. Vanderbei, Linear Programming: Foundations and Extensions, International Series in Operations Research & Management Science, vol. 37,2001. doi: 10.1007/978-1-4757-5662-3.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

S. Zhu, Z. Li and C. Zhang, Exponential stability analysis for positive systems with delays, IET Control Theory & Applications, 6 (2012), 761-767.   Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[3]

R. Bru and S. Romero-Vivo, Positive Systems, Lecture Notes in Control and Information Sciences, vol. 389, Berlin: Springer, 2009. doi: 10.1007/978-3-642-02894-6.  Google Scholar

[4]

S. L. V. Campbell, Singular Systems of Differential Equations, Boston, Mass. -London, 1980.  Google Scholar

[5]

L. Dai, Singular Control Systems, Berlin: Springer, 1989. doi: 10.1007/BFb0002475.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, New York: Wiley-Interscience, 2000. doi: 10.1002/9781118033029.  Google Scholar

[10]

A. Ilchmann and P. H. A. Ngoc, On positivity and stability of linear time-varying Volterra equations, Positivity, 13 (2009), 671-681.   Google Scholar

[11]

T. Kaczorek, Positive 1-D and 2-D Systems, Berlin: Springer, 2002. Google Scholar

[12]

J. Lam and S. Xu, Robust Control and Filtering of Singular Systems, Berlin: Springer, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

L. F. Shampine and P. Gahinet, Delay differential-algebraic equations in control theory, Applied Numerical Mathematics, 56 (2006), 574-588.   Google Scholar

[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.  Google Scholar

[21]

R. J. Vanderbei, Linear Programming: Foundations and Extensions, International Series in Operations Research & Management Science, vol. 37,2001. doi: 10.1007/978-1-4757-5662-3.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

S. Zhu, Z. Li and C. Zhang, Exponential stability analysis for positive systems with delays, IET Control Theory & Applications, 6 (2012), 761-767.   Google Scholar

Figure 1.  State response of the closed-loop system
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