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Design of a single window system for e-government services: the chilean case
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 |
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.
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. |
[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. |
[4] |
S. L. V. Campbell,
Singular Systems of Differential Equations, Boston, Mass. -London, 1980. |
[5] |
L. Dai,
Singular Control Systems, Berlin: Springer, 1989.
doi: 10.1007/BFb0002475. |
[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.
doi: 10.1002/9781118033029. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[4] |
S. L. V. Campbell,
Singular Systems of Differential Equations, Boston, Mass. -London, 1980. |
[5] |
L. Dai,
Singular Control Systems, Berlin: Springer, 1989.
doi: 10.1007/BFb0002475. |
[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.
doi: 10.1002/9781118033029. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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 |

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