Abstract
This paper investigates the ℓ∞-induced output-feedback controller synthesis problem for discrete-time positive systems. For positive linear systems, a necessary and sufficient condition for stability and ℓ∞-induced performance is first put forward using a group of linear inequalities. Then, on the basis of these inequalities, the conditions on which static output-feedback controllers exist are set up, and to solve these conditions, a method of iterative convex optimization is provided. Moreover, a complete solution to the synthesis problem of the controller is proposed based on linear programming for positive systems with single output or input. Finally, two examples are presented to illustrate and verify the effective and applicable methods derived in this paper.
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L. Caccetta, L. R. Foulds, and V. G. Rumchev, “A positive linear discrete-time model of capacity planning and its controllability properties,” Math. Comput. Model., vol. 40, pp. 217–226, 2004.
H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation, Sunderland, MA, Sinauer Assoc., 2001.
M. Ait Rami, B. Boulkroune, A. Hajjaji, and O. Pages, “Stabilization of LPV positive systems,” in 2014 IEEE 53rd Annual Conference on Decision and Control, pp. 4772–4776, 2014.
J. Wang, W. Qi, X. Gao, and Y. Kao, “Linear controllable systems,” International Journal of Control Automation & Systems,vol. 15, no. 2, pp. 640–646, 2017.
Y. Ebihara, D. Peaucelle, and D. Arzelier, “LMI approach to linear positive system analysis and synthesis,” Systems & Control Letters, vol. 63, no. 1, pp. 50–56, 2014.
X. Chen, M. Chen, and J. Shen, “A novel approach to L1-induced controller synthesis for positive systems with interval uncertainties,” Journal of the Franklin Institute, vol. 354, no. 8, pp. 3364–3377, 2017.
L. Liu, X. Cao, Z. Fu, S. Song, and H. Xing, “Input-output finite-time control of uncertain positive impulsive switched systems with time-varying and distributed delays,” International Journal of Control Automation & Systems, vol. 16, pp. 670–681, 2018.
J. Shen and J. Lam, “Input-output gain analysis for linear systems on cones,” Automatica, vol. 77, pp. 44–50, 2017.
X. Zhao, T. Wu, X. Zheng, and R. Li, “Discussions on observer design of nonlinear positive systems via T-S fuzzy modeling,” Neurocomputing, vol. 157, pp. 70–75, 2015.
X. Chen, M. Chen, J. Shen, and S. Shao, “ℓ 1-induced state-bounding observer design for positive Takagi-Sugeno fuzzy systems,” Neurocomputing, vol. 269, pp. 490–496, 2017.
L. Benvenuti and L. Farina, “A tutorial on the positive realization problem,” IEEE Trans. Automat. Control, vol. 49, no. 5, pp. 651–664, 2004.
Y. Ebihara, D. Peaucelle, D. Arzelier, and F. Gouaisbaut, “Dominant pole of positive systems with time-delays,” Proc. of European Control Conference, pp. 79–84, 2014.
M. Ait Rami and F. Tadeo, “Controller synthesis for positive linear systems with bounded controls,” IEEE Trans. Circuits and Systems (II), vol. 54, no. 2, pp. 151–155, 2007.
H. Gao, J. Lam, C. Wang, and S. Xu, “Control for stability and positivity: equivalent conditions and computation,” IEEE Trans. Circuits and Systems (II), vol. 52, pp. 540–544, 2005.
W. Haddad and V. Chellaboina, “Stability theory for nonnegative and compartmental dynamical systems with time delay,” Systems & Control Letters, vol. 51, no.5, pp. 355–361, 2004.
X. Zhao, X. Liu, S. Yin, and H. Li, “Improved results on stability of continuous-time switched positive linear systems,” Automatica, vol. 50, no. 2, pp. 614–621, 2014.
J. Feng, J. Lam, Z. Shu, and Q. Wang, “Internal positivity preserved model reduction,” Int. J. Control, vol. 83, no. 3, pp. 575–584, 2010.
G. Wang, B. Li, Q. Zhang, and C. Yang, “Positive observer design for discrete-time positive system with missing data in output,” Neurocomputing, vol. 168, pp. 427–434, 2015.
X. Liu, “Stability criterion of 2-D positive systems with unbounded delays described by Roesser model,” Asian Journal of Control, vol. 17, no. 2, pp. 544–553, 2015.
J. E. Kurek, “Stability of positive 2-D system described by the Roesser model,” IEEE Trans. Circuits and Systems (I), vol. 49, pp. 531–533, 2002.
Z. Duan, Z. Xiang, and H. R. Karimi, “Stability and ℓ 1-gain analysis for positive 2D T-S fuzzy state-delayed systems in the second FM model,” Neurocomputing, vol. 142, pp. 209–215, 2014.
M. E. Valcher, “Controllability and reachability criteria for discrete time positive systems,” Int. J. Control, vol. 65, no. 3, pp. 511–536, 1996.
M. P. Fanti, B. Maione, and B. Turchiano, “Controllability of multi-input positive discrete-time systems,” Int. J. Control, vol. 56, no. 6, pp. 1295–1308, 1990.
S. Sadraddini and C. Belta, “Formal synthesis of control strategies for positive monotone systems,” IEEE Transactions on Automatic Control, vol. 64, no. 2, pp. 480–495, Feb. 2019.
A. Muralidharan, R. Pedarsani, and P. Varaiya, “Analysis of fixed-time control,” Transportation Research Part B: Methodological, vol. 73, pp. 81–90, 2015.
E. S. Kim, M. Arcak, and S. A. Seshia, “Symbolic control design for monotone systems with directed specifications,” Automatica, vol. 83, pp. 10–19, 2017.
D. Wang, G. Li, and W. Wang, “Fault detection of switched positive systems based on lossy networks,” Proc. of the 33rd Chinese Control Conference, pp. 5644–5648, 2014.
S. Zhu, M. Meng, and C. Zhang, “Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization,” Journal of the Franklin Institute, vol. 350, no. 3, pp. 617–636, 2013.
X. Chen, J. Lam, and P. Li, “Positive filtering for continuous-time positive systems under L 1 performance,” Int. J. Control, vol. 87, no. 9, pp. 1–8, 2014.
C. Briat, “Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L 1-gain and L ∞ characterization,” Nature, vol. 135, no. 5, pp. 18–27, July 1990.
W. Qi and X. Gao, “Positive L1-gain filter design for positive continuous-time Markovian jump systems with partly known transition rates,” International Journal of Control Automation & Systems, vol. 14, no. 6, pp. 1413–1420, 2016.
L. Zhao, W. Qi, L. Zhang, Y. Kao, and X. Gao, “Stochastic stability, L 1-gain and control synthesis for positive semi-Markov jump systems,” International Journal of Control Automation & Systems, vol. 16, no. 5, pp. 2055–2062, 2018.
X. Chen, J. Lam, P. Li, and Z. Shu, “ℓ 1-induced norm and controller synthesis of positive systems,” Automatica, vol. 49, pp. 1377–1385, 2013.
X. Chen, J. Lam, and M. Meng, “Controller synthesis for positive Takagi-Sugeno fuzzy systems under ℓ 1 performance,” International Journal of Systems Science, vol. 48, pp. 515–524, 2017.
L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley-Interscience, 2000.
R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, 1997.
Y. Ebihara, D. Peaucelle, and D. Arzelier, “Optimal L 1-controller synthesis for positive systems and its robustness properties,” Proceedings of the American Control Conference, 2012.
W. Haddad, V. Chellaboina, and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, Princeton, New Jersey, 2010.
P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, LMI Control Toolbox User’s Guide, The MathWorks Inc., Natick, MA, 1995.
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Recommended by Associate Editor Young Ik Son under the direction of Editor Jessie (Ju H.) Park. This work is partially supported by NSFC 61503184, NSFC 61573184, NSFC 61503037, NSFC 61603180 and the Natural Science Foundation of Jiangsu Province under Grant BK20160810.
Xiaoming Chen received her B.S. degree in Automation from Qufu Normal University in 2008. She obtained an M.S. degree in Control Science and Engineering from Harbin Institute of Technology in 2010, and a Ph.D. degree in Control Engineering from the University of Hong Kong in 2014. She joined Nanjing University of Aeronautics and Astronautics, in 2014, taking a lectureship in College of Automation Engineering. Her research interests include positive systems, fuzzy systems, delay systems, 2-D systems, estimation and filtering, stochastic control and robust control.
Mou Chen received his B.Sc. degree in material science and engineering at Nanjing University of Aeronautics & Astronautics, Nanjing, China, in 1998, and his M.Sc. and Ph.D. degrees in automatic control engineering at Nanjing University of Aeronautics & Astronautics, Nanjing, China, in 2004. From June 2008 to September 2009, he was a research fellow in the Department of Electrical and Computer Engineering, the National University of Singapore. He is currently a professor in the College of Automation Engineering at Nanjing University of Aeronautics & Astronautics, China. His research interests include nonlinear control, artificial intelligence, imagine processing and pattern recognition, and flight control.
Liqun Wang was born in Hunan, China, in 1995. He received his B.S. degree in measurement and control technology and instrument from the Harbin Engineering University, Harbin, China, in 2017. He is currently pursuing an M.S. degree with the Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include filter design, 2-D systems, positive systems and fuzzy systems.
Jun Shen received his B.Sc. and M.Sc. degrees from Southeast University, Nanjing, China, in 2008 and 2011, respectively, and his Ph.D. degree from the Department of Mechanical Engineering, the University of Hong Kong, Hong Kong, in 2015. Since 2016 he is an Associate Professor in the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His current research interests include positive systems, monotone systems, fractional order systems, model reduction, and robust control and filtering.
Jiapan Hu was born in Zhejiang, China, in 1996. He received his B.S. degree in Electrical Engineering and Automation from the Ningbo University of Technology, Ningbo, China, in 2018. He is currently pursuing an M.S. degree with the Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include filter design, fault detection, positive systems and switched systems.
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Chen, X., Chen, M., Wang, L. et al. Static Output-feedback Controller Synthesis for Positive Systems under ℓ∞ Performance. Int. J. Control Autom. Syst. 17, 2871–2880 (2019). https://doi.org/10.1007/s12555-018-0581-0
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DOI: https://doi.org/10.1007/s12555-018-0581-0