Document Type : Original Research Paper


Department of Electrical and Electronic Engineering, Shiraz University of Technology, Shiraz, Iran


This paper considers solving optimization problem for linear discrete time systems such that closed-loop discrete-time system is positive (i.e., all of its state variables have non-negative values) and also finite-time stable. For this purpose, by considering a quadratic cost function, an optimal controller is designed such that in addition to minimizing the cost function, the positivity property of the optimal state trajectory of the closed-loop system is also guaranteed. Furthermore, state variables of the closed-loop system converge to the origin in finite steps (finite-time stability). In this regard, the LQR+(positive LQR) problem for the linear discrete time systems is stated. Once, the cost function with finite-time horizon is considered and another time the cost function with infinite-time horizon is assumed. In this regard, two theorems are given and proved which consider the problem of building positive and also optimize of the linear time-varying discrete time systems. Results can also be applied to linear time-invariant discrete time systems. Finally, computer simulations are given to illustrate effective performance of the designed controller and also verify the theoretical results.


[1] V. S. Bokharaie, “Stability analysis of positive systems with applications to epidemiology," Doctoral dissertation, National University of Ireland Maynooth Institute (NUIMI), Hamilton, 2012.
[2] A. Rantzer, “Scalable control of positive systems,” European Journal of Control, vol. 24, pp. 72-80, 2015.
[3] Y. Zheng and G. Feng, “Stabilisation of second-order LTI switched positive systems,” International Journal of Control, vol. 84, pp. 1387-1397, 2011.
[4] K. Loparo, J. Aslanis, and O. Hajek, “Analysis of switched linear systems in the plane, part 2: global behavior of trajectories, controllability and attainability,” Journal of Optimization Theory and Applications, vol. 52, no. 3, pp. 395-427, 1987.
[5] K. A. Loparo, J. Aslanis, and O. Hajek, “Analysis of switched linear systems in the plane, part 1: Local behavior of trajectories and local cycle geometry,” Journal of optimization theory and applications, vol. 52, no. 3, pp. 365-394, 1987. [6] A. Bemporad, G. Ferrari-Trecate, and M. Morari, “Observability and controllability of piecewise affine and hybrid systems,” IEEE Transactions on Automatic Control, vol. 45, no. 10, pp. 1864-1876, 2000.
[7] Z. Sun, S. S. Ge, and T. H. Lee, “Controllability and reachability criteria for switched linear systems,” Automatica, vol. 38, no. 5, pp. 775-786, 2002.
[8] R. Vidal, A. Chiuso, S. Soatto, and S. Sastry, “Observability of linear hybrid systems,” in International Workshop on Hybrid Systems: Computation and Control, Springer: Berlin, pp. 526- 539, 2003.
[9] P. Collins and J. H. Van Schuppen, “Observability of piecewiseaffine hybrid systems,” in International Workshop on Hybrid Systems: Computation and Control, Philadelphia, USA, pp. 265- 279, 2004.
[10] J. Hespanha and A. Morse, “Input-output gains of switched linear systems,” in Open problems in mathematical systems and control theory, ed: Springer, pp. 121-124, 1999.
[11] W. Xie, C. Wen, and Z. Li, “Input-to-state stabilization of switched nonlinear systems,” IEEE Transactions on Automatic Control, vol. 46, no. 7, pp. 1111-1116, 2001.
[12] L. Vu, D. Chatterjee, and D. Liberzon, “Input-to-state stability of switched systems and switching adaptive control,” Automatica, vol. 43, no. 4, pp. 639-646, 2007.
[13] J. Zhao, and D. J. Hill, “A notion of passivity for switched systems with state-dependent switching,” Journal of Control Theory And Applications, vol. 4, no. 1, pp. 70-75, 2006.
[14] J. Zhao and D. J. Hill, “Passivity and stability of switched systems: a multiple storage function method,” Systems & Control Letters, vol. 57, no. 2, pp. 158-164, 2008.
[15] J. C. Geromel, P. Colaneri, and P. Bolzern, “Passivity of switched linear systems: analysis and control design,” Systems & Control Letters, vol. 61, no. 4, pp. 549-554, 2012.
[16] A. Berman, M. Neumann, and R. J. Stern, “Nonnegative matrices in dynamic systems,” vol. 3: Wiley-Interscience, 1989.
[17] J. Shen and J. Lam, “Static output-feedback stabilization with optimal L1-gain for positive linear systems,” Automatica, vol. 63, pp. 248-253, 2016.
[18] A. Rantzer, “On the Kalman-Yakubovich-Popov lemma for positive systems,” IEEE Transactions on Automatic Control, vol. 61, no. 5, pp. 1346-1349, 2016.
[19] Z. Li, and J. Lam, “Dominant pole and eigenstructure assignment for positive systems with state feedback,” International Journal of Systems Science, vol. 47, no. 12, pp. 2901-2912, 2016.
[20] J. Liu, J. Lian, and Y. Zhuang, “Output feedback L1 finite-time control of switched positive delayed systems with MDADT,” Nonlinear Analysis: Hybrid Systems, vol. 15, pp. 11-22, 2015.
[21] J. Shen and J. Lam, “On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays,” International Journal of Systems Science, vol. 46, no. 11, pp. 1953-1960, 2015.
[22] T. Binazadeh and M.H. Shafiei, “Suboptimal stabilizing controller design for nonlinear slowly-varying systems: application in a benchmark system,” IMA Journal of. Math, control an Information, vol. 32, no. 3, pp. 471–483, 2015.
[23] T. Binazadeh and M.H. Shafiei, “Passivity-based optimal control of discrete-time nonlinear systems,” Control and Cybernetics, vol. 42, no. 3,.pp. 627-637, 2013.
[24] H. Behruz, M.H. Shafiei, and T. Binazadeh, “Design of optimal output sliding mode control for discrete-time systems and improving the response rate using the CNF method,” IEEE 3rd International Conference on Control, Instrumentation, and Automation (ICCIA) , Tehran, Iran, pp. 119-124, 2013.
[25] A. Modirrousta, M.S. Zeini, and T. Binazadeh, “Non-linear optimal fuzzy control synthesis for robust output tracking of uncertain micro-electro-mechanical systems,” Transactions of the Institute of Measurement and Control, DOI: [26] T. Binazadeh and M.H. Shafiei, “The design of suboptimal asymptotic stabilising controllers for nonlinear slowly varying systems,” International Journal of Control, vol. 87, no. 4, pp. 682-692, 2014.
[27] T. Binazadeh and M.H. Shafiei,. “Design of an optimal stabilizing control law for discrete-time nonlinear systems based on passivity characteristic,” Nonlinear Dynamics and Systems Theory, vol. 13, no. 4, pp. 359-366, 2013.
[28] F. M. Callier and C. A. Desoer, Realization theory in Linear System Theory, ed: Springer New York, pp. 295-314, 1991.
[29] B. D. Anderson, “Optimal Control: Linear Quadratic Methods,” Courier Corporation, 2007.
[30] R. Castelein and A. Johnson, “Constrained optimal control,” IEEE Transactions On Automatic Control, vol. 34, no. 1, pp. 122- 126, 1989.
[31] A. Johnson, “LQ state-constrained control,” in Computer-Aided Control System Design, in Proc., IEEE/IFAC Joint Symposium on, pp. 423-428, 1994.
[32] C. Beauthier and J. J. Winkin, “On the positive LQ-problem for linear discrete time systems,” in Positive Systems, ed: Springer, pp. 45-53, 2009.
[33] C. Beauthier and J. J. Winkin, “LQ‐optimal control of positive linear systems,” Optimal Control Applications and Methods, vol. 31, no. 6, pp. 547-566, 2010.
[34] R. F. Hartl, S. P. Sethi, and R. G. Vickson, “A survey of the maximum principles for optimal control problems with state constraints,” SIAM Review, vol. 37, no. 2, pp. 181-218, 1995.
[35] S. Kostova, I. Ivanov, L. Imsland, and N. Georgieva, “Infinite horizon LQR problem of linear discrete time positive systems,” Comptes Rendus De L Academie Bulgare Des Sciences, vol. 66, no. 8, pp. 1167-1174, 2013.
[36] S. Kostova, L. Imsland, and I. Ivanov, “LQR problem of linear discrete time systems with nonnegative state constraints,” in Proc. 7th International Conference for Promoting the Application of Mathematics in Technical and Natural SciencesAMTNS’15, p. 110003, 2015.
[37] L. Farina and S. Rinaldi, “Positive linear systems: Theory and applications,” John Wiley & Sons, vol. 50, 2011.
[38] A., Francesco and M. Ariola, “Finite-time control of discretetime linear systems,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 724-729, 2005.
[39] V. D. F. L. Lewis, and V. L. Syrmos, Optimal control: John Wiley & Sons, 2012.


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