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2015, 2015(special): 764-774. doi: 10.3934/proc.2015.0764

Decomposition of discrete linear-quadratic optimal control problems for switching systems

Galina Kurina 1, and  Sahlar Meherrem 2,

1. 

Voronezh State University, Universitetskaya pl., 1, Voronezh, 394006, Russian Federation
2. 

Yasar University, University aven., 35-37, Izmir, 35100, Turkey

Received  August 2014 Revised  March 2015 Published  November 2015

A discrete linear-quadratic optimal control problem for two controlled systems acting sequentially is considered. Matching conditions for trajectories at the switching point are absent, however the minimized functional depends on values of a state trajectory at the left and right sides from the switching point. State trajectories have fixed left and right points. We derive control optimality conditions in the maximum principle form. The unique solvability of the considered problem is established. The algorithm for solving the problem is given, which is based on solving sequentially some initial value problems. The formula for the minimal value of the performance index is also obtained. The transformation reducing the discrete problem with switching point to a problem without one is presented only in a special case.
Keywords: control optimality conditions, decomposition., Discrete linear-quadratic problems, switching systems.
Mathematics Subject Classification: Primary: 49N10, 93C30, 93C55; Secondary: 49K2.
Citation: Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764
References:
[1]

G. Zhai, H. Lin, X. Xu, J. Imae and T. Kobayashi, Analysis of switched normal discrete-time systems, Nonlinear Analysis. Theory Methods Appl., Ser. A, Theory Methods, 66 (2007), No. 8, 1788-1799.

[2]

S. S. Ge, Zhendong Sun and T. H. Lee, Reachability and controllability of switched linear discrete-time systems, IEEE Transactions on Automatic Control., 46 (2001), No. 9, 1437-1441.

[3]

Sh. F. Magerramov and K. B. Mansimov, Optimization of a class of discrete step control systems, Computational Mathematics and Mathematical Physics, 41 (2001), No. 3, 334-339.

[4]

A. Heydari and S. N. Balakrishnan, Optimal switching between autonomous subsystems, Journal of the Franklin Institute., 351 (2014), 2675-2690.

[5]

G. A. Kurina and Y. Zhou, Decomposition of linear-quadratic optimal control problems for two-steps systems, Doklady Mathematics, 83 (2011), No. 2, 275-277.

[6]

G. A. Kurina, On decomposition of linear-quadratic optimal control problems for two-steps descriptor systems, in 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, (2011), 6705-6710.

[7]

H. Abou - Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser Verlag, Basel-Boston-Berlin, 2003.

[8]

D. S. Naidu, Optimal Control Systems, CRC Press, Boca Raton London, New York, Washington, 2003.

[9]

E. R. Smolyakov, Unknown pages of optimal control history, Editorial URSS, Moscow, 2002 (Russian).

[10]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle, Systems and Control Letters, 57 (2008), No.11, 964-970.

[11]

G. A. Kurina and E. V. Smirnova, Asymptotics of solutions of optimal control problems with intermediate points in quality criterion and small parameters , Journal of Mathematical Sciences, 170 (2010), No. 2, 192-228.

show all references

References:
[1]

G. Zhai, H. Lin, X. Xu, J. Imae and T. Kobayashi, Analysis of switched normal discrete-time systems, Nonlinear Analysis. Theory Methods Appl., Ser. A, Theory Methods, 66 (2007), No. 8, 1788-1799.

[2]

S. S. Ge, Zhendong Sun and T. H. Lee, Reachability and controllability of switched linear discrete-time systems, IEEE Transactions on Automatic Control., 46 (2001), No. 9, 1437-1441.

[3]

Sh. F. Magerramov and K. B. Mansimov, Optimization of a class of discrete step control systems, Computational Mathematics and Mathematical Physics, 41 (2001), No. 3, 334-339.

[4]

A. Heydari and S. N. Balakrishnan, Optimal switching between autonomous subsystems, Journal of the Franklin Institute., 351 (2014), 2675-2690.

[5]

G. A. Kurina and Y. Zhou, Decomposition of linear-quadratic optimal control problems for two-steps systems, Doklady Mathematics, 83 (2011), No. 2, 275-277.

[6]

G. A. Kurina, On decomposition of linear-quadratic optimal control problems for two-steps descriptor systems, in 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, (2011), 6705-6710.

[7]

H. Abou - Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser Verlag, Basel-Boston-Berlin, 2003.

[8]

D. S. Naidu, Optimal Control Systems, CRC Press, Boca Raton London, New York, Washington, 2003.

[9]

E. R. Smolyakov, Unknown pages of optimal control history, Editorial URSS, Moscow, 2002 (Russian).

[10]

A. V. Dmitruk and A. M. Kaganovich, The hybrid maximum principle is a consequence of Pontryagin maximum principle, Systems and Control Letters, 57 (2008), No.11, 964-970.

[11]

G. A. Kurina and E. V. Smirnova, Asymptotics of solutions of optimal control problems with intermediate points in quality criterion and small parameters , Journal of Mathematical Sciences, 170 (2010), No. 2, 192-228.

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