American Institute of Mathematical Sciences

Advanced
2011, 1(2): 275-282. doi: 10.3934/naco.2011.1.275

An optimal impulsive control regulator for linear systems

Michael Basin 1, and  Pablo Rodriguez-Ramirez 2,

1. 

Department of Physical and Mathematical Science, Autonomous University of Nuevo Leon, Apdo postal 144-F, C.P. 66450, San Nicolas de los Garza, Nuevo Leon
2. 

Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, San Nicolas de los Garza, Nuevo Leon, Mexico

Received  January 2011 Revised  March 2011 Published  June 2011

This paper addresses the optimal control problem for a linear system with respect to a Bolza-Meyer criterion, where both integral and non-integral terms are of the first degree. The optimal solution is obtained as an impulsive control, whereas the conventional linear feedback control fails to provide a causal solution. The theoretical result is complemented with illustrative examples verifying performance of the designed control algorithm in cases of large and short control horizons.
Keywords: linear systems., Optimal impulsive control.
Mathematics Subject Classification: 49N2.
Citation: Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275
References:
[1]

A. Arutyunov, V. Jacimovic and F. Pereira, Second order necessary conditions of optimality for impulsive control systems, Proc. 41st IEEE Conference on Decision and Control, (2002), 1576-1581. doi: doi:10.1109/CDC.2002.1184744.

[2]

A. V. Arutyunov, D. Yu. Karamzin and F. Pereira, Pontryagin's Maximum Principle for Optimal Impulsive Control Problems, Doklady Mathematics, 81 (2010), 418-421. doi: doi:10.1134/S1064562410030221.

[3]

A. V. Arutyunov, D. Yu. Karamzin and F. L. Pereira, On constrained impulsive control problems, J. Mathematical Sciences, 165 (2010), 654-688. doi: doi:10.1007/s10958-010-9834-z.

[4]

M. V. Basin and M. A. Pinsky, On impulse and continuous observation control design in Kalman filtering problem, Systems and Control Letters, 36 (1999), 213-219. doi: doi:10.1016/S0167-6911(98)00094-2.

[5]

A. Blaquiere, Impulsive optimal control with finite or infinite time horizon, J. Optimization Theory and Applications, 46 (1985), 431-439. doi: doi:10.1007/BF00939148.

[6]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Kluwer, 1988.

[7]

T. F. Filippova, State estimation problem for impulsive control systems, Proc. 1oth Mediterranean Conference on Automation and Control, Lisbon, Portugal, 2002.

[8]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Springer, 1975.

[9]

H. Kwakernaak and R. Sivan, "Linear Optimal Control Systems," Wiley-Interscience, New York, 1972.

[10]

Z. G. Li, C. Y. Wen and Y. C. Soh, Analysis and design of impulsive control systems, IEEE Trans. Automatic Control, 46 (2001), 894-897. doi: doi:10.1109/9.928590.

[11]

X. Liu, Stability of impulsive control systems with time delay, Math. Computer Modelling, 39 (2004), 511-519. doi: doi:10.1016/S0895-7177(04)90522-5.

[12]

X. Liu and K. L. Teo, Impulsive control of chaotic system, Intern. J. Bifurcation and Chaos, 12 (2002), 1181-1190.

[13]

Y. Liu, K. L. Teo, L. S. Jennigns and S. Wang, On a class of optimal control problems with state jumps, J. Optimization Theory and Applications, 98 (1998), 65-82. doi: doi:10.1023/A:1022684730236.

[14]

G. N. Silva and R. B. Vinter, Necessary conditions for optimal impulsive control problems, Proc. 36th IEEE Conference on Decision and Control, (1997), 2085-2090. doi: doi:10.1109/CDC.1997.657074.

[15]

R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, SIAM J. Control, 3 (1965), 191-205.

[16]

J. Warga, "Optimal Control of Differential and Functional Equations," Academic Press, New York, 1972.

show all references

References:
[1]

A. Arutyunov, V. Jacimovic and F. Pereira, Second order necessary conditions of optimality for impulsive control systems, Proc. 41st IEEE Conference on Decision and Control, (2002), 1576-1581. doi: doi:10.1109/CDC.2002.1184744.

[2]

A. V. Arutyunov, D. Yu. Karamzin and F. Pereira, Pontryagin's Maximum Principle for Optimal Impulsive Control Problems, Doklady Mathematics, 81 (2010), 418-421. doi: doi:10.1134/S1064562410030221.

[3]

A. V. Arutyunov, D. Yu. Karamzin and F. L. Pereira, On constrained impulsive control problems, J. Mathematical Sciences, 165 (2010), 654-688. doi: doi:10.1007/s10958-010-9834-z.

[4]

M. V. Basin and M. A. Pinsky, On impulse and continuous observation control design in Kalman filtering problem, Systems and Control Letters, 36 (1999), 213-219. doi: doi:10.1016/S0167-6911(98)00094-2.

[5]

A. Blaquiere, Impulsive optimal control with finite or infinite time horizon, J. Optimization Theory and Applications, 46 (1985), 431-439. doi: doi:10.1007/BF00939148.

[6]

A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Kluwer, 1988.

[7]

T. F. Filippova, State estimation problem for impulsive control systems, Proc. 1oth Mediterranean Conference on Automation and Control, Lisbon, Portugal, 2002.

[8]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Springer, 1975.

[9]

H. Kwakernaak and R. Sivan, "Linear Optimal Control Systems," Wiley-Interscience, New York, 1972.

[10]

Z. G. Li, C. Y. Wen and Y. C. Soh, Analysis and design of impulsive control systems, IEEE Trans. Automatic Control, 46 (2001), 894-897. doi: doi:10.1109/9.928590.

[11]

X. Liu, Stability of impulsive control systems with time delay, Math. Computer Modelling, 39 (2004), 511-519. doi: doi:10.1016/S0895-7177(04)90522-5.

[12]

X. Liu and K. L. Teo, Impulsive control of chaotic system, Intern. J. Bifurcation and Chaos, 12 (2002), 1181-1190.

[13]

Y. Liu, K. L. Teo, L. S. Jennigns and S. Wang, On a class of optimal control problems with state jumps, J. Optimization Theory and Applications, 98 (1998), 65-82. doi: doi:10.1023/A:1022684730236.

[14]

G. N. Silva and R. B. Vinter, Necessary conditions for optimal impulsive control problems, Proc. 36th IEEE Conference on Decision and Control, (1997), 2085-2090. doi: doi:10.1109/CDC.1997.657074.

[15]

R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, SIAM J. Control, 3 (1965), 191-205.

[16]

J. Warga, "Optimal Control of Differential and Functional Equations," Academic Press, New York, 1972.

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