# American Institute of Mathematical Sciences

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

## An optimal impulsive control regulator for linear systems

 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.
Citation: Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & 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. doi: doi:10.1109/CDC.2002.1184744. Google Scholar [2] A. V. Arutyunov, D. Yu. Karamzin and F. Pereira, Pontryagin's Maximum Principle for Optimal Impulsive Control Problems,, Doklady Mathematics, 81 (2010), 418. doi: doi:10.1134/S1064562410030221. Google Scholar [3] A. V. Arutyunov, D. Yu. Karamzin and F. L. Pereira, On constrained impulsive control problems,, J. Mathematical Sciences, 165 (2010), 654. doi: doi:10.1007/s10958-010-9834-z. Google Scholar [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. doi: doi:10.1016/S0167-6911(98)00094-2. Google Scholar [5] A. Blaquiere, Impulsive optimal control with finite or infinite time horizon,, J. Optimization Theory and Applications, 46 (1985), 431. doi: doi:10.1007/BF00939148. Google Scholar [6] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Kluwer, (1988). Google Scholar [7] T. F. Filippova, State estimation problem for impulsive control systems,, Proc. 1oth Mediterranean Conference on Automation and Control, (2002). Google Scholar [8] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Springer, (1975). Google Scholar [9] H. Kwakernaak and R. Sivan, "Linear Optimal Control Systems,", Wiley-Interscience, (1972). Google Scholar [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. doi: doi:10.1109/9.928590. Google Scholar [11] X. Liu, Stability of impulsive control systems with time delay,, Math. Computer Modelling, 39 (2004), 511. doi: doi:10.1016/S0895-7177(04)90522-5. Google Scholar [12] X. Liu and K. L. Teo, Impulsive control of chaotic system,, Intern. J. Bifurcation and Chaos, 12 (2002), 1181. Google Scholar [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. doi: doi:10.1023/A:1022684730236. Google Scholar [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. doi: doi:10.1109/CDC.1997.657074. Google Scholar [15] R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures,, SIAM J. Control, 3 (1965), 191. Google Scholar [16] J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972). Google Scholar

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##### 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. doi: doi:10.1109/CDC.2002.1184744. Google Scholar [2] A. V. Arutyunov, D. Yu. Karamzin and F. Pereira, Pontryagin's Maximum Principle for Optimal Impulsive Control Problems,, Doklady Mathematics, 81 (2010), 418. doi: doi:10.1134/S1064562410030221. Google Scholar [3] A. V. Arutyunov, D. Yu. Karamzin and F. L. Pereira, On constrained impulsive control problems,, J. Mathematical Sciences, 165 (2010), 654. doi: doi:10.1007/s10958-010-9834-z. Google Scholar [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. doi: doi:10.1016/S0167-6911(98)00094-2. Google Scholar [5] A. Blaquiere, Impulsive optimal control with finite or infinite time horizon,, J. Optimization Theory and Applications, 46 (1985), 431. doi: doi:10.1007/BF00939148. Google Scholar [6] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Kluwer, (1988). Google Scholar [7] T. F. Filippova, State estimation problem for impulsive control systems,, Proc. 1oth Mediterranean Conference on Automation and Control, (2002). Google Scholar [8] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Springer, (1975). Google Scholar [9] H. Kwakernaak and R. Sivan, "Linear Optimal Control Systems,", Wiley-Interscience, (1972). Google Scholar [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. doi: doi:10.1109/9.928590. Google Scholar [11] X. Liu, Stability of impulsive control systems with time delay,, Math. Computer Modelling, 39 (2004), 511. doi: doi:10.1016/S0895-7177(04)90522-5. Google Scholar [12] X. Liu and K. L. Teo, Impulsive control of chaotic system,, Intern. J. Bifurcation and Chaos, 12 (2002), 1181. Google Scholar [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. doi: doi:10.1023/A:1022684730236. Google Scholar [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. doi: doi:10.1109/CDC.1997.657074. Google Scholar [15] R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures,, SIAM J. Control, 3 (1965), 191. Google Scholar [16] J. Warga, "Optimal Control of Differential and Functional Equations,", Academic Press, (1972). Google Scholar
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