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

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.  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

[1]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[2]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001

[3]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[4]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

[5]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026

[6]

Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021001

[7]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[8]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[9]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[10]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[11]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[12]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[13]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[14]

Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179

[15]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[16]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[17]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029

[18]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[19]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[20]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051

 Impact Factor: 

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]