ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED HYBRID DYNAMICAL SYSTEMS

doi:10.1016/S0252-9602(16)30081-9

Abstract

Abstract:

In this paper, we consider a class of optimal control problem for the singularly perturbed hybrid dynamical systems. By means of variational method, we obtain the necessary conditions of the hybrid dynamical systems. Meanwhile, the existence of solution for the hybrid dynamical system is proved by the sewing method and the uniformly valid asymptotic expansion of the optimal trajectory is constructed by the boundary function method. Finally, an example is presented to illustrate the result.

Key words: optimal control, hybrid dynamical system, singular perturbation

CLC Number:

34B15

Limeng WU, Juan ZHANG, Mingkang NI, Haibo LU. ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED HYBRID DYNAMICAL SYSTEMS[J].Acta mathematica scientia,Series B, 2016, 36(5): 1457-1466.

Trendmd

References

[1] Witsenhausen H S. A class of hybrid-state continuous-time dynamic systems. IEEE Trans Automatic Cont, 1966, 11(6):665-683
[2] Cellier F E. Combined Continuous/Discrete System Simulation by Use of Digital Computer:Techniques and Tools. Switzerland Zurich:Swiss Federal Institute of Technology, 1979
[3] Michel A N, Hu B. Towards a stability theory of general hybrid dynamical systems. Automatica, 1999, 35(3):371-384
[4] Yong J M. Systems governed by ordinary differential equations with continuous switching and impulse controls. Appl Math Opti, 1989, 20:223-235
[5] Branicky M S. Multiple Lyapunov functions and analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 1998, 34(4):475-482
[6] Mo Y W, Xiao D Y. Overview of hybrid dynamic system and its application. Contr Theory Appl, 2002, 19(1):1-8
[7] Guo L, Yu R L, Tian F Z. Optimal control of one kind general jump transition system. Journal of Shandong University, 2006, 41(1):35-40(in Chinese)
[8] Zhu F, Antsaklis P J. Optimal control of hybrid switched systems:a brief survey. Discrete Event Dynamics Systems:Theory and Applications, 2015, 25:345-364
[9] Marzbann H R, Hoseini S M. A hybrid approximation scheme for discretizing constrained quadratic optimal control problems. J Franklin Inst, 2014, 351:2640-2656
[10] Lee T. Optimal control of partitioned hybrid systems via discrete-time Hamilton-Jacobi theory. Automatica, 2014, 50:2062-2069
[11] Branicky M S, Borkar V S, Mirtter S K. A unified framework for hybrid control:model and optimal control theory. IEEE Trans Automatic Cont, 1998, 43(1):31-45
[12] Bensoussan A, Memaldi J L. Hybrid control and dynamic programming. Dynamics of Continuous, Discrete and Impulsive Systems, 1997, 3:395-442
[13] Xu X P, Antsaklis P J. Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans Automatic Cont, 2004, 49(1):2-16
[14] Fridman E. Exact slow-fast decomposition of the nonlinear singularly perturbed optimal control problem. Syst Control Lett, 2000, 40:121-131
[15] Ni M K, Dmitriev M G. Steplike contrast structure in an elementary optimal control problem. Comput Math Math Phys, 2010, 50(8):1312-1323
[16] Koskie S, Coumarbatch C, Gajic Z. Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation. Applied Mathematics and Computation, 2010, 216:1401-1411
[17] Ni M K, Lin W Z. Minimizing sequence of variational problems with small parameters. Applied Mathematics and Mechanics, English Edition, 2009, 30(6):695-701
[18] Ni M K, Dmitriev M G. Contrast structures in the simplest vector variational problem and their asymptotics. Avtomatika i Telemekhanika, 1998, 5:41-52
[19] Vasil'eva A B, Dmitriev M G, Ni M K. On a step-like contrast structure for a problem of the calculus of variations. Comput Math Math Phys, 2004, 44(7):1203-1212
[20] Kokotovic P, Khalil H, O'Reilly J. Singular Perturbation Methods in Control:Analysis and Design. Orlando:Academic Press, 1986
[21] Vasil'eva A B, Butuzov V F. Asymptotic Expansions of Singularly Perturbed Differential Equations. Moscow:Nauka, 1973
[22] Barbu L, Morosanu G. Singularly Perturbed Boundary-Value Problems. Basel, Boston, Berlin:Birkhauser, 2007
[23] Vasilieva A B, Butuzov V F, Kalachev L V. The Boundary Function Method for Singular Perturbation Problems. Philadelphia:SIAM, 1995
[24] Wang A F, Ni M K. The interior layer for a nonlinear singularly perturbed differential difference equation. Acta Mathematica Scientia, 2012, 32B(2):695-709
[25] Xie F. Singular perturbation for a class of reaction diffusion equation. Acta Mathematica Scientia, 2003, 23A(3):369-373(in Chinese)
[26] Mo J Q, Wang H, Lin W T. The solvability for a class of singularly perturbed quasi-linear differential system. Acta Mathematica Scientia, 2008, 28B(3):495-500