An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory

In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the stable Lagrangian submanifold. With this method, the closed loop stability is guaranteed and can be enhanced by taking higher order approximations. A numerical example shows the effectiveness of the method.

[1]  John Guckenheimer,et al.  A Survey of Methods for Computing (un)Stable Manifolds of Vector Fields , 2005, Int. J. Bifurc. Chaos.

[2]  E. T. Copson.  Partial Differential Equations: Partial differential equations of the first order , 1975 .

[3]  Arjan van der Schaft,et al.  Inner-outer factorization for nonlinear noninvertible systems , 2004, IEEE Transactions on Automatic Control.

[4]  S. Zagatti.  On viscosity solutions of Hamilton-Jacobi equations , 2008 .

[5]  P. Moylan,et al.  The stability of nonlinear dissipative systems , 1976 .

[6]  J. Cloutier,et al.  Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method , 1998 .

[7]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[8]  Jacquelien M. A. Scherpen,et al.  Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators , 2005, IEEE Transactions on Automatic Control.

[9]  Joseph A. Ball,et al.  J-inner-outer factorization, J-spectral factorization, and robust control for nonlinear systems , 1996, IEEE Trans. Autom. Control..

[10]  Y. Nishikawa,et al.  A method for suboptimal design of nonlinear feedback systems , 1971 .

[11]  Randal W. Beard,et al.  Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation , 1997, Autom..

[12]  Wei-Min Lu,et al.  Nonlinear optimal control: alternatives to Hamilton-Jacobi equation , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[13]  A. Astolfi.  Disturbance Attenuation and H,-Control Via Measurement Feedback in , 1992 .

[14]  J. Helton,et al.  H∞ control for nonlinear systems with output feedback , 1993, IEEE Trans. Autom. Control..

[15]  Pierpaolo Soravia.  ${\cal H}_\infty$ Control of Nonlinear Systems: Differential Games and Viscosity Solutions , 1996 .

[16]  J. Helton,et al.  H∞ control for nonlinear systems with output feedback , 1993, IEEE Trans. Autom. Control..

[17]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[18]  D. Lukes.  Optimal Regulation of Nonlinear Dynamical Systems , 1969 .

[19]  H. Hermes,et al.  Foundations of optimal control theory , 1968 .

[20]  A. Isidori,et al.  Disturbance attenuation and H/sub infinity /-control via measurement feedback in nonlinear systems , 1992 .

[21]  A. Rantzer,et al.  On approximate policy iteration for continuous-time systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[22]  Ruey-Wen Liu,et al.  Construction of Suboptimal Control Sequences , 1967 .

[23]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[24]  I. Norman Katz,et al.  An Iterative Algorithm for Solving Hamilton-Jacobi Type Equations , 2000, SIAM J. Sci. Comput..

[25]  Van,et al.  L2-Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H∞ Control , 2004 .

[26]  P J Fox,et al.  THE FOUNDATIONS OF MECHANICS. , 1918, Science.

[27]  W. Garrard.  Additional results on sub–optimal feedback control of non–linear systems , 1969 .

[28]  Arjan van der Schaft,et al.  An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[29]  A. P. S. Selvadurai,et al.  Partial differential equations of the first-order , 2000 .

[30]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[31]  A. Schaft.  On a state space approach to nonlinear H ∞ control , 1991 .

[32]  William L. Garrard,et al.  Design of nonlinear automatic flight control systems , 1977, Autom..

[33]  A. Schaft.  L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control , 1992 .

[34]  C. Robinson.  Dynamical Systems: Stability, Symbolic Dynamics, and Chaos , 1994 .

[35]  Noboru Sakamoto,et al.  Analysis of the Hamilton--Jacobi Equation in Nonlinear Control Theory by Symplectic Geometry , 2001, SIAM J. Control. Optim..