Analysis of the Hamilton-Jacobi equation in nonlinear control theory by symplectic geometry

Sakamoto, Noboru

doi:10.1109/cdc.2002.1184738

Cited by 12 publications

(18 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The papers [28] and [29] give a sufficient condition for the existence of the stabilizing solution using symplectic geometry. In [25], the geometric structure of the Hamilton-Jacobi equation is studied showing the similarity and difference with the Riccati equation. See also [30] for the treatment of the Hamilton-Jacobi equation as well as recently developed techniques in nonlinear control theory such as the theory of port-Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Sakamoto

Schaft

2007

2007 American Control Conference

View full text Add to dashboard Cite

Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract-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.

show abstract

Section: Introductionmentioning

confidence: 99%

“…In this paper, we attempt to develop a method to approximate the stabilizing solution of the Hamilton-Jacobi equation based on the geometric research in [28], [29] and [25]. The main object of the geometric research on the Hamilton-Jacobi equation is the associated Hamiltonian system.…”

Section: Introductionmentioning

confidence: 99%

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

Sakamoto

Schaft

2007

2007 American Control Conference

View full text Add to dashboard Cite

show abstract

“…CalculateX(t,X 0 ,P 0 ) andP (t,X 0 ,P 0 ) in (18) and restrict their initial points to the stable Lagrangian subspace of the linearized Riccati equation around the origin, {(X 0 , ΓX 0 ) | |X 0 | small}, where Γ is the stabilizing solution. Substitute them in (11) (or (12)) to get (19). Eliminate t and n − 1 variables fromX 01 , .…”

Section: Propositionmentioning

confidence: 99%

“…In this report, we attempt to develop a method to approximate the stabilizing solution of the Hamilton-Jacobi equation based on the geometric research in [21], [22] and [19]. The main object of the geometric research on the Hamilton-Jacobi equation is the associated Hamiltonian system.…”

Section: Introductionmentioning

confidence: 99%

“…In the second direction, [21] and [22] give a sufficient condition for the existence of the stabilizing solution using symplectic geometry. In [19], the geometric structure of the Hamilton-Jacobi equation is studied showing the similarity and difference with the Riccati equation. See also [23] for the treatment of the HamiltonJacobi equation as well as recently developed techniques in nonlinear control theory such as the theory of portHamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory

Sakamoto

Schaft²

2006

Proceedings of the 45th IEEE Conference on Decision and Control

View full text Add to dashboard Cite

An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory Sakamoto, Noboru; Schaft, Arjan J. van der Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract-In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical equation) that is derived from the theory of 1-st order partial differential equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati equation from the Hamilton-Jacobi equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained.

show abstract

A real‐time algorithm for nonlinear infinite horizon optimal control by time axis transformation method

Marutani

Ohtsuka

2012

Intl J Robust & Nonlinear

View full text Add to dashboard Cite

SUMMARYIn this paper, a real‐time algorithm is proposed for nonlinear infinite horizon optimal control. The time axis transformation of the horizon is combined with receding horizon control to realize state feedback by numerical optimization over an infinite horizon. Moreover, a descriptor equation is utilized in place of a state equation to avoid the numerical difficulty due to the infinite horizon. Then a nonlinear equation is obtained to determine the trajectories of the state, costate, and control input over the transformed horizon. The nonlinear equation is solved at each sampling time by a continuation method to trace a time‐varying solution. The proposed algorithm is applied to nonlinear systems to evaluate its accuracy and effectiveness. Copyright © 2012 John Wiley & Sons, Ltd.

show abstract

Analysis of the Hamilton-Jacobi equation in nonlinear control theory by symplectic geometry

Cited by 12 publications

References 26 publications

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

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

An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory

A real‐time algorithm for nonlinear infinite horizon optimal control by time axis transformation method

Contact Info

Product

Resources

About