Inner–Outer Factorization for Nonlinear Noninvertible Systems

Ball, Joseph A.; Petersen, Mark A.; Schaft, Arjan van der

doi:10.1109/tac.2004.825644

Cited by 13 publications

(6 citation statements)

References 44 publications

(90 reference statements)

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“…Closely related to optimal control and H ∞ control is the notion of dissipativity, which is characterized by a Hamilton-Jacobi inequality (see, e.g., [19], [42]). Some active areas of research in recent years are the factorization problem [6], [7] and the balanced realization problem [36], [15] and the solutions of these problems are again represented by Hamilton-Jacobi equations (or, inequalities). Contrary to the well-developed theory and computational tools for the Riccati equation, which are widely applied, the Hamilton-Jacobi equation is still an impediment to practical applications of nonlinear control theory.…”

Section: Introductionmentioning

confidence: 99%

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

Sakamoto

Schaft

2008

IEEE Trans. Automat. Contr.

135

125

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In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.

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Section: Introductionmentioning

confidence: 99%

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

Sakamoto

Schaft

2008

IEEE Trans. Automat. Contr.

135

125

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“…We have not yet been able to extend our analysis to the parametrization of more general (not necessarily affine, minimal or stable) nonlinear spectral factors. However, the assumption that can be removed has been discussed in [65] and further investigation is envisaged for the outer-spectral factorization case [11]. On the other hand, great difficulty has been experienced with finding an inner-outer factorization for nonstable systems.…”

Section: Discussionmentioning

confidence: 99%

“…In this regard, chemical process control is discussed in the joint paper by Ball, Petersen, and van der Schaft (cf. [11]) on noninvertible nonlinear systems. Also, it remains an open question whether our results are applicable to mechanical systems with Hamiltonian structure.…”

Section: Discussionmentioning

confidence: 99%

Nonsquare spectral factorization for nonlinear control systems

Petersen

Schaft²

2005

IEEE Trans. Automat. Contr.

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Abstract-This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton-Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system. Index Terms-Hamilton-Jacobi inequalities, invariant Lagrangian manifolds, nonlinear nonsquare spectral factors.

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“…In [9] this technique is extended to control systems with inputs and outputs and is known to be effective for fundamental control problems such as factorization [5], [4] and model reduction problems [11]. In this section we give a useful observation on a Hamiltonian lifted system when the original system is integrable.…”

Section: An Observation On Integrable Systems Andmentioning

confidence: 99%

“…Closely related to optimal control and H ∞ control is the notion of dissipativity, which is also characterized by a HamiltonJacobi equation (see, e.g., [13], [24]). Some active areas of research in recent years are the factorization problem [4], [5] and the balanced realization problem [11] and the solutions of these problems are again represented by Hamilton-Jacobi equations (or, inequalities).…”

Section: Introductionmentioning

confidence: 99%

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

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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.

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Inner–Outer Factorization for Nonlinear Noninvertible Systems

Cited by 13 publications

References 44 publications

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

Nonsquare spectral factorization for nonlinear control systems

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

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