An Iterative Algorithm for Solving Hamilton--Jacobi Type Equations

Markman, Jerry; Katz, Irving

doi:10.1137/s1064827598344315

Cited by 28 publications

(12 citation statements)

References 13 publications

(10 reference statements)

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“…In this section we consider a simple example from [20] that has an exact solution. This example is of particular interest because the numerical method in [7] failed to produce a stabilizing control based on the approximate SDRE solution.…”

Section: Example 1: Exact Solutionmentioning

confidence: 99%

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

Banks¹,

Lewis²,

Tran³

2003

103

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State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

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Section: Example 1: Exact Solutionmentioning

confidence: 99%

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

Banks¹,

Lewis²,

Tran³

2003

103

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“…The procedure we use is described in detail in [8], [9]. For each i, a starting estimate is taken to be p i−1 0 .…”

Section: Initial Guessmentioning

confidence: 99%

“…By applying the algorithm to sufficiently many points near the origin, an approximate solution over the entire neighborhood can be pieced together through, for example, polynomial interpolation. Details of implementation and numerical simulations are given in [8], [9]. This paper concentrates on the convergence of the algorithm.…”

Section: Introductionmentioning

confidence: 99%

Convergence of an iterative algorithm for solving Hamilton-Jacobi type equations

Markman

Katz

2001

Math. Comp.

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Abstract. Solutions of the optimal control and H∞-control problems for nonlinear affine systems can be found by solving Hamilton-Jacobi equations. However, these first order nonlinear partial differential equations can, in general, not be solved analytically. This paper studies the rate of convergence of an iterative algorithm which solves these equations numerically for points near the origin. It is shown that the procedure converges to the stabilizing solution exponentially with respect to the iteration variable. Illustrative examples are presented which confirm the theoretical rate of convergence.

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“…In this method, feedback control is given in a power series form and has a similar disadvantage to the series expansion technique in that it is useful only for simple nonlinearities. A technique that employs open-loop controls and their interpolation is used in [20]. The drawback is that the interpolation of openloop controls for each point in discretized state space is time-consuming and the computational cost grows exponentially with the state space dimension.…”

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

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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.

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An Iterative Algorithm for Solving Hamilton--Jacobi Type Equations

Cited by 28 publications

References 13 publications

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

Convergence of an iterative algorithm for solving Hamilton-Jacobi type equations

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

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