Patchy Solutions of Hamilton-Jacobi-Bellman Partial Differential Equations

Navasca, Carmeliza; Krener, Arthur J.

doi:10.1007/978-3-540-73570-0_20

Cited by 38 publications

(22 citation statements)

References 22 publications

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“…In both D 1 and D 2 , the TPBVP (10) is solved at each gridpoint in G q sparse using a method based on four-stage Lobatto IIIa formula in Kierzenka-Shampine. 9 The hierarchical surpluses for interpolation are computed using (12). Then we check the accuracy of V (0, v, ω).…”

Section: Examplesmentioning

confidence: 99%

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A Causality Free Computational Method for HJB Equations with Application to Rigid Body Satellites

Kang

Wilcox

2015

AIAA Guidance, Navigation, and Control Conference

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Solving Hamilton-Jacobi-Bellman (HJB) equations is essential in feedback optimal control. Using the solution of HJB equations, feedback optimal control laws can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, numerically solving HJB equations for general nonlinear systems is unfeasible due to the curse of dimensionality. In this paper, we develop a new computational method of solving HJB equations. The method is causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as 6-D HJB equations for the attitude control of rigid bodies. The method is applied to the optimal attitude control of a satellite system using momentum wheels. The accuracy of the numerical solution is verified at a set of randomly selected sample points.

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Section: Examplesmentioning

confidence: 99%

“…Then the hierarchical surpluses for the interpolation are computed using (12). Similar to the previous example, we check the accuracy at 500 random points in D 1 .…”

Section: Examplesmentioning

confidence: 99%

A Causality Free Computational Method for HJB Equations with Application to Rigid Body Satellites

Kang

Wilcox

2015

AIAA Guidance, Navigation, and Control Conference

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Section: (T Z(t)x) λ(T)xmentioning

confidence: 99%

“…But one drawback of this approach is that assumes the system is smooth with no active constraints. If these assumptions don't hold then one needs to go to a patchy approach [13], [5] which we will report on at a later date.…”

Section: Minimum Horizon Estimation Minimum Horizon Estimation (Mmentioning

confidence: 99%

Minimum Energy Estimation and Moving Horizon Estimation

Krener

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-Minimum Energy Estimation is a way of filtering the state of a nonlinear system from partial and inexact measurements. It is a generalization of Gauss' method of least squares. Its application to filtering of control systems goes back at least to Mortenson who called it Maximum Likelyhood Estimation [12]. For linear, Gaussian systems it reduces to maximum likelihood estimation (aka Kalman Filtering) but this is not true for nonlinear systems. We prefer the name Minimum Energy Estimation (MEE) that was introduced by Hijab [4]. Both Mortenson and Hijab dealt with systems in continuous time, we extend their methods to discrete time systems and show how power series techniques can lessen the computational burden.Moving Horizon Estimation (MHE) is a moving window version of MEE. It computes the solution to an optimal control problem over a past moving window that is constrained by the actual observations on the window. The optimal state trajectory at the end of the window is the MEE estimate at this time. The cost in the optimal control problem is usually taken to be an L2 norm of the three slack variables; the initial condion noise, the driving noise and the measurement noise. MHE requires the buffering of the measurements over the past window. The optimal control problem is solved in real time by a nonlinear program solver but it becomes more difficult as the length of the window is increased.The power series approach to MME can be applied to MHE and this permits the choice of a very short past window consisting of one time step. This speeds up MHE and allows its real time implementaion on faster processes.

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“…This theoretically elegant approach suffers some difficulties in computation due to the curse of dimensionality, a term that was coined by Richard E. Bellman when considering problems in dynamic optimization, which relates to the fact that the size of the discretized problem in solving HJB equations increases exponentially with the dimension. Finding an approximate solution to the HJB-type of equations in a local neighborhood of a trajectory has been extensively studied [1], [6], [14], [20], [21]. Some of them can be applied to systems in high dimensions.…”

Section: Introductionmentioning

confidence: 99%

An example of solving HJB equations using sparse grid for feedback control

Kang

Wilcox

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-It is well known that solving the Hamilton-JacobiBellman (HJB) equation in moderate and high dimensions (d > 3) suffers the curse of dimensionality. In this paper, we introduce and demonstrate an example of solving the 6-D HJB equation for the optimal attitude control of a rigid body equipped with two pairs of momentum wheels. The system is uncontrollable. To mitigate the curse-of-dimensionality, a computational method based on sparse grids is introduced. The method is causality free, which enjoys the advantage of perfect parallelism. The problem is solved using several hundred CPU cores in parallel. In the simulations, the solution of the HJB equation is integrated into a model predictive control for optimal attitude stabilization.

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Patchy Solutions of Hamilton-Jacobi-Bellman Partial Differential Equations

Cited by 38 publications

References 22 publications

A Causality Free Computational Method for HJB Equations with Application to Rigid Body Satellites

A Causality Free Computational Method for HJB Equations with Application to Rigid Body Satellites

Minimum Energy Estimation and Moving Horizon Estimation

An example of solving HJB equations using sparse grid for feedback control

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