Approximate solution of hyper-sensitive optimal control problems using finite-time Lyapunov analysis

Aykutlug, E.; Mease, Kenneth D.

doi:10.1109/acc.2009.5160681

Cited by 4 publications

(9 citation statements)

References 14 publications

(26 reference statements)

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“…Generally, the hypersensitive HJBs can be further divided into completely hypersensitive HJBs and partially hypersensetive HJBs, and here we will focus on the completely hypersensitive HJBs. It has been found that the solution of a completely hypersensitive HJB can be approximated by three phases along the time, including stable phase, unstable phase, and equilibrium point [21], expressed as…”

Section: Supervised Learning Based Hamiltonian (Slh)mentioning

confidence: 99%

Learning-based Hamilton-Jacobi-Bellman Methods for Optimal Control

You,

Dai,

2019

Preprint

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Many optimal control problems are formulated as two point boundary value problems (TPBVPs) with conditions of optimality derived from the HamiltonJacobiBellman (HJB) equations. In most cases, it is challenging to solve HJBs due to the difficulty of guessing the adjoint variables. This paper proposes two learning-based approaches to find the initial guess of adjoint variables in real-time, which can be applied to solve general TP-BVPs. For cases with database of solutions and corresponding adjoint variables of a TPBVP under varying boundary conditions, a supervised learning method is applied to learn the HJB solutions off-line. After obtaining a trained neural network from supervised learning, we are able to find proper initial adjoint variables for given boundary conditions in realtime. However, when validated solutions of TPBVPs are not available, the reinforcement learning method is applied to solve HJB by constructing a neural network, defining a reward function, and setting appropriate super parameters. The reinforcement learning based HJB method can learn how to find accurate adjoint variables via an updating neural network. Finally, both learning approaches are implemented in classical optimal control problems to verify the effectiveness of the learning based HJB methods.

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Section: Supervised Learning Based Hamiltonian (Slh)mentioning

confidence: 99%

Learning-based Hamilton-Jacobi-Bellman Methods for Optimal Control

You,

Dai,

2019

Preprint

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show abstract

“…2) cases, is our goal. Computationally, determining only low-dimensional manifolds may be feasible, but computing selected points on higher-dimensional manifolds is possible and useful (e.g., [4]). Figure 1: Geometry of a two-timescale 3D system with a 2D normally attracting center manifold.…”

Section: Overview Of Approachmentioning

confidence: 99%

“…For small values of m, the Hamiltonian system is in singularly perturbed standard form [33], and the system can be expected to evolve on disparate timescales. Using the twotimescale geometry to solve the boundary-value problem has been addressed in [4,23,49,58]. Here we focus on applying FTLA to the Hamiltonian system (51) to diagnose twotimescale behavior and locate points on the center manifold, which is in this case a slow manifold.…”

Section: D Hamiltonian System: Mass-spring-damper Systemmentioning

confidence: 99%

“…In [52], FTLA is used to identify the dimension of the attracting slow manifold along a trajectory. The application of FTLA to the solution of two-timescale boundary value problems related to optimal control is discussed in [4].…”

Section: Introductionmentioning

confidence: 99%

“…Because the intent is to analyze finite-time behavior, we first define two-timescale behavior in this context. Though we only directly consider two timescales and normally hyperbolic center manifolds in this paper, the discussion and results are relevant to systems with more than two timescales and also to additional manifold structure, such as the center-stable and center-unstable manifolds relevant to the solution of certain boundary-value problems [4,23,49,58]. We do not consider systems with persistent fast oscillations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and subspaces

Mease

Topcu

Aykutlug

et al. 2016

Communications in Nonlinear Science and Numerical Simulation

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Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior in the tangent linear dynamics and to determine the associated manifold structure in the flow of a finite-dimensional nonlinear autonomous dynamical system. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting is defined using finite-time Lyapunov exponents and vectors, guided by the asymptotic theory of partially hyperbolic sets, with important modifications for the finite-time case; for example, finite-time Lyapunov analysis relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. The splitting is used to locate points on normally hyperbolic center manifolds. Determining manifolds from tangent bundle structure is more generally applicable than approaches, such as the singular perturbation method, that require special normal forms or other a priori knowledge. The use, features, and accuracy of the approach are illustrated via several detailed examples.

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Hybrid solution of nonlinear stochastic optimal control problems using Legendre Pseudospectral and generalized Polynomial Chaos algorithms

Harmon

2017

2017 American Control Conference (ACC)

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Approximate solution of hyper-sensitive optimal control problems using finite-time Lyapunov analysis

Cited by 4 publications

References 14 publications

Learning-based Hamilton-Jacobi-Bellman Methods for Optimal Control

Learning-based Hamilton-Jacobi-Bellman Methods for Optimal Control

Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and subspaces

Hybrid solution of nonlinear stochastic optimal control problems using Legendre Pseudospectral and generalized Polynomial Chaos algorithms

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