Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation

Kunisch, Karl

doi:10.48550/arxiv.2002.08625

Cited by 3 publications

(7 citation statements)

References 21 publications

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“…Our work stems from the same framework as [23], which approximates the feedback control with an NN then optimizes the control cost on a distribution of initial states. The authors also provide a theoretical analysis of OC solutions via NN approximations.…”

Section: A High-dimensional Deterministic Optimal Controlmentioning

confidence: 99%

“…The HJ t penalizer arises from the first equation in (21), whereas HJ fin and HJ grad are direct results of the final-time condition in (21) and its gradient, respectively. Penalizers prove helpful in problems similar to (23) [8]- [10], [47]. These penalizers improve the training convergence (Sec.…”

Section: A Main Formulationmentioning

confidence: 99%

“…These penalizers improve the training convergence (Sec. V-B.3) without altering the solution of (23). The P HJt,x penalizer is accumulated along the trajectory similar to L. The scalar terms β 1 ,β 2 ,β 3 weight the importance of each HJB penalizer and are hyperparameters of the NN (Sec.…”

Section: A Main Formulationmentioning

confidence: 99%

“…3) Effect of the HJB Penalizers: We experimentally assess the effectiveness of the penalizers c HJt , c HJfin , c HJgrad in (23). To this end, we define six models (various combinations of the three HJB penalizers and one model with weight decay) and train three instances of each on the corridor problem.…”

Section: Figmentioning

confidence: 99%

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A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding

Onken,

Nurbekyan,

et al. 2021

Preprint

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We propose a neural network approach for solving high-dimensional optimal control problems arising in real-time applications. Our approach yields controls in a feedback form and can therefore handle uncertainties such as perturbations to the system's state. We accomplish this by fusing the Pontryagin Maximum Principle (PMP) and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizing the value function with a neural network. We train our neural network model using the objective function of the control problem and penalty terms that enforce the HJB equations. Therefore, our training algorithm does not involve data generated by another algorithm. By training on a distribution of initial states, we ensure the controls' optimality on a large portion of the state-space. Our grid-free approach scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate the effectiveness of our approach on several multi-agent collisionavoidance problems in up to 150 dimensions. Furthermore, we empirically observe that the number of parameters in our approach scales linearly with the dimension of the control problem, thereby mitigating the curse of dimensionality.

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Section: A High-dimensional Deterministic Optimal Controlmentioning

confidence: 99%

Section: A Main Formulationmentioning

confidence: 99%

Section: A Main Formulationmentioning

confidence: 99%

Section: Figmentioning

confidence: 99%

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A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding

Onken,

Nurbekyan,

et al. 2021

Preprint

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“…In the literature there are different techniques to tackle this situation. Related recent works include, polynomial approximation [28], deep neural technique [31], tensor calculus [18], Taylor series expansions [13] and graph-tree structures [5].…”

Section: Introductionmentioning

confidence: 99%

Policy iteration for Hamilton-Jacobi-Bellman equations with control constraints

Kundu¹,

Kunisch²

2020

Preprint

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Policy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases.

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A Neural Network Approach Applied to Multi-Agent Optimal Control

Onken,

Nurbekyan,

et al. 2020

Preprint

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We propose a neural network approach for solving high-dimensional optimal control problems. In particular, we focus on multi-agent control problems with obstacle and collision avoidance. These problems immediately become highdimensional, even for moderate phase-space dimensions per agent. Our approach fuses the Pontryagin's Maximum Principle and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizes the value function with a neural network. Our approach yields controls in a feedback form for quick calculation and robustness to moderate disturbances to the system.We train our model using the objective function and optimality conditions of the control problem. Therefore, our training algorithm neither involves a data generation phase nor solutions from another algorithm. Our model uses empirically effective HJB penalizers for efficient training. By training on a distribution of initial states, we ensure the controls' optimality is achieved on a large portion of the state-space. Our approach is grid-free and scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate our approach's effectiveness on a 150-dimensional multi-agent problem with obstacles.

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Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation

Cited by 3 publications

References 21 publications

A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding

A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding

Policy iteration for Hamilton-Jacobi-Bellman equations with control constraints

A Neural Network Approach Applied to Multi-Agent Optimal Control

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