Policy-Iteration-Based Finite-Horizon Approximate Dynamic Programming for Continuous-Time Nonlinear Optimal Control

The Hamilton-Jacobi-Bellman (HJB) equation serves as the necessary and sufficient condition for the optimal solution to the continuous-time (CT) optimal control problem (OCP). Compared with the infinite-horizon HJB equation, the solving of the finite-horizon (FH) HJB equation has been a long-standin...

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Published in:	IEEE transaction on neural networks and learning systems Vol. 34; no. 9; pp. 1 - 13
Main Authors:	Lin, Ziyu, Duan, Jingliang, Li, Shengbo Eben, Ma, Haitong, Li, Jie, Chen, Jianyu, Cheng, Bo, Ma, Jun
Format:	Journal Article
Language:	English
Published:	United States IEEE 01.09.2023 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Actor critic Algorithms approximate dynamic programming (ADP) Approximation algorithms Artificial neural networks Complex systems Dynamic programming finite-horizon (FH) Hamilton–Jacobi–Bellman (HJB) equation Heuristic algorithms Horizon Linear quadratic regulator Mathematical models Multilayers Neural networks Nonlinear control Nonlinear dynamical systems Optimal control Optimization policy iteration (PI) System effectiveness Tracking problem
ISSN:	2162-237X, 2162-2388, 2162-2388
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Abstract	The Hamilton-Jacobi-Bellman (HJB) equation serves as the necessary and sufficient condition for the optimal solution to the continuous-time (CT) optimal control problem (OCP). Compared with the infinite-horizon HJB equation, the solving of the finite-horizon (FH) HJB equation has been a long-standing challenge, because the partial time derivative of the value function is involved as an additional unknown term. To address this problem, this study first-time bridges the link between the partial time derivative and the terminal-time utility function, and thus it facilitates the use of the policy iteration (PI) technique to solve the CT FH OCPs. Based on this key finding, the FH approximate dynamic programming (ADP) algorithm is proposed leveraging an actor-critic framework. It is shown that the algorithm exhibits important properties in terms of convergence and optimality. Rather importantly, with the use of multilayer neural networks (NNs) in the actor-critic architecture, the algorithm is suitable for CT FH OCPs toward more general nonlinear and complex systems. Finally, the effectiveness of the proposed algorithm is demonstrated by conducting a series of simulations on both a linear quadratic regulator (LQR) problem and a nonlinear vehicle tracking problem.
AbstractList	The Hamilton–Jacobi–Bellman (HJB) equation serves as the necessary and sufficient condition for the optimal solution to the continuous-time (CT) optimal control problem (OCP). Compared with the infinite-horizon HJB equation, the solving of the finite-horizon (FH) HJB equation has been a long-standing challenge, because the partial time derivative of the value function is involved as an additional unknown term. To address this problem, this study first-time bridges the link between the partial time derivative and the terminal-time utility function, and thus it facilitates the use of the policy iteration (PI) technique to solve the CT FH OCPs. Based on this key finding, the FH approximate dynamic programming (ADP) algorithm is proposed leveraging an actor–critic framework. It is shown that the algorithm exhibits important properties in terms of convergence and optimality. Rather importantly, with the use of multilayer neural networks (NNs) in the actor–critic architecture, the algorithm is suitable for CT FH OCPs toward more general nonlinear and complex systems. Finally, the effectiveness of the proposed algorithm is demonstrated by conducting a series of simulations on both a linear quadratic regulator (LQR) problem and a nonlinear vehicle tracking problem. The Hamilton-Jacobi-Bellman (HJB) equation serves as the necessary and sufficient condition for the optimal solution to the continuous-time (CT) optimal control problem (OCP). Compared with the infinite-horizon HJB equation, the solving of the finite-horizon (FH) HJB equation has been a long-standing challenge, because the partial time derivative of the value function is involved as an additional unknown term. To address this problem, this study first-time bridges the link between the partial time derivative and the terminal-time utility function, and thus it facilitates the use of the policy iteration (PI) technique to solve the CT FH OCPs. Based on this key finding, the FH approximate dynamic programming (ADP) algorithm is proposed leveraging an actor-critic framework. It is shown that the algorithm exhibits important properties in terms of convergence and optimality. Rather importantly, with the use of multilayer neural networks (NNs) in the actor-critic architecture, the algorithm is suitable for CT FH OCPs toward more general nonlinear and complex systems. Finally, the effectiveness of the proposed algorithm is demonstrated by conducting a series of simulations on both a linear quadratic regulator (LQR) problem and a nonlinear vehicle tracking problem.The Hamilton-Jacobi-Bellman (HJB) equation serves as the necessary and sufficient condition for the optimal solution to the continuous-time (CT) optimal control problem (OCP). Compared with the infinite-horizon HJB equation, the solving of the finite-horizon (FH) HJB equation has been a long-standing challenge, because the partial time derivative of the value function is involved as an additional unknown term. To address this problem, this study first-time bridges the link between the partial time derivative and the terminal-time utility function, and thus it facilitates the use of the policy iteration (PI) technique to solve the CT FH OCPs. Based on this key finding, the FH approximate dynamic programming (ADP) algorithm is proposed leveraging an actor-critic framework. It is shown that the algorithm exhibits important properties in terms of convergence and optimality. Rather importantly, with the use of multilayer neural networks (NNs) in the actor-critic architecture, the algorithm is suitable for CT FH OCPs toward more general nonlinear and complex systems. Finally, the effectiveness of the proposed algorithm is demonstrated by conducting a series of simulations on both a linear quadratic regulator (LQR) problem and a nonlinear vehicle tracking problem.
Author	Duan, Jingliang Chen, Jianyu Li, Shengbo Eben Ma, Haitong Li, Jie Cheng, Bo Ma, Jun Lin, Ziyu
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SubjectTerms	Actor critic Algorithms approximate dynamic programming (ADP) Approximation algorithms Artificial neural networks Complex systems Dynamic programming finite-horizon (FH) Hamilton–Jacobi–Bellman (HJB) equation Heuristic algorithms Horizon Linear quadratic regulator Mathematical models Multilayers Neural networks Nonlinear control Nonlinear dynamical systems Optimal control Optimization policy iteration (PI) System effectiveness Tracking problem
Title	Policy-Iteration-Based Finite-Horizon Approximate Dynamic Programming for Continuous-Time Nonlinear Optimal Control
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