Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning

Jingang Zhao; Chi Zhang

doi:10.3934/jimo.2020030

Article Contents

2021, Volume 17, Issue 3: 1471-1483. Doi: 10.3934/jimo.2020030

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Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning

Jingang Zhao^, and
Chi Zhang

School of Automation, Beijing Institute of Technology, Beijing, 100081, China

^* Corresponding author: Jingang Zhao
^* Corresponding author: Jingang Zhao

Received: February 28, 2019

Revised: August 31, 2019

Early access: February 2020

Published: May 2021

The first author is supported by International Graduate Exchange Program of Beijing Institute of Technology; This paper is supported by the National Natural Science Foundation of China grant 61673065

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Abstract

This paper investigates finite-horizon optimal control problem of completely unknown discrete-time linear systems. The completely unknown here refers to that the system dynamics are unknown. Compared with infinite-horizon optimal control, the Riccati equation (RE) of finite-horizon optimal control is time-dependent and must meet certain terminal boundary constraints, which brings the greater challenges. Meanwhile, the completely unknown system dynamics have also caused additional challenges. The main innovation of this paper is the developed cyclic fixed-finite-horizon-based Q-learning algorithm to approximate the optimal control input without requiring the system dynamics. The developed algorithm main consists of two phases: the data collection phase over a fixed-finite-horizon and the parameters update phase. A least-squares method is used to correlate the two phases to obtain the optimal parameters by cyclic. Finally, simulation results are given to verify the effectiveness of the proposed cyclic fixed-finite-horizon-based Q-learning algorithm.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The flow chart of Algorithm 1

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Figure 2. Initial system state $x_0$ are randomly selected from a compact set $\Omega : = \{-1\le {{x}_{1}},{{x}_{2}}\le 1\}$

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Figure 3. The convergence process of $\hat W$

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Figure 4. The trajectories of system states

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Figure 5. The optimal control input $u$

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Figure 6. The convergence process of $\hat W$

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Figure 7. The trajectories of system states

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Figure 8. The optimal control input $u$

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