Abstract
In this article, a novel continuous-time optimal tracking controller is proposed for the single-input-single-output linear system with completely unknown dynamics. Unlike those existing solutions to the optimal tracking control problem, the proposed controller introduces an integral compensation to reduce the steady-state error and regulates the feedforward part simultaneously with the feedback part. An augmented system composed of the integral compensation, error dynamics, and desired trajectory is established to formulate the optimal tracking control problem. The input energy and tracking error of the optimal controller are minimized according to the objective function in the infinite horizon. With the application of reinforcement learning techniques, the proposed controller does not require any prior knowledge of the system drift or input dynamics. The integral reinforcement learning method is employed to approximate the Q-function and update the critic network on-line. And the actor network is updated with the deterministic learning method. The Lyapunov stability is proved under the persistence of excitation condition. A case study on a hydraulic loading system has shown the effectiveness of the proposed controller by simulation and experiment.
Highlights
Accurate tracking control has drawn great research interests in a number of application fields.1–3 Optimal control deals with problems of minimizing the prescribed objective function in the infinite or finite horizon
The optimal control policy is obtained by the method of deterministic learning.18,19. Different from those off-line deterministic policy gradient (DPG) methods, the deterministic learning method in this study enables an online policy update
We developed an adaptive optimal controller based on the deterministic learning technique
Summary
Accurate tracking control has drawn great research interests in a number of application fields. Optimal control deals with problems of minimizing the prescribed objective function in the infinite or finite horizon. The optimal control policy is regulated as a state feedback according to the gradient of the value function.. The integral compensation and feedforward part are introduced in the optimal controller. The optimal tracking control problem (OTCP) of the augmented system is formulated for minimizing the performance function in the infinite horizon. An integral of the linear quadratic function is calculated to obtain the Bellman error and update the critic weights. The OTCP of the system can be solved on-line with completely unknown dynamics, by employing Q-function approximation and deterministic learning method. The traditional optimal regulation method obtains a proportional–derivative (PD)-type controller with feedback only, which may cause steady-state error under uncertain dynamics. According to equation [20], the Bellman error with respect to the weights of the critic network W^ c can be written as. The objective function of the critic network can be written as EB e2B ð25Þ
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