1 Introduction
Trajectory tracking is a fundamental problem in control of mechatronic systems, e.g., robot manipulators and piezoelectric actuators (PEAs). In most cases, there are uncertainties and external disturbances, such as friction, sensor noise and variations of payload in the operations of these systems with nonlinear dynamics. A sequence of control methods were proposed to solve these issues, such as adaptive control [
1], sliding mode control [
2], learning control [
3,
4], and neural network control [
5,
6]. Moreover, mechatronic systems with imperfections is an universal control problem. In [
7], a control strategy to ensure the optimal working conditions was proposed, which focused on the effects of using chaotic vibrational signals to excite the hidden dynamics of the imperfect system. In [
8], the authors focused on a paradigmatic example of imperfect electromechanical structure and developed a control method to ensure coil rotation based on the excitation of the hidden dynamics induced by imperfections, characterizing its influence on the characteristics of the control signal and the power provided to the structure. Imperfections also play an important role in the realization of robust chaos generators based on simple circuits. In [
9], a strategy for estimating hidden dynamics parameters was designed and synchronization of imperfect chaotic circuits was achieved. Compared to active research on advanced control approaches in academia, classical linear controllers such as PID control are still playing a crucial part in industries for the sake of implementation simplicity. However, it is well known that PID controller behaves poorly on complex trajectories and on systems with nonlinear dynamics [
10]. Therefore, it is interesting to develop a control approach that is built on top of the off-the-shelf linear controllers but improves the tracking performance. Such a control approach has two significantly favorable features. First, most of the controllers provided by the manufacturers do not allow the users to modify low-level position controller but provides access to tunable parameters and a reference trajectory. In a position control task, the desired trajectory is the trajectory predefined for the mechatronic system to actually track. The reference trajectory can be obtained by using the trajectory generator module to modify the desired trajectory and is used as the input signal of the closed-loop mechatronic system. Second, without modifying the available control architecture, the system stability can be in general ensured. In this regard, many state-of-the-art controllers that design the control input such as [
11‐
19] are not applicable.
To cope with disturbances and imperfections of mechatronic systems, a lot of research effort has been made on modifying the reference trajectory to improve the tracking performance on top of an available feedback control system. A large group of these works is iterative learning control (ILC) that improves the performance of trajectory tracking with repetition of a same task and using knowledge from previous iterations [
20‐
22]. Although learning convergence can be proved in rigor, the information about the system learned by ILC cannot be transferred to another task, similar to adaptive control [
23].
As mechatronic systems generally have complicated dynamics, which are influenced by uncertainties [
24], there is ample motivation to investigate the effectiveness of machine learning in the control of mechatronic systems [
25]. Some researchers were attracted by the excellent capabilities of deep neural networks (DNNs) in function approximation and thus revisited the idea of constructing an NN model for mechatronic systems [
26], especially the inverse compensation control based on NN model [
27‐
29]. In [
30], a polynomial fitting model based on NN was proposed to describe the inverse dynamics of hysteresis in PEA. As a feedforward compensation module, the model is combined with a single neurogenic adaptive proportional integral differential controller to reduce the trajectory tracking error caused by hysteresis in piezoelectric drive mechatronic system. Different from the traditional control framework based on NN inverse model to approximate the open-loop dynamics and modify the control signal of the plant, the offline learning control framework proposed in this paper uses DNN to approximate the inverse dynamic characteristics of the closed-loop mechatronic system and uses the trained DNN as a trajectory generator to modify the reference trajectory, so that the tracking error can be compensated for in advance without changing the structure and stability of the baseline controller. Although the offline learning method can approximate the inverse dynamics of the closed-loop mechatronic system, the DNN model is still subject to modeling error, and the tracking control accuracy needs to be further improved based on online learning.
The online learning control framework based on iterative learning in this paper is suitable for repetitive tasks and can suppress unknown uncertainties. Compared to DNN, single-hidden-layer radial basis function neural networks (RBFNNs) have the advantages of simple structure and high computational efficiency. RBFNN is simple to implement in real time, and its learning convergence and the resultant closed-loop system stability can be strictly analyzed [
31,
32]. The control schemes based on RBFNN in the closed-loop control system mainly include supervisory control, model reference adaptive control, self-tuning control, etc. In [
33], for a class of nonlinear systems with unknown parameters and bounded disturbances, RBFNN combined with single-parameter direct adaptive control was designed to overcome the problems caused by unknown dynamics and external disturbances in nonlinear systems. The traditional control methods based on RBFNN modify the control signal of the controlled plant, and the parameters of the RBFNN need to be updated all the time [
34,
35]. Different from these works, the online learning control framework based on RBFNN proposed in this paper uses iterative method to update the parameters of the RBFNN and modify the reference trajectory until the tracking error is reduced below a target threshold. The advantages of the method proposed in this paper are as follows: (1) For the repetitive trajectory, the repetitive interference and error in the system can be suppressed. (2) It does not change the structure of the baseline controller and will not affect the stability of the closed-loop system. Thus, it can be easily applied to commercial control systems.
Based on the above discussions, this paper will investigate reference trajectory modification for mechatronic systems, by integrating a DNN for offline learning and a single-layer RBFNN for online learning. First of all, DNN is offline-trained to approximate the inverse dynamics model of mechatronic systems, and the trained DNN is used to obtain the modified reference trajectory as the input of the closed-loop mechatronic system or further modified by online learning of RBFNN. Then, we propose the online learning control framework-based RBFNN, and combined with Lyapunov function, we design the learning law of RBFNN and prove the stability of the system. The offline NN learning method learns the inverse dynamics of the closed-loop system and speeds up the online learning, which can compensate for tracking error in advance. The online learning NN method can deal with uncertainties and disturbances and thus achieve precise trajectory tracking control.
The main contribution of this paper is the hybrid offline/online learning control framework, which combines complementary advantages of DNN and a single-layer RBFNN. On the one hand, we propose the offline learning control framework with DNN as a reference trajectory generator, which is transferrable and can be used to conduct a new tracking task. Offline learning can provide an initial reference trajectory for online learning and speed up the convergence of RBFNN parameters; on the other hand, we propose online learning control framework with RBFNN to iteratively modify the reference trajectory generated by DNN as the input signal of the closed-loop mechatronic system, and prove its convergence.
The remaining structure of this paper is as below: Section
2 shows the system dynamics and transforms the control problem into mathematical models and introduces the proposed tracking control method based on hybrid offline/online NN. Sections
3 and
4 elaborate the processes of online and offline learning, respectively. Section
5 presents the results of the experiments. Section
6 concludes this work.
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