Active disturbance rejection control for electric cylinders with PD-type event-triggering condition☆
Introduction
Servo systems are important elements of industrial manufacturing involving the flight systems (Xue et al., 2015), shaking tables (Plummer, 2016), robotic systems (Ruiz-Carcel & Starr, 2018), water tank systems (Chen, Xue, & Huang, 2019), and forging manipulators (Ding, Deng, Xia, & Duan, 2017), and have received a lot of attention in the control community recently (Xue, Madonski, Lakomy, Gao, & Huang, 2017). Different control strategies of servo systems have been investigated for high performance in control accuracy and response speed, e.g., proportional–integral–derivative (PID) control (Puchta, Siqueira, & dos Santos Kaster, 2019), fuzzy logic control (Chaoui, Khayamy, & Aljarboua, 2017), sliding model control (Matraji, Al-Durra, Haryono, Al-Wahedi, & Abou-Khousa, 2018), adaptive robust control (Mendoza-Mondragon, Hernandez-Guzman, & Rodriguez-Resendiz, 2018), and active disturbance rejection control (ADRC) (Guo, Xue, & Hu, 2016). Proposed by Han (2009) and his co-workers (Zheng, Dong, Lee, & Gao, 2009) with a number of successful industrial applications, ADRC inherits from PID (Zhong, Huang, & Guo, 2020). Compared with the model-based control, little prior knowledge is required in the ADRC controller design, which maintains the practical advantage of PID. As the ignored dynamics and disturbances are treated as total disturbances, the ADRC scheme has an advantage in addressing the control problem of the nonlinear systems with disturbances.
With the development and application of communication networks (Wang, Wang, Liu, & Gu, 2015), the signals of servo systems are transmitted via wired or wireless communication networks (Liu, Shan, Chen, & Li, 2017). It becomes unnecessary to update control signals periodically or frequently in many applications (e.g., battery-powered robotic systems and embedded systems with limited communication bandwidth), due to the fact that (1) the reduction of communications helps to reduce data congestion and rate of packet loss, (2) reducing radio frequency wireless communication helps to save energy, which is meaningful for improving the battery life of wireless servo systems, (3) with the reduced usage of communication and computation resources, additional functions such as system status monitoring can be implemented. This motivates the need of adopting event-triggered control approaches for servo systems. Pioneered by the works in Åstrom and Bernhardsson (1999), numerous results have been obtained in event-based control (Årzen, Dec. 1999) and estimation (Shi, Shi, & Chen, 2016). For example, Tabuada (2007) revisited the problem of scheduling stabilizing control tasks on embedded processors and proposed an event-triggered real-time scheduling of stabilizing control tasks. Heemels and Donkers (2013) studied observer-based controllers for linear systems and proposed advanced event-triggering mechanisms (ETMs) that will reduce communication in both the sensor-to-controller channels and the controller-to-actuator channels. Li and Shi (2014) applied the event-triggered approach to model predictive control (MPC) for continuous-time nonlinear systems combined with disturbances. In Shi, Xue, Zhao, Wang, and Huang (2017), an event-triggered active disturbance rejection control approach was proposed to achieve position tracking of direct current (DC) torque motors. The authors considered event-triggered extended state observer (ESO) design for a continuous-time nonlinear system with uncertainty and disturbance in Huang, Wang, Shi, and Shi (2018). The event-triggering conditions of the results above are proportional-type. As the overall model has the distinguishing feature of asynchronous behavior with respect to different agents, an integral-type event-triggering condition is proposed in Wang, Mu, and Shi (2017). The integral-type event-triggering condition records the average effect of asynchronous sampling/broadcasting of information in the most recent event-detection period to some extent. However, research interests on the predictability of the event-triggering condition have rarely been seen for servo systems.
In this work, we consider a position tracking problem for electric cylinders used in our recently developed wheel-legged robotic system. In particular, each leg of this system is composed of six electric cylinders, which are controlled based on data from more than twelve sensors transmitted through a controller area network (CAN) and a user datagram protocol (UDP) network. To achieve satisfactory performance under limited communication resources, we investigate an event-triggered control approach for electric cylinders. On the one hand, the stable operation of the robotic system requires the reference signal of electric cylinders to change over a large range and a high-frequency band, which makes it difficult to design an event-triggering condition that can reduce triggering rate without compromising tracking performance. To deal with this challenge, a proportional–derivative-type (PD-type) event-triggering condition is designed. As the absolute value and the trend of the tracking error are taken into account, the event-triggering condition is predictive, which is an efficient improvement for tracking accuracy and response speed of the controller. Meanwhile, the triggering rate is effectively decreased when the tracking error is reducing even the absolute value is more than the set. On the other hand, although event-triggered PID algorithms were introduced in the literature (Årzen, Dec. 1999, Durand and Guerrero-Castellanos, 2015), the stability of the resultant closed-loop control system was not addressed. Meanwhile, as the electric cylinders use roller-screws to realize the linear motion of inertia loads when they drive the robotic system, the sliding friction (Yao, Deng, & Jiao, 2015), which is nonlinear related to the sliding velocity, and torque disturbance (Yang, Chen, Li, Guo, & Yan, 2017) are nonnegligible in such systems. These challenges motivate us to explore the convergence properties of event-triggered ADRC, which can be viewed as a nonlinear PID controller (Zhong et al., 2020) and has the advantages of dealing with internal uncertainties (Chu, Sun, Wu, & Sepehri, 2018) and external disturbances (Behzad & Amin, 2017). Since the Coulomb friction has defects in sliding friction analysis, the results in Shi et al. (2017), where the Coulomb friction model is the only concern in dynamic modeling for the DC motor control system, are not applicable to the electric cylinder control systems. As the LuGre friction model (Rubaai, Castro-Sitiriche, & Ofoli, 2008) takes viscous friction, Coulomb friction, static friction, and Stribeck friction into account, it can represent the most friction behaviors of the electric cylinders. The main contributions of our work are summarized as follows:
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To improve the predictability of the event-trigger, an improved event-triggering condition that considers both the size and the trend of tracking error is proposed. The absence of the Zeno phenomenon is guaranteed for this improved event-triggering condition. The stability of the ADRC-controlled closed-loop system is proved.
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The developed results are evaluated on an electric cylinder control system that can verify the control performance for the actuators of our quadruped robotic system. Position tracking of the system under different load conditions and reference inputs is obtained, which verifies the effectiveness of the proposed approach.
The rest of this paper is organized as follows. The dynamics analysis of an electric cylinder is proposed in Section 2. The event-triggered ADRC of the electric cylinders is shown in detail in Section 3. Experiment results are presented to evaluate the actual performance of the system in Section 4. The paper concludes with a brief discussion in Section 5.
Table 1 provides necessary explanation of the notations and alphabetic symbols used in the rest sections.
Section snippets
Nonlinear modeling and problem formulation
The basic structure of electric cylinders can be shown in Fig. 1. , where , , , and respectively represent the electromagnetic torque, friction torque, radial torque, and load force on the cylinder. As represents the lead of the screw, we have .
The torque equation holds where and are the inertia and rotation velocity of the screw.
The model of PMSM can be decoupled by using a vector control algorithm, such that it is easy to design the controller with
Event-triggered ADRC of electric cylinders
In this section, an event-triggered ADRC scheme for electric cylinders is introduced in detail. The boundedness of the event-triggered ESO and the position tracking error is guaranteed in theory.
The block diagram of the controller is as shown in Fig. 2. From (22), the control signal can be rewritten as For notational brevity, we define as From (7), (9), (29), we obtain that where
Experiment results
In this section, an electric cylinder control system shown in Fig. 3 is used to evaluate the proposed event-triggered control method. The system simulates the working state of electric cylinders during the movement of our recently developed quadruped robotic system. It mainly consists of a drive circuit, a control computer, two monitors and an execution and loading mechanism. The control method can be programmed by the control computer. The drive circuit can convert digital control commands
Discussion
In this paper, an event-triggered ADRC controller is investigated to control an electric cylinder, and a PD-type event-triggering condition considering both the size and trend of tracking error is proposed. With the proposed ET-ADRC controller, the observation error of the ESO and the tracking error of the system can be guaranteed to be asymptotically bounded. The performance of the controller is extensively evaluated through comparative experiments on an electric cylinder control system for
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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