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An adaptive backstepping sliding mode controller combined with a nonlinear disturbance observer is designed for trajectory tracking of the electrically driven hybrid conveying mechanism with mismatched disturbances. A nonlinear disturbance observer is constructed for estimation and compensation of the mismatched and matched disturbances. Then, a hybrid control scheme is designed by combining the adaptive backstepping sliding mode controller and the mentioned observer. The Lyapunov candidate functions are utilized to derive the control and adaptive law. According to the simulation and experimental results, superior tracking performance could be obtained through the presented control scheme compared with conventional backstepping sliding mode control. Meanwhile, the presented control scheme can effectively reduce the chattering problem and improve tracking precision.

The performance of the hybrid electrocoating conveying mechanism as the fundamental equipment of the coating process determines the quality and production efficiency of the electrocoating [

Many control strategies have been proposed to reject disturbances for parallel and hybrid mechanism, such as proportional integral derivative (PID) control [

Backstepping as an effective control strategy for rejecting matched and mismatched disturbances is a step-by-step recursive design method. However, the robustness could not be attained through the conventional backstepping control method. Recently, SMC has been widely used due to its fast response, easy implementation, insensitivity to parametric uncertainty, and disturbances [

In recent years, nonlinear disturbance observer (NDO) has received more and more attention [

In this study, a novel hybrid control scheme consisting of backstepping, SMC, and NDO is constructed for the trajectory tracking problem of HCM with matched and mismatched disturbances. In the design procedure, firstly, NDO is utilized to estimate matched and mismatched disturbances. Secondly, the backstepping method is used to design control laws. Then, in order to improve the robustness of the system, the mismatched disturbance estimation is introduced into the virtual control laws to compensate for the mismatched disturbance. Finally, the sliding mode control is introduced in the last step, and an adaptive law is designed to estimate the switching gain of the sliding mode control. The Lyapunov candidate functions of every step are designed to ensure the asymptotical stability of the whole system. The proposed an adaptive backstepping sliding mode control based on nonlinear disturbance observer (ABSMC + NDO) has two main advantages: First, the NDO is utilized to compensate for the mismatched disturbances in the virtual control law. Second, it not only alleviates the chattering problem but also improves tracking precision.

The organization of this paper is illustrated as follows. The dynamic model of the hybrid conveying mechanism is introduced in Section

The hybrid electrocoating conveying mechanism is shown in Figure

The mechanism schematics of HCM.

The Lagrangian–Euler method is employed to extract the dynamic of the HCM. The obtained dynamic model could be described as follows [

The model of the motor could be written as follows [

According to the physical characteristics of the motor, the torque vector

By combining the uncertainties and disturbances, the following lumped disturbances are obtained:

Considering the dynamic model of HCM given in (

The dynamic properties of the HCM are given in the following:

Consider that the state variables are defined as follows:

Now, the dynamics (

It could be concluded from equation (

In this section, an NDO-based adaptive backstepping SMC is developed for the HCM (

Schematic diagram of ABSMC + NDO for the hybrid conveying mechanism.

Consider that the dynamics (

Assume that the disturbances

Consider that the disturbance observer error

According to equation (

According to equation (

Consider a Lyapunov function as follows:

According to equations (

It could be concluded from (

In this subsection, the recursive procedure is employed to design a hybrid controller.

Define

Define a Lyapunov candidate function as follows:

Differentiating from both sides of (

Define

Substituting equations (

If

According to equations (

Consider the Lyapunov candidate function as follows:

Differentiating from both sides of (

According to the property of (

Define

Substituting equations (

Consider that the estimation error of the mismatched disturbances

According to (

According to the generic inequality

If

According to equations (

Define the sliding surface

Define a Lyapunov candidate function as follows:

Differentiating from both sides of (

Now, the following NDO based backstepping SMC could be designed:

Since obtaining the upper bound of

According to the adaptation mechanism (

Consider the hybrid conveying mechanism (

Consider a Lyapunov candidate function as follows:

By differentiating from both sides of (

Inserting equations (

Since

Substituting equations (

In order to evaluate the efficiency of the proposed ABSMC + NDO method, the numerical simulations are performed in this section. The parameters of the mechanism and motor for simulations are given in Tables

The mechanism parameters for simulation.

Parameter | Value |
---|---|

The motor parameters for simulation.

Parameters | Slider motors | Active wheel motor |
---|---|---|

Torque constant | ||

Inductance | ||

Resistance | ||

Antielectric torque constant | ||

Motor gear ratio |

In the simulations, both the suppression ability of the external disturbances and the robustness to system parameter uncertainties are considered. In practical applications, since the fluid resistance is random and large during the movement of the white body in the electrophoresis liquid, the control performance of the conveying mechanism may be degraded. Consider that the system parameters are deviated 10% from their nominal values. The external disturbances such as the fluid resistance, friction, and voltage disturbance are defined as follows:

The fluid resistance is defined as follows:

To illustrate the effectiveness of the constructed hybrid control scheme (ABSMC + NDO), the backstepping control (BC) and adaptive backstepping sliding mode control (ABSMC) are utilized for comparison analysis. For better comparison and analysis, the parameters of each controller are adjusted according to their optimal control performance. The controller gains of each control scheme are given as follows:

BC

ABSMC

ABSMC + NDO

The following two cases are considered in the simulation.

Compared with the backstepping control, the disturbances of the system are added to the system at

Each joint position tracking error.

Compared with the adaptive backstepping sliding mode control, the disturbances of the system are added to the system from the beginning. Figures

Each joint position tracking error.

Control input of each motor.

Estimated disturbances for each joint.

Estimated disturbances for each motor.

RMSEs for each joint.

Control scheme | ABSMC | ABSMC + NDO | |
---|---|---|---|

RMSE | First slider | 27.1 × 10^{−5} | 3.4 × 10^{−5} |

Second slider | 27.1 × 10^{−5} | 3.4 × 10^{−5} | |

Active wheel | 7.8 × 10^{−3} | 6.1 × 10^{−4} |

To further verify the effectiveness of the proposed control strategy, we use the HCM prototype shown as Figure

The hybrid conveying mechanism prototype.

The reference trajectory employed in the simulation is also utilized in the experiment. The experiment results are shown in Figure

The experimental results of each joint.

In this paper, a hybrid control scheme is constructed for trajectory tracking control of the electrically driven hybrid conveying mechanism with mismatched disturbances and parameter uncertainties. The main contribution of this paper is that the proposed controller not only compensates for mismatched disturbances but also effectively reduces sliding mode chattering based on NDO. The novel hybrid control scheme combines the NDO, backstepping, and SMC to reject mismatched disturbances. The Lyapunov stability theory is used to ensure the closed-loop system stability and tracking on the given trajectory. Simulation and experimental results are provided to demonstrate the validity of the proposed control scheme. Designing the adaptive NDO gain and the optimal disturbance rejection strategy could be considered as the future work.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was funded in part by National Natural Science Foundation of China, grant nos. 51375210 and 61503162, and Blue Project of Jiangsu Province.