1. Introduction
The motor-drive servo turntable is widely used in high-precision tracking radars [
1], radio telescopes [
2], inertial navigation systems [
3], and other equipment that require high tracking accuracy. It is of great importance to design a controller for turntable servo system with accurate tracking performance. In recent years, extensive research on the electromechanical servo systems have been conducted locally and internationally [
4,
5,
6]. Some advanced control strategies, such as robust method [
7], adaptive control [
8,
9], model predictive method [
10,
11], sliding control [
12,
13], and back-stepping control [
14], etc. are often employed together to acquire a better control performance. In order to describe the characteristics of servo turntable system in different working conditions, the switched theory based on Constrained Multi-objective Optimization Problem (CMOP) is introduced in [
15]. In this study, an improved Active Disturbance Rejection Controller (ADRC), mainly given by an improved Tracking Differentiator (TD) and the improved Extended State Observer (ESO), is analyzed in detail.
The ADRC, proposed by Han [
16], as a nonlinear control method with superior performance, are preferred for a wide application in industrial technical fields [
17,
18,
19,
20,
21]. In [
17], an adaptive linear ADRC is designed to acquire strong disturbance rejection performance and to reduce the noise sensitivity for electromechanical servo system. In [
18], it has been proved that the ADRC is with stronger anti-interference ability than PID, Fuzzy-PID, and BP-PID under the condition of 20% maximum control torque disturbance. To solve the output tracking problem of multi-input multi-output (MIMO) system with mismatched uncertainty effectively, a novel nonlinear ESO using nonsmoothed function [
19] is designed to estimate both state of system and uncertain disturbance. In [
20], the influence of input-gain uncertainty on tracking performance of ADRC is investigated based on a second-order plant. The results show that when the input gain is multiplied by a positive factor less than a certain threshold, the closed-loop system remains stable and the tracking error is reduced. For the purpose of improving the tracking accuracy of servo system effectively, an ADRC and feedback linearization-based control algorithm for the high-precision trajectory tracking is analyzed in [
21].
The turntable servo system often fails to achieve accurate tracking due to nonlinear disturbance factors such as friction [
22,
23,
24,
25,
26], backlash [
27], dead-zone [
28], and motor torque fluctuation [
29], etc. Friction is a common non-linear phenomenon, which exists in almost all electromechanical servo system between two contact surfaces with relative motion. Among the existing nonlinear friction models, the LuGre friction model [
22,
23] is most widely employed to describe nonlinear friction phenomenon because of its ability to capture most of the observed frictional behaviors. Nevertheless, some drifting behavior inevitably exists in the LuGre friction model. To deal with this challenge effectively, another single state friction model was proposed by Dupont in 2002, which is called the Elastoplastic (EP) friction model [
24]. It could be concluded that the EP friction model can overcome this drawback by incorporating a purely elastic area. In [
26], the dynamic parameters of EP friction model are identified and used for feed-forward compensation based on experimental measurement results.
However, it should not be ignored that most of the existing friction models are based on the deformation of rigid bristles, while the average deformation of rigid bristles is often too small to be measured. In this paper, to describe nonlinear friction disturbance more accurately, we consider regarding the immeasurable part of the EP friction model as a new state variable to constitute a novel ESO, which is the core of ADRC controller design.
It is not difficult to find out that the ESO used in the aforementioned studies were generally designed on the basis of traditional nonlinear function, e.g., fal. Although they are continuous, it is hard to guarantee the derivability on the intervals, and they have an inflection point around the origin, which may affect the control performance of ADRC. As a result, the ADRC controller can be improved by optimizing its structure, which is introduced in
Section 3.
Stability provides a prerequisite for the normal operation of turntable servo system. In [
30], the stability of traditional PID controller is investigated, then the extended PID is proposed to acquire more strong robustness. Results show that the extended PID controller is able to stabilize the nonlinear uncertain systems semi globally. In [
31], for the purpose of improving the tracking performance and robustness of robotic manipulators, the neural network (NN) algorithm is introduced to model predictive control (MPC). We have learned from previous studies that nonlinear ESO can obtain better performance than linear ESO [
32,
33,
34]. However, the stability analysis of nonlinear ESO is a challenging work. Most of the existing methods, such as circle criterion method [
35], describing function method [
36], are carried out in frequency domain. In this paper, the improved ADRC system is transformed into a Lurie system, then the extended circle criterion is employed to analyze the stability in frequency domain.
The main contributions of this study are as follows: (1) the immeasurable part of the Elastoplastic friction model is extended to a new state to achieve real-time estimation and compensation; (2) the fuzzy rules are introduced to realize intelligent tuning of the improved ESO gains based on the adjustable parameter systematic pole placement method; and (3) the designed control system above is transformed into Lurie system, then the extended circle criterion is adopted to analyze stability of the improved ADRC system.
The remaining part of this paper is organized as follows. The dynamic model of dual-axis servo tracking turntable system is shown in
Section 2. In
Section 3, the improved ADRC is designed based on improved TD and improved ESO. The gains of improved ESO are adjusted intelligently by fuzzy rules. In
Section 4, the improved ADRC system above is transformed into a Lurie system, then the stability of designed system is analyzed by extended circle criterion. Simulation analysis and experimental results are given in
Section 5 and
Section 6, respectively. The results illustrate that the improved ADRC control scheme can improve the tracking performance of dual-axis servo turntable system. Finally, some concluding remarks are drawn in
Section 7.
5. Simulation Results and Analysis
After completing the stability analysis of turntable tracking servo system, some simulations are conducted to demonstrate the performance of improved ADRC scheme. The second order differential equation model of turntable tracking servo system is given as Equation (29) in
Section 4. In order to compensate for the influence of nonlinear friction disturbance on the tracking performance better, a friction observer is designed based on the improved ESO, which regards the immeasurable part of EP friction model as a new state. In our previous work, the nonlinear friction parameter identification experiments using genetic algorithm (GA) is analyzed in [
23]. Similarly, in this study, the parameter identification results of EP friction model based on GA are listed in
Table 3.
In this section, a comparative simulation experiment based on improved ADRC, PD controller and traditional ADRC is analyzed to verify the tracking performance of the proposed control strategy. The same parameters are selected for the improved ADRC and traditional ADRC. We set the sinusoidal signal
(rad) as the expected trajectory signal.
Figure 9 gives the estimated results and error curve of friction observer. The angle and angle velocity tracking result of servo turntable is shown in
Figure 10 and
Figure 11, respectively. For the purpose of comparing tracking effects of different controllers further, corresponding tracking error indexes are given in
Table 4 and
Table 5.
As is shown from the above tables, the maximum angle velocity tracking error of improved ADRC is 3.5068 rad/s less than that of PD control method and 2.3979 rad/s less than that of traditional ADRC. The MEAN of angle tracking error of improved ADRC is 0.00064 rad, which is far lower than the corresponding values of PD and ADRC. The root mean square error (RMSE) of improved ADRC is also with the lowest value. It is not hard to find out that the improved ADRC is with better performance and stronger robustness than PD and traditional ADRC.
Finally, for the purpose of verifying the effectiveness of the improved ADRC against parameters uncertainties further, numerical simulation experiments are carried out for unknown parameters. The function
in Equation (29) is unknown here, which is chosen as plus minus 20% of the true value for experiments. The test signals are selected as
(rad) and
(rad), respectively. Simulation results are shown in
Figure 12, which illustrate the robustness and effectiveness of the proposed scheme.
7. Conclusions
In this study, an improved ADRC, which applies the improved TD and improved ESO, is proposed to realize high-precision tracking control of dual-axis servo tracking turntable. The mathematical model is established first, and the Elastoplastic (EP) friction model is adopted to describe friction nonlinear disturbance. Secondly, considering the properties of smooth and monotonically increasing, an improved TD is given based on hyperbolic tangent function. Thirdly, the improved ESO is analyzed in detail, which adopts a new nonlinear function. The immeasurable part of EP friction model is extended to a new state of improved ESO design. Additionally, the fuzzy algorithm is employed to tuning observer gains intelligently. Finally, the convergence of improved ESO is confirmed, and the improved ADRC system is transformed into a Lurie system, the stability of system is analyzed by extended circle criteria. It can be concluded that the maximum rotation tracking error of improved ADRC is 1.9598 rpm and 3.1611 rpm less than that of traditional ADRC for azimuth and pitch rotation, respectively. Simulation and experiment results demonstrate the effectiveness and robustness of the proposed control scheme.
To further improve the tracking and anti-disturbance performance of the turntable servo system in the future, one can focus on the compensation of other nonlinear disturbances, such as backlash, dead-zone and motor torque fluctuation, etc.