Abstract
An adaptive high-order neural network (HONN) control strategy is proposed for a hysteresis motor driving servo system with the Bouc-Wen model. To simplify control design, the model is rewritten as a canonical state space form firstly through coordinate transformation. Then, a high-gain state observer (HGSO) is proposed to estimate the unknown transformed state. Afterward, a filter for the tracking errors is adopted which converts the vector error into a scalar error . Finally, an adaptive HONN controller is presented, and a Lyapunov function candidate guarantees that all the closed-loop signals are uniformly ultimately bounded (UUB). Simulations verified the effectiveness of the proposed neural network adaptive control strategy for the hysteresis servo motor system.
1. Introduction
The servo motor system has been widely utilized in industry and military (vehicle systems [1–3], robotic systems [4–7], radar systems [8–10], and nonlinear turn table systems [11–15]). But hysteresis nonlinearity exists in these motor driving servo systems. The existence of the hysteresis can increase the control difficulty and decrease the controller performance; even more seriously, it can damage the stability of the motor servo system. Since the accuracy requirement becomes more and more strict today, hysteresis nonlinearity must be dealt with under a higher precision in these motor servo systems.
Hysteresis nonlinearity is a classic nonlinearity, and it has been investigated for many years. Plenty of results are obtained, and the hysteresis model can be roughly divided into two classes: the physical-based model and the mathematical-based model. The typical hysteresis physical-based model is the Jiles-Atherton (J-A) model [16–18]. The physical model is constructed with the physical parameters of the hysteresis systems. Different from the physical-based model, the mathematical-based model has no connection with real physical parameters, and it is described by mathematical arguments. The mathematical-based model includes the Preisach model [19–21], Bouc-Wen model [22–24], and Prandtl-Ishlinskii (P-I) model [25–27].
The Preisach model is the earliest mathematical-based model and has been used for system identification and control. Gao et al. [19] proposed new identification approaches and a controller for an unknown-order Hammerstein system where hysteresis nonlinearity was modeled through the Preisach model. The Hankel matrix and blind identification estimated the order and the linear section. Afterward, a new triangle matrix approach was proposed to identify the discrete Preisach model. Finally, a composite controller was designed for the hysteresis system. In [20], by utilizing the so-called Preisach plane, the Preisach model was reexpressed into a control-oriented form, in which the input signal was explicitly expressed. And a prescribed adaptive control approach was adopted to ensure the performance.
The P-I model is another popular hysteresis mathematical-based model. Many researchers focus on the P-I model investigation. Zhang et al. [26] proposed a fuzzy approximator for unknown time-delay asymmetric hysteresis nonlinear systems with a shifted Prandtl-Ishlinskii (ASPI) model through an adaptive dynamic surface control scheme, and the Krasovskii functionals have been adopted without the inverse ASPI model. An asymmetric hysteresis P-I model was presented in [27] for piezoelectric actuators, and two different hybrid controllers were compared to verify the control strategy where the proposed approach had better tracking precision.
The Bouc-Wen model was firstly proposed by Bouc in 1971 [28]. In 1976, Wen [29] expanded the model into a mathematical model. In recent years, the Bouc-Wen model has attracted more and more researchers’ interest. Ahmad [22] proposed a new robust controller with two degrees of freedom for a Bouc-Wen hysteresis system. The proposed controller had better performance and robustness. In [23], a Bouc-Wen model was identified by the bat-inspired algorithm for a piezoactuated hysteresis system. A sliding mode control method was designed suppressing the hysteresis nonlinearity. Comparing with another two mathematical-based hysteresis models, the Bouc-Wen model has more parameters, but it only needs differential calculation. Then, it is easier to implement. Thus, the Bouc-Wen hysteresis model should attract more attention for hysteresis investigations.
For nonlinear systems, the HONN has been focused on for many years [30–32]. Ge et al. [30] investigated a HONN-adaptive control strategy for the MIMO systems. A HONN controller with adaptive laws was designed, and the tracking errors were convergent to an adjustable compact set. In [33], a HONN control was designed by the back-stepping. However, to the author’s knowledge, HONN is rarely utilized to approximate the Bouc-Wen hysteresis nonlinearity for control systems.
This paper investigates an adaptive HONN control for hysteresis nonlinearity motor driving servo system with the Bouc-Wen model. The system with hysteresis is first reformulated as a canonical state space form, where the unknown dynamics are all lumped together to simplify the control design. In order to estimate the unknown transformed states, a HGSO is proposed and then the adaptive HONN controller is designed. Next, to simplify the controller design, a filter will be adopted to transform the tracking error into a scalar error , and a Lyapunov function candidate guarantees all the signals bounded in the closed-loop system. Finally, simulation results will demonstrate the effectiveness of the HONN-adaptive controller for hysteresis nonlinearity with the Bouc-Wen model of the motor driving servo system.
2. Problem Formulation
2.1. Hysteresis Servo Motor Driving System with the Bouc-Wen Model
Based on the literature [34], considering the hysteresis servo motor driving system as where and describe angle positions of the motor and load, respectively; and are their velocities, respectively; and represent the moments of inertia of the motor and load, respectively; is the viscous friction coefficient of the motor; is the input torque of the motor; is the friction torque of the load; and represents transmission torque, which can be formulated with the Bouc-Wen hysteresis model as where , , , and . The structure of the hysteresis servo motor system is illustrated in Figure 1.
In the Bouc-Wen model, the parameter decides the hysteresis shape and hysteresis amplitude, the parameter denotes the smoothness of the slope, and means the direction of the hysteresis nonlinearity. If the input is chosen as , the Bouc-Wen parameters are selected in Table 1. Then, the images of the Bouc-Wen model are illustrated by Figures 2(a) and 2(b).
(a) The Bouc-Wen positive curve
(b) The Bouc-Wen negative curve
2.2. Hysteresis Servo Motor Driving State Space Model
The hysteresis servo motor driving system (1) will be rewritten as a state space status to facilitate the controller design. The state space variables can be defined as
Then, the hysteresis servo motor driving system (1) is rewritten as where , , and .
Considering that represents angle positions of load, which is the output of the nonlinear system, we, therefore, have .
In order to simplify the design of the control strategy, the state space model (4) should be rewritten in a pure-feedback form. In view of reference [35], Na et al. proposed a new coordinate transformation to transform the pure-feedback system into a canonical system. Similar to the approach of [35], we also adopted a coordinate transformation to transform (4) into a canonical state space form.
Define the alternative state variables as
According (4) and (5), it has ; then, the follow equation can be deduced as
As an analogy, the system (4) is obtained as where .
3. High-Gain State Observer Design
Considering (7), the high-gain state observer (HGSO) is proposed as where are the designed parameters; is the estimation of , and and are the estimation of and , respectively. Based on the HGSO, the following Lemma is hold.
Lemma 1. Considering the servo motor system (4), it transformed into the canonical state space form (7) and then define the HGSO as (8); then, the HGSO can approximate the system with designed .
Proof 1. Define . When (7) subtracts (8), one has Rewriting (9) as where and all designed parameters should be properly chosen so that the polynomial is Hurwitz. Thus, for the matrix , select a positive definite symmetric matrix ; there exists a positive definite symmetric matrix such that the following inequality holds: Select the Lyapunov candidate for the state observer estimate errors as follows: Considering (10), (11), and (12), the derivative of is deduced as Then, based on the method of reference [36], we have where and . Therefore, the proposed high-gain state observer can estimate the unknown states of the hysteresis system.
4. Controller Design
Considering the transformed motor servo system, in order to estimate the unknown parameters, a new HONN will be designed for the servo motor system. Moreover, an adaptive controller will be presented based on the new HONN, and a Lyapunov function candidate will guarantee all the signals bounded in the closed-loop of the hysteresis system.
4.1. HONN
In this section, a new HONN will be designed through the states . To account for unknown friction and unknown torque , a NN approximation [35] is adopted over a compact set as where are the bounded NN weights; is a bounded error, that is, and where and are positive constants; and is the NN base vector. High-order functions , are used, where are -not-ordered subsets of , are nonnegative integers, and equals to , where and are positive bounded parameters and the real number is also bound.
4.2. Tracking Error Definition
In this section, we define a reference input as for the HONN-adaptive controller design, and the tracking error is selected as where is a filter vector and is Hurwitz. Therefore, is bounded when is bounded.
The estimation of states is defined as . Then, we have where the reference is bounded, that is, . Then, the errors and are denoted as
Substituting (4) into (17), we have
4.3. Adaptive NN Controller Design
For the hysteresis servo motor system (4), the designed HONN control is adopted as where and is the estimated weights.
The HONN weights are defined as where the adjustment coefficients and are the designed parameters.
Then, the Lemma is held.
Lemma 2 [35]. The in (22) are bounded with , where is bounded, that is, . If the error is defined as , then it has , where and is a positive constant which suits .
We define ; then, it has . The update law of is defined as
The control structure diagram is shown in Figure 3.
4.4. Stability Analysis
From Lemma 2, considering the designed HONN, the adaptive law (23), and the HONN weights (22), the following theorem is held.
Theorem 1. Considering the hysteresis servo motor system (4), the controller (21) with adaptive law (23), the HONN (16) with update law (22), all the signals in a closed loop, therefore, are UUB and the tracking errors are convergent to a small compact set as with defined in (25) and and defined in (30).
Proof 1. The derivative of the is deduced from (19), (20), and (21) as The selected Lyapunov function candidate is The derivative of can be deduced by (19) and (24): Substituting (19) and (23) into (26), one has
In view of the method of reference [35], considering Lemma 2, substituting (19) into (27), the derivative of in (27) can be deduced as
Since is bounded, considering (18) and (19), the error is bounded, that is, , where is a positive constant. Then, we define and as small positive constants. It follows where .
By applying Young’s inequality on the term , the following inequation can be deduced as where and are guaranteed positive. According to the Lyapunov theorem, the tracking error is bounded then is UUB.
Integrating both sides of (30) over , we have
Then, considering (25), it follows that
So that holds then is convergent to a small compact set. Therefore, all the signals in the closed loop are bounded.
5. Simulations
In this section, the proposed controller in a hysteresis servo motor system will be verified by simulations. The system parameters are shown in Table 2. To verify the applicability of the neural adaptive controller, reference sinusoidal signals and are employed and the parameters of the Bouc-Wen model are listed in Table 3.
In the simulation, the tracking error is defined in (18) and is adopted as . The controller (21) with adaptive law (23) and HONN update law (22) are implemented with parameters , , and ; the function is chosen as and .
The control results and the tracking errors are expressed in Figures 4–11. It is shown that the proposed neural adaptive controller has good control precision and shortened convergence time. But the control precision is concerned with the amplitude of the reference input. Different reference input amplitude has different control precision. The control results of motor position tracking and velocity tracking are illustrated in Figures 6 and 7 with . From Figure 6, one can obtain that the motor has tracked the reference position within 1 second and the mean absolute error (MAE) is equal to 0.0686. Then, the velocity of motor tracking has similar results from Figure 7. The control results of load position tracking and velocity tracking are shown in Figures 4 and 5 with . It is illustrated that the load position can track the reference signals rapidly as well as the velocity tracking of the load.
If the input signals have larger amplitude, for example, , the control results and errors are shown in Figures 8–11. From Figures 8 and 10, the maximum errors are 0.9566 and 1.0054, respectively. But in Figures 4 and 6, the maximum errors are 0.3121 and 0.2042, respectively. It is clearly illustrated that the amplitude of the reference input can influence the tracking precision of the adaptive control. But in velocity tracking, the accuracy of the control results are similar at a different reference input, and that can be seen from Figures 5, 7, 9, and 11. In short, the simulation results have verified the effectiveness of the proposed neural adaptive control.
6. Conclusion
This paper proposed an HGSO and an adaptive HONN controller for hysteresis nonlinearity of a motor driving servo system where the hysteresis nonlinearity was modeled through the Bouc-Wen model. The hysteresis motor driving servo system first has been transformed to a state space form by coordinate transformation. Then, a new HGSO was proposed, and in order to simplify the controller design, a filter was adopted for tracking errors and the adaptive HONN control was designed. Finally, a Lyapunov function candidate guaranteed all the signals of the bounded system. Simulations verified that the proposed approaches can accurately control the hysteresis nonlinearity motor driving servo system with Bouc-Wen model.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (61433003), Shandong Natural Science Foundation of China (ZR2017MF048), Shandong Key Research and Development Programme (2016GGX105013), Shandong Science and Technology Program of Higher Education (J17KA214), Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (2016RCJJ035), and Tai’an Science and Technology Development Program (2017GX0017).