# Model Parameter Identification of NC Machine Planar Motor Servo System Based on Adaptive Digital Sliding Mode Control

Authors: Erbao Peng; Hongying Wang; Changduo Chen
Journal: Security and Communication Networks
 (2022)

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## Abstract

An adaptive digital sliding mode control position controller was designed for a numerical control (NC) machine planar motor servo system with friction, parameter, and external disturbances. Firstly, the model of the NC machine planar motor servo system is discretised digitally. Secondly, the model parameters are identified using a recursive damping least squares method. Then, an adaptive digital sliding mode position controller is designed according to the state-space equation of the digitally discretised NC machine planar motor system, and the stability of the designed control system is analysed using Lyapunov’s theory. Finally, the effectiveness and superiority of the designed control method are verified by simulation experiments.


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## Body

## 1. Introduction

The traditional linear feed servo system is composed of a rotary motor with ball screw and nut pair. However, due to its large inertia and the large number of transmission parts, problems such as friction, elastic deformation, backlash, response lag, and insufficient stiffness are often faced, causing them to suffer in terms of high-speed and high-precision performance. As such, the direct-drive system composed of the linear motor is receiving more and more attention. In two-dimensional platforms, the direct drive of two-dimensional planar motors has become the ideal form of transmission for precision servo motion platform [1, 2].The establishment of an accurate mathematical model of the planar motor is difficult due to the coupling of flux, current, and velocity as well as the nonlinear factors such as end effect, parameter perturbation, and friction, making servo system control a challenging task [3, 4]. Planar motor control had been studied extensively by scholars. The PID control method is simple in structure and convenient in physical implementation but is highly dependent on the model parameters and has limited ability to suppress external disturbances [5, 6]. Based on the analysis of motor characteristics, a dual-ring PID control scheme has been proposed, which makes the motor suitable for operation in an environment with varying input frequencies [7]. A planar motor with a high-gain linear controller effectively suppressed low-frequency disturbances, but it amplified the problem of system noise [8]. To solve this, a variable gain nonlinear control method based on the amplitude of trajectory tracking error was proposed. This method effectively improved the tracking and positioning performance of the system and reduced the sensitivity to noise while enhancing the ability of the system to suppress low-frequency interference. Gan et al. [9] used a decoupling control to achieve the decoupling of six motion freedom degrees for the maglev planar motor of lithography processing and designed an X position servo controller to achieve the expected dynamic and static tracking performance. This method is based on the system model, which is sensitive to the change ratio of the system model parameters. For switched reluctance planar motors, Yang et al. [10] pointed out the shortcomings of a passivity control method for SRM based on energy dissipation theory. Combined with robust control techniques, a robust passive control method was designed to effectively overcome the sensitivity to static friction and sudden loads. Li et al. [11] introduced the adaptive mechanism into the planar motor control and proposed an adaptive PD control method. Based on the output of the proportional differential controller and the target acceleration dataset, the feed-forward coefficients are modified using a least squares method.Variable structure controls have become a popular method among scholars because of their simple structure, easy implementation, and insensitivity to system parameter changes and external disturbances [12–14]. For a three-degree-of-freedom planar motor, a planar motor with X-directional position servo controller based on an improved superhelix algorithm was designed and the control system was simulated [15]. However, the inherent buffeting phenomenon of the sliding mode control reduces the tracking performance of the system. In view of the phenomenon that the conventional discrete sliding mode control falls into oscillation when entering a steady state, an adaptive sliding mode control was proposed based on the analysis of its chattering mechanism. This control effectively improved the performance of the system, but its response speed still requires further improvement [16].Based on the above discussion, this paper proposes an adaptive sliding mode control method based on the planar motor of NC machine tools for electrochemical servo systems with uncertainties such as friction torque, load torque, and disturbance torque. Compared with the existing research results, the planar motor model is discretised and the parameters of the system model are identified by a recursive damping least squares method [17], and then the system model is represented as a spatial equation of state. The fractional-order adaptive digital sliding mode control method is used to suppress the perturbations of the system parameters and external disturbances to improve the dynamic tracking performance of the system. Finally, the sigmoid function is introduced to improve the steady-state performance of the system.


## 2. Structure and Mathematical Model of NC Machine Planar Motor

The NC machine planar motor consists of five parts: the stator platform, motion platform, winding promoter block, linear guide rail, and linear sensor. If the core loss is not considered and the counter-electromotive force is assumed to be the ideal sine wave, the dynamic model of the NC machine planar motor can be described as [18](1)Mnx¨+Dnv+FL+Ff+Fr=fxy=u,where Mn, x, and x˙=v represent mass, displacement, and moving velocity of the sliding mode control, respectively, fxy is the total thrust output for a motor shaft, u is the control input, Dn is the linear friction coefficient, and Fr, FL, and Ff are the thrust fluctuation, load disturbance, and nonlinear friction force of the system, respectively.When a phase winding of the rotor of the planar motor is electrified, the inductance in the winding changes. The equivalent circuit is shown in Figure1.Figure 1
Equivalent circuit.The equivalent circuit equation can be deduced from the figure above:(2)Uat=Raiat+dLax,iaiatdt=Raiat+Lax,iadiatdt+iatdLax,tdxdxdt,where Uat is the voltage, iat is the current, Rat is the equivalent inductance, and Ra is the equivalent resistance.


## 3. Parameter Identification of NC Machine Planar Motor Model

In general, we can use a linear model to describe this system. The model is discretised as follows:(3)yk=−den2yk−1−den3yk−2+num2uk−1+num3uk−1,where y(k) and u(k) denote the output quantity and num and den denote the discrete molecules and denominator, respectively.The defining parameter vectors are as follows:(4)θk=a1ka2kb1kb2kT=den2den3num2num3T.The defining vectors are as follows:(5)ψTk=−yk−1−yk−2uk−1uk−2T.The system model, equation (1), can be converted into(6)yk=ψTkθk.The identification model of measurement noise in the system is as follows:(7)yk=ψTkθk+υk.The recursive least squares damping algorithm is used to identify the parameters of the system (equation (6)) online. The method we use is the least squares method for parameter identification as follows:(8)θ^k=θ^k−1+ψkrkek,θ^0=1np0,ek=yk−ψTkθ^k−1,rk=rk−1+ψk2,where 1n=1,1,…,1T∈Rn is the nth vector and p0 is a large positive number. The identification flow when p0=106 is shown in Figure 2.Figure 2
Flowchart of parameter identification for the NC machine planar motor model.Model parametersMn and Dn are constants that can be obtained directly.


## 4. Digital Sliding Mode Controller Design for NC Machine Planar Motor

### 4.1. Discrete State Equation of Motion Platform for NC Machine Planar Motor

Whena=Dn/Mn and b=1/Mn, the system state model of the NC machine planar motor can be expressed as follows:(9)X.t=AXt+But+dt,where Xt=x1tx2t=xtx˙t, A=010−a, B=0b, dt=1/Mn0FL+Ff+Fr, and Ut=fxy.The discrete model of equation (8) can be expressed as follows:(10)Xk+1=A1Xk+B1Uk+D1k,where Xk=x1kx2k, Uk=iqk, and D1k are the state vector, control input, and disturbance, respectively, at time k. A1, B1, and D1 are the corresponding matrices after discretisation.A system with uncertain parameters can be expressed as follows:(11)Xk+1=A1+ΔAXk+B1Uk+D1k,(12)Xk+1=A1Xk+B1Uk+Πk,where Π=ΔAXk+D1k is the total disturbance of the system that satisfies Π≤κ.


### 4.2. Design of Adaptive Discrete Sliding Mode Controller

The position instruction isrk, and its change rate is drk. When Rk=rkdrk, the corresponding tracking error is ek=xk−Rk. Taking rk+1=rk+1drk+1, the details are as follows:(13)rk+1=2rk−rk−1,drk+1=2drk−drk−1.The discrete sliding die-cutting function is as follows:(14)sk=Ce=Cxk−rk,C=c1T.Theorem 1.
For a planar motor servo system as in equation (12), based on the fractional-order adaptive approach rate equation (16), the closed-loop system is asymptotically stable [16, 17].(15)uk=CTB−1CTk+1−CTAXkdsk,(16)dsk=sk+1=1−qTssk−Tsεsgnsk,where ε=kεskα, q>0, kε>0, 1>α>0, and 1−qTs>0.Proof.
(1)
Accessibility Analysis. When the sampling period is very small, the sliding mode of the planar motor servo system (equation (10)) must satisfy the following:(17)sk+1−sksgnsk<0,sk+1−sksgnsk>0,(18)sk+1−sksgnsk=−qTssk−Tskεskαsgnsksgnsk=−qTssk−Tskεskα<0,(19)sk+1+sksgnsk=2−qTssk−Tskεskαsgnsksgnsk=2−qTssk−Tskεskα>0.The control rate (equation (15)) satisfies the arrival condition (equation (17)).(2)
Stability Analysis. Suppose that(20)Fk=sk+1−sk=−qTssk−Tskεskαsgnsk.
Whensk=0, then sk+1=sk=0 and the system enters a steady state.
Whensk<0, then ∂Fk/∂sk=−qTs−Tskεα−skα−1<0 and Fk>−qTssk−Tskεskαsgnsk|sk=0, so sk+1>sk, that is, sk is incremental and tends to sk=0. When sk>0, then ∂Fk/∂sk=−qTs−Tskεα−skα−1<0 and Fk<−qTssk−Tskεskαsgnsk|sk=0, so sk+1<sk, that is, sk is regressive and tends to sk=0. So, the system is stable (sk=0), and the theorem is proved.
For reducing the inherent buffeting phenomenon of the switching function in equation (16), the sigmoid function χsk is used instead of the switching function, and the control rate can be expressed as(21)uk=CTB−1CTrk+1−CTAxk−dsk,dsk=1−qTssk−Tskεskαχsk.


## 4.1. Discrete State Equation of Motion Platform for NC Machine Planar Motor

Whena=Dn/Mn and b=1/Mn, the system state model of the NC machine planar motor can be expressed as follows:(9)X.t=AXt+But+dt,where Xt=x1tx2t=xtx˙t, A=010−a, B=0b, dt=1/Mn0FL+Ff+Fr, and Ut=fxy.The discrete model of equation (8) can be expressed as follows:(10)Xk+1=A1Xk+B1Uk+D1k,where Xk=x1kx2k, Uk=iqk, and D1k are the state vector, control input, and disturbance, respectively, at time k. A1, B1, and D1 are the corresponding matrices after discretisation.A system with uncertain parameters can be expressed as follows:(11)Xk+1=A1+ΔAXk+B1Uk+D1k,(12)Xk+1=A1Xk+B1Uk+Πk,where Π=ΔAXk+D1k is the total disturbance of the system that satisfies Π≤κ.


## 4.2. Design of Adaptive Discrete Sliding Mode Controller

The position instruction isrk, and its change rate is drk. When Rk=rkdrk, the corresponding tracking error is ek=xk−Rk. Taking rk+1=rk+1drk+1, the details are as follows:(13)rk+1=2rk−rk−1,drk+1=2drk−drk−1.The discrete sliding die-cutting function is as follows:(14)sk=Ce=Cxk−rk,C=c1T.Theorem 1.
For a planar motor servo system as in equation (12), based on the fractional-order adaptive approach rate equation (16), the closed-loop system is asymptotically stable [16, 17].(15)uk=CTB−1CTk+1−CTAXkdsk,(16)dsk=sk+1=1−qTssk−Tsεsgnsk,where ε=kεskα, q>0, kε>0, 1>α>0, and 1−qTs>0.Proof.
(1)
Accessibility Analysis. When the sampling period is very small, the sliding mode of the planar motor servo system (equation (10)) must satisfy the following:(17)sk+1−sksgnsk<0,sk+1−sksgnsk>0,(18)sk+1−sksgnsk=−qTssk−Tskεskαsgnsksgnsk=−qTssk−Tskεskα<0,(19)sk+1+sksgnsk=2−qTssk−Tskεskαsgnsksgnsk=2−qTssk−Tskεskα>0.The control rate (equation (15)) satisfies the arrival condition (equation (17)).(2)
Stability Analysis. Suppose that(20)Fk=sk+1−sk=−qTssk−Tskεskαsgnsk.
Whensk=0, then sk+1=sk=0 and the system enters a steady state.
Whensk<0, then ∂Fk/∂sk=−qTs−Tskεα−skα−1<0 and Fk>−qTssk−Tskεskαsgnsk|sk=0, so sk+1>sk, that is, sk is incremental and tends to sk=0. When sk>0, then ∂Fk/∂sk=−qTs−Tskεα−skα−1<0 and Fk<−qTssk−Tskεskαsgnsk|sk=0, so sk+1<sk, that is, sk is regressive and tends to sk=0. So, the system is stable (sk=0), and the theorem is proved.
For reducing the inherent buffeting phenomenon of the switching function in equation (16), the sigmoid function χsk is used instead of the switching function, and the control rate can be expressed as(21)uk=CTB−1CTrk+1−CTAxk−dsk,dsk=1−qTssk−Tskεskαχsk.


## 5. Digital Simulation and Analysis

The main parameters of the proposed method are motor quality ofX shaft (MX=8.75kg), motor quality of Y shaft (MY=15kg), and the friction coefficient (Bn=0.1N⋅m/s). To verify the effective line of this method, it is compared with the PID control and traditional sliding mode control.


### 5.1. Model Parameter Identification

The sampling period of the system isTs=1ms, and equation (1) is discretised. If the influence of external disturbance is not taken into account when identifying the parameters, the discrete X-axis model is given by(22)yk=2yk−1−k−2+5.714×10−7uk−1+5.714×10−7uk−2.The model of the discreteY axis is(23)yk=2yk−1−k−2+3.3333×10−8uk−1+3.3333×10−8uk−2.Considering the influence of measurement noise,ψTk=−yk−1−yk−2uk−1uk−2T is the parameter vector, θk=a1ka2kb1kb2kT is the parameter vector, and the true value of the X axis is θdX=−215.714×10−75.714×10−7T and θdY=−213.3333×10−83.3333×10−8T. υk is the zero mean and the white noise with square difference uniformly distributed. The test input and output and identification results are shown in Figure 3.Figure 3
Parameter identification curve of NC machine planar motor. (a) Input curve of identification test. (b) Output curve of identification test. (c) Parameter identification curve ofX axis. (d) Parameter identification curve of Y axis.
(a)(b)(c)(d)Figures3(a) and 3(b) show the test inputs and outputs generated by random signals, and Figures 3(c) and 3(d) show the identification curves of the X-axis and Y-axis parameters of NC machine planar motor, respectively. Figure 3 shows that the recursive damping least squares method can realise the identification of the model parameters of X axis and Y axis, and both converge to their true values.


### 5.2. Analysis of Control Performance

The system shows that theX axis and Y axis of NC machine planar motor have similar dynamic models. The following method is verified only for a single axis (X axis) simulation for which the total simulation time is 1 s, the instruction signal is r=1mm, and the system adds mutation load when t=0.5s. The PID controller parameters are KP=30, KI=0.2, KD=5, and α=7/9, while other parameters are the same as above. The position tracking and control input results are shown in Figures 4 and 5.Figure 4
Position tracking curve of the NC machine planar motor.Figure 5
Control input of NC machine planar motor.The expression of the PID controller is(24)Kpk=Kpk−1+γpk∇Kp,Kik=Kik−1+γik∇Ki,Kdk=Kdk−1+γDk∇Kd.The system state equation is(25)uk=Kpkek+Kikxk+KDkek−ek−1.Figures4 and 5 show that the PID controller with reasonable parameters has almost no overshoot when starting but deviates from the expected value at time t=0.5s with external interference and takes about 0.3 s to recover the expected value. The conventional sliding mode has a strong disturbance rejection ability when added with external disturbance, but the startup performance still needs improvement and the control input has a large vibration. This method can not only quickly and stably realise the tracking of position instructions without overshoot but also recover the expected value within 0.1 s when added with a load disturbance and has a strong anti-load buffeting ability.


## 5.1. Model Parameter Identification

The sampling period of the system isTs=1ms, and equation (1) is discretised. If the influence of external disturbance is not taken into account when identifying the parameters, the discrete X-axis model is given by(22)yk=2yk−1−k−2+5.714×10−7uk−1+5.714×10−7uk−2.The model of the discreteY axis is(23)yk=2yk−1−k−2+3.3333×10−8uk−1+3.3333×10−8uk−2.Considering the influence of measurement noise,ψTk=−yk−1−yk−2uk−1uk−2T is the parameter vector, θk=a1ka2kb1kb2kT is the parameter vector, and the true value of the X axis is θdX=−215.714×10−75.714×10−7T and θdY=−213.3333×10−83.3333×10−8T. υk is the zero mean and the white noise with square difference uniformly distributed. The test input and output and identification results are shown in Figure 3.Figure 3
Parameter identification curve of NC machine planar motor. (a) Input curve of identification test. (b) Output curve of identification test. (c) Parameter identification curve ofX axis. (d) Parameter identification curve of Y axis.
(a)(b)(c)(d)Figures3(a) and 3(b) show the test inputs and outputs generated by random signals, and Figures 3(c) and 3(d) show the identification curves of the X-axis and Y-axis parameters of NC machine planar motor, respectively. Figure 3 shows that the recursive damping least squares method can realise the identification of the model parameters of X axis and Y axis, and both converge to their true values.


## 5.2. Analysis of Control Performance

The system shows that theX axis and Y axis of NC machine planar motor have similar dynamic models. The following method is verified only for a single axis (X axis) simulation for which the total simulation time is 1 s, the instruction signal is r=1mm, and the system adds mutation load when t=0.5s. The PID controller parameters are KP=30, KI=0.2, KD=5, and α=7/9, while other parameters are the same as above. The position tracking and control input results are shown in Figures 4 and 5.Figure 4
Position tracking curve of the NC machine planar motor.Figure 5
Control input of NC machine planar motor.The expression of the PID controller is(24)Kpk=Kpk−1+γpk∇Kp,Kik=Kik−1+γik∇Ki,Kdk=Kdk−1+γDk∇Kd.The system state equation is(25)uk=Kpkek+Kikxk+KDkek−ek−1.Figures4 and 5 show that the PID controller with reasonable parameters has almost no overshoot when starting but deviates from the expected value at time t=0.5s with external interference and takes about 0.3 s to recover the expected value. The conventional sliding mode has a strong disturbance rejection ability when added with external disturbance, but the startup performance still needs improvement and the control input has a large vibration. This method can not only quickly and stably realise the tracking of position instructions without overshoot but also recover the expected value within 0.1 s when added with a load disturbance and has a strong anti-load buffeting ability.


## 6. Conclusion

The servo system of an NC machine planar motor is a complex and changeable nonlinear system. The system model is often difficult to express accurately by a mathematical model, and the existence of model parameter perturbation and external interference reduces the positioning accuracy of the system. In this paper, a digital discrete sliding mode control method based on model parameter identification is proposed, which can suppress parameter perturbations and external disturbances of the system and improve the buffeting influence of the traditional sliding mode control. By comparing numerical simulations with the PID control and conventional digital sliding mode control, the effectiveness of the proposed method is verified.



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