A novel high-precision motion control for permanent magnet linear synchronous motor servo system

Abstract

In this paper, a novel high-precision motion control is proposed for a permanent magnet linear synchronous motor (PMLSM) servo system, which is vulnerable to the influence of uncertainties. First, the dynamics of the PMLSM with uncertainties are derived. To cater for the parametric uncertainties, the model-based feedforward control is constructed so that the transient response of the system is improved. Moreover, the adaptive jerk control (AJC) scheme based on a robust integral of the sign of the error (RISE) feedback is adapted to restrain the uncertainties such as external disturbance and nonlinear friction in the system. To alleviate the chattering phenomenon, a novel exponential adaptive law is designed to bound the feedback gain of the jerk, under the condition of the initial term of control efforts considered. Then, the jerk signal is integrated to form the feedback control law, which generates continuous control input and ensures the stability of the system. Compared with sliding mode control (SMC), the experimental results indicate that both the robustness and tracking performance of the system are significantly improved without adding more control efforts. The position tracking error is reduced and high-frequency oscillation is effectively attenuated.

Keywords

Permanent magnet linear synchronous motor (PMLSM)adaptive jerk control (AJC)robust integral of the sign of the error (RISE) feedback robustness chattering

1. Introduction

A permanent magnet linear synchronous motor (PMLSM), which consists of primary and secondary components, is not required mechanical reductions and transmissions. In this regard, the PMLSM servo system has been designed and adopted in many applications due to its higher efficiency, lower thermal losses, better positioning accuracy, and quicker response [1,2]. Thus, it is widely used to operate in high-performance applications to yield high productivity such as industrial robots [3], XY platforms [4], CNC machine tools [5], and two-dimensional micro/nano manufacturing [6]. However, with the lack of transmission parts, the control performance of the PMLSM servo system is significantly degraded by the uncertainties including external load disturbance, nonlinear friction, parameter variations, and unmodeled dynamics [7]. Therefore, it is imperative to adopt a strategy to meet the requirements in the PMLSM servo system.

In the past decade, there are many control methods for the PMLSM presented in the kinds of literature. One of the traditional techniques to compensate for the effect of uncertainties is system identification with some adaptive methods, such as friction compensation and observation. In [8], LuGre model was presented to cope with the friction force. In [9] and [10], the Hsieh–Pan model and the DNLRX model were utilized to describe the nonlinear friction dynamics, respectively. However, there were no accurate models accessible due to the model uncertainties for different operating conditions.

On the other hand, the design of a robust controller has been a widely examined strategy that just requires the plant model of the nonlinear system used to suppress the lumped uncertainties. Sliding mode control (SMC) is a well-known control approach that provides enough robustness in different applications. Thus SMC mitigates the effect of the parameter variations when the trajectory reaches and switches on the sliding surface [11]. The control strategy often leads to a chattering phenomenon caused by the signum function and large switching control gain, which excites high-frequency mode. Also, the chattering phenomenon makes the control efforts switch back and forth, and PMLSM oscillates around in a small range, which may waste energy and damage the servo system. To overcome this problem, researchers look beyond the traditional SMC for some advanced versions. In [12], saturation function, instead of signum function, was in the part of switching laws. Complementary sliding mode control (CSMC) was provided with favorable tracking accuracy in [13]. The intelligent control approaches like fuzzy mechanisms and neural networks were developed in [14]. However, the difficulties of many calculations for the fuzzy neural networks and tuning of sliding mode controller have not been solved. Therefore, it is important to develop a continuous control scheme that balances the relationship between robustness and chattering.

Recently, a developed control scheme referred to a robust integral of the sign of the error (RISE), utilized the integral of the signum term to the jerk of the control as opposed to the control signal used in sliding mode directly, was proposed in [15–17]. RISE feedback is originated from super twisting sliding mode control. As long as the matched additive disturbance is smooth enough with known bounds of its time derivatives, the RISE feedback can achieve asymptotic tracking performance in [18,19]. More importantly, the resulting control efforts are always keeping continuous. In industrial applications, jerk reflects the change in acceleration. In order to achieve high-velocity machining, it is required to increase the feed rate in the presence of uncertainties. However, if there is an inflection point on the trajectory, the large jerk due to large feedback gain may cause serious mechanical shock and still lead to the chattering phenomenon, which the system cannot bear. Motivated by these problems, it is highly concerned to combine the jerk with other terms. In [20], a modular adaptive term of RISE feedback was developed. In [21] and [22], the RISE-based adaptive backstepping methods were designed for uncertain nonlinear systems. Multilayer neural network feedforward compensation and RISE feedback were proposed for a class of uncertain nonlinear systems in [23]. Above all the schemes with RISE feedback demonstrate that the adaptive feedforward term can reduce the effect of large feedback gain and the tracking performance of the system was improved. However, the feedback gain of RISE chosen to be a constant seems very prospective, it might still result in chattering. Although an exponential adaptive law was designed for the adaptive feedback gain [24], the initial term of control efforts, which may cause by large-gain feedback to compensate for the unmeasurable terms, should be considered.

Thus, in this paper, given the uncertainties of the PMLSM servo system, the parametric uncertainties can be estimated by a model-based adaptive feedforward while external load, nonlinear friction, and unmodeled dynamics can be compensated by the adaptive jerk control (AJC). The feedback control law generated by the adaptive jerk term can indirectly produce continuous and smooth control effects. The initial conditions of the control efforts are added. Adaptive feedback gain is bounded by an exponential updating law. It largely avoids the high-frequency oscillation while effectively compensating the time-varying position tracking error. In addition, the major contributions of this paper are listed as follows.

A two-degrees-of-freedom control structure is designed for the PMLSM servo system. Unlike most existing SMC, the unknown parameters are estimated via the model-based feedforward and the unmodeled disturbances are compensated via RISE feedback with the jerk adaptation, meanwhile ensuring the continuity of the control efforts.

A novel exponential adaptive law is designed to the adaptive feedback gain, which effectively reduces the large-gain feedback and chattering.

The convergence speed of the exponential adaptive law satisfies the fast convergence requirement so that the control law will response to regulate velocity error back to zero. AJC can achieve asymptotic tracking performance.

In the following part of this paper, the mathematic modeling for the PMLSM servo system is detailed in Section 2. The proposed control systems combined model-based adaptive feedforward and AJC are given in Section 3. Experimental results are provided in Section 4. Conclusions can be found in Section 5.

2. Modeling for PMLSM servo systems

2.1. Dynamics of PMSLM

The PMLSM is a device that directly generates linear motion, which can be seen as evolved from a permanent magnet synchronous motor. Figure 1(a) shows the PMLSM studied in this work. It can be obtained by cutting and expanding the permanent magnet synchronous motor in a radial direction into a parallel device. The stator of the permanent magnet synchronous motor is turned into a mover of PMLSM. To obtain electrical power from the power source, the mover should move with the feeder cable together. N, S pole permanent magnets are alternately mounted on a straight line along the forming direction over its entire length. Figure 1(b) is the block of construction for PMLSM. As a promising technology, a linear motor direct feed drive discards the transmission system required for a conventional feed drive and therefore there exists no transmission associated error such as backlash, pitch error, etc. Therefore, a linear direct feed drive is an excellent choice to meet the requirements of higher velocity, great accuracy, and improved reliability due to its mechanical simplicity.

The voltage equations of PMLSM in the synchronously rotating reference frame is described as follows $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V_{q}^{\ast }=R_{s}i_{q}^{\ast }+{\displaystyle \frac{d}{dt}}{\psi}_{q}+{\omega}_{r}{\psi}_{d}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V_{d}^{\ast }=R_{s}i_{d}^{\ast }+{\displaystyle \frac{d}{dt}}{\psi}_{d}-{\omega}_{r}{\psi}_{q}\end{eqnarray}$ (2) where 𝜓_d and 𝜓_q are the stator flux linkage and given by the following equations $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}_{q}=L_{q}i_{q}^{\ast }\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\psi}_{d}=L_{q}i_{q}^{\ast }+{\psi}_{f}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\omega}_{r}=P_{n}{\omega}.\end{eqnarray}$ (5) In Eqs (1)–(5), $V_{d}^{\ast }$ , $V_{q}^{\ast }$ , $i_{d}^{\ast }$ , $i_{q}^{\ast }$ , L _d and L _q are the d–q axis stator voltages, currents and inductances, respectively. R _s is the resistance of stator windings and 𝜓_f is the permanent magnet flux linkage. ω and ω_r are electrical angular velocity and angular velocity of the mover. P _n is the number of pole pairs.

Substituting (3), (4) into (1), (2) will yield $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V_{q}^{\ast }=R_{s}i_{q}^{\ast }+L_{q}{\displaystyle \frac{d}{dt}}i_{q}^{\ast }+{\omega}_{r}L_{d}i_{d}^{\ast }+{\omega}_{r}{\psi}_{f}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V_{d}^{\ast }=R_{s}i_{d}^{\ast }+L_{d}{\displaystyle \frac{d}{dt}}i_{d}^{\ast }-{\omega}_{r}L_{q}i_{q}^{\ast }.\end{eqnarray}$ (7) Moreover $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\omega}=({\pi}/{\tau})v\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle v_{r}=P_{n}v=2{\tau}f_{r}\end{eqnarray}$ (9) where v _r, v, f _r, and τ are electrical linear velocity, mover linear velocity, frequency, and the pole pitch, respectively.

Fig. 1.

(a) Process schematic diagram of turning rotating motor into linear motor and (b) the block of construction for PMLSM.

The electromagnetic thrust dynamic of PMSLM is expressed as $\begin{eqnarray}\displaystyle F_{e}={\displaystyle \frac{3}{2}}\cdot P_{n}\cdot {\displaystyle \frac{{\pi}}{{\tau}}}[{\psi}_{f}i_{q}^{\ast }+(L_{d}-L_{q})i_{d}^{\ast }i_{q}^{\ast }] & & \displaystyle\end{eqnarray}$ (10) where F _e is the electromagnetic thrust, $i_{q}^{\ast }$ is the command of thrust current. Then, setting d-axis current $i_{d}^{\ast }$ as zero for field-oriented control, the electromagnetic thrust is $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{e}=K_{f}i_{q}^{\ast }\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{f}={\displaystyle \frac{3}{2}}\cdot P_{n}\cdot {\displaystyle \frac{{\pi}}{{\tau}}}\cdot {\psi}_{f}\end{eqnarray}$ (12) where K _f is the thrust coefficient.

The dynamic equation of the mover motion with disturbance can be expressed by $\begin{eqnarray}\displaystyle (M+{\Delta}M)\ddot{x}+(B+{\Delta}B){\dot{x}}=F_{e}-F & & \displaystyle\end{eqnarray}$ (13) where $\ddot{x}$ is the acceleration and ${\dot{x}}$ is the velocity respectively. M is the total mass of the mover and B is the viscous friction coefficient. ΔM and ΔB are the uncertainties due to mechanical parameters M and B. F is the disturbance including external load disturbance, friction, unmodeled dynamics, etc.

From (12) and (13), the mechanical dynamics of PMLSM can be simplified as $\begin{eqnarray}\displaystyle {\theta}_{1}\ddot{x}=u-{\theta}_{2}{\dot{x}}-d & & \displaystyle\end{eqnarray}$ (14) where θ₁ = (M + ΔM)∕K _f, θ₂ = (B + ΔB)∕K _f, d = F∕K _f, and $u=i_{q}^{\ast }$ .

2.2. Problem formulation for PMLSM servo system

Although some favorable features are mentioned at the beginning of this section, the control task with challenges is presented. Uncertainties such as nonlinear friction mode and parameter variations are difficult to be accurately established. Therefore, a robust control scheme is required to cater to the uncertainties of this model. As a traditional robust control, SMC, which is insensitive to parametric uncertainties, has been proven to satisfy performance for the system. The control law of SMC contains a signum function to indicate the switching process, so SMC is a discontinuous control. If the initial condition of the nonlinear system is near the original point, it will cause high-frequency oscillation. The high frequency may excite nonlinear dynamics introduced by the uncertainties, which will consume energy and lead to instability. Hence, it is necessary to establish a control scheme to ensure the robustness of the system and accuracy of feedback control in the presence of model uncertainties, while avoiding the phenomenon of chattering.

3. Proposed control systems

To ensure minimal excitation of nonlinear dynamics, the control input signal should be smooth and continuous. Therefore, it is important not only to focus on the control law u but also to design the derivative of the control law, that is, the jerk signal. However, due to the model uncertainties, quick response to tracking error may still require a high jerk control signal. To satisfy stability and rapidity, a RISE feedback control scheme is proposed on the basis of the adaptive jerk control and model-based feedforward. The block diagram of the proposed two-degrees-of-freedom control system is shown in Fig. 2.

3.1. Design schemes

3.1.1. To design filtered error vector

Due to the existence of nonlinear dynamics, it is desirable to minimize the chattering of control signals. To quantify the design of the following controllers, the filtered error vector is defined as $\begin{eqnarray}\displaystyle \boldsymbol{z}=\left[\begin{array}{@{}ccc@{}}e_{1} & e_{2} & e_{3}\end{array}\right]^{\text{T}} & & \displaystyle\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}e_{1}=x_{d}-x\\ e_{2}={\dot{e}}_{1}+k_{1}e_{1}\\ e_{3}={\dot{e}}_{2}+k_{2}e_{2}\end{array}\right. & & \displaystyle\end{eqnarray}$ (16) where x _d is the desired trajectory. e ₁, e ₂ and e ₃ are the position tracking error, velocity error and accelerate error respectively. k ₁, $k_{2}\in \mathbb{R}$ denote positive control gains. The filtered error e ₃ is introduced to facilitate the stability analysis and is not measurable in the controller design since the expression in (16) depends on $\ddot{x}$ .

Fig. 2.

Block diagram of the control system.

3.1.2. To construct a two-degrees-of-freedom control structure

The open-loop tracking error system can be developed by the right multiplying ((16)) by θ₁ and utilizing the expressions in ((14)) to obtain the following expression $\begin{eqnarray}\displaystyle {\theta}_{1}e_{3}=\boldsymbol{Y }_{d}\boldsymbol{{\theta}}+S+d-u & & \displaystyle\end{eqnarray}$ (17) where $\boldsymbol{Y }_{d}=[\ddot{x}_{d},{\dot{x}}_{d}]$ refers to the desired trajectory vector and θ = [θ₁ θ₂]^T is the parameter vector. S is defined to be $\begin{eqnarray}\displaystyle S={\theta}_{1}(k_{1}{\dot{e}}_{1}+k_{2}e_{2})-{\theta}_{2}{\dot{e}}_{1}. & & \displaystyle\end{eqnarray}$ (18)

Different from the conventional robust control, the filtered error will introduce into the availability of additional design freedom. Therefore, a two-degrees-of-freedom control structure is adopted. The control law is structured (17) as $\begin{eqnarray}\displaystyle u=u_{1}+u_{2}. & & \displaystyle\end{eqnarray}$ (19) Here, u ₁ is the feedforward term to speed up the response and cater for parametric uncertainty, expressed as $\begin{eqnarray}\displaystyle u_{1}=\boldsymbol{Y }_{\text{d}}\hat{\boldsymbol{{\theta}}} & & \displaystyle\end{eqnarray}$ (20) where $\hat{\boldsymbol{{\theta}}}$ denotes the estimate of θ and u ₂ is the feedback control law to ensure the robustness of the closed-loop system when the system is suffered from external disturbances and unmodeled uncertainties. Since the desired trajectory vector Y _d and its time derivative depend on the desired trajectory only, they can be calculated offline so that the on-line computation time will be saved and the measurement noise is reduced. Thus, it is more suitable for practical applications.

3.1.3. To design model-based feedforward

To avoid the excitation of the resonant modes, it is important to ensure that the control signal u ₂ is continuous, the jerk signal $\dot{u}_{2}$ should be bounded. Hence, the time derivative is taken on the closed-loop tracking error system of (17) $\begin{eqnarray}\displaystyle {\theta}_{1}{\dot{e}}_{3}=-{\displaystyle \frac{1}{2}}\dot{{\theta}}_{1}e_{3}+\dot{\boldsymbol{Y }}_{d}\tilde{\boldsymbol{{\theta}}}+{\tilde{N}}+N_{d}-\dot{u}_{2}-e_{2} & & \displaystyle\end{eqnarray}$ (21) where $\tilde{\boldsymbol{{\theta}}}=\boldsymbol{{\theta}}-\hat{\boldsymbol{{\theta}}}$ refers to the parameter estimation error vector and the unmeasurable terms ${\tilde{N}}$ and N _d are defined to be. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\tilde{N}}(t)=-{\displaystyle \frac{1}{2}}\dot{{\theta}}_{1}e_{3}\boldsymbol{Y }_{d}\dot{\tilde{\boldsymbol{{\theta}}}}+{\dot{S}}+e_{2}\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle N_{\text{d}}(t)={\dot{d}}.\end{eqnarray}$ (23) It is a fact that the ${\tilde{N}}$ and N _d in (22) and (23) have different bounds, which facilitates the development of the $\hat{\boldsymbol{{\theta}}}$ update law and the subsequent stability analysis. Here, $\hat{\boldsymbol{{\theta}}}$ is designed based on a gradient-based adaptive update law, yielding $\begin{eqnarray}\displaystyle \dot{\hat{\boldsymbol{{\theta}}}}={\Gamma}\dot{\boldsymbol{Y }}_{d}^{\text{T}}e_{3} & & \displaystyle\end{eqnarray}$ (24) where ${\Gamma}\in \mathbb{R}$ is a positive scalar gain. To avoid utilizing $\ddot{x}$ for calculating e ₃, the feedforward control law u ₁ is integrated by parts as $\begin{eqnarray}\displaystyle \boldsymbol{Y }_{d}\hat{\boldsymbol{{\theta}}}=\boldsymbol{Y }_{d}\hat{\boldsymbol{{\theta}}}(0)+\boldsymbol{Y }_{d}{\Gamma}\dot{\boldsymbol{Y }}_{d}^{\text{T}}e_{2}({\sigma})|_{0}^{t}-\boldsymbol{Y }_{d}{\Gamma}\int _{0}^{t}[\ddot{\boldsymbol{Y }}_{d}^{\text{T}}e_{2}({\sigma})-k_{2}\dot{\boldsymbol{Y }}_{d}^{\text{T}}e_{2}({\sigma})]d{\sigma}. & & \displaystyle\end{eqnarray}$ (25)

3.1.4. To design feedback with AJC

The jerk $\dot{u}_{2}$ of the RISE feedback control term is designed to be $\begin{eqnarray}\displaystyle \dot{u}_{2}=(K_{s}+1)e_{3}+(\hat{{\beta}}_{1}+{\beta}_{2})\text{sgn}(e_{2}) & & \displaystyle\end{eqnarray}$ (26) where $\hat{{\beta}}_{1}>-{\beta}_{2}$ and 𝛽₂ > 0 are adaptive feedback gain and fixed feedback gain according to the jerk respectively. Accordingly, the feedback control law u ₂ with the integral term is given by $\begin{eqnarray}\displaystyle u_{2}(t)\text{} & =\text{} & \displaystyle (K_{s}+1)\displaystyle \int _{0}^{t}({\dot{e}}_{2}({\sigma})+k_{2}({\dot{e}}_{1}({\sigma})+k_{1}e_{1}({\sigma})))d{\sigma}+\displaystyle \int _{0}^{t}(\hat{{\beta}}_{1}+{\beta}_{2})\text{sgn}(e_{2}({\sigma}))d{\sigma}+u_{2}(0)\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle (K_{s}+1)\left[(e_{2}(t)-e_{2}(0))+k_{2}(e_{1}(t)-e_{1}(0))+k_{1}k_{2}\displaystyle \int _{0}^{t}e_{1}(t)d{\sigma}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\displaystyle \int _{0}^{t}(\hat{{\beta}}_{1}+{\beta}_{2})\text{sgn}(e_{2}({\sigma}))d{\sigma}+u_{2}(0)\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle (K_{s}+1)\left[{\dot{e}}_{1}+(k_{1}+k_{2})e_{1}+k_{1}k_{2}\displaystyle \int _{0}^{t}e_{1}({\tau})d{\tau}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\displaystyle \int _{0}^{t}(\hat{{\beta}}_{1}+{\beta}_{2})\text{sgn}(e_{2}({\tau}))d{\tau}+E_{0}+u_{2}(0)\end{eqnarray}$ (27) where K _s > 0, u ₂(0) is the value of u ₂(t) when t = 0, E ₀ is the offset due to initial conditions, expressed as $\begin{eqnarray}\displaystyle E_{0}=-(K_{\text{s}}+1)[k_{2}e_{1}(0)+e_{2}(0)]. & & \displaystyle\end{eqnarray}$ (28) Due to the structural complexity of the disturbance, it is very difficult to find its precise limits, even in practice. If boundaries are sometimes available, they are usually very conservative. Large-gain feedback will result in serious design conservativeness, while too small selection may result in performance degradation or even instability. To avoid the negative effect of excessive robust gain, $\dot{\hat{{\beta}}}_{1}$ is proposed as $\begin{eqnarray}\displaystyle \dot{\hat{{\beta}}}_{1}=-k_{3}\hat{{\beta}}_{1}+\overline{{\beta}}_{1}+e_{3}\text{sgn}(e_{2}) & & \displaystyle\end{eqnarray}$ (29) where 𝛽 ₁, k ₃ are constants.

TheoremA . (Kamaldin et al. 2019).

From (29), if e ₂ > 0 and ${\dot{e}}_{2}>$ 0, there will be a forced response of $\hat{{\beta}}_{1}$ with perturbation from ∥e ₃∥₁ ≥ 0, so that u will response to regulate e ₂ back to zero. ∥⋅∥_p ≥ 0 refers to the p-norm of a vector. Thus, an exponential adaptive law is proposed to automatically determine an optimal feedback gain.

Let $\hat{{\beta}}_{1}$ converge exponentially to 𝛽₁ = 𝛽 ₁∕k ₃, then $\hat{{\beta}}_{1}$ yields $\begin{eqnarray}\displaystyle \hat{{\beta}}_{1}={\beta}_{1}(1-{e_{2}}^{-k_{3}t})+|e_{2}|-{e_{2}}^{-k_{3}t}|e_{2}(0)|+{e_{2}}^{-k_{3}t}\hat{{\beta}}_{1}(0)+(k_{2}-k_{3}){e_{2}}^{-k_{3}t}\ast |e_{2}(t)| & & \displaystyle\end{eqnarray}$ (30) where (∗) denotes the convolution operator. Compared with the jerk law used the fixed RISE feedback gain, u will response more quickly to adjust the filtered error back to zero.

3.2. Stability analysis

TheoremB .
Uncertainty d is smooth enough such that $\begin{eqnarray}\displaystyle \Vert N_{d}(t)\Vert _{1}\leq {\varsigma}_{1},\quad \Vert {\dot{N}}_{d}(t)\Vert _{1}\leq {\varsigma}_{2} & & \displaystyle\end{eqnarray}$ (31) where 𝜍₁ > 0, 𝜍₂ > 0 are known constants.

Since ${\tilde{N}}$ is continuous, by the mean value theorem, its norm can be upper bounded by a positive, non-decreasing function 𝜌 as $\begin{eqnarray}\displaystyle \Vert {\tilde{N}}\Vert \leq {\rho}(\Vert \boldsymbol{z}\Vert )\Vert \boldsymbol{z}\Vert . & & \displaystyle\end{eqnarray}$ (32) For ${\beta}_{1}\geq {\varsigma}_{1}+\frac{1}{k_{2}}{\varsigma}_{2}$ and 𝛽₂ ≥ 0, auxiliary functions L ₁, L ₂ are defined as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L_{1}=e_{3}(N_{d}(t)-{\beta}_{1}\text{sgn}(e_{2}))\end{eqnarray}$ (33) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle L_{2}=-{\beta}_{2}{\dot{e}}_{2}\text{sgn}(e_{2})\end{eqnarray}$ (34) and L ₁, L ₂ satisfy $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \int _{0}^{t}L_{1}({\sigma})d{\sigma}\leq {\varsigma}_{b_{1}}\end{eqnarray}$ (35) $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle \displaystyle \int _{0}^{t}L_{2}({\sigma}){d}{\sigma}\leq {\varsigma}_{b_{2}} \end{eqnarray}$ (36) where 𝜍_{b
₁}, 𝜍_{b
₂} are positive constants. Reformulating ((35))–((36)), it yields $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{1}={\varsigma}_{b_{1}}-\displaystyle \int _{0}^{t}L_{1}({\sigma})d{\sigma}\end{eqnarray}$ (37) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{2}={\varsigma}_{b_{2}}-\displaystyle \int _{0}^{t}L_{2}({\sigma})d{\sigma}\end{eqnarray}$ (38) where P ₁ and P ₂ are always positive.

Fig. 3.
Photograph of PMLSM control system experiment platform based on DSP.

Fig. 4.
Structure diagram of PMLSM control system based on DSP.

Table 1
Parameters of PMLSM

Symbol Value Unit

K _f 50 N/A

L _d 41.4 mH

L _q 41.4 mH

𝜓_f 0.09 Wb

R _s 2.1 Ω

τ 32 mm

M 16 kg

B 8 N ⋅ s∕m

y (t) is defined as $\begin{eqnarray}\displaystyle \boldsymbol{y}(t)=\left[\begin{array}{@{}ccccc@{}}\boldsymbol{z}^{\text{T}}(t) & \tilde{\boldsymbol{{\theta}}}^{\text{T}}(t) & \tilde{{\beta}}_{1}(t) & \sqrt{P_{1}} & \sqrt{P_{2}}\end{array}\right]^{\text{T}}. & & \displaystyle\end{eqnarray}$ (39) Considering the Lyapunov function V ( y , t) be defined as [25] $\begin{eqnarray}\displaystyle V(\boldsymbol{y},t)={\displaystyle \frac{1}{2}}e_{1}^{2}+{\displaystyle \frac{1}{2}}e_{2}^{2}+{\displaystyle \frac{1}{2}}e_{3}^{2}{\theta}_{1}+{\displaystyle \frac{1}{2}}\tilde{\boldsymbol{{\theta}}}^{\text{T}}{\Gamma}^{-1}\tilde{\boldsymbol{{\theta}}}+{\displaystyle \frac{1}{2}}\tilde{{\beta}}_{1}^{2}+P_{1}+P_{2} & & \displaystyle\end{eqnarray}$ (40) where $\tilde{{\beta}}_{1}={\beta}_{1}-\hat{{\beta}}_{1}$ . V ( y , t) ≥ 0 since P ₁ ≥ 0, P ₂ ≥ 0 and other terms are quadratic. In addition, $\begin{eqnarray}\displaystyle {\eta}_{1}\Vert \boldsymbol{y}\Vert ^{2}\leq V(\boldsymbol{y},t)\leq {\eta}_{2}\Vert \boldsymbol{y}\Vert ^{2} & & \displaystyle\end{eqnarray}$ (41) where ${\eta}_{1}=\frac{1}{2}\min \{1,\text{}\underline{m},{\Gamma}^{-1}\}$ , ${\eta}_{1}=\frac{1}{2}\max \{2,\overline{M},{\Gamma}^{-1}\}$ , m = inf(θ₁), and M = sup(θ₁). Then it yields $\begin{eqnarray}\displaystyle \dot{V}(\boldsymbol{y},t)=e_{1}{\dot{e}}_{1}+e_{2}{\dot{e}}_{2}+e_{3}{\dot{e}}_{3}{\theta}_{1}+{\displaystyle \frac{1}{2}}e_{3}^{2}\dot{{\theta}}_{1}+\tilde{\boldsymbol{{\theta}}}^{\text{T}}{\Gamma}^{-1}\dot{\tilde{\boldsymbol{{\theta}}}}+{\dot{P}}_{1}+{\dot{P}}_{2}+\tilde{{\beta}}_{1}\dot{\tilde{{\beta}}}_{1}. & & \displaystyle\end{eqnarray}$ (42)

With substituting Eqs (15), (21), (24), (26), (37), and (38) into Eq. (42) $\begin{eqnarray}\displaystyle \dot{V}\text{} & =\text{} & \displaystyle e_{1}(e_{2}-k_{1}e_{1})+e_{2}(e_{3}-k_{2}e_{2})+e_{3}\left(-{\displaystyle \frac{1}{2}}\dot{{\theta}}_{1}e_{3}+\dot{\boldsymbol{Y }}_{d}\tilde{\boldsymbol{{\theta}}}+{\tilde{N}}+N_{d}-\dot{u}_{2}-e_{2}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼{\textstyle \frac{1}{2}}e_{3}^{2}\dot{{\theta}}_{1}+\tilde{\boldsymbol{{\theta}}}^{\text{T}}{\Gamma}^{-1}(-{\Gamma}\dot{\boldsymbol{Y }}_{d}^{\text{T}}e_{3})-e_{3}(N_{d}-{\beta}_{1}\text{sgn}(e_{2}))+{\beta}_{2}{\dot{e}}_{2}\text{sgn}(e_{2})+\tilde{{\beta}}_{1}(\dot{{\beta}}_{1}-\dot{\hat{{\beta}}}_{1})\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle e_{1}e_{2}-k_{1}e_{1}^{2}+e_{2}e_{3}-k_{2}e_{1}^{2}-{\displaystyle \frac{1}{2}}\dot{{\theta}}_{1}e_{3}^{2}+e_{3}\dot{\boldsymbol{Y }}_{d}\tilde{\boldsymbol{{\theta}}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼e_{3}({\tilde{N}}+N_{d}-\dot{u}_{2})-e_{2}e_{3}+{\displaystyle \frac{1}{2}}e_{3}^{2}\dot{{\theta}}_{1}-\tilde{\boldsymbol{{\theta}}}^{\text{T}}\dot{\boldsymbol{Y }}_{d}^{\text{T}}e_{3}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼e_{3}(N_{d}-{\beta}_{1}\text{sgn}(e_{2}))+{\beta}_{2}{\dot{e}}_{2}\text{sgn}(e_{2})-\tilde{{\beta}}_{1}\dot{\hat{{\beta}}}_{1}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle e_{1}e_{2}-k_{1}e_{1}^{2}-k_{2}e_{1}^{2}+e_{3} ({\tilde{N}}+N_{d}-(K_{s}+1)e_{3}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼(\hat{{\beta}}_{1}+{\beta}_{2})\text{sgn}(e_{2}))-e_{3}(N_{d}-{\beta}_{1}\text{sgn}(e_{2}))+{\beta}_{2}{\dot{e}}_{2}\text{sgn}(e_{2})-\tilde{{\beta}}_{1}\dot{\hat{{\beta}}}_{1}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle e_{1}e_{2}-k_{1}e_{1}^{2}-k_{2}e_{1}^{2}+e_{3}{\tilde{N}}+e_{3}N_{d}-K_{s}e_{3}^{2}-e_{3}^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼e_{3}(\hat{{\beta}}_{1}-{\beta}_{1})\text{sgn}(e_{2})+{\beta}_{2}({\dot{e}}_{2}-e_{3})\text{sgn}(e_{2})-e_{3}N_{d}-\tilde{{\beta}}_{1}\dot{\hat{{\beta}}}_{1}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle e_{1}e_{2}-k_{1}e_{1}^{2}-k_{2}e_{1}^{2}+e_{3}{\tilde{N}}-K_{s}e_{3}^{2}-e_{3}^{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad -∼e_{3}(-\tilde{{\beta}}_{1})\text{sgn}(e_{2})-{\beta}_{2}k_{2}e_{2}\text{sgn}(e_{2})-\tilde{{\beta}}_{1}\dot{\hat{{\beta}}}_{1}.\end{eqnarray}$ (43)

According to $\begin{eqnarray}\displaystyle e_{1}e_{2}\leq {\displaystyle \frac{1}{2}}e_{1}^{2}+{\displaystyle \frac{1}{2}}e_{2}^{2} & & \displaystyle\end{eqnarray}$ (44) substituting Eq. (44) into (43) and combining (32), it yields $\begin{eqnarray}\displaystyle \dot{V}\text{} & \leq \text{} & \displaystyle {\displaystyle \frac{1}{2}}e_{1}^{2}+{\displaystyle \frac{1}{2}}e_{2}^{2}-k_{1}e_{1}^{2}-k_{2}e_{2}^{2}+|e_{3}|{\rho}(\Vert \boldsymbol{z}\Vert )\Vert \boldsymbol{z}\Vert -K_{s}e_{3}^{2}-e_{3}^{2}-|e_{3}\tilde{{\beta}}_{1}|-|{\beta}_{2}k_{2}e_{2}|-|\tilde{{\beta}}_{1}\dot{\hat{{\beta}}}_{1}|\nonumber\\ \displaystyle \text{} & \leq \text{} & \displaystyle -\left(k_{1}-{\displaystyle \frac{1}{2}}\right)e_{1}^{2}-\left(k_{2}-{\displaystyle \frac{1}{2}}\right)e_{2}^{2}-e_{3}^{2}+{\rho}(\Vert \boldsymbol{z}\Vert )\Vert \boldsymbol{z}\Vert ^{2}-K_{s}e_{3}^{2}\nonumber\\ \displaystyle \text{} & \leq \text{} & \displaystyle -{\eta}_{3}\Vert \boldsymbol{z}\Vert ^{2}+{\rho}(\Vert \boldsymbol{z}\Vert )\Vert \boldsymbol{z}\Vert ^{2}-K_{s}e_{3}^{2}\nonumber\\ \displaystyle \text{} & \leq \text{} & \displaystyle -\Vert \boldsymbol{z}\Vert ^{2}\left({\eta}_{3}-{\displaystyle \frac{{\rho}^{2}(\Vert \boldsymbol{z}\Vert )}{4K_{s}}}\right)\end{eqnarray}$ (45) where ${\eta}_{3}=\min \{k_{1}-\frac{1}{2},k_{2}-\frac{1}{2},1\}$ , $k_{1}>\frac{1}{2}$ and $k_{2}>\frac{1}{2}$ .

Table 2
Parameters of controllers

Parameters Value

AJC k ₁ 1

k ₂ 60

k ₃ 5

$\hat{{\beta}}_{1}(0)$ 0.6

𝛽 ₁ 12.5

𝛽₁ 2.5

𝛽₂ 0.15

𝛤 10

K _s 50

SMC B _n 3.1

A _n −0.49

𝜆 50

𝜅 6

Φ 0.002

For $\begin{eqnarray}\displaystyle K_{\text{s}}>{\displaystyle \frac{{\rho}^{2}(\Vert \boldsymbol{z}\Vert )}{4{\eta}_{3}}} & & \displaystyle\end{eqnarray}$ (46) then it can guarantee $\dot{V}\leq 0$ . Thus, by Barbalat’s lemma, ∥ z ∥→0 and $\tilde{{\beta}}_{1}\rightarrow 0$ when t →∞, which ensures the stability of the system.

Fig. 5.
Trapezoid reference command.

Fig. 6.
External load disturbance.

Fig. 7.
Experimental results of trapezoid command.

Table 3
Result of RMS position tracking error and control efforts for trapezoid command

RMS error Value (μm)

SMC 2.3

AJC 0.701

% change −69.52%

RMS efforts Value (A)

SMC 1.0023

AJC 0.8469

% change −15.53%

4. Experimental verification

Symbol	Value	Unit
K _f	50	N/A
L _d	41.4	mH
L _q	41.4	mH
𝜓_f	0.09	Wb
R _s	2.1	Ω
τ	32	mm
M	16	kg
B	8	N ⋅ s∕m

	Parameters	Value
AJC	k ₁	1
	k ₂	60
	k ₃	5
	$\hat{{\beta}}_{1}(0)$	0.6
	𝛽 ₁	12.5
	𝛽₁	2.5
	𝛽₂	0.15
	𝛤	10
	K _s	50
SMC	B _n	3.1
	A _n	−0.49
	𝜆	50
	𝜅	6
	Φ	0.002

RMS error	Value (μm)
SMC	2.3
AJC	0.701
% change	−69.52%
RMS efforts	Value (A)
SMC	1.0023
AJC	0.8469
% change	−15.53%

4.1. Experimental system

To test the feasibility and the validity of the proposed scheme applied to the PMLSM control system in practical applications, some results such as tracking performance, control efforts as well as relevant statistic are introduced experimentally.

PMLSM control system experiment platform based on DSP is shown in Fig. 3. The structure of the PMLSM control system based on DSP is shown in Fig. 4. The control chip is TMS320F2812A produced by TI. IPM is utilized as the drive and protection module for the control system. Its switching frequency is 5 kHz, which can make sure the proposed control scheme operates in a near real-time control performance for the PMSLM servo system. It is equipped with a dual power drive mode of 1.9 V CPU voltage and 3.3 V I/O voltage. The sampling time of the system is 100 μs, and the A/D converter of the DSP is utilized to convert the current into a digital signal. Besides, the position and velocity are measured by a linear scale via QEP interface. The parameters of PMLSM are listed in Table 1.

To illustrate the advantage of the AJC with the novel jerk adaption instead of a conventional switching control law with signum function, SMC with switching law is introduced for comparison, which is given as $\begin{eqnarray}\displaystyle u_{\text{SMC}}=B_{n}^{-1}[\ddot{x}_{d}(t)-A_{n}{\dot{x}}(t)+{\lambda}{\dot{e}}_{1}+{\kappa}\text{sign}(s/{\Phi})]. & & \displaystyle\end{eqnarray}$ (47) The values of parameters in the above controllers are shown in Table 2.

Fig. 8.

Experimental results of sinusoidal command.

Table 4

Result of RMS position tracking error and control efforts for sinusoidal command

RMS error	Value (μm)
SMC	0.343
AJC	0.146
% change	−57.43%
RMS efforts	Value (A)
SMC	0.1022
AJC	0.0478
% change	−53.22%

4.2. Comparative experiments for trapezoid command

For the periodic trapezoid reference command in Fig. 5 with the varying external load in Fig. 6, it is to validate the effectiveness of the proposed AJC. Figure 7 depicts the experimental results of SMC and AJC with exponential adaptive law.

Fig. 9.

Welch power spectral density estimate of velocity error for sinusoidal command.

It can be seen that their performance in position tracking error is similar in Fig. 7(a)–(b). However, the SMC bears larger initial tracking error in the transient phase, and AJC can quickly force convergence of tracking error due to the model-based feedforward. When the system is affected by external load disturbance, the maximum error of the SMC is 7 μm, but AJC is 3.3 μm, which indicates the worse performance of SMC and the robustness of AJC. Furthermore, the transient response under varying external load disturbance is also rapid. The control efforts of the two control methods are shown in Fig. 4(c)–(d). In steady-state, the control efforts of SMC are within 0.5–0.7 A, and the steady-state error of SMC is within −1.2–1.1 μm. For AJC, the control efforts in steady-state are within 0.47–0.6 A, and the steady-state error is within −0.5–0.5 μm. The error distribution statistics of tracking trajectory and control efforts are given in Fig. 7(e)–(f). The root-mean-square (RMS) values of tracking error and control efforts by two control methods are summarized in Table 3. Here it is evident that AJC reduces the RMS tracking error and control efforts. The chattering phenomenon is improved and the energy of the system is saved.

Fig. 10.

Trapezoid reference command.

Fig. 11.

Experimental results of variable sinusoidal command.

Table 5

Result of RMS position tracking error and control efforts for variable sinusoidal command

RMS error	Value (μm)
SMC	0.409
AJC	0.239
% change	−41.56%
RMS efforts	Value (A)
SMC	0.2359
AJC	0.1393
% change	−40.94%

The evolution of adaptive feedback gain $\hat{{\beta}}_{1}$ and model parameter estimate $\hat{\boldsymbol{{\theta}}}$ are shown in Fig. 7(g)–(h). The evolution of robust gain is with ripples to cater for the variation of load disturbance and converges to 𝛽₁ = 2.5 following the designed response in the steady-state, which proves the applicability of the exponential adaptive law. Although the range of model parameter estimate $\hat{\boldsymbol{{\theta}}}$ is relatively obvious in the initial transient phase, $\hat{\boldsymbol{{\theta}}}$ converges in the steady-state. Therefore, it is concluded that the proposed method is provided with strong robustness under the varying external load. AJC can generate a more stable control signal and the chattering phenomenon of control efforts is effectively eliminated.

Fig. 12.

Experimental results of different robust gain.

Table 6

Result of RMS position tracking error and control efforts for variable robust gain

RMS error	Value (μm)
𝛽₁ = 1.5	0.844
𝛽₁ = 2	0.467
𝛽₁ = 2.5	0.146
𝛽₁ = 3	0.303
𝛽₁ = 5	0.683
RMS efforts	Value (A)
𝛽₁ = 1.5	0.0153
𝛽₁ = 2	0.0323
𝛽₁ = 2.5	0.0478
𝛽₁ = 3	0.069
𝛽₁ = 5	0.196

4.3. Comparative experiments for sinusoidal command

To further verify the tracking performance of AJC, the reference command is set as a sinusoidal trajectory described by x _d = 0.001sin(4πt). Figure 8 shows the results of the experiment. If the system is in the absence of the robust controller, the position tracking error appears as a large deviation at the peak of a sinusoidal trajectory where the velocity nears zero because of the uncertain nonlinear friction. It is shown in Fig. 8(a)–(b) that SMC and AJC can improve the position tracking performance. After comparison and analysis, the position tracking error of AJC is within −0.6–0.65 μm in steady-state and it is believed that AJC can track the sinusoidal command accurately. The results indicate that AJC can effectively compensate for the uncertain nonlinear friction and weaken the creeping phenomenon at low velocity.

The control efforts of different control methods are presented in Fig. 8(c)–(d). As shown in these two figures, the steady-state control efforts of SMC are within −0.21–0.22 A, and the efforts of AJC is within −0.09–0.08 A. In comparison with SMC, the proposed method improves the chatting of control efforts significantly. For the convenience of understanding, the RMS values of tracking error and control efforts for a sinusoidal command are summarized in Table 4. In the meantime, the error distribution statistics of tracking error and control efforts are shown in Fig. 8(e)–(f). All of them validate that the AJC scheme has comparable accuracy relative to the SMC in steady-state and performs better in the chattering of control efforts. In addition, Fig. 8(g)–(h) shows the evolution of adaptive feedback gain $\hat{{\beta}}_{1}$ and model parameter estimate $\hat{\boldsymbol{{\theta}}}$ . It can be seen that both of $\hat{{\beta}}_{1}$ and $\hat{\boldsymbol{{\theta}}}$ will converge in the steady-state.

The Welch power spectral density (PSD) estimate of the velocity tracking error for a sinusoidal command is shown in Fig. 9. The shaded area represents the 95% confidence bound. It is observed that the PSD of SMC is larger at a higher frequency because the sign function and the large robust gain cause the SMC to generate chattering phenomenon, which leads to the worse tracking performance. On the contrary, the velocity error of AJC is concentrated at 0–20 Hz, and it is gradually attenuated at high frequency, which indicates that the continuous control signal generated by AJC does not carry much power at high frequency. Hence, the control effort of AJC will not excite high-order terms in unmodeled dynamics, which means that high-frequency oscillation is effectively avoided.

4.4. Comparative experiments for variable sinusoidal command

A slower variable sinusoidal trajectory in Fig. 10 is also tested. The experimental results are present in Fig. 11 and the performance indexes are shown in Table 5. In this test case, the tracking error in SMC becomes chattering and its control efforts vary in a large level, about twice of those in AJC. While the tracking error in AJC presents rather smooth. As seen, for such a slow tracking command under nonlinear uncertainties, the proposed AJC is able to compensate for the unexpected effects, and achieves improved performance in comparison to the SMC. The uncertainties are mainly attenuated by the employed RISE feedback and parameter adaptation can help to solve the chattering. Thus, AJC can effectively suppress the uncertainties while it consumes weak control efforts, and is prone to be used in practice.

4.5. Effect of different feedback gain 𝛽₁

To investigate the effect of different feedback gain 𝛽₁ for sinusoidal trajectory, the experimental results of variable robust gain are shown in Fig. 12. In this test, the value of 𝛽₁ is chosen as 1.5, 2, 2.5, 3 and 5, respectively. To show compensation results clearly, Table 6 gives the RMS values of position tracking error and control efforts under different values of feedback gain 𝛽₁.

From these compensation results, it is found that there is an optimal choice to set feedback gain 𝛽₁ = 2.5. When the value of 𝛽₁ deviates away from 2.5, the position tracking errors are larger. Meanwhile, the chattering phenomenon of the control effect is limited in a rational range. Although the tracking error is convergence in steady-state when 𝛽₁ = 5, there is an overshoot in the transient phase and the serious chatting phenomenon is unacceptable. It is predictive that the larger value of 𝛽₁ leads to an increased jerk, which makes the proposed method is similar to the SMC with discontinuous control law. On the contrary, if the value of 𝛽₁ is selected to be 1.5, it is too small to cater to the uncertainties of the system so that the tracking performance is worse. On the basis of the above-mentioned results, it is suggested that an appropriate selection of the feedback gain 𝛽1 improves the tracking performance of the system without adding significant cost in control efforts.

5. Conclusion

In order to achieve the high-precision and high-velocity control performance PMLSM servo system, considering the existence of uncertainties such as external load disturbance, nonlinear friction, and unmodeled dynamics, a developed control scheme combined the model-based feedforward with AJC based on RISE feedback is proposed. The model feedforward control is utilized to compensate for the parametric uncertainties and AJC can form a stable and continuous control effort in the presence of external load and unmodelled dynamics. Adaptive law of robust term guarantees bounded robust gain in AJC. The stability of the control system is derived from the Lyapunov theorem. The experimental results show that the control scheme effectively suppresses the uncertainties and the chattering phenomenon is declined, which adequately verifies good control performance of the PMLSM servo system.

Footnotes

Acknowledgements

This work was supported in part by the Liaoning Provincial Natural Science Foundation of China, under grant 20170540677.

References

Wang

Yang

Zhang

Cao

and Li

, Inner-loop design for PMLSM drives with thrust ripple compensation and high-performance current control, IEEE Trans. Ind. Electron. 65(12) (2018), 9905–9915.

Ulu

N.G.

Ulu

and Cakmakci

, Design and analysis of a modular learning based cross-coupled control algorithm for multi-axis precision positioning systems, Int. J. Control Autom. Syst. 14(1) (2016), 272–281.

Lin

F.-J.

Teng

L.-T.

and Chu

, A robust recurrent wavelet neural network controller with improved particle swarm optimization for linear synchronous motor drive, IEEE Trans. Power Electron. 23(6) (2008), 3067–3078.

El-Sousy

F.F.M.

and Ali Abuhasel

, Self-organizing recurrent fuzzy wavelet neural network-based mixed H₂/H_∞ adaptive tracking control for uncertain two-axis motion control system, IEEE Trans. Ind. Appl. 52(6) (2016), 5139–5155.

Ying-Shieh

, Design and implementation of a high-performance PMLSM drives using DSP chip, IEEE Trans. Ind. Electron. 55(3) (2008), 1341–1351.

Wang

Liang

Tian

Zhao

and Zhang

, A flexure-based kinematically decoupled micropositioning stage with a centimeter range dedicated to micro/nano manufacturing, IEEE/ASME Trans. Mechatronics 21(2) (2016), 1055–1062.

Liu

T.-H.

Lee

Y.-C.

and Chang

Y.-H.

, Adaptive controller design for alinear motor control system, IEEE Trans. Aerosp. Electron. Syst. 40(2) (2004), 601–613.

Huang

C.-I.

and Fu

L.-C.

, Adaptive approach to motion controller of linear induction motor with friction compensation, IEEE/ASME Trans. Mechatronics 12(4) (2007), 480–490.

Lin

C.-J.

Yau

H.-T.

and Tian

Y.-C.

, Identification and compensation of nonlinear friction characteristics and precision control for a linear motor stage, IEEE/ASME Trans. Mechatronics 18(4) (2013), 1385–1396.

10.

Shen

J.-C.

Q.-Z.

C.-H.

and Jywe

W.-Y.

, Sliding-mode tracking control with DNLRX model-based friction compensation for the precision stage, IEEE/ASME Trans. Mechatronics 19(2) (2014), 788–797.

11.

Ali Masood Cheema

Edward Fletcher

Farshadnia

Xiao

and Faz Rahman

, Combined speed and direct thrust force control of linear permanent-magnet synchronous motors with sensorless speed estimation using a sliding-mode control with integral action, IEEE Trans. Ind. Electron. 64(5) (2017), 3489–3501.

12.

Beltran

Ahmed-Ali

and Benbouzid

, High-order sliding-mode control of variable-speed wind turbines, IEEE Trans. Ind. Electron. 56(9) (2009), 3314–3321.

13.

Lin

F.-J.

Hwang

J.-C.

Chou

P.-H.

and Hung

Y.-C.

, FPGA-based intelligent-complementary sliding-mode control for PMLSM servo-drive system, IEEE Trans. Power Electron. 25(10) (2010), 2573–2587.

14.

Lin

F.-J.

Chou

P.-H.

Chen

C.-S.

and Lin

Y.-S.

, DSP-based cross-coupled synchronous control for dual linear motors via intelligent complementary sliding mode control, IEEE Trans. Ind. Electron. 59(2) (2012), 1061–1073.

15.

Yao

Deng

and Jiao

, RISE-based adaptive control of hydraulic systems with asymptotic tracking, IEEE Trans. Autom. Sci. Eng. 14(3) (2017), 1524–1531.

16.

Patre

P.M.

MacKunis

Makkar

and Dixon

W.E.

, Asymptotic tracking for systems with structured and unstructured uncertainties, IEEE Trans. Control Syst. Technol. 16(2) (2008), 373–379.

17.

Shin

Jin Kim

Kim

and Dixon

W.E.

, Autonomous flight of the rotorcraft-based UAV using RISE feedback and NN feedforward terms, IEEE Trans. Control Syst. Technol. 20(5) (2012), 1392–1399.

18.

Deng

and Yao

, Adaptive integral robust control and application to electromechanical servo systems, ISA Trans. 67 (2017), 256–265.

19.

Xian

Dawson

D.M.

deQueiroz

M.S.

and Chen

, A continuous asymptotic tracking control strategy for uncertain nonlinear systems, IEEE Trans. Autom. Control 49(7) (2004), 1206–1211.

20.

Patre

P.M.

MacKunis

Dupree

and Dixon

W.E.

, Modular adaptive control of uncertain Euler–Lagrange systems with additive disturbances, IEEE Trans. Autom. Control 65(1) (2011), 155–160.

21.

Patre

P.M.

MacKunis

Johnson

and Dixon

W.E.

, Composite adaptive control for Euler–Lagrange systems with additive disturbances, Automatica 46(1) (2010), 140–147.

22.

Cai

de Queiroz

and Dawson

D.M.

, Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance, IEEE Trans. Autom. Control 51(3) (2006), 524–529.

23.

Yao

and Sun

, Adaptive RISE control of hydraulic systems with multilayer neural-networks, IEEE Trans. Ind. Electron. 66(11) (2019), 8638–8647.

24.

Kamaldin

Chen

S.-L.

Teo

C.S.

Lin

and Tan

K.K.

, A novel adaptive jerk control with application to large workspace tracking on a flexure-linked dual-drive gantry, IEEE Trans. Ind. Electron. 66(7) (2019), 5353–5363.

25.

Patre

P.M.

MacKunis

Makkar

and Dixon

W.E.

, Asymptotic tracking for systems with structured and unstructured uncertainties, IEEE Trans. Control Syst. Technol. 16(2) (2008), 373–379.

A novel high-precision motion control for permanent magnet linear synchronous motor servo system

Abstract

Keywords

1. Introduction

2. Modeling for PMLSM servo systems

2.1. Dynamics of PMSLM

3. Proposed control systems

3.1. Design schemes

3.1.1. To design filtered error vector

4.1. Experimental system

4.4. Comparative experiments for variable sinusoidal command

4.5. Effect of different feedback gain 𝛽1

5. Conclusion

Footnotes

Acknowledgements

References

4.5. Effect of different feedback gain 𝛽₁