Linearizing control of a permanent magnet linear synchronous motor with inverse system scheme plus an internal model controller

Abstract

Aimed at enhancing the control precision and speed tracking rapidity of a permanent magnet linear synchronous motor (PMLSM), a novel control scheme combining the inverse system method and the internal model control is proposed in this paper. A decoupling pseudo-linear system can be obtained by cascading the inverse model of the PMLSM with the original system. In addition, in order to improve the robustness of the whole system and to reject the influence of the unmodeled dynamics and various disturbances to the linearizing control accuracy, the internal model controller is used as the extra closed-loop controller for the pseudo-linear system. Therefore, the presented control scheme incorporates the advantages of both the inverse system method and the internal model control. The effectiveness of the proposed control scheme is verified by simulation and experimental results at various operations. Simulation studies are performed to prove the effectiveness of the proposed control scheme.

Keywords

Permanent magnet linear synchronous motor (PMLSM)inverse system linearizing control internal model control

1. Introduction

As demand increases for the fast and precise linear motion in the automatic control systems, the linear motor-driven motion systems have received significant attention for high-speed/high-accuracy motion control in renewable energy devices, transportation, automation of manufacturing processes, and pumping applications [1]. Compared with traditional rotary motors with ball screws and reduction gears [2], linear motors can directly produce thrust to loads without the mechanical transmission [3]. Among various kinds of linear motors, permanent magnet linear synchronous motors (PMLSMs) have some main merits of high speed and high precision in long-distance locations, high thrust force density, higher reliability, low thermal losses and low noise, and rapid dynamic response, and hence, PMLSMs are being actively researched and developed around the world [4, 5]. Thus, the increasing applications of the PMLSMs imply the vital importance of developing robust, simple, and fast control methods for PMLSMs to suit the primary requirements of these emerging industrial applications and are of prime concern.

The traditional proportional-integral-differential (PID) control scheme has been already widely applied in the PMLSMs. However, the PMLSM is a multivariable, nonlinear, and strongly coupling system with the parameter variations as well as unavoidable disturbances. Thus, it is very difficult for the PID control scheme to obtain a sufficiently high performance for this kind of nonlinear plants. In the past decades, with the rapid development of modern control theories [6, 7, 8] and motor control techniques, considerable works have been devoted to presenting a variety of nonlinear control schemes for PMLSMs, such as direct thrust force control [9, 10], adaptive disturbance compensation control [11], predictive control [12], backstepping control [13], fuzzy control [14, 15], sliding mode control [16, 17], etc. These nonlinear control schemes improve the control performance of BM systems in different aspects.

In addition to the aforementioned nonlinear control schemes, many researchers have concentrated on the decoupling and linearizing control approaches for PMLSMs and other linear motor drive systems in recent year. Generally, decoupling and linearizing control approaches mainly including intelligence-based approaches [18, 19, 20] as well as analytical-based decoupling and linearizing control approaches [21, 22]. The intelligent-based decoupling and linearizing control schemes are acceptable for the controlled plants where precise mathematical models are easy to obtain. Nevertheless, a large amount of data and large allocations of computer resources are needed that are often typically restricted in the PMLSM control system. Decoupling and linearizing control schemes mainly can be divided into the differential geometry approach and inverse system method [23, 24]. Compared with the differential geometry method, the inverse system method is relatively simple to implement in practical applications. Thus, the inverse system method is employed for the PMLSM in this paper.

However, the inverse system method may adversely influence the tracking, disturbance rejection and robustness properties because of the uncertainties, unmodeled dynamics, and parameters variation. In addition, the actual controlled plant of the PMLSM system is not dynamically equivalent to its ideal mathematical model since the model is obtained without consideration about the system noises, bandwidth of the amplifiers, computation delay, etc. Thus, to overcome the disadvantages, the internal model control theory is used to design the extra controllers to improve the robustness of the whole PMLSM system [25, 26, 27].

One objective of this paper is to show that the nonlinear PMLSM system with multiple coupled states and parameter variations can be successfully linearized by the inverse system method. The other objective of this paper is to demonstrate that the issues of control accuracy caused by the uncertainties, unmodeled dynamics, and parameters variation can be effectively resolved by employing the internal model controllers.

The paper is arranged in the following manner. In Section 2, the inverse system scheme is employed for linearizing control of the PMLSM system. And then, the internal model control theory is adopted to design the robust closed-loop controller in Section 3. After that, simulation researches are carried out to verify the effectiveness of the proposed control scheme in Section 4. Finally, Section 5 concludes the paper.

2. Inverse system modeling

2.1 Dynamic model of the PMLSM

The adopted PMLSM comprises a stationary long “secondary” which includes neodymium-iron-boron (NdFeB) PMs with guidance rail and linear scale, and a moving short “primary” which includes the core armature windings and Hall sensing elements. The electromagnetic thrust force of the PMLSM is generated by the coupling function of PM magnetic field and the magnetic field produced by the armature windings fed by a current-controlled pulse width modulation (PWM) voltage source inverter (VSI). Neglecting flux distortions, the voltage equations of the PMLSM can be given as

$\left\{{{\begin{array}[]{*{20}c}{u_{d}=R_{s}i_{d}+\frac{d\psi_{d}}{dt}-\frac{% \pi}{\tau}n_{p}v\psi_{q}}\\ {u_{q}=R_{s}i_{q}+\frac{d\psi_{q}}{dt}+\frac{\pi}{\tau}n_{p}v\psi_{d}}\\ \end{array}}}\right.$ (1)

with

$\left\{{{\begin{array}[]{ll}{\psi_{d}=L_{d}i_{d}+\psi_{f}}\\ {\psi_{q}=L_{q}i_{q}}\\ \end{array}}}\right.$ (2)

where $\psi_{d}$ and $\psi_{q}$ are the d- and q-axis flux linkages, respectively, $\psi_{f}$ is the primary PM flux linkage, $i_{d}$ and $i_{q}$ are the d- and q-axis primary currents, respectively, $u_{d}$ and $u_{q}$ are the d- and q-axis primary voltages, respectively, $R_{s}$ is the primary resistance, $\tau$ is the polar distance, $L_{d}$ and $L_{q}$ are the d- and q-axis primary inductances, respectively, $n_{p}$ is the number of pole pairs, $v$ is the primary linear speed.

The electromagnetic thrust force equation can be given by

$F_{e}=\frac{P_{e}}{v}=\frac{3\pi n_{p}\left[{\psi_{f}i_{q}+(L_{d}-L_{q})i_{d}i% _{q}}\right]}{2\tau}$ (3)

where $F_{e}$ is the electromagnetic thrust force and $P_{e}$ is the electromagnetic power. According to Newton’s second law, the motion equation of the PMLSM is

$F_{e}=M\frac{dv}{dt}+Bv+F_{l}+F_{d}$ (4)

where $M$ is the mass of the primary inertia, $B$ is the viscous friction factor, $F_{d}$ is the load force, and $F_{l}$ is the external disturbance term.

Therefore, a 3-order nonlinear model of the dynamic equation of the PMLSM can be given as the following explicit form:

$\left\{{{\begin{array}[]{l}{\frac{di_{d}}{dt}=-\frac{1}{L_{d}}\left({R_{s}i_{d% }-\frac{\pi}{\tau}L_{q}vi_{q}-u_{d}}\right)}\\ {\frac{di_{q}}{dt}=-\frac{1}{L_{q}}\left({\frac{\pi}{\tau}\psi_{f}v+\frac{\pi}% {\tau}L_{d}vi_{d}+R_{s}i_{q}-u_{q}}\right)}\\ {\frac{{d}v}{{d}t}=\frac{1}{M}\left\{{\frac{3\pi}{2\tau}n_{p}\left[{\left({L_{% d}-L_{q}}\right)i_{d}+\psi_{f}}\right]i_{q}-Bv-\left({F_{d}+F_{l}}\right)}% \right\}}\\ \end{array}}}\right.$ (5)

Based on the field-oriented control method with $i_{d}=$ 0, Eq. (5) can be approximately expressed as:

$\left\{{{\begin{array}[]{l}{\frac{di_{q}}{dt}=-\frac{1}{L_{q}}\left({\frac{\pi% }{\tau}\psi_{f}v+R_{s}i_{q}-u_{q}}\right)}\\ {\frac{{d}v}{{d}t}=\frac{1}{M}\left[{\frac{3\pi}{2\tau}n_{p}\psi_{f}i_{q}-Bv-% \left({F_{d}+F_{l}}\right)}\right]}\\ \end{array}}}\right.$ (6)

From Eq. (6), it can be seen that the dynamic model of the PMLSM is nonlinear.

State variables are chosen as

$x=\left[{x_{1},x_{2}}\right]=\left[{i_{q},v}\right]$ (7)

Input variable is chosen as

$u=u_{q}$ (8)

Output variable is chosen as

$y=v$ (9)

Substituting Eqs (7)–(9) into Eq. (6), we can get the 2-order state equation of the PMLSM drive system as

$\left\{{{\begin{array}[]{l}{\dot{x}_{1}=-\frac{1}{L_{q}}\left({\frac{\pi}{\tau% }\psi_{f}x_{2}+R_{s}x_{1}-u}\right)}\\ {\dot{x}_{2}=\frac{1}{M}\left[{\frac{3\pi}{2\tau}n_{p}\psi_{f}x_{1}-Bx_{2}-% \left({F_{d}+F_{l}}\right)}\right]}\\ \end{array}}}\right.$ (10)

2.2 Analysis of the Invertibility of the PMLSM

Based on the inverse system theory and Interactor algorithm, the outputs of the PMLSM drive system are differentiated until the derivatives contain input variable $u$ obviously

$\displaystyle\dot{y}=\dot{x}_{2}=\frac{1}{M}\left[{\frac{3\pi}{2\tau}n_{p}\psi% _{f}x_{1}-Bx_{2}-\left({F_{d}+F_{l}}\right)}\right]$ (11) $\displaystyle\ddot{y}=\ddot{x}_{2}=\frac{-3\pi}{2L_{q}\tau M}n_{p}\psi_{f}% \left({\frac{\pi}{\tau}\psi_{f}x_{2}+R_{s}x_{1}-u}\right)-\frac{B}{M^{2}}\left% [{\frac{3\pi}{2\tau}n_{p}\psi_{f}x_{1}-Bx_{2}-\left({F_{d}+F_{l}}\right)}\right]$ (12)

Then, the Jacobin matrix can be achieved as:

$\frac{\partial\ddot{y}}{\partial u}=\frac{3\pi n_{p}\psi_{f}}{2L_{q}\tau M}\neq 0$ (13)

The relative orders of the system is $\alpha=2=n$ , and it can be proved that the inverse of original system is existent.

2.3 Analytic solution of the inversion

In this part, the analytic solution of the inversion of the PMLSM drive system will be resolved. According to the theory of the inverse system, the new input variable is defined:

$\varphi=\ddot{y}_{d}$ (14)

where $y_{d}$ is used to express the expected output. By substituting the variable $\varphi$ for $\ddot{y}$ in Eq. (12), an equation with respect to $u$ can be obtained. By solving the equation, the inversion of original PMLSM drive system can be constructed as

$u=\left({R_{s}+\frac{BL_{q}}{M}}\right)x_{1}+\left({\frac{\pi}{\tau}\psi_{f}-% \frac{2B^{2}L_{q}\tau}{3\pi n_{p}\psi_{f}M}}\right)x_{2}+\frac{2L_{q}\tau}{3% \pi n_{p}\psi_{f}M}\left[{M^{2}\varphi-B\left({F_{d}+F_{l}}\right)}\right]$ (15)

By combining the inversion of the PMLSM drive system with the original system, a new pseudo-linear system consisting can be obtained, as shown in Fig. 1.

Figure 1.

Diagram of pseudolinear system.

3. Design of the internal model controllers

3.1 2-DOF internal model controller

The design and analysis of a 2-DOF internal model controller for the linearized PMLSM is developed in this section. The internal model control technique relies on the “internal model” principle including a model of the plant to be controlled in the control structure. In addition, a 2-DOF internal model controller includes two internal model controllers, i.e., a set point controller $G_{c1}(s)$ and a disturbance rejection one $G_{c2}(s)$ , as shown in Fig. 2. The set point controller is in an open loop form and the disturbance rejection one is in an inner-feedback loop structure. The two internal model controllers $G_{c1}(s)$ as well as $G_{c2}(s)$ consist of a filter and the inversion of the internal model.

From Fig. 1, the pseudo-linear system can be given as

$G_{v}(s)=1\mathord{\left/{\vphantom{1{s^{2}}}}\right.\kern-1.2pt}{s^{2}}$ (16)

The transfer function Eq. (16) is the nominal model of the primary linear speed $v$ of the PMLSM drive system. Owing to errors of the model and the uncertainty of the parameters, the composition of the actual control plant and its inversion Eq. (15) do not exactly equal to the linear system Eq. (16). Therefore, the actual system can be expressed as

$G(s)=G_{v}(s)+G_{d}(s)$ (17)

where $G_{d}(s)$ denotes an arbitrary uncertainty.

The schematic representation of the 2-DOF internal model controller for the pseudo-linear system of the primary linear speed $v$ is shown in Fig. 2, where $G_{c1}(s)$ and $G_{c2}(s)$ are internal model controllers, $G(s)$ and $G_{x}(s)$ are the real plants to be controlled and the internal model, respectively, $d$ is the outer disturbance input, $\bar{v}$ is the control input, $v^{\ast}$ and $v$ are the reference input and output of primary linear speed, respectively, and $e$ is the error which represents for the effect of disturbances and plant/model mismatch if exists.

Figure 2.

The 2-DOF internal model controller structure.

According to Fig. 2, the output can be calculated as

$v(s)=\frac{v^{\ast}(s)G_{c1}(s)G(s)+d(s)(1-G_{c2}(s)G_{v}(s))}{1+G_{c2}(s)G(s)}$ (18)

If $G_{x}(s)=G(s)$ , Eq. (18) can be rewritten as

$v(s)=v^{\ast}(s)G_{c1}(s)G_{v}(s)+d(s)(1-G_{c2}(s)G_{v}(s))$ (19)

From Eq. (19), it can be seen that the tracking characteristic only relies on $G_{c1}(s)$ , while the disturbance rejection property only depends on $G_{c2}(s)$ .To track the reference input $v$ without any steady-state error and to improve the system robustness, low-pass filters $Q_{1}(s)$ and $Q_{2}(s)$ are employed

$\left\{{{\begin{array}[]{*{20}c}{G_{c1}(s)=G_{v}^{-1}(s)Q_{1}(s)}\\ {G_{c2}(s)=G_{v}^{-1}(s)Q_{2}(s)}\\ \end{array}}}\right.$ (20)

The low-pass filters $Q_{1}(s)$ and $Q_{2}(s)$ are commonly chosen as

$\left\{{{\begin{array}[]{*{20}c}{Q_{1}(s)=(1+\lambda_{1}s)^{-2}}\\ {Q_{2}(s)=(1+\lambda_{2}s)^{-2}}\\ \end{array}}}\right.$ (21)

where $\lambda_{1}$ > 0 and $\lambda_{2}>$ 0.

Therefore, Fig. 2 is equivalent to Fig. 3,

$\left\{{{\begin{array}[]{l}{T(s)=\frac{G_{c1}(s)}{G_{c2}(s)}=\frac{(1+\lambda_% {2}s)^{2}}{(1+\lambda_{1}s)^{2}}}\\ {G_{c}(s)=\frac{G_{c2}(s)}{1-G_{c2}(s)G_{v}(s)}=\frac{s^{2}}{(1+\lambda_{2}s)^% {2}-1}}\\ \end{array}}}\right.$ (22)

Figure 3.

The closed-loop system of the 2-DOF internal model controller.

3.2 Characteristics of tracking, disturbance rejection, and robustness for the designed 2-DOF internal model controller

From Fig. 3, the output error transfer function of the closed-loop system can be defined as

$\displaystyle E(s)=v^{\ast}(s)-v(s)=\frac{(1+G_{c}(s)G(s)(1-T(s)))v^{\ast}(s)-% d(s)}{1+G_{c}(s)G(s)}$ (23)

According to the tracking performance, letting $d(s)=$ 0, Eq. (23) can be written as

$E(s)=\frac{1+G_{c}(s)G(s)(1-T(s))}{1+G_{c}(s)G(s)}v^{\ast}(s)$ (24)

Assuming $G(s)=G_{x}(s)$ , and substituting Eqs (16) and (22) into Eq. (24), we can obtain

$E(s)=(1-Q_{1}(s))v^{\ast}(s)=\left[{1-(1+\lambda_{1}s)^{-2}}\right]v^{\ast}(s)$ (25)

From Eq. (25), it is obvious that $E(s)$ is characterized with a high-pass characteristic. Moreover, a conclusion can also be drawn that the smaller $\lambda_{1}$ is, the faster the $E(s)$ approximates to zero, and the better tracking performance will be achieved.

Let $v^{\ast}(s)=0$ , we can get

$E(s)=v^{\ast}(s)-v(s)=-\frac{d(s)}{1+G_{c}(s)G(s)}$ (26)

Assuming $G(s)=G_{x}(s)$ , and substituting Eqs (16) and (22) into Eq. (26) yield

$E(s)=-(1-Q_{2}(s))d(s)=\left[{(1+\lambda_{2}s)^{-2}-1}\right]d(s)$ (27)

From Eq. (27), it is obvious that the output error equation $E(s)$ also has a high-pass characteristic. In addition, a conclusion could be drawn that the smaller the $\lambda_{2}$ , the better the performance of disturbance rejection.

Based on the analysis of system performance indexes and robust stability, it can be realized that the effective independent control of tracking and robustness performance by choosing optimal control parameter sets $\lambda_{1}$ and $\lambda_{2}$ .

Table 1

Parameters of the proposed method

Parameter	Value	Parameter	Value
$\lambda_{1}$	0.2 (0.11)	$\lambda_{2}$	0.14 (0.08)

Table 2

Parameters of the traditional PID method

Parameter	Value	Parameter	Value
$K_{p}$	2.5	$K_{i}$	1.03
$K_{d}$	2 (2.4)

Figure 4.

Control block diagram of the PMLSM drive system.

4. Simulation results

The control block diagram and the design flow chart of the entire PMLSM drive system using the proposed control scheme are shown in Figs 4 and 5, respectively. The parameters of the PMLSM drive system are as follows. The average electromagnetic thrust force is 275 N. The mass of the primary inertia is 3.2 kg. The viscous friction factor is 8 N $\cdot$ s/m. The flux linkage of permanent magnet is 0.172 Wb. The primary resistance is 2.6 $\Omega$ . The primary inductances of d- (q-) axial is 0.278 H. The polar distance is 26 mm. The pole-pair number is 3. The control parameters of the proposed control scheme and the traditional PID control method are given in Tables 1 and 2. The proposed control algorithm is implemented in the Matlab/Simulink.

Figure 5.

Flow chart of the control system.

Comparative simulation studies between the inverse system plus internal model control and the traditional PID control method have been performed to evaluate the effectiveness of the proposed control scheme for the BPMSM system. Some simulation and comparison results are presented here.

In order to test the tracking, robustness, and disturbance rejection performances of the PMLSM drive system, the system is subjected to the reference speed step, parameter variation and external disturbance, respectively. At $t=$ 0 s, the primary linear speed reference input increases from 0 m/s to 2 m/s. After 8 s, the d- and q-axis primary inductances $L_{d}$ and $L_{q}$ increase by 15%, and then a 60 N external step load disturbance is imposed on the PMLSM drive system at $t=$ 15 s. In addition, the comparative simulation results with different control values of $k_{d}$ in the PID control scheme as well as $\lambda_{1}$ and $\lambda_{2}$ in the proposed control scheme are also developed, as illustrated in Figs 6 and 7.

Figure 6.

Tracking, robustness, and disturbance rejection properties of the PID control scheme. (a) $k_{d}=$ 2. (b) $k_{d}=$ 2.4.

Figure 7.

Tracking, robustness, and disturbance rejection properties of the proposed control scheme. (a) $\lambda_{1}=$ 0.2 and $\lambda_{2}=$ 0.14. (b) $\lambda_{1}=$ 0.2 and $\lambda_{2}=$ 0.08. (c) $\lambda_{1}=$ 0.11 and $\lambda_{2}=$ 0.08.

According to Fig. 6a and b, as for the traditional PID control method, when $k_{d}$ increases from 2 to 2.4, the settling time the speed response increases, while the overshoots of the speed response decreases no matter whether there is an external disturbance or parameter variations. Therefore, a conclusion can be drawn that, by using the traditional PID control scheme, the control performances of the tracking and disturbance rejection cannot independently regulated. Therefore, it is difficult to seek the optimal control parameters of the PID control method since there are inevitable modeling errors. Even if the relative optimal control coefficients have been found sometimes, the control performance of the tracking, disturbance rejection and robustness with these fixed coefficients cannot be satisfied simultaneously.

On the contrary, from Fig. 7a–c, in regard to the proposed inverse system scheme plus internal model controller, the setting time and the overshoot of the speed all decrease when the values of $\lambda_{1}$ and $\lambda_{2}$ decrease. Specifically, from Fig. 7a and b, when $\lambda_{2}$ decreases, the properties of the robustness and disturbance rejection are enhanced without any influences on the tracking property; from Fig. 7b and c, when $\lambda_{1}$ decreases, the performance of tracking are enhanced without any influences on the properties of the robustness and disturbance rejection. Thus, a conclusion could be drawn that, by using the proposed control scheme, the control properties of the tracking and disturbance rejection (or robustness) can be independently adjusted, which agrees with the analysis in Section 3, that is, the control parameters can be simply adjusted by employing the proposed control scheme, and the control performances of the tracking, robustness and disturbance rejection could be simultaneously improved.

5. Conclusion

To achieve the high-performance control of the PMLSM drive system, this paper presents a linearizing control scheme consisting of inverse system plus an internal model controller. The simulation results demonstrate that, by using the proposed control scheme, linearizing control of the PMLSM drive system can be realized successfully, which can effectively avoid the disadvantages of the traditional PID scheme. In addition, by employing the 2-DOF internal model controller as the closed-loop controller for the pseudo-linear system, the control accuracy can be improved effectively, and the influence of the unmodeled dynamics can be eliminated as well. Moreover, we can adjust the control performances of the tracking, robustness, and disturbance rejection at the same time by regulating the values of control parameters $\lambda_{1}$ and $\lambda_{2}$ . Although the proposed control strategy requires sophisticated hardware, intensive computation and experiential knowledge, the rapid development of computer control technology will solve these problems. The control scheme proposed in this paper is still at the level of theoretical research. In the following study, a permanent magnet linear synchronous motor is designed and optimized, which will be used to verify the effectiveness of the proposed control method [28, 29]. In addition, through experimental studies the specific difference between the proposed control scheme and PID-controller could be carried out.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Projects 51305170, 51475213, and 51475214, the Natural Science Foundation of Jiangsu Province of China under Projects BK20170071 and BK20141301, the China Postdoctoral Science Foundation under Projects 2016M601726 and 2015T80508, the Six Categories Talent Peak of Jiangsu Province under Projects 2015-XNYQC-003, 2014-ZBZZ-017, and 2016-GDZB-096, the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Project KYCX17_1815, the Key Project of Natural Science Foundation of Jiangsu Higher Education Institutions under Project 17KJA460005, the “333 project” of Jiangsu Province under Project BRA2017441, Zhenjiang Key Research and Development Project under Project GY2016003, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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