Inverse system decoupling sliding mode control strategy of BLIM considering current dynamics

Abstract

Bearingless induction motor (BLIM), is a nonlinear, multivariable and strongly coupled object. In order to improve its dynamic decoupling control performance, and overcome the influences of the load disturbance and the motor parameter variation, under the conditions of considering the stator current dynamics of the torque system, the inverse system model of BLIM system is established firstly, and by the inverse system method, the BLIM system is decoupled into four second-order pseudo linear subsystems. And then, according to the sliding mode control (SMC) theory, the SMC regulator is designed for each subsystem, an exponential approach law is used to reduce the chattering of SMC system; the stability of SMC system is verified by the Lyapunov method. Simulation experimental results have shown that the inverse system decoupling SMC system not only has an excellent dynamic decoupling control performance, but also has a faster response speed and a stronger robustness.

Keywords

Bearingless induction motor current dynamics inverse system decoupling sliding mode control anti disturbance ability

1. Introduction

The motor supported by mechanical bearing can’t meet the requirements of long time and high-speed operation [1], then the AC motor supported by magnetic bearing is developed and widely used in high-speed drive field [2–5]. But the motor supported by magnetic bearing still has some disadvantages, such as the higher magnetic suspensions power waste, the over-speed difficulty, etc. [9,10]. Bearingless motor is a new type of AC motor that is suitable for high-speed operation, which is proposed based on the structure comparability between the magnetic bearing and the conventional AC motor [11–13]. The bearingless motor can simultaneously realize the rotation drive and radial suspension control of the rotor, while it has the advantages of compact structure and high critical speed [14–16], and then it has a broad application prospects. For common bearingless motor, there are two sets of stator windings, include a set of torque windings (with pole-pair p ₁ and current angular frequency ω₁) and a set of suspension control windings (with pole-pairs p ₂ and current angular frequency ω₂) [17,18]. When the two sets of stator windings meet the conditions of “p ₂ = p ₁ ±1, ω₂ = ω₁”, a controllable magnetic suspension force can be produced [9,19], which can be used to control the radial suspension motion of the rotor.

The bearingless control technology is applicable for all kinds of AC motor. Hereinto, for the robust and compact structure, the bearingless induction motor (BLIM) has become a hot research topic. But there exists a complicated coupling relationship within BLIM. To achieve its high performance dynamic control, it is necessary to achieve the dynamic decoupling between relevant variables. The inverse system method is a feedback linearization method, it is applicable for the decoupling control of the multi variable, nonlinear and strong coupling object. About the inverse system decoupling of BLIM system, the inverse system decoupling strategies under the rotor-, stator- and air gap-flux orientation conditions have been studied [20–22]; however, because the stator current dynamics is not considered, the exists a unpredictable load torque variable in the inverse system model. Under the conditions of considering the stator current dynamics of torque system, the related research on the rotor flux orientation inverse system decoupling method of BLIM has been done [23], and in the derived inverse system model, there is no load torque variable again; but because a traditional PID regulator is used for each subsystem, the dynamic control performance is limited. In the upper inverse system decoupling control methods, the influence of motor parameters variation is not discussed. In actual application, the motor parameter, especially the rotor resistance will change with the temperature rise of BLIM, which will influence the accuracy of the rotor flux orientation, and destroy the dynamic decoupling performance of BLIM system. In addition, the anti load disturbance performance of BLIM system is another important problem to be considered. The sliding mode control (SMC) method can design relevant sliding mode according to the actual need, which is independent of the object’s parameters and the external disturbance, and then it is more suitable for the nonlinear, multi-variable controlled object with time-varying parameters. In order to improve the control performance of BLIM system, the study on the SMC algorithm of speed regulator has been done [24]. Aiming at a bearingless permanent magnet synchronous motor, and on the basis of the inverse system decoupling of magnetic suspension force, the SMC strategy is studied in reference [25], and then an idea for SMC the control of bearingless motors based on inverse system decoupling has been provided.

In this paper, the BLIM is taken as a controlled object, an inverse system decoupling sliding mode control strategy is proposed. Under the conditions of considering the stator current dynamics of torque system, the BLIM system is decoupled by the inverse method; after then, aiming at the decoupled pseudo linear subsystem, the SMC regulator is designed, and the proposed control strategy has been verified by simulation results. The outline of this paper is as follows: considering the stator current dynamics of torque system, Section 2 gives the inverse system model and decoupling method of the BLIM system; Section 3 introduces the design method of the SMC regulator; the verification and analysis are described in Section 4; Section 5 summaries the full text and presents the conclusion.

2. Inverse system decoupling of BLIM

2.1. Mathematical model of BLIM system

Defining: 𝛼 −𝛽 is the stationary coordinate system; d − q is the synchronous coordinate system oriented by the rotor flux-linkage of torque system. Then the model of controllable magnetic suspension force can be expressed as follow [23]: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}F_{{\alpha}}=K_{m}(i_{s2d}{\psi}_{1d}+i_{s2q}{\psi}_{1q})\\ F_{{\beta}}=K_{m}(i_{s2d}{\psi}_{1q}-i_{s2q}{\psi}_{1d})\end{array}\right. & & \displaystyle\end{eqnarray}$ (1)

Where: K _m is the stiffness coefficient of controllable suspension force; i _s2d and i _s2q are the suspension control current components in d − q coordinate system; 𝜓_1d and 𝜓_1q are the air gap flux-linkage components in d − q coordinate system; F _𝛼 and F _𝛽 are controllable magnetic suspension force components in 𝛼 −𝛽 coordinate system.

From the kinetic principle, the radial suspension motion equations of rotor can be expressed as follows: $\begin{eqnarray}\displaystyle m\ddot{{\alpha}}=F_{{\alpha}}+f_{s{\alpha}},\quad m\ddot{{\beta}}=F_{{\beta}}+f_{s{\beta}} & & \displaystyle\end{eqnarray}$ (2)

Where: m is the rotor mass; f _s𝛼 and f _s𝛽 are the unbalanced unilateral magnetic force components in 𝛼 −𝛽 coordinate system, f _s𝛼 = k _s𝛼, f _s𝛽 = k _s𝛽, k _s is the radial displacement stiffness coefficient.

Considering the stator current dynamics of torque windings [23], the mathematical model of torque system can be derived as follow: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle \frac{di_{s1d}}{dt}=-\frac{R_{s1}L_{r1}^{2}+R_{r1}L_{m1}^{2}}{{\sigma}L_{s1}L_{r1}^{2}}i_{s1d}+{\omega}_{1}i_{s1q}+\frac{L_{m1}}{{\sigma}L_{s1}L_{r1}T_{r1}}{\psi}_{r1}+\frac{1}{{\sigma}L_{s1}}u_{s1d}\\[16.0pt] \displaystyle \frac{di_{s1q}}{dt}=-\displaystyle \frac{R_{s1}L_{r1}^{2}+R_{r1}L_{m1}^{2}}{{\sigma}L_{s1}L_{r1}^{2}}i_{s1q}-{\omega}_{1}i_{s1d}-\frac{L_{m1}}{{\sigma}L_{s1}L_{r1}}{\omega}{\psi}_{r1}+\frac{1}{{\sigma}L_{s1}}u_{s1q}\\[16.0pt] \displaystyle \frac{d{\psi}_{r1}}{dt}=-\frac{1}{T_{r1}}{\psi}_{r1}+\frac{L_{m1}}{T_{r1}}i_{s1d}\\[12.0pt] \displaystyle \frac{d{\omega}}{dt}=\displaystyle \frac{p_{1}^{2}L_{m1}}{JL_{r1}}{\psi}_{r1}i_{s1q}-\displaystyle \frac{p_{1}}{J}T_{L}\end{array}\right. & & \displaystyle\end{eqnarray}$ (3)

Where: ${\sigma}=1-L_{m1}^{2}/L_{s1}L_{r1}$ is the leakage coefficient; i _s1d and i _s1q are the current components of torque windings along the d- and q-coordinate axes; L _m1 is the exciting inductance of torque windings; L _s1 and L _r1 are the stator self-inductance and rotor self-inductance; T _r1 is the rotor time constant; p ₁ is the number of pole-pairs of torque windings; 𝜓_r1 is the rotor flux-linkage amplitude of torque system; J is the moment of inertia; T _L is the load torque; ω is the rotational angular velocity of rotor; ω₁ is the synchronous angular speed.

In (1), the air gap flux-linkage of torque system needed in the controllable magnetic levitation force calculation can be expressed as follow: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\psi}_{1d}=\displaystyle \frac{L_{m1}}{L_{r}}({\psi}_{r1}+L_{r1l}i_{s1d})\\[12pt] {\psi}_{1q}=\displaystyle \frac{L_{m1}}{L_{r}}L_{r1l}i_{s1q}\end{array}\right. & & \displaystyle\end{eqnarray}$ (4)

Selecting system state variable x, system input variable u and system output variable y as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle x=[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}]^{T}=[{\alpha},{\beta},\dot{{\alpha}},\dot{{\beta}},i_{s1d},i_{s1q},{\psi}_{r1},{\omega}]^{T}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u=[u_{1},u_{2},u_{3},u_{4}]^{T}=[u_{s1d},u_{s1q},i_{s2d},i_{s2q}]^{T}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle y=[y_{1},y_{2},y_{3},y_{4}]^{T}=[{\alpha},{\beta},{\psi}_{r1},{\omega}]^{T}\end{eqnarray}$ (7) Substituting (5)--(7) into (1) -(4), the state equation of BLIM system can be derived as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\dot{x}}_{1}=x_{3}\\ {\dot{x}}_{2}=x_{4}\\[9.60004pt] {\dot{x}}_{3}=\displaystyle \frac{K_{m}}{m}\cdot \frac{L_{m1}}{L_{r1}}[(x_{7}+L_{r1l}x_{5})u_{3}+L_{r1l}x_{6}u_{4}]+\frac{k_{s}}{m}x_{1}\\[12.0pt] {\dot{x}}_{4}=\displaystyle \frac{K_{m}}{m}\cdot \frac{L_{m1}}{L_{r1}}[L_{r1l}x_{6}u_{3}-(x_{7}+L_{r1l}x_{5})u_{4}]+\displaystyle \frac{k_{s}}{m}x_{2}\\[13.0pt] {\dot{x}}_{5}=-({\gamma}-{\delta})x_{5}+(x_{8}+L_{m1}{\delta}x_{6}/x_{7})x_{6}+{\xi}{\delta}{\eta}x_{7}+{\xi}u_{1}\\[5.0pt] {\dot{x}}_{6}=-({\gamma}-{\delta})x_{6}-(x_{8}+L_{m1}{\delta}x_{6}/x_{7})x_{5}-{\xi}{\eta}x_{7}x_{8}+{\xi}u_{2}\\[5.0pt] {\dot{x}}_{7}=L_{m1}{\delta}x_{5}-{\delta}x_{7}\\[5.0pt] {\dot{x}}_{8}={\mu}x_{6}x_{7}-p_{1}T_{L}/J\end{array}\right. & & \displaystyle\end{eqnarray}$ (8) Where: 𝛾 = R _s1∕σL _s1 + R _r1∕σL _r1, ${\sigma}=1-L_{m1}^{2}/L_{s1}L_{r1}$ , δ = R _r1∕L _r1, 𝜉 =1∕σL _s1, ${\mu}=p_{1}^{2}L_{m1}/JL_{r1}$ , 𝜂 = L _m1∕L _r1.

2.2. Inverse system decoupling of BLIM

From the Interactor arithmetic, the reversibility of BLIM system can be verified, and the specific processes can be expressed as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.\begin{array}{@{}l@{}}{\dot{y}}_{1}=x_{3}\\[6.0pt] \ddot{y} _{1}=\displaystyle \frac{K_{m}L_{m1}}{mL_{r1}}[(x_{7}+L_{r1l}x_{5})u_{3}+L_{r1l}x_{6}u_{4}]+\frac{k_{s}}{m}x_{1}\end{array}\right\}\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.\begin{array}{@{}l@{}}{\dot{y}}_{2}=x_{4}\\[6.0pt] \ddot{y} _{2}=\displaystyle \frac{K_{m}L_{m1}}{mL_{r1}}[L_{r1l}x_{6}u_{3}-(x_{7}+L_{r1l}x_{5})u_{4}]+\frac{k_{s}}{m}x_{2}\end{array}\right\}\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.\begin{array}{@{}l@{}}{\dot{y}}_{3}=L_{m1}{\delta}x_{5}-{\delta}x_{7}\\[6.0pt] \ddot{y} _{3}=L_{m1}{\delta}[-({\gamma}-{\delta})x_{5}+(x_{8}+L_{m1}{\delta}x_{6}/x_{7})x_{6}+{\xi}{\delta}{\eta}x_{7}+{\xi}u_{1}]-{\delta}^{2}(L_{m1}x_{5}-x_{7})\end{array}\right\}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.\begin{array}{@{}l@{}}{\dot{y}}_{4}={\mu}x_{6}x_{7}-\displaystyle \frac{p_{1}}{J}T_{L}\\[5.0pt] \ddot{y} _{4}={\mu}x_{7}[-({\gamma}-{\delta})x_{6}-(x_{8}+L_{m1}{\delta}x_{6}/x_{7})x_{5}-{\xi}{\eta}x_{7}x_{8}+{\xi}u_{2}]+{\mu}{\delta}x_{6}(L_{m1}x_{5}-x_{7})\end{array}\right\}\end{eqnarray}$ (12) Setting: $\begin{eqnarray}\displaystyle Y=[\ddot{y} _{1},\ddot{y} _{2},\ddot{y} _{3},\ddot{y} _{4}]^{T} & & \displaystyle\end{eqnarray}$ (13) From Eqs (9)--(12), the Jacobi matrix of BLIM system can be obtained as follow: $\begin{eqnarray}\displaystyle A=\displaystyle \frac{\partial Y}{\partial u}=\left[\begin{array}{@{}lccc@{}}0 & 0 & \displaystyle \frac{K_{m}}{m}{\psi}_{1d} & \displaystyle \frac{K_{m}}{m}{\psi}_{1q}\\[9.60004pt] 0 & 0 & \displaystyle \frac{K_{m}}{m}{\psi}_{1q} & \displaystyle \frac{K_{m}}{m}{\psi}_{1d}\\[9.60004pt] {\delta}{\xi}L_{m1} & 0 & 0 & 0\\[6.0pt] 0 & {\mu}{\xi}x_{7} & 0 & 0\end{array}\right] & & \displaystyle\end{eqnarray}$ (14)

In the normal operation of BLIM system, the rotor flux-linkage and the air gap flux-linkage along the d-coordinate axis don’t equal to zero. And then, det(A) ≠ 0, rank(A) = 4. The Jacobi matrix is non-singular. The relative order of BLIM system is: 𝛼 = (𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄) = (2, 2, 2, 2), and the sum of 𝛼_i (i =1, 2, 3 and 4) equals to eight. And then, the BLIM system described by Eq. (8) is reversible.

Selecting the input variables of inverse system as follow: $\begin{eqnarray}\displaystyle v=[{\nu}_{1},{\nu}_{2},{\nu}_{3},{\nu}_{4}]^{T}=[\ddot{y} _{1},\ddot{y} _{2},\ddot{y} _{3},\ddot{y} _{4}]^{T} & & \displaystyle\end{eqnarray}$ (15)

From implicit function theorem, the inverse system can be expressed as follow: $\begin{eqnarray}\displaystyle u={\phi}(x,{\nu}_{1},{\nu}_{2},{\nu}_{3},{\nu}_{4}) & & \displaystyle\end{eqnarray}$ (16)

According to the inverse system decoupling theory, the inverse system model of BLIM considering torque system current dynamics can be derived as follow: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}u_{1}=\displaystyle \frac{1}{{\xi}}\left[\frac{1}{{\delta}L_{m1}}v_{3}+{\gamma}x_{5}-{\delta}\left({\xi}{\eta}+\frac{1}{L_{m1}}\right)x_{7}-(x_{8}+\displaystyle \frac{L_{m1}{\delta}x_{6}}{x_{7}})x_{6}\right]\\[12.0pt] u_{2}=\displaystyle \frac{1}{{\xi}}\left[\displaystyle \frac{1}{{\mu}x_{7}}v_{4}+{\gamma}x_{6}+x_{5}x_{8}+{\xi}{\eta}x_{7}x_{8}\right]\\[5.0pt] u_{3}=\displaystyle \frac{L_{r1}}{L_{m1}K_{m}[(x_{7}+L_{r1l}x_{5})^{2}+(L_{r1l}x_{6})^{2}]}[L_{r1l}x_{6}\times (m{\nu}_{2}-k_{s}x_{2})+(x_{7}+L_{r1l}x_{5})(m{\nu}_{1}-k_{s}x_{1})]\\[12.0pt] u_{4}=\displaystyle \frac{L_{r1}}{L_{m1}K_{m}[(x_{7}+L_{r1l}x_{5})^{2}+(L_{r1l}x_{6})^{2}]}[L_{r1l}x_{6}\times (m{\nu}_{1}-k_{s}x_{1})-(x_{7}+L_{r1l}x_{5})(m{\nu}_{2}-k_{s}x_{2})]\end{array}\right. & & \displaystyle\end{eqnarray}$ (17)

From (17), it can be seen that after considering the stator current dynamics, there is no longer unpredictable load torque variable T _L in the derived inverse system model. And then it provide a convenience for improving the ability to resist load disturbance; meanwhile the identification link of load torque can be omitted, and the decoupling control system structure of BLIM can be simplified to a certain extent.

Figure 1 shows the decoupling principle of BLIM system. According to Fig. 1, connecting the inverse system in front of the BLIM original system in series, the BLIM system can be decoupled into four second-order pseudo linear integral subsystems, the transfer function of each subsystem can be expressed as 1/s².

Fig. 1.

Inverse system decoupling principle of BLIM.

3. Sliding mode Regulator design

In order to overcome the influence of the unknown nonlinear dynamics and modeling errors, and guarantee the robustness and stability of BLIM system, the closed-loop controller should be designed for each subsystem. Now, on the basis of the inverse system decoupling, the regulator is designed according to the sliding mode control (SMC) theory. The SMC system has a high adaptive ability to the system uncertainty, and it is applicable for the nonlinear object with parameter perturbation and external disturbance. The specific control method can be summarized as follow: The system state variable is attracted and retained in a region near the sliding mode surface firstly, and then gradually tends to the sliding mode surface; By limiting the influence of the unknown disturbance to a region near the sliding mode surface, the robustness of the SMC system is improved. Due to that the reaching speed of system state variable to the sliding mode surface doesn’t equal to zero, the system inertia would result in that the system state goes through the sliding mode surface constantly. The discontinuous switching would cause the chattering of SMC system. In this paper, an exponential reaching law is adopted, and by adjusting the parameters of the reaching law, the “arrival process” of sliding mode is optimized, thus the chattering problem can be effectively overcome to a certain extent.

Taking the 𝛼 displacement subsystem as an example, assumption: z ₁ = y ₁, $z_{2}={\dot{z}}_{1}$ . Then the state equation of 𝛼 displacement subsystem can be expressed as follow: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}z_{1}=y_{1}\\ {\dot{z}}_{1}={\dot{y}}_{1}=z_{2}\\ {\dot{z}}_{2}=\ddot{y} _{1}=v_{1}+d_{1}\end{array}\right. & & \displaystyle\end{eqnarray}$ (18) Where: d ₁ is the external disturbance, |d ₁| ≤ h ₁, h ₁ ≥ 0; v ₁ is the input variable of the 𝛼 displacement subsystem.

Assumption: r ₁ is the given instruction. Then the sliding mode surface can be expressed as follow: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}s_{0}=e_{1}=r_{1}-z_{1}\\ s_{1}={\dot{s}}_{0}+as_{0}\end{array}\right. & & \displaystyle\end{eqnarray}$ (19) Where: the a is a positive constant.

From (19), there is following equation: $\begin{eqnarray}\displaystyle {\dot{s}}_{1}=\ddot{s}_{0}+a{\dot{s}}_{0} & & \displaystyle\end{eqnarray}$ (20)

In order to eliminate the chattering of sliding mode controller [26], the following exponential approach law is adopted: $\begin{eqnarray}\displaystyle {\dot{s}}_{1}=-{\mu}s_{1}-{\eta}∼\text{sgn}(s_{1}) & & \displaystyle\end{eqnarray}$ (21) Where: 𝜇 is a constant determining the convergence rate of Lyapunov function, 𝜇 > 0; 𝜂 is the rate of the system moving point that approaches to the switching surface, 𝜂 = h ₁ + 𝜆; 𝜆 is the switch term gain, 𝜆 > 0.

Substituting (19), (20) into (21), then: $\begin{eqnarray}\displaystyle {\dot{s}}_{1}=-v_{1}-d_{1}-az_{2} & & \displaystyle\end{eqnarray}$ (22)

And then, the designed sliding mode controller can be expressed as follow: $\begin{eqnarray}\displaystyle v_{1}=-az_{2}+{\psi}s_{1}+{\eta}∼\text{sgn}(s_{1}) & & \displaystyle\end{eqnarray}$ (23)

Constructing Lyapunov function as follow: $\begin{eqnarray}\displaystyle V=s_{1}^{2}/2\geq 0 & & \displaystyle\end{eqnarray}$ (24)

Then from (22), (23) and (24), following equation can be derived: $\begin{eqnarray}\displaystyle \dot{V}_{1} & = & \displaystyle s_{1}{\dot{s}}_{1}=s_{1}(-v_{1}-d_{1}-az_{2})\nonumber\\ \displaystyle & \leq & \displaystyle -s_{1}[-az_{2}+{\mu}s_{1}+{\eta}∼\text{sgn}(s_{1})]+|s_{1}|h_{1}-s_{1}az_{2}\nonumber\\ \displaystyle & = & \displaystyle -{\mu}s_{1}^{2}-{\lambda}|s_{1}|<0\end{eqnarray}$ (25)

Because 𝜇 is larger than zero, then the value of (25) is smaller than zero. And then the system is asymptotically stable.

In order to avoid the case that the switch term gain 𝜆 being is oversized, in the sliding mode controller, a saturation function “sat(s)” is adopted to replace the symbolic function “sgn(s)” in (23). The saturation function can be expressed as follow: $\begin{eqnarray}\displaystyle \text{sat}(s)=\left\{\begin{array}{@{}ll@{}}1 & s>{\Delta}\\ ks\quad |s|\leq {\Delta}, & k=1/{\Delta}\\ -1 & s<-{\Delta}\end{array}\right. & & \displaystyle\end{eqnarray}$ (26) Where: Δ is the boundary layer (width). The usages of the saturation function: When the moving point is beyond the boundary layer, switching control is adopted, so that the system state can quickly enter into the sliding mode; when the moving point is within the boundary layer, the feedback control is used to reduce the chattering caused by the fast switch of sliding mode.

For 𝛽 displacement component, motor speed and rotor flux-linkage subsystems, their design methods of sliding mode controller are similar, and will not be described here.

4. System simulation analysis

Figure 2 is the inverse decoupling SMC system structure of a BLIM considering current dynamics. In order to verify the effectiveness of the presented control strategy, the system simulation experimental is carried out by Matlab/Simulink.

Fig. 2.

Inverse system decoupling SMC system of bearingless induction motor considering stator current dynamics.

Table 1

Parameters of BLIM

Parameter	Torque system	Suspension system
Power	2.2 kW	-
Rotor mass	8 kg
Auxiliary bearing air gap	-	0.2 mm
Moment of inertia	0.024 kg ⋅ m²	-
Stator resistance	1.6 Ω	2.7Ω
Rotor resistance (Ω)	1.423
Stator leakage inductance	0.0043 H	0.0398 H
Rotor leakage inductance	0.0043 H	0.0398 H
Magnetizing inductance	0.0859 H	0.23 H
Number of pole pair	2	1

The parameters of BLIM are as shown in Table 1, and the simulation conditions are set as follows:

(1) Initial displacement 𝛼₀ = −0.12 mm, 𝛽₀ = −0.16 mm.

(2) Given signal of each variable: motor speed ω^∗ = 1500 r/min, given rotor flux-linkage ${\psi}_{\text{r1}}^{\ast }=∼0.95$ Wb, given displacement 𝛼^∗ = 𝛽^∗ = 0; load torque T _L = 0.

(3) In order to verify the decoupling performance and the system stability, some given signals are suddenly changed at different times, the specific circumstances are as follows: at 1.0 s, the given rotor flux-linkage is reduced to 0.5 Wb, and the given speed is suddenly increased to 2500 r/min; the given load is suddenly increased to 7 N ⋅ m at 2.0 s, and returned to zero at t = 2.3 s; the given 𝛼 displacement is stepped to 0.05 mm at t = 2.5 s, recovered to zero at 2.8 s; the given 𝛽 displacement is stepped to −0.05 mm at t = 2.9 s, recovered to zero at 3.2 s.

Figure 3 to Fig. 6 present the responses curves of the inverse decoupling control system with SMC regulator. For the convenience of comparative analysis, the response curves of the traditional inverse decoupling control system are also given, the PD regulator used for each subsystem is additionally equipped with an inertial filter link, its transfer function can be expressed as “G _c(s) = K (τ₁ s +1)∕(τ₂ s + 1)”; the open-loop transfer function and phase margin of each subsystem can be expressed as “G _ol(s) = K (τ₁ s +1)∕s ²(τ₂ s + 1)” and “𝛾_m = arctan τ₁ω_c2 − arctan τ₂ω_c2”; by reasonably setting several parameters, include the regulator gain K and differential lead time constant τ₁ of the PD regulator, the inertial filtering time constant τ₂, and the subsystem cutoff frequency ω_c2, a certain stability margin and a fast response speed of each subsystem can be ensured.

Fig. 3.

Speed response curve of inverse decoupling control system.

Fig. 4.

Rotor flux-linkage response curve of inverse decoupling control system.

The response curves of the motor speed and rotor flux-linage are shown in Fig. 3 and Fig. 4 respectively. From Fig. 3 and Fig. 4, there are following results:

(1) During the starting process, when the proposed inverse system decoupling SMC strategy is adopted, there exists no speed overshoot basically; the overshoot of rotor flux-linkage is about 3.1%, the response time is about 0.02 s. But when the traditional inverse system decoupling control method is adopted, there exists an speed overshoot about 6.7%; the overshoot of rotor flux-linkage is about 6.7%, the response time is about 0.16 s.

(2) During the sudden loading and unloading process, when the proposed inverse system decoupling SMC strategy is adopted, there are almost no changes in the motor speed and rotor flux-linkage response curves. But when the traditional inverse system decoupling control method is adopted, a speed fluctuation about 2% is occurred, there is no obvious rotor flux-linkage fluctuation.

(3) During the sudden change processes of the speed and the rotor flux-linage, when the proposed inverse system decoupling SMC strategy is adopted, there is almost no speed overshoot; the rotor flux-linkage only has a slightly fluctuation about 2.5%, and restores steady-state value in very short time. But when the traditional inverse system decoupling control method is adopted, there is a speed fluctuation about 2%; in addition, there is an obvious fluctuation about 25% in the rotor flux-linkage response curve.

(4) The simulation experimental results have shown that when the proposed inverse system decoupling SMC strategy is adopted, the BLIM control system has a smaller overshoot, a stronger stability, a quicker response speed, and a stronger anti load disturbance ability.

The response curves of the 𝛼 and 𝛽 displacement components are presented in Fig. 5 and Fig. 6 respectively. From Fig. 5 and Fig. 6, there are following results:

(1) If the traditional inverse system decoupling control method is adopted, when the given 𝛼 displacement signal is suddenly changed, the response time is about 0.03 s, the overshoot is about 62%; when the given 𝛽 displacement signal is suddenly changed, the response time is about 0.04 s, the overshoot is about 62%. However, if the proposed inverse system decoupling SMC strategy is adopted, when the given 𝛼 displacement signal is suddenly changed, the response time is about 0.005 s; when the given 𝛽 displacement signal is suddenly changed, the response time is about 0.025 s; there are almost no overshoots in the response curves of the 𝛼- and 𝛽-displacement components.

(2) The simulation experimental results have shown that when the proposed inverse system decoupling SMC strategy is adopted, the system has a quicker response speed, a smaller radial displacement overshoot, and a better displacement tracking performance.

Fig. 5.

Displacement response curve of inverse decoupling control system in 𝛼 direction.

Fig. 6.

Displacement response curve of inverse decoupling control system in 𝛽 direction.

By comprehensive comparison between the response waveforms in Fig. 3--Fig. 6, it can be concluded that based on the proposed inverse system decoupling SMC strategy, the dynamic decoupling between the motor speed, rotor flux-linkage and two radial displacement components has been achieved.

In order to verify the adaptive robustness of the BLIM control system to the motor parameter variation, the rotor resistance is increased by 50%, i.e. the rotor resistance is increased to 2.1345 Ω, and the other conditions remain unchanged. Figure 7 to Fig. 12 present the response curves of relevant variables after the rotor resistance change.

Fig. 7.

Speed response curves of inverse decoupling control system after resistance change.

Fig. 8.

Speed contrast response curves of inverse decoupling SMC system before and after the rotor resistance change.

After the rotor resistance change, Fig. 7 presents the contrast response curves of motor speed between the inverse decoupling SMC system and the traditional inverse decoupling control system. Figure 8 presents the speed contrast response curves of the inverse decoupling SMC system before and after the rotor resistance change. Figure 9 presents the speed response curves of the traditional inverse decoupling control system before and after the rotor resistance change.

Fig. 9.

Speed contrast response curves of traditional inverse decoupling control system before and after the rotor resistance change.

From Fig. 7--Fig. 9, there are following results:

(1) After the rotor resistance change, the system response speed is decreased somewhat, the response time of the inverse decoupling SMC system and that of the traditional inverse decoupling control system are all delayed about 0.01 s. But compared to the traditional inverse decoupling control system, the inverse decoupling SMC system still has a quicker response speed and a smaller overshoot of motor speed.

(2) During the sudden loading and unloading process, when the proposed inverse system decoupling SMC strategy is adopted, there is almost no change in the motor speed; But when the traditional inverse system decoupling control method is adopted, compared to the case before resistance change, the speed fluctuation increased by about 0.8%.

(3) The simulation result has further shown that the inverse decoupling SMC system has a stronger anti disturbance ability, a stronger robustness to the variation of motor parameter.

Figure 10 presents the contrast response curves of the rotor flux-linkage before and after the rotor resistance variation. From Fig. 10, after the rotor resistance change, because the speed starts slightly slower, the overshoot of the rotor flux-linkage has been reduced, hereinto:

(1) When the proposed inverse system decoupling SMC strategy is adopted, the rotor flux-linkage overshoot in starting stage is about 1.0%, the response time is about 0.014 s; when the traditional inverse system decoupling control method is adopted, the rotor flux-linkage overshoot in starting stage is about 6.3%, the response time is about 0.12 s.

Fig. 10.

Rotor flux-linkage contrast response curve of inverse decoupling control system after the rotor resistance change.

(2) During the motor speed jumping process, when the traditional inverse system decoupling control method is adopted, the rotor flux-linkage fluctuates about 17.5%; but when the proposed inverse system decoupling SMC strategy is adopted, the rotor flux-linkage waveform only has a very small fluctuation about 0.8%.

(3) During the sudden loading and unloading process, when the traditional inverse system decoupling control method is adopted, a rotor flux-linkage ripple about 3.5% is generated. But when the proposed inverse system decoupling SMC strategy is adopted, the rotor flux-linkage has no fluctuation basically.

(4) The simulation results have shown that compared with the traditional inverse decoupling control system, the inverse decoupling SMC system has the advantages of faster flux-linkage response and stronger robustness.

After the rotor resistance change, Fig. 11 and Fig. 12 present the response curves of the 𝛼- and 𝛽-displacement components. From Fig. 11 and Fig. 12, when the proposed inverse system decoupling SMC strategy is adopted, there exist almost no overshoots in the 𝛼- and 𝛽-displacement response curves; in addition, the system still has a faster displacement response speed, a smaller displacement overshoot, and a better displacement tracking performance.

Fig. 11.

Displacement response curve of inverse decoupling control system in 𝛼 direction after the rotor resistance change.

Fig. 12.

Displacement response curve of inverse decoupling control system in 𝛽 direction after the rotor resistance change.

5. Conclusion

Bearingless induction motor (BLIM) is a nonlinear, multi-variable and strongly coupled object. Aiming at the influences of the motor parameter variation and the load disturbance on the control performance, under the conditions of considering the stator current dynamics, an inverse system decoupling sliding mode control (SMC) strategy of BLIM is proposed. Firstly, the inverse system mathematical model of BLIM is presented, and by inverse system method, the BLIM system is decoupled into four independent second-order pseudo-linear integral subsystems. After then, aiming at the poor adaptability of traditional regulator to the parameter variation, based on the SMC theory, a SMC regulator is designed for each decoupled subsystem; By introducing the “exponential reaching law” to overcome the chattering problem of inverse decoupling SMC system; based on Lyapunov stability principle, the system stability is verified. Finally, the simulation experimental analysis and verification of the proposed inverse system decoupling SMC strategy is carried out. The specific research conclusions can be summarized as follows:

(1) Under the conditions of considering the stator current dynamics of torque windings, the inverse decoupling SMC system of bearingless induction motor is stable.

(2) Based on the proposed inverse system decoupling SMC strategy, not only the excellent dynamic decoupling control between the motor speed, rotor flux-linkage and two radial displacement components can be achieved, but also the BLIM control system has a series of advantages, such as a quicker response speed, a smaller overshoot, a stronger stability and a better tracking performance.

(3) Adopting the proposed inverse system decoupling SMC strategy, the BLIM control system has a stronger anti disturbance ability, a stronger adaptability and a stronger robustness to the motor parameter variation.

(4) The proposed inverse system decoupling SMC strategy of BLIM is effective and feasible.

Funding

The support of National Natural Science Foundation of China (51277053) is acknowledged.

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