Optimal control strategy of state feedback control for surface-mounted PMSM drives based on auto-tuning of seeker optimization algorithm

Abstract

This paper focus on a state feedback controller (SFC)-based optimal control scheme for surface-mounted permanent-magnet synchronous motor (SPMSM) with auto-tuning of controller built on seeker optimization algorithm (SOA). First, based on the nonlinear state-space model of SPMSM, voltage feedforward compensation is used to design a linear SFC. Then in order to guarantee the steady performance in speed and current, integral models considering the errors of rotor speed and current response in d-axis are added in the state space model of SPMSM. Furthermore, by statically decoupling the load torque in the state equation, feedforward compensation is implemented on the load torque to improve the dynamic performance of the controller. The load torque is estimated by using disturbance observer with reasonable parameter selection. Then, with the consideration of the search capacity of seeker optimization algorithm (SOA), it is adopted to acquire matrix coefficient of the presented controller. Furthermore, in order to suppress the speed overshoot, a penalty term is introduced to the fitness index. The performance of the proposed method has been validated experimentally and compared with the conventional method under different conditions.

Keywords

Surface-mounted permanent magnet synchronous motor (SPMSM)feedforward compensation state feedback control disturbance observer seeker optimization algorithm (SOA)

1. Introduction

Due to the inherent advantages of high torque-inertia ratio, high efficiency and excellent control performance, permanent magnet synchronous motor (PMSM) has been widely used in industrial application such as hybrid and electrical vehicles and industrial robots [1–7]. In conventional motor system control, field-oriented control (FOC) strategy is most commonly applied to achieve the dynamic performance of PMSM thanks to its constant switching frequency [8–14].

In the conventional vector control system, cascade control structure is often employed including two cascade proportional-integral (PI) loops. The outermost loop is used for speed control and the inner loops are used for dq-axes current control [15–19]. The classical PI control technique is widely used thanks to its simple implementation and high reliability [20–25]. However, because of the inherent inner loop structure, the dynamic control performance of cascade closed loop structure will be affected for cascade [26–32]. And tuning process in order from inner current loops to outer speed loop needs analysis and experiment. Additionally, in order to guarantee robustness against disturbance and system uncertainties, some nature-inspired optimization algorithms have been employed for the parameters auto-tuning of cascade controllers [33–37]. However, despite the introduction of parameter adjustment algorithms, the existence of inner loop control in the cascade structure has affected the dynamic performance of control system [38–44].

In order to improve the system dynamic property with satisfactory stability, state feedback control strategy is an alternative method for speed control of PMSM drive [45–49]. In SFC structure, only one controller is required, in which all state variables in the control plant can be handled in synchronization [50–58]. Therefore, the cascade structure of speed loop and current loops can be avoided and the drawback of cascade structure is eliminated naturally. In [59], the application of SFC has obtained better dynamic characteristics, especially in disturbance compensation.

However, the process in which SFC parameters are determined is time-consuming and relatively complicated. It is known that the conventional tuning approach of SFC parameters is based on trial-and-error method [60–65]. In [66], a gain-scheduled linear-quadratic (LQR) control system based on linearization of PMSM model has been presented, in which an analytical method is employed for determining elements of weighting matrices. In contrast, some computer-aided optimization techniques have been adopted for the determination of SFC gain coefficients. In [67], a state feedback current controller with an oscillation term is designed for the application of three-phase grid-connected voltage source converter under grid voltage distortion condition, in which a particle swarm optimization (PSO) algorithm is proposed for the offline tuning of the controller with less optimization time. In [68], an optimal control strategy for PMSM drive has been proposed based on state feedback control method with an artificial bee colony (ABC) algorithm and a flower pollination (FP) algorithm, which shows the superiority of coefficients tuning in LQR optimization process.

In this paper, seeker optimization algorithm (SOA) is used for auto-tuning of the controller, which is a new population-based heuristic search algorithm [69–74]. It seeks the optimal solution based on human thinking, so it can simulate human search behaviour to optimize parameters. Because of the properties of high solution precision and fast convergence capability [75–81], SOA is considered as a suitable to solve optimal problems.

In order to achieve speed control of the drive system with error-free in steady state, it is necessary to consider internal models of reference input (i.e., speed reference and d-axis current reference), which are used to establish extended state-space model [82–86]. Moreover, static decoupling from disturbance is used for improvement of the system dynamic performance. In order to estimate the unmeasured disturbance, a sign function-based disturbance observer is proposed and proved to be of asymptotic stability, which is further used to design feedforward load torque compensation for SFC. The specific performance of the disturbance observer is determined by the designed parameters of the observer [87–89].

This paper is organized as follows. Section 2 describes the mathematical model of SPMSM and further develops the linear state-space model in a standard form. In Section 3, discrete SFC is designed for the extended state-space model with introduction of internal models and feedback compensation on load torque. In Section 4, SOA is introduced and analyzed. Auto tuning of SFC based SOA is discussed in Section 5. Experimental results are provided in Section 6. Finally, Section 7 gives the summary and draws the main conclusions of the present study.

2. Linearization of PMSM model

Assuming that the iron core of the SPMSM is unsaturated and the eddy currents and hysteresis are negligible [90–94], the mathematical model of SPMSM in the synchronous reference frame can be presented by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left\{\begin{array}{@{}l@{}}u_{d}=Ri_{d}+\dot{{\psi}}_{d}-{\omega}_{e}{\psi}_{q}\\ u_{q}=Ri_{q}+\dot{{\psi}}_{q}+{\omega}_{e}{\psi}_{d}\end{array}\right.\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left\{\begin{array}{@{}l@{}}\dot{{\omega}}_{m}=1/J_{m}(T_{e}-T_{l}-{\nu}{\omega}_{m})\\ T_{e}=1.5p_{n}{\psi}_{f}i_{q}.\end{array}\right.\end{eqnarray}$ (2)

The flux linkage equation is described as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\psi}_{d}=Li_{d}+{\psi}_{f}\\ {\psi}_{q}=Li_{q},\end{array}\right. & & \displaystyle\end{eqnarray}$ (3) where u _d, u _q and i _d, i _q are the stator voltages and currents in d-q coordinates, respectively. R and L represent the armature winding resistance and inductance, respectively. It is assumed the d-axis inductance is equal with q-axis inductance for SPMSM. 𝜓_f, J _m, and v are the flux linkage, rotor inertia and the viscous friction coefficient, respectively. ω_m is the rotor mechanical speed, p _n is the number of pole pairs, and ω_e = p _nω_m is the rotor electrical speed. T _e is the electromagnetic torque, and T _l is the load torque.

The typical block diagram of the SPMSM based on field-oriented control scheme is shown in Fig. 1. The scheme adopts the structure of cascade control loops including a speed loop and two current loops [95–99]. Since the dq-axis inductances of SPMSM are equal, limit circles of voltage and current intersect on the 𝛽 axis. Therefore i _d = 0 strategy can be adopted to achieve the maximum torque current ratio under no flux-weakening.

Fig. 1.

Block diagram of the SPMSM based on field-oriented control scheme under i _d = 0 strategy.

The model given above can be written in state-space form in d-q reference frame as follow: $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}\boldsymbol{x}(t)=\boldsymbol{A}\boldsymbol{x}(t)+\boldsymbol{B}\boldsymbol{u}(t)+\boldsymbol{E}d(t), & & \displaystyle\end{eqnarray}$ (4) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{A}=\left[\begin{array}{@{}ccc@{}}-{\displaystyle \frac{R}{L}} & p_{n}{\omega}_{m} & 0\\[6.0pt] -p_{n}{\omega}_{m} & -{\displaystyle \frac{R}{L}} & -p_{n}{\displaystyle \frac{{\psi}_{f}}{L}}\\[6.0pt] 0 & {\displaystyle \frac{3p_{n}{\psi}_{f}}{2J_{m}}} & -{\displaystyle \frac{B_{m}}{J_{m}}}\end{array}\right],\quad \boldsymbol{B}=\left[\begin{array}{@{}cc@{}}{\displaystyle \frac{1}{L}} & 0\\ 0 & {\displaystyle \frac{1}{L}}\\ 0 & 0\end{array}\right],\quad \boldsymbol{E}=\left[\begin{array}{@{}c@{}}0\\ 0\\ -1/J_{m}\end{array}\right],\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{x}(t)=\left[\begin{array}{@{}c@{}}i_{d}(t)\\ i_{q}(t)\\ {\omega}_{m}(t)\end{array}\right],\quad \boldsymbol{u}(t)=\left[\begin{array}{@{}c@{}}u_{d}\\ u_{q}\end{array}\right],\quad d(t)=T_{l}(t).\nonumber\end{eqnarray}$

It can be observed that the angular velocity is existed in the first and the second row of matrix A , therefore the above state space equation is cross coupled [100–105]. In order to apply linear SFC for PMSM drive control, it is necessary to linearize the nonlinear model. The block diagram of proposed control strategy with SFC is illustrated in Fig. 2.

Fig. 2.

Block diagram of PMSM drive with proposed SFC under i _d = 0 strategy.

In order to achieve the linearization of the motor model [106–110], new variables are defined as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}u_{d0}=-{\omega}_{e}{\psi}_{q}\\ u_{q0}={\omega}_{e}{\psi}_{d}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (5) Combining (5) and (1), the voltage equation can be obtained as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}u_{ld}=u_{d}-u_{d0}=Ri_{d}+L{\displaystyle \frac{d}{dt}}i_{d}\\[6.0pt] u_{lq}=u_{q}-u_{q0}=Ri_{q}+L{\displaystyle \frac{d}{dt}}i_{q}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (6)

It can be seen that the nonlinear and cross-couple terms are eliminated because the right side of the first row of above equation includes only of d-axis variables and the right side of the second row of above equation includes only of q-axis variables [111–116].

Combing (1), (2) and (3), a state-space equation in standardization can be obtained as follows: $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}\boldsymbol{x}_{l}(t)=\boldsymbol{A}_{l}\boldsymbol{x}_{l}(t)+\boldsymbol{B}_{l}\boldsymbol{u}_{\mathbf{l}}(t)+\boldsymbol{E}d(t), & & \displaystyle\end{eqnarray}$ (7) where $\begin{eqnarray}\displaystyle \boldsymbol{A}_{l}=\left[\begin{array}{@{}ccc@{}}-{\displaystyle \frac{R}{L}} & 0 & 0\\[6.0pt] 0 & -{\displaystyle \frac{R}{L}} & 0\\[6.0pt] 0 & {\displaystyle \frac{3p_{n}{\psi}_{f}}{2J_{m}}} & -{\displaystyle \frac{B_{m}}{J_{m}}}\end{array}\right],\quad \boldsymbol{u}_{l}(t)=\left[\begin{array}{@{}l@{}}u_{ld}\\ u_{lq}\end{array}\right],\quad \boldsymbol{x}_{l}(t)=\boldsymbol{x}(t),\quad \boldsymbol{B}_{l}=\boldsymbol{B}. & & \displaystyle \nonumber\end{eqnarray}$

3. SFC controller design

Due to the proposed system model is type 0 control system, steady state performance will be affected under step change of reference input. In order to enhance the steady performance of the controller, internal models of reference values are taken into consideration and added into the system model to design a type 1 control system. Thus, integral models considering the errors of rotor speed and current response in d-axis are added in control model. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}e_{id}=\displaystyle \int (i_{d}-{i_{d}}^{ref})dt\\ e_{{\omega}}=\displaystyle \int ({\omega}_{m}-{{\omega}_{m}}^{ref})dt,\end{array}\right. & & \displaystyle\end{eqnarray}$ (8) where ${\omega}_{m}^{ref}$ and $i_{d}^{ref}$ are the corresponding references of mechanical speed and current response in d-axis.

Therefore the state variable matrix can be extended as follows: $\begin{eqnarray}\displaystyle \overline{\boldsymbol{x}}(t)=\left[\begin{array}{@{}cccc@{}}i_{d}(t) & i_{q}(t) & {\omega}_{m}(t) & {\boldsymbol{x}_{e}}^{T}(t)\end{array}\right]^{T}, & & \displaystyle\end{eqnarray}$ (9) where x _e(t) = [e _id(t) e _ω(t)]^T.

Then the extended state-space equation with the proposed integral models can be obtained as $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}\overline{\boldsymbol{x}}(t)=\overline{\boldsymbol{A}}\overline{\boldsymbol{x}}(t)+\overline{\boldsymbol{B}}\overline{\boldsymbol{u}}(t)+\boldsymbol{H}\boldsymbol{g}(t), & & \displaystyle\end{eqnarray}$ (10) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{\boldsymbol{A}}=\left[\begin{array}{@{}ccccc@{}}-{\displaystyle \frac{R}{L}} & 0 & 0 & 0 & 0\\[8.39996pt] 0 & -{\displaystyle \frac{R}{L}} & 0 & 0 & 0\\[8.39996pt] 0 & {\displaystyle \frac{3p_{n}{\psi}_{f}}{2J_{m}}} & -{\displaystyle \frac{B_{m}}{J_{m}}} & 0 & 0\\[12.0pt] 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\end{array}\right],\quad \overline{\boldsymbol{B}}=\left[\begin{array}{@{}cc@{}}{\displaystyle \frac{1}{L}} & 0\\[8.39996pt] 0 & {\displaystyle \frac{1}{L}}\\[8.39996pt] 0 & 0\\ 0 & 0\\ 0 & 0\end{array}\right],\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{H}=\left[\begin{array}{@{}ccc@{}}0 & 0 & 0\\ 0 & 0 & 0\\ -1/J_{m} & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1\end{array}\right],\quad \mathbf{g}(t)=\left[\begin{array}{@{}c@{}}T_{l}\\ i_{dref}\\ {\omega}_{ref}\end{array}\right],\quad \overline{\boldsymbol{u}}(t)=\boldsymbol{u}_{l}(t).\nonumber\end{eqnarray}$

Ignoring disturbance and assuming that reference signals are equal to zero, the system state equation with full state feedback is described as follows: $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}\overline{\boldsymbol{x}}(t)=\overline{\boldsymbol{A}}\overline{\boldsymbol{x}}(t)+\overline{\boldsymbol{B}}\overline{\boldsymbol{u}}(t). & & \displaystyle\end{eqnarray}$ (11)

Since the motor control model has been extended by the internal models of the reference signals, the control law can be implemented to ensure steady-state error-free operation under step changes in the reference speed and load torque conditions. $\begin{eqnarray}\displaystyle \overline{\boldsymbol{u}}(t)=-\boldsymbol{K}_{d}\overline{\boldsymbol{x}}(t), & & \displaystyle\end{eqnarray}$ (12) where x (t) represents system state, u (t) is control variables and K _d is the coefficient matrix.

After the determination of the feedback controller structure, the gain coefficients of controller need to be simultaneously selected. In the proposed method, a discrete form of SFC will be designed to obtain a form suitable for execution. $\begin{eqnarray}\displaystyle \overline{\boldsymbol{u}}(k)=-\boldsymbol{K}_{\boldsymbol{d}}\overline{\boldsymbol{x}}(k)=-\boldsymbol{K}_{dx}\boldsymbol{x}(k)-\boldsymbol{K}_{de}\boldsymbol{x}_{e}(k), & & \displaystyle\end{eqnarray}$ (13) where $\begin{eqnarray}\displaystyle \boldsymbol{K}_{d}=[\boldsymbol{K}_{dx}∼\boldsymbol{K}_{de}]=\left[\begin{array}{@{}ccccc@{}}K_{dx1} & K_{dx2} & K_{dx3} & K_{de1} & K_{de2}\\ K_{dx4} & K_{dx5} & K_{dx6} & K_{de3} & K_{de4}\end{array}\right], & & \displaystyle \nonumber\end{eqnarray}$ K _dxi and K _dei are the gain coefficients of the SFC, k is discrete sample time index. The discrete form of variables x _e (k) is calculated by (8) based on backward Euler integration method as follows: $\begin{eqnarray}\displaystyle \boldsymbol{x}_{e}(k)=\left[\begin{array}{@{}c@{}}e_{id}(k)\\ e_{{\omega}}(k)\end{array}\right]=\left[\begin{array}{@{}c@{}}e_{id}(k-1)+T_{s}(i_{d}(k)-{i_{d}}^{ref}(k))\\ e_{{\omega}}(k-1)+T_{s}({\omega}_{m}(k)-{\omega}^{ref}(k)),\end{array}\right] & & \displaystyle\end{eqnarray}$ (14) where T _s is sampling period.

In order to enhance the transient control performance of the controller, feedforward compensation is implemented on the load torque. Then the state-space equation is written as $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}\overline{\boldsymbol{x}}(t)=\overline{\boldsymbol{A}}\overline{\boldsymbol{x}}(t)+\overline{\boldsymbol{B}}\overline{\boldsymbol{u}}(t)+(\boldsymbol{H}_{1}-\overline{\boldsymbol{B}}\boldsymbol{K}_{co})T_{l}, & & \displaystyle\end{eqnarray}$ (15) where H ₁ is the first column of matrix H , and K _co = [ K _co1 K _co2]^T.

By statically decoupling the load torque in the equation of state, gain matrix K _co for load torque compensation can be obtained as $\begin{eqnarray}\displaystyle \boldsymbol{K}_{co}={\displaystyle \frac{-2}{3p_{n}{\psi}_{f}}}\left[\begin{array}{@{}c@{}}R+K_{dx1}\\ K_{dx4}.\end{array}\right] & & \displaystyle \nonumber\end{eqnarray}$

Then state feedback controller is modified as $\begin{eqnarray}\displaystyle \overline{\boldsymbol{u}}(k)=-\boldsymbol{K}_{\boldsymbol{d}}\overline{\boldsymbol{x}}(k)-\boldsymbol{K}_{co}\hat{T}_{l}. & & \displaystyle\end{eqnarray}$ (16)

The block diagram of proposed state feedback controller is illustrated in Fig. 3.

Fig. 3.

Block diagram of the proposed state feedback controller.

In order to obtain the external load torque, disturbance observer is designed to accurately estimate disturbance. Based on the mechanical Eq. ((2)), the following state equation can be obtained $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}y=ay+bi_{q}+r, & & \displaystyle\end{eqnarray}$ (17) where $y=\omega_{m}, a=-\frac{\nu}{J_m},b=\frac{3p_n\psi_f}{2J_m},r=-\frac{T_l}{J_m}$ and ${\dot{r}}=d$ represents the variation rate of disturbance.

The disturbance observer is constructed with introduction of sign function as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\displaystyle \frac{d}{dt}}{\hat{y}}=a{\hat{y}}+bi_{q}+\hat{r}+{\zeta}sign(e_{y})\\[6.0pt] {\displaystyle \frac{d}{dt}}\hat{r}={\lambda}{\zeta}sign(e_{y}),\end{array}\right. & & \displaystyle\end{eqnarray}$ (18) where $e_{y}=y-{\hat{y}}$ , 𝜆 and ζ are designed parameters of the observer, which need to be reasonably selected to guarantee observer performance.

Combining (17) and (18), error equation is obtained as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\displaystyle \frac{d}{dt}}e_{y}=ae_{y}+e_{r}-{\zeta}sign(e_{y})\\[6.0pt] {\displaystyle \frac{d}{dt}}e_{r}=d-{\lambda}{\zeta}sign(e_{y}),\end{array}\right. & & \displaystyle\end{eqnarray}$ (19) where $e_{r}=r-\hat{r}$ .

Based on the constructed disturbance observer, parameter ζ should be suitably determined to guarantee stability of the observer. The observer is stable if $\begin{eqnarray}\displaystyle \dot{V}=e_{y}\cdot {\dot{e}}_{y}\leq 0. & & \displaystyle\end{eqnarray}$ (20)

Substituting (19) in (20) yields $\begin{eqnarray}\displaystyle \dot{V}\text{} & =\text{} & \displaystyle e_{y}\cdot {\dot{e}}_{y}=e_{y}\cdot (ae_{y}+e_{r}-{\zeta}sign(e_{y}))\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \left\{\begin{array}{@{}l@{}}e_{y}\cdot (ae_{y}+e_{r}-{\zeta})e_{y}>0\\ e_{y}\cdot (ae_{y}+e_{r}+{\zeta})e_{y}>0.\end{array}\right.\end{eqnarray}$ (21)

It can be obtained that ${\zeta}\geq \max \{ae_{y}+e_{r},-(ae_{y}+e_{r})\}$ . ζ is the designed parameter of the disturbance observer, which need to be reasonably selected to guarantee observer performance. When the value of the designed parameter is larger, the stability can be improved, but the chattering of the observed value will be larger. Therefore, it is necessary to balance the stability and transient performance.

When the disturbance observer reaches sliding mode surface, it satisfies that $e_{y}={\dot{e}}_{y}=0$ .

Therefore, it can be obtained that $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}0=e_{r}-{\zeta}sign(e_{y})\\ {\displaystyle \frac{d}{dt}}e_{r}=d-{\lambda}{\zeta}sign(e_{y}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (22)

Simplifying (20) yields $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}e_{r}+{\lambda}e_{r}-d=0. & & \displaystyle\end{eqnarray}$ (23)

Solving the differential equation to obtain disturbance estimation as $\begin{eqnarray}\displaystyle e_{r}=e^{-{\lambda}t}\left(l-\displaystyle \int d\cdot e^{{\lambda}t}dt\right), & & \displaystyle\end{eqnarray}$ (24) where 𝜆 is positive and l is an arbitrary constant. Additonally, the convergence rate of disturbance estimation error is directly related with the value of parameter 𝜆.

After determining SFC structure as described above, it is necessary to determine the controller coefficients. Thanks to the characteristics of SFC without cascading structure, all controller coefficients need to be selected simultaneously. which is important for multi-state variable systems. In conventional method for closed loop optimal control, linear quadratic optimization method can be adopted to determine the coefficients by selecting matrices Q and R to minimize discrete performance indicator [117–121].

The discrete performance indicator is described as $\begin{eqnarray}\displaystyle I_{lqr}=\displaystyle \mathop{\sum }_{k=0}^{\infty }\left[\overline{\boldsymbol{x}}^{T}(k)\boldsymbol{Q}\overline{\boldsymbol{x}}(k)+\overline{\boldsymbol{u}}^{T}(k)\boldsymbol{R}\overline{\boldsymbol{u}}(k)\right], & & \displaystyle\end{eqnarray}$ (25) with Q = diag ([q ₁ q ₃ q ₃ q ₄ q ₅]) R = diag (r ₁ r ₂).

The weighting matrices are symmetric positive definite matrices, and the dynamic performance is seriously affected by the value of the weighting matrix. Since it is related to the control signal, the larger value of the matrix R is set to give more attention to the control signal. Therefore, the overshoot can be suppressed to generate less system energy. Since the coefficients of penalty matrices Q and R are determined by manual selection, which is a very time-consuming and complex project for control system with strict control requirements.

4. Seeker optimization algorithm

The SOA is based on the research of human random intelligent search behaviour, which can be applied to the search for problem-optimized solutions obtained by seeker population. The total population is randomly divided into three subpopulations, which are searched in several different domains in the search space. The selection the search direction is based on the empirical gradient while the search step is determined by the uncertainty of human search, so as to complete the searcher position update to optimize the problem. In the implement process of the SOA algorithm, search direction and step length 𝛼_ij are calculated independently for each seeker on each variable at each time step.

In SOA, each searcher chooses the search direction according to the empirical gradient by evaluating the current and historical location of him and his neighbours. Empirical gradients involve egotistic behaviours, altruistic behaviours, and active behaviours to generate their own directions. Generally, there are two extreme cooperative behaviours in group dynamics. One selfish person is completely kissing myself, and the other selfless person is completely a family of relatives. Each seeker, as a single senior agent, has a unified self-awareness. He believes that he should move towards the best position in history based on his own judgment. The attitude of the ith searcher can be simulated by the egotistical direction vector $\vec{r}_{i,\mathit{ego}}$ .

In altruistic behaviour, seekers expect to cooperate explicitly and respond to another seeker in the same neighbourhood to adjust their behaviour to achieve the desired goal. This means that searchers exhibit pro-group behaviour. Therefore, the population exhibits self-organizing aggregation behaviour, and its positive feedback usually takes the form of attraction to a given signal source. The alternative altruistic directions can be modeled as altruistic direction $\vec{r}_{i,\mathit{alt}}$ .

In addition, seekers do not just respond to the environment, they can make goal-oriented behaviours. Moreover, future behaviours can be predicted and guided by past behaviours. Therefore, the searcher can actively change his search direction according to his past behaviour and exhibit goal-oriented behaviour. Therefore, each searcher is associated with an proactive direction $\vec{r}_{i,\mathit{pro}}$ .

Aforementioned proactive direction $\vec{r}_{i,\mathit{pro}}$ , egotistical direction $\vec{r}_{i,\mathit{ego}}$ and altruistic direction $\vec{r}_{i,\mathit{alt}}$ are expressed as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\vec{r}_{i,\mathit{ego}}(t)=\vec{m}_{i,\mathit{best}}-\vec{x}_{i}(t)\\[6.0pt] \vec{r}_{i,\mathit{alt}}(t)=\vec{n}_{i,\mathit{best}}-\vec{x}_{i}(t)\\[6.0pt] \vec{r}_{i,\mathit{pro}}(t)=\vec{x}_{i}(t_{1})-\vec{x}_{i}(t_{2}),\end{array}\right. & & \displaystyle\end{eqnarray}$ (26) where $\vec{m}_{i,\mathit{best}}$ is the best position for the seeker m _i, $\vec{n}_{i,\mathit{best}}$ is the best position for the neighbour seekers of seeker m _i, while $\vec{x}_{i}(t_{1}),\vec{x}_{i}(t_{2})$ represent the two better positions in $\{\vec{x}_{i}(t-1),\vec{x}_{i}(t-2),\vec{x}_{i}(t)\}$ .

The direction update of SOA algorithm is determined by altruistic direction, egotistical direction and proactive direction, which is expressed as follows: $\begin{eqnarray}\displaystyle \vec{r_{i}}(t)=\text{sign}({\alpha}\vec{r}_{i,pro}+{\varphi}_{1}\vec{r}_{i,ego}+{\varphi}_{2}\vec{r}_{i,alt}), & & \displaystyle\end{eqnarray}$ (27) where 𝛼 is the inertia weight, which decreases from 0.9 to 0.1. φ₁, φ₂ are random weights between 0 and 1, sign() is signum function.

The uncertain inference behaviour of SOA adopts fuzzy system to approximate human intelligent search behaviour, so as to build the relationship between step size and objective function. The fuzzy variables of the search step are expressed by Gaussian membership function: $\begin{eqnarray}\displaystyle u_{A}(x)=e^{-(x-u)^{2}/2{\delta}^{2}}, & & \displaystyle\end{eqnarray}$ (28) where u _A is Gaussian membership, x is the system input; δ and u are parameters of membership function.

In order to make the membership degree proportional to the order of the function, a linear membership function is used, that is, the maximum membership value u _max = 1, which is in the best position. The worst position corresponds to u _min = 0.0111. In order to introduce variable randomness and improve local search ability, the following formula is introduced as follows: $\begin{eqnarray}\displaystyle u_{ij}=rand(u_{i},1),\quad j=1,2,\ldots ,K, & & \displaystyle\end{eqnarray}$ (29) where u _i is the membership of the objective function, u _ij is the membership of the objective function u _i in the j-dimensional search space, K is the dimension of search space.

5. Parameter tuning of SFC

First, the fitness index needs to be designed for determination of the gain matrix using SOA. Considering the two control objectives of zero d-axis current and zero speed error, the fitness index is designed as follows: $\begin{eqnarray}\displaystyle G_{1}={\displaystyle \frac{1}{N}}\displaystyle \mathop{\sum }_{k=0}^{N}[|{\Delta}e_{{\omega}}(k)|kT_{s}+|{\Delta}e_{id}(k)|kT_{s}], & & \displaystyle\end{eqnarray}$ (30) where k is discrete sample time index. The penalty term 𝜂 is introduced to the function for the suppression of overshoot as follows: $\begin{eqnarray}\displaystyle G_{2}={\displaystyle \frac{1}{N}}\displaystyle \mathop{\sum }_{k=0}^{N}[|{\Delta}e_{{\omega}}(k)|kT_{s}+|{\Delta}e_{id}(k)|kT_{s}+{\eta}{\Delta}e_{{\omega}}(k)kT_{s}], & & \displaystyle\end{eqnarray}$ (31) with ${\rm\Delta}e_{{\omega}}(k)={\omega}_{m}(k)-{\omega}_{m}^{ref}(k)$ and Δe _id(k) = i _d(k) − i _d ^ref(k).

Based on multiple simulations implemented with different penalty term, 𝜂 = 14 is selected which can suppress speed overshoot effectively. Obviously, in the process of parameter tuning, the fitness value will decrease as the iterations increases. At the same time, the variation of fitness value shows the convergence of SOA. As defined by the fitness function, the smaller the speed error and the i _d error, the smaller the fitness value, which means better controller performance.

Fig. 4.

Evolution of G1 and G2 during parameter tuning procedure.

The overall trend of G ₁ and G ₂ during parameter tuning process with SOA are shown in Fig. 4. It can be observed that, the introduction of penalty term 𝜂 results in rather large initial value of G ₂, however the value of G ₂ decreases rapidly with increasing iterations. When iterating 16 times, the difference between G2 and G1 becomes very small. After 32 iterations, the fitness indexes corresponding to the two fitness functions are G1 = 8.53 and G2 = 7.63. Figure 5–6 shows the automatic tuning process curve in G1 and G2.

Fig. 5.

Evolution of speed during parameter tuning process with G1.

Fig. 6.

Evolution of speed during parameter tuning process with G2.

The final gain coefficient matrices can be acquired after 32 iterations as follows $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{K}_{dG_{1}}=\left[\begin{array}{@{}ccccc@{}}263.2 & 0 & 0 & 0 & 73.17\\ 0 & 0.5 & 0.036 & 0.65 & 0\end{array}\right],\quad \boldsymbol{K}_{dG_{2}}=\left[\begin{array}{@{}ccccc@{}}160.53 & 0 & 0 & 0 & 60.65\\ 0 & 1.75 & 0.043 & 0.44 & 0\end{array}\right],\quad \nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \boldsymbol{K}_{coG_{1}}=\left[\begin{array}{@{}c@{}}-37.2\\ 0\end{array}\right],\quad \boldsymbol{K}_{coG_{2}}=\left[\begin{array}{@{}c@{}}-22.74\\ 0\end{array}\right].\nonumber\end{eqnarray}$

It should be noted that the element values of K _{dG
₁} [2,2] and K _{dG
₂} [2,2] are related to the sensitivity of the controllers to variable i _q. A larger value indicates controller will attach more importation to the variable. Therefore, when the equal output voltage is requisite for controller with K _{dG
₁} (SFC₁) and SFC with K _{dG
₂} (SFC₂), i _q in SFC₂ will be smaller than that in SFC₁, which will be validated in experiment section.

6. Experimental results

For the purpose of verifying the effectiveness of the proposed controller, experiments for comparison between the proposed method with different fitness index have been conducted at a laboratory platform.

The experiments are performed on the platform of dSPACE 1401 test bench, through which the experimental measurements can be exported to MATLAB and plotted. The experimental setup includes a SPMSM, a magnetic powder brake and a torque sensor, as shown in Fig. 7. And the SPMSM and system parameters are given in Table 1.

Fig. 7.

Experimental setup used to verify the purposed SFC method.

Table 1

SPMSM and system parameters

Parameter	Symbol	Value
Pole pairs	P	22
Stator resistance	R	0.8 $\Omega$
Stator inductance	L	4.5 mH
Permanent-magnet flux linkage	𝜓_f	0.215 Wb
Inertia	J	0.03 kgm²
Viscous friction coefficient	𝜈	0.0006 N⋅m⋅s
Rated speed	N	360 rpm
Rated power	P _N	30 kW
DC-link voltage	u _dc	420 V

For the comparison of the proposed SFCs, three test cases have been taken into consideration among SFC₀ (i.e. conventional state feedback method), SFC₁ (i.e. proposed method with index fitness G ₁) and SFC₂ (i.e. proposed method with index fitness G ₂). In all cases, the reference speed is set to 350 rpm. The difference of the three cases is that in case 1, there is no load during operation. In case 2, a step change from 0 to 10 Nm in load torque is applied to the system at 0.2 s. In test case 3, the initial load torque is set as 10 Nm and remove at 0.2 s. The results of speed and current response with different controllers are shown in different conditions.

It can be observed from Fig. 8 that three controllers achieve zero error in stable condition. While different dynamic performances are shown thanks to different choice of weighting matrices. Obviously, SFC₀ method fails to track the reference speed exactly in initial 0.1 s, which produces large tracking error. In contrast, SFC₁ and SFC₂ produce quick response with smaller rising time. Furthermore, with penalty term of speed overshoot in index fitness G ₂, SFC₂shows great dynamic tracking performance with smallest speed overshoot. Additionally, SFC₁ and SFC₂ produce smaller ripples current response due to the error term of d-axis current introduced in index fitness. In Table 2 dynamic performance such as rise time, peak overshoot and settling time, is given under different load conditions.

Fig. 8.

Experimental results under no load condition. (a) Speed, (b) d-axis current response and (c) q-axis current response.

Figures 9–10 show the system response with different controllers under load torque step increase condition and step decrease condition, respectively. When step changes in load torque are applied in system, the speed responses of SFC₁ and SFC₂ converge to reference value quickly, while SFC₁ fails to make the speed response error converge to zero, which is caused by the different weighting matrix R . Great values in R produce more constraints on control signals, restricting the energy for control system. In contrast, smaller values of weighting matrix R in SFC₁ and SFC₂ have better performance against perturbance. Furthermore, since internal models of reference input are introduced in the extended state equation of the proposed SFC, SFC₁ and SFC₂ perform better capacity for load variation. Meanwhile, the penalty term in SFC₂ performs better in speed response overshoot.

In terms of current response, SFC₁ and SFC₂ produce smaller oscillation, which can be explained as the greater values in matrix Q attach more attention to corresponding state variable. Therefore greater q ₁ and q ₂ values make the control system give more emphasis on dq-axes current response.

Table 2

Dynamic response under different constant load conditions

Method	Load condition (Nm)	Rise time (ms)	Peak overshoot (%)	Settling time (ms)
SFC₀	0	52	/	112
	10	79	/	>200
SFC₁	0	11	3.7	84
	10	16	3.2	52
SFC₂	0	8	/	18
	10	18	/	29

Fig. 9.

Experimental results under a step increase in load torque condition. (a) Speed, (b) d-axis current response and (c) q-axis current response.

Figure 11 illustrates the estimated disturbance r based on the proposed disturbance observer, which is used to improve the dynamic performance of the feedback controller. The performance of the disturbance observer can be validated by substituting T _l and J _m into − T _l∕J _m. Consequently, the observed disturbance tracking the real value accurately despite ripples under non-zero load torque due to the introduction of the sign function. The observer not only maintains zero steady-state tracking error, but has rapid dynamic response performance. Meanwhile the ripples of observed value are acceptable and can be further suppressed by substituting sign function with saturation function.

Fig. 10.

Experimental results under a step decrease in load torque condition. (a) Speed, (b) d-axis current response and (c) q-axis current response.

In addition, Table 3 shows different dynamic performance of different methods in speed variation and transient time in variable load condition. And these performance indicators exhibit the superiority of the proposed method.

Table 3

Dynamic response under different variable load conditions

Method	Load condition (Nm)	Speed variation (%)	Transient time (ms)
SFC₀	Case 2	4.8	>200
	Case 3	4.5	>200
SFC₁	Case 2	6.7	18
	Case 3	6.4	16
SFC₂	Case 2	4.2	12
	Case 3	4.0	9

In all test conditions, it can be observed that SFC₁ and SFC₂ track reference speed with smaller steady-state error. Compared with SFC₂, the response of SFC₁ is slightly faster while there exists a relatively large speed overshoot. However, thanks to the introduction of penalty term in G ₂, it allows tracking reference speed almost with no overshoot for SFC₂. Furthermore, as shown in Fig. 8(b), Fig. 9(b) and Fig. 10(b), the q-axis current response based on SFC₁ reaches the peak, which may deteriorate the controller dynamic performance.

Fig. 11.

Estimated disturbance based on the proposed disturbance observer. (a) Case 2 condition (b) Case 3 condition.

7. Conclusion

In this paper, an extended state feedback control method has been proposed for speed control of PMSM drive. In order to ensure satisfactory dynamic performance of the drive system, integral models considering the errors of rotor speed and current response in d-axis has been taken into consideration in linear SFC. Disturbance observer has been designed to estimate load torque for implementation of feedforward compensation on load torque, which improve dynamic performance under variable load condition. With reasonable selection of designed parameter, the disturbance observer shows satisfactory performance. SFC parameters are obtained through auto-tuning of SOA. Furthermore, for the suppression of overshoot in speed and improvement of dynamic performance, a penalty term is introduced in the fitness index. And the experimental results show the superiority in the control performance of the proposed SFC₂ compared with SFC₁. Detailed analysis to verify the proposed SFC scheme is shown in the experimental results.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Project 51875261, the Natural Science Foundation of Jiangsu Province of China under Project BK20180046, the “Qinglan project” of Jiangsu Province, and the Postgraduate Research & the Practice Innovation Program of Jiangsu Province under Project SJCX19_0574.

References

Chen

Sun

and Cai

, Three-vector-based model predictive torque control for a permanent magnet synchronous motor of EVs, IEEE Transactions on Transportation Electrification (2021), doi:10.1109/TTE.2021.3053256.

Sun

Lei

Guo

and Zhu

, An improved model predictive current control for PMSM drives based on current track circle, IEEE Trans. Ind. Electron. 68(5) (2021), 3782–3793.

Ruan

Nian

and Sun

, An improved modulation technique with minimum switching actions within one PWM cycle for open-end winding PMSM system with isolated DC bus, IEEE Trans. Ind. Electron. 67(5) (2020), 4259–4264.

Zheng

Zhao

Zhu

and Lee

C.H.T.

, Sleeve design of permanent-magnet machine for low rotor losses, Chinese Journal of Electrical Engineering 6(4) (2020), 86–96.

Sun

Shi

Lei

Guo

and Zhu

, Analysis and design optimization of a permanent magnet synchronous motor for a campus patrol electric vehicle, IEEE Trans. Veh. Technol. 68(11) (2019), 10535–10544.

Tian

Cai

Sun

Zhu

and Xu

, An adaptive ECMS with driving style recognition for energy optimization of parallel hybrid electric buses, Energy 189 (2019), 116151.

Chen

Zhou

Wang

and Liu

, Unified decoupling vector control of five-phase permanent-magnet motor with double-phase faults, IEEE Access 8 (2020), 152646–152658.

Sun

Shi

Cai

Lei

Guo

and Zhu

, Driving-cycle oriented design optimization of a permanent magnet hub motor drive system for a four-wheel-drive electric vehicle, IEEE Trans. Transport. Electrific. 6(3) (2020), 1115–1125.

Roetzer

Vollmer

and Kennel

R.M.

, Demodulation approach for slowly sampled sensorless field-oriented control systems enabling multiple-frequency injections, IEEE Trans. Ind. Appl. 54(1) (2018), 732–744.

10.

Zhao

and Wang

, Active disturbance rejection position servo control of PMSLM based on reduced-order extended state observer, Chinese Journal of Electrical Engineering 6(2) (2020), 30–41.

11.

Sun

Shi

Lei

Guo

and Zhu

, Multi-objective design optimization of an IPMSM based on multilevel strategy, IEEE Trans. Ind. Electron. 68(1) (2021), 139–148.

12.

and Zhu

, Backstepping control of three-pole radial hybrid magnetic bearing, IET Electric Power Applications 14(8) (2020), 1405–1411.

13.

Zhou

Sun

and Song

, High performance three-phase PMSM open-phase fault-tolerant method based on reference frame transformation, IEEE Trans. Ind. Electron. 66(10) (2019), 7571–7580.

14.

Shi

Sun

Cai

Yang

Lei

Guo

, Torque analysis and dynamic performance improvement of A PMSM for EVs by skew angle optimization, IEEE Trans. Appl. Supercon. 29(2) (2019), 0600305.

15.

Sun

Jin

Wang

Yang

Fan

, Performance improvement of torque and suspension force for a novel five-phase BFSPM machine for flywheel energy storage systems, IEEE Trans. Appl. Supercon. 29(2) (2019), 0601505.

16.

Xia

Wang

Yan

and Shi

, Direct torque control for VSI-PMSM using vector evaluation factor table, IEEE Trans. Ind. Electron. 63(7) (2016), 4571–4583.

17.

Yang

Sun

Zhu

and Zhao

, Active disturbance rejection control for bearingless induction motor based on hyperbolic tangent tracking differentiator, IEEE Journal of Emerging and Selected Topics in Power Electronics 8(3) (2020), 2623–2633.

18.

and Zhu

, Active disturbance rejection control of 5-degree-of-freedom bearingless permanent magnet synchronous motor based on fuzzy neural network inverse system, ISA Transactions 101 (2020), 295–308.

19.

Sun

Chen

Jiang

Yang

Chen

and Zhang

, High-performance control for a bearingless permanent magnet synchronous motor using neural network inverse scheme plus internal model controllers, IEEE Trans. Ind. Electron. 63(6) (2016), 3479–3488.

20.

Qin

Chen

Meng

Huang

and Fu

, Computationally efficient coordinate transformation for field-oriented control using phase shift of linear hall-effect sensor signals, IEEE Trans. Ind. Electron. 67(5) (2020), 3442–3451.

21.

Shi

Sun

Lei

Yang

Guo

and Zhu

, Analysis and optimization of radial force of permanent magnet synchronous Hub motors, IEEE Trans. Magn. 56(2) (2020), 7508804.

22.

Zhou

Chen

Tao

Wang

and Liu

, Virtual-stator-flux-based direct torque control of five-phase fault-tolerant permanent-magnet motor with open-circuit fault, IEEE Trans. Power Electron. 35(5) (2020), 5007–5017.

23.

Hannan

M.A.

Ali

J.A.

Mohamed

Amirulddin

U.A.U.

Tan

N.M.L.

and Uddin

M.N.

, Quantum-behaved lightning search algorithm to improve indirect field-oriented fuzzy-PI control for IM drive, IEEE Trans. Ind. Appl. 54(4) (2018), 3793–3805.

24.

Sun

Diao

Lei

Guo

and Zhu

, Study on segmented-rotor switched reluctance motors with different rotor pole numbers for BSG system of hybrid electric vehicles, IEEE Trans. Veh. Technol. 68(6) (2019), 5537–5547.

25.

Zhang

Wang

and Zhang

, Novel PMSM control for anti-lock braking considering transmission properties of the electric vehicle, IEEE Trans. Veh. Technol. 67(11) (2018), 10378–10386.

26.

Costa

B.L.G.

Bacon

V.D.

da Silva

S.A.O.

and Angélico

B.A.

, Tuning of a PI-MR controller based on differential evolution metaheuristic applied to the current control loop of a Shunt-APF, IEEE Trans. Ind. Electron. 64(6) (2017), 4751–4761.

27.

Sun

Diao

Lei

Guo

and Zhu

, Real-time HIL emulation for a segmented-rotor switched reluctance motor using a new magnetic equivalent circuit, IEEE Trans. Power Electron. 35(4) (2020), 3841–3849.

28.

Wang

Zhang

Zhao

and Xu

, Review of parameter identification and sensorless control methods for synchronous reluctance machines, Chinese Journal of Electrical Engineering 6(2) (2020), 7–18.

29.

Cheng

Sun

Zhao

and Yang

, Direct torque and suspension force control for bearingless induction motors based on active disturbance rejection control scheme, IEEE Access (2019), 86989–87001.

30.

Zhu

and Huang

, Compensation control of suspension force for LS-BLPMSM, IET Trans. Ind. Appl. 11(4) (2017), 622–630.

31.

Sun

Chen

Yang

and Zhu

, Speed-sensorless vector control of a bearingless induction motor with artificial neural network inverse speed observer, IEEE/ASME Trans. Mechatron 18(4) (2013), 1357–1366.

32.

Neath

M.J.

Swain

A.K.

Madawala

U.K.

and Thrimawithana

D.J.

, An optimal PID controller for a bidirectional inductive power transfer system using multiobjective genetic algorithm, IEEE Trans. Power Electron. 29(3) (2014), 1523–1531.

33.

Zhao

and Zhu

, Design and analysis of a novel bearingless flux-switching permanent magnet motor, IEEE Trans. Ind. Electron. 64(8) (2017), 6127–6136.

34.

Sun

Chen

Yang

and Li

, Design and analysis of interior composite-rotor bearingless permanent magnet synchronous motors with two layer permanent magnets, Bulletin of the Polish Academy of Sciences: Technical Sciences 65(6) (2017), 833–843.

35.

Zhang

Yang

and Ma

, Improved initial rotor position estimation for PMSM drives based on HF pulsating voltage signal injection, IEEE Trans. Ind. Electron. 65(6) (2018), 4702–4713.

36.

Ling

Sun

Zhao

and Yang

, Reactive-power-based MRAC for rotor resistance and speed estimation in bearingless induction motor drives, International Journal of Applied Electromagnetics and Mechanics 62(1) (2020), 127–143.

37.

Chen

Wang

Sun

Cai

Diao

, Development of digital control system for a belt-driven starter generator SSRM for HEVs, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 234(9) (2020), 975–984.

38.

Sun

Cao

Lei

Guo

and Zhu

, Speed sensorless control for permanent magnet synchronous motors based on finite position set, IEEE Trans. Ind. Electron. 67(7) (2020), 6089–6100.

39.

Preindl

and Bolognani

, Model predictive direct speed control with finite control set of PMSM drive systems, IEEE Trans. Power Electron. 28(2) (2013), 1007–1015.

40.

Apte

Joshi

V.A.

Mehta

and Walambe

, Disturbance-observer-based sensorless control of PMSM using integral state feedback controller, IEEE Trans. Power Electron. 35(6) (2020), 6082–6090.

41.

Sun

Wang

Yang

Lei

Zhu

, Performance analysis of suspension force and torque in an IBPMSM with V-shape PMs for flywheel batteries, IEEE Trans. Magn. 54(11) (2018), 8105504.

42.

Sun

and Sun

, Optimal design and performance analysis for interior composite-rotor bearingless permanent magnet synchronous motors, IEEE Access (2019), 7456–7465.

43.

Yang

Chen

Sun

Bao

and Lu

, Rotor radial disturbance control for a bearingless induction motor based on improved active disturbance rejection control, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 38(1) (2019), 138–152.

44.

Hao

Zhu

Cheng

and Huang

, Speed control of bearingless permanent magnet synchronous motor based on flux strengthening and voltage regulation, IEEE Access (2018), 72392–72401.

45.

Sun

Jin

Chen

and Yang

, Disturbance rejection based on iterative learning control with extended state observer for a four-degree-of-freedom hybrid magnetic bearing system, Mechanical Systems and Signal Processing 153 (2021), 107465.

46.

Shi

Sun

Cai

and Yang

, Robust design optimization of a five-phase PM hub motor for fault-tolerant operation based on taguchi method, IEEE Trans. Energy Convers. 35(4) (2020), 2036–2044.

47.

Yang

Sun

Bao

and Chen

, Direct levitation force control for a bearingless induction motor based on model prediction, IEEE Access 7(1) (2016), 65368–65378.

48.

Sun

Lei

Gang

Guo

and Zhu

, State feedback control for a PM hub motor based on grey wolf optimization algorithm, IEEE Trans. Power Electron. 35(1) (2020), 1136–1146.

49.

Yang

Sun

and Ding

, Study on active disturbance rejection control of a bearingless induction motor based on an improved particle swarm optimization-genetic algorithm, IEEE Transactions on Transportation Electrification (2021), doi:10.1109/TTE.2020.3031338.

50.

Tarczewski

and Grzesiak

L.M.

, An application of novel nature-inspired optimization algorithms to auto-tuning state feedback speed controller for PMSM, IEEE Trans. Ind. Appl. 54(3) (2018), 2913–2925.

51.

Aghili

, Optimal feedback linearization control of interior PM synchronous motors subject to time-varying operation conditions minimizing power loss, IEEE Trans. Ind. Electron. 65(7) (2018), 5414–5421.

52.

Sun

Shi

and Zhu

, Multi-objective design optimization of an IPMSM for EVs based on fuzzy method and sequential Taguchi method, IEEE Trans. Ind. Electron. (2021), doi:10.1109/TIE.2020.3031534.

53.

Tarczewski

and Grzesiak

L.M.

, Constrained state feedback speed control of PMSM based on model predictive approach, IEEE Trans. Ind. Electron. 63(6) (2016), 3867–3875.

54.

Jin

Sun

Yang

Wang

Chen

and Li

, A novel four degree-of-freedoms bearingless permanent magnet machine using modified cross feedback control scheme for flywheel energy storage systems, International Journal of Applied Electromagnetics and Mechanics 60(3) (2019), 379–392.

55.

Sun

Yin

Wang

and Tian

, Multiple iterations search sensorless control for linear motor in vehicle regenerative suspension, IEEE Transactions on Transportation Electrification .2021, doi:10.1109/TTE.2021.3052989.

56.

Hua

Zhu

Gao

and Ji

, Design and analysis of two permanent-magnet-assisted bearingless synchronous reluctance motors with different rotor structure, Energies 14(4) 2021.

57.

Ling

Sun

Cai

Zhao

and Yang

, Displacement sensorless control for bearingless induction motor drives based on the MRAS method, International Journal of Applied Electromagnetics and Mechanics 62(4) (2020), 787–805.

58.

Sun

Feng

Diao

and Yang

, An improved direct instantaneous torque control based on adaptive terminal sliding mode for a segmented-rotor SRM, IEEE Transactions on Industrial Electronics (2021), accepted. doi:10.1109/TIE.2020.3029463.

59.

Meirinho

C.J.

Bartsch

de Oliveira

and Cavalca

M.S.M.

, An optimal MIMO control approach for PMSM drives, in: Proc. Brazilian Power Electron. Conf., 2017, pp. 1–6.

60.

Sun

Zhu

Wang

Zhou

Yang

, MPTC for PMSMs of EVs with multi-motor driven system considering optimal energy allocation, IEEE Trans. Magn. 55(7) (2019), 8104306.

61.

Diao

Sun

Lei

Bramerdorfer

Guo

and Zhu

, System-level robust design optimization of a switched reluctance motor drive system considering multiple driving cycles, IEEE Trans. Energy Convers. (2020).

62.

Zhou

Sun

and Zeng

, PMSM open-phase fault-tolerant control strategy based on four-leg inverter, IEEE Trans. Power Electron. 35(3) (2020), 2799–2808.

63.

Sun

Lei

Cai

Chen

and Guo

, Torque modeling of a segmented-rotor SRM using maximum-correntropy-criterion-based LSSVR for torque calculation of EVs, IEEE J. Emerg. Sel. Topics Power Electron. (2021), doi:10.1109/JESTPE.2020.2977957.

64.

Ruan

Nian

and Sun

, Simplified modulation scheme for open-end winding PMSM system with common DC bus under open-phase fault based on circulating current suppression, IEEE Trans. Power Electron. 35(1) (2020), 10–14.

65.

Yang

Ding

Sun

Zhao

and Luo

, Analysis and optimization of a bearingless induction motor’s suspension force and unbalanced magnetic pulling force mathematical model, IET Electric Power Applications 14(7) (2020), 1247–1255.

66.

Sun

Cao

Lei

Guo

and Zhu

, A composite sliding mode control for SPMSM drives based on a new hybrid reaching law with disturbance compensation, IEEE Transactions on Transportation Electrification (2021), doi:10.1109/TTE.2021.3052986.

67.

Sun

Lei

Guo

and Zhu

, Torque ripple reduction of SRM drive using improved direct torque control with sliding mode controller and observer, IEEE Transactions on Industrial Electronics (2021), accepted. doi:10.1109/TIE.2020.3020026.

68.

Galccki

Michalczuk

Ufnalski

and Grzesiak

L.M.

, Particle swarm optimization of the state feedback current controller with oscillatory terms for a three-phase grid-tie converter, in: Proc. 20th European Conf. Power Electron. Appl., 2018, pp. 1–10.

69.

Sun

Diao

and Yang

, Performance improvement of a switched reluctance machine with segmental rotors for hybrid electric vehicles, Computers and Electrical Engineering (2019), 244–259.

70.

Brasel

, A gain-scheduled multivariable LQR controller for permanent magnet synchronous motor, in: Proc. 19th Int. Conf. Methods Models Auto. Robot., 2014, pp. 722–725.

71.

Yang

Zhang

Sun

and Chen

, Nonsingular fast terminal sliding mode control for a bearingless induction motor, IEEE Access (2017), 16656–16664.

72.

Jin

Sun

Cai

Zhu

Lei

and Guo

, Comprehensive sensitivity and cross-factor variance analysis-based multi-objective design optimization of a 3-DOF hybrid magnetic bearing, IEEE Trans. Magn. 57(2) (2021), 8000204.

73.

Yang

Wan

Sun

Chen

and Chen

, Sliding mode control for bearingless induction motor based on a novel load torque observer, Journal of Sensors (2016), 8567429.

74.

Sun

Chen

Yang

and Shi

, Precise control of a four degree-of-freedom permanent magnet biased active magnetic bearing system in a magnetically suspended direct-driven spindle using neural network inverse scheme, Mechanical Systems and Signal Processing 88 (2017), 36–48.

75.

Tarczewski

and Grzesiak

L.M.

, An application of novel nature-inspired optimization algorithms to auto-tuning state feedback speed controller for PMSM, IEEE Trans. Ind. Appl. 54(3) (2018), 2913–2925.

76.

Dai

Chen

and Zhu

, Seeker optimization algorithm for digital IIR filter design, IEEE Trans. Ind. Electron. 57(5) (2010), 1710–1718.

77.

Diao

Sun

Lei

Guo

and Zhu

, Multiobjective system level optimization method for switched reluctance motor drive systems using finite element model, IEEE Trans. Ind. Electron. 67(12) (2020), 10055–10065.

78.

Yang

Zhang

Sun

and Ye

, Adaptive exponential sliding mode control for a bearingless induction motor based on a disturbance observer, IEEE Access (2018), 35425–35434.

79.

Sun

Chen

Jiang

Yang

Chen

and Zhang

80.

Yang

Ding

Sun

and Zhao

, Design and analysis of a novel wound rotor for a bearingless induction motor, International Journal of Electronics 106(12) (2016), 1829–1844.

81.

Shaw

Mukherjee

and Ghoshal

S.P.

, Seeker optimisation algorithm: Application to the solution of economic load dispatch problems, IET. Gener. Transm. Distrib. 5(1) (2011), 81–91.

82.

Sun

Jin

Cai

Yang

and Chen

, Grey wolf optimization algorithm based state feedback control for a bearingless permanent magnet synchronous machine, IEEE Trans. Power Electron. 35(12) (2020), 13631–13640.

83.

Hua

and Zhu

, Rotor radial displacement sensorless control of bearingless permanent magnet synchronous motor based on MRAS and suspension force compensation, ISA Transactions 103 (2020), 306–318.

84.

Sun

Shi

Wang

Chen

Yang

and Li

, Linearizing control of a permanent magnet linear synchronous motor with inverse system scheme plus an internal model controller, International Journal of Applied Electromagnetics and Mechanics 55(4) (2017), 523–534.

85.

Sun

Zhu

Lei

and Guo

, Improved model predictive torque control for PMSM drives based on duty cycle optimization, IEEE Trans. Magn. 57(2) (2021), 8200505.

86.

Cheng

Sun

Chen

and Nian

, Model predictive current control for an open-winding PMSM system with a common DC bus in 3-D space, IEEE Trans. Power Electron. 35(9) (2020), 9597–9607.

87.

Cheng

Sun

and Yang

, A nonlinear flux linkage model for bearingless induction motor based on GWO-LSSVM, IEEE Access (2019), 36558–36567.

88.

Sun

Yang

Lei

and Guo

, High-performance control for a permanent-magnet linear synchronous generator using state feedback control scheme plus grey wolf optimization, IET Electr. Power Appl. 14(5) (2020), 771–780.

89.

Shi

Sun

Cai

Tian

and Chen

, Design optimization of an outer-rotor permanent magnet synchronous hub motor for a low-speed campus patrol EV, IET Electr. Power Appl. (2020), doi:10.1049/iet-epa.2020.0130.

90.

Huang

Rong

Zhang

and Liao

, Robust predictive torque control of N*3-phase PMSM for high-power traction application, IEEE Trans. Power Electron. 35(10) (2020), 10799–10809.

91.

Sun

Zhou

Chen

Yang

and Han

, Performance analysis of segmented rotor switched reluctance motors with three types of winding connections for belt-driven starter generators of hybrid electric vehicles, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 37(3) (2018), 1258–1270.

92.

Cho

Ahn

Liu

H.c.

Hong

Lee

and Go

, Analysis of inductance according to the applied current in spoke-type PMSM and suggestion of driving mode, IEEE Trans. Magn. 53(6) (2017), 1–4.

93.

Zhu

and Liu

, Rotor displacement self-sensing modeling of six-pole radial hybrid magnetic bearing using improved particle swarm optimization support vector machine, IEEE Transactions on Power Electronics 35(11) (2020), 12296–12306.

94.

Zhao

Liu

Chen

and Sui

, Research on the torque ripple and scanning range of an arc-structure PMSM used for scanning system, IEEE Trans. Magn. 50(11) (2014), 1–4.

95.

Sun

Shen

Zhou

Yang

and Chen

Long

, Modeling and control of a bearingless permanent magnet synchronous motor, International Journal of Applied Electromagnetics and Mechanics 53(1) (2017), 151–165.

96.

Zhou

and Chen

, Predictive DTC strategy with fault-tolerant function for six-phase and three-phase pmsm series-connected drive system, IEEE Trans. Ind. Electron. 65(11) (2018), 9101–9112.

97.

Chung

Kim

Chun

Woo

and Hong

, Fractional slot concentrated winding pmsm with consequent pole rotor for a low-speed direct drive: Reduction of rare earth permanent magnet, IEEE Trans. Energy Convers. 30(1) (2015), 103–109.

98.

Lara

and Chandra

, Effects of rotor position error in the performance of field-oriented-controlled PMSM drives for electric vehicle traction applications, IEEE Trans. Ind. Electron. 63(8) (2016), 4738–4751.

99.

Sun

Lei

Yang

Guo

and Zhu

, Speed sensorless control of SPMSM drives for EVs with a binary search algorithm-based phase-locked loop, IEEE Trans. Veh. Technol. 69(5) (2020), 4968–4978.

100.

Kim

, Simple online fault detecting scheme for short-circuited turn in a PMSM through current harmonic monitoring, IEEE Trans. Ind. Electron. 58(6) (2011), 2565–2568.

101.

Wang

Chen

Cheng

and Chau

K.T.

, Field-oriented control and direct torque control for paralleled VSIs Fed PMSM drives with variable switching frequencies, IEEE Trans. Power Electron. 31(3) 2417–2428.

102.

Sun

Shen

Wang

Lei

Yang

and Han

, Core losses analysis of a novel 16/10 segmented rotor switched reluctance BSG motor for HEVs using nonlinear lumped parameter equivalent circuit model, IEEE/ASME Trans. Mechatron. 23(2) (2018), 747–757.

103.

and Gu

, Fuzzy adaptive internal model control schemes for PMSM speed-regulation system, IEEE Trans. Ind. Inform. 8(4) (2012), 767–779.

104.

Zhou

Sun

and Song

, High performance three-phase PMSM open-phase fault-tolerant method based on reference frame transformation, IEEE Trans. Ind. Electron. 66(10) (2019), 7571–7580.

105.

Diao

Sun

Lei

Guo

and Zhu

, Multimode optimization of switched reluctance machines in hybrid electric vehicles, IEEE Transactions on Energy Conversion (2020), doi:10.1109/TEC.2020.3046721.

106.

Wang

Kuang

Zhao

Zhang

and Xu

, Rotor position estimation of PMSM in low-speed region and standstill using zero-voltage vector injection, IEEE Trans. Power Electron. 33(9) (2018), 7948–7958.

107.

Kazerooni

Hamidifar

and Kar

N.C.

, Analytical modelling and parametric sensitivity analysis for the PMSM steady-state performance prediction, IET Electr. Power Appl. 7(7) (2013), 586–596.

108.

Sun

Chen

Yang

Chen

and Zhang

, Nonlinear flux linkage modeling of a bearingless permanent magnet synchronous motor based on AW-LSSVM regression algorithm, International Journal of Applied Electromagnetics and Mechanics 51(2) (2016), 151–159.

109.

Nian

and Zheng

, Torque ripple suppression method with reduced switching frequency for open-winding PMSM drives with common DC bus, IEEE Trans. Ind. Electron. 66(1) (2019), 674–684.

110.

Lai

, PMSM drive system efficiency optimization using a modified gradient descent algorithm with discretized search space, IEEE Trans. Transport. Electrific. 6(3) (2020), 1104–1114.

111.

Ping

Wang

Huang

Wang

and Li

, Internal model control of PMSM position servo system: Theory and experimental results, IEEE Trans. Ind. Inform. 16(4) (2020), 2202–2211.

112.

Sun

Wang

Diao

and Yang

, Analysis of torque ripple and fault-tolerant capability for a 16/10 segmented switched reluctance motor in HEV applications, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 38(6) (2019), 1725–1737.

113.

Feng

Lai

and Kar

N.C.

, An analytical solution to optimal stator current design for PMSM torque ripple minimization with minimal machine losses, IEEE Trans. Ind. Electron. 64(10) (2017), 7655–7665.

114.

Lee

Ahmed

and Lukic

S.M.

, Universal restart strategy for high-inertia scalar-controlled PMSM drives, IEEE Trans. Ind. Appl. 52(5) (2016), 4001–4009.

115.

Zhan

Zhu

and Odavic

, Analysis and suppression of zero sequence circulating current in open winding PMSM drives with common DC bus, IEEE Trans. Ind. Appl. 53(4) (2017), 3609–3620.

116.

Sun

Diao

Yang

Lei

Guo

and Zhu

, Direct torque control based on a fast modeling method for a segmented-rotor switched reluctance motor in HEV application, IEEE J. Emerg. Sel. Topics Power Electron. 9(1) (2021), 232–241.

117.

Huangfu

Wang

Di Rienzo

and Zhu

, Radiated EMI modeling and performance analysis of a PWM PMSM drive system based on field-circuit coupled FEM, IEEE Trans. Magn. 53(11) (2017), 1–4, Art no. 8208804.

118.

Mohamed

Y.A.I.

and El-Saadany

E.F.

, a current control scheme with an adaptive internal model for torque ripple minimization and robust current regulation in PMSM drive systems, IEEE Trans. Energy Convers. 23(1) (2008), 92–100.

119.

Xia

Zhao

Yan

and Shi

, A novel direct torque and flux control method of matrix converter-fed PMSM drives, IEEE Trans. Power Electron. 29(10) (2014), 5417–5430.

120.

Sun

Chen

Yang

Qiao

, A high-performance control scheme for reluctance type bearingless motors, International Journal of Applied Electromagnetics and Mechanics 53(3) (2017), 537–549.

121.

Feng

Lai

and Kar

N.C.

, Practical testing solutions to optimal stator harmonic current design for PMSM torque ripple minimization using speed harmonics, IEEE Trans. Power Electron. 33(6) (2018), 5181–5191.