Torque Oscillation Attenuation in PMSM Using Equivalent-Input-Disturbance-Based Sliding-Mode Control

Abstract

This paper presents a torque oscillation attenuation method for permanent magnet synchronous motors (PMSMs) based on the combination of sliding-mode control (SMC) and the equivalent input disturbance (EID) approach. To deal with the changes in PMSM parameters, we explored a continuous-domain ant colony optimization (CDACO) method to design a control system for such a plant. This is the first application of SMC-EID to uncertain PMSM plants, with CDACO enabling robust parameter tuning in continuous spaces. First, we designed an EID estimator to estimate the disturbance caused by torque oscillation. Next, we added the estimated disturbance to the sliding-mode controller to improve disturbance attenuation performance. Then, we extended an ant colony optimization (ACO) algorithm to the continuous domain to optimize controller parameters for an uncertain plant. Finally, a speed control experiment was carried out on a two-mass experimental system for PMSMs to demonstrate the validity of the method. The experimental results show that our method yields better control performance than the SMC.

1. Introduction

A permanent magnet synchronous motor (PMSM) has many advantages, including a simple structure, a large torque, and high efficiency and reliability [1]. It has been widely used in industrial robots [2], conveyors [3], drilling machines [4], and many other industrial applications. However, it is difficult to achieve high-precision control of a PMSM servo system because it is affected by disturbances [5].

Torque oscillations are critical for PMSMs because they degrade the control performance of PMSM servo systems [6]. Thus, suppressing torque oscillations is an important figure of merit for PMSMs in the application of electrical vehicle traction, especially in low-speed operations [7]. It has been extensively investigated in the literature. Approaches are mainly divided into two categories: design-based and control-based [8]. The former focuses on optimizing the geometric structures of stators and rotors to minimize torque oscillations [9]. The latter focuses on designing control strategies to smooth the PMSM torque [10].

Control-based approaches can be used to further mitigate torque oscillations for a suitably designed physical structure. Sliding-mode control (SMC) [11], active disturbance rejection methods [12], and many other advanced control strategies have been applied to PMSM control systems to achieve high control precision. The SMC system has been widely used in control practice due to its simple structure, fast dynamic response, and strong disturbance attenuation [13]. However, such a system requires disturbance information in the design of a sliding-mode controller. Moreover, there is a trade-off between disturbance attenuation performance and chattering for SMC systems. More specifically, the high gain of a sliding-mode controller guarantees disturbance attenuation performance, but it leads to a chattering phenomenon. Thus, SMC has been modified to reduce chattering from various angles, such as high-order SMC [14], integral SMC [15], and a reaching-law method [16]. However, these methods have complex structures and relatively difficult design methods. Recent advanced SMC methods, including disturbance-observer-based SMC for synchronous reluctance motors [17] and linear-extended-state-observer-based terminal SMC for PMSMs [18], manage to actively estimate and attenuate disturbances, but these lack handling of uncertainties.

Active disturbance rejection methods use observers to estimate and compensate for disturbances actively. The equivalent input disturbance (EID) approach is one of these methods. An EID is a signal on the control input channel of a plant that produces the same effect on the output as actual disturbances do [19]. Compared to active disturbance rejection control (ADRC), disturbance-observer-based control (DOBC), and other active disturbance rejection methods, the EID approach has its unique advantages. Unlike ADRC, the configuration of an EID system is relatively simple; unlike DOBC, it does not require an inverse model of a plant. It yields satisfactory disturbance attenuation performance in many applications, such as vibration depression of buildings [20], quadrotor control [21], and servo systems [22]. It is worth mentioning that the system stability and control performance of an SMC system are ensured by a low gain of the sliding-mode controller if the system uses the information of a disturbance.

Several studies have demonstrated the potential to improve the oscillation attenuation performance of PMSMs by combining SMC and EID approaches. Refs. [23,24] incorporated EID into a conventional SMC system. These two studies allowed SMC to reject unmatched disturbances and calculate the stability domain of SMC switching gains. However, they did not explain how to choose specific SMC switching gains. Refs. [25,26] used SMC as the controller in an EID system. These two methods improved the dynamic performance of the control system. Nevertheless, they did not give the method for designing SMC parameters. Ref. [27] used an SMC controller to suppress the estimation error that the conventional EID approach neglected. This study managed to ensure system stability with a low switching gain, which produced little chattering compared to conventional SMC. However, it used the method of enlarging and reducing to design the switching gain. There was conservativeness in the process. These studies proved that the combination of SMC and EID improved the control performance and produced little chattering with a low switching gain. In PMSM servo systems, the moment of inertia and the viscous friction coefficient may change with time and operating environments. However, up to now, the SMC-EID method has only been used to handle plants without uncertainties. It is challenging to extend SMC-EID to plants with uncertainties.

Intelligent optimization algorithms provide a way to address the design problem of uncertain systems. Among these algorithms, the ant colony optimization (ACO) algorithm is a reliable population-based algorithm mimicking ant foraging behavior for solving discrete problems [28]. It is a multi-agent meta-heuristic optimization method. ACO algorithms involve simple agents (ants) that cooperate with each other to achieve an emergent unified behavior for a system. They are capable of finding high-quality solutions for problems with a large search space [29]. Note that the solution of the ACO is obtained from a finite set of discrete combinations [30]. The optimization of SMC parameters requires finding a solution in a continuous solution space. How to extend the ACO method to a continuous solution space is a critical problem.

In this study, we combine the EID and SMC methods to attenuate torque oscillation in an uncertain plant. We extend ACO to a continuous solution space and name it continuous-domain ant colony optimization (CDACO) for designing control systems for such plants. Unlike these prior works [23,24,25,26,27], which only handled plants without uncertainties and focused on the switch gain design, we address uncertainties via CDACO, providing a systematic multi-parameter design for real-world variations in PMSM servo systems, where the moment of inertia and the viscous friction coefficient may change with time and operating environments. The basic idea was presented in [31]. This paper is a significant improvement over that paper: the plant is uncertain, and we developed a CDACO algorithm to design the control system for such a plant. In the rest of this paper, first, we analyze the disturbances in PMSM servo systems. Next, we present a system configuration that combines the EID and SMC methods for a PMSM. Then, we develop a CDACO algorithm to design the SMC-EID-based control system for an uncertain plant. Finally, experiments verify its validity. The main contributions of this article are as follows:

(1)

We applied the SMC-EID method to an uncertain system. SMC-EID attenuates a disturbance effectively and achieves a fast response while having little chattering.

(2)

We extended the discrete ACO to a continuous solution space and developed a CDACO algorithm.

(3)

We used the CDACO algorithm to design the control system for an uncertain plant. CDACO optimizes the integral of timed square error (ITSE) of an uncertain system to design the SMC parameters.

In this paper, ${\left|\right|D\left|\right|}_{\infty }:={\sup }_{0\le \omega \le \infty }{ϵ}_{\max }\left[D\left(j\omega \right)\right]$, and ${ϵ}_{\max }\left(D\right)$ means the maximum singular value of D. $D\left(s\right)$ is the Laplace transform of $d\left(t\right)$ and ${\mathcal{L}}^{-1}$ is the inverse Laplace transform.

2. Torque Oscillations in PMSM Servo Systems

A basic control strategy for a PMSM drive is vector control. The following model of a PMSM (Table 1) is used for this purpose:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{i}_{\mathit{d}}}{\mathrm{d}t}}& ={\displaystyle \frac{1}{L_{\mathit{d}}}}\left(u_{\mathit{d}}-R{i}_{\mathit{d}}+n_p\omega L_q{i}_q\right)\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}& ={\displaystyle \frac{1}{L_q}}\left(u_q-R{i}_q-n_p\omega L_{\mathit{d}}{i}_{\mathit{d}}-n_p\omega {\psi }_f\right)\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}\omega }{\mathrm{d}t}}& ={\displaystyle \frac{1}{J}}\left({\tau }_e-B_m\omega +{\tau }_o\right)\hfill \\ \hfill {\tau }_e& ={\displaystyle \frac{3}{2}}{n}_p\left[{\psi }_f{i}_q+\left(L_{\mathit{d}}-L_q\right)i_{\mathit{d}}{i}_q\right],\hfill \end{array}\right.$$

(1)

where the subscripts d and q mean the direct and quadrature axes, respectively.

Table 1. PMSM parameters ($j=d,q$).

Parameter	Meaning
$i_j$	Stator current of the j axis
$u_j$	Stator voltage of the j axis
$L_j$	Inductor of the j axis stator
R	Reluctance of stator
$B_m$	Viscous damping coefficient
J	Moment of the inertia
$\omega$	Angular speed of rotor
${\psi }_f$	Permanent magnet flux
$n_p$	Number of pole pairs
${\tau }_e$	Electrical torque
${\tau }_o$	Oscillation torque

The oscillation torque, ${\tau }_o$, is mainly caused by a cogging torque and a torque ripple. A cogging torque results from the structure of the motor, also known as a position-dependent torque. A torque ripple is caused by the flux linkage between the permanent magnet and the stator currents. Based on [9], we analyze and model the torque oscillation in PMSMs.

2.1. Cogging Torque

The interaction between the permanent magnet flux and the magnetic conductance generated by the stator slot is the source of cogging torque, which is determined by its fundamental and harmonic waves. The period of the torque is

$$T_1={\displaystyle \frac{2n_p}{N_1}},$$

(2)

where $N_1$ is the least common multiple between the number of stator slots and pole pairs $n_p$.

The interaction between the direct current component of permanent magnet flux and the harmonic interaction of magnetic conductance is another source of the cogging torque. Similarly, the period of the torque is

$$T_2={\displaystyle \frac{2n_p}{N_2}},$$

(3)

where $N_2$ is the least common multiple between the number of stator slots and rotor slots.

As a result, the cogging torque is

$$d_{cog}=d_1^{cog}sin{\displaystyle \frac{\pi N_1}{n_p}}{\theta }_e+d_2^{cog}sin{\displaystyle \frac{\pi N_2}{n_p}}{\theta }_e,$$

(4)

where $d_1$ and $d_2$ are the amplitudes of the cogging torque and ${\theta }_e$ is the electrical angle.

2.2. Torque Ripple

Due to a non-sinusoidal flux density distribution in air gaps, the resultant flux linkage between the permanent magnet and the stator currents contains the harmonics in the a-b-c frame.

The corresponding flux linkage appears as the 6th and its multiple harmonics, which are expressed as

$${\psi }_{dm}={\psi }_{d0}+{\psi }_{d6}cos6{\theta }_e+{\psi }_{d12}cos12{\theta }_e+\cdots ,$$

(5)

where ${\psi }_{d0}$, ${\psi }_{d6}$, and ${\psi }_{d12}$ are the DC, the 6th, and the 12th harmonic terms of the d-axis flux linkage and ${\theta }_e$ is the electrical angle.

The effect of a torque ripple is

$$d_{tr}={\displaystyle \frac{3}{2}}{n}_p{i}_q\sum _{j=0}^{\infty }{\psi }_{d6j}cos6j{\theta }_e.$$

(6)

The cogging torque and the torque ripple are both functions of the angle ${\theta }_e$. Thus, the oscillation torque contains both of them:

$$\begin{array}{cc}\hfill d_{to}=& d_1^{cog}sin{\displaystyle \frac{\pi N_1}{n_p}}{\theta }_e+d_2^{cog}sin{\displaystyle \frac{\pi N_2}{n_p}}{\theta }_e+{\displaystyle \frac{3}{2}}{n}_p{i}_q\sum _{j=0}^{\infty }{\psi }_{d6j}cos6j{\theta }_e.\hfill \end{array}$$

(7)

The disturbance model in (7) is derived from physical principles underlying PMSM torque oscillations. Cogging torque arises from the interaction between the permanent magnet flux and stator slot reluctance, with periods $T_1$ and $T_2$ determined by the least common multiple of stator slots and rotor poles, leading to periodic harmonics. Torque ripple originates from non-sinusoidal flux and current harmonics, typically dominated by 6th-order multiples in three-phase systems. It is a disturbance that relates to the angles of the motor. Its frequencies vary with the speed of the PMSM. As the EID approach attenuates disturbances in a wide frequency range, we combine the SMC and EID methods to achieve a fast response and satisfy torque oscillation attenuation.

3. SMC-EID System for PMSM

The configuration of an SMC-EID control system (Figure 1) has four parts: a plant, an SMC, a state observer, and a disturbance estimator. This section explains the basic structure and configuration of the EID approach.

Figure 1. Configuration of SMC-EID.

This study considers a surface-mounted PMSM, in which $L_d=L_q$. Let $i_d=0$, which is widely used to control such a PMSM. Thus, (1) becomes

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{i}_q\left(t\right)}{\mathrm{d}t}}& ={\displaystyle \frac{1}{L_q}}\left[u_q\left(t\right)-R{i}_q\left(t\right)-n_p\omega \left(t\right){\psi }_f\right]\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}\omega \left(t\right)}{\mathrm{d}t}}& ={\displaystyle \frac{1}{J}}\left[{\displaystyle \frac{3}{2}}{n}_p{\psi }_f{i}_q\left(t\right)-B_m\omega \left(t\right)+{\tau }_o\left(t\right)\right]\hfill \end{array}\right..$$

(8)

Define

state variables and the control input as

$$\left\{\begin{array}{cc}\hfill x_1\left(t\right)& =\omega \left(t\right)\hfill \\ \hfill x_2\left(t\right)& =i_q\left(t\right)\hfill \\ \hfill u\left(t\right)& =u_q\left(t\right)\hfill \end{array}\right.$$

(9)

respectively. Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{x}_1\left(t\right)}{\mathrm{d}t}}& ={\displaystyle \frac{1}{J}}\left[{\displaystyle \frac{3}{2}}{n}_p{\psi }_f{x}_2\left(t\right)-B_m{x}_1\left(t\right)+{\tau }_o\left(t\right)\right],\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{x}_2\left(t\right)}{\mathrm{d}t}}& ={\displaystyle \frac{1}{L_q}}\left[u-R{x}_2\left(t\right)-n_p{\psi }_f{x}_1\left(t\right)\right].\hfill \end{array}$$

(10)

The inductor of the q-axis is usually small for a small-size PMSM (less than 10 mH). Thus, we assume that

$$u-R{x}_2\left(t\right)-n_p{\psi }_f{x}_1\left(t\right)=L_q{\displaystyle \frac{\mathrm{d}{x}_2\left(t\right)}{\mathrm{d}t}}=0.$$

(11)

This allows us to write (10) as

$$x_2\left(t\right)={\displaystyle \frac{1}{R}}\left[u\left(t\right)-K{x}_1\left(t\right)\right].$$

(12)

Since a voltage compensator is usually used to compensate for $K{x}_1\left(t\right)$, the model of the PMSM, (10), with a voltage compensator becomes

$${\displaystyle \frac{\mathrm{d}{x}_1\left(t\right)}{\mathrm{d}t}}=-{\displaystyle \frac{B_m}{J}}{x}_1\left(t\right)+{\displaystyle \frac{3K}{2JR}}u\left(t\right)+{\displaystyle \frac{1}{J}}{\tau }_o\left(t\right).$$

(13)

Define the sliding surface as

$$\sigma \left(t\right)=c\left[r\left(t\right)-x_1\left(t\right)\right],$$

(14)

An SMC controller is given as

$$u_f\left(t\right)={\epsilon }_1·\mathrm{sgn}\left[{\sigma }_1\left(t\right)\right]+\lambda \sigma \left(t\right).$$

(15)

Combining with (21) yields

$$u\left(t\right)={\epsilon }_1·\mathrm{sgn}\left[\sigma \left(t\right)\right]+{\lambda }_1\sigma \left(t\right)-{\tilde{\tau }}_{oe}\left(t\right).$$

(16)

The control input of a conventional SMC system without EID is

$$u\left(t\right)={\epsilon }_2·\mathrm{sgn}\left[{\sigma }_2\left(t\right)\right]+\lambda \sigma \left(t\right).$$

(17)

3.1. Basic Structure of EID Approach

A linear plant with a disturbance

$$\left\{\begin{array}{cc}\hfill \dot{x}\left(t\right)& =Ax\left(t\right)+Bu\left(t\right)+B_d{\tau }_o\left(t\right)\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right)\hfill \end{array}\right.$$

(18)

can be interpreted as a plant with an EID [32], ${\tau }_{de}\left(t\right)$, that is,

$$\left\{\begin{array}{cc}\hfill \dot{x}\left(t\right)& =Ax\left(t\right)+B\left[u\left(t\right)+{\tau }_{de}\left(t\right)\right]\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right).\hfill \end{array}\right.$$

(19)

She et al. devised a method to produce an estimate of $d_e\left(t\right)$ in a simple way (Figure 2) [19]. A full-order observer is used to estimate it.

Figure 2. Basic configuration of EID approach.

The observer is

$$\left\{\begin{array}{cc}\hfill \dot{\widehat{x}}\left(t\right)& =A\widehat{x}\left(t\right)+B{u}_f\left(t\right)+L\left[y\left(t\right)-C\widehat{x}\left(t\right)\right]\hfill \\ \hfill \widehat{y}\left(t\right)& =C\widehat{x}\left(t\right),\hfill \end{array}\right.$$

(20)

where $\widehat{x}\left(t\right)$ is the reconstructed state of $x\left(t\right)$, $\widehat{y}\left(t\right)$ is the output of the observer, $u_f\left(t\right)$ is the control input of the observer, and L is the observer gain. Then

$$u\left(t\right)=u_f\left(t\right)-{\tilde{\tau }}_{de}\left(t\right).$$

(21)

An estimate of the EID [19] is

$${\widehat{\tau }}_{de}\left(t\right)=-B^{+}LC\Delta x\left(t\right)+u_f\left(t\right)-u\left(t\right),$$

(22)

where

$$\Delta x\left(t\right)=\widehat{x}\left(t\right)-x\left(t\right)$$

(23)

and

$$B^{+}={\left(B^{T}B\right)}^{-1}{B}^T.$$

(24)

A low-pass filter $F\left(s\right)$ is used to select the estimation frequency band. It satisfies

$$\left|F\left(j\omega \right)\right|\approx 1,0\le \omega \le {\omega }_r,$$

(25)

where ${\omega }_r$ is the highest angular frequency for disturbance attenuation. ${\tilde{\tau }}_{de}\left(t\right)$ is the filtered signal of ${\widehat{\tau }}_{de}\left(t\right)$:

$${\tilde{\tau }}_{de}\left(t\right)={\mathcal{L}}^{-1}\left[F\left(s\right){\widehat{T}}_{de}\left(s\right)\right],$$

(26)

Substituting (21) into (20) yields

$$\dot{\widehat{x}}\left(t\right)=A\widehat{x}\left(t\right)+B\left[u\left(t\right)+{\tilde{\tau }}_{de}\left(t\right)\right]+L[y-C\widehat{x}\left(t\right)].$$

(27)

Note that the difference between $x\left(t\right)$ and $\widehat{x}\left(t\right)$ is equivalent to the difference between ${\tau }_{de}\left(t\right)$ and ${\tilde{\tau }}_{de}\left(t\right)$, according to (19) and (27). Thus, a properly designed observer gain, L, can ensure $|\Delta x(t\left)\right|\approx 0$, which makes $|{\tau }_{de}\left(t\right)-{\tilde{\tau }}_{de}\left(t\right)|$ small enough.

3.2. Stability Analysis

We should design the low-pass filter and the observer gain so that they do not destroy the stability of the whole system. Let

$${\tau }_o\left(t\right)=0.$$

(28)

The plant (18) is

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right).$$

(29)

Substituting (21) into (22) yields

$${\widehat{\tau }}_{de}\left(t\right)=-B^{+}LC\Delta x\left(t\right)+{\tilde{\tau }}_{de}\left(t\right).$$

(30)

Combining (27), (29), and (30), and redrawing Figure 1, yields Figure 3.

Figure 3. Block diagram of SMC-EID.

The stability conditions for the improved servo system are broken down into two parts. First, the control law (21) guarantees the stability of the EID system. Second, the feedback control system is stable.

The transfer function from $\tilde{d}\left(t\right)$ to $\hat{d}\left(t\right)$ is

$$\begin{array}{c}\hfill G\left(s\right)=1-B^{+}LC{[sI-(A-LC)]}^{-1}B=B^{+}(sI-A){[sI-(A-LC)]}^{-1}B.\end{array}$$

(31)

Theorem 1.

When the system is under the following conditions, (21) guarantees the stability of the control system.

(1)

$A-LC$ is stable.

(2)

${\left|\right|GF\left|\right|}_{\infty }<1$. Here, ${\left|\right|GF\left|\right|}_{\infty }:={\sup }_{0\le \omega \le \infty }{ϵ}_{\max }\left[G\left(j\omega \right)F\left(j\omega \right)\right]$, and ${ϵ}_{\max }\left(G\right)$ means the maximum sigular value of G.

The feedback system is an SMC system. The sliding surface is defined in (14). The parameters of the state-space Formula (18) for the PMSM (13) are

$$A=-{\displaystyle \frac{B_m}{J}},B={\displaystyle \frac{3K}{2JR}},B_d={\displaystyle \frac{1}{J}},C=1.$$

(32)

The derivative of $\sigma \left(t\right)$ with time is

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}\sigma \left(t\right)}{\mathrm{d}t}}=& {\displaystyle \frac{c{B}_m}{J}}{x}_1\left(t\right)-{\displaystyle \frac{3cK}{2JR}}u\left(t\right)\hfill \\ \hfill =& {\displaystyle \frac{B_m}{J}}[cr\left(t\right)-\sigma \left(t\right)]-{\displaystyle \frac{3cK}{2JR}}u\left(t\right)\hfill \\ \hfill =& -{\displaystyle \frac{B_m}{J}}\sigma \left(t\right)-{\displaystyle \frac{3cK}{2JR}}u\left(t\right)+{\displaystyle \frac{c{B}_m}{J}}r\left(t\right)\hfill \end{array}.$$

(33)

The controller is designed as (15). Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}\sigma \left(t\right)}{\mathrm{d}t}}=& -{\displaystyle \frac{B_m}{J}}\sigma \left(t\right)-{\displaystyle \frac{3cK}{2JR}}\left[{\epsilon }_1·\mathrm{sgn}\left(\sigma \left(t\right)\right)+{\lambda }_1\sigma \left(t\right)\right]+{\displaystyle \frac{c{B}_m}{J}}r\left(t\right)\hfill \\ \hfill =& -\left({\displaystyle \frac{B_m}{J}}+{\displaystyle \frac{3cK}{2JR}}{\lambda }_1\right)\sigma \left(t\right)-{\displaystyle \frac{3cK}{2JR}}{\epsilon }_1·\mathrm{sgn}\left[\sigma \left(t\right)\right]+{\displaystyle \frac{c{B}_m}{J}}r\left(t\right)\hfill \end{array}$$

(34)

Define a Lyapunov function

$$V={\displaystyle \frac{1}{2}}{\sigma }^2.$$

(35)

The system is stable when

$$\dot{V}=\sigma ·\dot{\sigma }<0,\sigma \ne 0.$$

(36)

Then

$$\begin{array}{c}\hfill \dot{V}=-\left({\displaystyle \frac{B_m}{J}}+{\displaystyle \frac{3cK}{2JR}}{\lambda }_1\right){\sigma }^2-\left[{\displaystyle \frac{3cK}{2JR}}{\epsilon }_1·\mathrm{sgn}\left(\sigma \right)-{\displaystyle \frac{c{B}_m}{J}}r\left(t\right)\right]\sigma .\end{array}$$

(37)

In (37), when $\sigma \ne 0$ and ${\lambda }_1>0$, $-\left({\displaystyle \frac{B_m}{J}}+{\displaystyle \frac{3cK}{2JR}}{\lambda }_1\right){\sigma }^2<0$. We need to design ${\epsilon }_1$ to ensure that

$$\left[{\displaystyle \frac{3cK}{2JR}}{\epsilon }_1·\mathrm{sgn}\left(\sigma \right)-{\displaystyle \frac{c{B}_m}{J}}r\left(t\right)\right]\sigma >0.$$

(38)

It is obvious that (38) holds when the control gain ${\epsilon }_1$ satisfies

$$\begin{array}{cc}\hfill {\epsilon }_1>& {\displaystyle \frac{2R{B}_m}{3K}}{\parallel r\left(t\right)\parallel }_{\infty }\hfill \end{array}.$$

(39)

Thus, the system is stable when Theorem 1 and (39) hold.

4. Design of EID Estimator and SMC

This section explains the design of the EID estimator and the parameters of the SMC-EID system (c, $\lambda$, k, and $\epsilon$).

4.1. EID Estimator Design

Two elements need to be selected for the EID estimation and compensation: the low-pass filter $F\left(s\right)$ and the observer gain L.

An appropriate low-pass filter $F\left(s\right)$ ensures disturbance attenuation performance. A first-order low-pass filter is designed:

$$F\left(s\right)={\displaystyle \frac{1}{T_{F}s+1}},$$

(40)

where $T_F$ is used to regulate the angular-frequency band for disturbance attenuation. The cutoff angular frequency ${\omega }_c={\displaystyle \frac{1}{T_F}}$.

To investigate the relationship between the cutoff angular frequency and the disturbance attenuation performance, we chose a set of cutoff angular frequencies of 100, 63, 32, 16, 8, and 2 rad/s and drew Bode diagrams (Figure 4). When the cutoff angular frequency of the low-pass filter is five times higher than a disturbance component, the component of the disturbance estimate passes the filter with virtually no gain attenuation or phase lag.

Figure 4. Bode plot of $F\left(s\right)$: (a) magnitude and (b) phase.

But if the angular-frequency band is too wide, it cannot attenuate high-frequency noise in the system. Thus, $T_F$ is chosen as [32]

$$T_F={\displaystyle \frac{1}{5\sim 10}}{\displaystyle \frac{1}{{\omega }_r}},$$

(41)

where ${\omega }_r$ in (41) is the highest angular frequency of disturbances. Thus,

$$F\left(j\omega \right)\approx 1,\forall \omega \in {\Omega }_r,{\Omega }_r=\left\{\omega \right|0\le \omega \le {\omega }_r\},$$

(42)

where ${\Omega }_r$ is the angular-frequency band for disturbance attenuation.

After we choose an appropriate $F\left(s\right)$, we need to design an observer gain L that ensures $|{\tau }_{de}\left(t\right)-{\tilde{\tau }}_{de}\left(t\right)|$ is small enough.

Letting $u_f\left(t\right)=0$ and redrawing Figure 2 yields Figure 5.

Figure 5. EID configuration for disturbance estimate analysis.

The system is a unity feedback system and $F\left(s\right)={\displaystyle \frac{1}{T_{F}s+1}}$. The open-loop transfer function is

$$\begin{array}{c}\hfill G_o\left(s\right)={\displaystyle \frac{CB}{s-A}}·{\displaystyle \frac{L(s-A)}{s+(LC-A)}}·B^{-1}·{\displaystyle \frac{1}{T_{F}s}}={\displaystyle \frac{LC}{T_F{s}^2+T_F(LC-A)s}}.\end{array}$$

(43)

Thus, the closed-loop transfer function from ${\tau }_{oe}\left(t\right)$ to ${\tilde{\tau }}_{oe}\left(t\right)$ is

$$G_c\left(s\right)={\displaystyle \frac{LC}{T_F{s}^2+T_F(LC-A)s+LC}}.$$

(44)

Then, the transfer function from ${\tau }_{oe}\left(t\right)$ to ${\tau }_{oe}\left(t\right)-{\tilde{\tau }}_{oe}\left(t\right)$ is

$$\begin{array}{cc}\hfill G_e\left(s\right)=1-{\Phi }_{\tau }\left(s\right)=& {\displaystyle \frac{T_F{s}^2+T_F(LC-A)s}{T_F{s}^2+T_F(LC-A)s+LC}}\hfill \\ \hfill =& {\displaystyle \frac{s^2+(LC-A)s}{s^2+(LC-A)s+\frac{LC}{T_F}}}.\hfill \end{array}$$

(45)

According to the Routh Criterion, it is stable when

$$LC>0.$$

(46)

$G_e\left(s\right)$ can be interpreted as

$$\left\{\begin{array}{cc}\hfill G_e\left(s\right)& ={\displaystyle \frac{s[s+(LC-A\left)\right]}{\left(s+\frac{(LC-A)-\sqrt{\Delta }}{2}\right)\left(s+\frac{(LC-A)+\sqrt{\Delta }}{2}\right)}}\hfill \\ \hfill \Delta & ={(LC-A)}^2-4{\displaystyle \frac{LC}{T_F}}.\hfill \end{array}\right.$$

(47)

We choose an L that ensures

$$\Delta >0.$$

(48)

$\Delta$ can be represented by

$$\begin{array}{cc}\hfill \Delta =& {\left(LC\right)}^2-2\left(A+{\displaystyle \frac{2}{T_F}}\right)LC+{\left(A+{\displaystyle \frac{2}{T_F}}\right)}^2-4{\displaystyle \frac{A{T}_F+1}{T_F^2}}\hfill \\ \hfill =& {\left[LC-\left(A+{\displaystyle \frac{2}{T_F}}\right)\right]}^2-4{\displaystyle \frac{A{T}_F+1}{T_F^2}}.\hfill \end{array}$$

(49)

When $LC$ is large enough that

$${\left[LC-\left(A+{\displaystyle \frac{2}{T_F}}\right)\right]}^2\gg 4{\displaystyle \frac{A{T}_F+1}{T_F^2}}.$$

(50)

Then,

$$\sqrt{\Delta }\approx LC-\left(A+{\displaystyle \frac{2}{T_F}}\right).$$

(51)

Thus,

$$\begin{array}{cc}\hfill G_e\left(s\right)=& {\displaystyle \frac{s[s+(LC-A\left)\right]}{\left(s+\frac{1}{T_F}\right)\left[s+\left(LC-A-\frac{1}{T_F}\right)\right]}}.\hfill \end{array}$$

(52)

As the zero $-(LC-A)$ and the pole $-(LC-A-{\displaystyle \frac{1}{T_F}})$ are close to each other, and they are both far from other zeros and poles, they result in zero-pole cancellation. Then,

$$G_e\left(s\right)\approx {\displaystyle \frac{s}{s+\frac{1}{T_F}}}.$$

(53)

According to (41),

$${\displaystyle \frac{1}{T_F}}=m·{\omega }_r,m\in [5,10].$$

(54)

From the Bode diagram of $G_e\left(s\right)$ is shown as Figure 6.

Figure 6. Bode diagram of $G_e\left(s\right)$. The solid line represents the exact frequency response, while the dashed line denotes the asymptotic magnitude characteristic indicating the slope and cutoff frequency.

When the disturbance frequency satisfies

$${\omega }_{\tau }\le {\omega }_r,$$

(55)

the inequality below holds.

$$\parallel G_e·{\tau }_{oe}{\parallel }_{\infty }\le {\displaystyle \frac{1}{m}}\left|\right|{\tau }_{oe}{\left|\right|}_{\infty }.$$

(56)

We choose the observer gain L that follows (48) and (50).

4.2. Ant Colony Optimization-Based SMC Design

We replaced the sign function in SMC with a saturation function to further suppress chattering. As a result, the SMC-EID in this study is

$$u\left(t\right)={\epsilon }_1·\mathrm{sat}\left[{\displaystyle \frac{\sigma \left(t\right)}{k}}\right]+{\lambda }_1\sigma \left(t\right)-{\tilde{\tau }}_{de}\left(t\right),$$

(57)

where

$$\mathrm{sat}\left({\displaystyle \frac{\sigma }{k}}\right)=\left\{\begin{array}{c}1,\sigma <-k\hfill \\ \sigma /k,-k\le \sigma \le k\hfill \\ 1,\sigma >k,\hfill \end{array}\right.$$

(58)

and the positive constant k is the linear region of the saturation function. As shown in (14), c should be a positive real number. Thus, $c=1$ is chosen for the design of ${\epsilon }_1$, $k_1$, and ${\lambda }_1$ in (57). As there are uncertainties in the system, we set up a group of systems with different parameters. The number of systems is N. The optimal set of control parameters should show great control performance in all systems.

${\epsilon }_1$ is the switching gain of the SMC and ${\epsilon }_1/k_1$ is the gradient of the saturation function. The smaller $k_1$ is, the closer to a sign function the saturation function is. While the combination of a larger ${\epsilon }_1$ and a smaller $k_1$ yields better disturbance attenuation performance, it produces larger chattering. The chattering is bounded by the saturation function $\mathrm{sat}\left({\displaystyle \frac{\sigma }{k}}\right)$, limiting amplitude to $|{\epsilon }_1/k|$. ${\lambda }_1$ is a proportional control gain. The larger it is, the faster the dynamic response is and the larger the overshoot is.

This study developed a method of designing the SMC parameters based on CDACO to solve the problem.

A conventional ACO solution space is usually a discrete solution space. However, this study needs to find a solution in a continuous solution space.

Let an ant be $[{\epsilon }_1,k_1,{\lambda }_1]$. The pheromone of Ant n in the ith iteration is

$$\left\{\begin{array}{cc}{\mathrm{phe}}_m^{\left(i\right)}=0,\hfill & \mathrm{f}\mathrm{o}\mathrm{r}i=0\hfill \\ {\mathrm{phe}}_m^{\left(i\right)}=(1-\zeta ){\mathrm{phe}}_m^{(i-1)}+\zeta e^{-p\overline{E}},\hfill & \mathrm{f}\mathrm{o}\mathrm{r}i>0\&\zeta <1,\hfill \end{array}\right.$$

(59)

where p is a weight of the performance index, which is a positive constant indicating the transition performance; $\zeta$ is a forgetting factor; and $\overline{E}$ is the weighted average of $E_n(n=1,2,\cdots ,N)$. $E_n$ is the ITSE of the n-th system:

$$E_n={\int }_0^{2}t{(\overline{\omega }-\omega )}^2\mathrm{d}t.$$

(60)

$E_n$ is small when

(1)

Settling time is short;

(2)

Overshoot is small;

(3)

Chattering is little;

(4)

Static error is eliminated.

$E_n$ evaluates the control performance comprehensively. The smaller $E_n$ is, the better the overall control performance is. The optimal ant has the smallest performance index $E_n$ and the largest pheromone (${\mathrm{phe}}_{\mathrm{opt}}$), that is, a small speed fluctuation and small chattering. It searches around itself and looks for a better solution.

Other ants calculate their transition probabilities. For example, the probability of Ant n is

$$P_n^{\left(i\right)}={\displaystyle \frac{{\mathrm{phe}}_{\mathrm{opt}}-{\mathrm{phe}}_n^{\left(i\right)}}{{\mathrm{phe}}_{\mathrm{opt}}}}.$$

(61)

Thus, a low pheromone means the ant has a high transition probability of moving toward the optimal solution. If it decides not to do so, then it searches around itself.

We condense the above discussion as follows:

• CDACO algorithm:
  (Step 1) Set the number of ants M, evaporation rate $\rho$, initial pheromone $\tau$, and max iteration times I. Generate initial ants uniformly in the search space justified by the lower bound from (39) to ensure stability and the upper bound to avoid excessive gain.
• (Step 2) Update the pheromone for each ant using (59).
• (Step 3) Calculate the transition probability for each ant using (61) and make a decision by sampling from Gaussian probability.
• (Step 4) Go to (Step 2) and repeat until reaching the iteration times.

The flow chart of CDACO (Figure 7) further clarifies the CDACO procedure.

Figure 7. Flow chart of CDACO.

The performance of CDACO depends on the number of ants M, the number of iteration times I, the weight of the performance index p, the forgetting factor $\zeta$, and the performance index $E_n$. Note that a large $\zeta$ means that the pheromone depends strongly on the current state, and a large p means a high transition probability $P_n^{\left(i\right)}$.

5. Experimental Verification

The values of state-space (32) for Table 2 are

$$A=-25,B=123,B_d=250,C=1.$$

(62)

Table 2. Nominal PMSM specifications.

Variable	Value
Viscous friction coefficient, $B_m$	0.001 N/rad$·{\mathrm{s}}^{-1}$
Moment of inertia, J	0.004 kg · m2
Flux of permanent magnet, ${\psi }_f$	0.072 Wb
Number of pole pairs, $n_p$	5
Reluctance of stator, R	0.875 $\mathrm{\Omega }$
Inductor of q axis stator, $L_q$	8.5 mH

Let the reference input be

$$r\left(t\right)=500\times 1\left(t\right)\mathrm{rpm},$$

(63)

and the disturbance model is derived from PMSM torque oscillations (7)

$$\begin{array}{c}\hfill d\left(t\right)=0.1+0.4sin{\displaystyle \frac{\theta \left(t\right)}{10}}+0.4sin{\displaystyle \frac{\theta \left(t\right)}{5}}+0.4cos6\theta \left(t\right)\mathrm{N}·\mathrm{m},\end{array}$$

(64)

where $\theta \left(t\right)$ is the mechanical angle of the PMSM. Since the target angular velocity of the PMSM is 500 rpm, which is approximately 50 rad/s, the angular frequencies in the disturbance are about 5 rad/s, 10 rad/s, and 300 rad/s. Note that torque oscillations at 300 rad/s or higher have little influence on the PMSM system. ${\omega }_r$ was chosen to be 10 rad/s and the low-pass filter to be

$$F\left(s\right)={\displaystyle \frac{1}{0.01s+1}}.$$

(65)

According to (48) and (50) and ensuring observer performance, the time constant of the observer should be less than one-tenth of that of the plant. Thus, the observer gain L is chosen to be

$$L=2000.$$

(66)

The CDACO algorithm was established in MATLAB 2020b/Simulink and the parameters optimized offline. We set up different systems with different moments of inertia and viscous friction coefficients, which simulate the uncertainties in the system. Among them, the moment of inertia $J\in \{0.002,0.004,0.006\}$ kg$·{\mathrm{m}}^2$ and the viscous friction coefficients $B_m\in \{0.0005,0.0010,0.0015\}$ N/rad$·{\mathrm{s}}^{-1}$. Thus, there are nine different systems with different parameters. The system whose $J=0.004$ kg$·{\mathrm{m}}^2$ and $B_m=0.0010$ N/rad$·{\mathrm{s}}^{-1}$ is the nominal system. Its performance index is $E_1$. The performance indexes of other systems are $E_n,n\in \{2,3,\dots ,9\}$. E is calculated as follows:

$$E=0.2E_1+0.1\sum _{n=2}^9{E}_n.$$

(67)

It guarantees the robustness of the system.

The SMC-EID, (57), has three parameters, ${\epsilon }_1$, $k_1$, and ${\lambda }_1$, that need designing. According to (39), we choose $B_m=0.0015$. Thus, the range of ${\epsilon }_1$ is

$${\epsilon }_1>1.13.$$

(68)

Then, we set the search space to be

$${\epsilon }_1\in [1.13,8],k_1\in (0,3],{\lambda }_1\in (0,15].$$

(69)

Setting the parameters of CDACO (Table 3), uniformly generating the first generation of ants in the search space, (69), and optimizing ${\epsilon }_1$, $k_1$, and ${\lambda }_1$ in (57) yields

$${\epsilon }_1=1.29,k_1=0.60,{\lambda }_1=3.73.$$

(70)

Table 3. Parameters of CDACO.

Parameter	Value
Number of ants M	50
Maximum iteration I	100
Forgetting constant $\zeta$	0.9
Weight of performance index p	2

For comparison, we used CDACO to optimize the parameters in a conventional SMC

$$u\left(t\right)={\epsilon }_2\mathrm{sat}\left[{\displaystyle \frac{\sigma \left(t\right)}{k_2}}\right]+{\lambda }_2\sigma \left(t\right).$$

(71)

The conventional SMC is designed with the same CDACO, sliding surface (14), and nine identical uncertain systems as the SMC-EID method. Optimizing $[{\epsilon }_1,k_1,{\lambda }_1]$ gives

$${\epsilon }_2=6.40,k_2=1.52,{\lambda }_2=5.01.$$

(72)

A two-mass experimental system (Figure 8) was used to demonstrate the validity of the method. It has two identical PMSMs: one is used as a plant, and the other is a disturbance generator. It has a clutch that controls the engagement and disengagement of the flywheel, thereby changing the system’s rotational inertia. The nameplate values of the PMSMs are summarized in Table 4.

Figure 8. Two-mass experimental system.

Table 4. PMSM nameplate values.

Name	Value
Manufacturer	INOVANCE
Model	ISMH1-75B30CB
Rated power	750 W
Rated voltage	220 V
Rated current	4.8 A
Rated rev.	3000 rpm
Rated torque	2.39 Nm
Max rev.	6000 rpm

MATLAB R2020b was used to design a control system. TwinCAT 3 was used to perform system control with a sampling period of 500 μs.

We conducted four sets of comparative experiments with different moments of inertia J and reference speeds r. The four experimental conditions are shown as follows:

• Exp. 1: $J_1$ = 0.004 $\mathrm{kg}·{\mathrm{m}}^2$ and $r_1$ = 500 rpm;
• Exp. 2: $J_2$ = 0.004 $\mathrm{kg}·{\mathrm{m}}^2$ and $r_2$ = 250 rpm;
• Exp. 3: $J_3$ = 0.006 $\mathrm{kg}·{\mathrm{m}}^2$ and $r_3$ = 500 rpm;
• Exp. 4: $J_4$ = 0.006 $\mathrm{kg}·{\mathrm{m}}^2$ and $r_4$ = 250 rpm.

The speed response (Figure 9) and speed error (Figure 10) show that the disturbance was attenuated to a lower level for the SMC-EID than for the SMC. A comparison between performance indexes (Table 5) reveals that in all four experiments, the peak-to-peak (PP) speed error and ITSE of the SMC-EID are all less than those of the conventional SMC. The average PP speed error of the SMC-EID is about 37% of that of the conventional SMC, and the average ITSE of the SMC-EID is about 35% of that of the conventional SMC.

Figure 9. Comparison of speed response between SMC-EID and conventional SMC in different sets of experiments: (a) Exp. 1, (b) Exp. 2, (c) Exp. 3, and (d) Exp. 4.

Figure 10. Comparison of speed error between SMC-EID and conventional SMC in different sets of experiments: (a) Exp. 1, (b) Exp. 2, (c) Exp. 3, and (d) Exp. 4.

Table 5. Control performance comparison (PP: peak-to-peak).

		PP Speed Error (rpm)	ITSE
Exp (1)	SMC	25.1	1.35
	SMC-EID	11.6	0.96
Exp (2)	SMC	31.5	1.09
	SMC-EID	13.8	0.31
Exp (3)	SMC	56.3	4.60
	SMC-EID	19.5	1.96
Exp (4)	SMC	52.9	3.53
	SMC-EID	15.8	0.50
Average	SMC	41.5	2.64
	SMC-EID	15.2	0.93

We take Exp. 1 as an example. The torque comparison (Figure 11) between SMC-EID and conventional SMC shows that the chattering is much smaller for the SMC-EID than for the SMC.

Figure 11. Torque comparison between SMC and SMC-EID in Exp (1).

6. Conclusions

This paper explained a control method that combines the SMC and EID methods for uncertain systems. We added an EID estimate to a sliding-mode controller, not only to improve the disturbance attenuation performance but also to reduce chattering. Then, CDACO solves the design problem of uncertain systems. This method has significant advantages:

(1)

Our method extends the SMC-EID method to uncertain plants, demonstrated by stability analysis and optimization over nine uncertain systems.

(2)

The novel CDACO algorithm handles continuous optimization spaces and solves the multi-parameter design problem for uncertain systems.

(3)

Experimental results validate superior control performance compared to conventional SMC. In the four experiments, the average PP speed error of the SMC-EID method was about 37% of that of the conventional SMC, and the ITSE of SMC-EID was about 35% of that of the conventional SMC.