Angle-Based RGN-Enhanced ADRC for PMSM Compressor Speed Regulation Considering Aperiodic and Periodic Disturbances

Abstract

Achieving excellent speed control in permanent magnet synchronous motors (PMSMs) relies on the simultaneous suppression of both aperiodic and periodic disturbances. This paper presents an enhanced Active Disturbance Rejection Control (ADRC) strategy specifically designed to address these disturbances in single-rotor compressors (SRCs). To achieve simultaneous suppression, a Recursive Gauss–Newton (RGN) algorithm is implemented in parallel with the conventional extended state observer (ESO) to enhance the ADRC framework. The RGN algorithm iteratively estimates the amplitude and phase information of periodic disturbances, while the ESO primarily observes the system’s aperiodic disturbances. In contrast to existing methods, the proposed angle-based approach demonstrates superior performance during speed transients. Detailed convergence and decoupling analyses are provided to facilitate parameter tuning. The effectiveness of the proposed method is validated through simulations and experiments conducted on a 650 W SRC, demonstrating its superiority over proportional–integral (PI) control, conventional ADRC, and quasi-resonant controller-based ADRC (QRC-ADRC) under both steady-state and dynamic conditions.

1. Introduction

Permanent magnet synchronous motor (PMSM) compressors are widely employed in air conditioning systems due to their high power density and efficiency [1]. In particular, single-rotor compressors (SRCs) are extensively employed in low-power systems due to their cost-effectiveness. However, these drive systems are often subjected to both aperiodic and periodic disturbances. Aperiodic disturbances primarily stem from sources such as variations in load torque, parameter mismatches, and frictional torque [2,3]. Periodic disturbances mainly arise from inherent load fluctuations [4,5]. Figure 1 illustrates the load torque of an SRC over one mechanical cycle. As shown, the load torque exhibits significant fluctuations that vary with the mechanical angle. This variation arises from the cyclic process of refrigerant inhalation, compression, and exhalation that occurs during each mechanical rotation. Given the substantial speed fluctuations caused by these disturbances, achieving excellent speed control in SRCs demands the simultaneous suppression of both aperiodic and periodic disturbances.

The conventional proportional–integral (PI) controller, despite its common application in speed control, experiences performance deterioration under off-nominal operating conditions. In contrast, Active Disturbance Rejection Control (ADRC) has emerged as an effective alternative to the PI controller, garnering widespread research attention due to its significant advantages in both performance and practicality [6]. The extended state observer (ESO), integrated within ADRC, treats external and internal disturbances as a total disturbance, enabling effective suppression of aperiodic disturbances. However, the ESO is fundamentally a low-pass filter and exhibits limitations in handling higher-frequency periodic signals [7,8].

To simultaneously suppress both aperiodic and periodic disturbances, a straightforward approach involving increasing the order of the ESO has been proposed in [9,10]. While the estimation accuracy of the time-varying disturbance can be improved with a higher order, the effective frequency range for periodic disturbance rejection remains limited, making it more suitable for low-frequency periodic disturbances.

Consequently, numerous researchers have integrated dedicated periodic control algorithms with ADRC to enhance its capabilities in handling periodic disturbances. Methods such as the Generalized Integrator (GI) [11,12,13] and Resonant Control (RC) [14,15,16,17,18,19] have been explored. For instance, GI-based ADRC (GI-ADRC) [11] has demonstrated effective suppression of significant fast-varying sinusoidal disturbances in grid-connected converters. However, practical applications often encounter variations in the disturbance frequency, leading to the development of the Quasi-Generalized Integrator (QGI) [12,13]. By introducing a cutoff frequency, the QGI improves the robustness of GI-ADRC against these inaccuracies while retaining the high-gain properties of the GI. Similarly, Resonant Control (RC) has been combined with ADRC to mitigate voltage and current disturbances [14,15]. However, like GI, RC is susceptible to frequency drift in real-world scenarios, necessitating the common adoption of Quasi-Resonant Control (QRC). QRC offers high gain at the target frequency and within a neighboring band. QRC-based ADRC (QRC-ADRC) has been investigated for the simultaneous suppression of both periodic and aperiodic disturbances in PMSM speed control [16,17,18]. Furthermore, the quasi-resonant principle has been incorporated into fixed-time convergent ESOs [19], enhancing their ability to estimate periodic disturbances in PMSM current regulation. Despite the improved tolerance to frequency variations offered by QGI and QRC, they inevitably amplify undesired frequencies during transient states.

Repetitive Control (RPC) and Iterative Learning Control (LLC) integrated with ADRC [20,21,22,23,24] present an alternative strategy capable of simultaneously suppressing multiple disturbance frequencies. Specifically, RPC-based ADRC has been employed in [21,22] to suppress current and speed disturbances, respectively. Furthermore, ref. [21] proposes the use of low-pass filtering to further enhance the ESO’s observation capability, while [22] introduces improved embedded structures to simplify parameter selection. Sampled-data iterative learning controllers have also been proposed for suppressing uncertain periodic and aperiodic disturbances [24]. Notably, angle-based implementations of RPC and LLC [25,26] offer a significant advantage for systems where disturbances are correlated with the mechanical angle, such as the single-rotor compressor studied in this paper, as they can effectively handle disturbance frequency variations. However, a key limitation of RPC and LLC is their substantial memory requirement, which can restrict their practical implementation, especially considering that for single-rotor compressors, compensating only the dominant first harmonic is often a sufficient and efficient approach [27,28].

To overcome the limitations of existing methods, particularly the memory demands of angle-based RPC/LLC and the frequency sensitivity of GI/RC, this paper proposes an angle-based ADRC enhanced with the Recursive Gauss–Newton (RGN) algorithm. The main contributions are as follows:

• An angle-based RGN is employed to enhance the periodic disturbance observation capability of the ESO, offering an improved solution for addressing variations in disturbance frequency while requiring minimal memory resources.
• A detailed convergence and coupling analysis is conducted, and the parameter tuning methodology is also provided.
• Experimental and simulation results demonstrate that, compared to PI, conventional ADRC, and QRC-ADRC, the proposed method achieves superior disturbance rejection in both steady-state and dynamic conditions.

The remainder of this paper is structured as follows: Section 2 details the mathematical models and the disturbance analysis for the compressor. Section 3 introduces the principle of the proposed method. The convergence and decoupling analyses are presented in Section 4. Section 5 describes a series of comparative experiments conducted to verify the algorithm’s performance. Finally, Section 6 concludes the paper.

2. Dynamics Models and Disturbance Analysis

2.1. Dynamics Model of PMSM

The mathematical dynamics of PMSM is written as follows:

$${\displaystyle \frac{d}{dt}}{\omega }_m={\displaystyle \frac{1}{J}}{T}_e-{\displaystyle \frac{1}{J}}{T}_L-{\displaystyle \frac{1}{J}}B{\omega }_m$$

(1)

where $J$ is the inertia, ${\omega }_m$ is the mechanical angular speed, $T_e$ is the electromagnetic torque, $T_L$ is the motor load torque, and $B$ is the viscous friction coefficient.

The electromagnetic torque $T_e$ can be expressed as

$$T_e\left(t\right)={\displaystyle \frac{3}{2}}P[{\lambda }_f{I}_s\left(t\right)\sin \beta +\left(L_d-L_q\right)I_s^2\left(t\right)\sin \beta \cos \beta ]$$

(2)

where $P$ is the number of the pole pairs, ${\lambda }_f$ is the flux linkage, $I_s$ is the amplitude of the stator current, $\beta$ is the electrical angle between the stator flux linkage and the permanent magnet flux linkage, and $L_d$ and $L_q$ are the inductances in the dq-axes.

Substituting (2) into (1) yields

$${\displaystyle \frac{d}{dt}}{\omega }_m\left(t\right)=b{I}_s\left(t\right)+{\displaystyle \frac{1}{J}}\Delta T_e\left(t\right)-{\displaystyle \frac{1}{J}}{T}_L\left(t\right)-{\displaystyle \frac{1}{J}}B{\omega }_m\left(t\right)$$

(3)

where $b={\displaystyle \frac{3}{2J}}P{\lambda }_f\sin \beta$ and $\Delta T_e\left(t\right)={\displaystyle \frac{3}{2}}P\left(L_d-L_q\right)I_s^2\left(t\right)\sin \beta \cos \beta .$

2.2. Disturbance Analysis

In practical applications, the speed loop of the PMSM is subjected to disturbances that can degrade performance. These disturbances can be classified into two main categories: aperiodic disturbances and periodic disturbances.

• Aperiodic disturbances: These disturbances arise from factors such as parameter mismatches, frictional torque, and non-periodic variations in $T_L$.
• Periodic disturbances: These disturbances arise from factors such as flux harmonics, inverter nonlinearities, current measurement errors, and periodic variations in $T_L$. However, among all periodic disturbances, only the first harmonic of $T_L$ is compensated for in this research. This is because, in SRCs, the first harmonic of $T_L$ is dominant, making other periodic disturbances negligible.

In subsequent analysis, the first harmonic of $T_L$ is defined as $T_{L,p}$, and the aperiodic component is $T_{L,ap}$. Considering the aforementioned aperiodic and periodic disturbances, (3) is updated to (4).

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\omega }_m\left(t\right)=\left(b_0+\Delta b\right)I_s\left(t\right)-{\displaystyle \frac{1}{J}}(T_{L,ap}\left(t\right)+{\displaystyle \frac{1}{J}}B{\omega }_m\left(t\right)\\ +T_{L,p}\left(t\right)+T_u\left(t\right)-\Delta T_e\left(t\right))\end{array}$$

(4)

where $\Delta b=b-b_0$, $b_0$ denotes the nominal value of $b$, and $T_u$ represents the unknown aperiodic disturbance.

Let $d\left(t\right)$ denote the total disturbance; (4) can be simplified as

$${\displaystyle \frac{d}{dt}}{\omega }_m\left(t\right)=b_0{I}_s\left(t\right)+d\left(t\right)$$

(5)

$$d\left(t\right)=d_{ap}\left(t\right)+d_p\left(t\right)$$

(6)

where $d_{ap}\left(t\right)=\Delta b{I}_s\left(t\right)-{\displaystyle \frac{1}{J}}[T_{L,ap}\left(t\right)+B{\omega }_m\left(t\right)+T_u(t)-\Delta T_e\left(t\right)]$ is the total aperiodic disturbance, and $d_p\left(t\right)=T_{L,p}\left(t\right)$ is the total periodic disturbance. Here, $d\left(t\right)$ is assumed to be differentiable, and its differential value $h$ is bounded, as indicated in [29].

3. Proposed Method

To suppress disturbances and achieve smooth speed regulation, the proposed RGN-based ADRC (RGN-ADRC) is introduced in this section. The ESO is employed to estimate aperiodic disturbances, while the RGN estimates periodic disturbances.

3.1. ADRC

Taking the total disturbance $d\left(t\right)$ as an extended state variable, the extended state equation can be written as

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{dt}}{\omega }_m\left(t\right)=b_0{I}_s\left(t\right)+d\left(t\right)\\ {\displaystyle \frac{d}{dt}}d\left(t\right)=h\end{array}\right.$$

(7)

And the ESO is designed as

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{dt}}{\widehat{\omega }}_m\left(t\right)=b_0{I}_s\left(t\right)+\widehat{d}\left(t\right)+l_1({\omega }_m\left(t\right)-{\widehat{\omega }}_m\left(t\right))\\ {\displaystyle \frac{d}{dt}}\widehat{d}\left(t\right)=l_2({\omega }_m\left(t\right)-{\widehat{\omega }}_m\left(t\right))\end{array}\right.$$

(8)

where ${\widehat{\omega }}_m\left(t\right)$ and $\widehat{d}\left(t\right)$ represent the estimated values of ${\omega }_m\left(t\right)$ and $d\left(t\right)$ by the ESO, respectively. And $l_1$ and $l_2$ represent the observer gains. Typical values are $l_1=2{\omega }_0$ and $l_2={\omega }_0^2$, where ${\omega }_0$ is the bandwidth of the ESO.

The speed loop block diagram with ADRC is shown in Figure 2, where ${\omega }_{m,ref}$ is the reference speed, $k_p$ is the gain coefficient, $y_2(t)$ is the output of the RGN part, $y_1(t)=\widehat{d}(t)$ is the output of the ESO part, and $T_s$ is the algorithm execution period. Assuming the ESO can accurately estimate the total disturbance, i.e., $\widehat{d}(t)=d(t)$, then the control law is designed as (9).

$$I_s(t)={\displaystyle \frac{u_0(t)-y_1(t)}{b_0}}$$

(9)

where $u_0(t)=k_p({\omega }_{m,ref}-{\omega }_m(t))$. Substituting (9) into (5) yields the desired speed dynamics:

$${\displaystyle \frac{d}{dt}}{\omega }_m\left(t\right)=u_0\left(t\right)-\widehat{d}\left(t\right)+d\left(t\right)=u_0\left(t\right)$$

(10)

Combining Equations (7) and (8), the following transfer function can be obtained:

$$G_d\left(s\right)={\displaystyle \frac{\widehat{d}\left(s\right)}{d\left(s\right)}}={\displaystyle \frac{{\omega }_0^2}{s^2+2{\omega }_{0}s+{\omega }_0^2}}$$

(11)

The low-pass filter form of (11) enables the ESO to effectively estimate constant or slowly varying aperiodic disturbances but introduces distortion for the high-frequency periodic disturbances. While increasing the ESO bandwidth enhances the observation of high-frequency signals, it simultaneously introduces additional noise, thereby degrading performance. Consequently, in practical implementations utilizing a standard ESO, aperiodic disturbances are effectively mitigated, whereas high-frequency periodic disturbances remain within the system. To overcome this inherent limitation, this paper proposes the use of the RGN algorithm.

3.2. RGN-ADRC

The general form of $T_{L,p}$ can be expanded as follows:

$$T_{L,p}\left(k\right)=A\sin \left({\theta }_m\left(k\right)+\phi \right)=B\sin {\theta }_m\left(k\right)+C\cos {\theta }_m\left(k\right)$$

(12)

where $k=1,2,\dots ,N$, ${\theta }_m\left(k\right)$ is the rotor mechanical angle ${\theta }_m$ at execution times k. $A$ and $\phi$ are the amplitude and phase of $T_{L,p}$. For convenience, $T_{L,p}$ can be further written as the sum of a sine signal with amplitude $B$ and a cosine signal with amplitude $C$. Furthermore, due to the slow temperature change, $B$ and $C$ are treated as constants in the subsequent analysis.

Then, the estimate of $T_{L,p}$ can be computed as

$${\widehat{T}}_{L,p}(k)=\widehat{B}\left(k-1\right)\sin {\theta }_m\left(k\right)+\widehat{C}\left(k-1\right)\cos {\theta }_m\left(k\right)$$

(13)

where $\widehat{B}$ and $\widehat{C}$ represent the amplitudes of the sine and cosine components of ${\widehat{T}}_{L,p}$, respectively. And the estimate error is

$$\begin{array}{c}{e}_1\left(k\right)=T_{L,p}(k)-{\widehat{T}}_{L,p}(k)\\ =\left[\begin{array}{cc}\sin {\theta }_m\left(k\right)& \cos {\theta }_m\left(k\right)\end{array}\right]\left[\begin{array}{c}B-\widehat{B}\left(k-1\right)\\ C-\widehat{C}\left(k-1\right)\end{array}\right]\end{array}$$

(14)

The cost function is selected as

$$\epsilon \left(k\right)={\sum }_{i=0}^k{\lambda }^{k-i}{e}_1^2\left(k\right)$$

(15)

where $0<\lambda <1$ is the forgetting factor. When $\epsilon \left(k\right)$ has a minimum value, the optimal estimation of ${\widehat{T}}_{L,p}$ is achieved. Let the parameter vector be $\widehat{v}\left(k\right)={ \left[\begin{array}{cc}\widehat{B}\left(k\right)& \widehat{C}\left(k\right)\end{array}\right]}^T$. With the RGN method [30], the updating equations are given by (16).

$$\widehat{v}\left(k\right)=\widehat{v}\left(k-1\right)-H^{-1}\left(k\right)\Phi \left(k\right)e_1\left(k\right)$$

(16)

$$H\left(k\right)={\sum }_{i=0}^k{\lambda }^{k-i}\Phi \left(i\right){\Phi \left(i\right)}^T$$

(17)

where $H$ is the Hessian matrix, and $\Phi$ is the gradient vector.

In this case, the gradient vector is given by (18), and the Hessian matrix is shown in (19).

$$\Phi \left(k\right)={\displaystyle \frac{\partial e_1\left(k\right)}{\partial \widehat{v}}}=\left[\begin{array}{c}-\sin {\theta }_m\left(k\right)\\ -\cos {\theta }_m\left(k\right)\end{array}\right]$$

(18)

$$H\left(k\right)={\sum }_{i=0}^k{\lambda }^{k-i}\left[\begin{array}{cc}{\sin }^2{\theta }_m\left(k\right)& {\displaystyle \frac{\sin {2\theta }_m\left(k\right)}{2}}\\ {\displaystyle \frac{\sin {2\theta }_m\left(k\right)}{2}}& {\cos }^2{\theta }_m\left(k\right)\end{array}\right]$$

(19)

$H\left(k\right)$ can be approximated as (19) when ${\theta }_m\left(k\right)-{\theta }_m\left(k-1\right)$ is not close to 0 or $\pi$ [30,31].

$$H\left(k\right)\approx {\sum }_{i=0}^k{\lambda }^{k-i}\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}& 0\\ 0& {\displaystyle \frac{1}{2}}\end{array}\right]={\displaystyle \frac{1-{\lambda }^{k+1}}{2\left(1-\lambda \right)}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

(20)

The inverse Hessian matrix can be therefore determined as

$${H\left(k\right)}^{-1}=1/c\left(k\right)\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

(21)

$$c\left(k\right)={\displaystyle \frac{1-{\lambda }^{k+1}}{2\left(1-\lambda \right)}}$$

(22)

The updating equation of $c\left(k\right)$ can be written as

$$c\left(k\right)=(\lambda c\left(k-1\right)+{\displaystyle \frac{1}{2}})$$

(23)

Substituting (18) and (21)–(22) into (16) yields the updating Equation (24).

$$\widehat{v}\left(k\right)=\widehat{v}\left(k-1\right)+{\displaystyle \frac{1}{c\left(k\right)}}\left[\begin{array}{c}\sin {\theta }_m\left(k\right)e_1\left(k\right)\\ \cos {\theta }_m\left(k\right)e_1\left(k\right)\end{array}\right]$$

(24)

Substituting the iterative result of (24) into (13) yields the output $y_2(k)={\widehat{T}}_{L,p}(k)$. And the control law (9) is updated to (25).

$$I_s\left(k\right)={\displaystyle \frac{u_0\left(k\right)-y_2(k)-y_1(k)}{b_0}}$$

(25)

However, the $e_1\left(k\right)$ required as input by the RNG cannot be directly measured. The solution is illustrated in Figure 3. From (5), (6) and (25), the following expression can be obtained:

$$\begin{array}{cc}{\omega }_m\left(k\right){\displaystyle \frac{z-1}{T_s}}-u_0\left(k\right)& =[b_0{I}_s\left(k\right)+d\left(k\right)]-[b_0{I}_s\left(k\right)+y_1\left(k\right)+y_2\left(k\right)]\\ & =d_p(k)-{\widehat{T}}_{L,p}(k)+O_1\\ & =e_1(k)+O_1\end{array}$$

(26)

where $O_1=d_{ap}(k)-\widehat{d}(k)$. In the implementation shown in Figure 3, the influence of $O_1$ is neglected. This simplification is justified by the frequency selectivity of the RGN. The specific proof is as follows:

Let the initial values be zero:

$$\widehat{v}\left(0\right)=\left[\begin{array}{c}\widehat{B}\left(0\right)\\ \widehat{C}\left(0\right)\end{array}\right]=0$$

(27)

Equation (24) can be written in the following accumulated form:

$$\widehat{v}\left(k\right)=\widehat{v}\left(0\right)+{\sum }_{i=1}^k\left[\begin{array}{c}\sin {\theta }_m\left(i\right)e_1\left(i\right)/c\left(i\right)\\ \cos {\theta }_m\left(i\right)e_1\left(i\right)/c\left(i\right)\end{array}\right]$$

(28)

Equation (28) resembles a discrete Fourier transform; therefore, the convergence of $\widehat{v}\left(k\right)$ is determined solely by the first harmonic component. Furthermore, the RGN output (13) is inherently a first harmonic sinusoidal output. This frequency selectivity allows the RGN to disregard non-first harmonic components within $O_1$. The impact of the first harmonic in $O_1$ is treated as coupling and discussed separately in Section 4.3. And the whole block diagram of the proposed RGN-ADRC is shown in Figure 4.

4. Convergence Analysis

4.1. Convergence Analysis of ESO

Define the ESO estimate error, $e_m\left(k\right)={\omega }_m\left(k\right)-{\widehat{\omega }}_m\left(k\right)$, $e_d\left(k\right)=d(k)-\widehat{d}\left(k\right)$. According to (7) and (8), the discrete Equation (29) is obtained.

$$\left[\begin{array}{c}{e}_m\left(k\right)\\ e_d\left(k\right)\end{array}\right]=A_{eso}\left[\begin{array}{c}{e}_m\left(k-1\right)\\ e_d\left(k-1\right)\end{array}\right]+\left[\begin{array}{c}0\\ T_{s}h\end{array}\right]$$

(29)

$$A_{eso}=\left[\begin{array}{cc}1-T_s{l}_1& T_s\\ -T_s{l}_2& l_1\end{array}\right]$$

(30)

By substituting the typical values $l_1=2{\omega }_0$ and $l_2={\omega }_0^2$, the eigenvalues of the system matrix $A_{eso}$ are found to be repeated roots, specifically ${\mu }_1={\mu }_2=1-T_s{\omega }_0$. According to the system stability condition $\left|\mu \right|<1$, it can be deduced that $T_s{\omega }_0<2$, where $T_s$ and ${\omega }_0$ are both greater than 0 in practical applications. The influence of $h$ on the ESO’s steady-state error is specifically represented by the vector ${ \left[\begin{array}{cc}h/{\omega }_0^2& 2h/{\omega }_0\end{array}\right]}^T$. Consequently, employing a sufficiently large ${\omega }_0$ can mitigate the impact of $h$.

4.2. Convergence Analysis of RGN

Define the parameter vector error of RGN $e_v\left(k\right)={ \left[\begin{array}{cc}B& C\end{array}\right]}^T-{ \left[\begin{array}{cc}\widehat{B}(k)& \widehat{C}(k)\end{array}\right]}^T$. According to (14) and (22)–(24), the updating equations of $e_v\left(k\right)$ are (31) and (32).

$$\begin{array}{cc}{e}_1\left(k\right)& =\left[\begin{array}{cc}\sin {\theta }_m\left(k\right)& \cos {\theta }_m\left(k\right)\end{array}\right]\left[\begin{array}{c}B-\widehat{B}\left(k-1\right)\\ C-\widehat{C}\left(k-1\right)\end{array}\right]\\ & ={x\left(k\right)}^T{e}_v\left(k-1\right)\end{array}$$

(31)

$$\begin{array}{cc}{e}_v\left(k\right)& =\left[\begin{array}{c}B\\ C\end{array}\right]-\widehat{v}\left(k\right)\\ & =\left[\begin{array}{c}B\\ C\end{array}\right]-\widehat{v}\left(k-1\right)-{\displaystyle \frac{1}{c\left(k\right)}}\left[\begin{array}{c}\sin {\theta }_m\left(k\right)e_1\left(k\right)\\ \cos {\theta }_m\left(k\right)e_1\left(k\right)\end{array}\right]\\ & =e_v\left(k-1\right)-{\displaystyle \frac{1}{\gamma }}x\left(k\right)e_1\left(k\right)\\ & =\left[I-{\displaystyle \frac{1}{\gamma }}x\left(k\right){x\left(k\right)}^T\right]e_v\left(k-1\right)\end{array}$$

(32)

where $x\left(k\right)={[\sin {\theta }_m\left(k\right) \cos {\theta }_m\left(k\right)]}^T$. Given a constant $\lambda$, $c\left(k\right)={\displaystyle \frac{1-{\lambda }^{k+1}}{2\left(1-\lambda \right)}}$ converges to a constant value $\gamma =1/(2-2\lambda )$ as $k$ becomes large, according to (22). Consequently, $c\left(k\right)$ in (32) is substituted with $\gamma$ for computational simplicity.

Construct the Lyapunov function $V\left(k\right)$ as (33).

$$\begin{array}{cc}V\left(k\right)& ={e_v\left(k\right)}^T{e}_v\left(k\right)\\ & ={e_v\left(k-1\right)}^T{F\left(k\right)}^{T}F\left(k\right)e_v\left(k-1\right)\end{array}$$

(33)

where $F(k)=I-x\left(k\right){x\left(k\right)}^T/\gamma$. Then $\Delta V\left(k\right)$ can be written as (34).

$$\begin{array}{cc}\Delta V\left(k\right)& =V\left(k\right)-V\left(k-1\right)\\ & ={e_v\left(k-1\right)}^T{F\left(k\right)}^{T}F\left(k\right)e_v\left(k-1\right)-{e_v\left(k-1\right)}^T{e}_v\left(k-1\right)\\ & ={e_v\left(k-1\right)}^T\left[{F\left(k\right)}^{T}F\left(k\right)-I\right]e_v\left(k-1\right)\end{array}$$

(34)

Expand ${F\left(k\right)}^{T}F\left(k\right)$ for simplification.

$$\begin{array}{cc}{F\left(k\right)}^{T}F\left(k\right)& ={\left[I-x\left(k\right){x\left(k\right)}^T/\gamma \right]}^T\left[I-x\left(k\right){x\left(k\right)}^T/\gamma \right]\\ & =I-{\displaystyle \frac{2}{\gamma }}x\left(k\right){x\left(k\right)}^T+{\displaystyle \frac{1}{{\gamma }^2}} x\left(k\right){x\left(k\right)}^{T}x\left(k\right){x\left(k\right)}^T\end{array}$$

(35)

Since ${x\left(k\right)}^{T}x\left(k\right)={\sin }^2{\theta }_m\left(k\right)+{\cos }^2{\theta }_m\left(k\right)=1$, $x\left(k\right){x\left(k\right)}^{T}x\left(k\right){x\left(k\right)}^T=x\left(k\right){x\left(k\right)}^T$, then gets (36).

$${F\left(k\right)}^{T}F\left(k\right)=I-\left({\displaystyle \frac{2}{\gamma }}-{\displaystyle \frac{1}{{\gamma }^2}}\right)x\left(k\right){x\left(k\right)}^T$$

(36)

Furthermore, (34) can be simplified as follows:

$$\begin{array}{cc}\Delta V\left(k\right)& ={e_v\left(k-1\right)}^T\left[-\left({\displaystyle \frac{2}{\gamma }}-{\displaystyle \frac{1}{{\gamma }^2}}\right)x\left(k\right){x\left(k\right)}^T\right]e_v\left(k-1\right)\\ & =-\left({\displaystyle \frac{2}{\gamma }}-{\displaystyle \frac{1}{{\gamma }^2}}\right){\left[{x\left(k\right)}^T{e}_v\left(k-1\right)\right]}^T\left[{x\left(k\right)}^T{e}_v\left(k-1\right)\right]\\ & =-\left({\displaystyle \frac{2}{\gamma }}-{\displaystyle \frac{1}{{\gamma }^2}}\right){\left[{x\left(k\right)}^T{e}_v\left(k-1\right)\right]}^2\\ & =-4\lambda \left(1-\lambda \right){e_1\left(k\right)}^2\end{array}$$

(37)

It is worth noting that ${x\left(k\right)}^T{e}_v\left(k-1\right)=e_1\left(k\right)$ is a scalar quantity; therefore its transpose is identical to itself. Since $-4\lambda \left(1-\lambda \right)<0$ for $0<\lambda <1$, it follows from the discrete Lyapunov theorem that $\Delta V\left(k\right)$ fulfills the negative definiteness condition $(\Delta V\left(k\right)<0)$, thereby confirming the asymptotic stability of the RGN algorithm.

4.3. Coupling Analysis and Parameter Tuning

While the proposed RGN-ADRC strategy employs the ESO and RGN in parallel to address aperiodic and periodic disturbances, respectively, it is crucial to analyze potential coupling effects between these two components. The ESO, acting as a low-pass filter, and the RGN, selectively responding to the first harmonic, form the basis of the disturbance rejection.

The control block diagram presented in Figure 4 ensures that the RGN’s estimation process does not affect the ESO’s convergence. This is because the calculation for (11) can be derived by subtracting (8) from (7). In this derivation, $I_s$ is eliminated, making the result independent of the specific expression for $I_s$. Consequently, even though the control law in Figure 4 updates from (9) to (25), the form of (11) remains unchanged. This means the RGN does not influence the ESO’s convergence.

To analyze the coupling in more detail, Figure 5 illustrates the simplified coupling analysis block diagram of the proposed method, where $y=y_1+y_2$ represents the total disturbance estimation. The total disturbance acting on the system is $d=T_{L,p}+d_{ap}$. The ESO aims to estimate $d_{ap}$, while the RGN aims to estimate $T_{L,p}$. However, due to the inherent low-pass characteristic of (11), its output, $y_1$, will contain an attenuated and phase-shifted first harmonic component, denoted as ${\widehat{d}}_p$, in addition to the aperiodic estimate. Thus, the ESO’s output can be expressed as $y_1={\widehat{d}}_{ap}+{\widehat{d}}_p$.

From Figure 5, it is evident that the additional ${\widehat{d}}_p$ component within $y_1$ is compensated by adjusting the RGN’s equilibrium point, shifting it from its original ${\widehat{T}}_{L,p}$ to ${\widehat{T}}_{L,p}-{\widehat{d}}_p$. By substituting $y_1={\widehat{d}}_{ap}+{\widehat{d}}_p$ into (26), (38) is obtained:

$$\begin{array}{cc}{\omega }_m\left(k\right){\displaystyle \frac{z-1}{T_s}}-u_0\left(k\right)& =[b_0{I}_s\left(k\right)+d\left(k\right)]-[b_0{I}_s\left(k\right)+y_1\left(k\right)+y_2\left(k\right)]\\ & =T_{L,p}(k)+d_{ap}(k)-{\widehat{T}}_{L,p}(k)-{\widehat{d}}_{ap}(k)-{\widehat{d}}_p(k)\\ & =(T_{L,p}(k)-{\widehat{d}}_p(k))-{\widehat{T}}_{L,p}(k)+O_2\end{array}$$

(38)

where $O_2=d_{ap}(k)-{\widehat{d}}_{ap}(k)$, which does not contain the first harmonic component that the RGN is designed to track. Although the RGN’s reference signal changes from $T_{L,p}(k)$ to $T_{L,p}(k)-{\widehat{d}}_p$, in (38), it remains a first harmonic signal. As analyzed in Section 4.2, this change does not affect the RGN’s convergence characteristics but rather causes the RGN’s equilibrium point to shift from ${\widehat{T}}_{L,p}$ to $y_2={\widehat{T}}_{L,p}-{\widehat{d}}_p$. The combined disturbance estimate is then $y=\left({\widehat{d}}_{ap}+{\widehat{d}}_p\right)+\left({\widehat{T}}_{L,p}-{\widehat{d}}_p\right)={\widehat{d}}_{ap}+{\widehat{T}}_{L,p}$, thus achieving an estimation of the total disturbance.

However, transient coupling may still occur during ESO convergence. If the ESO generates significant first harmonics while settling its aperiodic estimate, it could temporarily affect RGN estimation. Minimizing this requires significantly faster ESO convergence than the RGN.

The convergence rate of the ESO is governed by its bandwidth ${\omega }_0$, which also subsequently influences the system’s anti-disturbance performance. Optimizing ${\omega }_0$ is crucial for achieving a fast and accurate estimation of aperiodic disturbances. Similarly, the proportional gain $k_p$ largely influences the dynamic response performance of the speed control loop and serves as a key parameter for its tuning. A common guideline is to set ${\omega }_0=(5-10)k_p$ to ensure a sufficiently fast ESO response.

The convergence rate of the RGN observer is determined by $\lambda$. As indicated by Equation (35), a rapid convergence rate is attainable when $\lambda$ approaches 0.5. Conversely, $\lambda$ values approaching 1 result in a slower convergence rate. Therefore, it is recommended to set $\lambda$ within the range of 0.9 to 1 to avoid coupling with the ESO. Regarding the initial value setting of $\widehat{B}\left(0\right)$ and $\widehat{C}\left(0\right)$ for the RGN, while theoretically any random initial values are permissible, practical considerations for the studied compressor system suggest setting them to 0. Substantially incorrect initial values can result in system instability.

5. Experimental Verification

The experimental setup is depicted in Figure 6. The control scheme, as detailed in Figure 4, is implemented on a Renesas R5F24T8ADFM chip. The switching frequency of the power devices, the algorithm execution frequency, and the system sampling frequency are all set at 8 kHz. The compressed parameters are detailed in

Table 1. Compressor parameters.

Symbol	Parameter	Value with Unit
$P_{rated}$	Rated power	650 W
$V_{rated}$	Rated voltage	AC 220 V
$P$	Number of pole pairs	3
${\omega }_{m\_rated}$	Rated rotor speed	3600 rpm
${\omega }_{m\_min}$	Minimum speed	900 rpm
$R_s$	Stator resistance	$1.2 \Omega$
$L_d$	d-axis inductance	13.2 mH
$L_q$	q-axis inductance	18.5 mH
$J$	Inertia constant	0.000286 kg m2
$K_t$	Torque constant	0.6 Nm/A

. An air conditioning system utilizing R410a refrigerant provides a practical load environment for the compressor. The position and speed observer in Figure 4 employs the method described in [32]. All experimental data are transmitted to a PC via serial communication tools. For comprehensive validation, experimental verification was combined with complementary simulations conducted in the MATLAB/Simulink R2023a environment. In the subsequent experiments, speed control algorithms based on the conventional PI control, the conventional ADRC, the QRC-ADRC [16], and the proposed RGN-ADRC are tested, respectively. To ensure the fairness of the comparative experiments, the parameters of the four control schemes are tuned to achieve a speed loop bandwidth of 30 rad/s. All specific control parameters are documented in

Table 2. Control parameters of different methods.

Method	Symbol	Value with Unit
$f_s$	Execution frequency	8k Hz
PI	$k_{sp}$	0.0286
	$k_{si}$	43
ADRC	$k_p$	50
	${\omega }_0$	180 rad/s
	$b_0$	2000
QRC	$k_r$	200
	${\omega }_{\mathrm{c}1}$	0.0015${\omega }_{\mathrm{m}}$
RGN	$\lambda$	0.96

. The parameters for the conventional PI control are set to $k_{sp}=0.0286$ and $k_{si}=43$. For the conventional ADRC, QRC-ADRC, and the proposed scheme, the parameters $k_p$, ${\omega }_0$, and $b_0$ are consistently selected as 50, 180 rad/s, and 2000, respectively. The parameters of the QRC are set to $k_r=200$, ${\omega }_{\mathrm{c}1}=0.0015{\omega }_{\mathrm{m}}$. For the RGN, the parameter $\lambda$ is set to 0.96.

5.1. Disturbance Rejection Performance

Compared to conventional ADRC, the proposed RGN-ADRC speed controller effectively suppresses periodic load disturbances in the system. Figure 7 and Figure 8 present the experimental results of ADRC and RGN-ADRC at 1800 rpm. As shown in Figure 7a, under ADRC, the speed fluctuation reaches 912 rpm. Upon activation of the RGN, the speed fluctuation smoothly decreases exponentially to 65 rpm. Figure 7b also illustrates the A phase current waveform. Given the motor has three pole pairs, one mechanical cycle corresponds to three electrical cycles, resulting in a clearly observable mechanical periodicity in the current waveform. Furthermore, Figure 8 shows the Fast Fourier Transform (FFT) results of speed. As observed, with the proposed scheme, the first harmonic component of the speed is significantly reduced from 25.25% to 0.02%.

Figure 9 and Figure 10 illustrate the experimental results of different control methods during speed step-up and step-down transients. As shown in Figure 9a, the traditional PI controller takes 0.65 s to stabilize when the target speed steps up from 1800 rpm to 2400 rpm. Due to its inability to effectively suppress periodic disturbances, the speed ripple at 1800 rpm and 2400 rpm are 737 rpm and 619 rpm, respectively. Furthermore, noticeable overshoot and oscillations are present. In Figure 9b, the traditional ADRC controller achieves stabilization in 0.24 s for the same speed step-up. The speed ripple at 1800 rpm and 2400 rpm are 879 rpm and 817 rpm, respectively, but overshoot and oscillations are absent. Figure 9c demonstrates that the QRC-ADRC controller stabilizes within 0.47 s for the 1800 rpm to 2400 rpm speed step-up. The speed ripple at the lower and upper steady-state speeds is reduced to 195 rpm and 107 rpm, respectively, indicating smaller speed fluctuations; however, the ripple increases significantly during the step change, reaching a maximum of 386 rpm. Finally, Figure 9d shows that the proposed RGN-ADRC controller stabilizes in 0.28 s for the speed step-up from 1800 rpm to 2400 rpm. The speed ripple at the two target speeds is significantly suppressed to 71 rpm and 34 rpm, respectively, demonstrating effective speed fluctuation suppression. Moreover, the ripple during the step change is considerably smaller than that of QRC-ADRC, with a maximum of 116 rpm.

As shown in Figure 10, when the speed command steps down from 3600 rpm to 1800 rpm, the proposed scheme remains the best-performing among the four methods. The proposed scheme takes 0.69 s to complete the speed regulation during the downward step, with speed fluctuations kept at a consistently low level throughout the dynamic process and no overshoot observed. Its speed ripple at the steady-state speed of 3600 rpm is 39 rpm, lower than the 423 rpm and 486 rpm of PI and ADRC, and comparable to the 41 rpm of QRC-ADRC. It is worth noting that the steady-state speed ripple at 1800 rpm in Figure 9 and Figure 10 is slightly different due to the time required for the air conditioning system to stabilize.

To further verify the impact of aperiodic disturbances, particularly load step changes that are challenging to perform precisely on the physical test bench, a simulation is conducted in MATLAB/Simulink. This simulation utilizes the identical control block diagram (Figure 4) and parameters (Table 1 and Table 2) as the hardware setup, with the simulated motor’s load torque precisely matching the 25 °C load-angle curve shown in Figure 1. Figure 11 presents the Simulink results of different methods under a sudden change of 100% rated load torque. Notably, PI control exhibited the poorest anti-disturbance capability, resulting in a significant speed fluctuation of 891 rpm. While ADRC possesses anti-disturbance capability, its limited periodic disturbance suppression led to a considerable speed fluctuation of 640 rpm. QRC-ADRC, despite incorporating periodic disturbance rejection, showed a speed fluctuation of 460 rpm due to the impact of the sudden load change on its QRC component. In contrast, the proposed scheme’s periodic disturbance suppression remained largely unaffected by the aperiodic disturbance, achieving the best performance with a minimal speed fluctuation of 288 rpm. Therefore, the proposed scheme has good aperiodic disturbance suppression capability.

The Total Harmonic Distortion (THD) results of different control methods at various steady-state speeds are presented in Figure 12, with specific numerical values tabulated in

Table 3. FFT results at different speeds.

Speed (rpm)	PI	ADRC	QRC-ADRC	Proposed
1200	37.85%	39.73%	9.87%	5.73%
1800	20.19%	25.68%	4.52%	1.44%
2400	12.58%	16.49%	1.74%	0.31%
3600	5.98%	6.95%	0.43%	0.27%

. As observed, the PI and ADRC controllers, lacking periodic disturbance suppression capability, exhibit similar and high THD values, reaching 37.85% and 39.73%, respectively, at 1200 rpm. Although the THD decreases with increasing speed, it remains around 6% at 3600 rpm for both. In contrast, QRC-ADRC and the proposed method, which incorporate periodic disturbance suppression, show significantly lower THD values of 9.87% and 5.73%, respectively, at 1200 rpm. The proposed method demonstrates superior periodic disturbance suppression across the entire speed range compared to QRC-ADRC. While the THD values of both methods converge at 3600 rpm, with QRC-ADRC at 0.43% and the proposed method at 0.27%, the proposed method consistently achieves lower THD. Therefore, the proposed method effectively suppresses periodic disturbances.

5.2. Decoupling Performance Analysis

To verify the effectiveness of the decoupling strategy, Figure 13 illustrates the estimated results $y_1$ and $y_2$ in the proposed RGN-ADRC. Figure 13a displays the waveforms from the test shown in Figure 7, which corresponds to a periodic disturbance step condition. During the 0.8 s convergence of $y_2$, $y_1$ exhibits only a gradual overall decrease due to the reduction in speed fluctuation, without any significant abnormal fluctuations. Conversely, Figure 13b displays the waveforms from the test shown in Figure 11d, which corresponds to an aperiodic disturbance step condition. It takes 0.12 s for $y_1$ to converge. while the waveform of $y_2$ remains essentially unchanged during this period. Therefore, the proposed method effectively achieves the designed decoupled control.

Experimental results under the speed step-up with ${0.5b}_0$, ${1.0b}_0$, and ${1.5b}_0$ are presented in Figure 14a, Figure 9d and Figure 14b, respectively. At the steady-state speed of 1800 rpm, the speed ripple for these three cases is 78 rpm, 71 rpm, and 65 rpm, respectively, demonstrating comparable performance. Similarly, at 2400 rpm, the speed fluctuation exhibits close proximity, with values of 25 rpm, 34 rpm, and 27 rpm, respectively. Furthermore, the speed regulation time for all three scenarios is 0.28 s, and the maximum speed deviation during the dynamic process is also quite similar, with values of 119 rpm, 116 rpm, and 103 rpm. This indicates that the proposed method exhibits strong robustness against parameter mismatch.

6. Conclusions

This paper introduces a novel RGN-ADRC speed controller for effective aperiodic and periodic disturbance suppression. The proposed method integrates an angle-based Gauss–Newton model parallel with conventional ADRC. Comprehensive experiments on a 650 W PMSM single-rotor compressor validated its superior disturbance rejection and enhanced speed regulation compared to PI, ADRC, and QRC-ADRC. The angle-based periodic disturbance compensation makes it well-suited for angle-dependent disturbances like those in single-rotor compressors, achieving smoother control and better steady-state performance. Quantitatively, the RGN-ADRC demonstrated significantly improved dynamic responses, achieving a speed step-up regulation time of 0.28 s (vs. 0.65 s for PI and 0.47 s for QRC-ADRC) and a speed step-down regulation time of 0.69 s (vs. 1.86 s for PI and 0.80 s for QRC-ADRC). More critically, it exhibited superior disturbance rejection: the maximum speed drop during a 100% rated load step was reduced by over 37% compared to QRC-ADRC (from 460 rpm to 288 rpm), and the speed’s THD at 1800 rpm was drastically suppressed to 0.31%, representing a 93% reduction compared to QRC-ADRC (4.52%). Furthermore, the proposed method effectively decoupled periodic and aperiodic disturbance suppression and demonstrated strong robustness to parameter mismatch. The successful application of the RGN-ADRC to single-rotor compressors suggests its potential for broader use in similar angle-dependent load systems, a direction worthy of future study.