Design of Lumped Disturbance Observer in Current Loop of IPMSM Based on Recursive Integral Sliding Mode Surface

Abstract

To overcome the problem of current control effect being reduced by unideal factors in a motor control system, such as motor parameter variation, inverter dead time, nonlinearity of the system, etc., a sliding mode disturbance observer for an interior permanent magnet synchronous motor is proposed in this paper. The model of an interior permanent magnet synchronous motor with unideal factors is designed, and the unideal factors are unified into lumped disturbances of motor stator voltage. Then, the observer for lumped disturbance is designed. A recursive integral sliding surface is used to replace the terminal sliding surface to avoid the noise sensitivity and singularity problem of the traditional terminal sliding mode observer. The observer can estimate the lumped disturbance of the current loop without relying on the accurate system model in finite time. Moreover, the structure of the current loop does not need to be adjusted while using the observer to observe and compensate for disturbances. Experiments are carried out to verify the effectiveness of the proposed observer.

1. Introduction

The interior permanent magnet synchronous motor (IPMSM) has been widely utilized due to its simple structure, high efficiency, high power density, fast response and other excellent features [1,2]. Among the IPMSM control methods, the field-oriented control (FOC) is most widely applied due to the features of low torque ripple, fast response and high control precision. At the same time, the double-loop structure of FOC can control the torque and magnetic flux independently [3,4,5]. The inner current loop determines the dynamic and steady-state performance of stator currents [6,7]. However, IPMSM is a nonlinear and strong coupling system and contains unavoidable and unmeasured disturbances such as possible parameter variations in operating [8,9,10]. Moreover, the inverter includes nonlinearity such as dead time, voltage drop and other factors [11,12]. These unideal factors, which are usually ignored in motor control system modeling, will affect the current control accuracy and cause controller parameter mismatch. It is difficult to meet the high performance requirements of the IPMSM control system using traditional control methods.

With the development of power electronics and digital signal processing, many methods to improve current control accuracy have been proposed. In general, these methods can be divided into two groups. One is to improve the current control performance by introducing new control laws. Study [13] presents a method based on an adaptive linear neuron to suppress the current harmonics by a self-tuning voltage feedforward, which does not need to extract the current harmonics. Study [14] proposes a model predictive current control (MPCC) method with adaptive-adjusting timescales based on the motor operation conditions, which can improve the system dynamic response and the current prediction accuracy. Study [15] presents a fixed switching period sliding mode control (SMC) method which combines the advantages of FOC and direct torque control (DTC). This method can allow the control law to be transferred to the inverter directly by a novel decoupling algorithm to find a set of decoupled sliding surfaces. Study [16] analyzes the relationship between the low-pass filter in the feedback loop and the controller parameters, and proposes a modified model compensation active disturbance rejection control (ADRC) method, which can make the current response closer to that of the ideal closed-loop control system. Study [17] proposes a complex vector decoupling method considering the time delay caused by sampling, calculation and PWM modulation, which quantifies and compensates the phase delay to improve the dynamic performance.

Although the above methods improved the current control performance by introducing new control laws, the parameters’ perturbation and other unideal factors are ignored, and the system disturbances have not been eliminated yet.

The other methods obtained system information through sensors or observers and improve the controller [18,19,20,21,22,23,24,25,26]. Among them, disturbance observer (DO) gains the favor of many scholars and becomes a research hotspot because of its simple structure, lack of dependence on system model accuracy and lack of an additional hardware cost. At the same time, compensation based on the disturbance observer does not change the original structure of the controller. In [21], a multiple-models adaptive disturbance observer is proposed to solve the steady-state current error by estimating the lumped disturbances, which can improve the robustness to parameter perturbations and model mismatch. Study [22] proposes an adaptive observer based on the model reference adaptive system (MRAS), which improves the control performance, anti-interference ability and robustness of the control system. Study [23] presents a novel super-twisting algorithm associated with extended state observer (ESO) for FOC, which can estimate system disturbances, eliminate chattering, improve the system anti-interference ability, and obtain a fast dynamic response. Study [24] introduces an adaptive internal model observer (AIMO) to observe system disturbances quickly without static difference, which avoid disturbances caused by parameter variations. Study [25] proposes a sliding-mode observer based on adaptive super-twisting algorithm (ASTA-SMO) to improve the performance of deadbeat predictive current control (DPCC), which can predict future values of stator current and track system disturbance caused by parameter mismatch. Study [26] presents an adaptive stator current disturbance observer considering the discrete characteristics of the current prediction error, which structures a parallel strategy, observes the discrete disturbance related to each voltage vector separately, and then eliminates the prediction error.

The above disturbance observation methods can effectively estimate and compensate for system disturbances. Among them, the SMO has simple structure and is easy to implement. Other methods have disadvantages such as difficulty with observer parameter adjustment, a poor observation effect on nonlinear disturbances, etc. However, due to the discontinuous switching characteristics, the ordinary sliding mode method has a chattering problem [27]. The terminal sliding mode method introduces nonlinear terms into the sliding mode surface, which can make the observation error converge to zero in finite time, and its dynamic performance is better than the ordinary sliding mode method with state asymptotic convergence under linear sliding mode surface conditions [28]. And the terminal sliding mode method does not have switching items, which can effectively eliminate chattering. However, the introduction of nonlinear terms results in the existence of negative exponential term in the sliding mode control function which will bring about singularity problems [29].

In this research, a terminal sliding mode disturbance observer (SMDO) based on a recursive integral sliding mode surface for lumped disturbances of IPMSM is proposed. The observer can realize the lumped estimation of the electromagnetic parameter variation and external disturbances of the control system without relying on the accurate system model in finite time. At the same time, the noise sensitivity caused by the forward Euler derivative and the singularity problem of the traditional terminal sliding mode observer are avoided. In this paper, a novel IPMSM model is presented, and all the system disturbances as the lumped stator voltage variation are considered in Section 2. The recursive integral sliding mode disturbance observer is analyzed and designed in Section 3. The experimental results are given to verify the feasibility of the proposed observer in Section 4.

2. Modeling of IPMSM with Parameter Variation and Unknown Disturbance

According to study [2], the state equation of the IPMSM dynamic model in a synchronously rotating reference frame can be denoted as (1):

$$\left\{\begin{array}{c}\frac{\mathrm{d}{i}_{\mathrm{d}}}{\mathrm{d}t}=-\frac{R_{\mathrm{s}}}{L_{\mathrm{d}}}{i}_{\mathrm{d}}+\frac{L_{\mathrm{q}}}{L_{\mathrm{d}}}{\omega }_{\mathrm{e}}{i}_{\mathrm{q}}+\frac{1}{L_{\mathrm{d}}}{u}_{\mathrm{d}}\hfill \\ \frac{\mathrm{d}{i}_{\mathrm{q}}}{\mathrm{d}t}=-\frac{R_{\mathrm{s}}}{L_{\mathrm{q}}}{i}_{\mathrm{q}}-\frac{L_{\mathrm{d}}}{L_{\mathrm{q}}}{\omega }_{\mathrm{e}}{i}_{\mathrm{d}}+\frac{1}{L_{\mathrm{q}}}{u}_{\mathrm{q}}-\frac{{\psi }_{\mathrm{f}}}{L_{\mathrm{q}}}{\omega }_{\mathrm{e}}\hfill \end{array}\right.$$

(1)

where u_d, u_q, i_d and i_q are the stator voltage and current components in d- and q-axis, respectively; R_s is the nominal value of stator resistance; L_d and L_q are the nominal values of the inductance in the d- and q-axis, respectively; and ψ_f is the nominal value of permanent magnet flux linkage.

The system state variables are defined as x = [i_d i_q]^T, the input variables are defined as u = [u_d u_q]^T, then (1) can be expressed as (2):

$$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}+\mathit{C}$$

(2)

where

$$\mathit{A}=\left[\begin{array}{cc}-\frac{R_{\mathrm{s}}}{L_{\mathrm{d}}}& \frac{{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}}{L_{\mathrm{d}}}\\ -\frac{{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}}{L_{\mathrm{q}}}& -\frac{R_{\mathrm{s}}}{L_{\mathrm{q}}}\end{array}\right],\mathit{B}=\left[\begin{array}{cc}\frac{1}{L_{\mathrm{d}}}& 0\\ 0& \frac{1}{L_{\mathrm{q}}}\end{array}\right],\mathit{C}=\left[\begin{array}{c}0\\ -\frac{{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}}{L_{\mathrm{q}}}\end{array}\right].$$

The motor parameters including stator resistance, stator inductance, and permanent magnet flux linkage vary with the varying operating condition, then the controller performance can be affected by the varying motor parameters. Meanwhile, the system nonlinearity, dead time and other factors of the inverter can also affect the motor voltage and current. After considering the above, the system state equation can be expressed as [2]:

$$\left\{\begin{array}{c}\frac{\mathrm{d}{i}_{\mathrm{d}}}{\mathrm{d}t}=-\frac{R_{\mathrm{s}}+\Delta R_{\mathrm{s}}}{L_{\mathrm{d}}+\Delta L_{\mathrm{d}}}{i}_{\mathrm{d}}+\frac{L_{\mathrm{q}}+\Delta L_{\mathrm{q}}}{L_{\mathrm{d}}+\Delta L_{\mathrm{d}}}{\omega }_{\mathrm{e}}{i}_{\mathrm{q}}+\frac{1}{L_{\mathrm{d}}+\Delta L_{\mathrm{d}}}(u_{\mathrm{d}}+\Delta u_{\mathrm{d}})\hfill \\ \frac{\mathrm{d}{i}_{\mathrm{q}}}{\mathrm{d}t}=-\frac{R_{\mathrm{s}}+\Delta R_{\mathrm{s}}}{L_{\mathrm{q}}+\Delta L_{\mathrm{q}}}{i}_{\mathrm{q}}-\frac{L_{\mathrm{d}}+\Delta L_{\mathrm{d}}}{L_{\mathrm{q}}+\Delta L_{\mathrm{q}}}{\omega }_{\mathrm{e}}{i}_{\mathrm{d}}+\frac{1}{L_{\mathrm{q}}+\Delta L_{\mathrm{q}}}(u_{\mathrm{q}}+\Delta u_{\mathrm{q}})-\frac{{\psi }_{\mathrm{f}}+\Delta {\psi }_{\mathrm{f}}}{L_{\mathrm{q}}+\Delta L_{\mathrm{q}}}{\omega }_{\mathrm{e}}\hfill \end{array}\right.$$

(3)

where ΔR, ΔL_d, ΔL_q, Δψ_f are variations of R_s, L_d, L_q, and ψ_f, respectively; Δu_d and Δu_q are the voltage variations caused by the other unideal factors. After considering the above issues, (2) can be obtained:

$$\dot{\mathit{x}}=(\mathit{A}+\mathsf{\Delta }\mathit{A})\mathit{x}+(\mathit{B}+\mathsf{\Delta }\mathit{B})(\mathit{u}+\mathsf{\Delta }\mathit{u})+(\mathit{C}+\mathsf{\Delta }\mathit{C})$$

(4)

From (4), the variation in current response will induce additional errors, and cause parameter mismatch of the current controller, resulting in a decrease in the current control effect.

Considering the effects above, all the disturbances can be regarded as the lumped disturbance f = [f_d f_q]^T in the d- and q-axis stator voltage.

$$\mathit{f}={\mathit{B}}^{-1}\left(\mathsf{\Delta }\mathit{A}\mathit{x}+\mathit{B}\mathsf{\Delta }\mathit{u}+\mathit{u}\mathsf{\Delta }\mathit{B}+\mathsf{\Delta }\mathit{B}\mathsf{\Delta }\mathrm{u}+\mathsf{\Delta }\mathit{C}\right)$$

(5)

Then (4) can be expressed as:

$$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}(\mathit{u}+\mathit{f})+\mathit{C}$$

(6)

3. Design and Analysis of Recursive Integral Sliding Mode Disturbance Observer

Disturbance observer is an effective method to solve the current controller mismatch. After the lumped disturbance f is estimated by the disturbance observer, disturbance compensation controller can eliminate the disturbance influence efficaciously. The block diagram of the IPMSM control scheme with disturbance compensation is shown in Figure 1, where ω_e* is the reference value of the rotating speed speed, x* is the reference value of current, $\widehat{\mathit{f}}$ is the estimated value of the lumped disturbances f, and f_c is the disturbance compensation value obtained by $\widehat{\mathit{f}}$.

The controlled system with the disturbance observation and compensation can be equivalent to Figure 2, then the system state equation can be shown as (7):

$$\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}(\mathit{u}+\mathit{f}-{\mathit{f}}_{\mathbf{c}})+\mathit{C}$$

(7)

According to (7), when f − f_c = 0, the effects of system disturbances can be compensated and the actual system state can be equivalent to the ideal state without disturbances. Then, the current loop parameters can be designed according to (2) under ideal conditions.

3.1. Design of Recursive Integral SMDO

The SMDO is constructed based on (6), and the observation equation is as follows:

$$\dot{\hat{\mathit{x}}}=A\hat{\mathit{x}}+\mathbf{B}\mathit{u}+C+{\mathit{u}}_{\mathrm{smo}}$$

(8)

where $\dot{\hat{\mathit{x}}}$ = [${\widehat{i}}_{\mathrm{d}}$${\widehat{i}}_{\mathrm{q}}$]^T, ${\widehat{i}}_{\mathrm{d}}$ and ${\widehat{i}}_{\mathrm{q}}$ are the observed values of i_d and i_q, respectively; u_smo = [u_smod u_smoq]^T, u_smod and u_smoq are the d- and q-axis control functions, respectively.

The current observation error e = [e_id e_iq]^T is defined as (9):

$$\mathit{e}=\mathit{x}-\widehat{\mathit{x}}$$

(9)

By substituting (6) and (8) into (9), the state equation of the current observation error is derived as:

$$\dot{\mathit{e}}=\mathit{A}\mathit{e}+\mathit{B}\mathit{f}-{\mathit{u}}_{\mathrm{smo}}$$

(10)

To ensure the observation error can converge to 0 in finite time, and the observer control functions u_smod and u_smoq are non-singular, an integral fast terminal sliding mode surface is selected as [27]:

$${\mathit{s}}_{\mathrm{i}}=\left[\begin{array}{c}{s}_{\mathrm{id}}\\ s_{\mathrm{iq}}\end{array}\right]=\left[\begin{array}{c}{e}_{\mathrm{id}}+{\displaystyle {\int }_0^{\mathrm{t}}[{\alpha }_{\mathrm{id}}{e}_{\mathrm{id}}+{\beta }_{\mathrm{id}}{\left|e_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{id}}}\mathrm{sign}(e_{\mathrm{id}})]\mathrm{d}\tau }\\ e_{\mathrm{iq}}+{\displaystyle {\int }_0^{\mathrm{t}}[{\alpha }_{\mathrm{iq}}{e}_{\mathrm{iq}}+{\beta }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{iq}}}\mathrm{sign}(e_{\mathrm{iq}})]\mathrm{d}\tau }\end{array}\right]$$

(11)

where α_id, α_iq, β_id, β_iq, γ_id, and γ_iq are sliding mode observer coefficients; in general, α_id > 0, α_iq > 0, β_id > 0, β_iq > 0, 0 < γ_id < 1, and 0 < γ_iq < 1.

When the control functions u_smo satisfy the SMDO reachability condition and the observer is in sliding mode, the first-order derivative of the sliding mode surface is 0:

$${\dot{\mathit{s}}}_{\mathrm{i}}=\left[\begin{array}{c}{\dot{s}}_{\mathrm{id}}\\ {\dot{s}}_{\mathrm{iq}}\end{array}\right]=\left[\begin{array}{c}{\dot{e}}_{\mathrm{id}}+{\alpha }_{\mathrm{id}}{e}_{\mathrm{id}}+{\beta }_{\mathrm{id}}{\left|e_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{id}}}\mathrm{sign}(e_{\mathrm{id}})\\ {\dot{e}}_{\mathrm{iq}}+{\alpha }_{\mathrm{iq}}{e}_{\mathrm{iq}}+{\beta }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{iq}}}\mathrm{sign}(e_{\mathrm{iq}})\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

(12)

The equivalent observer control function u_eq can be obtained according to (10) and (12):

$${\mathit{u}}_{\mathrm{eq}}=\mathit{A}\mathit{e}+\left[\begin{array}{c}{\alpha }_{\mathrm{id}}{e}_{\mathrm{id}}+{\beta }_{\mathrm{id}}{\left|e_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{id}}}\mathrm{sign}(e_{\mathrm{id}})\\ {\alpha }_{\mathrm{iq}}{e}_{\mathrm{iq}}+{\beta }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{iq}}}\mathrm{sign}(e_{\mathrm{iq}})\end{array}\right]$$

(13)

Since there are nonlinear and unmeasurable factors in the lumped disturbances f, and the observer cannot obtain the estimated values $\widehat{\mathit{f}}$ directly by only using the equivalent control functions u_eq, it is necessary to design the disturbance tracking functions u_f = [u_fd u_fq]^T to estimate the lumped disturbances. According to (6) and (10), (14) can be obtained as follows:

$$\dot{\hat{\mathit{x}}}=\mathit{A}\hat{\mathit{x}}+\mathit{B}\mathit{u}+\mathit{C}+\mathit{B}{\mathit{u}}_{\mathrm{f}}+{\mathit{u}}_{\mathrm{eq}}$$

(14)

The state equation of error e is rewritten as:

$$\dot{\mathit{e}}=\mathit{A}\mathit{e}+\mathit{B}(\mathit{f}-{\mathit{u}}_{\mathrm{f}})-{\mathit{u}}_{\mathrm{eq}}$$

(15)

On the basis of an integral fast terminal sliding mode surface s_i, the recursive sliding mode surface s_f is adopted to realize the estimation of lumped disturbances f, and the SMDO based on the recursive terminal sliding mode surface is obtained. The s_f is as (16):

$${\mathit{s}}_{\mathrm{f}}=\left[\begin{array}{c}{s}_{\mathrm{fd}}\\ s_{\mathrm{fq}}\end{array}\right]=\left[\begin{array}{c}{\dot{s}}_{\mathrm{id}}+{\alpha }_{\mathrm{fd}}{s}_{\mathrm{id}}+{\beta }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}}\mathrm{sign}(s_{\mathrm{id}})\\ {\dot{s}}_{\mathrm{iq}}+{\alpha }_{\mathrm{fq}}{s}_{\mathrm{iq}}+{\beta }_{\mathrm{fq}}{\left|s_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{fq}}}\mathrm{sign}(s_{\mathrm{iq}})\end{array}\right]$$

(16)

where α_fd, α_fq, β_fd, β_fq, γ_fd, and γ_fq are sliding mode observer coefficients of s_f; in general, α_fd > 0, α_fq > 0, β_fd > 0, β_fq > 0, 0 < γ_fd < 1, and 0 < γ_fq < 1.

The first-order derivative of s_f is given in (17):

$${\dot{\mathit{s}}}_{\mathrm{f}}=\left[\begin{array}{c}{\dot{s}}_{\mathrm{fd}}\\ {\dot{s}}_{\mathrm{fq}}\end{array}\right]=\left[\begin{array}{c}{\ddot{s}}_{\mathrm{id}}+[{\alpha }_{\mathrm{fd}}{s}_{\mathrm{id}}+{\beta }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}}\mathrm{sign}(s_{\mathrm{id}}){]}^{\prime }\\ {\ddot{s}}_{\mathrm{iq}}+[{\alpha }_{\mathrm{fq}}{s}_{\mathrm{iq}}+{\beta }_{\mathrm{fq}}{\left|s_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{fq}}}\mathrm{sign}(s_{\mathrm{iq}}){]}^{\prime }\end{array}\right]$$

(17)

In order to solve the disturbance tracking function u_f in SMDO, (15) is substituted into (11) to obtain the first and second order derivative of s_i:

$${\dot{\mathit{s}}}_{\mathrm{i}}=\mathit{B}(\mathit{f}-{\mathit{u}}_{\mathrm{f}})$$

(18)

$${\ddot{\mathit{s}}}_{\mathrm{i}}=\mathit{B}(\dot{\mathit{f}}-{\dot{\mathit{u}}}_{\mathrm{f}})$$

(19)

The disturbance tracking function u_f is selected as (20):

$${\mathit{u}}_{\mathrm{f}}=\left[\begin{array}{c}{L}_{\mathrm{d}}[{\alpha }_{\mathrm{fd}}{s}_{\mathrm{id}}+{\beta }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}}\mathrm{sign}(s_{\mathrm{id}})]\\ L_{\mathrm{q}}[{\alpha }_{\mathrm{fq}}{s}_{\mathrm{iq}}+{\beta }_{\mathrm{fq}}{\left|s_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{fq}}}\mathrm{sign}(s_{\mathrm{iq}})]\end{array}\right]$$

(20)

According to Equations (13), (14) and (20), the observer control function u_smo can be obtained as (21):

$$\begin{array}{c}{\mathit{u}}_{\mathrm{smo}}={\mathit{u}}_{\mathrm{eq}}+\mathit{B}{\mathit{u}}_{\mathrm{f}}\hfill \\ \hspace{1em}\hspace{1em}=\mathit{A}\mathit{e}+\left[\begin{array}{c}{\alpha }_{\mathrm{id}}{e}_{\mathrm{id}}+{\beta }_{\mathrm{id}}{\left|e_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{id}}}\mathrm{sign}(e_{\mathrm{id}})\\ {\alpha }_{\mathrm{iq}}{e}_{\mathrm{iq}}+{\beta }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{iq}}}\mathrm{sign}(e_{\mathrm{iq}})\end{array}\right]+\left[\begin{array}{c}{\alpha }_{\mathrm{fd}}{s}_{\mathrm{id}}+{\beta }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}}\mathrm{sign}(s_{\mathrm{id}})\\ {\alpha }_{\mathrm{fq}}{s}_{\mathrm{iq}}+{\beta }_{\mathrm{fq}}{\left|s_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{fq}}}\mathrm{sign}(s_{\mathrm{iq}})\end{array}\right]\hfill \end{array}$$

(21)

When the observer is in the balance state e = 0, the equivalent observer control function u_eq in (21) is equal to zero and the disturbance tracking function u_f in (21) is equal to the lumped disturbance f, so u_f is the observed value $\widehat{\mathit{f}}$ of the lumped disturbance.

The block diagrams of the disturbance observer are shown in Figure 3.

While using the traditional terminal sliding mode method to observe the system states, the system state equation is usually in the form of an integral chain [28]. At this time, the system state equation as follows:

$$\left\{\begin{array}{c}\dot{\mathit{e}}={\mathit{e}}_{\mathrm{f}}\\ {\dot{\mathit{e}}}_{\mathrm{f}}=\dot{\mathit{f}}-{\mathit{u}}_{\mathrm{smoT}}\end{array}\right.$$

(22)

where e_f = f − $\widehat{\mathit{f}}$= [e_fd e_fq]^T, and u_smoT is the sliding mode control function and is also the observed value of $\widehat{\mathit{f}}$.

The sliding mode surface is selected as (23) according to study [29], and the recursive sliding mode surfaces is selected as (24):

$${\mathit{s}}_1=\left[\begin{array}{c}{s}_{1\mathrm{d}}\\ s_{1\mathrm{q}}\end{array}\right]=\left[\begin{array}{c}{\dot{e}}_{\mathrm{id}}+{\alpha }_{1\mathrm{d}}{e}_{\mathrm{id}}+{\beta }_{1\mathrm{d}}{e_{\mathrm{id}}}^{{\gamma }_{1\mathrm{d}}}\\ {\dot{e}}_{\mathrm{iq}}+{\alpha }_{1\mathrm{q}}{e}_{\mathrm{iq}}+{\beta }_{1\mathrm{q}}{e_{\mathrm{iq}}}^{{\gamma }_{\mathrm{iq}}}\end{array}\right]$$

(23)

$${\mathit{s}}_2=\left[\begin{array}{c}{s}_{\mathrm{fd}}\\ s_{\mathrm{fq}}\end{array}\right]=\left[\begin{array}{c}{\dot{s}}_{1\mathrm{d}}+{\alpha }_{2\mathrm{d}}{s}_{1\mathrm{d}}+{\beta }_{2\mathrm{d}}{s_{1\mathrm{d}}}^{{\gamma }_{2\mathrm{d}}}\\ {\dot{s}}_{1\mathrm{q}}+{\alpha }_{2\mathrm{q}}{s}_{1\mathrm{q}}+{\beta }_{2\mathrm{q}}{s_{1\mathrm{q}}}^{{\gamma }_{2\mathrm{q}}}\end{array}\right]$$

(24)

where α_ij > 0, β_ij > 0, and 0 < γ_ij < 1 (i =1, 2; j = d, q) are sliding mode coefficients.

According to (22)–(24), the control function u_smoT is selected as:

$${\mathit{u}}_{\mathrm{smoT}}=\left[\begin{array}{c}{\alpha }_{1\mathrm{d}}{e}_{\mathrm{fd}}+{\beta }_{1\mathrm{d}}{\gamma }_{1\mathrm{d}}{e_{\mathrm{id}}}^{{\gamma }_{1\mathrm{d}}-1}{e}_{\mathrm{fd}}+{\alpha }_{2\mathrm{d}}{s}_{1\mathrm{d}}+{\beta }_{2\mathrm{d}}{s_{1\mathrm{d}}}^{{\gamma }_{2\mathrm{d}}}\\ {\alpha }_{1\mathrm{q}}{e}_{\mathrm{fq}}+{\beta }_{1\mathrm{q}}{\gamma }_{1\mathrm{q}}{e_{\mathrm{iq}}}^{{\gamma }_{1\mathrm{q}}-1}{e}_{\mathrm{fq}}+{\alpha }_{2\mathrm{q}}{s}_{1\mathrm{q}}+{\beta }_{2\mathrm{q}}{s_{1\mathrm{q}}}^{{\gamma }_{2\mathrm{q}}}\end{array}\right]$$

(25)

According to (25), the control function u_smoT contains the current observation error e and the disturbance observation error e_f. e can be obtained directly based on the actual current of the system and observed current of the observer, but e_f needs to be obtained by derivation of e. In common, e_f is calculated as ${\mathit{e}}_{\mathrm{f}}(k)=\dot{\mathit{e}}(k)\approx [\mathit{e}(k)-\mathit{e}(k-1)]/{\mathrm{T}}_{\mathrm{c}}$, which is sensitive to noise. T_c is the control period of the current loop. At the same time, since γ_1d − 1 < 0 and γ_1q − 1 < 0, the observer has a singular problem, while e = 0 and e_f ≠ 0.

Compared with traditional terminal SMDO, the proposed recursive integral SMDO considers lumped disturbance as a motor stator voltage variation. The state equation of observer is the first-order differential equation shown in (10). The observer control function u_smo as shown in (21) only requires the current observation error e and does not contain the derivative s_i or e. And the observer control function u_smo does not contain the negative exponential term, there is no singular problem when the observer is in sliding mode. Therefore, the proposed recursive integral SMDO can avoid the noise sensitivity and singularity problem while tracking the system lumped disturbance in finite time.

The SMDO only demands the input and output information of a system, and the lumped disturbance can be eliminated by disturbance observation and compensation.

3.2. Stability Analysis

The Lyapunov function V_i is defined as (26), and its first-order differential is given as (27):

$$V_{\mathrm{i}}=\frac{1}{2}{\dot{\mathit{s}}}_{\mathrm{i}}^{\mathrm{T}}{\dot{\mathit{s}}}_{\mathrm{i}}$$

(26)

$$\begin{array}{ll}{\dot{V}}_{\mathrm{i}}={\dot{\mathit{s}}}_{\mathrm{i}}^{\mathrm{T}}{\ddot{\mathit{s}}}_{\mathrm{i}}& =\frac{1}{L_{\mathrm{d}}}{\dot{s}}_{\mathrm{id}}({\dot{f}}_{\mathrm{d}}-{\alpha }_{\mathrm{fd}}{\dot{s}}_{\mathrm{id}}-{\beta }_{\mathrm{fd}}{\gamma }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}-1}{\dot{s}}_{\mathrm{id}})+\frac{1}{L_{\mathrm{q}}}{\dot{s}}_{\mathrm{iq}}({\dot{f}}_{\mathrm{q}}-{\alpha }_{\mathrm{fq}}{\dot{s}}_{\mathrm{iq}}-{\beta }_{\mathrm{fq}}{\gamma }_{\mathrm{fq}}{\left|s_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{fq}}-1}{\dot{s}}_{\mathrm{iq}})\\ & \le \frac{1}{L_{\mathrm{d}}}\left|{\dot{s}}_{\mathrm{id}}\right|(\left|{\dot{f}}_{\mathrm{d}}\right|-{\alpha }_{\mathrm{fd}}\left|{\dot{s}}_{\mathrm{id}}\right|-{\beta }_{\mathrm{fd}}{\gamma }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}-1}\left|{\dot{s}}_{\mathrm{id}}\right|)+\frac{1}{L_{\mathrm{d}}}\left|{\dot{s}}_{\mathrm{id}}\right|(\left|{\dot{f}}_{\mathrm{d}}\right|-{\alpha }_{\mathrm{fd}}\left|{\dot{s}}_{\mathrm{id}}\right|-{\beta }_{\mathrm{fd}}{\gamma }_{\mathrm{fd}}{\left|s_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{fd}}-1}\left|{\dot{s}}_{\mathrm{id}}\right|)\end{array}$$

(27)

When the control period of the current loop is short enough, ${\dot{f}}_{\mathrm{d}}$ and ${\dot{f}}_{\mathrm{q}}$ can be considered as zero. ${\dot{V}}_{\mathrm{i}}$ < 0 is drawn while the observer parameter ranges are α_fd > 0, α_fq > 0, β_fd > 0, β_fq > 0, 1 > γ_fd > 0, and 1 > γ_fq > 0. It can be derived that ${\dot{\mathit{s}}}_{\mathrm{i}}$ is asymptotically stable at itsorigin, which implies that the current observation error e can converge to 0 in finite time with the control function u_smo.

Define the disturbance observation error e_f = f − u_f, and define the Lyapunov function V_f as follows:

$$V_{\mathrm{f}}=\frac{1}{2}{\mathit{e}}_{\mathrm{f}}^{\mathrm{T}}{\mathit{e}}_{\mathrm{f}}$$

(28)

According to (18) and (19), the first-order derivative of V_f is given as:

$${\dot{V}}_{\mathrm{f}}={\mathit{e}}_{\mathrm{f}}^{\mathrm{T}}{\dot{\mathit{e}}}_{\mathrm{f}}={\left({\mathit{B}}^{-1}{\dot{\mathit{s}}}_{\mathrm{i}}\right)}^{\mathrm{T}}\left({\mathit{B}}^{-1}{\ddot{\mathit{s}}}_{\mathrm{i}}\right)$$

(29)

It can be derived that e_f is also asymptotically stable at its origin. Thus, the current observation results $\dot{\hat{\mathit{x}}}$ of SMDO can converge to the actual currents, and the disturbance tracking functions u_f converge to the lumped disturbance f in finite time.

The trajectory of the observation error e is denoted as (30):

$$\left\{\begin{array}{c}{\dot{e}}_{\mathrm{id}}+{\alpha }_{\mathrm{id}}{e}_{\mathrm{id}}+{\beta }_{\mathrm{id}}{\left|e_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{id}}}\mathrm{sign}(e_{\mathrm{id}})=0\\ {\dot{e}}_{\mathrm{iq}}+{\alpha }_{\mathrm{iq}}{e}_{\mathrm{iq}}+{\beta }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{iq}}}\mathrm{sign}(e_{\mathrm{iq}})=0\end{array}\right.$$

(30)

When the current observation error is close to the balance state e = 0, the convergence speed is mainly determined by (31):

$$\left\{\begin{array}{c}{\dot{e}}_{\mathrm{id}}=-{\beta }_{\mathrm{id}}{\left|e_{\mathrm{id}}\right|}^{{\gamma }_{\mathrm{id}}}\mathrm{sign}(e_{\mathrm{id}})\\ {\dot{e}}_{\mathrm{iq}}=-{\beta }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}\right|}^{{\gamma }_{\mathrm{iq}}}\mathrm{sign}(e_{\mathrm{iq}})\end{array}\right.$$

(31)

When the current observation error is far from its origin, the convergence speed is mainly determined by (32):

$$\left\{\begin{array}{c}{\dot{e}}_{\mathrm{id}}=-{\alpha }_{\mathrm{id}}{e}_{\mathrm{id}}\\ {\dot{e}}_{\mathrm{iq}}=-{\alpha }_{\mathrm{iq}}{e}_{\mathrm{iq}}\end{array}\right.$$

(32)

By solving (30), the time to arrive at e = 0 can be derived as:

$$\left\{\begin{array}{c}{t}_{\mathrm{id}}=\frac{1}{{\alpha }_{\mathrm{id}}(1-{\gamma }_{\mathrm{id}})}\ln \frac{{\alpha }_{\mathrm{id}}{\left|e_{\mathrm{id}}(0)\right|}^{1-{\gamma }_{\mathrm{id}}}+{\beta }_{\mathrm{id}}}{{\beta }_{\mathrm{id}}}\\ t_{\mathrm{iq}}=\frac{1}{{\alpha }_{\mathrm{iq}}(1-{\gamma }_{\mathrm{iq}})}\ln \frac{{\alpha }_{\mathrm{iq}}{\left|e_{\mathrm{iq}}(0)\right|}^{1-{\gamma }_{\mathrm{iq}}}+{\beta }_{\mathrm{iq}}}{{\beta }_{\mathrm{iq}}}\end{array}\right.$$

(33)

4. Experimental Results

An experimental setup was built to validate the proposed algorithm, as shown in Figure 4. An IPMSM, an asynchronous induction motor (ACIM) and a torque sensor are connected via diaphragm couplings, where the IPMSM is used to test the algorithm proposed in this paper, and the ACIM is used to provide the load torque. The parameters of IPMSM are listed in

Table 1. Parameters of IPMSM.

Descriptions	Parameters	Values
Stator resistance	Rs	12.4 mΩ
d-axis inductance	Ld	190 μH
q-axis inductance	Lq	400 μH
Flux	ψf	0.0712 Wb
Pole pairs	p	6
Inertia	J	0.09615 kg·m2

.

The control algorithm of IPMSM is implemented through DSP TI TMS320F28377D. The CPU1 of DSP communicates with the PC while running the basic motor control algorithm. The CPU2 is responsible for realizing the disturbance observer. The interfaces of multiple peripherals including resolver-to-digital converter (R/D Converter), analog–digital converter (ADC), and digital–analog converter (DAC), the external I/O interface, and the hardware protection logic are implemented in FPGA Xilinx XC6SLX16. Hall sensors are used to obtain voltage and current information of IPMSM. A rotating transformer is used to measure the speed and position of IPMSM. The pulse width modulation (PWM) switching frequency is 20 kHz, and the DC bus voltage is 200 V. An oscilloscope (DLM5054), a recorder (DL850) and a power analyzer (WT5000) manufactured by Yokogawa were used to record the system data during the experiments. The experimental setup is shown in Figure 5.

Experiments are carried out to validate the proposed recursive integral SMDO. During the experiments, the same parameters are applied to the speed loop and current loop. After IPMSM reaches 1400 rpm, the load torque increases from 0 N·m to 20 N·m, then decreases to 0 N·m.

Figure 6 shows the experimental results of currents without disturbance compensation. In Figure 6a,b, the top waveforms in each figure are the currents i_dn and i_qn, calculated by DSP based on the current motor currents i_d and i_q and the voltage given values u_d and u_q of the current loop through the state Equation (1) under ideal conditions. The middle waveforms in each figure are the actual i_d and i_q obtained by park transformation of the measured phase currents. The waveforms at the bottom of each figure are the current observation values ${\widehat{i}}_{\mathrm{d}}$ and ${\widehat{i}}_{\mathrm{q}}$ of the observer. i_d varies between −1.4 A and −5.7 A, and i_q varies between 34.2 A and 38.2 A when speed is 1400 rpm and torque is 20 N·m.

Due to the disturbances shown in (3), the actual currents i_d and i_q are not equal to the theoretical currents i_dn and i_qn calculated by DSP. The difference between the actual currents and the theoretical currents error_id= i_dn − i_d and error_iq = i_qn − i_q are shown in Figure 7. The absolute values of error_id and error_iq increase with speed and load increasing. The error_id varies between −3.8 A and −1.8 A, and the error_iq varies between −0.2 A and −0.9 A when speed is 1400 rpm and torque is 20 N·m.

Figure 8 shows the difference between the observed currents and the actual currents, that is, the current observation error e in (9). Although the fluctuation range of observation error increases with the increase in speed and load, it still remains within a small range. Both e_id and e_iq fluctuate between −0.2 and 0.2.

Figure 9 shows the derivative of the current observation error calculated by DSP based on the formula $\dot{\mathit{e}}(k)\approx [\mathit{e}(k)-\mathit{e}(k-1)]/{\mathrm{T}}_{\mathrm{c}}$. The derivation based on the backward Euler method will amplify the quantization errors and noise in the error e, and also bring about truncation error. These factors will affect the accuracy of disturbance observation while using the traditional terminal SMDO.

The observed lumped disturbance $\widehat{\mathit{f}}$ is shown in Figure 10. The absolute values of observed disturbance values also increase with speed and load increasing. ${\widehat{\mathit{f}}}_{\mathrm{d}}$ varies between 8.1 V and 13.4 V, and ${\widehat{\mathit{f}}}_{\mathrm{q}}$ varies between 2.2 V and 5.5 V when speed is 1400 rpm and torque is 20 N·m.

Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 show the experimental results when the observed lumped disturbance $\widehat{\mathit{f}}$ is utilized in the system.

Figure 11 shows the currents with the disturbance compensation. i_d varies between −1.8 A and −5.2 A, and i_q varies between −34.6 A and 38.1 A under 1400 rpm and 20 N·m. Compared with the current waveforms in Figure 6, the fluctuations in the motor currents i_d and i_q become smaller.

error_id and error_iq after adding the disturbance compensation are shown in Figure 12, which indicates that the absolute values of steady-state differences no longer increase with the increase in speed and load, but remain close to zero. The error_id varies between −0.9 A and 0.8 A, and the error_iq varies between −0.3 A and −0.3 A under 1400 rpm and 20 N·m.

Figure 13 shows the current observation error e. e_id and e_iq still fluctuate between −0.2 and 0.2. The introduction of the disturbance compensation does not affect the accuracy of current observation.

The observed lumped disturbance value $\widehat{\mathit{f}}$ with the disturbance compensation is shown in Figure 14. At this time, $\widehat{\mathit{f}}$ is no longer the observed values of f, but f − f_c in (7), that is, the remaining disturbances after compensation. The steady-state of $\widehat{\mathit{f}}$ also become close to zero. ${\widehat{\mathit{f}}}_{\mathrm{d}}$ varies between −2.2 V and 2.1 V, and ${\widehat{\mathit{f}}}_{\mathrm{d}}$ varies between −1.2 V and 1.2 V when speed is 1400 rpm and torque is 20 N·m.

Figure 15 shows the compensation value f_c of the lumped disturbance. The absolute value of f_c also increases with the increase in speed and load torque. f_cd varies between 9.8 V and 12.0 V, and f_cd varies between 3.1 V and 4.5 V when speed is 1400 rpm and torque is 20 N·m. Compared with Figure 10, f_c is close to the observed value $\widehat{\mathit{f}}$ when the system is not compensated.

Table 2. Comparison of system states.

System States	Peak-to-Peak Value without Compensation	Peak-to-Peak Value with Compensation	Reduced Proportion
id	4.3 A	3.4 A	20.9%
iq	4.0 A	3.5 A	12.5%
errorid	2.0 A	1.7 A	15.0%
erroriq	0.7 A	0.6 A	14.2%
${\widehat{f}}_{\mathrm{d}}$	5.3 V	4.3 V	18.9%
${\widehat{f}}_{\mathrm{q}}$	3.3 V	2.4 V	27.3%

shows the comparison of the peak-to-peak values of system states when the lumped disturbances are compensated and uncompensated under 1400 rpm and 20 N·m where the current fluctuation and control error are at the maximum. It can be seen from Table 2 that the fluctuation in the system state becomes smaller after compensation, which means that the current loop control precision is higher and the motor output torque is more stable.

5. Conclusions and Future Studies

This article proposed a sliding mode disturbance observer for IPMSM. An IPMSM model which unifies the unideal factors in the current loop into the lumped disturbance of the motor stator voltage is considered. Furthermore, a terminal SMDO based on the recursive integral sliding mode surface is used to observe the lumped disturbances of the system. The integral sliding mode surface avoids the noise sensitivity and singularity problems of the traditional terminal sliding mode surface, and the introduction of the recursive sliding mode surface enables the estimation of lumped disturbances. The stability analysis shows that the recursive integral SMDO proposed in this article can realize the estimated values of currents and lumped disturbances to converge to the real values in finite time. Compensation based on observation values of lumped disturbance can improve current control accuracy and eliminate controller current loop parameter mismatch without changing the structure of the current loop. The experimental results are presented to validate the proposed observer. With the introduction of disturbance observation and compensation, the accuracy current control is improved, the current control errors are maintained near 0, and the fluctuations in error are reduced by about 15%. At the same time, the fluctuation in current in the d-axis is reduced by 20.9%, and the fluctuation in current in the q-axis is reduced by 12.5%. The proposed terminal sliding mode SMDO only considers the voltage equation of IPMSM to observe lumped disturbance of the current loop, and considers the motor speed to be a constant rather than a variable. Future work will combine the IPMSM torque equation to estimate the disturbance of speed loop disturbance while observing the lumped disturbance of the current loop, and improve the performance of the IPMSM control system by compensating for disturbances in the current loop and speed loop.