# Output-Feedback Sliding Mode Control for Permanent Magnet Synchronous Motor Servo System Subject to Unmatched Disturbances

Authors: Wei Jiang; Lu Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper aims to investigate the speed regulation problem for permanent magnet synchronous motor (PMSM) servo systems subject to unknown load torque disturbances. The proposed method utilizes sliding mode control (SMC), invariant manifold theory, and disturbance observation technique. In the PMSM servo systems, the unknown load torques will affect the control performance to a large extent, which is unmatched. In addition, compared with full-state measurement, the output-feedback framework is easy to implement and reduces the sensor costs. However, it is difficult to handle unmatched disturbance and unmeasured states simultaneously. To this end, this paper specifically combines the sliding mode control theory with the invariant manifold theory and puts forward an output-feedback disturbance rejection control method. The key idea is that the unmatched disturbance in the PMSM servo systems is transformed into matched one by taking advantage of the invariant manifold, which is different from existing results. The transformation maintains most of dynamics of the PMSM system for control design, which improves the accuracy. In addition, an extended state observer is designed to estimate the current and lumped disturbance simultaneously; then, the output-feedback SMC method is proposed by introducing the estimations. Besides, the switching gain in the proposed sliding mode controller can change with estimation errors adaptively, and the chattering reduces. Simulation results on a PMSM system validate the effectiveness of the proposed control strategy.


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## Body

## 1. Introduction

Permanent magnet synchronous motor (PMSM) has been widely applied to various practical systems, such as robotics, aerospace, and power generations [1–4], due to the high-efficiency, high air-gap flux density, large torque-to-inertia ratio, and high power density [1]. High-accuracy tracking performance and satisfactory dynamic response are significant in the aforementioned applications. However, it should be noted that the servo control performance is significantly affected by nonlinearities, uncertainties, and disturbances in PMSM systems, and the traditional linear control strategies including the proportional-integral (PI) controller [5] are unable to provide satisfactory control performance [6]. In order to obtain better performance, many advanced nonlinear control methods have been developed for PMSM servo systems in recent years, such as adaptive control [2, 7], robust control [8, 9], linearization control [10], disturbance observer-based control [2, 11], fuzzy-logic based control [6, 12], finite time control [13, 14], fractional order control [15], sliding mode control [16, 17], and neuro-network based control [5, 12]. These control strategies improve control performance for PMSM servo systems from different aspects.In industrial applications, sliding mode control (SMC) is deemed as one of the most effective control techniques [18, 19] due to its conceptual simplicity and powerful ability to reject matched disturbances/uncertainties [20]. The SMC is able to force the system dynamics to reach a predesigned sliding manifold in a finite time [21], and then the system trajectory tracking error will converge to zero along the sliding manifold [22, 23]. It is well known that the load torque disturbance in PMSM system is unmatched one if we combine the speed control loop and current control loop together to get fast speed regulation performance [8]. In [24], a disturbance observer-based SMC strategy is developed to handle unmatched disturbances effectively. In [25], an adaptive sliding mode control method is proposed to attenuate the inductance disturbance of the PMSM. In order to estimate the unmeasured mechanical parameters of PMSM, a terminal sliding mode observer is proposed in [26], which can estimate the control performance with a finite-time convergence rate. The work in [27] employs the sliding mode control technique and the extended state observer for PMSM system to improve the robustness against load disturbance and parameter variations. These results are mostly implemented in the condition that all the system states are available. However, in PMSM servo systems, there are measurement noises in current sensors, which will bring adverse effects to the control performance. Toward that end, it is of great importance to investigate output-feedback SMC approaches for PMSM systems, which only require the interested output measurement rather than the full states.The widely used output-feedback SMC method in PMSM is the observer-based SMC, which develops a state observer [28, 29] to reconstruct all the unmeasured states and then designs a controller by utilizing the estimation values. Most of the existing observer-based SMC methods need large control gains to reject bounded equivalent input disturbances [30], which will cause serious chattering. In recent years, the output-feedback SMC based upon the extended state observer (ESO) [31] provides an intuitive solution for chattering attenuation [32]. In [33], a higher-order ESO-based SMC method is proposed to reject the unknown higher-order lumped disturbances.It is noticed that the existing ESO-based SMC method for PMSM system requires the second-order derivatives of the angular velocity to obtain a standard form for control design [33]. Under the standard ESO-based framework, the dynamics except for the integral chain are lumped for estimation. The lumped disturbances and uncertainties may change in a wide range due to the variation of operation condition. Therefore, to achieve satisfactory estimation and control performance, it is generally required to assign relatively large observer poles. For a PMSM servo system, the increase of observer poles will result in a drastic increase of observer gains, and the measurement noises will be significantly amplified in practice. Therefore, there will be serious switching action in the control signal, which will bring adverse effects to actuators and even excites the unmodelled dynamics of the system [34]. Besides, the derivatives of the reference signal may be unavailable. The ESO-based methods solve this problem by adding tracking differentiators for online estimation of the unknown information, which will considerably increase the complexity for implementation. Taking all the aforementioned facts into account, it is imperative to investigate some new output-feedback SMC strategies for PMSM system subject to unmatched disturbances to address the above problems and to achieve good disturbance rejection ability and satisfactory control performance.Actually, the output-feedback SMC for PMSM servo system subject to unknown load torque disturbances is a challenging problem, since it is difficult to estimate both the unmeasured states and the unmatched disturbance at the same time. Reference [35] gives a new idea for output-feedback SMC design, which combines SMC with the basic idea of output regulation theory [36, 37]. Inspired by [35], this paper aims to solve the output-feedback SMC problem for PMSM servo system subject to unmatched disturbances. Under the proposed controller, the angular velocity is driven to track the reference signal asymptotically. The main contributions and benefits of the proposed method are summarized as follows: (1) full dynamics of the PMSM system are thoroughly exploited in the invariant manifold based output-feedback SMC design process, which admits higher bandwidth without higher observer gains, attenuating measurement noises to a large extent; (2) the switching gain changes with estimation errors adaptively, and thus the chattering will be reduced; (3) the proposed output-feedback SMC method can compensate the influences caused by unknown derivatives of the desired reference signal without resorting to tracking differentiators, which results in a simpler control structure and saves the implementation burden of the algorithm. To further demonstrate the effectiveness of the output-feedback SMC method proposed in this paper, simulations on a PMSM system are carried out by utilizing MATLAB/Simulink.The paper is organized as follows. Section2 shows the mathematical model of the PMSM system and formulates the problem. In Section 3, the output-feedback sliding mode controller is developed and stability analysis is shown in detail. Then, Section 4 gives the simulation results and the conclusion is shown in Section 5.


## 2. System Model and Problem Formulation

### 2.1. Mathematical Model of PMSM

The magnetic circuit is assumed to be unsaturated, the eddy current loss and hysteresis are ignored, and the distribution of the magnetic field is supposed to be in sine space. A three-phase surface-mounted PMSM system can be modeled ind−q coordinate, and the mathematical model is given by [2](1)diddt=−RLid+npωiq+1Lud,diqdt=−npωid−RLiq−1Lnpωψf+1Luq,dωdt=3npψf2Jiq−BJω−TLJ,where iq and id are the q−axis and d−axis stator currents, respectively; uq and ud are the q−axis and d−axis stator voltages, respectively; ψf is the rotor flux linkage; np is the number of pole pairs; J is the rotor inertia; L is the stator inductance; R is the stator resistance; B is the viscous friction coefficient; ω is the angular velocity; and TL is the unknown load torque.The field-oriented control is one of the widely used control frameworks in PMSM, and Figure1 shows the schematic diagram. Under this framework, the flux-producing and torque-producing components of the stator current are decoupled such that flux controls and the independent torque are possible as those in DC motors. In PMSM, the controllers take advantage of the cascade control loop, which includes two current loops and a speed loop. In the two current loops, two PI controllers are used. Here, we pay our attention to developing a controller for speed loop to reject the unknown load torque disturbance.Figure 1
Schematic diagram of field-oriented control for PMSM.In a PMSM servo system, it is always expected to obtain maximum torque; thus, there should beid=0. Therefore, the mathematical model (1) can be decoupled as the following form:(2)diqdt=−RLiq−1Lnpωψf+1Luq,dωdt=3npψf2Jiq−BJω−TLJ,where uq is the control signal and TL is the unknown unmatched disturbance. This paper mainly explores the control strategy for the decoupled system given by (2).


#### 2.1.1. Control Objective

This paper attempts to utilize the sliding mode control theory to address speed regulation problem for PMSM servo systems subject to unmatched disturbances. For system (2), an output-feedback sliding mode controller combining invariant manifold with disturbance observer is designed to drive the angular velocity ω to track the reference signal ω∗ asymptotically.


### 2.2. Traditional Output-Feedback SMC Design

Define the system output tracking error asy=ω−ω∗. Taking the second-order derivative of y, one obtains [33](3)y˙=ω˙−ω˙∗=3npψf2Jiq−BJω−TLJ−ω˙∗,y¨=3npψf2Ji˙q−BJω˙−T˙LJ−ω¨∗=−3npψfR2JL+3npψfB2J2iq−3np2ψf22JL−B2J2ω+BTLJ−T˙LJ+3npψf2JLuq.DenotingD=−3npψfR/2JL+3npψfB/2J2iq−3np2ψf2/2JL−B2/J2ω+BTL/J−T˙L/J−ω¨∗ as the lumped disturbances and κ=3npψf/2JL as the control coefficient, one has the following input-output formulation:(4)y¨=D+κuq.Denoteν1=y, ν2=y˙, and ν3=D. Then we have the following controllable state-space model:(5)ν˙1ν˙2ν˙3=010001000ν1ν2ν3+0κ0+00ϖ.It is assumed that the lumped disturbanceD and its derivative ϖ are bounded and satisfy limt⟶∞ϖ=0. On the basis of (5), the observer is constructed as(6)ν^˙1=ν^2+l1ν1−ν^1,ν^˙2=ν^3+l2ν1−ν^1+κuq,ν^˙3=l3ν1−ν^1,where l1, l2, and l3 are observer gains. Introducing the estimations, the sliding manifold is developed as(7)s=ν^2+βν^1,where β>0. The output-feedback controller is(8)uq=−1κν^3+βν^2+k1sgns+k2s,where k1=ρ+βl1+l2ν1−ν^1, ρ>0, k2>0.


### 2.3. Motivations

For the PMSM system (2) subject to unmatched disturbance, the effects caused by unmatched disturbance can be removed by the traditional output-feedback sliding mode controller effectively. Nevertheless, it is seen from coordinate transformation given by (3) and (4) that all unmeasured states are lumped into D, which includes q-axis current and load torques. In order to obtain satisfactory performance, we have to choose large observer poles; thus, large observer gains are required. As a result, measurement noises are amplified, and high-frequency fluctuations will be brought in control signal, which may excite the unmodelled dynamics of PMSM system, cause adverse effects on actuator, and even destabilize the system.


## 2.1. Mathematical Model of PMSM

The magnetic circuit is assumed to be unsaturated, the eddy current loss and hysteresis are ignored, and the distribution of the magnetic field is supposed to be in sine space. A three-phase surface-mounted PMSM system can be modeled ind−q coordinate, and the mathematical model is given by [2](1)diddt=−RLid+npωiq+1Lud,diqdt=−npωid−RLiq−1Lnpωψf+1Luq,dωdt=3npψf2Jiq−BJω−TLJ,where iq and id are the q−axis and d−axis stator currents, respectively; uq and ud are the q−axis and d−axis stator voltages, respectively; ψf is the rotor flux linkage; np is the number of pole pairs; J is the rotor inertia; L is the stator inductance; R is the stator resistance; B is the viscous friction coefficient; ω is the angular velocity; and TL is the unknown load torque.The field-oriented control is one of the widely used control frameworks in PMSM, and Figure1 shows the schematic diagram. Under this framework, the flux-producing and torque-producing components of the stator current are decoupled such that flux controls and the independent torque are possible as those in DC motors. In PMSM, the controllers take advantage of the cascade control loop, which includes two current loops and a speed loop. In the two current loops, two PI controllers are used. Here, we pay our attention to developing a controller for speed loop to reject the unknown load torque disturbance.Figure 1
Schematic diagram of field-oriented control for PMSM.In a PMSM servo system, it is always expected to obtain maximum torque; thus, there should beid=0. Therefore, the mathematical model (1) can be decoupled as the following form:(2)diqdt=−RLiq−1Lnpωψf+1Luq,dωdt=3npψf2Jiq−BJω−TLJ,where uq is the control signal and TL is the unknown unmatched disturbance. This paper mainly explores the control strategy for the decoupled system given by (2).


### 2.1.1. Control Objective

This paper attempts to utilize the sliding mode control theory to address speed regulation problem for PMSM servo systems subject to unmatched disturbances. For system (2), an output-feedback sliding mode controller combining invariant manifold with disturbance observer is designed to drive the angular velocity ω to track the reference signal ω∗ asymptotically.


## 2.1.1. Control Objective

This paper attempts to utilize the sliding mode control theory to address speed regulation problem for PMSM servo systems subject to unmatched disturbances. For system (2), an output-feedback sliding mode controller combining invariant manifold with disturbance observer is designed to drive the angular velocity ω to track the reference signal ω∗ asymptotically.


## 2.2. Traditional Output-Feedback SMC Design

Define the system output tracking error asy=ω−ω∗. Taking the second-order derivative of y, one obtains [33](3)y˙=ω˙−ω˙∗=3npψf2Jiq−BJω−TLJ−ω˙∗,y¨=3npψf2Ji˙q−BJω˙−T˙LJ−ω¨∗=−3npψfR2JL+3npψfB2J2iq−3np2ψf22JL−B2J2ω+BTLJ−T˙LJ+3npψf2JLuq.DenotingD=−3npψfR/2JL+3npψfB/2J2iq−3np2ψf2/2JL−B2/J2ω+BTL/J−T˙L/J−ω¨∗ as the lumped disturbances and κ=3npψf/2JL as the control coefficient, one has the following input-output formulation:(4)y¨=D+κuq.Denoteν1=y, ν2=y˙, and ν3=D. Then we have the following controllable state-space model:(5)ν˙1ν˙2ν˙3=010001000ν1ν2ν3+0κ0+00ϖ.It is assumed that the lumped disturbanceD and its derivative ϖ are bounded and satisfy limt⟶∞ϖ=0. On the basis of (5), the observer is constructed as(6)ν^˙1=ν^2+l1ν1−ν^1,ν^˙2=ν^3+l2ν1−ν^1+κuq,ν^˙3=l3ν1−ν^1,where l1, l2, and l3 are observer gains. Introducing the estimations, the sliding manifold is developed as(7)s=ν^2+βν^1,where β>0. The output-feedback controller is(8)uq=−1κν^3+βν^2+k1sgns+k2s,where k1=ρ+βl1+l2ν1−ν^1, ρ>0, k2>0.


## 2.3. Motivations

For the PMSM system (2) subject to unmatched disturbance, the effects caused by unmatched disturbance can be removed by the traditional output-feedback sliding mode controller effectively. Nevertheless, it is seen from coordinate transformation given by (3) and (4) that all unmeasured states are lumped into D, which includes q-axis current and load torques. In order to obtain satisfactory performance, we have to choose large observer poles; thus, large observer gains are required. As a result, measurement noises are amplified, and high-frequency fluctuations will be brought in control signal, which may excite the unmodelled dynamics of PMSM system, cause adverse effects on actuator, and even destabilize the system.


## 3. Main Results

### 3.1. Output-Feedback Sliding Mode Controller Design

Motivated by the basic idea of output regulation theory [36, 38], the steady states of the PMSM system (2) are calculated from regulator equations, which are shown as follows:(9)dπiqdt=−RLπiq−1Lnpψfπω+1Lπuq,dπωdt=3npψf2Jπiq−BJπω−TLJ,0=πω−ω∗,where πiq, πω, and πuq represent the steady states of the PMSM system states iq, ω, and the control input uq, respectively. It is obvious that the solutions of (9) are impossible to be obtained due to the presence of unknown load torque disturbance TL. This section will take advantage of the observation technique and sliding mode control to develop an output-feedback controller to solve this problem, which will only need the computable πω=ω∗.Denoteε1=iq−πiq and ε2=ω−πω. Combining the PMSM system model (2) with the regulator equations given in (9), the invariant manifold is introduced as E¯=ε1,ε2|ε1,ε2T=0,0T. On the basis of the invariant manifold, the PMSM system (2) is transformed into(10)ε.1=−RLε1−1Lnpψfε2+1Luq−πuq,ε.2=3npψf2Jε1−BJε2.From (10), it is seen that the unmatched disturbance in (2) is lumped into the matched disturbance πuq, which needs to be compensated.Denoteε3=πuq as an extended state. Then an augmented system is(11)ε.1=−RLε1−1Lnpψfε2+1Luq−1Lε3,ε.2=3npψf2Jε1−BJε2,ε.3=ς,e=ε2,where ς=π˙uq.Assumption 1.
The lumped disturbanceπuq in system (10) and its derivative π˙uq are assumed to be bounded and satisfy the condition that limt⟶∞π˙uq=0.
An extended state observer for augmented system (11) is designed as(12)ε^.1=−RLε^1−1Lnpψfε^2+1Luq−1Lε^3+l1e−e^,ε^.2=3npψf2Jε^1−BJε^2+l2e−e^,ε^.3=l3e−e^,e^=ε^2,where l1,l2,l3 are observer gains to be determined and ε^1,ε^2,ε^3 are estimations of ε1,ε2,ε3 in (11). Introducing the estimations, the sliding manifold is designed as(13)s=ε^1+βε^2,where β>0 is the parameter to be determined. The output-feedback sliding mode controller is constructed as(14)uq=LRL−3npψfβ2Jε^1+npψfL+BβJε^2+1Lε^3−k1sgns−k2s,where k1=ρ+l1+βl2ε2−ε^2, k2>0, and ρ>0.


### 3.2. Stability Analysis

To begin with, we give a lemma as follows, which will play an essential role in analysis.Lemma 1 (see [39]).
If the matrixA∈ℝn×n is Hurwitz, then there is a scalar c>0 such that eAt≤ceλmaxA/2t, where λmaxA=maxReλiA.Taking the time derivative of the designed sliding manifolds along (13) and substituting the extended state observer (12) and the output-feedback controller (14), we have(15)s˙=ε^.1+βε^.2=−RLε^1−1Lnpψfε^2−1Lε^3+1Luq+l1e−e^+β3npψf2Jε^1−BJε^2+l2e−e^=−k1sgns−k2s+βl2+l1e−e^.Define the candidate Lyapunov function asV=1/2s2. Taking the derivative of V and considering (15), one obtains(16)V˙=ss˙≤−k1s−k2s2+sβl2+l1e−e^<−ρs−k2s2=−2ρV1/2−2k2V.As a result, the system trajectories will reach the sliding manifold in a finite timeTr with [18](17)Tr≤1k2ln1+2k2ρV1/2s0.Denote the estimation errors ase˜1=ε1−ε^1 and e˜2=ε2−ε^2. On sliding motion, there is(18)s=ε^1+βε^2=0⟹ε^1=−βε^2.Then, we have(19)ε1=e˜1+ε^1=e˜1−βε^2=e˜1−βε2−e˜2.Defineξ=ε2,e˜1,e˜2,e˜3T. On sliding motion, the reduced-order closed-loop system can be written in a compact form, which is shown as(20)ξ˙=Θξ+Ψς,where Ψ=0,0,0,1T, and(21)Θ=−BJ+3npψfβ2J3npψf2J3npψfβ2J00−RL−npφfL+l1−1L03npψfβ2J−BJ+l2000−l30.The main theorem of this paper is summarized as follows.Theorem 1.
Under Assumption1, if the observer gains l1, l2, and l3 in (12) are selected such that the matrix,(22)M=−RL−npφfL+l1−1L3npψf2J−BJ+l200−l30,is Hurwitz and the controller parameter β in (13) and (14) is chosen such that −B/J+3npψfβ/2J<0, then the estimation errors and the output tracking error of the PMSM system (2) will converge to a bounded neighbourhood of the origin, and the ultimate bound can be made arbitrarily small. Furthermore, if π˙uq tends to zero as t⟶∞, then the closed-loop system (20) is globally asymptotically stable.For better understanding of the main design idea of the proposed control strategy, the detailed proof for Theorem1 is given in the Appendix. Figure 2 shows the schematic diagram of the proposed output-feedback SMC strategy in implementation.Figure 2
Schematic diagram of the proposed output-feedback SMC strategy for PMSM system.Remark 1.
In parameter tuning, there are usually a number of conflicts/constraints, such as tracking versus disturbance rejection and nominal performance versus robustness. There are three parametersl1, l2, and l3 to be determined in the extended state observer given in (12). For tuning the observer gains, we usually choose the observer poles ωo and calculated the values of l1, l2, and l3 based on the observer expressions (12). If the observer poles are larger, the settling time will be shorter, but the overshoot will be larger, and the measurement noise will be amplified more seriously. In the controller given in (13) and (14), there are three parameters to be assigned, which are β, k1, and k2. The detailed expression of k1 is given by k1=ρ+l1+βl2ε2−ε^2, and ρ should be positive. The larger ρ means that system states will reach the sliding manifold in a shorter time, but the chattering will be more seriously. The parameters of β and k2 should be positive. Large values of control gains will reduce the settling time, but the overshoot and the control efforts will be large. Generally, when choosing the parameters, one needs to balance the overshoot, the settling time, and the control efforts.


## 3.1. Output-Feedback Sliding Mode Controller Design

Motivated by the basic idea of output regulation theory [36, 38], the steady states of the PMSM system (2) are calculated from regulator equations, which are shown as follows:(9)dπiqdt=−RLπiq−1Lnpψfπω+1Lπuq,dπωdt=3npψf2Jπiq−BJπω−TLJ,0=πω−ω∗,where πiq, πω, and πuq represent the steady states of the PMSM system states iq, ω, and the control input uq, respectively. It is obvious that the solutions of (9) are impossible to be obtained due to the presence of unknown load torque disturbance TL. This section will take advantage of the observation technique and sliding mode control to develop an output-feedback controller to solve this problem, which will only need the computable πω=ω∗.Denoteε1=iq−πiq and ε2=ω−πω. Combining the PMSM system model (2) with the regulator equations given in (9), the invariant manifold is introduced as E¯=ε1,ε2|ε1,ε2T=0,0T. On the basis of the invariant manifold, the PMSM system (2) is transformed into(10)ε.1=−RLε1−1Lnpψfε2+1Luq−πuq,ε.2=3npψf2Jε1−BJε2.From (10), it is seen that the unmatched disturbance in (2) is lumped into the matched disturbance πuq, which needs to be compensated.Denoteε3=πuq as an extended state. Then an augmented system is(11)ε.1=−RLε1−1Lnpψfε2+1Luq−1Lε3,ε.2=3npψf2Jε1−BJε2,ε.3=ς,e=ε2,where ς=π˙uq.Assumption 1.
The lumped disturbanceπuq in system (10) and its derivative π˙uq are assumed to be bounded and satisfy the condition that limt⟶∞π˙uq=0.
An extended state observer for augmented system (11) is designed as(12)ε^.1=−RLε^1−1Lnpψfε^2+1Luq−1Lε^3+l1e−e^,ε^.2=3npψf2Jε^1−BJε^2+l2e−e^,ε^.3=l3e−e^,e^=ε^2,where l1,l2,l3 are observer gains to be determined and ε^1,ε^2,ε^3 are estimations of ε1,ε2,ε3 in (11). Introducing the estimations, the sliding manifold is designed as(13)s=ε^1+βε^2,where β>0 is the parameter to be determined. The output-feedback sliding mode controller is constructed as(14)uq=LRL−3npψfβ2Jε^1+npψfL+BβJε^2+1Lε^3−k1sgns−k2s,where k1=ρ+l1+βl2ε2−ε^2, k2>0, and ρ>0.


## 3.2. Stability Analysis

To begin with, we give a lemma as follows, which will play an essential role in analysis.Lemma 1 (see [39]).
If the matrixA∈ℝn×n is Hurwitz, then there is a scalar c>0 such that eAt≤ceλmaxA/2t, where λmaxA=maxReλiA.Taking the time derivative of the designed sliding manifolds along (13) and substituting the extended state observer (12) and the output-feedback controller (14), we have(15)s˙=ε^.1+βε^.2=−RLε^1−1Lnpψfε^2−1Lε^3+1Luq+l1e−e^+β3npψf2Jε^1−BJε^2+l2e−e^=−k1sgns−k2s+βl2+l1e−e^.Define the candidate Lyapunov function asV=1/2s2. Taking the derivative of V and considering (15), one obtains(16)V˙=ss˙≤−k1s−k2s2+sβl2+l1e−e^<−ρs−k2s2=−2ρV1/2−2k2V.As a result, the system trajectories will reach the sliding manifold in a finite timeTr with [18](17)Tr≤1k2ln1+2k2ρV1/2s0.Denote the estimation errors ase˜1=ε1−ε^1 and e˜2=ε2−ε^2. On sliding motion, there is(18)s=ε^1+βε^2=0⟹ε^1=−βε^2.Then, we have(19)ε1=e˜1+ε^1=e˜1−βε^2=e˜1−βε2−e˜2.Defineξ=ε2,e˜1,e˜2,e˜3T. On sliding motion, the reduced-order closed-loop system can be written in a compact form, which is shown as(20)ξ˙=Θξ+Ψς,where Ψ=0,0,0,1T, and(21)Θ=−BJ+3npψfβ2J3npψf2J3npψfβ2J00−RL−npφfL+l1−1L03npψfβ2J−BJ+l2000−l30.The main theorem of this paper is summarized as follows.Theorem 1.
Under Assumption1, if the observer gains l1, l2, and l3 in (12) are selected such that the matrix,(22)M=−RL−npφfL+l1−1L3npψf2J−BJ+l200−l30,is Hurwitz and the controller parameter β in (13) and (14) is chosen such that −B/J+3npψfβ/2J<0, then the estimation errors and the output tracking error of the PMSM system (2) will converge to a bounded neighbourhood of the origin, and the ultimate bound can be made arbitrarily small. Furthermore, if π˙uq tends to zero as t⟶∞, then the closed-loop system (20) is globally asymptotically stable.For better understanding of the main design idea of the proposed control strategy, the detailed proof for Theorem1 is given in the Appendix. Figure 2 shows the schematic diagram of the proposed output-feedback SMC strategy in implementation.Figure 2
Schematic diagram of the proposed output-feedback SMC strategy for PMSM system.Remark 1.
In parameter tuning, there are usually a number of conflicts/constraints, such as tracking versus disturbance rejection and nominal performance versus robustness. There are three parametersl1, l2, and l3 to be determined in the extended state observer given in (12). For tuning the observer gains, we usually choose the observer poles ωo and calculated the values of l1, l2, and l3 based on the observer expressions (12). If the observer poles are larger, the settling time will be shorter, but the overshoot will be larger, and the measurement noise will be amplified more seriously. In the controller given in (13) and (14), there are three parameters to be assigned, which are β, k1, and k2. The detailed expression of k1 is given by k1=ρ+l1+βl2ε2−ε^2, and ρ should be positive. The larger ρ means that system states will reach the sliding manifold in a shorter time, but the chattering will be more seriously. The parameters of β and k2 should be positive. Large values of control gains will reduce the settling time, but the overshoot and the control efforts will be large. Generally, when choosing the parameters, one needs to balance the overshoot, the settling time, and the control efforts.


## 4. Simulations

Simulations, using MATLAB/Simulink, are carried out to validate the control performance of the proposed output-feedback sliding mode control method, especially in rejecting unmatched disturbances. Table1 lists the nominal values of the PMSM parameters.Table 1
Parameters of PMSM system in simulations.
MeaningsParametersNominal valuePole pairsnp4Rotor flux linkageφf0.402wbStator inductanceL4mHRotor inertiaJ1.78×10−4kg⋅m2Viscous friction coefficientB7.4×10−5N⋅m⋅s/radStator resistanceR1.74ΩIn simulations, to demonstrate the superiority of the proposed output-feedback sliding mode controller, the traditional output-feedback SMC given by (6)–(8) is compared as benchmark method. The observer poles of the proposed method and the traditional method are chosen as ωo=−800 and ωo=−1000, respectively (the reason why we choose different poles for the two observers will be discussed in the following Case 1). The parameters of the two controllers are shown in Table 2.Table 2
Control parameters of two methods.
MethodsController parametersObserver parametersSymbolsValuesSymbolsValuesProposedβ1l17116.5Output-feedbackρ1l21964.6SMCk25l3−151.1376Traditionalβ80l13000Output-feedbackρ1l23×106SMCk25l31×109In this section, we will present three simulation scenarios, which are listed in Table3.Table 3
Simulation scenarios.
CasesScenariosDetailsIConstant referenceω∗=150rad/sSpeed trackingIISudden loadω∗=150rad/sTorque disturbanceTL=0⟶1N⋅mIIITime-varying loadω∗=150rad/sTorque disturbanceTL=0⟶2+sin2πtN⋅mCase 1.
Constant Reference Speed Tracking. In this case, the reference angular velocity is set as ω∗=150rad/s. The response curves of the angular velocity ω, the q-axis current iq, and the q-axis stator voltage uq are shown in Figure 3. It is observed that larger observer poles are needed in the traditional output-feedback SMC method to achieve similar control performance with the output-feedback sliding mode controller proposed in this paper. This is the main reason why we choose different poles for the two observers. Moreover, the changes of the switching gains in the two controllers are shown in Figure 4, from which it can be seen that the switching gain k1 in the proposed controller is much smaller than that in the traditional one during the transient period.Figure 3
Response curve of the proposed output-feedback SMC and traditional SMC (Case1): (a) angular velocity ω; (b) q-axis current iq; (c) control input uq.
(a)(b)(c)Figure 4
Switching gain of the two controllers: (a) proposed output-feedback SMC; (b) traditional output-feedback SMC.
(a)(b)Case 2.
Sudden Load Torque Disturbance. When unknown load torques are suddenly imposed, the simulation results are shown by Figure 5. It can be observed from Figure 5(a) that the angular velocity fluctuations of the PMSM are much smaller under the proposed controller than in the traditional method, although the observer poles of the traditional method are larger than those of the proposed one.Figure 5
Response curve of the proposed output-feedback SMC and traditional SMC in the presence step load torque disturbance (Case2): (a) angular velocity ω; (b) q-axis current iq; (c) control input uq.
(a)(b)(c)Case 3.
Sinusoidal Load Torque Disturbance. Different from Case 3, where constant load torque disturbance is considered; in this case, we aim to evaluate the influences of period time-varying external disturbances on the steady-state performance. Response curves of the angular velocity, the q-axis current, and the q-axis voltage are shown in Figure 6. It can be seen from Figure 6 that the proposed control method shows better disturbance rejection ability than the traditional output-feedback SMC method.Figure 6
Response curve of the proposed output-feedback SMC and traditional SMC in the presence sinusoidal load torque disturbance (Case3): (a) angular velocity ω; (b) q-axis current iq; (c) control input uq.
(a)(b)(c)


## 5. Conclusion

This paper has investigated the speed regulation problem for PMSM system subject to unmatched disturbance. Taking advantage of the invariant manifold, the unmatched disturbance has been transformed into matched one and an augmented system has been obtained. An extended state observer has been developed for the augmented system to reconstruct unmeasured states. An output-feedback sliding mode controller, based on invariant manifold, has been proposed to achieve asymptotic tracking. Under the proposed control method, the angular velocity of the PMSM system can track the desired signal asymptotically even in the presence of the unmatched disturbance. Simulations have been carried out to validate the superiority of the controller proposed in this paper.



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