Speed control for permanent magnet synchronous motor based on an improved extended state observer

Abstract

An improved extended state observer is designed to eliminate the influences of speed control for a permanent magnet synchronous motor. The improved extended state observer is designed based on a new nonlinear function. This function exhibits better continuity and derivability than previously available functions and can effectively reduce the high-frequency flutter phenomenon. The nonlinear dynamics, model uncertainty, and external disturbances of the permanent magnet synchronous motor speed control system are extended to a new state. The improved extended state observer is utilized to observe this state. The overtime variation of the permanent magnet synchronous motor speed control system can be predicted and compensated in real time by the improved extended state observer. Therefore, the improved extended state observer, which is designed based on the new nonlinear function, can eliminate the disturbances on the permanent magnet synchronous motor speed control system. Finally, simulation experiments are performed and results show that the permanent magnet synchronous motor speed control system with improved extended state observer exhibits better performances.

Keywords

Improved extended state observer permanent magnet synchronous motor speed control nonlinear function

Introduction

Speed control of the permanent magnet synchronous motor (PMSM) is the core of the PMSM control system. Controlling the speed of PMSM is conducted under complicated industrial situations. Therefore, small interferences will result in serious errors. Thus, the control system needs a control strategy with stronger anti-interference ability. The control strategy based on an improved extended state observer (ESO) does not depend on the precise model of a control object, and the disturbances of the system can be eliminated in real time by the improved ESO. Moreover, the control quantity can be compensated for promptly. Therefore, this control strategy has a stronger anti-interference ability. The control strategy based on the improved ESO, which was proposed by Han,¹ was used in this study.

Extensive research on the control strategy based on ESO for PMSM speed control has been conducted locally and internationally, and the research has demonstrated several achievements. For example, Hebertt et al.² presented an ESO for the angular velocity trajectory tracking task on a substantially perturbed PMSM and disturbances, such as time variations, the presence of unknown, load-torque inputs, and unknown system parameters were observed. Du et al.³ applied a linear active disturbance rejection controller based on ESO for sensorless control of internal PMSM. The high performance of position estimation was achieved using line ESOs. Compared to a conventional sliding-model observer, the phase delay and speed chattering were obviously reduced. A sensorless estimator based on the technique of the ESO is constructed to create rapid and accurate speed identification.⁴ Liu and Li⁵ developed an improved speed control method, called the predictive functional control (PFC)+ESO method, to optimize the control performance of the PMSM. In the improved speed control method, ESO is introduced to estimate lumped disturbances and add a feed-forward compensation item based on the estimated disturbances to the controller. A composite control method, that is, the extended-state-observer-based control method, is employed to ensure the performance of the PMSM closed-loop system. The ESO can estimate both the states and the disturbances simultaneously, so the composite speed controller can use a corresponding part to compensate for the disturbances.⁶ A sliding mode and active disturbance rejection control based on ESO was employed to regulate the speed of a PMSM. This proposed control scheme ensures speed control accuracy and improves the robustness and anti-load disturbance ability of the system.⁷ However, noise of the speed control for PMSM was not eliminated sufficiently when the differentiation was estimated. Some ESO performances, such as the frequency-domain performance, the measurement noise restrain performance, and the absolute stability, were studied and analyzed.^8–14 However, the parameters of the studied ESO were extensive to set; they did not provide specific setting rules. Furthermore, ESO is widely used in many industrial occasions. For instance, Kang et al.¹⁵ proposed a robust tracking controller for a wheeled mobile robot with unknown skidding and slipping, and the controller was designed based on a generalized ESO. Mokhtari et al.¹⁶ used an ESO to estimate the state and the unknown aerodynamic disturbances of a coaxial-rotor unmanned aerial vehicle. Ma et al.¹⁷ proposed an ESO according to the slow convergence and control input chattering phenomenon in traditional nonsingular terminal sliding mode control. Cai et al.¹⁸ designed an ESO based on the back-stepping theory to enhance the control ability of rotor current and the fault ride-through capacity of a doubly-fed induction generator. Guo and colleagues^19,20 designed a fuzzy active disturbance rejection controller based on ESO and improved the performances of a system. A weight-transducerless rollback mitigation strategy, which adopts an enhanced model predictive control with ESO for speed-loop, was proposed to attenuate the car sliding of the gearless elevator installed with an ordinary resolution encoder.²¹ Yang et al.²² designed an ESO control scheme and used it on the trajectory tracking. An active disturbance rejection controller with an adaptive ESO was adopted in air–fuel ratio control in gasoline engines.²³ A high-order ESO-enhanced controller was proposed to address the problem of velocity and altitude tracking for a hypersonic flight vehicle with strong nonlinearity, parameter uncertainty, and external disturbance.²⁴ Sun et al.²⁵ used an ESO applied to the fast terminal sliding mode control for swing nozzle of anti-aircraft missile. Luis et al.²⁶ described an adaptive control design based on ESO to solve the trajectory tracking problem of a Delta robot with uncertain dynamical model. Zhu et al.²⁷ used ESO to estimate the external disturbances and uncertain items of the model of airship horizontal model, and the dynamic feedback was compensated in real time. An adaptive ESO-based active disturbance rejection control was proposed to address the uncertainties, both in the plant and in the sensors.²³ However, the ESO used in the aforementioned works was traditional one and was designed based on a nonlinear function fal. The traditional ESO cannot effectively eliminate the high-frequency flutter of a system because the nonlinear function is ideal. The function is continuous but non-derivable. Therefore, designing an improved ESO based on a new nonlinear function is crucial to eliminate the high-frequency flutter of the system.

This study proposes an improved ESO based on a new nonlinear function, and a control strategy based on the improved ESO is designed. The control strategy is applied to the speed control system of the PMSM. Simulations are conducted in the PMSM control system for the cases that adopts proportional–integral (PI), traditional ESO, and the improved ESO control strategy. Finally, the simulation results are discussed.

Control model

The schematic diagram of the mathematical model of PMSM is shown in Figure 1.

Figure 1.

Mathematical model schematic diagram of PMSM.

In the model, $U_{q}$ is the armature voltage, the motor input, R is the armature resistance of motor, L is the armature inductance, $K_{t}$ is the torque sensitivity, $K_{e} = p_{n} \cdot φ_{f}$ , $p_{n}$ is the pole logarithmic of motor, $φ_{f}$ is the flux linkage of motor, J is the rotary inertia that is equivalent to the motor shaft by the rotary inertia of the gear and load of the lateral swing, $T_{d}$ is the load torque, and $ω$ is the angular speed output of motor.

The transfer function between $ω$ and $U_{q}$ , $T_{d}$ , can be calculated by Figure 1, and shown as

ω = G_{1} U_{q} + G_{2} T_{d}

(1)

The following equation can be obtained by equation (1)

\frac{ω}{U_{q}} = G_{1} + \frac{G_{2} T_{d}}{U_{q}}

(2)

where $G_{1} = \frac{K_{t}}{(Ls + R) Js + K_{t} K_{e}} and G_{2} = \frac{Ls + R}{(Ls + R) Js + K_{t} K_{e}}$ .

Several definitions are given as $\frac{RJ}{LJ} = \frac{R}{L} = 2 ξ ω_{n}$ , $\frac{K_{t} K_{e}}{LJ} = ω_{n}^{2}$ , and $\frac{K_{t}}{LJ} = b$ .

Equation (3) can be obtained according to the expression of $G_{1}$

\frac{ω}{U_{q}} = \frac{b}{s^{2} + 2 ξ ω_{n} s + ω_{n}^{2}} + \frac{G_{2} T_{d}}{U_{q}}

(3)

Equation (4) can be obtained by equation (3)

\overset{\cdot\cdot}{ω} = - 2 ξ ω_{n} \dot{ω} - ω_{n}^{2} ω + b U_{q} + f

(4)

where $f = G_{2} \cdot T_{d} \cdot (s^{2} + 2 ξ ω_{n} s + ω_{n}^{2}) = \frac{L {\overset{\cdot}{T}}_{d} + R T_{d}}{LJ}$ .

Design of improved ESO for PMSM

Improved ESO design

The design of the improved ESO for the PMSM speed control can be performed by equation (4).

Assuming that $y = x_{1}$ , $x_{1} = ω$ and $x_{2} = \overset{\cdot}{ω}$ , the state space of equation (4) can be expressed as follows

{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - 2 ξ ω_{n} x_{2} - ω_{n}^{2} x_{1} + b U_{q} + Δ b U_{q} + f \\ y = x_{1} \end{matrix}

(5)

where f is regarded as the disturbance of the system, and the definition $w = - 2 ξ ω_{n} x_{2} - ω_{n}^{2} x_{1} + Δ b U_{q} + f$ is given. w is treated as the total disturbance of the system and is set to a new state variable $x_{3}$ . The state space can be expressed as follows

{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = b U_{q} + w \\ y = x_{1} \end{matrix}

(6)

The improved ESO of the PMSM speed control system can be designed as follows

{\begin{matrix} e_{1} = z_{1} - y (t) \\ {\dot{z}}_{1} = z_{2} - β_{1} n e w f a l (e_{1}, α_{1}, δ, γ) \\ {\dot{z}}_{2} = z_{3} - β_{2} n e w f a l (e_{1}, α_{2}, δ, γ) + b_{0} U_{q} \\ {\dot{z}}_{3} = - β_{3} n e w f a l (e_{1}, α_{3}, δ, γ) \end{matrix}

(7)

where $α_{1}, α_{2}, and α_{3}$ are the nonlinear factors, which are set to $α_{1} = 0.5, α_{2} = 0.25, and α_{3} = 0.125$ , respectively. $δ$ is a filtering factor with a value of $δ = 0.01$ . $β_{1}, β_{2}, and β_{3}$ are the gain coefficients of the improved ESO. $b_{0}$ is the estimation of the compensation factor. If the values of $β_{1}, β_{2}, and β_{3}$ are selected appropriately, then the following tracking can be realized: $Z_{1} \to x_{1}, Z_{2} \to x_{2}, Z_{3} \to x_{3}$ . Thus, the improved ESO of the PMSM speed control system can be obtained.

A proportion differentiation (PD) controller and a compensation device of the total disturbance is designed as follows

u = k_{p} (v - Z_{1}) - k_{d} Z_{2} - \frac{1}{b_{0}} Z_{3}

(8)

where v is the given signal. $k_{p} and k_{d}$ are the gains of the PD controller. The values of $k_{p} and k_{d}$ in this article are two constants. The structure chart of the controller based on the improved ESO for the PMSM speed control system can be designed according to equations (6)–(8) and shown in Figure 2.

Figure 2.

Structure chart of controller based on improved ESO for PMSM speed control system.

Calculation of nonlinear function for improved ESO

The calculation of the nonlinear function is the core of the improved ESO design. The nonlinear function was first calculated as a $fan (e, α)$ function. The expression of the $fan (e, α)$ function is shown as follows

f a n (e, α) = {| e |}^{α} s i g n (e)

(9)

The expression of the $fan (e, α)$ function indicates that the derivation of the $fan (e, α)$ function on the origin of the coordinates is calculated to be infinity. Thus, the $fan (e, α)$ function can cause the high-frequency flutter phenomenon around the origin of the coordinates. Therefore, the nonlinear function is improved to $fal (e, α, δ)$ function. The expression of the $fal (e, α, δ)$ function is shown as follows

f a l (e, α, δ) = {\begin{matrix} {| e |}^{α} s i g n (e), | e | > δ \\ e / δ^{1 - α}, | e | \leq δ \end{matrix}, δ > 0

(10)

The $fal (e, α, δ)$ function is a continuous and non-derivable function. If the value of $δ$ is significantly small, the $fal (e, α, δ)$ function can cause high-frequency flutter phenomenon. Adjusting the value of $δ$ is difficult because the control performance of the $fal (e, α, δ)$ function is sensitive to the value of $δ$ . Therefore, the nonlinear function is improved to $newfal (e, α, δ, γ)$ function. The requirements of nonlinear function $newfal (e, α, δ, γ)$ are (1) odd function, (2) continuously differentiable, (3) monotonically increasing from negative to positive infinity, (4) value of $newfal' (e, α, δ, γ)$ is not constant 0, (5) the neighborhood of e (e is very small) can be approximated as a linear function, that is, $newfal (e, α, δ, γ) = e$ , and (6) the function value is close to saturation when the absolute value of e is very large. The $newfal (e, α, δ, γ)$ function is expressed as follows

n e w f a l (e, α, δ, γ) = {\begin{matrix} (α - 1) δ^{α - 3} e^{3} - (α - 1) δ^{α - 2} e^{2} s i g n (e) + δ^{α - 1} e, & | e | \leq δ \\ {| e |}^{α} s i g n (e), & δ < | e | < γ \\ γ^{α - 1}, & γ \leq | e | \end{matrix}

(11)

Given that $α = 1$ , the function $newfal (e, α, δ, γ)$ turns into a linear function $newfal (e, α, δ, γ) = e$ . Features of this functions are as follows: (1) function $fal (e, α, δ)$ is unsmooth switching when $| e | = δ$ , therefore smooth switching is introduced into function $newfal (e, α, δ, γ)$ to avoid oscillation; (2) setting $φ (e, α, δ, γ) = γ^{α - 1}$ when $γ \leq | e |$ is proposed to avoid the state tracking error e to be very large and the tracking performance of ESO deteriorated sharply.

Simulations are performed on the three nonlinear functions, and results demonstrate that the $newfal (e, α, δ, γ)$ function exhibits better continuity and differentiability than the $fan (e, α)$ and $fal (e, α, δ)$ functions. Thus, the $newfal (e, α, δ, γ)$ function is insensitive to the selection of the $δ$ value and can avoid high-frequency flutter phenomenon. This study used the $newfal (e, α, δ, γ)$ function in the improved ESO to obtain good performances.

Convergence and absolute stability analysis

Convergence analysis

The nonlinear parts of the improved ESO are divided into many areas to analyze the convergence of the improved ESO. All areas are replaced by piecewise linear functions. The convergence of the entire area can be ensured when the convergence of each area is ensured. This study used the linear expression $z_{1} - y (t)$ to replace the nonlinear function $newfal (e, α, δ, γ)$ to analyze the convergence of the improved ESO.

In this paper, h is defined as the bandwidth of system. The relationship between β₁,β₂,β₃ and h is expressed as β₁ = 3h, β₂ = 3h², β₃ = h³. Therefore, the following expressions can be obtained.

{\begin{matrix} Z_{1} (s) = \frac{3 h s^{2} + 3 h^{2} s + h^{3}}{{(s + h)}^{3}} \cdot y (s) + \frac{b_{0} s}{{(s + h)}^{3}} \cdot U_{q} (s) \\ Z_{2} (s) = \frac{(3 h^{2} s + h^{3}) s}{{(s + h)}^{3}} \cdot y (s) + \frac{b_{0} s (s + 3 h)}{{(s + h)}^{3}} \cdot U_{q} (s) \\ Z_{3} (s) = \frac{h^{3} s^{2}}{{(s + h)}^{3}} \cdot y (s) - \frac{b_{0} h^{3}}{{(s + h)}^{3}} \cdot U_{q} (s) \end{matrix}

(12)

The errors are defined as follows

e_{1} (s) = Z_{1} (s) - y (s)

(13)

e_{2} (s) = Z_{2} (s) - \overset{\cdot}{y} (s)

(14)

\begin{matrix} e_{3} (s) = Z_{3} (s) - x_{3} (s) \\ = Z_{3} (s) - (- 2 ξ ω_{n} x_{2} (s) - ω_{n}^{2} x_{1} (s) + f (s)) \\ = Z_{3} (s) - ({\overset{\cdot}{x}}_{2} (s) - b_{0} U_{q} (s)) = Z_{3} (s) - \overset{\cdot\cdot}{y} (s) + b_{0} U_{q} (s)) \end{matrix}

(15)

The following expressions can be obtained as follows

{\begin{matrix} e_{1} (s) = - \frac{s^{3}}{{(s + h)}^{3}} \cdot y (s) + \frac{s}{{(s + h)}^{3}} \cdot b_{0} \cdot U_{q} (s) \\ e_{2} (s) = - \frac{(s + 3 h) s^{3}}{{(s + h)}^{3}} \cdot y (s) + \frac{s}{{(s + h)}^{3}} \cdot b_{0} \cdot U_{q} (s) \\ e_{3} (s) = - \frac{s^{3} + 3 h^{2} s^{2} + 3 h^{2} s}{{(s + h)}^{3}} \cdot s^{2} \cdot y (s) \\ + \frac{s^{3} + 3 h^{2} s^{2} + 3 h^{2} s}{{(s + h)}^{3}} \cdot b_{0} \cdot U_{q} (s) \end{matrix}

(16)

This study selected y and $U_{q}$ as unit step input, thus, $y (s) = 1 / s$ and $U_{q} (s) = 1 / s$ . The steady-state errors can be calculated as follows

{\begin{matrix} e_{s 1} = lim_{s \to 0} s e_{1} (s) = 0 \\ e_{s 2} = lim_{s \to 0} s e_{2} (s) = 0 \\ e_{s 3} = lim_{s \to 0} s e_{3} (s) = 0 \end{matrix}

(17)

Therefore, the improved ESO has good convergence and estimating ability.

Absolute stability analysis

In the rotor reference frame, speed-loop model of PMSM can also be expressed as follows

\overset{\cdot}{ω} = - \frac{B}{J} ω - \frac{1}{J} T_{d} + \frac{1.5 p_{n} φ_{f}}{J} i_{q}

(18)

where B is the friction coefficient, $T_{e}$ is the electromagnetic torque of the motor, and $i_{q}$ is the control signal. Assume that $y = x = ω$ , $a = - \frac{B}{J}$ , $b = \frac{1.5 p_{n} φ_{f}}{J}$ , and $w (t) = - \frac{T_{d}}{J}$ , therefore

{\begin{matrix} y = x \\ \overset{\cdot}{x} = ax + w (t) + b i_{q} \end{matrix}

(19)

The ESO of speed-loop control system can be designed as follows

{\begin{matrix} e_{1} = Z_{1} - y \\ {\overset{\cdot}{Z}}_{1} = Z_{2} - β_{1} \cdot φ (e, α_{1}, δ_{1}, γ) + b_{0} i_{q} \\ {\overset{\cdot}{Z}}_{2} = - β_{2} \cdot φ (e, α_{2}, δ_{1}, γ) \end{matrix}

(20)

The compensation device is designed as follows

i_{q} = \frac{i_{q 0} - Z_{2}}{b_{0}}

(21)

where $i_{q 0} = k φ (e', α_{1}, δ_{1}, γ)$ , k is the compensation coefficient, $k = ω_{0}^{2} = 100$ , and $ω_{0}$ is the bandwidth of ESO, where $ω_{0} = 10$ .

While ignoring the total disturbance influence, the following expression can be obtained by equations (19) and (21)

{\begin{matrix} y = x \\ \overset{\cdot}{x} = ax + b \times \frac{i_{q 0} - Z_{2}}{b_{0}} = ax - k Z_{1} - Z_{2} \end{matrix}

(22)

Some definitions are given such that $A_{11} = a, A_{12} = - k, A_{13} = - 1$ .

The following expression can be obtained by equations (20) and (21)

{\begin{matrix} {\overset{\cdot}{Z}}_{1} = - β_{1} \cdot φ (e, α_{1}, δ_{1}, γ) - k Z_{1} \\ {\overset{\cdot}{Z}}_{2} = - β_{2} \cdot φ (e, α_{2}, δ_{1}, γ) \end{matrix}

(23)

Some definitions are given such that $A_{21} = - k, B_{11} = - β_{1}, B_{12} = - β_{2}$ .

According to equations (22) and (23), the following expression can be obtained

{\begin{matrix} \overset{\cdot}{X} = A_{11} X + A_{12} Z + A_{13} Z_{2} \\ \overset{\cdot}{Z} = A_{21} Z + B_{11} u' \\ {\overset{\cdot}{Z}}_{2} = B_{12} u' \\ u' = - φ (e_{1}) \\ e_{1} = C_{1}^{T} X + C_{2}^{T} Z \end{matrix}

(24)

Some definitions are given such that $C_{1} = - 1, C_{2} = 1$ , and $X = x, Z = Z_{1}$ . Assuming that $a \neq 0$ , $A_{11}$ is an invertible matrix. Defining that $Y = A_{11} X + Z_{2}$ , the following expression can be obtained

{\begin{matrix} \overset{\cdot}{Y} = A_{11} Y + A_{11} A_{12} Z + A_{13} B_{12} u' \\ \overset{\cdot}{Z} = A_{21} Z + B_{11} u' \\ u' = - φ (σ) \\ σ = C_{1}^{T} {A_{11}}^{- 1} Y + C_{2}^{T} Z - C_{1}^{T} {A_{11}}^{- 1} A_{13} Z_{2} \end{matrix}

(25)

The Lurie system general expression for equation (25) is as follows

{\begin{matrix} \overset{\cdot}{x} = Ax + bu' \\ \overset{\cdot}{ξ} = u' \\ σ = c^{T} x + ρ ξ \\ u' = - φ (σ) \end{matrix}

(26)

where $A = [\begin{matrix} A_{11} A_{11} A_{12} \\ 0 A_{21} \end{matrix}] = [\begin{matrix} a - ak \\ 0 - k \end{matrix}]$ , $B = [\begin{matrix} B_{12} \\ - B_{11} \end{matrix}] = [\begin{matrix} - β_{2} \\ β_{1} \end{matrix}]$ , $C = [C_{1}^{T} A_{11}^{- 1} C_{2}^{T}] = [- \frac{1}{a} 1]$ , and $ρ = - C_{1}^{T} A_{11}^{- 1} A_{13} B_{12} = - \frac{β_{2}}{a}$ . The linear dynamic transfer function of Lurie system is $C (s) = c^{T} (sI - A)^{- 1} B + ρ / s$ . The value of a is set to $a = 1$ , and the ESO’s coefficients can be calculated as $β_{1} = 3 ω_{0} = 30$ and $β_{2} = 3 ω_{0}^{2} = 300$ , therefore, $C (s) = {[- 1 1]}^{T} {[\begin{matrix} s - 1 100 \\ 0 s + 100 \end{matrix}]}^{- 1} [\begin{matrix} - 300 \\ 30 \end{matrix}] - \frac{300}{s}$ . Therefore, the system’s absolute stability can be verified by the Nyquist stability criterion.

The entire principle diagram of the PMSM speed control system based on the improved ESO is shown in Figure 3.

Figure 3.

Principle diagram of the PMSM speed control system based on improved ESO.

Simulation experiments and discussions

This section conducts simulation experiments based on the PI control strategy, a traditional ESO control strategy, and the improved ESO control strategy. The improved ESO is designed based on the new nonlinear function newfal, and the traditional ESO is designed based on the traditional nonlinear function fal. The parameters of the experiment PMSM are given in Table 1. The parameters of the traditional ESO and the improved ESO are the same parameters, which are given in Table 2.

Table 1.

Parameters of experiment PMSM.

Parameter name	Dimension	Numerical value
Maximum non-load speed	r/min	2750
Continuous current	A	5.1
Continuous torque	N·m	6.11
Motor pole logarithmic		16
Resistance	Ω	2.9
Inductance	mH	6.8

Table 2.

Parameters of ESO.

Parameter name	Symbol	Numerical value
Bandwidth	$ω_{0}$	10
Nonlinear factor	$α_{1}$	0.75
	$α_{2}$	0.5
	$α_{3}$	0.25
Filtering factor	$δ$	0.01
Gain coefficient	$β_{1}$	30
	$β_{2}$	300
	$β_{3}$	1000
Compensation factor	$b_{0}$	2.485

First, simulations are performed to verify the performance of the improved ESO. The input current is chosen as $i_{q} = 0.5 A$ and the position signal is chosen as $θ = 1 (t)$ . The output response curves of the improved ESO are shown in Figure 4(a). Figure 4(b) shows the output response curves of the improved ESO in the case of the position signal is chosen as sine signal, $θ = \sin (2 π t)$ . Figure 4 shows that $Z_{1}$ , $Z_{2}$ , and $Z_{3}$ can successfully tracking the output signal, the output’s differential signal, and the total disturbance. Therefore, the improved ESO exhibits good tracking performance.

Figure 4.

Tracking curves of improved ESO.

Second, simulations are performed when the PMSM starts without load. Three given speeds of 1, 1000, and 2750 r/min are applied on the PMSM at 0 s, respectively. The speed response curves are shown in Figure 5. The figure shows that the improved ESO control strategy has a shorter adjustment time and smaller overshoot amount than the PI control strategy and the traditional ESO control strategy. Figure 5(a) demonstrates that the overshoot amount of the improved ESO control strategy is 9.92% less than that of the traditional ESO control strategy and 14.89% less than that of the PI control strategy.

Figure 5.

Speed response curves in the case that PMSM starts without load.

Third, simulations are conducted when external load disturbance is suddenly applied on the PMSM. The speed response curves are shown in Figure 6. In Figure 6(a)–(c), the initial speeds of the PMSM are 1, 1000, and 2750 r/min. At 5 and 2 s, three different external load disturbances of 0.5, 200, and 1500 r/min are applied on the PMSM, respectively. The simulation results demonstrate that the improved ESO control strategy has smaller response curve fluctuation, shorter recovery time, and smaller load disturbance influence than the traditional ESO control strategy and the PI control strategy. Figure 6(a) demonstrates that the overshoot amount of the improved ESO control strategy is 8.62% less than that of the traditional ESO control strategy and 10.21% less than that of the PI control strategy. The recovery time of the improved ESO control strategy is 1.23 s less than that of the traditional ESO control strategy and 2.65 s less than that of the PI control strategy.

Figure 6.

Speed response curves in the case that external load disturbance is suddenly applied on PMSM.

Fourth, simulations are conducted when the PMSM speed changes suddenly. The initial PMSM speed is 1 r/min, which decreases to 0.5 r/min at 5 s. The speed response curves are shown in Figure 7. The figure shows that the improved ESO control strategy has smaller response curve fluctuations and a better recovery capability than the traditional ESO control strategy and the PI control strategy when the PMSM speed changes suddenly. The simulation results demonstrate that the amount of overshoot of the improved ESO control strategy is 9.13% less than that of the traditional ESO control strategy and 14.58% less than that of the PI control strategy. The response time of the improved ESO control strategy is 1.63 s less than that of the traditional ESO control strategy and 2.87 s less than that of the PI control strategy.

Figure 7.

Speed response curves in the case that PMSM speed changes suddenly.

Fifth, simulations are performed for the speed error. The speed error response curves are shown in Figure 8. The figure shows that the improved ESO control strategy has shorter error vanish time than the traditional ESO control strategy and the PI control strategy. The simulation results demonstrate that the error vanish time of the improved ESO control strategy is 1.80 s less than that of the traditional ESO control strategy and 8.49 s less than that of the PI control strategy. The detailed parameters for the above-mentioned four simulation cases are presented in Table 3.

Figure 8.

Response curves of speed error.

Table 3.

Response parameters.

Simulation case	Response name	PI control strategy	Traditional ESO control strategy	Improved ESO control strategy
Starts without load	Overshoot amount	15.08%	10.11%	0.19%
Disturbance applying	Overshoot amount	10.30%	8.91%	0.09%
Disturbance applying	Recovery time	2.89 s	1.47 s	0.24 s
Speed changes	Overshoot amount	15.49%	10.04%	0.91%
Speed changes	Response time	3.21 s	1.97 s	0.34 s
Vanish time of speed error	−	9.01 s	2.32 s	0.52 s

Finally, a white noise signal is applied on the PI control strategy, traditional ESO control strategy, and improved ESO control strategy to verify their elimination capacity for the measurement noise, respectively. The input signal is chosen as step signal, $θ = 1 (t)$ . The noise power is 0.001 and the noise sample time is 0.001 t. Simulations are carried out and the outputs are shown in Figure 9. Figure 9 shows that the improved ESO control strategy exhibits better elimination capacity for the measurement noise than the PI control strategy and the traditional ESO control strategy.

Figure 9.

Output of PI control strategy, traditional ESO control strategy and improved ESO control strategy with white noise.

Conclusion

This study introduces an improved ESO control strategy for PMSM speed control based on a new nonlinear function. The non-differentiable and discontinuous characteristics of the nonlinear function of the traditional ESO nonlinear state error feedback are addressed using the improved ESO. The high-frequency flutter phenomenon is reduced significantly. Simulations were performed, and the results indicate that the system with the improved ESO control strategy exhibits better dynamic performance, static performance, and robustness than the system with the traditional ESO control strategy or PI control strategy. In future studies, the proposed design technique will be extended to the improved ESO control strategy of current and position controls.

Footnotes

Appendix 1 Acknowledgements

The author is grateful to the anonymous reviewers for their helpful comments.

Handling Editor: Ivo Bukovský

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the Key Project of Natural Science by Education Department of Anhui Province (grant no. KJ2015A316) and Open Research Fund of Anhui Province Key Laboratory of Detection Technology and Energy Saving Devices (grant no. 2017070503B026-A02).

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