Control method for permanent magnet synchronous motor based on fractional active disturbance rejection

Abstract

To improve the response speed and anti-interference ability of a permanent magnet synchronous motor (PMSM), a PMSM control method based on fractional-order active disturbance rejection control (FOADRC) is proposed in this study. First, a fractional-order tracking differentiator (FOTD) and fractional-order extended state observer (FOESO) are designed using the fractional calculus. According to the response speed of the FOTD and the observation effect of the FOESO, a new fastest function and new nonlinear function are designed. Second, a fractional-order active disturbance rejection (FOADR) controller is built based on the new FOTD, new FOESO, and nonlinear feedback control law, and its stability is proven. Finally, the superiority and effectiveness of the proposed strategy are verified through a simulation and experiments.

Keywords

Permanent magnet synchronous motor fractional-order tracking differentiator fractional-order state observer fractional active disturbance rejection

Introduction

A permanent magnet synchronous motor (PMSM) uses a permanent magnet to replace the excitation winding of a wound synchronous motor and is widely used in various situations because of its simple control, high power density, high precision, and high efficiency. A PMSM is a multivariable, strongly coupled, and time-varying nonlinear system, and its application environment is generally complex and subject to varying degrees of external interference, including motor parameter changes, load disturbances, and other uncertain factors (Gao et al., 2020; Liu et al., 2021; Qi et al., 2015; Yi et al., 2016). Therefore, the control strategy of a PMSM is important. The active disturbance rejection control (ADRC) strategy is widely used in the field of PMSM control because of its strong robustness, high control accuracy, strong anti-interference ability, and other advantages (Meng et al., 2019; Song et al., 2021; Wang et al., 2019; Xi et al., 2003).

ADRC technology is a nonlinear control method first proposed by Han Jingqing, which does not rely on the mathematical model of the controlled object (Chen et al., 2004). The internal and external disturbances of a system are observed in real-time mainly through the extended state observer (ESO), and feedforward compensation is carried out to eliminate the influence of the disturbances on the system (Alhelou et al., 2020; Jin et al., 2020; Zhong et al., 2020). Owing to the increasing number of objects controlled by the ADRC strategy, many of which are a noninteger order, the traditional ADRC technology has encountered some problems, such as low tracking progress, weak anti-interference ability, and slow response speed for fractional-order systems (FOSs). As a result, satisfying the control and tracking of the high-precision, high-accuracy, and strong anti-interference ability of FOSs is difficult for the traditional control strategy. Considering the limitations of traditional ADRC for FOSs, this study proposes a new fractional-order active disturbance rejection control (FOADRC) strategy that can overcome the limitations and enhance control accuracy (Cui et al., 2020; Huang et al., 2020; Liu et al., 2019).

Fractional calculus mainly converts integer order into fractional order for control, so as to reflect the physical object more closely. FOADRC combines ADRC and FOSs to improve the application value of ADRC. Many researchers have studied this direction and achieved many research results. For example, Cortes-Romero et al. (2020) designed a fractional active disturbance rejection control (FADRC) for integer-order systems, which improves the observation ability and control effect of the control system, but there is less research on nonlinear control systems.Al-Saggaf et al. (2020) proposed a fractional-order proportional integral (FPI) controller, which was designed by replacing the nonlinear state error feedback controller. Using the combination of nonlinear function and FPI, an ADRC scheme was designed, which can realize the stable control of the target, but the improvement of the response speed of the whole control system can be further studied. Wen et al. (2013) proposed a fractional-order extended state observer (FESO), and used fractional-order proportional differential controller to replace the traditional tracking differentiator (TD) and nonlinear state feedback, which enhanced the observation ability and tracking performance of the system. Li et al. (2016) proposed a fractional-order nonlinear active disturbance rejection controller (FO-NADRC), which is mainly based on the combination of fractional-order state error feedback law (FO-NLESF) and a new ESO. It controls the Gas turbine plant and further improves the accurate control of the system on the basis of meeting the stable control conditions. Although the fractional-order improvement methods proposed by Li et al. (2016) and Wen et al. (2013) are effective, the overall stability proof required for the application of the FOADRC control strategy for FOSs is less, and the theoretical analysis of the system can be further studied.

Based on the above analysis, this study further examines the FOADRC control strategy and proposes a new control strategy for the stable and rapid control of the PMSM system position angle. Furthermore, this study provides the stability proof of the FOADRC strategy. The main contributions of this study are threefold. First, this study designs a new fastest function and applies it to a TD and designs a fractional-order tracking differentiator (FOTD) using the fractional calculus. The response speed of the whole system is improved. At the same time, the new nonlinear function is designed and applied to fractional-order extended state observer (FOESO). The function proof shows that the new nonlinear function is continuously derivable at the origin, so as to reduce the system jitter. Second, FOADRC controller is designed by FOTD, fractional-order extended state observer (FOESO) and nonlinear state error feedback (NLSEF). The feasibility and stability of the designed system are proved by theoretical derivation, and a new proof method is provided for the stability proof of fractional-order system. Finally, this study verifies the proposed scheme through a simulation and experiments. The results show that when the controller controls the motor, the motor can respond quickly, and has strong anti-interference ability, and has better robustness.

PMSM mathematical model

To facilitate the analysis, the mathematical model of the PMSM is simplified as follows: the spatial distribution of the rotor permanent magnetic field is sinusoidal, the saturation characteristic of the stator core is ignored, the magnetic circuit is linear, the motor parameters remain unchanged, and no core eddy current and hysteresis loss exist (Sun et al., 2020). The voltage equation of the PMSM in the d-p rotating coordinate system can be expressed as

[\begin{matrix} {\overset{\cdot}{i}}_{q} \\ \overset{\cdot}{w} \end{matrix}] = [\begin{matrix} - \frac{R}{L_{d}} - \frac{p ψ_{f}}{L_{d}} \\ \frac{1.5 p ψ_{f}}{J} - \frac{B}{J} \end{matrix}] \cdot [\begin{matrix} i_{q} \\ w \end{matrix}] + [\begin{matrix} \frac{u_{q}}{L_{d}} \\ - \frac{T_{L}}{J} \end{matrix}]

(1)

[\begin{matrix} u_{d} \\ u_{q} \end{matrix}] = [\begin{matrix} R + \frac{d L_{d}}{d_{t}} - w L_{q} \\ \frac{d L_{q}}{d_{t}} \end{matrix}] \cdot [\begin{matrix} 0 \\ i_{q} \end{matrix}] + [\begin{matrix} 0 \\ w ψ_{f} \end{matrix}]

(2)

where $u_{d}$ and $u_{q}$ are the stator voltage components of the d-p axis; $i_{d}$ and $i_{q}$ represent the stator current components of the current d-q axis; $L_{d}$ and $L_{q}$ represent the direct and quadrature axis inductance, respectively; $T_{L}$ is the load torque; $J$ is the moment of inertia of the motor; $p$ is the number of motor pole pairs; $R$ is the stator winding resistance of the motor; $w$ is the angular frequency of the rotor; and $ψ_{f}$ is the flux linkage between the permanent magnet and stator.

The torque equation of the PMSM is

T_{e} = \frac{3}{2} p ψ_{f} i_{q}

(3)

The equation of the motion of the PMSM is

\frac{d^{2} θ}{d t^{2}} = - \frac{B}{J} \frac{d θ}{dt} - \frac{1}{J} T_{L} + \frac{3 p ψ_{f}}{2 J} i_{q}

(4)

where $B$ is the viscous friction coefficient, $θ$ is the rotor angular displacement, and $d θ / dt = w_{a}$ , where $w_{a}$ is the angular velocity.

Consider that the maximum torque output can be achieved under the vector control mode of $i_{d} = 0$ in equation (4), and suppose that the rotor angular displacement $θ$ of the PMSM is the state variable $x_{1}$ , the mechanical angular velocity $w_{a}$ is the state variable $x_{2}$ , and the total disturbance $a_{(t)}$ is the new state variable $x_{3}$ . Then, the expression is

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = bu + x_{3} \\ {\overset{\cdot}{x}}_{3} = a (t) \\ y = x_{1} \end{matrix}

(5)

where $b$ is the compensation coefficient, $b = 3 p ψ_{f} / 2 J$ , $a (t) = - \frac{1}{J} [\frac{3}{2} p ψ_{f} ({\bar{i}}_{q} - i_{q}) + T_{L} + B w_{a}]$ , $u$ is the orthogonal current given value $u = {\bar{i}}_{q}$ , and $a (t)$ represents the total interference.

FOADRC

As traditional fractional calculus is the basis of derivatives and integrals of any order, it can be described by the calculus operator represented by equation (6)

{ϑ^{χ}}_{t} = {\begin{matrix} \begin{matrix} \frac{d^{χ}}{d t^{χ}}, R (χ) > 0 \end{matrix} \\ \begin{matrix} 1, R (χ) = 0 \end{matrix} \\ \int_{0}^{t} {(d^{o})}^{- χ}, R (χ) < 0 \end{matrix}

(6)

where $χ$ is the calculus order, and $R (χ)$ represents a real number. When $χ$ is an integer, it represents integer-order calculus; when $χ$ is a real number greater than zero, ${D^{χ}}_{t}$ represents a fractional differential; and when $χ$ is a real number less than zero, ${D^{χ}}_{t}$ represents a fractional integration.

In this study, the concept of fractional-order control is applied to a TD and ESO by combining ADRC and fractional calculus theory, and an FOADRC framework is built, as shown in Figure 1.

The figure shows that the FOADRC is composed mainly of three components: the FOTD, which is mainly used for tracking signals and differential pair tracking signals; FOESO, which is used to observe the internal and external disturbances; and the system state. The nonlinear state error feedback control law is employed mainly to improve the control effect of the system, where X represents the system input signal, and Y represents the output signal of the system response.

Nonlinear function design

As the core of the ADRC, the nonlinear function has a considerable impact on the convergence and stability of the whole controller. As the traditional nonlinear function is not differentiable at the origin and breakpoint and lacks satisfactory continuity and smoothness, in this study, the traditional $fal (e, a, δ)$ function is improved to a new nonlinear function $ϕ (e, a, δ)$ , which is differentiable at the origin and breakpoint and has better continuity and smoothness.

When $| e | > δ$ , the expression of the $ϕ (e, b, δ)$ function is $ϕ (e, a, δ) = {| e |}^{a} sign (e)$ . When $| e | \leq δ$ , the expression of the $ϕ (e, a, δ)$ function is in the following form, which can be seen by fitting, with interpolation

ϕ (e, a, δ) = m \tanh (e) + n e^{2} + z \tan (e)

(7)

The fitting process should meet the differentiable continuity condition, then

{\begin{matrix} ϕ (e, a, δ) = δ^{a}, e = δ \\ ϕ (e, a, δ) = - δ^{a}, e = - δ \\ ϕ^{'} (e, a, δ) = a δ^{a - 1}, e = \pm δ \end{matrix}

(8)

Solving the equation shows that

ϕ (e, a, δ) = {\begin{matrix} {| e |}^{a} \cdot sign (e), | e | > δ \\ \frac{δ^{a} \cdot s^{2} (δ) - a \cdot δ^{a - 1} \tan (δ)}{\tanh (δ) \cdot s^{2} (δ)} \tanh (e) + \frac{a \cdot δ^{a - 1} \cdot \tan (δ) \cdot \tanh (δ) - δ^{a} \cdot \tan (δ) \cdot (1 - \tanh^{2} (δ))}{\tan (δ) \cdot \tanh (δ) \cdot s (δ) - \tan^{2} (δ) \cdot (1 - \tanh^{2} (δ))} \cdot \tan (e), | e | \leq δ \end{matrix}

(9)

FOTD

To meet the rapidity of the system response and reduce the jump and jitter in the process, in this study, for the traditional second-order nonlinear differential tracker, the Caputo fractional calculus is used to improve the expression as shown in equation (10). First, the tanh function is used to replace the sign function, and then the n-times square root operation is performed on the whole. Furthermore, FOTD expression is shown in equation (11)

_{c}^{C} ϑ_{t}^{a_{E}} f (x) =_{c}^{R} ι_{t}^{m - a_{E}} [ϑ^{m} f (x)] = \frac{1}{Γ (m - a_{E})} \int_{c}^{t} \frac{f^{(m)} (τ)}{{(t - τ)}^{a_{E} - m + 1}} d τ

(10)

where $m$ is a positive integer satisfying $m - 1 < a_{E} < m$ , $f (x)$ is a smooth function, and $Γ (x) = \int_{0}^{\infty} x^{β - 1} e^{- x} dx$ , and $a$ is the order of the micro-product

{\begin{matrix} ϑ^{a} v_{1} (k) = v_{2} (k) \\ ϑ^{a} v_{2} (k) = - r_{1} \sqrt[n]{(\tanh [(v_{1} - r) + \frac{v_{2} | v_{2} |}{r_{1}} W])} \end{matrix}

(11)

where $r$ is the set point, $v_{1}$ and $v_{2}$ represent the transition process and its differential signal, $ϑ$ is the derivative symbol, and $ϑ^{a_{E}} =_{c}^{C} ϑ_{t}^{a_{E}} f (x)$ is the Caputo fractional calculus. To reduce the response jitter and improve the response speed, in this study, $n$ is an odd number, and 3 is selected. And $W = 1 - Γ^{2} (a_{E} + 1) / Γ (2 a_{E} + 1)$ .

Since the fractional calculus $0 < a_{E} < 1$ , under $| ϑ^{a_{E}} v (t) | \leq r$ condition, the state $v_{1} (t)$ can track the input signal $v (t)$ , while $v_{2} (t)$ reflects the fractional differential of $v_{1} (t)$ , which can be regarded as $v_{1} (t) \approx v (t)$ and $v_{2} (t) \approx ϑ^{a_{E}} v (t)$ .

FOESO

The ESO is the core ADRC component, which realizes the observation of internal and external disturbances. The function of the ESO is to expand the disturbance affecting the output of the system into a new state variable. This process does not depend on the model of the controlled object, and the disturbance in the system can be observed without direct measurement.

FOESO is designed based on the fractional system of PMSM, and then expression (5) is transformed into

{\begin{matrix} ϑ^{a_{E}} {\overset{\cdot}{x}}_{1} = x_{2} \\ ϑ^{a_{E}} {\overset{\cdot}{x}}_{2} = bu + x_{3} \\ ϑ^{a_{E}} {\overset{\cdot}{x}}_{3} = a (t) \\ y = x_{1} \end{matrix}

so FOESO is designed as follows

{\begin{matrix} e (t) = z_{1} (t) - y (t) \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{1} (t) = z_{2} (t) - β_{01} ϕ_{1} (e (t), a_{1}) \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{2} (t) = z_{3} (t) - β_{02} ϕ_{2} (e (t), a_{2}) + bu (t) \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{3} (t) = - β_{03} ϕ_{3} (e (t), a_{3}) \end{matrix}

(12)

where $ϕ_{1} (e, a_{1})$ represents the nonlinear function, as shown in equation (8); $β_{01}$ , $β_{02}$ , and $β_{03}$ are the gain coefficients; $e$ is the error signal; and $a_{i} (i = 1, 2, 3)$ is the nonlinear factor.

The bandwidth method proposed by Gao is used in this study to adjust the gain coefficients. According to the characteristic equation of the ESO, the following can be known (Ai et al., 2021)

F (s) = s^{3 a_{E}} + β_{01} s^{2 a_{E}} + β_{02} s^{a_{E}} + β_{03}

(13)

Through pole assignment, the bandwidth of the observer is set as $q$ , then $β_{01} = 3 q$ , $β_{02} = 3 q^{2}$ , and $β_{03} = q^{3}$ .

FOADRC stability analysis

Owing to the difficult problem of nonlinear controlled object stability analysis, a linear nominal model is generally used to examine the main dynamics of a system or dynamics near the equilibrium state. The linear error feedback control law is designed as equation (14)

u = \frac{u_{0} - z_{3}}{b_{0}}, u_{0} = \sum_{i = 1}^{n} k_{i} (v_{i} - z_{i})

(14)

Assumption: When the input r of the control system is zero and the transition process is not considered or the initial value of the transition process, vi (i = 1,2,3, …., n) is assumed to be zero. In addition, the control gain b₀ = b (Wan, 2001).

To examine the stability of the closed-loop nonlinear system, the research system is transformed. If the initial input value of the system is zero, then $v_{1}$ and $v_{2}$ are zero, thereby meeting the assumption.

According to $e_{i} = v_{i} - z_{i}$ , the equivalent gain of the linear function is

η (e_{i}, a_{i}) = \frac{ϕ (e_{i}, a)}{e_{i}} = \frac{ϕ (v_{i} - z_{i}, a)}{v_{i} - z_{i}}

(15)

When $v_{i} = 0$

η (z_{i}, a_{i}) = - \frac{ϕ (- z_{i}, a)}{- z_{i}}

(16)

It is known that

u_{0} = \sum_{i = 1}^{n} k_{i} {(v_{i} - z_{i}) |}_{v_{i} = 0} = \sum_{i = 1}^{n} k_{i} (z_{i}) (- z_{i})

(17)

Let $η_{0 i} (e) = ϕ_{i} (e) / ϕ_{1} (e), i = 1, 2, \dots, n + 1$ . It can be known that

{\begin{matrix} e (t) = z_{1} (t) - y (t) \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{1} (t) = z_{2} (t) - β_{01} η_{01} (e) ϕ_{1} (e) \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{2} (t) = z_{3} (t) - β_{02} η_{03} (e) ϕ_{1} (e) + bu (t) \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{3} (t) = - β_{03} η_{03} (e) ϕ_{1} (e) \end{matrix}

(18)

The equation below can be obtained by substituting equations (14) and (15) into equation (5)

{\begin{matrix} ϑ^{a_{E}} \overset{\cdot}{X} = A_{01} X + A_{02} \bar{Z} + A_{03} z_{3} \\ y = x_{1} \end{matrix}

(19)

where

\begin{matrix} X = {[\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}]}^{T}, A_{01} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a_{3} & a_{2} & a_{1} \end{matrix}] \\ A_{02} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - η_{01} (z_{1}) k_{1} & - η_{02} (z_{2}) k_{2} & - η_{03} (z_{3}) k_{3} \end{matrix}] \\ A_{03} = {[0 0 - 1]}^{T}, and \bar{Z} = {[z_{1} z_{2} z_{3}]}^{T} \end{matrix}

The equation below can be obtained by substituting equations (14) and (17) into equation (18)

{\begin{matrix} ϑ^{a} \overset{\cdot}{\bar{Z}} = A_{11} \bar{Z} + A_{12} \hat{u} \\ ϑ^{a} {\overset{\cdot}{z}}_{3} = β_{03} η_{03} \hat{u} \\ \hat{u} = - ϕ_{1} (e) \end{matrix}

(20)

in equation (20), $A_{11}$ and $A_{12}$ are shown as follows

\begin{matrix} A_{11} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ - η_{01} (z_{1}) k_{1} & - η_{02} (z_{2}) k_{2} & - η_{03} (z_{3}) k_{3} \end{matrix}] \\ A_{12} = {[η_{01} (e) β_{01} η_{02} (e) β_{02} η_{03} (e) β_{03}]}^{T} \end{matrix}

The equation below can be known by combining equations (19) and (20)

{\begin{matrix} ϑ^{a_{E}} \overset{\cdot}{X} = A_{01} X + A_{02} \bar{Z} + A_{03} z_{3} \\ ϑ^{a_{E}} \overset{\cdot}{\bar{Z}} = A_{11} \bar{Z} + A_{12} \tilde{u} \\ ϑ^{a_{E}} {\overset{\cdot}{z}}_{3} = β_{03} η_{03} \hat{u} \\ e = c_{01} X + c_{02} \bar{Z} \\ \hat{u} = - ϕ_{1} (e) \end{matrix}

(21)

where $c_{01} = [\begin{matrix} - 1 & 0 & 0 \end{matrix}]^{T}$ and $c_{02} = [\begin{matrix} 1 & 0 & 0 \end{matrix}]^{T}$ . In practical research, $a_{n}$ can be replaced by any small number, and matrix $A_{01}$ is reversible. $Y = A_{01} X + A_{03} z_{n + 1}$ is set to obtain

{\begin{matrix} ϑ^{a_{E}} \overset{\cdot}{Y} = A_{01} Y + A_{01} A_{02} \bar{Z} + A_{03} β_{03} η_{03} \hat{u} \\ ϑ^{a_{E}} \overset{\cdot}{\bar{Z}} = A_{11} \bar{Z} + A_{12} \tilde{u} \\ e = {c_{01}}^{T} {A_{01}}^{- 1} Y + {c_{02}}^{T} \bar{Z} - {c_{01}}^{T} {A_{01}}^{- 1} A_{13} z_{n + 1} \\ \hat{u} = - ϕ_{1} (e) \end{matrix}

(22)

which can be further converted to

{\begin{matrix} \overset{\cdot}{\overset{⌢}{x}} = \overset{⌢}{A} \overset{⌢}{x} + b \hat{u} \\ \overset{\cdot}{\overset{⌢}{ν}} = \hat{u} \\ e = c^{T} \overset{⌢}{x} + κ \overset{⌢}{ν} \\ \hat{u} = - ϕ_{1} (e) \end{matrix}

(23)

where

\begin{array}{l} \overset{⌢}{x} = [ϑ^{a_{E}} Y, ϑ^{a_{E}} Z], \overset{⌢}{A} = [\begin{matrix} A_{01} & A_{01} A_{02} \\ 0 & A_{11} \end{matrix}] \\ \overset{⌢}{ν} = \int_{0}^{t} \hat{u} d t, b = [\begin{matrix} A_{03} β_{03} η_{03} \\ A_{12} \end{matrix}] \\ c = [\begin{matrix} {c_{01}}^{T} {A_{01}}^{- 1} & {c_{2}}^{- 1} \end{matrix}], and κ = - {c_{01}}^{T} {A_{01}}^{- 1} \\ A_{03} β_{03} η_{03} = - \frac{β_{03} η_{03}}{a_{2}} \end{array}

Let $y = \int_{0}^{t} \overset{⌢}{x} dt$ . Then, $\overset{\cdot}{y} = \overset{⌢}{x}$ and $\overset{\cdot\cdot}{y} = \overset{\cdot}{\overset{⌢}{x}}$ are substituted into equation (23) and can be known by

{\begin{array}{l} y = \overset{⌢}{A} y + b \overset{⌢}{ν} \\ \dot{\overset{⌢}{ν}} = \hat{u} \\ e = c^{T} \overset{⌢}{A} \overset{⌢}{x} + c^{T} b \overset{⌢}{ν} + κ \overset{⌢}{ν} \\ \hat{u} = - ϕ_{1} (e) \end{array}

(24)

det [\begin{matrix} \overset{⌢}{A} & b \\ c^{T} A & κ + c^{T} b \end{matrix}] = det [\begin{matrix} \overset{⌢}{A} & b \\ 0 & κ \end{matrix}] \neq 0

According to control theory, when the matrix is a full-rank matrix, the represented system is stable.

Consider that the system stability shown in equation (24) is equivalent to that shown in equation (25)

{\begin{matrix} \overset{\cdot}{\overset{⌢}{x}} = \overset{⌢}{A} \overset{⌢}{x} + b \overset{⌢}{ν} \\ \overset{\cdot}{\overset{⌢}{ν}} = \hat{u} \\ \overset{\cdot}{e} = {\tilde{c}}^{T} \overset{⌢}{x} + c^{T} b \overset{⌢}{ν} + \overset{⌢}{κ} \overset{⌢}{ν} \\ \hat{u} = - ϕ_{1} (e) \end{matrix}

(25)

It can be expressed as follows

{\begin{matrix} \overset{\cdot}{\overset{⌢}{x}} = \tilde{A} \overset{⌢}{x} + \tilde{b} e \\ \overset{\cdot}{e} = {\tilde{c}}^{T} \overset{⌢}{x} + re + \overset{⌢}{κ} \hat{u} \\ \hat{u} = - ϕ (e) \end{matrix}

(26)

where $\tilde{A} = \overset{⌢}{A} - \frac{b c^{T} \overset{⌢}{A}}{κ + c^{T} b}$ , $\tilde{b} = \frac{b}{κ + c^{T} b}$ , ${\tilde{c}}^{T} = c^{T} \overset{⌢}{A} (\frac{\overset{⌢}{A} - b c^{T} \overset{⌢}{A}}{κ + c^{T} b})$ , $r = \frac{c^{T} \overset{⌢}{A} b}{κ + c^{T} b}$ , and $\overset{⌢}{κ} = κ + c^{T} b$ . Further transformation can be achieved, as follows

\overset{\cdot}{l} = Wl + O ϕ (e)

(27)

where

\begin{matrix} O = {[\begin{matrix} 0 & \dots & 0 & - \overset{⌢}{κ} \end{matrix}]}^{T} \in R^{2 n + 1}, \\ W = [\begin{matrix} \tilde{A} & \tilde{b} \\ {\tilde{c}}^{T} & r \end{matrix}] \in R^{(2 n + 1) \times (2 n + 1)} \\ l = {[\overset{⌢}{x} e]}^{T} \in R^{2 n + 1} \end{matrix}

The nonlinear system equation (27) is linearized at the equilibrium point, and the Lyapunov indirect method is combined. Let $Q (l) = Wl + O ϕ (e)$ and $L = {\frac{\partial Q}{\partial l} |}_{l = 0}$ .

Theorem: The sufficient and necessary condition for the asymptotic stability of the zero solution of systematic equation (27) is that matrix L is a Hurwitz matrix.

The nonlinear system is linearized at the equilibrium point so the matrix coefficient L can be expressed as

\begin{matrix} L = {\frac{\partial Q}{\partial l} |}_{l = 0} = [\begin{matrix} \frac{\partial Q_{1}}{\partial l_{1}} & \frac{\partial Q_{1}}{\partial l_{2}} & \dots & \frac{\partial Q_{1}}{\partial l_{2 n + 1}} \\ \frac{\partial Q_{2}}{\partial l_{1}} & \frac{\partial Q_{2}}{\partial l_{2}} & \dots & \frac{\partial Q_{2}}{\partial l_{2 n + 1}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial Q_{2 n + 1}}{\partial l_{1}} & \frac{\partial Q_{2 n + 1}}{\partial l_{2}} & \dots & \frac{\partial Q_{2 n + 1}}{\partial l_{2 n + 1}} \end{matrix}] \end{matrix}

(28)

According to the Lyapunov indirect method, if matrix L is a Hurwitz matrix, then the equilibrium point of system equation (27) is asymptotically stable.

According to equation (28), relevant parameters are included for simplification, and the following equation can be obtained

L = [\begin{matrix} 0 & 1 & - k_{1} {δ^{'}}^{{a^{'}}_{1} - 1} & - k_{2} {δ^{'}}^{{a^{'}}_{2} - 1} & 0 \\ a_{02} - \frac{β_{03}}{β_{01}} δ^{a_{3} - a_{1}} & a_{01} & - a_{01} k_{1} {δ^{'}}^{{a^{'}}_{1} - 1} & - a_{01} k_{2} {δ^{'}}^{{a^{'}}_{2} - 1} + \frac{β_{03}}{β_{01}} δ^{a_{3} - a_{1}} & - \frac{β_{03}}{β_{01}} δ^{a_{3} - a_{1}} \\ 1 & 0 & 0 & 0 & 1 \\ \frac{β_{02}}{β_{01}} δ^{a_{2} - a_{1}} & 0 & - k_{1} {δ^{'}}^{{a^{'}}_{1} - 1} & - \frac{β_{02}}{β_{01}} δ^{a_{2} - a_{1}} - k_{2} {δ^{'}}^{{a^{'}}_{2} - 1} & \frac{β_{02}}{β_{01}} δ^{a_{2} - a_{1}} \\ \frac{β_{02}}{β_{01}} δ^{a_{2} - a_{1}} & - 1 & 0 & - \frac{β_{02}}{β_{01}} δ^{a_{2} - a_{1}} & \frac{β_{02}}{β_{01}} δ^{a_{2} - a_{1}} - β_{01} δ^{a_{1} - 1} \end{matrix}]

(29)

The same motor parameters are selected as follows: $p = 2$ , $J = 2.2 \times 10^{- 5} kg m^{2}$ , $ψ_{f} = 0.07147 Wb$ , $a_{02} = 0$ , $B = 3 \times 10^{- 3} N m s$ , $a_{01} = B / J \approx 13.64$ , and $T_{L} = 0.2 N m s$ , where $a_{01}$ and $a_{02}$ represent the first-order and constant term coefficients, respectively. In this study, a particle swarm optimization algorithm is used to examine the nonlinear ADRC parameter tuning problem, combined with the empirical formula method (Miklosovic et al., 2006) and pole assignment method. The observer bandwidth $w_{0}$ is generally larger than the controller bandwidth $w_{c}$ , and the bandwidth adjustment is closely related to the controlled object. Based on the empirical method $w_{0} \approx 3 ~ 5 w_{c}$ (Gao, 2003; Han, 2008), the following parameters are selected according to their optimal selection range. Since the order is between 0 and 1, according to the root trajectory, it can be known that the root of the transfer function is located in the left half-plane of s. Using the bandwidth method of integer-order expansion state device, it can be known that

S^{3 a_{E}} + β_{01} S^{2 a_{E}} + β_{02} S^{a_{E}} + β_{03} = {(S^{a_{E}} + w_{0})}^{3}

(30)

It can be seen that the selection of fractional parameters is mainly related to bandwidth, and the fractional parameters are selected as follows: $η = 0.5$ , $k_{1} = {w_{c}}^{2}$ , $k_{2} = 2 w_{c}$ , $w_{c} = 20$ , $w_{0} = 60$ , $β_{01} = 3 w_{0}$ , $β_{02} = 3 {w_{0}}^{2}$ , $β_{03} = {w_{0}}^{3}$ , $a_{1} = 1$ , $a_{2} = 0.5$ , $a_{3} = 0.25$ , $δ = 0.01$ , and $a_{E} = 0.3$ .

The verification results are shown in equation (31)

L_{PMSM} = [\begin{matrix} 0 & 1 & - 400 & - 40 & 0 \\ - 37, 920 & 13.64 & - 5456 & 37, 374.4 & - 37920 \\ 1 & 0 & 0 & 0 & 1 \\ 600 & 0 & - 400 & - 640 & 600 \\ 600 & - 1 & 0 & - 600 & 420 \end{matrix}]

(31)

Similarly, when the observer bandwidth $w_{0} > 0$ is satisfied and greater than the controller bandwidth $w_{c}$ and the PMSM L matrix shown in equation (29) is a Hurwitz matrix, the closed-loop system will be asymptotically stable.

Simulation and experiments

Simulation

To verify the performance of each component of the FOADRC, the parameters of the motor are selected, and the traditional ADRC, improved ADRC (Liu et al., 2017), and FOADRC are used to control the motor to test its control performance under the same conditions. The parameters of the traditional ADRC, improved ADRC, and FOADRC are the same, and the traditional ADRC parameters are the best adjustment parameters. As the larger the $w_{0}$ , the smaller the error disturbance amplitude that the ESO can track, an appropriate increase in $β_{03}$ can optimize the overall performance. The parameter selection is presented in Table 1.

(a) To verify the performance of the improved FOTD, a step input signal with a value of 1 is used for the verification under the same parameter selection. The response curve is illustrated in Figure 2.

Table 1.

FOADRC parameters.

FOADRC component	Parameter name	Symbol representation	Numerical value
FOTD	Set point	r	400
	Fractional order	a_E	0.3
	Simulation step size	a_T	0.07
FOESO	Nonlinear factors	a ₁	1
	Nonlinear factors	a ₂	0.5
	Nonlinear factors	a ₃	0.25
	Filter factor	δ	0.01
	Gains	β ₀₁	60
	Gains	β ₀₂	120.0
	Gains	β ₀₃	800.0
	Fractional order	a_E	0.3
NLSEF	Gains	k ₁	400
	Gains	k ₂	40
	Filter factor	δ	1

Figure 2(a) shows the response curve of the improved FOTD and that the stable tracking time of the traditional TD and FOTD is 0.424 and 0.199 seconds, respectively. The traditional TD can stably track the differential signal of the input signal in 0.443 seconds, and the FOTD can stably track the differential signal of the input signal in 0.025 seconds. Compared with traditional TD, FOTD improves the tracking response speed by 50%. Figure 2(b) demonstrates the response results of the step input signal with a value of 1 for the TD in the ADRC. It can be seen that the time required for the traditional ADRC to reach stability is 0.602 seconds, whereas the time required for the improved FOADRC to reach stability is 0.442 seconds. It can be seen that under the same conditions, the response speed of the ADRC built by the improved FOTD is about 30% higher than that built by the traditional TD to track the input signal, and shows a good tracking effect.

(b) To verify the performance of the FOESO, an ideal second-order system is observed under the same parameters. The superior performance of the FOESO is verified by comparing its results with those of the traditional ESO observation. The response curve is illustrated in Figure 3.

The waveforms of the actual position and observation position shown in Figure 3 reveal that the FOESO has a higher observation accuracy than the ESO.

(c) To examine the response speed and anti-interference ability of the FOADRC, an ideal motor model is used for the simulation. Different input signals are observed, and the results are shown in Figures 4 and 5.

Figure 4(a)–(c) demonstrates the response curves of the three controllers under different input conditions. The comparison shows that the response speed of the FOADRC is significantly higher than that of the improved ADRC and traditional ADRC. As can be seen from Figure 4(a), the improved ADRC improves the speed by 10% compared with the traditional ADRC, while the FOADRC improves the response speed by 13% compared with the traditional ADRC. Similarly, according to Figure 4(b) and (c), the FOADRC response curve is closer to the expected curve, which improves the response speed and the tracking accuracy.

Figure 5(a) and (b) illustrates the simulation diagram of the anti-interference curves of the three ADRC strategies under a step signal with an input value of 100, a pulse signal with an input value of 1, and a step interference signal with a value of 50 and 20. It can be seen from Figure 5(a) that under the step input signal with a value of 100, the traditional ADRC fluctuation amplitude is 23, the improved ADRC fluctuation amplitude is 17, and the FOADRC fluctuation amplitude is 10. Thus, the anti-interference ability of the FOADRC is improved by 13% compared with that of the traditional control strategy. Figure 5(b) reveals that under the pulse input signal with a value of 1, the traditional ADRC fluctuation amplitude is 0.08, the improved ADRC fluctuation amplitude is 0.07, and the FOADRC fluctuation amplitude is 0.05, thereby further verifying the anti-interference performance of the FOADRC. Table 2 presents all the characteristic response values.

Table 2.

Comparison of results under different signal inputs.

Input signal	External interference	Characteristic	Traditional ADRC	Improved ADRC	FOADRC
Step	No	Rise time (seconds)	0.50	0.37	0.25
Step	No	Overshoot (%)	2.3	1.1	0
Sinusoidal	No	The first time the peak was reached (seconds)	0.55	0.54	0.51
Sinusoidal	No	Error value (rad)	3.6	3.1	2.1
Square wave	No	Rise time (seconds)	0.61	0.58	0.35
Square wave	No	Overshoot (%)	2.2	1.2	0
Step	Step interference	Error value (rad)	2.3	1.6	0
Step	Step interference	Fluctuation (%)	23	17	11
Square wave	Step interference	Error value (rad)	2.7	1.3	0
Square wave	Step interference	Fluctuation (%)	8.1	8.0	4.1

When the results in Figures 3 and 4 are compared, it can be seen that the tracking curve of the FOADRC has smaller and fewer fluctuations and a faster response speed than the tracking curve of the traditional ADRC, which is in line with the input signal curve.

Experiments

To verify the performance of the proposed FOADRC, a 4000-W Yaskawa servo motor is used. The experimental platform of the system is composed of the Yaskawa servo motor, a PC, a servo driver, and a load controller, as shown in Figure 6.

The selected motor parameters are presented in Table 3.

Table 3.

Motor parameters.

Parameter	Numerical value
Rated voltage	200 V
Rated current	2.5 A
Working frequency	250 Hz
Rated power	400 W
Torque	1.27 N m
Rated speed	3000 r/min
Maximum speed	6000 r/min

When the input value of the motor is 1000 cts/s, the experiment results of the traditional ADRC and FOADRC under a step signal, sinusoidal signal, and square wave signal are shown in Figures 7 –9. It can be seen from Figures 7 and 8 that the time for the two control strategies to enter the stable state is about 45 ms, but the FOADRC has a faster response performance than the traditional ADRC control strategy. Figure 9 reveals that in the presence of interference, the FOADRC has a superior anti-interference effect and can be controlled stably, thereby improving the anti-interference ability of the model. The experimental results show that the FOADRC has the advantages of fast response, high control accuracy, no overshoot, and strong robustness.

Summary

This study examines a new ADRC strategy applied to an FOS and presents the feasibility and superiority of the strategy through theoretical and experimental verification. The experimental comparison shows that the proposed control strategy can improve the response speed, enhance the anti-interference ability, and reduce the overshoot of the system on the basis of the stable control of the motor. In addition, this study realizes the application of fractional-order and integer-order target. On the basis of integer order, the stability analysis method for fractional-order controller is explored, increases the application range of the ADRC strategy, and provides a new control method for the selection of a motor control strategy and new research idea to future studies on complex objects, thereby demonstrating high research and application value.

Footnotes

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: Anhui Province University Discipline (Professional) Top Talent Academic Funding Project (gxbjZD2021065), The key R&D project of Wuhu City “R&D and Application of Key Technologies of Robot Intelligent Detection System Based on 3D Vision” (2021yf32), and the Anhui Engineering University Jiujiang Industrial collaborative innovation special fund project (2021cyxtb2).

ORCID iD

Zhenben Wu

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