Corrigendum

Abstract

Rongrong Qian, Minzhou Luo, and Peng Sun. Improved nonlinear sliding mode control based on load disturbance observer for permanent magnet synchronous motor servo system. Advances in Mechanical Engineering 2016; 8(4): 1-12. DOI: 10.1177/1687814016642670

The Advances in Mechanical Engineering Editorial Office became aware that the peer review process organised by the Guest Editors of the Special Collection Advanced Control and Analysis of Mechatronics Systems with Modelling Uncertainty did not meet the journal’s usual rigorous standards of peer review. Due to this, all articles in the Special Collection were re-reviewed by a new set of independent referees.

Based on the feedback from the new independent referees, the following corrections are to be made to the above-referenced article.

Vector u(t) in equations after (11) should be revised as $u(t)=\left[\begin{array}{c}{v}_d\hfill \\ \multicolumn{1}{c}{v_q}\end{array}\right]$

The error system is a time-varying system. In this error system, since $\psi (y,r)$ is a continuously differentiable with respect to y, it is also continuously differentiable with respect to x. The authors then consider a positive continuous function $\Pi (x)=-2\psi (y,r)M^{\mathrm{T}}M$ . Therefore, below (22), the following is acquired, which satisfies the condition of asymptotic convergence:

$$\begin{array}{c}{\stackrel{.}{V}}_1(t)={\stackrel{.}{e}}_1^{\mathrm{T}}(t)P{e}_1(t)+e_{\mathrm{1}}^{\mathrm{T}}(t)P{\stackrel{.}{e}}_1(t)\\ =e_{\mathrm{1}}^{\mathrm{T}}(t)(A_{11}-A_{12}F)\mathrm{T}P{e}_1(t)+e_{\mathrm{1}}^{\mathrm{T}}(t)P(A_{11}-A_{12}F)e_1(t)+2e_{\mathrm{1}}^{\mathrm{T}}(t)P{A}_{12}\psi (y,r)A_{12}^{\mathrm{T}}P{e}_1(t)\\ =-e_{\mathrm{1}}^{\mathrm{T}}(t)W{e}_1(t)+2\psi (y,r)M^{\mathrm{T}}M<-\Pi (x)\end{array}$$

Assumption (29) should be removed. Assuming that (27) is bounded means to assume the boundedness of the sliding variable, which is exactly what is being proved with the first term of the Lyapunov function (31). This is a tautology. In order to avoid the tautology, the paragraph above (29) should be revised to:

“Given the system (10) with the controller as (25), the gain ${\hat{K}}_1(t)$ and ${\hat{K}}_2(t)$ has upper bounds as follows:”

Equations (20) and (21) are acquired under the sliding surface of the control system states reach the sliding surface s(x, t) = 0. They are valid not only for x_ref (t) = 0.

The expressions (36)-(37) appear to use the variable s as the Laplace variable. to avoid the misunderstanding, we define a new expression for the Laplace variable. Then (36)-(37) are modified as follows:

“The transfer function between $s_i(\vartheta )$ and ${\rho }_{\mathrm{di}}(\vartheta )$ , i = 1,2 can be deduced from (35) as follows

$s_i(\vartheta )=\frac{{\rho }_{di}(\vartheta )}{\vartheta +k_1+\frac{{\widehat{K}}_i(t)}{\ell }},i=1,2$

(36)

where $\vartheta$ is Laplace variable.

According to the final value theorem, it yields

$\begin{array}{c}\underset{t\to \infty }{\lim }{s}_i(t)=\underset{\vartheta \to 0}{\lim }\vartheta s_i(\vartheta )=\underset{\vartheta \to 0}{\lim }\frac{\vartheta {\rho }_{\mathrm{di}}(\vartheta )}{\vartheta +k_1+\frac{{\hat{K}}_i(t)}{\ell }}\hfill \\ =\frac{1}{k_1+\frac{{\hat{K}}_i(t)}{\ell }}\underset{t\to \infty }{\lim }{\rho }_{\mathrm{di}}(t)\le \frac{{\rho }_{\mathrm{di}}^{*}}{k_1+\frac{{\hat{K}}_i(t)}{\ell }},i=1,2\hfill \end{array}$

(37)

In equations (40), (41), and (42) the authors did not adequately explain the meaning of the variable z or the introduced quantities $p({\omega }_e)$ and $l({\omega }_e)$ . Equation (40) – (43) are the construction process of nonlinear observer and this method is quoted in reference (27). The variable z and quantities $p({\omega }_e)$ and $l({\omega }_e)$ are the auxiliary variables.