Fixed-Time Adaptive Chaotic Control for Permanent Magnet Synchronous Motor Subject to Unknown Parameters and Perturbations

Abstract

It is well known that the permanent magnet synchronous motor (PMSM) exhibits chaotic characteristics when its parameters fall within a certain range, which can lead to system instability. This article proposes an adaptive control strategy for achieving the fixed-time chaotic stabilization of PMSM, even in the presence of unknown parameters and perturbations. The developed controller is synthesized by combining a parametric adaptive mechanism with a fixed-time control technique. The stability analysis demonstrates that the system states under the developed controller can converge to small neighborhoods around the equilibrium point within a fixed time. Thanks to the adoption of the parametric adaptive mechanism, the developed controller is not only insensitive to unknown parameters but also robust against perturbations. Finally, simulated studies are conducted to verify and emphasize the effectiveness of the developed control strategy.

1. Introduction

Permanent magnet synchronous motors (PMSMs) play a crucial role in various industrial applications. It is well-known that PMSMs can exhibit chaotic behavior when their parameters fall within a certain range, which may lead to system instability. Ensuring the safe operation of PMSMs requires the significant consideration of chaotic stabilization [1,2,3]. Chaotic systems exhibit irregular and unpredictable dynamic behavior, which is highly insensitive to initial conditions. To control or manipulate chaotic systems, two main approaches are commonly used. The first approach involves using feedback control techniques to regulate the system states from chaos to order. The second approach involves estimating the system states using filtering techniques and then utilizing these estimates for regulation. The chaotic stabilization of PMSMs falls under the first approach, where appropriate feedback control is added to achieve the desired behavior. Various methods have been extensively investigated for chaotic control, including adaptive control [4,5,6], backstepping control [7,8,9,10,11,12,13,14], sliding mode control [15,16,17,18,19,20,21,22], iterative learning control [23], and intelligent control [24,25,26,27,28,29,30,31,32,33,34,35].

Notably, most of the above controllers can only achieve asymptotic stabilization or, at best, exponential stabilization. In contrast, finite-time control can ensure that the system states stabilize to zero or small neighborhoods around zero within a bounded settling time. Finite-time control is mainly designed based on either the homogeneous system theory [36] or the finite-time Lyapunov stability theory [37]. Finite-time controllers have been utilized for the stabilization of chaotic systems in [38,39,40,41,42,43,44,45,46]. In [47], a finite-time adaptive control scheme was implemented for the chaotic stabilization of PMSM with uncertain parameters. Moreover, ref. [48] developed a terminal sliding mode controller for the finite-time synchronization of fractional-order PMSM.

However, finite-time control has the disadvantage of having the settling time heavily determined by the initial system states, which limits its practical application. Fortunately, fixed-time control can overcome this weakness, as its settling time is bounded and independent of the initial system states. Fixed-time control is primarily designed based on the bilimit homogeneous system theory [49] or the fixed-time Lyapunov stability theory [50,51]. Fixed-time controllers have been developed for the stabilization of chaotic systems in [52,53,54,55,56,57]. For instance, ref. [58] proposed a fixed-time adaptive control approach for the stabilization of the Lorenz system, while a similar controller was applied to the stabilization and synchronization of hyperchaotic Lü systems in [59]. However, these studies [58,59] did not consider perturbations, and the performance of the controllers in the presence of perturbations cannot be guaranteed.

Inspired by the aforementioned content, in this article, we propose an adaptive control strategy for achieving the fixed-time chaotic stabilization of PMSM in the presence of unknown parameters and perturbations. The developed controller is a combination of the parametric adaptive mechanism and fixed-time control technique. The major contributions of this work are two-fold.

• The developed controller is designed within the fixed-time control framework. Stability analysis demonstrates that the developed controller can ensure the system states stabilize within a fixed time to small neighborhoods around the equilibrium point.
• The parametric adaptive mechanism is incorporated into the developed controller to estimate the unknown parameters and perturbations, respectively. Unlike the controllers in [58,59], this design ensures that the developed controller is not only insensitive to unknown parameters but also robust against perturbations.

The remainder of this paper is organized as follows: Section 2 describes the problem and presents some preliminaries. Section 3 presents the main results. Section 4 conducts the simulated studies. Finally, Section 5 summarizes the main conclusions.

2. Preliminaries and Problem Description

2.1. Preliminaries

The following lemmas can support to obtain the main results.

Lemma 1

(Ref. [51]). Consider the nonlinear system:

$$\dot{x}=f\left(x\right),x\left(0\right)=0,x\in {ℝ}^n,$$

(1)

where $f(\cdot )$ is a continuous function vector and $ℝ$ stands for the set of real numbers. If there is a positive definite function $V\left(x\right)$, such that $\dot{V}\left(x\right)\le -{\rho }_1{V}^p\left(x\right)-{\rho }_2{V}^q\left(x\right)+\Delta$, where ${\rho }_1>0$, ${\rho }_2>0$, $0<p<1$, $q>1$, and $\Delta >0$, then system (1) is practically fixed-time stable and $V\left(x\right)$ can stabilize to the following small neighborhood about zero:

$$V\left(x\right)\le \min \left\{{\left(\frac{\Delta }{{\rho }_1\left(1-\epsilon \right)}\right)}^{\frac{1}{p}},{\left(\frac{\Delta }{{\rho }_2\left(1-\epsilon \right)}\right)}^{\frac{1}{q}}\right\},$$

(2)

where $0<\epsilon <1$, in fixed time $T_c\le \frac{1}{{\rho }_1\epsilon \left(1-p\right)}+\frac{1}{{\rho }_2\epsilon \left(q-1\right)}$.

Lemma 2

(Ref. [60]). For $x_1\in ℝ$, $x_2\in ℝ$, $p>0$, $q>0$, $\xi >0$, the following inequality is true:

$${\left|x_1\right|}^p{\left|x_2\right|}^q\le \frac{p}{p+q}\xi {\left|x_1\right|}^{p+q}+\frac{q}{p+q}{\xi }^{-\frac{p}{q}}{\left|x_2\right|}^{p+q}.$$

(3)

Lemma 3

(Ref. [60]). For $x_i\in ℝ$, $i=1,2,\dots ,n$, $0<p\le 1$, and $q>1$, the following inequalities are true:

$${\left({\displaystyle \sum _{i=1}^n\left|x_i\right|}\right)}^p\le {\displaystyle \sum _{i=1}^n{\left|x_i\right|}^p},n^{1-q}{\left({\displaystyle \sum _{i=1}^n\left|x_i\right|}\right)}^q\le {\displaystyle \sum _{i=1}^n{\left|x_i\right|}^q}.$$

(4)

2.2. Problem Description

Referring to [1,2], the PMSM system can be expressed as

$$\{\begin{array}{l}{\dot{i}}_d=-i_d+i_{q}w,\\ {\dot{i}}_q=-i_q-i_{d}w+\gamma w,\\ \dot{w}=\sigma \left(i_q-w\right),\end{array}$$

(5)

where $i_d$ is the quadrature-axis current, $i_q$ is the direct-axis current, $w$ is the motor angular frequency, and $\gamma$ and $\sigma$ are the system parameters. When $\gamma =20$ and $\sigma =5.46$, the PMSM system has the chaotic characteristics. The chaotic phenomena are illustrated in Figure 1 with the initial states set as $i_d\left(0\right)=5$, $i_q\left(0\right)=1$, and $w\left(0\right)=-1$. Let ${\dot{i}}_d={\dot{i}}_q=\dot{w}=0$. It is easy to find that the PMSM system has three equilibrium points $O\left(0,0,0\right)$, $E^{+}\left(19,\sqrt{19},\sqrt{19}\right)$, and $E^{-}\left(19,-\sqrt{19},-\sqrt{19}\right)$.

The PMSM system in the presence of controls and perturbations can be described as

$$\{\begin{array}{l}{\dot{i}}_d=-i_d+i_{q}w+u_1+d_1,\\ {\dot{i}}_q=-i_q-i_{d}w+\gamma w+u_2+d_2,\\ \dot{w}=\sigma \left(i_q-w\right)+u_3+d_3,\end{array}$$

(6)

where $u_1$, $u_2$, and $u_3$ are the control inputs and $d_1$, $d_2$, and $d_3$ are the perturbations. The parameters $\gamma$ and $\sigma$ and the perturbations $d_1$, $d_2$, and $d_3$ are supposed to be unknown in the control design.

The objective is to develop an appropriate controller such that the system states can stabilize to the small neighborhoods about the specific equilibrium point in fixed time.

3. Main Results

The main results are provided in this section. First, the controller is developed for the chaotic stabilization about the equilibrium point $O\left(0,0,0\right)$. Then, the controllers for the chaotic stabilization about the equilibrium points $E^{+}\left(19,\sqrt{19},\sqrt{19}\right)$ and $E^{-}\left(19,-\sqrt{19},-\sqrt{19}\right)$ are designed in a similar way.

3.1. Chaotic Control about $O\left(0,0,0\right)$

Theorem 1.

For PMSM system (6), if the fixed-time adaptive controller is developed as

$$\{\begin{array}{l}{u}_1=-k_{11}{\mathrm{sig}}^p\left(i_d\right)-k_{12}{\mathrm{sig}}^q\left(i_d\right)+i_d-i_{q}w-{\widehat{d}}_1,\\ u_2=-k_{21}{\mathrm{sig}}^p\left(i_q\right)-k_{22}{\mathrm{sig}}^q\left(i_q\right)+i_q+i_{d}w-\widehat{\gamma }w-{\widehat{d}}_2,\\ u_3=-k_{31}{\mathrm{sig}}^p\left(w\right)-k_{32}{\mathrm{sig}}^q\left(w\right)-\widehat{\sigma }\left(i_q-w\right)-{\widehat{d}}_3,\end{array}$$

(7)

where $k_{11}>0$, $k_{12}>0$, $k_{21}>0$, $k_{22}>0$, $k_{31}>0$, $k_{32}>0$, $0<p<1$, $q>1$, $\widehat{\gamma }$, $\widehat{\sigma }$, ${\widehat{d}}_1$, ${\widehat{d}}_2$, and ${\widehat{d}}_3$ are the estimations of $\gamma$, $\sigma$, $d_1$, $d_2$, and $d_3$, ${\mathrm{sig}}^p(\cdot )$ is defined as ${\mathrm{sig}}^p\left(x\right)={\left|x\right|}^p\mathrm{sgn}\left(x\right)$, and the parametric adaptive mechanism is provided as

$$\{\begin{array}{l}\dot{\widehat{\gamma }}=-{\eta }_{11}\widehat{\gamma }+{\eta }_{12}w{i}_q,\\ \dot{\widehat{\sigma }}=-{\eta }_{21}\widehat{\sigma }+{\eta }_{22}\left(i_q-w\right)w,\\ {\dot{\widehat{d}}}_1=-{\eta }_{31}{\widehat{d}}_1+{\eta }_{32}{i}_d,\\ {\dot{\widehat{d}}}_2=-{\eta }_{41}{\widehat{d}}_2+{\eta }_{42}{i}_q,\\ {\dot{\widehat{d}}}_3=-{\eta }_{51}{\widehat{d}}_3+{\eta }_{52}w,\end{array}$$

(8)

where ${\eta }_{11}>0$, ${\eta }_{12}>0$, ${\eta }_{21}>0$, ${\eta }_{22}>0$, ${\eta }_{31}>0$, ${\eta }_{32}>0$, ${\eta }_{41}>0$, ${\eta }_{42}>0$, ${\eta }_{51}>0$, and ${\eta }_{52}>0$, then the system states can stabilize to the small neighborhoods about the equilibrium point $O\left(0,0,0\right)$ in fixed time.

Proof. 

The Lyapunov function is involved as

$$V=\frac{1}{2}{i}_d^2+\frac{1}{2}{i}_q^2+\frac{1}{2}{w}^2+\frac{1}{2{\eta }_{12}}{\tilde{\gamma }}^2+\frac{1}{2{\eta }_{22}}{\tilde{\sigma }}^2+\frac{1}{2{\eta }_{32}}{\tilde{d}}_1^2+\frac{1}{2{\eta }_{42}}{\tilde{d}}_2^2+\frac{1}{2{\eta }_{52}}{\tilde{d}}_3^2,$$

(9)

where $\tilde{\gamma }$, $\tilde{\sigma }$, ${\tilde{d}}_1$, ${\tilde{d}}_2$, and ${\tilde{d}}_3$ are the estimation errors of $\tilde{\gamma }=\gamma -\widehat{\gamma }$, $\tilde{\sigma }=\sigma -\widehat{\sigma }$, ${\tilde{d}}_1=d_1-{\widehat{d}}_1$, ${\tilde{d}}_2=d_2-{\widehat{d}}_2$, and ${\tilde{d}}_3=d_3-{\widehat{d}}_3$. Evaluating the time differentiation of $V$ yields

$$\begin{array}{ll}\dot{V}=& i_d\left(-i_d+i_{q}w+u_1+d_1\right)+i_q\left(-i_q-i_{d}w+\gamma w+u_2+d_2\right)+w\left(\sigma \left(i_q-w\right)+u_3+d_3\right)\\ & -\frac{1}{{\eta }_{12}}\tilde{\gamma }\dot{\widehat{\gamma }}-\frac{1}{{\eta }_{22}}\tilde{\sigma }\dot{\widehat{\sigma }}-\frac{1}{{\eta }_{32}}{\tilde{d}}_1{\dot{\widehat{d}}}_1-\frac{1}{{\eta }_{42}}{\tilde{d}}_2{\dot{\widehat{d}}}_2-\frac{1}{{\eta }_{52}}{\tilde{d}}_3{\dot{\widehat{d}}}_3.\end{array}$$

(10)

Substituting the fixed-time adaptive controller (7) and the parametric adaptive mechanism (8) into (10), we have

$$\begin{array}{ll}\dot{V}=& i_d\left(-k_{11}{\mathrm{sig}}^p\left(i_d\right)-k_{12}{\mathrm{sig}}^q\left(i_d\right)+{\tilde{d}}_1\right)+i_q\left(-k_{21}{\mathrm{sig}}^p\left(i_q\right)-k_{22}{\mathrm{sig}}^q\left(i_q\right)+\tilde{\gamma }w+{\tilde{d}}_2\right)\\ & +w\left(-k_{31}{\mathrm{sig}}^p\left(w\right)-k_{32}{\mathrm{sig}}^q\left(w\right)+\tilde{\sigma }\left(i_q-w\right)+{\tilde{d}}_3\right)-\frac{1}{{\eta }_{12}}\tilde{\gamma }\left(-{\eta }_{11}\widehat{\gamma }+{\eta }_{12}{i}_{q}w\right)\\ & -\frac{1}{{\eta }_{22}}\tilde{\sigma }\left(-{\eta }_{21}\widehat{\sigma }+{\eta }_{22}w\left(i_q-w\right)\right)-\frac{1}{{\eta }_{32}}{\tilde{d}}_1\left(-{\eta }_{31}{\widehat{d}}_1+{\eta }_{32}{i}_d\right)\\ & -\frac{1}{{\eta }_{42}}{\tilde{d}}_2\left(-{\eta }_{41}{\widehat{d}}_2+{\eta }_{42}{i}_q\right)-\frac{1}{{\eta }_{52}}{\tilde{d}}_3\left(-{\eta }_{51}{\widehat{d}}_3+{\eta }_{52}w\right)\\ =& -k_{11}{\left|i_d\right|}^{p+1}-k_{12}{\left|i_d\right|}^{q+1}-k_{21}{\left|i_q\right|}^{p+1}-k_{22}{\left|i_q\right|}^{q+1}-k_{31}{\left|w\right|}^{p+1}-k_{32}{\left|w\right|}^{q+1}\\ & +\frac{{\eta }_{11}}{{\eta }_{12}}\tilde{\gamma }\widehat{\gamma }+\frac{{\eta }_{21}}{{\eta }_{22}}\tilde{\sigma }\widehat{\sigma }+\frac{{\eta }_{31}}{{\eta }_{32}}{\tilde{d}}_1{\widehat{d}}_1+\frac{{\eta }_{41}}{{\eta }_{42}}{\tilde{d}}_2{\widehat{d}}_2+\frac{{\eta }_{51}}{{\eta }_{52}}{\tilde{d}}_3{\widehat{d}}_3.\end{array}$$

(11)

It is not difficult to derive the following inequalities:

$$\frac{{\eta }_{11}}{{\eta }_{12}}\tilde{\gamma }\widehat{\gamma }\le -\frac{{\eta }_{11}}{2{\eta }_{12}}{\tilde{\gamma }}^2+\frac{{\eta }_{11}}{2{\eta }_{12}}{\gamma }^2,$$

(12)

$$\frac{{\eta }_{21}}{{\eta }_{22}}\tilde{\sigma }\widehat{\sigma }\le -\frac{{\eta }_{21}}{2{\eta }_{22}}{\tilde{\sigma }}^2+\frac{{\eta }_{21}}{2{\eta }_{22}}{\sigma }^2,$$

(13)

$$\frac{{\eta }_{31}}{{\eta }_{32}}{\tilde{d}}_1{\widehat{d}}_1\le -\frac{{\eta }_{31}}{2{\eta }_{32}}{\tilde{d}}_1^2+\frac{{\eta }_{31}}{2{\eta }_{32}}{d}_1^2,$$

(14)

$$\frac{{\eta }_{41}}{{\eta }_{42}}{\tilde{d}}_2{\widehat{d}}_2\le -\frac{{\eta }_{41}}{2{\eta }_{42}}{\tilde{d}}_2^2+\frac{{\eta }_{41}}{2{\eta }_{42}}{\tilde{d}}_2^2,$$

(15)

$$\frac{{\eta }_{51}}{{\eta }_{52}}{\tilde{d}}_3{\widehat{d}}_3\le -\frac{{\eta }_{51}}{2{\eta }_{52}}{\tilde{d}}_3^2+\frac{{\eta }_{51}}{2{\eta }_{52}}{d}_3^2,$$

(16)

Substituting the above inequalities into (11), we have

$$\begin{array}{ll}\dot{V}=& -k_{11}{\left|i_d\right|}^{p+1}-k_{12}{\left|i_d\right|}^{q+1}-k_{21}{\left|i_q\right|}^{p+1}-k_{22}{\left|i_q\right|}^{q+1}-k_{31}{\left|w\right|}^{p+1}-k_{32}{\left|w\right|}^{q+1}\\ & -{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}-{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-{\left(\frac{{\eta }_{21}}{4{\eta }_{22}}{\tilde{\sigma }}^2\right)}^{\frac{p+1}{2}}-{\left(\frac{{\eta }_{21}}{4{\eta }_{22}}{\tilde{\sigma }}^2\right)}^{\frac{q+1}{2}}\\ & -{\left(\frac{{\eta }_{31}}{4{\eta }_{32}}{\tilde{d}}_1^2\right)}^{\frac{p+1}{2}}-{\left(\frac{{\eta }_{31}}{4{\eta }_{32}}{\tilde{d}}_1^2\right)}^{\frac{q+1}{2}}-{\left(\frac{{\eta }_{41}}{4{\eta }_{42}}{\tilde{d}}_2^2\right)}^{\frac{p+1}{2}}-{\left(\frac{{\eta }_{41}}{4{\eta }_{42}}{\tilde{d}}_2^2\right)}^{\frac{q+1}{2}}\\ & -{\left(\frac{{\eta }_{51}}{4{\eta }_{52}}{\tilde{d}}_3^2\right)}^{\frac{p+1}{2}}-{\left(\frac{{\eta }_{51}}{4{\eta }_{52}}{\tilde{d}}_3^2\right)}^{\frac{q+1}{2}}+\Delta ,\end{array}$$

(17)

where $\Delta$ is defined as

$$\begin{array}{ll}\Delta =& {\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{2{\eta }_{12}}{\tilde{\gamma }}^2+{\left(\frac{{\eta }_{21}}{4{\eta }_{22}}{\tilde{\sigma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{21}}{4{\eta }_{22}}{\tilde{\sigma }}^2\right)}^{\frac{q+1}{2}}\\ & -\frac{{\eta }_{21}}{2{\eta }_{22}}{\tilde{\sigma }}^2+{\left(\frac{{\eta }_{31}}{4{\eta }_{32}}{\tilde{d}}_1^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{31}}{4{\eta }_{32}}{\tilde{d}}_1^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{31}}{2{\eta }_{32}}{\tilde{d}}_1^2+{\left(\frac{{\eta }_{41}}{4{\eta }_{42}}{\tilde{d}}_2^2\right)}^{\frac{p+1}{2}}\\ & +{\left(\frac{{\eta }_{41}}{4{\eta }_{42}}{\tilde{d}}_2^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{41}}{2{\eta }_{42}}{\tilde{d}}_2^2+{\left(\frac{{\eta }_{51}}{4{\eta }_{52}}{\tilde{d}}_3^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{51}}{4{\eta }_{52}}{\tilde{d}}_3^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{51}}{2{\eta }_{52}}{\tilde{d}}_3^2.\end{array}$$

(18)

Consider the item ${\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{2{\eta }_{12}}{\tilde{\gamma }}^2$ in the above inequality. Two cases are discussed in the sequel. For the case that $\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\ge 1$, we have

$${\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{2{\eta }_{12}}{\tilde{\gamma }}^2\le {\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2.$$

(19)

For the case that $\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2<1$, recalling Lemma 2, we have

$${\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{2{\eta }_{12}}{\tilde{\gamma }}^2\le {\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}-\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\le \left(1-\overline{p}\right){\overline{p}}^{\frac{\overline{p}}{1-\overline{p}}},$$

(20)

where $\overline{p}=\frac{p+1}{2}$. Construct a compact set ${\Theta }_1$ satisfying ${\Theta }_1=\left\{\tilde{\gamma }\in ℝ|\left|\tilde{\gamma }\right|\le {\theta }_1\right\}$, where ${\theta }_1>0$. Combining (19) and (20), we have the following inequality:

$${\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\tilde{\gamma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{2{\eta }_{12}}{\tilde{\gamma }}^2\le {\mu }_1,$$

(21)

where ${\mu }_1$ is defined as

$${\mu }_1=\{\begin{array}{l}\left(1-\overline{p}\right){\overline{p}}^{\frac{\overline{p}}{1-\overline{p}}},{\theta }_1<2\sqrt{\frac{{\eta }_{12}}{{\eta }_{11}}},\hfill \\ {\left(\frac{{\eta }_{11}}{4{\eta }_{12}}{\theta }_1^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{11}}{4{\eta }_{12}}{\theta }_1^2,{\theta }_1\ge 2\sqrt{\frac{{\eta }_{12}}{{\eta }_{11}}}.\hfill \end{array}$$

(22)

Likewise, the following inequalities can be easily derived:

$${\left(\frac{{\eta }_{21}}{4{\eta }_{22}}{\tilde{\sigma }}^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{21}}{4{\eta }_{22}}{\tilde{\sigma }}^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{21}}{2{\eta }_{22}}{\tilde{\sigma }}^2\le {\mu }_2,$$

(23)

$${\left(\frac{{\eta }_{31}}{4{\eta }_{32}}{\tilde{d}}_1^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{31}}{4{\eta }_{32}}{\tilde{d}}_1^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{31}}{2{\eta }_{32}}{\tilde{d}}_1^2\le {\mu }_3,$$

(24)

$${\left(\frac{{\eta }_{41}}{4{\eta }_{42}}{\tilde{d}}_2^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{41}}{4{\eta }_{42}}{\tilde{d}}_2^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{41}}{2{\eta }_{42}}{\tilde{d}}_2^2\le {\mu }_4,$$

(25)

$${\left(\frac{{\eta }_{51}}{4{\eta }_{52}}{\tilde{d}}_3^2\right)}^{\frac{p+1}{2}}+{\left(\frac{{\eta }_{51}}{4{\eta }_{52}}{\tilde{d}}_3^2\right)}^{\frac{q+1}{2}}-\frac{{\eta }_{51}}{2{\eta }_{52}}{\tilde{d}}_3^2\le {\mu }_5.$$

(26)

Substituting the above inequalities into (17) and recalling Lemma 3, we have

$$\dot{V}\le -{\rho }_1{V}^{\frac{p+1}{2}}-{\rho }_2{V}^{\frac{q+1}{2}}+\overline{\Delta },$$

(27)

where ${\rho }_1$, ${\rho }_2$, and $\overline{\Delta }$ are defined as

$${\rho }_1=\min \left\{2^{\frac{p+1}{2}}{k}_{11},2^{\frac{p+1}{2}}{k}_{21},2^{\frac{p+1}{2}}{k}_{31},{\left(\frac{{\eta }_{11}}{2}\right)}^{\frac{p+1}{2}},{\left(\frac{{\eta }_{21}}{2}\right)}^{\frac{p+1}{2}},{\left(\frac{{\eta }_{31}}{2}\right)}^{\frac{p+1}{2}},{\left(\frac{{\eta }_{41}}{2}\right)}^{\frac{p+1}{2}},{\left(\frac{{\eta }_{51}}{2}\right)}^{\frac{p+1}{2}}\right\},$$

(28)

$${\rho }_2=8^{\frac{1-q}{2}}\min \left\{2^{\frac{q+1}{2}}{k}_{12},2^{\frac{q+1}{2}}{k}_{22},2^{\frac{q+1}{2}}{k}_{32},{\left(\frac{{\eta }_{12}}{2}\right)}^{\frac{q+1}{2}},{\left(\frac{{\eta }_{22}}{2}\right)}^{\frac{q+1}{2}},{\left(\frac{{\eta }_{32}}{2}\right)}^{\frac{q+1}{2}},{\left(\frac{{\eta }_{42}}{2}\right)}^{\frac{q+1}{2}},{\left(\frac{{\eta }_{52}}{2}\right)}^{\frac{q+1}{2}}\right\},$$

(29)

$$\overline{\Delta }={\mu }_1+{\mu }_2+{\mu }_3+{\mu }_4+{\mu }_5+\frac{{\eta }_{11}}{2{\eta }_{12}}{\gamma }^2+\frac{{\eta }_{21}}{2{\eta }_{22}}{\sigma }^2+\frac{{\eta }_{31}}{2{\eta }_{32}}{d}_1^2+\frac{{\eta }_{41}}{2{\eta }_{42}}{d}_2^2+\frac{{\eta }_{51}}{2{\eta }_{52}}{d}_3^2.$$

(30)

By Lemma 1, the closed-loop system is practically fixed-time stable and $V$ can stabilize to the following small neighborhood about zero:

$$V\le \min \left\{{\left(\frac{\overline{\Delta }}{{\rho }_1\left(1-\epsilon \right)}\right)}^{\frac{p+1}{2}},{\left(\frac{\overline{\Delta }}{{\rho }_2\left(1-\epsilon \right)}\right)}^{\frac{q+1}{2}}\right\},$$

(31)

where $0<\epsilon <1$, in fixed time, $T_c\le \frac{2}{{\rho }_1\epsilon \left(1-p\right)}+\frac{2}{{\rho }_2\epsilon \left(q-1\right)}$. Recalling the definition of $V$, it follows that all error variables $i_d$, $i_q$, $w$, $\tilde{\gamma }$, $\tilde{\sigma }$, ${\tilde{d}}_1$, ${\tilde{d}}_2$, and ${\tilde{d}}_3$ can stabilize to the small neighborhoods about zero in fixed time. This completes the proof. □

3.2. Chaotic Control about $E^{+}\left(19,\sqrt{19},\sqrt{19}\right)$ and $E^{-}\left(19,-\sqrt{19},-\sqrt{19}\right)$

Similarly, the controllers for the chaotic stabilization about the equilibrium points $E^{+}\left(19,\sqrt{19},\sqrt{19}\right)$ and $E^{-}\left(19,-\sqrt{19},-\sqrt{19}\right)$ are provided in Theorems 2 and 3, respectively.

Theorem 2.

For PMSM system (6), if the fixed-time adaptive controller is developed as

$$\{\begin{array}{l}{u}_1=-k_{11}{\mathrm{sig}}^p\left(i_d-19\right)-k_{12}{\mathrm{sig}}^q\left(i_d-19\right)+i_d-i_{q}w-{\widehat{d}}_1,\\ u_2=-k_{21}{\mathrm{sig}}^p\left(i_q-\sqrt{19}\right)-k_{22}{\mathrm{sig}}^q\left(i_q-\sqrt{19}\right)+i_q+i_{d}w-\widehat{\gamma }w-{\widehat{d}}_2,\\ u_3=-k_{31}{\mathrm{sig}}^p\left(w-\sqrt{19}\right)-k_{32}{\mathrm{sig}}^q\left(w-\sqrt{19}\right)-\widehat{\sigma }\left(i_q-w\right)-{\widehat{d}}_3,\end{array}$$

(32)

and the parametric adaptive mechanism is provided as

$$\{\begin{array}{l}\dot{\widehat{\gamma }}=-{\eta }_{11}\widehat{\gamma }+{\eta }_{12}w\left(i_q-\sqrt{19}\right),\\ \dot{\widehat{\sigma }}=-{\eta }_{21}\widehat{\sigma }+{\eta }_{22}\left(i_q-w\right)\left(w-\sqrt{19}\right),\\ {\dot{\widehat{d}}}_1=-{\eta }_{31}{\widehat{d}}_1+{\eta }_{32}\left(i_d-19\right),\\ {\dot{\widehat{d}}}_2=-{\eta }_{41}{\widehat{d}}_2+{\eta }_{42}\left(i_q-\sqrt{19}\right),\\ {\dot{\widehat{d}}}_3=-{\eta }_{51}{\widehat{d}}_3+{\eta }_{52}\left(w-\sqrt{19}\right),\end{array}$$

(33)

then the system states can stabilize to the small neighborhoods about the equilibrium point $E^{+}\left(19,\sqrt{19},\sqrt{19}\right)$ in fixed time.

Theorem 3.

For PMSM system (6), if the fixed-time adaptive controller is developed as

$$\{\begin{array}{l}{u}_1=-k_{11}{\mathrm{sig}}^p\left(i_d-19\right)-k_{12}{\mathrm{sig}}^q\left(i_d-19\right)+i_d-i_{q}w-{\widehat{d}}_1,\\ u_2=-k_{21}{\mathrm{sig}}^p\left(i_q+\sqrt{19}\right)-k_{22}{\mathrm{sig}}^q\left(i_q+\sqrt{19}\right)+i_q+i_{d}w-\widehat{\gamma }w-{\widehat{d}}_2,\\ u_3=-k_{31}{\mathrm{sig}}^p\left(w+\sqrt{19}\right)-k_{32}{\mathrm{sig}}^q\left(w+\sqrt{19}\right)-\widehat{\sigma }\left(i_q-w\right)-{\widehat{d}}_3,\end{array}$$

(34)

and the parametric adaptive mechanism is provided as

$$\{\begin{array}{l}\dot{\widehat{\gamma }}=-{\eta }_{11}\widehat{\gamma }+{\eta }_{12}w\left(i_q+\sqrt{19}\right),\\ \dot{\widehat{\sigma }}=-{\eta }_{21}\widehat{\sigma }+{\eta }_{22}\left(i_q-w\right)\left(w+\sqrt{19}\right),\\ {\dot{\widehat{d}}}_1=-{\eta }_{31}{\widehat{d}}_1+{\eta }_{32}\left(i_d-19\right),\\ {\dot{\widehat{d}}}_2=-{\eta }_{41}{\widehat{d}}_2+{\eta }_{42}\left(i_q+\sqrt{19}\right),\\ {\dot{\widehat{d}}}_3=-{\eta }_{51}{\widehat{d}}_3+{\eta }_{52}\left(w+\sqrt{19}\right),\end{array}$$

(35)

then the system states can stabilize to the small neighborhoods about the equilibrium point$E^{-}\left(19,-\sqrt{19},-\sqrt{19}\right)$ in fixed time.

Remark 1.

It is worth noting that perturbations were not considered in [58,59], and as a result, the performance of the controllers in those studies in the presence of perturbations cannot be guaranteed. In contrast, in this article, we have embedded the parametric adaptive mechanism (Equation (8)) to estimate the unknown parameters and perturbations. This design ensures that the developed controller is not only insensitive to unknown parameters, but also robust against perturbations.

Remark 2.

The dynamic behavior of chaotic systems, such as the PMSM system, is irregular, unpredictable, and highly insensitive to initial conditions. Due to this inherent nature, achieving complete stabilization of chaotic systems is impossible. In fact, the developed controller cannot stabilize the attractor itself but can only guide the system states to achieve the desired behavior.

4. Simulated Studies

Simulated studies are carried out to demonstrate the developed control strategy in this section. Without loss of generality, we take account of the chaotic stabilization about the equilibrium point $O\left(0,0,0\right)$. The following simulations are deployed though two scenarios. Scenario 1 is the performance comparisons and Scenario 2 is the fixed-time stability tests.

4.1. Performance Comparisons

In Scenario 1, the performance comparisons are made to show the advantages of the developed controller. In the simulations, the parameters are set as $\gamma =20$ and $\sigma =5.46$, and the perturbations are given as $d_1=0.2\sin \left(0.8t\right)$, $d_2=0.3\cos \left(0.9t\right)$, and $d_3=0.2\sin \left(0.9t\right)$. The parameters and perturbations are supposed to be unknown in the control design. The initial system states are chosen as $i_d\left(0\right)=1$, $i_q\left(0\right)=3$, and $w\left(0\right)=-2$.

Besides the developed controller (7), the existing linear feedback controller is also utilized for comparisons, which is presented as

$$\{\begin{array}{l}{u}_1=-k_1{i}_d,\\ u_2=-k_2{i}_q,\\ u_3=-k_{3}w,\end{array}$$

(36)

where $k_1>0$, $k_2>0$, and $k_3>0$.

In the simulations, the parameters of the developed controller (7) are made as $k_{11}=10$, $k_{12}=10$, $k_{21}=10$, $k_{22}=10$, $k_{31}=10$, $k_{32}=10$, $p=9/11$, $q=13/11$, ${\eta }_{11}=0.01$, ${\eta }_{12}=1$, ${\eta }_{21}=0.01$, ${\eta }_{22}=1$, ${\eta }_{31}=0.01$, ${\eta }_{32}=1$, ${\eta }_{41}=0.01$, ${\eta }_{42}=1$, ${\eta }_{51}=0.01$, and ${\eta }_{52}=1$. The initial values of the adaptive parameters are set as $\widehat{\gamma }\left(0\right)=18$, $\widehat{\sigma }\left(0\right)=5$, ${\widehat{d}}_1\left(0\right)=0$, ${\widehat{d}}_2\left(0\right)=0$, and ${\widehat{d}}_3\left(0\right)=0$. The parameters of the existing controller (36) are made as $k_1=12$, $k_2=12$, and $k_3=12$.

The simulation results of Scenario 1 are given in Figure 2, Figure 3 and Figure 4. Specifically, Figure 2 shows the time history of the system states. It is desirable that both the developed and existing controllers can fulfill the chaotic stabilization successfully. Quantitatively, the steady-state state errors $i_d$, $i_q$, and $w$ under the developed controller are within the ranges of $\pm 0.008$, $\pm 0.016$, and $\pm 0.01$, and under the existing controller are within the ranges of $\pm 0.02$, $\pm 0.06$, and $\pm 0.04$. The existing controller can attain the much larger steady-state state errors than the developed controller. This implies that the stabilization performance of the developed controller is better than that of the existing controller in terms of higher control accuracy. The time history of the control inputs is presented in Figure 3. The control inputs under the existing controller even have a larger magnitude than those under the developed controller. This means that comparisons are quite fair or at least not partial to the developed controller. Figure 4 gives the time history of the adaptive parameters. These adaptive parameters are changing smoothly with time. Owing to the utilization of the parametric adaptive mechanism, the developed controller is not only insensitive to unknown parameters but also robust against perturbations. By contrast, the disturbance rejection property of the existing linear feedback controller is really poor.

4.2. Fixed-Time Stability Tests

In Scenario 2, the tests with different initial system states are provided to examine the fixed-time stability capability of the developed controller. Three groups of initial system states are considered. In Group 1, the initial system states are chosen as $i_d\left(0\right)=1$, $i_q\left(0\right)=3$, and $w\left(0\right)=-2$. In Group 2, the initial system states are chosen as $i_d\left(0\right)=-1.5$, $i_q\left(0\right)=-0.5$, and $w\left(0\right)=1$. In Group 3, the initial system states are chosen as $i_d\left(0\right)=3$, $i_q\left(0\right)=-2.5$, and $w\left(0\right)=2.5$. In Group 4, the initial system states are chosen as $i_d\left(0\right)=-2.5$, $i_q\left(0\right)=2$, and $w\left(0\right)=0.5$. The perturbations and parameters of the developed controller (7) are chosen in the same way as those in Scenario 1.

The simulation results of Scenario 2 are provided in Figure 5 and Figure 6. It is clearly seen that the developed controller can realize the chaotic stabilization within the similar settling time $T_c=0.25\mathrm{s}$ in the presence of different initial system states. This means that the settling time is bounded and particularly the upper bound is irrelevant to the initial system states. Therefore, it can be derived that the developed controller has the fixed-time stability capability.

5. Conclusions

This article addresses the fixed-time chaotic stabilization of PMSM subject to unknown parameters and perturbations by developing an adaptive control strategy. The developed controller is synthesized by combining it with the parametric adaptive mechanism under the fixed-time control framework. Stability analysis demonstrates that the system states under the developed controller can stabilize within fixed time to small neighborhoods around the equilibrium point. A distinctive advantage of the developed controller is its insensitivity to unknown parameters and robustness against perturbations. Finally, the developed control strategy is verified and highlighted through simulated studies.