Fixed-Time Consensus Multi-Agent-Systems-Based Speed Cooperative Control for Multiple Permanent Magnet Synchronous Motors with Complementary Sliding Mode Control

Abstract

To improve the tracking performance and robustness of traditional multi-motor speed cooperative control, this paper proposes a speed cooperative control method for multiple permanent magnet synchronous motors (multi-PMSMs) based on the fixed-time consensus protocol for multi-agent systems (MASs) combined with CSMC. Firstly, the speed regulation system of multi-PMSMs is regarded as a MAS. By designing a distributed consensus protocol based on an undirected communication topology, the system achieves fixed-time consensus convergence. Then, a terminal integral sliding mode observer (TISMO) is designed to estimate disturbances, and feedforward compensation is introduced into the consensus protocol to obtain the desired q-axis current. Furthermore, within the framework of the vector control speed cooperative system of PMSMs, a CSMC is designed to track the q-axis reference current. Meanwhile, the stability of the above controllers and observers is theoretically proven using the Lyapunov functions. Finally, comparative experiments are conducted on a multi-PMSM speed regulation experimental platform to verify the proposed control method against the traditional deviation coupling control (DCC) method. The results indicate that under the new control method proposed in this paper, the chattering phenomenon is reduced by about 2 r/min compared to the traditional DCC method. During sudden load and sudden relief load conditions, the speed fluctuation is reduced by approximately 4%, demonstrating good tracking performance and strong robustness.

1. Introduction

Due to the advantages of high efficiency, high power density, and excellent dynamic response, permanent magnet synchronous motors (PMSMs) are widely applied in industrial production [1,2,3]. For modern complex industrial production processes such as electric vehicles, textiles, and aerospace, the speed synchronization control accuracy of multi-motor control systems is a crucial performance indicator determining production quality.

Traditional multi-motor speed control systems include master–slave control, cross-coupling control (CCC), deviation coupling control (DCC), and others. In master–slave control, due to the slave motor following the master motor’s speed instead of the reference speed, the slave motor’s speed always lags behind the master motor, resulting in significant synchronization errors during startup and when the slave motors experience load disturbances, leading to chattering in the slave motors [4]. Therefore, the concept of CCC strategy is proposed, using the speed difference between two motors as the feedback signal to achieve speed control of multiple motors. Reference [5] designed a CCC system based on model predictive, effectively solving chattering in the speed regulation system. However, when the CCC method is extended to more than two motors, it significantly impacts the synchronous performance of the motors. To address this issue, DCC was proposed based on CCC. Reference [6] proposed a mean DCC strategy for multi-motor speed regulation systems, which can improve the synchronization performance of the system. Traditional coupling control strategies provide an effective method for achieving speed coordination control in multi-motor systems. However, the introduction of the coupling structure causes mutual influence between tracking control and synchronization control, making it difficult to balance both performances. Additionally, it lacks flexibility, as changes in the number of motors require modifications to the original coupling structure. To achieve synchronized control, the rotational speed information of all motors in the system must be obtained, which leads to a complex structure. As the scale of the multi-motor system increases, traditional coordinated control methods struggle to meet real-time computational requirements [7].

Nowadays, multi-agent systems (MASs) are widely applied because of the advantages of flexibility, reliability, and strong self-organizing capabilities. MASs composed of simple agents are often used to address distributed control problems, such as aerial vehicle formations [8], multi-robot systems [9], and rail transit [10]. Considering the fundamental similarity between the consensus control in MASs and the speed cooperative control in multi-motor systems, each motor system can be regarded as an agent, and graph theory can be employed to depict communication among agents. Introducing the concept of consensus into multi-motor speed cooperative control systems and leveraging the superior synchronization capability of multi-agents to achieve speed synchronization control for multi-motor speed cooperative control systems. This provides a novel approach and means for addressing the speed collaborative control problem in multi-motor speed cooperative control systems.

Based on the convergence time of MASs, it can be categorized into various types, such as the asymptotic consensus, the finite-time consensus, and the fixed-time consensus. Reference [11] introduces an asymptotic exponential consensus to ensure convergence in handling MASs with uncertain agent dynamics. Compared to the asymptotic exponential consensus, the finite-time consensus offers advantages such as faster convergence speed and better robustness to external uncertainties [12]. The finite-time convergence implies that a MAS reaches the desired consensus or synchronization state within a finite amount of time. This is typically applicable to applications that require a MAS to rapidly converge to a consensus state within a specified time frame [13], such as obstacle avoidance conditions, decision-making in emergencies [14], and so on. Reference [15] introduces and analyzes two distributed control protocols based on finite-time stability to address the first-order nonlinear MASs problem with rapid convergence. The initial state typically affects the convergence time of the finite-time consensus. When the initial state of the system is large, the convergence time will correspondingly increase, and the effectiveness of achieving finite-time convergence may not be as pronounced [16]. Therefore, the fixed-time consistency theory was developed to address this issue. The fixed-time consensus requires that a MAS reaches a consensus synchronization state within a predetermined fixed time, regardless of the initial state. The fixed-time consensus is typically applicable to applications with strict performance and robustness requirements for MASs, such as formation control of aircraft, and synchronization in satellite networks. Reference [17] proposed a fixed-time consistency protocol to achieve convergence of the system within a fixed time. The results show that the system performance is significantly better than the finite-time consistency protocol.

In multi-motor speed regulation systems, the accuracy of speed synchronization control depends not only on the speed loop controller structure and control algorithms but also heavily on the design of the current loop controller. Sliding mode control (SMC) is widely utilized in the design of controllers for speed regulation systems due to its significant advantages such as high precision, strong robustness, and simple structure [18]. In traditional SMCs, a high switching control gain is typically employed to ensure system robustness. However, this high switching gain strategy often leads to chattering phenomena [19]. Complementary sliding mode control (CSMC), compared to other control methods, not only effectively reduces the chattering phenomenon but also offers improved control precision [20]. Through this approach, it can be ensured that the velocity/position error of the motor is reduced by at least 50% when compared to traditional SMC [21]. Reference [22] proposes a disturbance-observer-based CSMC, which endows the entire PMSM control system with excellent capabilities for suppressing chattering and ensuring tracking performance. Additionally, the conventional SMC method cannot guarantee the robustness of the servo system in the approaching phase [23]. Therefore, the TISMC method has been developed [24]. Compared with traditional SMC, TISMC can set the initial value of the integral term to place the system in the sliding phase at the beginning, eliminating the approaching phase and enhancing system robustness.

The main contributions are as follows:

• This paper innovatively considers the multi-PMSM speed control system as a MAS, transforming the problem of coordinated speed control of multi-PMSMs into a consensus problem of MASs.
• The designed consensus protocol enables the system to converge within a fixed time. Additionally, a terminal integral sliding mode observer (TISMO) is designed to observe disturbances and compensate for them, thereby obtaining the desired q-axis current.
• Within the framework of PMSM vector control, a CSMC current loop controller is designed for each PMSM system, effectively suppressing system chattering and enhancing the robustness of the system.

The structure of this paper is as follows: Section 2 presents the mathematical model of the multi-PMSM system, and some used lemmas are given. In Section 3, a consensus protocol and a disturbance observer are designed, and their stability is proven using the Lyapunov function. In Section 4, a CSMC current loop controller is designed. In Section 5, comparative experimental validation is carried out and the results are analyzed. This paper is summarized in Section 6.

2. Preliminaries

2.1. Mathematical Model of Multi-PMSMs System

The mathematical model of the multi-PMSM system is described as follows:

$$\left\{\begin{array}{l}{\dot{\omega }}_{r,i}=-{\displaystyle \frac{B_i}{J_i}}{\omega }_{r,i}-{\displaystyle \frac{n_p}{J_i}}{T}_{L,i}+{\displaystyle \frac{1.5n_p^2{\psi }_{f,i}}{J_i}}{i}_{q,i}\\ {\dot{i}}_{d,i}=-{\displaystyle \frac{R_{s,i}}{L_i}}{i}_{d,i}+{\omega }_{r,i}{i}_{q,i}+{\displaystyle \frac{1}{L_i}}{u}_{d,i}\\ {\dot{i}}_{q,i}=-{\omega }_{r,i}{i}_{d,i}-{\displaystyle \frac{R_{s,i}}{L_i}}{i}_{q,i}-{\displaystyle \frac{{\psi }_{f,i}}{L_i}}{\omega }_{r,i}+{\displaystyle \frac{1}{L_i}}{u}_{q,i}\end{array}\right.$$

(1)

${\omega }_{r,i}$ represents the electrical angular velocity; ${\psi }_{f,i}$ is the permanent magnet flux chain; $n_p$ is the pole pairs; $L_i$ is the dq-axis inductance; $B_i$ represents the friction coefficient; $T_{L,i}$ represents the load torque; $J_i$ represents the rotor inertia; $u_{d,i}$, $u_{q,i}$, $i_{d,i}$, $i_{q,i}$ are the dq-axis voltages and currents; $R_{s,i}$ represents the stator resistance.

2.2. Graph Theory

In a multi-agent system, $G(A)=(V,E,A)$ is defined as an undirected graph, $V(G)=\{v_1,v_2,\dots ,v_N\}$ as a node set, and $E(G)\subseteq V\times V$ as an edge set. Edge $(v_i,v_i)\in E$ means that the node $j$ can obtain the state of node $i$, and also $(v_i,v_i)\in E$ in an undirected graph. A sequence of edges $\left\{\left(v_i,v_k\right),\left(v_k,v_l\right),\dots ,\left(v_p,v_j\right)\right\}$ is called a path from node $i$ to node $j$. If the adjacency matrix is $A=[a_{ij}]\in {ℝ}^{N\times N}$, $a_{ij}>0\iff (j,i)\in E$, otherwise it is $a_{ij}=0$. It is used to represent the communication relationships among multiple agents, and for $i\in V$, there is $a_{ii}=0$. The Laplacian matrix $L(A)$ is defined as $l_{ij}=-a_{ij},i\ne j,i,j=1,2,\dots ,N$ and $l_{ii}={\displaystyle {\sum }_{i=1}^N{a}_{ij},i}\ne j$.

2.3. Problem Formulation

In this article, the multi-PMSM system is considered a first-order MAS with disturbances. We investigate the speed synchronization issues of multi-PMSM systems based on the first equation of (1). Then, based on the PMSM vector control strategy with $i_d=0$, a current controller based on the second equation of (1) is designed to implement the desired q-axis current.

Regarding the speed synchronization problem, the dynamic of the multi-PMSMs system is described as follows:

$$\left\{\begin{array}{l}{\dot{\omega }}_{r,i}={\vartheta }_i{u}_i+f_i\\ {\dot{\omega }}_{r,0}=u_0\end{array}\right.$$

(2)

${\vartheta }_i=3n_p^2{\psi }_{f,i}/2J_i$, ${\vartheta }_i$ is the control coefficient of the PMSM; $u_i$ is the control input of the PMSM, which is ${i_{q,i}}^{\ast }$; $f_i=-n_p{T}_{L,i}/J_i-B_i{\omega }_{r,i}/J_i$.

This article adopts a static leader, so $u_0=0$. The consensus error is defined as:

$$e_{r,i}={\omega }_{r,i}-{\omega }_{r,0}$$

(3)

2.4. Related Lemmas

Lemma 1 ([25]).

For $\forall {\xi }_i\in ℝ$, $0<\mu <1$ and $\upsilon >1$. Then,

$$\begin{array}{l}{\displaystyle \sum _{i=1}^N{\left|{\xi }_i\right|}^{\mu }}\ge {\left({\displaystyle \sum _{i=1}^N\left|{\xi }_i\right|}\right)}^{\mu }\\ {\displaystyle \sum _{i=1}^N{\left|{\xi }_i\right|}^{\upsilon }}\ge n^{1-\upsilon }{\left({\displaystyle \sum _{i=1}^N\left|{\xi }_i\right|}\right)}^{\upsilon }\end{array}$$

(4)

Lemma 2 ([26]).

(1)

The Laplacian matrix $G$ of an undirected graph is semi-definite, with one eigenvalue equal to 0. If the undirected graph is connected, all other eigenvalues are positive.

(2)

The second smallest eigenvalue $L(A)$ of the Laplacian matrix of an undirected graph satisfies

$$L(A)=\underset{x\ne 0,{\displaystyle \sum _{i=1}^N{x}_i=0}}{\min }{\displaystyle \frac{x^{T}L(A)x}{x^{T}x}}>0$$

and if ${\displaystyle {\sum }_{i=1}^N{x}_i=0}$, then $x^{T}L(A)x\ge {\lambda }_2(L(A))x^{T}x$.

(3)

For any $x={(x_1,x_2,\cdots ,x_N)}^T\in {ℝ}^N$, have

$$x^{T}L(A)x={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{a}_{ij}}}{(x_j-x_i)}^2$$

(5)

Lemma 3 ([27]).

Suppose $V(·):{ℝ}^n\to {ℝ}^{+}\cup \left\{0\right\}$ satisfies

$$\left\{\begin{array}{l}(\mathrm{i})V(x)=0\iff x=0\\ (\mathrm{ii})\forall x,\mathrm{The}\mathrm{following}\mathrm{inequality}\mathrm{holds}\end{array}\right.$$

$$\stackrel{·}{V}(x)\le -a{V}^{\xi }(x)-b{V}^{\eta }(x)-cV(x)a,b,c>0,0<\xi <1,\eta >1$$

In that case,$V(x)$ is fixed-time stable, and

$$T_{\max }\triangleq {\displaystyle \frac{1}{c(1-\xi )}}\ln (1+{\displaystyle \frac{c}{a}})+{\displaystyle \frac{1}{c(\eta -1)}}\ln (1+{\displaystyle \frac{c}{b}})$$

(6)

2.5. Traditional DCC Method

The schematic diagram of the traditional DCC method for multi-PMSMs is shown in Figure 1. $F_i(s)$ is the synchronization error compensator, and $G_{s,i}(s)$ is the speed tracking controller. The synchronization error compensator $F_i(s)$ is designed as:

$$F_i\left(s\right)=K{\displaystyle \sum _{j=1,j\ne i}^N\frac{J_i}{J_j}\left({\omega }_{m,i}-{\omega }_{m,j}\right)}$$

(7)

$K$ is the compensation coefficient, with $K=1$ in the experiment. $J_i$ and $J_j$ represent the moments of inertia of motors $i$ and $j$, respectively. Since the parameters of the three motors used in the experiment are identical, it follows that $J_i/J_j=1$. The speed-tracking controller uses a PI control algorithm:

$$G_{s,i}(s)=K_{P,i}+{\displaystyle \frac{K_{I,i}}{s}}$$

(8)

Traditional multi-PMSM DCC control lacks advantages in terms of flexibility. To achieve synchronized control, the speed information of all motors in the system must be obtained. Additionally, as the number of motors in the speed regulation system increases, modifications to the original coupling structure are required. Therefore, traditional multi-PMSM DCC control struggles to meet real-time computational requirements.

3. Design of Fixed-Time Consensus Protocol

This section presents a cooperative control scheme for the coordinated speed control of multi-PMSMs based on an undirected topology graph. A fixed-time consensus protocol $u_i$ is designed to replace the traditional speed loop controller. Communication within the established topology structure between the PSMSs ensures that the state variables ${\omega }_{r,i}$ in Equation (1) converge within the fixed time. At the same time, a TISMO to estimate disturbances and compensate for the disturbance $f_i$ in the control protocol. The main block diagram is illustrated in Figure 2.

3.1. Design of Consistency Protocol

The fixed-time consensus protocol is designed as follows:

$$\begin{array}{ll}{u}_i& ={\displaystyle \frac{k_1}{{\vartheta }_i}}{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}({\omega }_{r,j}-{\omega }_{r,i})}\right]}^{{\displaystyle \frac{p}{q}}}+{\displaystyle \frac{k_2}{{\vartheta }_i}}{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}({\omega }_{r,j}-{\omega }_{r,i})}\right]}^{{\displaystyle \frac{r}{s}}}\\ & +{\displaystyle \frac{k_3}{{\vartheta }_i}}\left[{\displaystyle \sum _{j=1}^N{a}_{ij}({\omega }_{r,j}-{\omega }_{r,i})}-m_i({\omega }_{r,i}-{\omega }_{r,0})\right]-{\displaystyle \frac{{\widehat{f}}_i}{{\vartheta }_i}}\end{array}$$

(9)

$k_1,k_2,k_3>0$ represent the control gain and $A=\left[a_{ij}\right]\in R^{N\times N}$ represents the adjacency matrix of the information transmission network among the MASs. $p,q,r,s$ are all controller gains, they are all positive odd numbers, and they satisfy $p<q,r>s$.

Theorem 1.

In an undirected connected multi-agent network, under the influence of consensus protocol (9), for any solution satisfying conditions on

$$\dot{V}\le -a{V}^{\xi }-b{V}^{\eta }-cVa,b,c>0,0<\xi <1,\eta >1$$

it enables the multi-PMSM system (1) to achieve speed synchronization control within a fixed time, and the fixed convergence time is:

$$T_{\max }\triangleq {\displaystyle \frac{1}{c(1-\xi )}}\ln (1+{\displaystyle \frac{c}{a}})+{\displaystyle \frac{1}{c(\eta -1)}}\ln (1+{\displaystyle \frac{c}{b}})$$

Proof. 

Define the system error equation as:

$$\begin{array}{ll}{\dot{e}}_{r,i}& ={\vartheta }_i{u}_i+f_i\\ & =k_1{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}({\omega }_{r,j}-{\omega }_{r,i})}\right]}^{{\displaystyle \frac{p}{q}}}+k_2{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}({\omega }_{r,j}-{\omega }_{r,i})}\right]}^{{\displaystyle \frac{r}{s}}}\\ & +k_3{\displaystyle \sum _{j=1}^N{a}_{ij}({\omega }_{r,j}-{\omega }_{r,i})}-k_3{m}_i({\omega }_{r,i}-{\omega }_{r,0})\end{array}$$

(10)

Define the semi-positive definite function as:

$$V={\displaystyle \frac{1}{2}}{e}^{T}L(A)e,A={[a_{ij}]}_{N\times N}$$

(11)

Based on the symmetry of $A$, it can be determined that:

$${\displaystyle \frac{\partial V}{\partial e_{r,i}}}=-{\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})}$$

(12)

The derivative of $V$ versus time is:

$$\begin{array}{ll}\dot{V}& ={\displaystyle \sum _{i=1}^N\frac{\partial V}{\partial e_{r,i}}}{\dot{e}}_{r,i}\\ & =-k_1{\displaystyle \sum _{i=1}^N{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})}\right]}^{\frac{q+p}{q}}}-k_2{\displaystyle \sum _{i=1}^N{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})}\right]}^{\frac{s+r}{s}}}\\ & -k_3{\displaystyle \sum _{i=1}^N{\left[{\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})}\right]}^2}-k_3{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})m_i{e}_{r,i}}}\end{array}$$

(13)

$k_3{\displaystyle {\sum }_{i=1}^N{\displaystyle {\sum }_{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})m_i{e}_{r,i}=}}-k_3{e}^{T}L(M)e$, $M=diag\left\{m_1,m_2,\cdots m_n\right\}$, and $e^{T}L(M)e/e^{T}e\ge {\lambda }_{\min }L(M)$. According to ${\lambda }_{\min }L(A)\le e^{T}L(A)e/e^{T}e\le {\lambda }_{\max }L(A)$, one has:

$$\begin{array}{ll}-k_3{e}^{T}L(M)e& \le -k_3{\lambda }_{\min }L(M)e^{T}e\\ & =-k_3{\lambda }_{\min }L(M){\displaystyle \frac{e^{T}L(A)e}{{\lambda }_{\max }L(A)}}\\ & =-k_3{\displaystyle \frac{{\lambda }_{\min }L(M)}{{\lambda }_{\max }L(A)}}2V\le 0\end{array}$$

(14)

According to Lemma 1, if $p<q$, $r>s\ge 0$, then $0<(q+p)/2q<1$, $(r+s)/2s>1$. The following inequalities can be obtained:

$$\begin{array}{l}\stackrel{·}{V}\le -k_1{\left({\displaystyle \sum _{i=1}^N{({\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})})}^2}\right)}^{{\displaystyle \frac{q+p}{2q}}}\\ -k_2{N}^{{\displaystyle \frac{s-r}{2s}}}{\left({\displaystyle \sum _{i=1}^N{({\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})})}^2}\right)}^{{\displaystyle \frac{s+r}{2s}}}-k_3{\displaystyle \sum _{i=1}^N{({\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})})}^2}\end{array}$$

(15)

There exists a semi-positive matrix $Q\in R^{N\times N}$ such that $L(A)=Q^{T}Q$. Using Lemma 2, one has

$$\begin{array}{c}{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{\left({\displaystyle \sum _{j=1}^N{a}_{ij}(e_{r,j}-e_{r,i})}\right)}^2}}{V}}={\displaystyle \frac{e^{T}L{(A)}^{T}L(A)e}{\frac{1}{2}{e}^{T}L(A)e}}\\ ={\displaystyle \frac{2e^T{Q}^{T}Q{Q}^{T}Qe}{e^T{Q}^{T}Qe}}\ge 2{\lambda }_{2}L(A)\end{array}$$

(16)

From (15) and (16) can be obtained

$$\dot{V}\le -k_1{[{\lambda }_{2}L(A)]}^{{\displaystyle \frac{q+p}{2q}}}{V}^{{\displaystyle \frac{q+p}{2q}}}-k_2{N}^{{\displaystyle \frac{s-r}{2s}}}{[{\lambda }_{2}L(A)]}^{{\displaystyle \frac{s+r}{2s}}}{V}^{{\displaystyle \frac{s+r}{2s}}}-2k_3{\lambda }_{2}L(A)V$$

(17)

Defining $\xi =(q+p)/2q$; $\eta =(s+r)/2s$; $a=-k_1{[{\lambda }_{2}L(A)]}^{(q+p)/2q}$; $b=-k_2{N}^{(s-r)/2s}{[{\lambda }_{2}L(A)]}^{(s+r)/2s}$; $c=-2k_3{\lambda }_{2}L(A)$ the following inequalities can be obtained:

$$\dot{V}\le -a{V}^{\xi }-b{V}^{\eta }-cV$$

(18)

According to Lemma 3, the setting time is

$$T_{\max }={\displaystyle \frac{1}{c(1-\xi )}}\ln (1+{\displaystyle \frac{c}{a}})+{\displaystyle \frac{1}{c(\eta -1)}}\ln (1+{\displaystyle \frac{c}{b}})$$

(19)

□

3.2. Disturbance Observer Design

This paper designs a TISMO to estimate the comprehensive disturbances in the system. The TISMO is established as follows:

$$\left\{\begin{array}{l}{\dot{\widehat{\omega }}}_{r,i}=-{\displaystyle \frac{B_i}{J_i}}{\widehat{\omega }}_{r,i}+{\displaystyle \frac{1.5{n_P}^2{\psi }_{f,i}}{J_i}}{u}_i-{\displaystyle \frac{n_p}{J_i}}{\widehat{T}}_{L,i}-{\displaystyle \frac{n_p}{J_i}}h\\ \stackrel{·}{{\widehat{T}}_{L,i}}=lh\end{array}\right.$$

(20)

where $u_i$ represents the system control input; $T_{L,i}$ represents the load torque; ${\widehat{T}}_{L,i}$ is the estimated value of $T_{L,i}$; $l$ is a constant, and $h$ is the yet-to-be-designed sliding mode control input term.

Define the estimation error of the system as follows:

$$\left\{\begin{array}{l}{e}_{\omega ,i}={\omega }_{r,i}-{\widehat{\omega }}_{r,i}\\ e_{T_{L,i}}=T_{L,i}-{\widehat{T}}_{L,i}\end{array}\right.$$

(21)

where ${\widehat{\omega }}_{r,i}$ is the estimated value of ${\omega }_{r,i}$; $e_{\omega ,i}$ and $e_{T_{L,i}}$ are estimated errors.

Obtained from (20) and (21)

$${\dot{e}}_{\omega ,i}=-{\displaystyle \frac{B_i}{J_i}}{e}_{\omega ,i}-{\displaystyle \frac{n_p}{J_i}}{e}_{T_{L,i}}+{\displaystyle \frac{n_p}{J_i}}h$$

(22)

Define the TISMO surface $S_0$ as follows:

$$S_0=e_{\omega ,i}+{\displaystyle {\int }_0^t(k_4{e}_{\omega ,i}+k_{5}sig{n}^{\gamma }(e_{\omega ,i}))}{d}_{\tau }$$

(23)

$k_4,k_5$ represent the given positive constant; $\gamma$ is a positive fraction less than 1; $sign(e_{\omega ,i})$ represents the sign function, defined as:

$$sign(e_{\omega ,i})=\left\{\begin{array}{l}1,e_{\omega ,i}>0\\ 0,e_{\omega ,i}=0\\ -1,e_{\omega ,i}<0\end{array}\right.$$

The designed sliding mode reaching law is as follows:

$${\dot{S}}_0=-{\varpi }_1{S}_0-{\varpi }_{2}sig{n}^{\sigma }(S_0)-{\varpi }_{3}sig{n}^{\delta }(S_0)$$

(24)

${\varpi }_1>0,{\varpi }_2>0,{\varpi }_3>0$; $0<\sigma <1$; $1<\delta$.

Choose the Lyapunov function $V_1$ as follows:

$$V_1={\displaystyle \frac{1}{2}}{S_0}^2$$

(25)

The derivation $V_1$ can be obtained as follows:

$$\begin{array}{ll}{\dot{V}}_1& =S_0{\dot{S}}_0\\ & =S_0(-{\varpi }_1{S}_0-{\varpi }_{2}sig{n}^{\sigma }(S_0)-{\varpi }_{3}sig{n}^{\delta }(S_0))\\ & =-{\varpi }_1{S_0}^2-{\varpi }_2{\left|S_0\right|}^{\sigma +1}-{\varpi }_3{\left|S_0\right|}^{\delta +1}\le 0\end{array}$$

(26)

Obtained from (20) and (26)

$$h=-{\displaystyle \frac{J_i}{n_p}}(-{\displaystyle \frac{B_i}{J_i}}{e}_{\omega ,i}+k_4{e}_{\omega ,i}+k_{5}sig{n}^{\gamma }(e_{\omega ,i})+{\varpi }_1{S}_0+{\varpi }_{2}sig{n}^{\sigma }(S_0)+{\varpi }_{3}sig{n}^{\delta }(S_0))$$

(27)

The design of the TISMO is as follows:

$$\left\{\begin{array}{l}{e}_{\omega ,i}={\omega }_{r,i}-{\widehat{\omega }}_{r,i}\\ S_0=e_{\omega ,i}+{\displaystyle {\int }_0^t(k_4{e}_{\omega ,i}+k_{5}sig{n}^{\gamma }(e_{\omega ,i}))}{d}_{\tau }\\ h=-{\displaystyle \frac{J_i}{n_p}}(-{\displaystyle \frac{B_i}{J_i}}{e}_{\omega ,i}+k_4{e}_{\omega ,i}+k_{5}sig{n}^{\gamma }(e_{\omega ,i})+{\varpi }_1{S}_0+{\varpi }_{2}sig{n}^{\sigma }(S_0)+{\varpi }_{3}sig{n}^{\delta }(S_0))\\ {\widehat{\dot{\omega }}}_{r,i}=-{\displaystyle \frac{B_i}{J_i}}{\widehat{\omega }}_{r,i}+{\displaystyle \frac{1.5n_p^2{\psi }_{f,i}}{J}}{u}_i-{\displaystyle \frac{n_p}{J}}{\widehat{T}}_{L,i}-{\displaystyle \frac{n_p}{J}}{h}_{\omega ,i}\\ \stackrel{·}{{\widehat{T}}_{L,i}}=lh\\ f_i=-{\displaystyle \frac{n_p}{J_i}}{\widehat{T}}_{L,i}-{\displaystyle \frac{B_i}{J_i}}{\widehat{\omega }}_{r,i}\end{array}\right.$$

(28)

4. Design of the Current Loop for the Multi-PMSM Speed Control System

Defined the CSMC tracking error as follows:

$$\mathcal{l}=\left[\begin{array}{l}{e}_{d,i}\\ e_{q,i}\end{array}\right]=\left[\begin{array}{l}{i}_{d,i}^{*}-i_{d,i}\\ i_{q,i}^{*}-i_{q,i}\end{array}\right]={\varsigma }^{\ast }-\varsigma$$

(29)

${\varsigma }^{*}=\left[\begin{array}{l}{i}_{d,i}^{*}\\ i_{q,i}^{*}\end{array}\right]$; $\varsigma =\left[\begin{array}{l}{i}_{d,i}\\ i_{q,i}\end{array}\right]$.

By using the mathematical model of PMSM (1), the matrix from incorporating disturbances can be obtained as follows:

$$\dot{\varsigma }=X\varsigma +Y{u}^{\ast }+ZF$$

(30)

$u^{\ast }=\left[\begin{array}{l}{u}_{d,i}\\ u_{q,i}\end{array}\right]$; $X=-R_{s,i}/L_i$; $Y=1/L_i$; $Z=1$; $F=\left[\begin{array}{c}{\omega }_{r,i}{i}_{q,i}\\ -{\omega }_{r,i}{i}_{d,i}-{\displaystyle \frac{{\psi }_{f,i}}{L_i}}\end{array}\right]$.

The PMSM system can be described as follows:

$$\begin{array}{c}\dot{\varsigma }=\left(X_n+\Delta X\right)\varsigma +\left(Y_n+\Delta Y\right)u^{\ast }+\left(Z_n+\Delta Z\right)F\\ =X_n\varsigma +Y_n{u}^{\ast }+\zeta \end{array}$$

(31)

$X_n,Y_n,Z_n$ are the nominal values of $X,Y,Z$; $\Delta X,\Delta Y,\Delta Z$ denote the uncertainties introduced by system parameters; $\zeta =\Delta X\varsigma +\Delta Y{u}^{\ast }+\left(Z_n+\Delta Z\right)F$ is the lumped uncertainty; and $\zeta$ is assumed to be bounded.

$$\left|\zeta \right|\le \rho$$

(32)

$\rho$ represents a control constant.

The integral sliding surface $S_1$ and the complementary sliding surface $S_2$ are as follows:

$$S_1=\mathcal{l}+\lambda {\displaystyle {\int }_0^t(\mathcal{l}}+sig{n}^{\alpha }(S_2-S_1))d\tau$$

(33)

$$S_2=\mathcal{l}-\lambda {\displaystyle {\int }_0^t(\mathcal{l}}+sig{n}^{\alpha }(S_2-S_1))d\tau$$

(34)

$\lambda$ and $\alpha$ represent the given positive constant; $sign(S_2-S_1)$ represent the sign function, defined as:

$$sign(S_2-S_1)=\left\{\begin{array}{l}1,(S_2-S_1)>0\\ 0,(S_2-S_1)=0\\ -1,(S_2-S_1)<0\end{array}\right.$$

By substituting (29)–(31) into (33) and (34) and taking the derivatives, the following equations can be obtained:

$$\begin{array}{ll}{\dot{S}}_1& =\dot{\mathcal{l}}+\lambda \left[\mathcal{l}+sig{n}^{\alpha }(S_2-S_1)\right]\\ & ={\dot{\varsigma }}^{\ast }-X_n\varsigma -Y_n{u}^{\ast }-\zeta +\lambda \mathcal{l}+\lambda sig{n}^{\alpha }(S_2-S_1)\end{array}$$

(35)

$$\begin{array}{ll}{\dot{S}}_2& =\dot{\mathcal{l}}-\lambda \left[\mathcal{l}+sig{n}^{\alpha }(S_2-S_1)\right]\\ & ={\dot{\varsigma }}^{\ast }-X_n\varsigma -Y_n{u}^{\ast }-\zeta -\lambda \mathcal{l}-\lambda sig{n}^{\alpha }(S_2-S_1)\end{array}$$

(36)

The relationship between $S_1$ and $S_2$ is defined as follows:

$$S(\mathcal{l})=S_1(\mathcal{l})+S_2(\mathcal{l})$$

(37)

The following equation can be obtained by using (33)–(36)

$$\begin{array}{ll}{\dot{S}}_2& ={\dot{S}}_1-\lambda (S_1+S_2)-2\lambda sig{n}^{\alpha }(S_2-S_1)\\ & ={\dot{S}}_1-\lambda S-2\lambda sig{n}^{\alpha }(S_2-S_1)\end{array}$$

(38)

The control law $u^{\ast }$ of the CSMC is defined as follows:

$$u^{*}=u_{eq}^{*}+u_{hit}^{*}$$

(39)

$$u_{eq}^{*}=\left[\begin{array}{l}{u}_{deq}^{*}\\ u_{qeq}^{*}\end{array}\right]=\left[\begin{array}{l}{\displaystyle \frac{1}{Y_n}}\left({\dot{\varsigma }}^{*}-X_n\varsigma +\lambda \mathcal{l}+\lambda S_1\right)\\ {\displaystyle \frac{1}{Y_n}}\left({\dot{\varsigma }}^{*}-X_n\varsigma +\lambda \mathcal{l}+\lambda S_1\right)\end{array}\right]$$

(40)

$$u_{hit}^{*}=\left[\begin{array}{l}{u}_{dhit}^{*}\\ u_{qhit}^{*}\end{array}\right]=\left[\begin{array}{l}{\displaystyle \frac{1}{Y_n}}\left[\rho sat({\displaystyle \frac{S}{\phi }})\right]\\ {\displaystyle \frac{1}{Y_n}}\left[\rho sat({\displaystyle \frac{S}{\phi }})\right]\end{array}\right]$$

(41)

$$sat({\displaystyle \frac{S}{\phi }})=\left\{\begin{array}{l}1,S\ge \phi \\ {\displaystyle \frac{S}{\phi }},-\phi <S<\phi \\ -1,S\le -\phi \end{array}\right.$$

(42)

Define $u_{eq}^{*}$ as the equivalent control law; $u_{hit}^{*}$ as the switching control rate; $sat(·)$ as the saturation function; and $\phi$ as the boundary layer thickness.

Define the Lyapunov function $V_2$ as:

$$V_2={\displaystyle \frac{1}{2}}{S}_1^2+{\displaystyle \frac{1}{2}}{S}_2^2$$

(43)

The time derivative of $V_2$ can be obtained as:

$$\begin{array}{ll}{\dot{V}}_2& =S_1{\dot{S}}_1+S_2{\dot{S}}_2\\ & =S_1{\dot{S}}_1+S_2\left[{\dot{S}}_1-\lambda S-2\lambda sig{n}^{\alpha }(S_2-S_1)\right]\\ & ={\dot{S}}_{1}S-\lambda S_{2}S-2\lambda S_{2}sig{n}^{\alpha }(S_2-S_1)\\ & =S\left[-\lambda S_1-\rho sat({\displaystyle \frac{S}{\phi }})-\zeta +\lambda sig{n}^{\alpha }(S_2-S_1)\right]-\lambda S{S}_2-2\lambda S_{2}sig{n}^{\alpha }(S_2-S_1)\\ & =S\left[-\lambda S-\rho sat({\displaystyle \frac{S}{\phi }})-\zeta \right]-\lambda \left(S_2-S_1\right)sig{n}^{\alpha }(S_2-S_1)\\ & =-\lambda S^2+S\left(-Y_n{u}_{hit}^{*}\right)+S(-\zeta )-\lambda \left(S_2-S_1\right)sig{n}^{\alpha }(S_2-S_1)\\ & \le -\lambda S^2+\left|S\right|\left(\left|\zeta \right|-\rho \right)-\lambda |S_2-S_1{|}^{\alpha +1}\\ & \le -\lambda S^2+\left|S\right|\left(\left|\zeta \right|-\rho \right)\end{array}$$

(44)

Based on $\left|\zeta \right|\le \rho$; ${\dot{V}}_2(S_1,S_2)\le 0$, the system satisfies the Lyapunov stability conditions, ensuring that the state trajectory from any position can reach the boundary layer within a finite time and eventually reach the sliding surface, thereby guaranteeing the stability of the system [28]. The schematic diagram of the CSMC is illustrated in Figure 3.

5. Experimental Validation

Comparative experiments are conducted with the traditional DCC method on the semi-physical simulation experimental platform shown in Figure 4. The experimental platform includes a host computer, three controllers, three PMSM drivers, three load PMSM drivers, three PMSMs, and three load PMSMs. The host computer sends control signals to the three controllers, and the controller applies control algorithms to the three PMSM drivers and three load PMSM drivers to drive the PMSM to rotate. The three controllers form a local area network with the host computer through a switch and transmit data through the user datagram protocol (UDP).

Table 1. Parameters of PMSM.

Name and Symbol	Value
Phase resistance $R_s/\Omega$	0.5
dq-axis inductance $L/\mathrm{H}$	0.01
Friction coefficient $B/(\mathrm{N}·\mathrm{m}·\mathrm{s})$	0.0043
Moment of inertia $J/(\mathrm{kg}·{\mathrm{m}}^2)$	0.00194
Flux linkage ${\psi }_f/(\mathrm{V}·\mathrm{s})$	0.1
Polo pairs $n_p$	2

presents the PMSM parameters on the experimental platform.

Table 2. Parameters of control methods.

Control Method	Parameter	Value
Traditional DCC method	$K_{P,i}$	1.1
	$K_{I,i}$	3
New control method	$k_1$	5
	$k_2$	0.9
	$k_3$	10
	$p$	3
	$q$	5
	$r$	7
	$s$	5

presents the parameters of the traditional DCC method and the new control method proposed in this paper.

Considering practical engineering applications, the new control method is compared with the traditional DCC method on the multi-motor speed control and load integration experimental platform. Comparative experiments are conducted on operations such as speed-up and speed-down as well as forward and reverse operations, sudden load, and sudden relief load, as well as on low speed.

5.1. Comparative Experiments on Speed-Up and Speed-Down as Well as Forward and Reverse Operation

To verify the synchronous performance of the new control method, a comparative experiment on speed-up and speed-down as well as forward and reverse operation is conducted. The multi-motor speed cooperative control system is started from an initial speed of 0, accelerated to 200 r/min, then to 500 r/min, and further to 700 r/min. Subsequently, it decelerates to 300 r/min and finally reverses to −300 r/min. The results of the comparative experiment on speed-up and speed-down as well as forward and reverse operation are shown in Figure 5.

From Figure 5a, the new control method exhibits no obvious overshoot phenomena during speed-up, speed-down, forward, and reverse operation conditions. The transition process is relatively smooth, with a settling time of around 2.5 s. In contrast, the traditional DCC method shows around 8% overshoot during speed-up, speed-down, forward, and reverse operation conditions, with a settling time of around 5 s. In steady-state, the new control method exhibits a chattering amplitude of around 3 r/min, significantly lower than the approximately 5 r/min steady-state process of the traditional DCC method. From Figure 5b, there is no significant overshoot phenomenon during forward and reverse operation conditions. From Figure 5c, the synchronization error of the new control method is around 3 r/min, which is lower than the approximately 8 r/min synchronization error of the traditional DCC method. Therefore, the new control method effectively reduces the system’s chattering, while ensuring both the system’s rapid performance and tracking capability.

5.2. Comparative Experiments on a Sudden Load and Sudden Relief Load

To validate the robustness of the new control method, a comparison experiment involving sudden load and sudden relief load is conducted. Different loads are applied to or removed from different motors at different times while the multi-motor speed control system is running at 300 r/min. The loads are applied or removed by the load motor group depicted in Figure 3. The results of the sudden load and sudden relief load comparison experiment are shown in Figure 6.

When subjected to a sudden load disturbance, there is little difference in response time between the two methods, as both can restore the speed to the set 300 r/min within a short period. From Figure 6a, after the sudden load disturbance, the chattering amplitude of the new control method is around 3 r/min, while the chattering amplitude of the traditional DCC method is around 5 r/min. The recovery time of the traditional DCC method is 0.8 s, which is shorter than the 1.5 s recovery time of the new control method. At the moment of loading, from Figure 6b, the speed increases by around 4% with the new control method, while it increases by around 8% with the traditional DCC method. Additionally, from Figure 6c, when the speed of one motor changes, the speeds of other motors also adjust accordingly to reduce the system’s synchronization error. After the sudden load, the synchronization error of the new control method is around 3 r/min, which is better than the synchronization error of approximately 5 r/min in the traditional DCC method. Therefore, the new control method can effectively guarantee the robustness of the system.

5.3. Comparative Experiments on Low-Speed

To verify the low-speed performance of the control method, the system is decelerated from an initial speed of 100 r/min to 50 r/min, and then further decelerated to 20 r/min. The results of the low-speed comparison experiment are shown in Figure 7.

From Figure 7a, the new control method exhibits a chattering amplitude of around 3 r/min under low-speed conditions, significantly lower than the chattering amplitude of around 6 r/min seen with the traditional DCC method. From Figure 7b, the settling time of the new control method under low-speed conditions is around 2.5 s, while the settling time of the traditional DCC method is around 5 s. From Figure 7c, the new control method has a significant advantage in synchronous error of around 3 r/min, while the synchronous error of the traditional DCC method is around 7 r/min. Therefore, the new control method can still maintain good responsiveness and tracking performance under low-speed conditions, and effectively reduce the chattering phenomenon of the system.

6. Conclusions

This paper proposes a speed-coordinated control method for multi-PMSMs based on the fixed-time consensus protocol for MASs combined with CSMC. The speed cooperative system of multi-PMSMs is regarded as a MAS, and an undirected topology structure based on communication between each PMSM is established. Under the framework of vector control for PMSMs, a fixed-time consensus protocol is designed to replace the traditional speed loop controller. TISMO is incorporated into the protocol to estimate disturbances and introduce feedforward compensation, obtaining the desired q-axis current, thereby achieving speed consistency among multi-PMSMs within a fixed time. Simultaneously, CSMCs are designed in the current loop, effectively improving the system’s performance. The experimental results compared with the traditional DCC method are shown in

Table 3. Comparison of performances.

Control Method	Operating Condition	Overshoot Phenomena	Settling Time	Chattering Amplitude
Traditional DCC method	No-load condition	Yes	5 s	5 r/min
	Loading condition	Yes	0.8 s	5 r/min
	Low speed condition	Yes	5 s	6 r/min
New control method	No-load condition	No	2.5 s	3 r/min
	Loading condition	No	1.5 s	3 r/min
	Low speed condition	No	2.5 s	3 r/min

. The results indicate that the new control method exhibits good tracking and synchronization performance, along with faster convergence and enhanced robustness. The next step will be to study the coordinated speed control of multiple motors under network delays, data packet loss, and other issues that may arise in suboptimal network conditions.