Practical discontinuous tracking control for a permanent magnet synchronous motor

1.   Introduction

In recent years, the control problem of time-varying systems has been widely considered ^[1,2,3,4], and most control protocols of time-varying systems are based on continuous control methods ^[5,6,7,8,9], which require continuous control input. However, discontinuous control ^[10,11] is more effective in some specific cases, and it has attracted more and more attention from researchers. Although the theoretical research on discontinuous control strategy is abundant, few researchers have applied it to practical systems, which motivates us to use this method in actual scenarios. In addition, the input of the actual system is limited, and the control input based on the traditional approach is likely to exceed the limit and affect the control performance of the controller. Thus, input saturation can-not be ignored in the process of controller design. According to the above analysis, we study the practical discontinuous control of PMSM and complete the design of an intermittent controller by the sliding mode control method.

Discontinuous control strategy includes impulsive control ^[12,13], intermittent control, and sampling control. According to the division of the control cycle, intermittent control is divided into periodic and non-periodic intermittent control. In ^[14,15], a novel control strategy consisting of an event-triggering mechanism and impulsive control showed its effectiveness in stabilizing nonlinear systems and can effectively reduce the cost of signal transmission. To solve the leader-follower synchronization problem in complex networks ^[16,17], a batch control scheme based on being dual event-driven was proposed. In ^[18], the observer-based state feedback non-periodic intermittent control for uncertain systems with structural uncertainty was studied. A class of finite-time synchronous control of reaction-diffusion nervous systems coupled with small regions with interaction diagrams on spatial boundaries was researched in ^[19]. In ^[20], the intermittent control strategy for synchronous analysis of time-varying complex dynamic networks was studied, and some numerical simulations verified the validity of the results. Although discontinuous control has received much attention from researchers, most of the above research remains at the stage of theoretical research. To improve the practical ability of the algorithm, we carry out the research of applying the discontinuous control algorithm to motor control.

For a real system, the limit of input saturation can not be ignored. Input saturation ^[21,22] often seriously affects the performance of the system and even leads to system instability. In ^[23], the global stabilization problem of discrete-time linear systems with input saturation and delay was studied, and three examples demonstrated the effectiveness of the proposed method. In ^[24], the researchers studied the consistency of a class of linear multi-agent systems with input saturation under a self-triggering mechanism. The adaptive control strategy of the flexible manipulator with a double pendulum structure was analyzed, and the effectiveness of the saturation controller in tracking control was verified in ^[25]. The tracking control problem for nonlinear stochastic systems with time-varying state constraints and input saturation was studied in ^[26]. In ^[27], adaptive fuzzy tracking control for high-order nonlinear time-delay systems with full state constraints and input saturation was studied. Considering that the input is bounded in the real system, the input saturation auxiliary system is used to compensate for the influence of input saturation on the control performance, which makes our algorithm more valuable in the application.

Currently, many control methods are applied to controller design, such as the backstepping method ^[28,29], model predictive control and sliding mode control ^[30,31]. As one of the common control methods, sliding mode control has a wide range of application scenarios ^[32,33]. In ^[34], the sliding mode control of uncertain linear systems with impulsive effects was studied, and the stability of the system was analyzed by piecewise discontinuous Lyapunov function. A robust fault-tolerant compensation control method based on the integral sliding mode method was proposed for uncertain linear systems on networks ^[35]. In ^[36], the researchers considered unknown asymmetric input saturation and control singularity and designed a fuzzy terminal sliding mode controller based on the disturbance observer. In ^[37], a new sliding mode switching approximation law was proposed to improve sliding mode control performance for a class of high-order self-organizing systems. The problem of event-triggered fuzzy sliding mode control for networked control systems with semi-Markov process adjustment was studied in ^[38]. To design a practical discontinuous controller with input saturation, sliding mode control is used in the design of the control algorithm to achieve the desired control objective.

Through the above analysis, we consider the practical discontinuous tracking control problem of PMSM under input saturation and use the sliding mode control method to complete the design of the controller. The main contributions of this paper are described as follows:

1) To improve the applicability of the discontinuous control method, we apply the discontinuous control method to the tracking control of PMSM and carry out the experimental verification. However, most of the previous discontinuous control is at the level of theoretical research and rarely applied to the actual system.

2) Due to physical constraints, we consider the input saturation in the actual scenario. By introducing the input saturation auxiliary system, we reduce the influence of input saturation on the system and calculate the theoretical control input of the system.

3) To realize the discontinuous control of PMSM, sliding mode control is introduced into the design of the discontinuous controller, which ensures the stability and robustness of the system under the input of discontinuous control.

The remaining part of the paper is described as follows. Some essential knowledge and a necessary lemma are introduced in Section 2. We use the sliding mode control method to complete the design of the discontinuous controller and provide the stability analysis in Section 3. We resort to the simulation results to verify the availability of the controller design in Section 4. In Section 5, the experimental platform demonstrates the effectiveness of the practical discontinuous control algorithm. In Section 6, we summarize the whole paper.

2.   Preliminaries and problem formulation

This section introduces the definitions of intermittent control, systems model, and input saturation, which will be used in the rest of the work.

2.1.   Discontinuous control

Intermittent control is one of the strategies of discontinuous control. The division of the control cycle can be divided into periodic intermittent control and aperiodic intermittent control. In this paper, we adopt the periodic intermittent control strategy. The main design idea is to divide the whole control cycle into equal intervals $T$ of the control cycle. Each control period is divided into the controlled period and the uncontrolled period. The two continuous control periods, $n$ and $n+1$, are described in Figure 1, and $n = 0, 1, 2, \cdots$.

When $t\in [nT, nT+\eta]$, at $nT$ moment, the system receives instructions from the controller, and at $nT+\eta$ moment, the control instruction ends. Thus, $[nT, nT+\eta]$ is the controlled period of the system. During the period of $[nT+\eta, (n+1)T]$, the system does not receive any instructions from the controller. Thus, $[nT+\eta, (n+1)T]$ is the uncontrolled period of the system. In addition, in periodic intermittent control, the control amplitude $\eta = nT+\eta-nT > 0$ is a constant value.

Lemma 1. (Young's inequality) For arbitrary positive real numbers $x$ and $y$, the following inequality holds.

where $a > 0$ and $b > 0$, with $(a-1)(b-1) = 1$. If and only if $x^a = y^b$, the equality sign holds.

Definition 1. Considering the impact of input saturation, the actual control input $u\left(i_q\right)$ is the nonlinear function of ideal control input $i_q$. The $u\left(i_q\right)$ is defined as follows

where $i_{M}$ is the boundary of control input saturation.

2.2.   The dynamic model of the PMSM and system descriptions

The dynamic model of a 3-phase PMSM can be described as ^[39]

where $J$ is the moment of inertia, $\omega$ is the angular velocity, and $T_e$ and $T_l$ are the electromagnetic torque and load torque, respectively. $f$ is the damping coefficient. $p$ represents the number of pairs of the permanent magnet rotor poles. $L_d$ and $L_q$ are the inductance of the d-q axis, and $i_d$ and $i_q$ represent the d-q axis parts of stator current. $\Phi$ is the permanent magnet flux linkage.

Normally, the space vector pulse-width modulation control method of $i_d$ = 0 is used for the 3-phase PMSM with the surface-mounted permanent magnet rotor. Hence, we can the following formula ^[40].

where $L_s$ is the inductance of the stator.

According to Definition 1, (2.3) and (2.4), the dynamic model under input saturation can be described as the following:

where $\theta_1$ and $\omega_1$ are angle and angular velocity, respectively.

Let $g = \frac{3}{2} \frac{p \Phi}{J}$, $d = -T_l/J$, $c = -f/J$, and (2.5) can be redescribed as

where $i_q$ and $y_1$ are the ideal control input and output, respectively.

The dynamics of the reference signal are described as

where $\theta_2$ and $\omega_2$ are angle and angular velocity of the reference signal.

Control objective: In this paper, the control objective is to design the discontinuous controller with the sliding mode control method under the constraint of input saturation, such that the error variables $\varepsilon_1$ and $\varepsilon_2$ converge to zero asymptotically.

3.   Discontinuous controller design and stability analysis

3.1.   Discontinuous controller design

In this section, we design the discontinuous controller through sliding mode control to realize the tracking control to reference signal.

The structure diagram of the practical discontinuous controller is shown in Figure 2. The controller is mainly composed of the discontinuous controller, input saturation auxiliary system, input saturation actuator and PMSM. The dynamic control model of PMSM can be expressed by Eq (2.6), and the constraint effect of the input saturation actuator on the control input signal can be expressed by Eq (2.2). The design of other parts will be given in detail below.

For any $n \in \mathbb{Z}_{+} \cup\{0\}, t \in(n T, (n+1) T)$, the error variables are described as follows :

where $\varepsilon_1$ and $\varepsilon_2$ are error variables. $\xi$ is the input saturation auxiliary system, which will be designed in subsequent processes.

We define

The time derivative of (3.1) is written as

Case 1: For any $n \in \mathbb{Z}_{+} \cup\{0\}, t \in[n T, n T+\eta)$, we construct the following Lyapunov function

The time derivative of (3.6) is written as

Substituting (3.5) into (3.7), we have

The input saturation auxiliary system $\xi$ and ideal control input $i_q$ are designed as follows:

Substituting (3.9) into (3.8), we can obtain

Substituting (3.10) into (3.11), we have

Case 2: For $t \in[n T+\eta, (n+1) T)$ and there holds that

The input saturation auxiliary system $\xi$ and control input $u$ are designed as follows

Substituting (3.14) and (3.15) into (3.13), we can obtain

Substituting (3.4) into (3.16), we can obtain

According to Lemma 1, we have

Substituting (3.18) into (3.17), we can obtain

where $k_{0} = |1-(k+c) k|+(k+c)$, $2k+c > 0$.

3.2.   Stability analysis

Case 1: For any $n \in \mathbb{Z}_{+} \cup\{0\}, t \in[n T, n T+\eta)$, we have

Then, we have

Proof : Multiplying both sides by $e^{2kt}$, (3.20) becomes

Integrating it over $[nT, nT+\eta]$, we have

Case 2: For any $n \in \mathbb{Z}_{+} \cup\{0\}, t \in[n T+\eta, (n+1) T)$, we have

Similarly, we have

According to (3.23) and (3.25)

Therefore, when the inequality

holds the function $V(t)$ is a monotonically decreasing continuous function.

According to (3.6), we have

This indicates that

Thus, when $t\rightarrow \infty$, we have

Then, at the moment $0 < t_1 < \infty$, $t_1 \in[(n-1) T, n T)$, $i_q\leq i_M$, $u\left(i_q\right) = i_q$, the following equality holds:

Further, we can obtain

where $t_2 > nT$.

Thus, when $t\rightarrow \infty$, we have

According to the above analysis, when $t\rightarrow \infty$, the tracking control target is accomplished. Simultaneously, there are $\theta_1 = \theta_2$ and $\omega_1 = \omega_2$.

4.   Simulation example

The simulation design parameters are designed as $c = -2.1, d = 0, k = 2, g = 200, \eta/T = 0.5 > 0.22$ and $i_{M} = 5$. The initial values of the system are selected as $\theta_1(0) = 0, \omega_1(0) = 0$, $\theta_2(0) = 50, \omega_2(0) = 800$ and $\xi(0) = 0$.

Figures 3–10 show the simulation results of two kinds of control methods. Figure 3 displays the trajectories of $\theta_1$ and $\theta_2$. Figure 4 displays the trajectories of $\omega_1$ and $\omega_2$. The trajectories of error variables $\varepsilon_{1}$ and $\varepsilon_{2}$ are shown in Figures 5 and 6, respectively. From Figures 3–6, the simulation results indicate the proposed discontinuous control algorithm is valid. The trajectories of the control input $i_q$ are exhibited in Figure 7. Figure 8 exhibits the trajectories of saturated input $u(i_q)$. The trajectories of input saturation auxiliary system $\xi$ are shown in Figure 9. The trajectories of signals $\int_{0}^{t} \xi d t-\int_{0}^{n T} \xi d t$ are exhibited in Figure 10. The comparison results show that the discontinuous controller designed in this paper can achieve similar control effects to the continuous controller.

5.   Experimental results

The bicycle robot is a high-performance robot that integrates bicycles and intelligent control. It has both the convenience and flexibility of bicycles and the high intelligence of integrated control systems. Bicycle robots can replace human work on specific occasions, and they have broad application prospects in rescue and disaster relief, industrial production and resource exploration. The intermittent control algorithm designed in this paper has great application value in the balance control problem of the balance bicycle. Based on this, we carry out the experimental test of the intermittent control algorithm for a single balance control motor of the balance bike, which is shown in Figure 11. It can be seen that the system consists of a DC power source, a controller and a motor. The communication network is based on the controller area network (CAN). Figure 12 displays the trajectories of $\theta_1$ and $\theta_2$. Figure 13 shows the trajectories of $\omega_1$ and $\omega_2$. Figure 14 exhibits the trajectories of saturated input $u(i_q)$.

6.   Conclusions

In this paper, we complete the PMSM tracking controller design by the practical discontinuous control algorithm. An input saturation auxiliary system is introduced to ensure the practicality of the controller, which overcomes the influence of input saturation on controller design and control performance. Then, the desired control goal is achieved by the sliding mode control method. Through stability analysis, the proposed controller can achieve the desired control goal. Finally, the effectiveness of the proposed algorithm is verified by simulation results and the experimental platform.

Acknowledgments

This work was funded by the National Natural Science Foundation of China under Grant 62173215, Grant 62006192, and Major Basic Research Program of the Natural Science Foundation of Shandong Province in China (ZR2021ZD04).

Conflict of interest

No potential conflict of interest was reported by the author(s).