Model-Free Hybrid Control with Intelligent Proportional Integral and Super-Twisting Sliding Mode Control of PMSM Drives

In this research, based on the ultra-local model, a novel compound model-free control strategy with an intelligent Proportional-Integral and super-twisting Sliding Mode Control (MFiPISTSMC) strategy for permanent magnet synchronous motor (PMSM) drives is proposed. Firstly, an intelligent Proportional-Integral (iPI) control strategy is designed for motor speed regulation. Secondly, a super-twisting Sliding Mode Control (STSMC) strategy is constructed based on the ultra-local model of PMSM. At the same time, the unknown term of the ultra-local model of PMSM is estimated by a Linear Extended State Observer (LESO). The stability of the compound MFiPISTSMC strategy is proved by the Lyapunov stability theorem. As a result of the compound MFiPISTSMC strategy integrating the STSMC strategy, the iPI control strategy and the LESO is proposed to have excellent performance. Finally, the static characteristic, dynamic characteristic and robustness of the novel compound MFiPISTSMC strategy are verified by simulation and experimental results.


Introduction
Due to some advantages, such as simple structure, small volume, high efficiency, and high power factor, the permanent magnet synchronous motor (PMSM) has been widely used in many applications such as in aerospace actuation [1], electric vehicles [2,3], and suspended positioning systems [4].
Because of the invention of the PMSM, a multitude of scholars have not stopped studying control strategies. Many control theories have been widely and effectively applied to the speed regulation system of the PMSM, which can be roughly divided into two categories: model-related control and model-free control (MFC). More precisely, for the model-related control, a host of scholars used the mathematical model of the PMSM to deeply analyze various advanced controllers; examples of such algorithms include, model predictive control [5], backstepping control [6], nonlinear adaptive control [7], H∞ robust control [8], and so on. However, the external loads and internal parameters are variable in the actual working environment. Consequently, the PMSM cannot be accurately represented by the mathematical model, which bring challenges and troubles to the model-related control. In recent years, the MFC strategy has been more and more popular. Although the MFC strategy was introduced only a few years ago, the traditional Proportional-Integral (PI) control strategy, as the most widely used MFC strategy, has been widely applied in the PMSM speed regulation system [9,10]. However, the traditional PI control strategy may cause the PMSM speed control system to deviate 1.
A compound MFiPISTSMC strategy is proposed, which integrates the STSMC strategy, the iPI control strategy, and the MFC strategy. The STSMC strategy and the iPI control strategy are all constructed based on the ultra-local model of the PMSM.

2.
A novel LESO based on the compound MFiPISTSMC strategy is proposed for the PMSM drives. The stability of the proposed control strategy is proved by the Lyapunov stability theorem. It will be shown from the theoretical discussions that the proposed MFiPISTSMC control strategy can ensure the tracking error of the system tends to zero. 3.
The simulation and experimental results show the effectiveness of the proposed model-free hybrid control strategy.
The remaining parts of this paper are arranged as: In Section 2, the mathematical model of PMSM and the ultra-local model of PMSM are briefly presented. The control strategy and stability analysis are given in in Section 3. The results of simulation and experimental results are shown in Section 4. Finally, concluding observations are contained in Section 5.

Problem Formulation
This section presents the mathematical model of PMSM and the ultra-local model of PMSM. The flux linkage equation, electromagnetic torque equation and voltage equation of salient PMSM are as follows [31]: When L d = L q = L are in this paper, mechanical torque equation is expressed in (4), the dynamic equation of PMSM can be denoted in (5) [31]. .
The key parameters of PMSM are listed in Table 1. According to [11], the general single input single output system can be replaced by the ultra-local model as: where y(t) and are the output and input of the ultra-local model, respectively; a is the non-physical constant parameter; F is the unknown term of the system but also any disturbances. In this paper n = 1 is satisfied, the first-order system (5) can be selected to describe the dynamics of the controlled system. The system tracking error is usually defined as: where y r (t) is the desired output of the system; e(t) is the system tracking error. The purpose of this paper is to design a stable controller to make the tracking error tend to zero. In this paper, y(t) is the actual value of the mechanical speed; y r (t) is the given value of the mechanical speed; e(t) is the tracking error of the mechanical speed.

Main Results
In this section, the proposed control strategy will be given. Furthermore, the stability of the proposed control strategy will be proved by the Lyapunov stability theorem. A novel control block diagram of PMSM speed regulation system is based on the proposed MFiPISTSMC strategy and the LESO with d axis current i d (t) = 0 is manifested in Figure 1. The control goal is to make the actual output y(t) tracking the desired output y r (t). The novel control block diagram of PMSM speed regulation system consists of a proposed MFiPISTSMC strategy, an LESO, a current controller, a voltage source inverter, and a PMSM. this paper is to design a stable controller to make the tracking error tend to zero. In this paper, ) (t y is the actual value of the mechanical speed; ) (t y r is the given value of the mechanical speed; ) (t e is the tracking error of the mechanical speed.

Main Results
In this section, the proposed control strategy will be given. Furthermore, the stability of the proposed control strategy will be proved by the Lyapunov stability theorem.
A novel control block diagram of PMSM speed regulation system is based on the proposed MFiPISTSMC strategy and the LESO with d axis current Figure 1. The control goal is to make the actual output ) (t y tracking the desired output ) (t y r . The novel control block diagram of PMSM speed regulation system consists of a proposed MFiPISTSMC strategy, an LESO, a current controller, a voltage source inverter, and a PMSM.

Design of Unknown Terms Observer
The unknown term of the ultra-local model is estimated by the LESO. For the sake of efficiency and effectiveness, a second order LESO is selected in this paper.

Design of Unknown Terms Observer
The unknown term of the ultra-local model is estimated by the LESO. For the sake of efficiency and effectiveness, a second order LESO is selected in this paper.
The second order LESO can be described as [22,23]: where β 1 and β 2 are positive constants; b 0 is the design parameter; Z 21 and Z 22 are the estimates of x 1 and x 2 , respectively.

Design of Novel Controller
Theorem 1 [15]. The iPI control strategy is defined as: where K p > 0, K i > 0, u 1 (t) is the control law of the iPI control strategy.
Substituting (10) into (6), the error equation can be obtained: where ∆d(t) is as follows: Electronics 2020, 9, 1427 5 of 16 According to References [15,32], we can get the iPI control strategy that cannot guarantee that the tracking error of the system tends to zero quickly. A detailed description can be found in References [15,32].
The following sliding mode surface is chosen in this paper [33][34][35][36]: where η 1 and η 2 are the positive parameters of the sliding mode surface.

Theorem 2.
The proposed MFiPISTSMC strategy is designed in the following format: where u 2 (t)is the control law of the STSMC strategy; u 21 is the equivalent control law; u 22 is the switching control law.
In addition, in order to make the control error of the system rapidly zero, according to (6), (7) and (14), the following equation can be obtained: .
Taking derivative of (13), it can be obtained that: .
Ideally, according to (17) and ∆d(t) = 0, the equivalent control law can be obtained as: In this paper, the switching control strategy of the MFiPISMC strategy is chosen as: where k 1 ∈ R + , k 2 ∈ R + . For compensating external disturbance and eliminating the chattering phenomenon caused by the SMC strategy, the following ST scheme is usually chosen as [17]: .
where k 1 and k 2 are the positive parameters of the ST scheme. Consequently, the switching control law is obtained as: Figure 2 describes the structure diagrams of the proposed MFiPISTSMC strategy.
where 1 k and 2 k are the positive parameters of the ST scheme.
Consequently, the switching control law is obtained as: Figure 2 describes the structure diagrams of the proposed MFiPISTSMC strategy.
Step 1: The stability of the LESO will be discussed. Laplace transform is carried out for (9), and we can get:
Step 1: The stability of the LESO will be discussed. Laplace transform is carried out for (9), and we can get: where λ is the bandwidth of the system.
Step 2: In what follows, the stability of the proposed MFiPISTSMC strategy will be discussed.
d(t) has an upper limit, and the estimation error of the LESO satisfies the following condition: where Ψ is a positive constant. The condition (26) shows the estimation error of the LESO is a bounded perturbation.
In order to guarantee the stability of the proposed MFiPISTSMC strategy, we do the following steps: By substituting (14) into (17), we obtain the following: .
Equation (27) can be shown as: Defining the following new variables as: The following Lyapunov function is chosen: where P is a positive definite matrix which can be selected in accordance with the procedure given in [39]; V is a quadratic, strict, and robust Lyapunov function. Actually, according to Assumption 1, we can get: where X is a symmetric and positive definite matrix. We can get .
V is negative semi-definite. A detailed description can be found in the Reference [39].
Accordingly, the proposed controller can ensure the stability of the system. Consequently, the proof of the closed-loop system under the proposed MFiPISTSMC strategy is completed.

Simulink Results
We implement the simulation, which are based on the Matlab/Simulink. The key PMSM parameters used in the simulation are listed as follows: the stator phase resistance R is 2.875 Ω; the inductance L is 8.5 mH; the magnetic chain of permanent magnets φ is 0.175 Wb; the moment of inertia J is 0.003 kg·m 2 ; the viscous damping B is 0.008 N·m·s; the pole pairs p n is 4. The parameters of the MFiPISMC strategy used in the simulation are listed as follows: K p = 1; K i = 1; η 1 = 10; η 2 = 1; k 1 = 10; k 2 = 12; a = 1000. The parameters of the PI strategy used in the simulation are listed as follows: the value of proportional coefficient set to 0.1; the value of integral coefficient set to 0.5. The parameters of the proposed MFiPISTSMC strategy used in the simulation are listed as follows: K p = 1; K i = 1; η 1 = 10; η 2 = 1; k 1 = 300; k 2 = 100; a = 1000. The parameters of the LESO used in the simulation are listed as follows: β 1 = 20000; β 2 = 1500000; b 0 = 1000. In order to compare more accurately, the stable simulation data from 0.2 to 0.5 s are used to perform the following calculations: Maximum Absolute Error (MAE) = Max e i (t) The PI control strategy and the MFiPISMC strategy are compared with the control strategy proposed in this study. In order to compare the control performance of various controllers, the reference speed is set to 100 r/s and the simulation time is set to 0.5 s. Figure 3 and Table 2 compare the speed response curves of the motor under different control strategies without load starting and stable operation. Moreover, Figure 4 and Table 3 compare the speed response curves of the motor under different control strategies with 0.5 N·m load starting and stable operation. As shown in Figure 3a or Figure 4a, compared with the PI control strategy and MFiPISMC strategy, the adjustment time controlled by the proposed MFiPISTSMC strategy is the shortest, which indicates that the proposed MFiPISTSMC strategy has the optimal transient performance. As shown in Tables 2 and 3, compared with the PI control strategy and MFiPISMC strategy, the RMSE and MAE controlled by the proposed MFiPISTSMC strategy also are the smallest, which reveal that the proposed MFiPISTSMC strategy has the optimal control accuracy. From Figure 3b or Figure 4b, we also can obviously conclude that the chattering phenomenon of the proposed MFiPISTSMC strategy has been eliminated significantly. Consequently, the proposed MFiPISTSMC strategy has the best steady-state performance.
Electronics 2020, 9, x FOR PEER REVIEW 9 of 17 stable operation. Moreover, Figure 4 and Table 3 compare the speed response curves of the motor under different control strategies with m N ⋅ 5 . 0 load starting and stable operation. As shown in Figure 3a or Figure 4a, compared with the PI control strategy and MFiPISMC strategy, the adjustment time controlled by the proposed MFiPISTSMC strategy is the shortest, which indicates that the proposed MFiPISTSMC strategy has the optimal transient performance. As shown in Tables 2 and 3, compared with the PI control strategy and MFiPISMC strategy, the RMSE and MAE controlled by the proposed MFiPISTSMC strategy also are the smallest, which reveal that the proposed MFiPISTSMC strategy has the optimal control accuracy. From Figure 3b or Figure 4b, we also can obviously conclude that the chattering phenomenon of the proposed MFiPISTSMC strategy has been eliminated significantly. Consequently, the proposed MFiPISTSMC strategy has the best steady-state performance.      In order to further verify the anti-disturbance ability of the proposed control strategy, load disturbances are suddenly changed at 0.5 and 1 s, respectively. Tables 4 and 5 show the comparative results of load changed suddenly under different control strategies, including speed perturbation, speed recovery time, and torque adjustment time. Figures 5 and 6 compare the speed response curves and the torque response curves under different control strategies with the load changed suddenly. As shown in Tables 4 and 5 and Figures 5 and 6, compared with the PI control strategy and MFiPISMC strategy, the speed recovery time and torque adjustment time controlled by the proposed MFiPISTSMC strategy are the shortest, the speed perturbation amplitude controlled by the proposed MFiPISTSMC strategy is also the smallest, which indicates that the proposed MFiPISTSMC strategy has stronger robustness and better reliability when subjected to external uncertainties.    In order to further verify the anti-disturbance ability of the proposed control strategy, load disturbances are suddenly changed at 0.5 and 1 s, respectively. Tables 4 and 5 show the comparative results of load changed suddenly under different control strategies, including speed perturbation, speed recovery time, and torque adjustment time. Figures 5 and 6 compare the speed response curves and the torque response curves under different control strategies with the load changed suddenly. As shown in Tables 4 and 5 and Figures 5 and 6, compared with the PI control strategy and MFiPISMC strategy, the speed recovery time and torque adjustment time controlled by the proposed MFiPISTSMC strategy are the shortest, the speed perturbation amplitude controlled by the proposed MFiPISTSMC strategy is also the smallest, which indicates that the proposed MFiPISTSMC strategy has stronger robustness and better reliability when subjected to external uncertainties.

Experimental Results
To further verify the effectiveness of the proposed method, a cSPACE (Control signal process and control engineering) based PMSM speed control experimental platform has been applied in this paper. The cSPACE experimental platform of the PMSM drive system is depicted in Figure 7. The cSPACE experimental platform consists of a TI TMS320F28335 DSP, a Matlab/Simulink, a SM060R20B30M0AD PMSM, and a MY1016 DC generator. The cSPACE experimental platform is the software and hardware platform of fast control prototype and hardware in loop real-time simulation.
To further verify the effectiveness of the proposed method, a cSPACE (Control signal process and control engineering) based PMSM speed control experimental platform has been applied in this paper. The cSPACE experimental platform of the PMSM drive system is depicted in Figure 7. The cSPACE experimental platform consists of a TI TMS320F28335 DSP, a Matlab/Simulink, a SM060R20B30M0AD PMSM, and a MY1016 DC generator. The cSPACE experimental platform is the software and hardware platform of fast control prototype and hardware in loop real-time simulation.
The key PMSM parameters used in the experimentation are listed as follows: the stator phase resistance R is Ω 16 . 0 ; the inductance L is mH 44 . 0 ; the magnetic chain of permanent magnets ϕ is Wb 0077 . 0 ; the moment of inertia J is ; The parameters of the LESO used in the experimentation are listed as follows:  Figure 8 shows the actual speed response curves of tracking 100 r/s and 300 r/s under the PI control strategy based on the cSPACE experimental platform. Figure 9 shows the actual speed response curves of tracking 100 r/s and 300 r/s under the proposed MFiPISTSMC strategy based on the cSPACE experimental platform. Figure 10 and Table 6 show the comparative results of the dynamic experiments. Because the control strategy is written as discrete form on the experimental platform, the horizontal coordinates of the experimental results are the sampling points. From the experimental results presented in Table 6 and Figures 8-10, we can clearly find that the actual speed response curves of the PMSM using the proposed MFiPISTSMC strategy can track the set speed value more quickly. The experimental results also reveal that the proposed MFiPISTSMC strategy has the optimal control accuracy. It is a fact that the proposed MFiPISTSMC strategy has the superior dynamic and static characteristics. The key PMSM parameters used in the experimentation are listed as follows: the stator phase resistance R is 0.16 Ω; the inductance L is 0.44 mH; the magnetic chain of permanent magnets φ is 0.0077 Wb; the moment of inertia J is 0.342 kg·m 2 ·10 −4 ; the viscous damping B approximately is zero; the pole pairs p n is 4. The parameters of the proposed MFiPISTSMC strategy used in the experimentation are listed as follows: K p = 1; K i = 0.01; η 1 = 1; η 2 = 1; k 1 = 500; k 2 = 10; a = 1000. The parameters of the LESO used in the experimentation are listed as follows:β 1 = 20000; β 2 = 1500000; b 0 = 1000. Besides, the saturation limit of u(t) is set to ±10A. Figure 8 shows the actual speed response curves of tracking 100 r/s and 300 r/s under the PI control strategy based on the cSPACE experimental platform. Figure 9 shows the actual speed response curves of tracking 100 r/s and 300 r/s under the proposed MFiPISTSMC strategy based on the cSPACE experimental platform. Figure 10 and Table 6 show the comparative results of the dynamic experiments. Because the control strategy is written as discrete form on the experimental platform, the horizontal coordinates of the experimental results are the sampling points. From the experimental results presented in Table 6 and Figures 8-10, we can clearly find that the actual speed response curves of the PMSM using the proposed MFiPISTSMC strategy can track the set speed value more quickly. The experimental results also reveal that the proposed MFiPISTSMC strategy has the optimal control accuracy. It is a fact that the proposed MFiPISTSMC strategy has the superior dynamic and static characteristics.    When the PMSM is subjected to the external disturbance, the abilities of the PI control strategy and the proposed MFiPISTSMC strategy to resist the external disturbance are shown in Figures 11 and 12, respectively. In detailing, we sample 16,000 points in the experiment. The external load current of the MY1016 DC generator suddenly becomes 200 mA from 0 mA at sampling point 7000 in Figures 11 and 12, Electronics 2020, 9,1427 13 of 16 respectively. The external load current of the MY1016 DC generator suddenly becomes 0 mA from 200 mA at sampling point 13,000 in Figures 11 and 12, respectively. From the experimental results, it can be concluded that the proposed MFiPISTSMC strategy has strong anti-disturbance ability and better reliability for suddenly increases and suddenly decreases of load disturbance. It can be clearly observed that the proposed MFiPISTSMC strategy can conveniently suppress the external disturbances. When the PMSM is subjected to the external disturbance, the abilities of the PI control strategy and the proposed MFiPISTSMC strategy to resist the external disturbance are shown in Figures 11  and 12, respectively. In detailing, we sample 16,000 points in the experiment. The external load current of the MY1016 DC generator suddenly becomes 200 mA from 0 mA at sampling point 7000 in Figures 11 and 12, respectively. The external load current of the MY1016 DC generator suddenly becomes 0 mA from 200 mA at sampling point 13,000 in Figures 11 and 12, respectively. From the experimental results, it can be concluded that the proposed MFiPISTSMC strategy has strong antidisturbance ability and better reliability for suddenly increases and suddenly decreases of load disturbance. It can be clearly observed that the proposed MFiPISTSMC strategy can conveniently suppress the external disturbances.
In summary, the experimental results demonstrate the proposed MFiPISTSMC strategy achieves superior control performance in the sense of the dynamic characteristic, static characteristic, robustness, and reliability.

Conclusions
In this research, a novel compound MFiPISTSMC strategy based on the ultra-local model is proposed for the control of the PMSM speed regulation system. The unknown term of the ultra-local model of PMSM is estimated by the LESO. The Lyapunov approach is used to demonstrate the stability of the closed-loop system. The proposed compound MFiPISTSMC strategy consists of the In summary, the experimental results demonstrate the proposed MFiPISTSMC strategy achieves superior control performance in the sense of the dynamic characteristic, static characteristic, robustness, and reliability.

Conclusions
In this research, a novel compound MFiPISTSMC strategy based on the ultra-local model is proposed for the control of the PMSM speed regulation system. The unknown term of the ultra-local model of PMSM is estimated by the LESO. The Lyapunov approach is used to demonstrate the stability of the closed-loop system. The proposed compound MFiPISTSMC strategy consists of the STSMC strategy, the iPI control strategy, and the MFC strategy, which has excellent performance. The static characteristic, dynamic characteristic, robustness, and reliability of the novel compound MFiPISTSMC strategy are verified by simulation and experimental results.
In the future, we will introduce the Fractional-order theory into the compound MFiPISTSMC strategy, which can further improve the control effect.