Design of an adaptive terminal sliding mode to control the PMSM chaos phenomenon

This paper schemes a new nonlinear method for controlling and suppressing of chaos instability in the PMSM. First, a nonlinear model of PMSM is considered that is able to describe the dynamic behaviour of the PMSM in different functional conditions, including the occurrence of chaos. As an integral part of most physical systems, disturbance with an unknown upper limit is considered in the design of the control scheme. Parametric uncertainty is also included in the PMSM system model and a new control technique is presented based on robust adaptive and sliding mode approaches to prevent chaos. In the proposed method, adaptive technique is used to estimate the upper bound of uncertain terms, disturbances and nonlinear terms, and the robust sliding mode scheme is designed to stabilize the closed-loop PMSM system in contradiction of chaos. Using comparisons, the efficiency of the proposed method is shown in different working conditions.


Introduction
Due to the improved efficiency, permanent magnet synchronous motors (PMSM) have become increasingly important for industrial applications, including wind turbines and renewable energy (Sant & Harish, 2021).Low volume, light weight, high power factor, fast dynamics, high efficiency and high output torque are some of the advantages of PMSMs and their remarkable reliability makes them suitable for applications such as industrial propulsion systems and electric vehicle batteries (Babaghorbani et al., 2021;Lee & Lim, 2021;Pasqualotto & Zigliotto, 2021).In addition, for wind energy conversion, PMSMs act as the preferred system (Babaghorbani et al., 2021).However, ensuring the safe and stable operation of PMSMs, which is essential for industrial automation, faces many challenges.These problems are mostly due to the nonlinear and multivariate nature of the PMSM dynamic model, which may even show bifurcation, limit cycles, and disordered behaviours at specific parameter values.
Chaotic PMSMs show irregular movements that have a detrimental effect on their performance stability, leading to intermittent fluctuations in torque and velocity, unstable performance, electromagnetic noise (Yang, Yang, Sheng, et al., 2021;Babaei et al., 2021;Rongyun et al., 2021), and external disturbances and system failure (Siddique & Rehman, 2021;Takhi et al., 2022).Chaos CONTACT Hamidreza Akbari h.akbari@iauyazd.ac.ir has been observed in most synchronous motors including reluctance motor (Harb, 2004), hysteresis motor (Zribi et al., 2009), brushless DC motor (Wei et al., 2007) and permanent magnet motor (Harb & Ahmad, 2002).At present, due to theoretical and practical importance, chaos control and analysis has become an important research topic (Chen et al., 2021;Hafsa et al., 2021;Liu et al., 2021;Takhi et al., 2022).Chaos was first reported in 1989 in engine propulsion systems.Since then, many efforts have been made to identify and control chaos in the performance of electric machines (Harb, 2004;Lu & Wang, 2021;Rongyun et al., 2021;Siddique & Rehman, 2021;Takhi et al., 2022;Wei et al., 2007;Zribi et al., 2009).Chaos in PMSM is linked with adverse consequences such as instability and failure of the whole system (Liu et al., 2017;Souhail et al., 2019).In the event of chaos, the output power of PMSM will have severe fluctuations, which will have a great impact on the entire system and even lead to the collapse of the overall system (Yang, Yang, Sheng, et al., 2021).Therefore, chaos prevention in PMSMs using effective control schemes is extremely important and is the main purpose of this study.
PMSM performance is strongly influenced by nonlinear dynamic disturbances, system parameters and external load.Numerous studies have shown the chaos phenomenon in PMSMs for certain values of parameters and under certain operating conditions Li et al. (2014) and Hu et al. (2016) which have had detrimental effects on the stable performance of PMSMs.Due to the ability of nonlinear analysis in chaos control applications, further considerations are aimed in this study at more accurate analysis of nonlinear dynamic behaviour and ensuring effective and stable performance of PMSMs in the presence of parameter uncertainty and disturbance.
The parameter perturbation technique, which is the most well-known OGY method (OGY stands for the names of its inventors, 'Ott, Grebogi, and Yorke'), is considered as one of the first tools for chaos control (Rahimi et al., 2016), but choosing an adjustable parameter was a complex task.Feedback control for chaotic PMSM also faces challenges in real-world applications (Xie et al., 2018).Neural fuzzy control (NFC) shows superior performance in the presence of uncertainties and nonlinearities (Qiang et al., 2013).However, despite NFC's learning capacity, its ability to gain control is limited due to its expensive time cost (Li et al., 2017).Sliding mode control (SMC) also offers effective performance against nonlinear effects, uncertainty and limited disturbance, but the chattering phenomenon is a major drawback in the application (Loria, 2009;Yang et al., 2019).A robust control scheme using SMC and backstepping is also designed in Sun et al. (2016).However, the proposed method was performed under some limited conditions.The backstepping-based nonlinear adaptive method (Coban, 2019;Gopaluni et al., 2003;Karagiannis & Astolfi, 2008;Zhou & Wang, 2005) has been considered in the control of nonlinear systems, where uncertainties do not satisfy matching conditions.However, the 'complexity explosion' due to the repeated derivation of the virtual control function remains an inherently challenging negative aspect in the standard backstepping method and should be avoided by using appropriate schemes (Hou & Han, 2010;Sun & Zhu, 2013).To effectively improve the performance of PMSM chaos control, a nonlinear proportional control technique is proposed in Zeng and Liu (2014).Using a neural network-based adaptive dynamic programming method, tracking the maximum energy point and overcoming disturbances is pursued in Zhong- Qiang et al. (2015).The use of the T-S fuzzy model to handle the nonlinear system and the design of a new fuzzy tracker have been done in Wang et al. (2011) for effective removal of chaotic motion.The investigation of dynamic characteristics and the development of a new predictive control method for suppressing chaotic behaviour have been carried out in Borah and Roy (2017), and reference (Rajagopal et al., 2017) provides a self-adaptive approach for suppressing the phenomenon of chaos in the device.
This study aims to control and avoid chaos in PMSM, while disturbance and parameter uncertainty are present.
For this purpose, a nonlinear robust approach has been developed to suppress chaotic behaviour in the PMSM system.In this study, the system is affected by a boundary disturbance with an unknown upper bound.In addition, uncertainty is considered in the design of the controller and the stability of the closed-loop system, while its upper bound is unspecified.We also apply adaptive law to the controller design process for the uncertainty and disturbance estimation of the PMSM drive system.In addition, a nonlinear robust controller is used to ensure the stability of the system against the uncertainty of the parameters and disturbances applied to the model.Finally, the performance of the control system is evaluated by comparison.Therefore, it can be said that the most important innovations of this study are the design of a hybrid robust control scheme using sliding mode and adaptive approaches by meeting the following goals: (1) Ensuring the prevention of chaos in the strongly nonlinear dynamics of the PMSM system.(2) Covering of uncertain parametric effects with an unknown upper bound.
(3) Covering the effects of matched and unmatched disturbances.(4) Ensuring stability in Lyapunov's concept.
(5) Improving transient and permanent response of the system states.
So, this article has been compiled as follows: Section 2 describes the PMSM model and Section 3 presents the proposed control approach, while Section 4 presents simulation and comparison results for the proposed method.Finally, the conclusion is given in Section 5.

Mathematical model of the chaotic PMSM drive system
In this section, a model used for describing the dynamic behaviour of the PMSM system is presented.The transformed model of PMSM with the smooth air gap is as follows (Liu et al., 2021): The description of each of the parameters used in Equations ( 1) and ( 2) is as follows: u d , and u q are the direct and quadrature-axis stator voltages, respectively.i d , and i q signify the direct-and quadrature-axis stator currents, respectively.R is the stator winding resistance.L d , and L q indicate the direct and quadrature axis stator inductance, respectively.ψ fd = 1.5ψ f is the direct-axis permanent magnet flux, p n signifies the number of polepairs, ω signifies motor angular frequency, J equ denotes the equivalent polar moment of inertia, T L specifies the external load torque, and B equ denotes the equivalent viscous damping coefficient.By analyzing Equation (1), valuable conclusions can be drawn that bifurcation and chaos occur when the parameters of PMSM or external inputs are located in a specific region (Liu et al., 2021) and during that, the phase path is out of order in a range and shows chaotic behaviour as shown in Figure 1.Since the chaotic motion drastically reduces PMSM performance, it needs to be removed.From the dynamic equation of the system (1), it can be seen that ũq and ũd can be used as two accessible manipulated variables.That is, there are control inputs that must be properly designed to eliminate chaos and guide system states.

The proposed controller implementation
In this section, a robust nonlinear adaptive sliding mode method is designed and presented for chaos suppression in PMSM.The robust controller designed in this study is placed in the same structure and schema as Iqbal and Singh (2019).The proposed method cannot only handle the effects of nonlinear terms, but also counteract the effects of uncertainties and disturbances.The controller design process is as follows.First, the dynamics equations of the system are rewritten considering the disturbances as follows: (2) and d 3 (t) are disturbances in the system that have a boundary but the upper bound is indistinct.u 1 and u 2 are also defined as follows: Now, according to the state of the system, u 2 is used to control x 3 .Also, u 1 is scheduled to control x 1 and x 2 .The reason for this is due to the placement of the state variable x 2 in the equations of ẋ1 and here x 2 present in ẋ1 is intended to control the behaviour of x 1 .In general, u 1 is used to control x 2 and control the behaviour of x 1 .So, for this purpose, we have three steps to control the state of x 1 .We define the sliding surface as follows: By taking the first order derivative of Equation ( 4), it is obtained.
With the definition of w 1 (t) as follows: where L 1 is the upper bound of the unknown w 1 (t).So Equation ( 5) can be rewritten as below in which x 2 is used as a virtual control input.Now, for the stability of the state x 1 , we apply the Lyapunov function as follows.
where L1 is the upper bound estimate of w 1 (t) or L 1 .Now by deriving the Lyapunov function, it is obtained.
In order to establish stability in relation 10, we define x 2 as the virtual control input as follows.
where c 1 and k 1 are the positive gains of the controller and 0 < q < 1 is a constant number.Substituting Equation (11) in Equation ( 9), we will have Now by choosing the adaptive control rule in the form below where λ 1 is the adjustment gain of the adaptive law, we will have To control the state of x 2 , we define the sliding surface as follows.
Now, taking the first-order derivative of Equation ( 15), we will have With the definition of w 2 (t) as follows: where L 2 is the upper bound of the unknown w 2 (t).So Equation ( 16) can be rewritten as below Now, for the stability of the state x 2 , we apply the Lyapunov function as follows.
where L2 is the upper bound estimate of w 2 (t) or L 2 .Now by deriving the Lyapunov function V 2 , it is obtained.
By selecting u 1 as below where c 2 and k 2 are the positive gains of the controller and the placement in Equation ( 21), it is obtained Now by choosing the adaptive control rule in the form below where λ 2 is the adaptive adjustment gain.By substituting, it is obtained In the last step, to control the state of x 3 , we define the sliding surface as follows.
By taking the first-order derivative of Equation ( 26), we will have By defining w 3 (t) as follows: where L 3 is the upper bound of the unknown w 3 (t).So Equation ( 27) can be rewritten as below For the stability of the state x 3 , we apply the Lyapunov function as follows. Where and L3 is the upper bound estimate of w 3 (t) or L 3 .Now by deriving the Lyapunov function V 3 , it is obtained.
(32) By selecting u 2 as below (33) Where c 3 and k 3 are the positive gains of the controller and the placement in Equation ( 32), it is obtained Now by choosing the adaptive control rule in the form below where λ 3 is the adaptive adjustment gain.By substituting, it is obtained Finally, we candidate the overall Lyapunov function of the system as follows.It is obtained by deriving from Equation ( 37) Which shows the stability of the closed loop system under the planned control scheme and the obtained virtual and adaptive signals.

Simulation results
In this section, the efficiency of the proposed control method to prevent chaos in the PMSM system is shown through simulation in MATLAB environment.For better evaluation, two scenarios with different operating conditions have been considered and in addition, a complete comparison has been performed against the adaptive fuzzy backstepping, finite time and sliding mode methods (Zhang et al., 2021).All simulation parameters are considered similar to Zhang et al. (2021).The simulation was done using an Intel ® Core TM i7-9700 K Processor computer (12M Cache, up to 4.90 GHz) in the MATLAB 2018a environment.The parameters of the proposed controller are all obtained by trial and error and include By applying the aforementioned controller, the simulation scenarios are described in the following.

The first simulation scenario
In this case, the regulation of PMSM system states is followed to the desired values, while no disturbance enters the system.The simulation results for scenario 1 are shown in Figures 2-4.For better physical understanding, the PMSM drive variables of ω, i d and i q which denote the angular speed and the d-q axis currents, are shown in the figures instead of the state variables, i.e. x 1 = ω, x 2 = i q , x 3 = i d .Figure 2 shows the ω curve of PMSM under the four adaptive fuzzy backstepping, finite time, sliding mode and the proposed nonlinear adaptive sliding mode approaches, while Figures 3 and 4 show the i d and i q states under these controllers.
As can be seen from the Figure 2, ω converges to zero value in the fastest time without any overshoot and undershoot under the proposed controller, while under the adaptive fuzzy backstepping method, ω experiences fluctuations and undershoot to reach the final   value.Under the finite time method, although there is no overshoot or undershoot in reaching the final value, the time spent to achieve this goal is higher than the two previously mentioned methods.Under the sliding mode method, the ω state does not reach the final value at all.Under the sliding mode method, the i d and i q states never reach their final desired values.Under the finite time technique, this action happens at a very low speed, and of course, the adaptive fuzzy backstepping scheme shows a lower speed and more overshooting than the planned technique.
Rapid regulation of states to desired values, low response oscillations and short settling time are among the advantages of the proposed nonlinear adaptive sliding mode method.To more accurately demonstrate the capability of the proposed control method, the regulatory error is measured under the four controllers used in the simulation and based on different error principles, i.e.Integral square error (ISE), Integral absolute error (IAE), Integral time absolute error (ITAE) and Integral time square error (ITSE) criteria is given in Tables 1-3.As can be seen from these tables, the least amount of deviation under all four error criteria is obtained for the states by the proposed nonlinear adaptive sliding mode method.

The second simulation scenario
In this case, the regulation of PMSM system states is followed despite the presence of disturbances in the system.The form of the disturbances is as follows In this scenario, the simulation results are shown in .Figure 5 shows the ω state, and Figures 6 and 7 show the tracking the desired values in the i d and i q states, respectively.
As it is clear from Figure 5, ω tends to the final value with the highest speed and the lowest fluctuation under the proposed method.Under the finite time technique, this action has been done at a low speed although the regulatory goal has been achieved.Disturbances have made the adaptive fuzzy backstepping scheme unable to perform the regulatory action, and in addition, a very large fluctuation can be seen on ω.In the case of i q shown in Figure 6, the sliding mode technique is not able to regulate.Under finite time and adaptive fuzzy backstepping approaches, there is always fluctuation on i q , and the fastest regulatory action with the least fluctuations is obtained by the planned method.Regarding the i d shown in Figure 7, both the finite time and sliding mode techniques are not able to perform regulatory action.Under the adaptive fuzzy backstepping scheme, there is a fluctuation on the i d response, and the regulatory action is obtained with the fastest speed and the lowest fluctuation under the proposed method.As can be seen from these figures, despite the presence of disturbances in this scenario, the results obtained for scenario one are valid here as well.That is, high speed in the regulation of desirable values, less fluctuations with a more limited amplitude and shorter settling time are among the advantages of the proposed nonlinear adaptive sliding mode method.In addition, Tables 4-6, which indicate the amount of tracking error under the four control methods in the simulation according to the ISE, ITSE, IAE and ITAE definitions, show that the lowest error rate is obtained in the proposed nonlinear adaptive sliding mode method.

Conclusion
In this paper, the issue of chaos in the PMSM system was considered and due to its adverse effects on system performance and the possibility of failure, a new control method was proposed to prevent chaos.At first, a nonlinear model of the PMSM system was considered, which in addition to handling nonlinear terms was studied the    uncertainty and disturbance effects.Using adaptive and sliding mode techniques, a new combined method was proposed to prevent chaos instability and limit cycles in the PMSM system.By comparing and simulating in MATLAB environment, the efficiency of the proposed nonlinear adaptive sliding mode method was achieved in preventing the occurrence of chaos and providing

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

Figure 1 .
Figure 1.The phase trajectory and chaotic behaviour of the system PMSM.(a) xy phase (b) xz phase (a) yz phase (a) xyz phase.

Figure 2 .
Figure 2. The ω curve of the chaotic PMSM under different controllers in scenario 1.

Figure 3 .
Figure 3.The i q curve of the chaotic PMSM under different controllers in scenario 1.

Figure 4 .
Figure 4.The i d curve of the chaotic PMSM under different controllers in scenario 1.

Figure 5 .
Figure 5.The ω curve of the chaotic PMSM under different controllers in scenario 2.

Figure 6 .
Figure 6.The i q curve of the chaotic PMSM under different controllers in scenario 2.

Figure 7 .
Figure 7.The i d curve of the chaotic PMSM under different controllers in scenario 2.

Table 1 .
Comparison of performance index in terms of different error criteria for ω state in scenario 1.

Table 2 .
Comparison of performance index in terms of different error criteria for i q state in scenario 1.

Table 3 .
Comparison of performance index in terms of different error criteria for i d state in scenario 1.

Table 4 .
Comparison of performance index in terms of different error criteria for ω state in scenario 2.

Table 5 .
Comparison of performance index in terms of different error criteria for i q state in scenario 2.

Table 6 .
Comparison of performance index in terms of different error criteria for i d state in scenario 2.Adaptive terminal sliding mode 0.0026 5.8864e-4 0.0399 0.0753 the desired transient and permanent dynamic responses.Subsequent studies can also consider the optimization of the control signal as well as the effects of noise.