Adaptive Robust Backstepping Control of Permanent Magnet Synchronous Motor Chaotic System with Fully Unknown Parameters and External Disturbances

The chaotic behavior of permanent magnet synchronous motor is directly related to the parameters of chaotic system. The parameters of permanentmagnet synchronousmotor chaotic system are frequently unknown.Hence, chaotic control of permanent magnet synchronous motor with unknown parameters is of great significance. In order to make the subject more general and feasible, an adaptive robust backstepping control algorithm is proposed to address the issues of fully unknown parameters estimation and external disturbances inhibition on the basis of associating backstepping control with adaptive control. Firstly, the mathematical model of permanent magnet synchronous motor chaotic system with fully unknown parameters is constructed, and the external disturbances are introduced into the model. Secondly, an adaptive robust backstepping control technology is employed to design controller. In contrastwith traditional backstepping control, the proposed controller ismore concise in structure and avoids many restricted problems. The stability of the control approach is proved by Lyapunov stability theory. Finally, the effectiveness and correctness of the presented algorithm are verified through multiple simulation experiments, and the results show that the proposed scheme enablesmaking permanentmagnet synchronousmotor operate away from chaotic state rapidly and ensures the tracking errors to converge to a small neighborhood within the origin rapidly under the full parameters uncertainties and external disturbances.


Introduction
In recent years, the permanent magnet synchronous motor (PMSM) is utilized widely in various industrial fields due to its constantly dropping production cost, simple structure, high torque, and high efficiency.However, Hemati found that PMSM would generate chaotic behavior with system parameters entering into a certain region [1].Previous studies have shown that the chaotic movement of PMSM will produce irregular oscillations of torque and speed, exacerbate current noise, and worsen operation performance and may even damage the entire drive system.Therefore, research on PMSM chaos phenomenon has attracted extensive attention worldwide [2][3][4][5], and further studying on the control method of PMSM chaos is of extreme significance [6][7][8].
The nonlinear characteristics of PMSM, such as multivariability, strong coupling, and high dimension, make it difficult to control for traditional linear control theory.Hence, a variety of modern and nonlinear control algorithms are introduced to suppress PMSM chaotic behavior.In terms of these control algorithms whether or not relying on the model parameters, the previous control methods can primarily be divided into two categories.The first type is on the basis of accurate model parameters, such as entrainment and migration control [9], exact feedback linearization control [10], and decoupling control [11].However, the accuracy of these control methods directly depends on PMSM model parameters; if the system parameters deviate from the rated values, the control performance will go bad.The second type is based on unknown parameters, which have become the research focus of PMSM chaos suppression recently, mainly including sliding mode variable structure control [12,13], fuzzy control [14], and  ∞ control [15].However, sliding mode variable structure control requires uncertain parameters to satisfy certain matching conditions, fuzzy control is 2 Mathematical Problems in Engineering dependent on the fuzzification of Takagi-Sugeno, and  ∞ control is inclined to ignore the operating states under special conditions [16].In essence, PMSM chaotic system is highly sensitive to initial states and parameters, and PMSM model parameters are susceptible to the temperature and humidity of the surrounding environment.Therefore, PMSM chaotic repression with unknown model parameters has applicability to a broader field and is more in line with reality [17].Actually, the adaptive control (AC) provides a natural routine for PMSM chaotic control with unknown parameters, which has been presented in literatures [12,13,18].
Backstepping control (BC) is one of the most popular nonlinear control methods newly proposed to address parameter uncertainty, specifically the uncertainty not satisfying matching condition, which has been successfully applied to many engineering fields such as motor drive, temperature control of boiler main steam, and rocket location tracking.The core idea of BC is that complex highdimensional nonlinear systems are decomposed into many simple low-dimensional subsystems and virtual control variables are introduced to backstepping process to design concrete controllers.In addition, BC has been successfully applied to suppress Liu chaotic system [19] and Chen chaotic system [20].Therefore, the idea of combining BC with AC provides a useful and feasible train of thought to control PMSM chaotic system with unknown parameters.Literatures [21,22] have exactly practiced this idea.
However, the conventional backstepping approach is confronted with two major problems of solving complicated "regression matrix" [23] and encountering "explosion of terms" [24].In [25], the complexity of regression matrix is sufficiently manifested, which almost occupies one full page.Nevertheless, explosion of terms is an inherent shortcoming and is induced by repeated differentiations of virtual variables, particularly in design of adaptive backstepping controller [26].Additionally, integration of BC with AC is frequently faced with the singularity arising from any estimation term emerging as a denominator of any control input.The overparameterization caused by the number of estimations larger than actual system parameters hinders the conventional adaptive backstepping control.
In addition to the above problems, to the extent of our knowledge, mostly existing literatures on PMSM chaotic control only concentrate on the cases of single unknown parameter and partial unknown parameters [21,22], and there is no way to address the issue of fully unknown parameters.Furthermore, the existing researches mainly aim at the situation of sudden power failure during PMSM operation [16]; the existing conclusions lack the generality.Hence, through combination of BC and AC, not only does this paper study the control issue of PMSM chaos suppression with fully unknown parameters, but also the external disturbances are taken into account in PMSM chaos model.Newly adaptive updating laws of unknown parameters are designed to totally estimate unknown parameters of PMSM chaotic model, and adaptive robust backstepping controllers on the basis of adaptive estimations and external disturbances are developed to drive PMSM to escape out of chaotic state quickly, inhibit the external disturbances, and accomplish the given signals tracking rapidly.The method proposed in this paper expands the applied range of backstepping control theory in PMSM chaotic system.Moreover, the study of chaos control problem with totally unknown parameters and external disturbances is more general and practical, and the results and conclusions obtained are more applicable.
In reality, the three parameters , , and   in (2) tend to be unknown or to have uncertainties resulting from operating conditions.In other words, when all the parameters , , and   cannot be determined, (2) actually represents PMSM chaotic system model with fully unknown parameters.

Design of Adaptive Robust Controller with Backstepping Approach
Taking a more general situation into account, the PMSM chaotic model described in ( 2) is immersed by external disturbances.The model can be rewritten as follows: where Δ 1 (x, ) and Δ 2 (x, ) represent the external disturbances, x indicates the system states, and x = ( 1 ,  2 ,  3 ) = (,   ,   ).

Control Objective and Assumptions.
Control problem in the paper can be described as follows: for PMSM chaotic system (3) with fully unknown parameters , , and   and external disturbances Δ 1 and Δ 2 , adaptive laws of unknown parameters , , and   are designed and adaptive robust controllers   and   are constructed to ensure PMSM breaks away from chaos rapidly and runs into an expected orbit.Simultaneously, the fully unknown parameters , , and   can be estimated accurately and the external disturbances can be inhibited effectively.
For convenience of controller design, the control system is supposed to hold some reasonable assumptions as follows.
Assumption 1.The state variables for PMSM chaotic system (,   ,   ) are observable.The estimated values of unknown system parameters are described as δ, γ, and τ ; then, the estimation errors δ, γ, and τ can be expressed as follows: 3.2.Controller Design.The essence of adaptive robust backstepping controller is to design controller through combination of backstepping method and adaptive approach; then, a reasonably stable function is built in accordance with Lyapunov stability theory to guarantee error variables to be effectively stabilized and meanwhile ensure the output of closed loop system tracks reference signals quickly.On the basis of this, the adaptive robust backstepping controller is designed as follows.
Step 1.For the speed reference signal  * , define the tracking error   as follows: Taking PMSM chaotic system model (3) into account, the derivative of ( 5) can be written as Define the tracking error   of  axis stator current   as follows: where  *  is the expected output value of   .For the  axis current reference signal  *  , its tracking error   is defined as follows: By substitution of ( 7) into (6), we can obtain Let where  1 represents the positive control gain.Through substitution of ( 10) into ( 9), ( 12) can be obtained: Lyapunov function  1 is selected as follows: Then, the derivative of  1 can be described as Step 2. To stabilize the output  axis current of PMSM, the derivative of   is conducted as follows: By substitution of ( 12) into (15), we can get Combined with the mathematical model of PMSM chaotic system, ( 16) can be further calculated as follows: By substitution of δ = δ +  into (17), the following equation can be obtained: Lyapunov function  2 is chosen as follows: Then, the derivative of  2 can be described as The first control variable is selected as where   and   are the model compensation and robust control inputs, respectively.Then,   and   can be, respectively, chosen as where  2 is another positive control gain and  1 is a positive number chosen arbitrarily.By substitution of ( 21) and ( 22) into (20), we can acquire Additionally, Step 3. Differentiating the tracking error   of  axis current   , we can get Lyapunov function  3 is chosen as Then, the derivative of  3 can be represented as In terms of (28),  axis output stator voltage   can be calculated: where where  3 is the positive control gain and  2 is a positive number chosen arbitrarily.By substitution of (30) and (31) into ( 26) and (28), respectively, the following equations can be acquired: (34) Step 4. Lyapunov function  of PMSM chaotic system with fully unknown parameters and external disturbances is selected as follows: where  1 ,  2 , and  3 represent positive adaptive gains.
Combined with equations δ = δ , γ = γ , and τ  = τ  , derivative of selected Lyapunov function  can be calculated as follows: In terms of (36), the adaptive laws of unknown parameters , , and   can be selected, respectively, as follows: By substitution of (37), (38), and (39) into (36), (36) can be simplified as follows:  21) and ( 29) can ensure the tracking error signals ( 5), (7), and ( 8) of PMSM chaotic systems are asymptotically stable.That is to say, PMSM chaotic system can run out of chaos quickly through the proposed controllers ( 21) and ( 29) and track the given reference signals.
Through stability analysis, we want to verify the correctness of the theorem.
According to (40), new expression can be obtained as follows through some mathematical computations: Appropriate controller gains  1 and  2 are selected as follows: Equation ( 42) can be replaced by the following: Then, by substitution of ( 43) into (41), we can obtain where where () = (  ,   ,   ).By integration of (45), we can get  and  3 can make the inequalities of  ≥ 0 and V < 0 hold.In addition, equation of  = 0 is not satisfied until   =   =   = τ = γ = δ = 0.In summary, PMSM chaotic system is globally asymptotically stable at the equilibrium point of (  ,   ,   ) = (0, 0, 0).

Numerical Simulation and Discussions
In order to illustrate the superiority of the proposed approach adequately, the simulation is carried out in MATLAB environment for three cases under the initial condition of (,   ,   ) = (0, 0, 0).Let (,   ,   ) = ( 1 ,  2 ,  3 ) and the control parameters are selected as  1 = 10,  2 = 30000,  3 = 5, and  1 =  2 = 0.01; the adaptive gains are chosen as  1 = 6.2,  2 = 100, and  3 = 0.06.The simulation time is chosen as 100 s and the designed controller is put into effect at the time of 20 s.

Test-I.
The PMSM chaotic system is tested with the parameters  = 0.2,  = 50, and   = 3.2.In order to be consistent with the reality better, we assume that the three parameters of PMSM chaotic system are all unknown with the initial condition of (, ,   ) = (0, 0, 0), and the expected reference signals are set as  * = 10 and  *  = 1.Furthermore, the external disturbances Δ 1 (x, ) = 20 3 sin(5) and Δ 2 (x, ) = 10 sin(5) are injected into the PMSM chaotic system.The simulation results given in Figures 2-4 apparently show PMSM runs in a chaotic state with no control inputs.Therefore, introduction of the presented control approach to suppress chaos in PMSM system will be of great importance and necessity.Figures 5-12 show that the proposed controller is utilized to control the PMSM chaotic system, where Figures 5-7 display the curves of state variables changing over time for PMSM chaotic system, which demonstrate the PMSM system stays away from the previous chaotic state when the designed controller is added to PMSM chaotic system, and track the desired signals accurately and rapidly.Furthermore,        Figures 10-12 show the estimated errors δ, γ, and τ of unknown parameters , , and   for PMSM chaotic system, which testify the effectiveness of constructed adaptive laws and demonstrate the proposed approach has a good robustness against the uncertainties in system parameters.controller inputs   and   shown in Figures 16-17, which demonstrate that the designed controller can guarantee the outputs track references well and suppress the external disturbances effectively.Figures 18-20 indicate that the designed adaptive law can estimate the fully unknown parameters precisely even if the fully unknown parameters are changed.

Test-III.
For Test-III, the external disturbances are enlarged in addition to changing the unknown motor parameters and expected reference signals on the basis of Test-II, which are described as Δ 1 (x, ) = 40 3 sin(5) and Δ 2 (x, ) = 20 sin(5).The control difficulty in Test-III is larger than the previous two experiments and Test-III is a more general instance to verify the controller's performance.The simulation results are shown in Figures 21-28.Figures 21-23 give the curves of the state variables ,   , and   , which manifest these variables are controlled to their references and chaos is eliminated when adding the proposed controllers Remark 5. Previous researches on parameter estimation of PMSM chaotic system mostly assumed that only partial parameters of the system are unknown.The paper takes fully nondeterministic parameters , , and   into account; it undoubtedly extends the theory of parameter estimation for PMSM chaotic system.Remark 6.The action time of control inputs is 20 s in the simulation.The aim of doing this is to observe the effect of the control approach better.In reality, as long as the chaos occurs, the controller will be put into effect.Remark 7. On the basis of considering fully unknown parameters, the external disturbances are introduced into the PMSM chaotic model.Hence, the designed control consists of two parts.One is to guarantee the state variables to track the reference signals; another is to suppress the external disturbances.In general, the simultaneous consideration of fully unknown parameters and external disturbances makes the research results more general and practical.

Conclusions
In this paper, a control approach is proposed to address the control issue of chaos in PMSM system with fully unknown parameters and external disturbances.Main conclusions are acquired as the following: (1) Through combination of adaptive control with backstepping control, the presented adaptive robust backstepping control scheme resolves the main problems of the conventional backstepping algorithm encountered.And the stability of the designed controller is proved by Lyapunov theory.
(2) The simulation results show that the designed controller is able to make the PMSM operate out of chaotic state quickly, and the adaptive laws are established to estimate the unknown parameters accurately.Furthermore, the proposed algorithm can ensure the unknown parameters converge to the actual values fast and restrain the external disturbance effectively.
(3) The design method in this paper is simple and effective.For PMSM chaotic system with fully unknown parameters, the control variables in proposed approach can be self-adjusted with the changing of system parameters.Therefore, our findings are more practical and more convenient for engineering applications.Future research will discuss the application of the proposed control approach into practical implementation.

Figure 2 :
Figure 2: The  curve of PMSM chaotic system with no control inputs   and   .

Figure 3 :
Figure 3: The   curve of PMSM chaotic system with no control inputs   and   .

Figure 4 :Figure 5 :
Figure 4: The   curve of PMSM chaotic system with no control inputs   and   .

Figure 6 :Figure 7 :
Figure 6: The   curve of PMSM chaotic system added the controller inputs   and   .

Figure 12 :Figure 13 :
Figure 12: The τ curve of estimation error of   .

Figures 5 -
Figures 5-7 indicate the proposed controller shown in Figures 8-9 can inhibit the external disturbances.Figures10-12show the estimated errors δ, γ, and τ of unknown parameters , , and   for PMSM chaotic system, which testify the effectiveness of constructed adaptive laws and demonstrate the proposed approach has a good robustness against the uncertainties in system parameters.

Figure 20 :Figure 21 :
Figure 20: The τ curve of estimation error of   .
rotor, ȋ and ȋ are  axis and  axis currents of stator winding, respectively, ȗ and ȗ are  axis and  axis voltages of stator winding,  is the number of rotor pole pairs,   is the flux generated by permanent magnets,  is the moment of inertia,  is the viscous damping coefficient, τ is the load torque,  is the phase resistance of the stator windings, and   and   are  axis and  axis inductances of stator winding, respectively.For a PMSM with uniform air gap,   =   .Hence, we use  to substitute   and   in the following paper.