Nonlinear Dynamic Surface Control of Chaos in Permanent Magnet Synchronous Motor Based on the Minimum Weights of RBF Neural Network

and Applied Analysis 3 0 5 10 0 5 10 15 0 5 10 15 20 25 30


Introduction
The permanent magnet synchronous motor is widely used in the industrial applications [1,2].But PMSM with nonuniform breath appears to be the chaotic behavior at specific parameters and working conditions.This behavior leads to the intermittent oscillation of torque and speed, irregular current noise of the system, and unstable control performance.Furthermore, it intensively influences the stability and safety of the system [3].
For ameliorating the performance of the PMSM system, a large amount of literatures and control methods have been attempted to apply to the motor.For example, to improve the error convergence rate, the nonsingular fast terminal sliding mode control (SMC) [4] which can reach finite-time stability is applied.In [5], a high-order SMC method via backstepping is presented to attain finite-time tracking control regardless of mismatched disturbance.However, the controller appears to be discontinuous phenomenon in dynamic sliding manifolds.In [6], an adaptive backstepping based terminal sliding mode controller for parameter strict-feedback system is proposed, where the finite-time convergence of the error is achieved.Furthermore, the OGY method is a fundamental technology for controlling chaos [7,8].Unfortunately, choosing an adjustable parameter usually becomes very difficult in real practice.Control of chaos by using the time-delay feedback control technology is introduced to the real applications, but it suffers from some problems as the control objective must be the equilibrium [9].The dynamic surface control developed by Swaroop et al. [10] is a control technique by introducing a filter at each recursive step of the backstepping design procedure, so the differentiation items on the virtual function can be avoided.DSC has been pioneered by the work of Swaroop et al. about 10 years ago, but it does not consider nonlinear plant with uncertain time delays and disturbances.Also, nonlinear items are assumed to be completely known and the control gain is equal to one.In 2005, by incorporating DSC into a neural network based adaptive control design framework, Wang and Huang proposed a backstepping based control design for a class of nonlinear systems in strictfeedback form with arbitrary uncertainty [11].However, uncertain time delays and disturbances are not involved in the model and the control gain is equal to one.[16] and solved the recursive filtering problem for a class of discrete-time nonlinear stochastic systems with random parameter matrices, multiple fading measurements, and correlated noises [17].He further investigated the recursive finite-horizon filtering problem for a class of nonlinear time-varying systems subject to multiplicative noises, missing measurements, and quantisation effects [18,19].The issues discussed in these literatures offer significant references to the research of chaos control in PMSM.
Inspired by the work above, a new approach to design the nonlinear dynamic surface controller based on the minimum weights of RBF neural network is proposed for permanent magnet synchronous motor with the unknown parameters, disturbances, and chaos.During the controller design process, RBF neural network is employed to approximate unknown nonlinear functions.The main difficulty encountered in the controller design process is how to deal with the unknown control gain in the system.To overcome this difficulty, the adaptive method was also introduced to handle the problem.The proposed controller guarantees a good tracking performance and the boundedness of all the signals in the closed-loop system.Furthermore, the suggested controller contains the minimum weights of RBF neural network.As a result, the computational burden of the scheme is greatly alleviated.This makes our design controller more suitable for practical applications.

Mathematic Model
It is well known that the PMSM is applied widely in the motor drives, servo systems, and household appliances owing to advantages, for instance, simple structure, high efficiency, high power density, and low manufacturing cost [20].However, the PMSM is experiencing chaotic behavior when the system parameters are falling into a special area, which can lead to the enormous destruction.The mathematic model of the PMSM based on the - axis is given as follows [21,22]: The denotations of the PMSM parameters are shown in Table 1.Suppose the direct and the quadrature-axis winding inductances are equal; that is,  =   =   .Meanwhile assume the time scale  to be such that  = / and define the normalized time  to be such that  = / and the scalar  to be such that  = /(    ).Finally, the scaled state variables ,   , and   are defined as follows: where ,   , and   are the normalized motor angular speed and the normalized quadrature-axis and direct-axis currents, respectively.
Then, the new normalized model for the PMSM is rewritten as where  = −  /(),  = /,   =  2   /,   =   /(),   =   /(),   and   denote the normalized quadrature-axis and direct-axis stator voltage, respectively,   presents the normalized load torque, and  and  are previously defined system parameter.It is obvious that the model of the PMSM has high nonlinearity because of the coupling between the speed and the currents.In addition, the indeterminate system parameters  and  are enormously impacted by realistic conditions.So as to control efficiently the PMSM,   and   are used as the manipulated variables.Then, a nonlinear dynamics surface control approach based on RBF neural network is proposed to restrain the chaos, parameters variation, and external disturbance in the PMSM.
For the sake of simplicity, the following symbols are introduced: Then, the mathematic model of the PMSM can be represented as follows: where Δ  ,  = 1-3, denote the external disturbance.

Controller Design
3.1.RBF Neural Network.The type of RBF neural network is considered as a two-layer network, which contains a hidden layer and an output layer.In this paper, the RBF neural network will be used to approximate the unknown continuous function () :   →  as follows: where  ∈ Ω ⊂   is the input vector with  being the neural network input dimension,  * = [ * 1 ,  * 2 , . . .,  *  ]  ∈   is the weight vector,  > 1 is the node number of neuron, and () = [ 1 (),  2 (), . . .,   ()]  ∈   is a basic function vector with   () chosen as the commonly used Gaussian function in the following form: where   = [ 1 ,  2 , . . .,   ]  is the center of the receptive field and   is the width of   ().
For given scalar  > 0, by choosing sufficiently large , the RBF neural network can approximate any continuous function over a compact set Ω ∈   to arbitrary accuracy as follows: where () is the approximation error and  is an unknown ideal constant weight vector, which is an artificial quantity required for analytical purpose.Typically,  is chosen as the value of  * that minimizes |()|, for all  ∈ Ω; that is,

The Controller of Dynamics Surface Control Approach
Based on RBF Neural Network.In this section, the controller of dynamics surface control approach will be developed based on the minimum weights of RBF neural network.The design procedure consists of three steps.Then, the detail process will be given.
Step 1.The first dynamic surface is defined as Then, the time derivative of  1 can be obtained as follows: where The operating parameter  is usually unknown due to the effect of the work environment.It is difficult for traditional methods to deal with the problem.In order to solve it, the adaptive technique is introduced to estimate the , and the adaptive RBF neural network is used to approximate the uncertain item  1 with little error.Therefore, for any given  1 , there exists a RBF neural network    1  1 such that where  1 is the approximation error and satisfies | 1 | ≤   .Substituting ( 11) into (10) yields the following: The virtual control and related adaptive laws can be designed as follows: where λ1 = ‖ θ1 ‖ 2 belongs to a minimum weights of RBF neural network which greatly speeds up the solution speed,  1 ,  1 ,  1 ,  1 ,  1 , and Γ 1 are the design constant, and  is a small positive constant.
Introduce variables λ1 and σ as follows: Let  2 be passed through a first-order filter with a time constant  2 as follows: Define the filter error as  2 =  2 −  2 .With (17) and  2 , it yields that Then, the time derivative of  2 can be obtained as follows: It is obtained that Using Young's inequality, one has where  2 ( 1 ,  2 ,  2 , λ1 , σ,   , ẏ  , ÿ  ) is the continuous function.
Substituting ( 13)-( 21) into ( 12), (12) becomes One has Consider the Lyapunov function candidate as follows: Then, the time derivative of  1 is obtained as follows: For the terms − 1 σσ and −(1/ 1 ) 1 λ1 λ1 , one has − (26) Step 2. The second dynamic surface is given as Then, differentiating  2 gives where To facilitate engineering application, a minimumweights-based RBF neural network will be employed to approximate the nonlinear function  2 again.Therefore, there exists a RBF neural network system such that where  2 is the approximation error and satisfies | 2 | ≤   .Substituting ( 29) into (28), one has Similarly, the relevant control law and adaptive law are provided in the following forms: where  2 ,  2 ,  2 , and  2 are the design constant and λ2 = ‖ θ2 ‖

2
. With (31) and ( 32), ( 30) is written as follows: One has Choose the Lyapunov function candidate as follows: Then, the time derivative of  2 is For the term Step 3. Choose the last dynamic surface as  3 =  3 .Then, the time derivative of  3 is calculated as follows: where In the same way, there is a minimum-weights-based RBF neural network such that where  3 is the approximation error and satisfies | 3 | ≤   .Substituting (38) into (37) gives  At the present stage, the control input is designed as follows: where  3 is the positive constant and λ3 = ‖ θ3 ‖ 2 .According to the mention above, the adaptive law is chosen as follows: where  3 ,  3 , and  3 are the design constant.Similarly, (37) is given as follows: There exists Choose the Lyapunov function candidate as follows: Using the equality in (44), it can be verified easily that For the term −(1/ 3 ) 3 λ3 λ3 , one has −(1/ 3 ) 3 λ3 λ3 ≤ −(1/2)( 3 / 3 ) λ2 3 + (1/2)( 3 / 3 ) 2 3 .Up to now, the design procedure of proposed controller of the PMSM is completed.The proposed controller significantly reduces the computation complexity compared with traditional backstepping control and dynamics surface control.Based on previous procedure, the configuration of the proposed control system is depicted in Figure 3.The overall configuration consists of the PMSM with load, the space vector pulse width modulation (SVPWM), the voltagesource inverter, the power source rectifier, the automatic current regulator of the motor, the encoder used to detect speed and position, -axis and -axis controllers.

Stability Analysis
For any given  > 0, the closed sets can be defined as follows:   uniformly ultimately bounded, and the output tracking error converges to a neighborhood of zero.
Proof.Define the Lyapunov function candidate as follows: Consequently, one can obtain where
To take into account the disturbance, the corresponding expressions are given as follows: (50) The simulation results are shown in Figures 4(a)-4(f).Figures 4(a) and 4(b) explicitly illustrate that the state error of angular speed of the PMSM is gradually converged to zero within a short time.Meanwhile, these pictures show that the system successfully escapes from the chaotic behavior within 0.1 s and tracks the given trajectory with a great performance in spite of uncertainty, nonlinearity, and external disturbance.In Figures 4(c)-4(f), three kinds of curves basically overlap without disturbance on the whole time.Obviously, these pictures show the performance when the system parameters  and  have a perturbation.Furthermore, the indicators of the PMSM can still converge quickly when the model suffers from the disturbance.From the simulation results, it can clearly be seen that the proposed controller guarantees the boundedness of all the signals in the closed-loop system and also achieves the good tracking performance.

Conclusion
In this paper, an adaptive dynamic surface control method of chaos is applied to the permanent magnet synchronous motor based on the minimum weights of RBF neural network.The proposed controller guarantees the boundedness of all the signals in the closed-loop system, while the tracking error eventually converges to a small neighborhood of the origin.Moreover, the suggested controller contains minimum weights of RBF neural network.This makes the design scheme easier to be implemented in practical applications.Simulation results are given to show the effectiveness and robustness of the proposed controller.Finally, some potential future research works are pointed out, such as the recursive filtering for time-varying nonlinear systems and sliding mode design for time-invariant nonlinear systems.
current (A) t (s) (c) The normalized -axis current

Figure 2 :
Figure 2: The chaotic time series of the PMSM.

Table 1 :
The denotation of the PMSM parameters.
[15]urbed uncertainties using neural networks[12], Na et al. in 2011[13]presented adaptive neural dynamic surface control for servo systems with unknown dead zone, Li et al. in 2013 presented an adaptive fuzzy DSC output feedback approach for a single-link robotic manipulator coupled to a brushed direct current motor with a nonrigid joint[14], and Tong et al. in 2013 presented an adaptive fuzzy decentralized backstepping output feedback control approach for a class of uncertain large-scale stochastic nonlinear systems without the measurements of the states[15].In their works, it is assumed that there exists positive constant which satisfies specified constraint condition.But it is very difficult to define the boundedness of unknown control gain in real practice.Incidentally, the PMSM is known to exhibit chaotic behavior under certain conditions.Whether the latter methods can suppress the chaos oscillation in PMSM needs further research since DSC with RBF has been pioneered by the work of Wang et al.In addition, Hu et al. overcame the gain-constrained recursive filtering challenge for a class of time-varying nonlinear stochastic systems with probabilistic sensor delays and correlated noises Assumption 1.The unknown disturbance terms Δ  satisfy |Δ  (, )| <   ,  = 1-3, and the parameter  is unknown and bounded.Moreover, it is assumed that || ≤ .