Abstract
This paper focuses on an adaptive dynamic surface control based on the Radial Basis Function Neural Network for a fourth-order permanent magnet synchronous motor system wherein the unknown parameters, disturbances, chaos, and uncertain time delays are presented. Neural Network systems are used to approximate the nonlinearities and an adaptive law is employed to estimate accurate parameters. Then, a simple and effective controller has been obtained by introducing dynamic surface control technique on the basis of first-order filters. Asymptotically tracking stability in the sense of uniformly ultimate boundedness is achieved in a short time. Finally, the performance of the proposed control has been illustrated through simulation results.
1. Introduction
Recently, the permanent magnet synchronous motor (PMSM) is the most widely used driven mechanism because of the advantageous merits of cost, reliability, and performances. The PMSM is characterized by complexity, high nonlinearity, time-varying dynamics, inaccessibility of some states, and output for measurements; hence, it can be considered as a challenging engineering problem [1, 2]. It is found that the PMSM is experiencing chaotic behavior at specific parameters and working conditions [3, 4]. Then, the intermittent oscillation of torque and rotational speed, irregular current noise of the system, and unstable control performance appear in the PMSM, which seriously affect the stability and safety. Thus, it is difficult to accomplish the high-performance control of PMSM by using classic PID-type control methods.
A neuron-fuzzy controller (NFC) [5] is suitable for control of systems with uncertainties and nonlinearities. The NFC approach can also achieve self-learning; however, it is unsuitable for online learning real-time control due to the drawback of time consuming [6, 7]. The sliding mode control (SMC) [8] can guarantee the robustness only under the bounds of the uncertainties and it has a shortage named chattering. The terminal sliding mode control (TSMC) method can assure convergence to the origin in finite time. Hence, the TSMC is successfully applied to PMSM driver system to improve control performance [9]. A position tracking control method via adaptive fuzzy backstepping is presented for the induction motors with unknown parameters [10]. Unfortunately, the traditional backstepping suffers from the “explosion of complexity” caused by the repeated differentiation of virtual control functions [11]. In order to overcome the above shortcomings, a backstepping approach combined with SMC technique is presented to realize synchronization of uncertain fractional-order strict-feedback chaotic system [12].
The dynamic surface control (DSC) developed by Swaroop et al. [13] is a control technique by introducing a first-order filter at each recursive step of the backstepping design procedure, so the differentiation items on the virtual function can be avoided. By incorporating DSC into a neural network-based adaptive control design framework, Wang and Huang [14] proposed a backstepping-based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. Zhang and Ge [15] further studied the control design for some special nonlinear systems with unknown dead-zone using the DSC technique. To control the undesirable chaos in PMSM, an adaptive DSC controller was designed by introducing first-order low-pass filters [16], whereas in [17] DSC method was extended to time-delay uncertain nonlinear systems in parametric strict-feedback form. The new developments referring to DSC for different nonlinear systems and the applications to various engineering fields can be found in [18–20].
In this paper, motivated by the previous studies reported in the literature, the following problems will be addressed. An adaptive DSC based on the Radial Basis Function Neural Network (RBFNN) is discussed for the chaotic PMSM system. The whole process of the system design is performed in a step-by-step manner. For each step, a first-order filter is used to gain the information of derivative of the virtual control function and RBFNN is adopted to approximate the nonlinearity. The uncertain parameters in the system are updated by adaptive laws, which realize accurate parameters estimate. Then, an adaptive virtual control law is designed to stabilize each subsystem in the former steps. Finally, an adaptive DSC is employed via RBFNN in the last step to stabilize the whole system. Further contributions include the design of controller to handle uncertain time delays and disturbances.
2. Dynamics of the PMSM System
The model of a PMSM can be described in the well-known - frame through the Park transformation as follows. The stator , equations in the rotor frame are expressed as follows [21, 22]: where and are the - axis voltages, which denote the system control input. , , , and are state variables, which stand for the - axis currents, the rotor position, and rotor angular velocity, respectively. , , and mean the pole pair, moment of inertia, and viscous friction coefficient. and denote the inductance of the stator in the - frame. , , and stand for the load torque, stator resistance, and magnet flux linkage of inertia.
The general schematic of the PMSM is depicted in Figure 1. The overall system consists of a PMSM with load, space vector pulse width modulation (SVPWM), field-orientation mechanism, voltage-source inverter (VSI), and three controllers. The controllers employ a structure of cascade control loop including a speed loop and two current loops. PI controller, which is used to stabilize the -axis current errors of the vector controlled drive, is adopted in the -axis current loop. From Figure 1, the rotor angular velocity can be obtained from the position and speed sensor. The currents and can be calculated from and by Clarke and Park transformations. However, in real applications, this method suffers from serious problems, including internal disturbance such as torque ripples, parameters variation, friction forces, and unmodeled dynamics and external disturbance such as load disturbance. Thus, an adaptive dynamic surface controller via RBFNN is developed to solve these problems which can degrade the performance of closed-loop system if the controller does not have enough ability to reject them.
Equation (1) are rewritten as follows: where , , , , , , , , , , , , .
The chaotic behavior is demonstrated in PMSM system when its parameters fall into a certain area. Figures 2 and 3 show the strange attractor and the chaotic time series of the PMSM under the condition of , , , , , , , and , in which the PMSM appears an aperiodic, random, sudden, or intermittent morbid oscillation.
(a)
(b)
(c)
(d)
For simplicity, the following notations are introduced:
By using these notations, the dynamic model of PMSM with uncertain time delays and disturbance can be described by the following differential equations: where , , , , denote uncertain nonlinear function owing to the effect of time delay, , , represent time-delay constants, and , , mean the disturbances.
Define the tracking errors where , will be given later by the first-order filter and is given continuous tracking signal.
Then, for the uncertain nonlinear time-delay functions, the following inequalities can be satisfied: where and are known continuous functions, .
In order to facilitate the design, the following assumption is given.
Assumption 1. , , the desired trajectory is continuous, and its first-order derivative and second-order derivative are bounded and available.
In this paper, the control objective is to force the tracking error asymptotical stable in the sense of uniformly ultimate boundedness, which also ensures that the system state can track a reference trajectory .
3. Adaptive Dynamic Surface Controller Based on RBFNN
3.1. RBFNN System
The type of RBFNN shown in Figure 4 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 where is the input vector with being the neural network input dimension, is the weight vector, is the node number of neuron, and is a basic function vector with chosen as the commonly used Gaussian function in the following form: where is the center of the receptive field and is the width of .
For given scalar , by choosing sufficiently large , the RBFNN can approximate any continuous function over a compact set to arbitrary accuracy as where is the approximation error, satisfying , 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 ; that is,
3.2. Controller Design
In this section, an adaptive dynamic surface control approach of the permanent magnet synchronous motor based on RBF neural network will be developed. The DSC technique based on first-order filters is used to design the controller instead of the traditional backstepping method, and in sequence the explosion of terms problem is avoided.
In order to design dynamic surface controller, the boundary layer errors are defined as follows: where virtual control input , will be designed in the subsequent content.
The designing process is as follows.
Step 1. Consider the tracking error ; the time derivative of is given by
Choose Lyapunov-Krasovskii functional as
The derivative of is
Using Young’s inequality and inequality (6), the following inequalities can be obtained:
Thus, it follows immediately from substituting (15) into (14) that
where .
The virtual control function is constructed as
where design parameter .
It is obtained that
Step 2. Let be passed through a first-order filter as follows:
where is a time constant.
With (5) and (19), one has
The derivative of is
It is obtained that
where , , is the continuous function.
Using Young’s inequality, one has
Differentiating gives
where , .
In the realistic model of PMSM, limited to the working conditions, the parameter is unknown. So it cannot be used to construct the control signal. Thus, let be the estimation of . Meanwhile, notice that the nonlinear function contains uncertain parameters, such as , , , and . This will make the traditional adaptive DSC design very troublesome. To avoid this trouble and simplify the control structure, the RBFNN will be employed to approximate .
According to the description above, for any given , there exists a RBFNN system such that
where is the approximation error and satisfies .
With (24) and (25), one has
Consider the Lyapunov-Krasovskii functional candidate as
Then, the derivative of is given by
Using Young’s inequality and inequality Assumption 1, the following inequalities can be obtained:
Substituting (29) into (28) gives
where .
Choose the virtual control function and the corresponding parameters adaptive laws as
where , , , , , and are design parameters.
Substituting (31)–(33) into (30), (30) becomes
where is a continuous function.
Step 3. Filter through the following first-order filter
where is a time constant.
With (5) and (35), one has
The derivative of is
It is obtained that
where , is the continuous function.
Using Young’s inequality, one has
Differentiating results in the following equation:
where .
Notice that the unknown parameters, such as , , and , appear in the expression of . To facilitate the design of controller, the RBFNN will be employed to approximate the nonlinear function .
According to (9), for any given , there exists a neural network such that
where is the approximation error and satisfies .
Thus, (40) is rewritten as follows:
Define Lyapunov-Krasovskii functional
Then, the derivative of is computed by
where .
Using Young’s inequality and inequality (6), one has
Thus, it follows immediately from substituting (45) into (44) that
where .
At the present stage, the control law and the related parameters adaptive laws are designed as
where , , , , , and are design parameters.
By similar manipulations previously done in Step 2, one has
where ≥ is a continuous function.
Step 4. At this step, the control law will be constructed. In this end, choose Lyapunov-Krasovskii functional as = + + + + . Then, the derivative of is given by
where , .
Similarly, the RBFNN system is utilized to approximate the nonlinear function with unknown parameters and . For any given , one has
where is the approximation error and satisfies .
Note the fact that
Substituting (52)-(53) into (51) gives
where .
Now design , , and as
where , , , , , and are design parameters.
By similar manipulations previously done in Step 2, one has
where is a continuous function.
Up to now, the whole design process of the controller of PMSM is completed, and the configuration of the proposed control system is depicted in Figure 5.
4. Stability Analysis
For any given , the closed sets can be defined as follows:
Theorem 2. Suppose that the control law in (47) and (55) with adaptive law in (32), (33), (48), (49), (56), and (57) is applied to the PMSM (4), Assumption 1 stands, and if there exists a positive constant that the initial condition satisfies and the design constants , and satisfy the following conditions: then the closed-loop control system is semiglobal uniformly ultimately bounded, and the output tracking error converges to a neighborhood of zero.
Proof. Define the following Lyapunov function candidate for the fourth step:
For the term , one has . Similarly, holds. Consequently, the derivative of is
where .
Furthermore, (62) implies that
By Lyapunov function, it is proved that all the signals , , , and in the closed-loop system are uniformly ultimately bounded. Consequently, , , and are also uniformly ultimately bounded.
5. Simulation
To illustrate the effectiveness of the proposed controller, the simulation will be done for the PMSM with the parameters: = , = , = , = , = , = , = 3, = 1.5, , = , = , = , = , = 1, = 1, = = = = , = , = , = , = , = = = = .
The simulation is running under the assumption that the system parameters and nonlinear functions are unknown. The initial conditions of system . Then, the reference signal is taken as , the control parameters are chosen as follows: , , , , , , , , , and RBFNN is chosen in the following way. Neural network ,, contains nine nodes with centers spaced evenly in the interval and width being equal to two.
With the proposed RBFNN-based adaptive dynamic surface control, from Figures 6 and 7, it is observed that the tracking performance is satisfactory and the tracking error quickly converges to zero in a short time for PMSM system in the presence of unknown parameters, disturbances, and uncertain time delays. The final controllers and are shown in Figures 8 and 9. Moreover, the robustness of the proposed controllers against uncertainty in system parameters and disturbance is analyzed by simulation. According to Figure 10, it is seen clearly that all state variables are bounded and convergent. Meanwhile, by contrast with Figure 3, the proposed controller can suppress the chaos of PMSM. Therefore, these simulation results demonstrate the effectiveness of the proposed control in achieving its objective.
6. Conclusion
A novel adaptive DSC based on the RBFNN is presented for the fourth-order chaotic PMSM system with uncertain time delays so that the first state can track a given trajectory with little time. The unknown PMSM parameters, disturbances, and uncertain time delays are all considered. Compared with the traditional backstepping approach, the main advantage of the proposed method is its simplicity without repetitive differentiation of the nonlinear functions under the condition. It is noted that the system has excellent tracking performance and antidisturbing by choosing appropriate design parameters. Finally, the simulation results are provided to demonstrate the effectiveness and robustness against the unknown parameters, disturbances, chaos, and uncertain time delays.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the Natural Science Foundation of China (Grants nos. 51375506, 50905189), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20100191110008), China Postdoctoral Science Foundation funded project (Grant no. 2013M542258), and the Par-Eu Scholars Program Special Foundation of Chongqing. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.