Adaptive Chebyshev Neural Network Control for Ventilator Model under the Complex Mine Environment

Ventilator is important equipment for mines as it safeguards the lives under the shaft and ensures other equipment’s proper functioning by providing fresh air. %erefore, how to effectively control the ventilator system becomes more significant. In order to acquire the commonly used model and control strategy for ventilator systems, a new universal ventilator model is established based on the blast capacity differential pressure in the ventilating duct and the ventilator motor model. %en, an adaptive Chebyshev neural network (ACNN) controller is proposed to effectively control the ventilator system where the unknown load torque and the unknown disturbance caused by the complex environment under the shaft are approximated by the Chebyshev neural network (CNN). Afterwards, an appropriate Lyapunov function candidate is designed to guarantee the stability of the proposed controller and the closed-loop ventilator system. Finally, the ACNN controller has been demonstrated to be effective in terms of validity and precision for the new proposed ventilator model through the simulations.


Introduction
e ventilator motor is very important for the mine since it is the core equipment supporting the fresh air under the shaft [1][2][3][4][5][6][7]. Controlling the ventilator motor is always the most important issue in the process of mining. In order to precisely control the ventilator for saving the energy, the ventilator model must be investigated firstly. It has several kinds of models (ARMA model, computational fluid dynamics model, etc.) for ventilator systems so far, but all are suiting special conditions in special systems. en, investigating the universal ventilator system model for mines is of significance. Zhang et al. [2] investigated a train tunnel model based on big data. e ventilation system was adjusted and regulated by iterative learning control algorithm and the results demonstrated the proposed control method was superior to active control. Roubłk et al. [6] investigated a mechanical ventilation for lung injury aimed at preservation of spontaneous breathing when using high-frequency oscillatory (HFO) ventilators, who designed a linear quadratic Gaussian state feedback controller to compensate for the changes in mean airway pressure (MAP) in the ventilator circuit caused by spontaneous breathing. Different from Zhang et al. [2] and Roubłk et al. [6], Yan et al. [7] investigated the high-altitude ventilator motor model for stratospheric airship airbags. ey extended the traditional minimum stator current operation to minimize conduction below the rated speed concept, where an extended minimum stator current operation (EMSCO) was designed at any speed to formalize as constrained optimization problem. en, they proposed optimal duty cycle model predictive current control (ODC-MPCC) to reduce the current ripple and computation burden. From these references, one can find that most control strategies were based on optimal control. But the mine environment even under the shaft is so intricate that many elements can disturb the optimal control strategy. erefore, the optimal controller will not acquire the desired results under the complex disturbance in the mine. It needs a new control strategy for the ventilator motor suiting the complex environment of the mine, especially under the shaft.
ere are a lot of researchers applying the adaptive control to solve the nonlinear issues. Chen et al. [19] proposed a fixedtime adaptive control for buck DC/DC converters. An adaptive update law was adopted to estimate all the unknown parameters of the buck converters, and the fixedtime sliding mode surface (SMS) and corresponding control were developed for the buck converters with known parameters such that the output error would converge to the equilibrium point within a fixed time. Gao et al. [14] investigated the discrete Hammerstein system with Preisach hysteresis nonlinearity and unknown order linear dynamics. ey proposed a lower triangular matrix to identify the Preisach density function and adopted Hankel matrix estimating the linear dynamics order. en, a composite control consisting of discrete inverse model-based controller (DIMBC) and discrete adaptive sliding mode controller (DASMC) was proposed to deal with the Hammerstein system which can reduce the reaching time of DASMC and improve the robustness of DIMBC. Sun et al. [16] designed a feedback adaptive controller for controlling the backlash servo system. Firstly, an extended state observer (ESO) was adopted to estimate the system states and the vibration torque.
en, considering the feedback and feedforward signals, an adaptive robust compensation controller was designed for this backlash servo system and the experiment results have demonstrated the effectiveness for the systems. Different from former researchers, Wang et al. [17] applied the adaptive method to estimate the unknown parameters rather than using it as controller.
en, considering the problem for the sluggish convergence caused by the online learning, this study proposed a new adaptive law to achieve optimal parameter estimation. In order to achieve this purpose, an auxiliary filter was designed to compel the adaptive law with a time-varying gain under a cost function. Finally, an adaptive nonsingular terminal sliding mode control (ANTSMC) was proposed for a considered servo system to obtain tracking control and parameter estimation simultaneously. Na et al. [20] considered an energy maximization problem for wave energy converters (WECs) subject to nonlinearities and constraints. e adaptive approach was used to solve the Hamilton-Jacobi-Bellman equation and a critic neural network (NN) to approximate the time-dependent optimal cost value. ey combined the adaptive control and optimal control to deal with the wave energy converters (WECs). From these studies, it shows that the adaptive control can perfectly be adopted for the nonlinear systems with adaptive update law; even so, some unknown nonlinear sections or disturbances or models will also be estimated by other methods, such as fuzzy logic [21] and neural network [22][23][24][25].
e Chebyshev neural network (CNN) is a functionallink neural network that is a single-layer neural structure with less computation to estimate the nonlinear function [26][27][28][29][30]. It has been utilized to estimate most kinds of uncertain linear or nonlinear sections. Zou et al. [26] used the Chebyshev neural network for approximating the spacecraft attitude motion in uncertain spacecraft systems. ey proposed Chebyshev neural network controller-I and adaptive NN controller-II for global representation without singularities of the systems. Different from Zou et al. [26], Gao and Liu [27] investigated a backlash-like hysteresis nonlinear system using the multiscale Chebyshev neural network (MSCNN) method to identify the unknown hysteresis nonlinearity. Firstly, the unknown hysteresis backlash-like nonlinearity would be approximated by the new multiscale Chebyshev neural network. en, the tracking error was transformed into scalar error by Laplace transformation to simplify the computation. Finally, an adaptive control was designed for controlling the backlash-like hysteresis nonlinear system, and the simulations verified the effectiveness of the multiscale Chebyshev neural network identification and the adaptive controller. Sun et al. [28] focused on the multimotor servomechanism with unmodeled dynamics. An extended state observer based on highorder sliding mode (HOSM) differentiator was applied for unmeasured velocity, and the Chebyshev neural network was adopted to deal with the friction and disturbances. e experiments proved that the proposed approach was useful. To sum up, the Chebyshev neural network was utilized for different nonlinear systems and obtained good results.
erefore, we will utilize the CNN to deal with the unknown nonlinear complex uncertain sections in the ventilator system in this paper.
Firstly, a new ventilator model will be established based on the blast capacity differential pressure of the ventilating duct and the ventilator motor, that is, a nonlinear dynamic model where some load torques and some complex disturbances are unknown. en, a Chebyshev neural network will be designed to estimate the unknown nonlinear section and an adaptive CNN controller will be proposed for the closed-loop ventilator system. Finally, the appropriate Lyapunov function guarantees the convergence of the closed-loop ventilator system and the simulations also affirm that the proposed approach is precise and effective.
is paper is organized as follows. Section 2 gives the problem formulations, which introduces the proposed ventilator model based on the blast capacity differential pressure and the ventilator motor and also introduces the structure of the Chebyshev neural network. Section 3 presents the controller design, that is, the process to the design control. e stability of the ventilator system by Lyapunov function is also given in this section. en, simulations are given in Section 4, and Section 5 concludes the paper.

e Ventilator Model.
e most important index of the ventilator model is the flux of the ventilating duct. Considering the complicated and dusty environment of the ventilating duct in the mine, the approach of the blast capacity differential pressure is selected to solve the fresh air velocity measurement hole blocking in order to realize the real-time accuracy measurement and monitor the wind pressure. e schematic diagram structure of the blast capacity differential pressure method is illustrated in Figure 1. From Figure 1, the energy equations of sections I and II can be deduced as follows: 2 Complexity where p s1 , p s2 , ρ 1 , ρ 2 , Z 1 , Z 2 , v 1 , v 2 represent the static pressure, air density, elevation, and air speed of sections I and II, respectively, h r is the local resistance of section II, and ξ is the coefficient of local resistance of section II. Since sections I and II have close distance and same elevation, ρ � ρ 1 � ρ 2 � Z 1 � Z 2 holds. Considering the equation (1), we have Since sections I and II have close distance, the air volume of ventilating duct Q can be deduced as where Q 1 , Q 2 represent the air volume of sections I and II, respectively.
Substituting (2), we obtain where s 1 , s 2 represent the areas with the different definition in (4), respectively, and d 1 , d 2 mean outer diameter and insider diameter of the fairing, respectively. From equation (4), the ventilator air volume flow can be calculated where ρ, s 1 , s 2 , p s1 , p s2 is measured by the different sensors. Based on reference [7], it is obvious that the effective control for ventilator motor will save energy and control the air volume flow Q. erefore, an effective control strategy for ventilator motor will effectively control the ventilator air volume flow. e ventilator motor model in D-Q coordinates is represented as follows: where R, L represent the stator resistance and inductance, respectively. K � nψ, n is the number of the pole pairs, ψ is the rotor flux linkage, K t , J i , u q are the torque constant, inertia, and q-axis stator voltages, respectively. T n is the torque caused by some nonlinear portion such as friction, disturbance, and so on. T l � f(Q) is the load torque which is the unknown function about ventilator air volume flow in this paper. Define x � [x 1 , x 2 ] T � [q, _ q] T ; based on reference [31], ventilator motor model (5) can be rewritten as

Chebyshev Neural Network.
In this section, a CNN will be designed to estimate the unknown sections in equation (6). CNN is one of the functional-link neural networks whose structure is illustrated in Figure 2. From Figure 2, CNN is a single-layer neural network which relies on the Chebyshev polynomials. erefore, it has lower computation and is widely used for function estimation.
e CNN can be defined as follows: where W * is the weight matrix of the CNN and ζ is the bounded approximation error for the CNN.
where [x 1 , x 2 , . . . , x n ] T is the n-dimensional input vector, m represents the order of the Chebyshev polynomial, and the Chebyshev polynomial d k (x l ) is defined as the two-term recursive formula as follows: where the original value of d(x) is selected as e CNN update law is defined as follows: where ϱ 3 is the designed positive constant, e 2 is defined in (16), and W is the estimation of the weight W * .

Controller Design
Considering the structure of the ventilator models (1) and (6), we should design a recursive control strategy to solve the problem. e recursive controller method is designed as follows.
Step 1: in order to design the U-model CNN adaptive controller, the output error is defined as follows: Complexity where y d is a known signal which has (m + 1)th derivative such as a sinusoidal signal. Considering equation (6), the derivative of e can be deduced as Assume T n and f(Q) are unknown, where T n R and Rf(Q) are estimated by the CNN. en, the 2th derivative of e is obtained as Define the following Lyapunov function candidate as en, the derivate of V 1 can be deduced as In this step, x 2 is treated as a virtual input; therefore, we define an error as e 2 � x 2 − a 2 , where a 2 is a virtual control law. Substitute x 2 � e 2 − a 2 into (15), and we have Define the virtual control law a 2 as follows: where l 1 is a positive feedback control gain, and we can go to the next step.
Activation function Sign()

Update law Error
Output Desired output x 2 x n Chebyshev expansion

Complexity
Step 2: the actual controller u will be acquired in this step. Firstly, we define the derivative of e 2 as Based on (17), the derivation of a 2 is deduced as and then substituting (19) into (18), one can get Define the adaptive update errors as where a 2 , x 2 are the estimations of a 2 , x 2 . e adaptive update law is defined as follows: where ϱ 1 , ϱ 2 are designed positive constants. We select the true controller input as Select the Lyapunov function candidate as follows: e derivative of V is deduced as Substituting (20) into (25), the derivative of V can be deduced as Substituting the true controller (23) into (26), one can obtain Substituting (10) and (22) into (27), according to Lyapunov theory, one can obtain that the closed loop is stable with the designed CNN adaptive controller.

Simulation
e simulations is designed to verify the effectiveness of the ventilator motor in the mine. In the simulations, the ventilator model parameters are listed in Table 1. e parameters of the ventilator motor model in D-Q coordinates are listed in Table 2.
e reference input is selected as y d � sin(t).
e simulation results are shown in Figures 3-12. From Figures 3-7, it is clearly shown that the proposed control strategy is effective. Figure 3 shows the control results with the proposed neural network adaptive controller. e controlled output can quickly track the reference input with smaller and convergent error which is also shown in Figure 4. From Figure 4, it is very obviously illustrated that the error is converged and the mean absolute error (MAE) is less than 0.02. Figure 5 shows the controller input, and it shows the same conclusions compared with Figures 3 and 4. From Figure 5, the dynamic process can be clearly shown less than 1 second in these simulations, and then the neural network and the adaptive update law can estimate the unknown nonlinear section and parameters quickly. e tracking result is clearly shown in Figure 3. Figures 6 and 7 show the trajectories for the estimation x 1 , x 2 of x 1 and x 2 . From these two diagrams, it can be concluded that the proposed method can estimate the system state rapidly and the tracking controller is effective, which are in line with the results in  To magnify the amplitude of the input signal, the results are shown in Figures 8-12. From Figures 8 and 9, one can have the same conclusions compared with Figures 3 and 4. But comparing Figure 10 with Figure 5, it is clearly illustrated that Figure 10 has more drastic change.
en, comparing Figures 3-5 with Figures 8-10, it is illustrated that magnifying the signal amplitude will influence the controlled input, but the proposed controller also retains the control precision with strong control input. Figures 11 and  12 also show the same conclusion.

Complexity
Remark 1. In this paper, since the controller is designed based on the output tracking error, the tracking control results are excellent in Figures 3 and 8. But in Figures 7 and  12, the system states x 1 , x 2 have a chattering. e reason is that we did not consider the state control and we also did not utilize the state observer. en, the tracking error can affect the system states by the feedback structure that makes some chattering in the system states. We will further investigate this phenomenon in the future work.

Conclusion
A new adaptive Chebyshev neural network (ACNN) control was presented to precisely control the ventilator system in this paper. Considering that the optimal controller does not suit the complex environment under the shaft in the mine which is easily affected by the strong disturbance caused by the hostile environment, the ACNN replaced the optimal control compared with other traditional control strategies. Firstly, the new ventilator model was proposed which consisted of the blast capacity differential pressure equation in the ventilator duct and the ventilator motor equation. en, the CNN estimated the unknown nonlinear sections such as the unknown load torque and the unknown disturbance caused by the hostile environment under the shaft. Finally, the stability of the closed-loop ventilator system was guaranteed by a designed appropriate Lyapunov function candidate and the simulation results verified the effectiveness and the precision of the established new ventilator model and the presented ACNN control strategy.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.