Adaptive Neural Network Control of an Uncertain Nuclear Refueling Machine With Input Backlash and Asymmetric Output Constraint

This paper studies the control problem of an uncertain nuclear refueling machine (RM) system in the presence of uncertainty, external disturbance, input backlash, and asymmetric output constraint. Taking into account the effects of external disturbance and input backlash, the RM system with fuel rod is modeled as a coupling partial differential equation-ordinary differential equation (PDE-ODE). Using the backstepping method, an adaptive neural network control scheme is designed to drive the RM and bridge to the desired positions and simultaneously reduce the vibration of the fuel rod. A novel asymmetric tangent-type barrier Lyapunov function (Tan-BLF) is constructed to restrict the position tracking error into the given range. The formulation of backlash nonlinearity is transformed into the expected input and nonlinear error. Then the combination of input backlash error and boundary disturbance is defined as a disturbance-like item. Applying the robust control strategy and adaptive technique, two auxiliary input signals are proposed to offset the impact of the disturbance-like item. The adaptive neural network (NN) is employed to compensate for the system uncertainty. The practical stability of the RM system with the proposed control is demonstrated via the theoretical Lyapunov analysis. Simulation verifies the performance of the designed control scheme.


I. INTRODUCTION
Nuclear energy is widely applied in power generation to satisfy the growing energy requirement. The fuel rod is installed in the reactor core of the nuclear plant to produce the energy. RM transports the fuel rod to the given position along the bridge to complete the maintenance and installation tasks. However, both external disturbance and movement of RM would induce the vibration of the fuel rod. These undesired vibrations could result in fatigue damage to fuel rods and even lead to serious nuclear safety accidents. Therefore, the vibration and position controls of the RM with the fuel rod are fundamental to ensure the safe operation of the fuel assembly.
Many researchers have developed different techniques to achieve vibration reduction and position tracking for the RM The associate editor coordinating the review of this manuscript and approving it for publication was Min Wang . with fuel rob. In [1], a theoretical method is proposed to analyze the vibration characteristics of the fuel rod subjected to axial flow. In [2], the authors provide a solution to compute the flow-induced vibration of the fuel rod by analyzing the operational modal. In [3], the simulation experiments are conducted to show the nonlinear vibration responses of the nuclear fuel assembly, where the system boundaries are clamped. Nevertheless, for convenience, the fuel rod is simplified as the finite-dimensional ODE in previous works, which only describes the finite system modes. These results would bring inaccurate system response, and the un-modeled system modes could destabilize the system. Mathematically, the fuel rod is a typical distributed parameter system governing PDE. Control design of the PDE system is more difficult than the classical ODE system [4], [5], [6], [7], [8]. In this paper, considering the coupling between the motion of RM and the vibration of the fuel rod, the nuclear RM system is modeled as a coupling PDE-ODE. VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Numerous control methods have been employed to stabilize distributed parameter systems [9], [10], [11], [12], [13], [14], [15]. In [16], the authors propose a distributed control to track the given signal for a constrained flexible structural system. The mixed control scheme consisting of boundary control and distributed control is employed to stabilize the spacecraft in the presence of external disturbance and actuator fault in [17]. For a flexible gantry crane system with nonuniform tension, the authors design two boundary control laws to regulate the deformation of cable and simultaneously move the tip payload to the desired position in [18]. In [19], by installed boundary control, the elastic deformation of the rigid-flexible wing is suppressed. Among these control methods, distributed control and boundary control are the two chief control methods. From the view of practical engineering, compared with distributed control, boundary control possesses better economy and realizability since its implementation only installs sensors and actuators at the system boundary [20], [21], [22], [23]. Therefore, we design two boundary control laws to offset the transverse and lateral vibration displacements of the fuel rod and simultaneously drive RM and bridge to the desired positions.
In [24], the authors propose a PD control scheme to control the RM system without any constraints. To guarantee the efficient and safe operation of the nuclear plant, the position tracking errors of RM and bridge are generally restricted in the desired ranges. Hence, the proposed control scheme in [24] is not suitable for the RM system with asymmetric output constraint. In this paper, by embedding a Tan-BLF into the process of control design, the proposed control scheme can ensure that the asymmetric constraint of position tracking error is never violated. Nonlinear input backlash generally occurs in actuators. The unknown backlash nonlinearity would produce the input error and even affect the stability of the control system [25], [26], [27]. For designing control, the expression of backlash nonlinearity is decomposed into the desired input and the bounded input error. Additionally, the disturbance rejection should be considered in control design. The sum of input backlash error and boundary disturbance is defined as a disturbance-like item. Using the robust control strategy and adaptive technique, two auxiliary input signals are developed to estimate the disturbancelike item. Then the disturbance-like item will be eliminated in the negative feedback loop. The precise values of the physical structural parameters are greatly difficult to measure in practical applications. The radial basis function neural network (RBFNN) is used to approximate the system uncertainty, where adaptive weight law is proposed to estimate the weight of NN. In this paper, an adaptive neural network control scheme is developed to stabilize the RM system with uncertainty, external disturbance, input backlash, and asymmetric output constraint.
Different from the previous results, this paper gives a coupling PDE-ODE model to describe the dynamical feature of the RM system with a fuel rod. Applying the backstepping method, two adaptive neural network boundary control laws are developed to reduce the lateral and transverse vibrations of the fuel rod and simultaneously drive the bridge and RM to the desired positions. The asymmetric Tan-BLF is constructed to prevent asymmetric constraint violation. The expression of backlash nonlinearity is divided into the expected input and the nonlinear error. The combination of boundary disturbance and input backlash error is defined as a disturbance-like item. Two auxiliary input signals are designed to handle the impacts of input backlash and external disturbance. RBFNN is employed to attenuate the system uncertainty. The designed control scheme can guarantee practical stability.

II. PROBLEM FORMULATION
In this part, we propose the system dynamical equations. Under Hamilton's principle, the system governing equation is expressed as PDE and the boundary condition is expressed as ODE.
In this paper, we consider a moving nuclear RM that transports the fuel rod along the bridge in a nuclear reactor. The simplified system configuration is displayed in Fig.1. w(x, t) and s(x, t) represent the transverse and lateral vibration displacements of the fuel rod respectively. F y (x, t), F z (x, t), d y (t) and d z (t) define the distributed and boundary disturbances respectively. y(t) and z(t) represent the positions of bridge and RM along the j and k axes respectively. u y (t) and u z (t) define the control inputs. The system structural parameters are listed as follows: EI , l, d and ρ are the bending stiffness, length, diameter, and mass per unit length of fuel rod respectively; m b and m r are the masses of bridge and RM respectively.
For convenience, note that( ·) = ∂(·) ∂t ,( ·) = ∂ 2 (·) ∂t 2 , (·) = ∂(·) ∂x , (·) = ∂ 2 (·) ∂x 2 , (·) = ∂ 3 (·) ∂x 3 and (·) = ∂ 4 (·) ∂x 4 . In order to obtain the dynamical equations of the nuclear RM system, the fuel rod with a large aspect ratio can be regarded as an Euler-Bernoulli beam. The system kinetic energy, potential energy, and virtual work are given by Applying the Hamilton's principle: and the corresponding boundary conditions as In this paper, the following nonlinear backlash models are investigated where v y (t) and v z (t) are the expected control commands, η y , η z > 0 are the slope, B ry , B rz > 0 and B ly , B lz < 0 are the constant parameters, u y (t ) and u z (t ) imply that u y (t) and u z (t) have not changed.
To facilitate control design, (10) and (11) are transformed as where d(v y ) and d(v z ) denote the nonlinear backlash errors and can be expressed as To proceed, we employ the following lemmas and assumption.
Lemma 1 [28]: Let y(x, t) : [0, l] × R + → R be a continuously differentiable function, we have [28]: For any |ζ | < 1, the following inequality holds Assumption 1: In this paper, we suppose that all external Remark 1: To handle the effects of unknown backlash and external disturbance, we define two disturbance-like terms as According to (14) and (15), it is obviously observed that From (12), (13) and (16), the boundary conditions (6) and (7) can be rewritten as

III. CONTROL DESIGN
In this part, two adaptive neural network boundary control laws are designed to achieve the following targets: (1) suppress the transverse and lateral vibrations of the fuel rod and simultaneously drive RM and bridge to the desired positions; (2) regulate the position tracking errors into the given ranges; (3) offset the impacts of uncertainty, external disturbance, and input backlash.
For the desired positions y d and z d , we define the tracking errors as e y (t) = y(t) − y d and e z (t) = z(t) − z d . In order to use the backstepping method to design control, the following transformation is developed where τ y and τ z are virtual controls.
Step 1: To avoid the violation of asymmetric constraint, construct an asymmetric Tan-BLF as where α > 0, and S 1 and S 2 are defined as ))(y 2 + τ y ) From the backstepping method, we chose the virtual control as where k 1 and k 2 are positive control gains.
Substituting (23) into (22) yieldṡ The first four terms of (24) are negative definite, the other terms will be handled in the next step.
Step 2: To stabilize the unmanned nuclear RM system, the lyapunov function is adjusted as The derivative of V 2 (t) is computed aṡ Substituting (17) and (18) into the above equation leads tȯ From the backstepping method, we propose the expected boundary control laws as Substituting the above control laws,V 2 (t) becomeṡ Nevertheless, in practice, both of the system parameters EI , m b , m r and the disturbance-like terms d 1 (t), d 2 (t) are unknown. The unknown terms of the proposed control laws will be handled in the next step.
Step 3: To deal with the impact of the disturbance-like term, the following auxiliary input signals are developed by robust control strategy where ι 1 , ι 2 > 0, andd 1 (t) andd 2 (t) are the estimates of d 1 and d 2 respectively. The adaptive disturbance laws are defined aṡ where θ y , θ z > 0 are the accommodation coefficients. In this step, we employ the radial basis function neural network (RBFNN) to approximate the system uncertainty. According to the property of RBFNN, the following equations hold where * y and * z are the ideal weight vectors, Y = [s (0, t),τ y ] T and Z = [w (0, t),τ z ] T are the input vectors, the approximation errors y and z satisfy that | y | < 1 and z < 2 , and the Gaussian functions h y ℵ yi and ℵ zj are the center vectors, ω yi and ω zj are the widths of Gaussian function. Now, the actual adaptive neural network boundary control laws are proposed as whereˆ y andˆ z are the estimates of * y and * z respectively. The adaptive weight laws of NNs are defined aṡ where γ 1 , γ 2 > 0 are the modification coefficients which can improve the system robustness, 1 and 2 are the positive diagonal matrixes. The augmented Lyapunov function is further modified as where λ max ( 1 ) and λ max ( 2 ) are the maximum eigenvalues of 1 and 2 respectively.
Theorem 1: For a nuclear RM system subject to external disturbance, input backlash, and asymmetric output constraint, under Assumption 1, the proposed control laws (39) and (40) Form (60), it can get that V (t) is bounded. Applying Lemma 1,(20) and (46) yields Furthermore, taking the limit of (62) leads to According to (45) and (60), we can obtain that V 1 (t) is bounded. From the expression of V 1 (t), it can conclude that V 1 (t) → ∞ as y 1 → a 1 or −a 2 , z 1 → a 3 or −a 4 . Using Lemma 1 of [29] implies that the tracking errors y 1 and z 1 satisfy that −a 2 < y 1 < a 1 and −a 4 < z 1 < a 3 .

V. SIMULATIONS
From the above analysis, the practical stability of the studied refueling machine system is demonstrated theoretically. To verify the performance of the proposed control scheme further, simulations are carried out by using the finite difference method. The physical parameters of the studied refueling machine system are given as EI = 10 Nm 2 , ρ = 0.038 kg/m, m r = 3.     of tracking errors are a 1 = 0.05, a 2 = 0.65, a 3 = 0.1 and a 4 = 0.25. The external disturbances are given by d y (t) = 0.3sin(t), d z (t) = 0.1sin(t), F y (x, t) = 0.5(1 + sin(xt) + cos(2xt)) and F z (x, t) = 0.5(2 + sin(0.5xt) + cos(xt)).     cooperative control scheme can move RM and bridge to the desired positions. According to Fig. 8, it is clear that the tacking errors satisfy the given constrained conditions, i.e., −a 2 < y 1 < a 1 and −a 4 < z 1 < a 3 . By analyzing these results, the designed adaptive control scheme is efficient.

VI. CONCLUSION
This paper has designed two adaptive neural network boundary control laws to stabilize a nuclear RM system with uncertainty, external disturbance, input backlash, and asymmetric output constraint. The system dynamic equation has been given by a coupling PDE-ODE. Under the proposed control scheme, the transverse and lateral vibrations of the fuel rod have been completely reduced, whereas the RM and bridge positions have been moved to the expected positions. The position tracking errors have been restricted in the desired ranges by using an asymmetric Tan-BLF. RBFNN has been employed to compensate for the uncertainty of control system. Both the effects of input backlash and boundary disturbance have been handled by constructing two auxiliary input signals. The system practical stability has been proven. Simulation has verified that the developed control scheme can realize all control targets. He is currently a Lecturer with the School of Information Engineering, Nanchang University. His current interests include vibration control, intelligent control, and distributed parameter systems.
ZUOXUAN GAN is currently pursuing the B.S. degree with Nanchang University. His current research interests include adaptive control and flexible systems.
BIN CHEN received the bachelor's degree in automation from Nanchang University, Nanchang, China, in 2020, where he is currently pursuing the M.S. degree. His current research interests include distributed parameter systems and boundary control.
YUSHUI HUANG received the Ph.D. degree from Zhejiang University, Zhejiang, China, in 2004. He is currently a Professor with Nanchang University. His research interests include nonlinear control and fault diagnosis. VOLUME 10, 2022