Abstract
In this paper, the tracking control of the air-breathing hypersonic vehicle with model parameter uncertainties and actuator faults is studied. Firstly, a high-order linearization model is used to build an adaptive terminal sliding mode that eliminates chattering and provides increased robustness for unknown disturbances in the system. Second, a fault-tolerant control method mixing the radial basis function neural network with adaptive sliding mode control is suggested, with the addition of a hyperbolic tangent function to avoid controller input saturation. Finally, the stability of the controller is proved strictly by the Lyapunov theory, and the robustness and effectiveness of the controller are further verified by numerical simulations of the longitudinal model of the hypersonic vehicle.
1. Introduction
Air-breathing hypersonic vehicles (ABHV) are an appealing technology for military and low-cost space access needs. The aerodynamic properties of ABHV make flight control more susceptible to parameter errors and actuator faults than traditional flying aircraft at high attitude and Mach numbers. Strong fault-tolerant control systems should therefore be created for ABHVs to guarantee flight safety [1]. Various researchers have been researching hypersonic vehicle modelling since the 1990s, which is used to describe the dynamic characteristics of hypersonic vehicles. Marrison and Stengel suggested a high-fidelity longitudinal dynamics model based on the universal hypersonic vehicle developed at NASA Langley Research Center [2]. Strong nonlinear, time-varying, and nonminimum phase characteristics are the main features of a hypersonic vehicle. The longitudinal dynamics of ABHV are linearized to higher order [3]. Many sophisticated control techniques [4], including adaptive control [5], backstepping control [6], predictive control [7], fuzzy control [8], and feedback linearization control [9], can be employed to form fault-tolerant controllers for ABHV.
Sliding mode control is one of the most appealing nonlinear control tools among all types of control approaches. The feedback linearization model is used to construct an adaptive sliding mode controller with model uncertainty [10]. Because the linear sliding mode surface can only converge exponentially, the integral sliding mode controller is employed to achieve finite-time state convergence [11]. Based on this finite-time integral sliding mode surface, a simplified smooth second-order sliding mode approach law can realize synchronous convergence to zero [12]. In addition, finite-time convergence of states can be realized by terminal sliding mode, and unknown actuator faults can be estimated by adaptive law [13]. In the meantime, the sliding mode observer (SMO) may be used to assess actuator fault information and model uncertainty, and the estimation error converges in a finite amount of time [14] .
In actuality, system uncertainty and actuator faults are difficult to model. For dynamics that cannot be modeled, neural networks have been shown to be able to approximate any continuous function [15]. The typical way is always to build the feedback controller according to the model and then use the neural network to estimate the system disturbance to adjust the control input [16] . It is easy to convert model parameter uncertainties into system disturbances using neural network compensation to increase the robustness of the feedback controller [17]. In some cases [18], neural networks may approximate the model function and gain function for a nonlinear system, hence increasing robustness [19]. Further, the control input may be defined as a new object function [20], and a radial basis function neural network may also be employed to approximate it. As a result, the uncertainty induced by actuator failure may be predicted using a neural network [21, 22].
Despite substantial advances in fault-tolerant hypersonic vehicle management, the accompanying literature analysis demonstrates that there is still room for improvement [23, 24]. For example, just the actuator failure was analyzed, ignoring bias and outage failures. Although the failure observer’s assistance was used to estimate failure information, the overall system’s stability was not revealed, making it impossible to assess the process’s condition at any time. We investigate the failure tracking control of a hypersonic vehicle employing terminal sliding mode theory and a neural network approach in this research [25–29]. In this paper, a fault-tolerant control (FTC) method combining radial basis function neural network (RBFNN) and adaptive terminal sliding mode (ATSM) is proposed, which can track the ABHV trajectory of an ABHV in the presence of air density, mass, and moment of inertia uncertainties as well as actuator faults. The main contributions of this paper are as follows: (1)This scheme resolves the coupling problem in MIMO systems like ABHV, which has theoretical implications(2)The RBFNN may approximate the actuator’s fault characteristics and provide the hyperbolic tangent function to prevent saturation of the control input, thus assuring the control system’s practical application value(3)This fault-tolerant control scheme ensures the ABHV closed-loop control system’s stability and allows for a fast and effective reaction to the reference trajectory
The arrangement of this paper is as follows:
In Section 2, the longitudinal dynamics model of ABHV is input and output linearized, as well as actuator fault modelling and control input antisaturation processing. Then, Section 3 introduces the prior knowledge of RBFNN and the relevant lemma needed for controller design. The FTC scheme based on RBFNN is designed for the control model in Section 2. In Section 4, the ABHV model with and without faults is numerically simulated. Finally, this paper is summarised in Section 5.
2. Problem Statements
The longitudinal model of the ABHV is described as follows [8]: where is the flight velocity, is the angle of attack, is the flight altitude, is the flight path angle (FPA), and is the pitch rate. is the moment of inertia, is the ABHV’s mass, is the gravitational acceleration, and is the earth’s radius. is the lift force, is the thrust of engine, is the resistance, and is the pitching moment. These forces, torques, and coefficients of ABHV are as follows: where is the elevator angular deflection and is fuel-to-air ratio. In fact, the actuator output is bounded and can be expressed as and , where , , , and are known constants. A second-order system describes the dynamics of the engine [11]: where signifies the control input demand, means the engine dynamics’ natural frequency undamped, and is the damping ratio. Because the control input is bound by Equation (4), the following hyperbolic tangent function is used to avoid control input saturation:
Actuator faults can generally be described as a linear model: where is the control signals under faults, is the saturation input, is the fault gain matrix, and is the vector of fault deviation.
3. Controller Design
Lemma 1 (approximation based on RBFNN). The following RBFNNs can estimate an unknown smooth nonlinear function on a compact set: where is the input of network, is the ideal weight matrix, is the convolution vector, and is the approximation error. is chosen as the convolution formula: where and are the center and width of the -th neuron, respectively, and is the node number of the neural networks.
In Figure 1, the FTC scheme framework combining RBFNN and ATSM proposed in this paper is shown.
In order to track ABHV’s reference velocity and reference altitude , the state variable is set to , the input variable to , and the output variable to . The tracking errors can be defined as and .
The high-order linearization model of ABHV is shown as follows: where , , , , , and are defined in [11]. The auxiliary variables [13] are introduced as
Considering the influence of parameter uncertainty and external disturbance, the system can be defined as follows: where , , and are known in [13] as follows:
Assumption 2. There exists a positive value with an unknown upper bound, such that the system disturbance satisfies:
Because the state variables and aerodynamic parameters are practically bounded, the disturbance is a function of them and has a certain range of values.
In order to satisfy Lemma 1, a terminal sliding manifold is chosen as follows: where , , , , and are positive constants. When and are positive odd numbers, and , the sliding mode is Hurwitz.
The law of the sliding mode approach is as follows: where and are positive constants.
Since the system disturbance cannot be accurately obtained, a robust term is introduced in the form of where is the estimated value of and is a positive constant. The estimated error of is defined as .
Considering system (13), create a Lyapunov function by following these steps:
The derivative of can be obtained:
Applying Assumption 2 and the approach law (14), the ATSM system’s input is constructed as follows:
By substituting the derivative from Equation (18) into Equation (17), the following may be deduced:
The adaptive law of parameters is chosen as where is positive constant.
Substituting Equation (20) into Equation (19), the following inequality relation can be obtained:
Therefore, Equation (19) can be simplified as
Taking into account the actuator faults and saturations, the system (10) can be represented as follows: where are the errors between the control inputs and the control signals under faults.
The RBFNN is used to estimate the disturbance term given by the following:
The estimate error of can be described as where is the estimate value of and is the estimated error.
Assumption 3. The approximation error is supposed to be bound by an unknown positive constant as follows:
According to Equation (18), the actual input of the FTC scheme is constructed as follows:
For the system (23), the Lyapunov function corresponding to Equation (16) is created:
The derivative of can be found as follows:
Applying Equations (20), (23), and (27), the time derivative of can be written as
The adaptive adjustment law for is chosen as follows: where is a positive constant and is a diagonal positive matrix.
Substituting Equation (31) into Equation (30), the time derivative of can be written as When , Equation (32) can be rewritten as:
According to the unequal relationship,
Substitute into Equation (32) to obtain
When is the largest eigenvalue of the matrix , , the following relationships exist: where and .
The solution to formula (35) is as follows:
According to Equations (22) and (37), it can be seen that all signals in the closed-loop control system are bounded. The state tracking errors , , , and , as well as the estimation errors of the disturbance’s upper bound and the neural network weights , are bounded. Based on Lyapunov theory, the stability of this closed-loop system is uniformly ultimately stable and exponentially converge to a bounded neighbourhood of the system original point.
4. Simulation Results
In order to verify the effectiveness of the proposed control strategy, according to the dynamics models of Equations (1), (2), and (3), and by referring to the parameters of reference [2], simulation is carried out to make the aircraft track the given altitude command and speed command. The initial values of the states are set as , , , , , , and . The step command is and . The reference commands of and are calculated as follows: with and . To verify the robustness of the controller, the model parameters are processed as follows: , , , , , and . The external disturbances [11–13] are given as and .
The design parameters of the ATSM controller method in this paper are, respectively, set as , , , , , , , , and . The design parameters of the FTC method in this paper are, respectively, set as , , , , , , , , , , and .
The center value matrix of the Gaussian basis function is chosen as
Example 1 (the failure-free simulation analysis between ATSM and FTC). In the example, the parameters of Equation (5) are set as and . The range of control input is set as and . The simulation results are shown in Figure 2.
(a) Velocity
(b) Altitude
(c) Flight path angle, attack angle, and pitch rate
(d) Velocity error and altitude error
(e) Fuel-to-air ratio
(f) Elevator deflection
(g) Adaptive parameters
(h) NN weight adaption
(i) NN estimate values
The tracking curves of velocity and altitude can be obtained from Figures 2(a) and 2(b), allowing the ATSM controller to track the reference command with a small tracking error. It is visible from Figures 2(a)–2(c) that the states converge to steady-state values in a brief period. Figures 2(d) and 2(e) show the control inputs of fuel-to-air ratio and deflection in the limit range, which are smooth. Figure 2(f) describes the curves of the adaptive parameters in the case of unknown uncertainties. The NN weight adaption and estimate values fluctuate quickly owing to tracking errors, as seen in Figures 2(g) and 2(h), and the variables converge to a constant in a limited amount of time.
Example 2 (the failure-tolerant simulation analysis between ATSM and FTC). In the example, the range of control input is set as in Example 1, and the parameters of Equation (5) are set as follows: The simulation results are shown in Figure 3.
(a) Velocity
(b) Altitude
(c) Flight path angle, attack angle, and pitch rate
(d) Velocity error and altitude error
(e) Fuel-to-air ratio
(f) Elevator deflection
(g) Adaptive parameters
(h) NN weights adaption
(i) NN estimate values
The hypersonic vehicle is marginally impacted by velocity and altitude in time-varying fault conditions, as seen in Figures 3(a)–3(c). Figures 3(e) and 3(f) illustrate the curves of the input signal. From Figure 3(d), when simulating for 40 s, the actuator faults occur, the ATSM controller adjusts the control gain rapidly, and the control input is out of the lower bound of the fuel-to-air ratio. The results of simulation show that the amplitude of deviation can be reduced with the method and improve the convergence of the states. From Figures 3(g)–3(i), it can be seen that the FTC scheme has a strong fault-tolerant capacity to adjust the adaptive parameters and the NN estimate values online and handle failure effectively under the occurrence of failure.
5. Conclusions
In this paper, the ABHV longitudinal model’s parameter uncertainties, external disturbances, and actuator defects are investigated using radial basis function neural networks (RBFNN) and adaptive sliding mode (ATSM). The input-output linearization approach is used to create an adaptive terminal sliding mode controller based on the simplified nonlinear dynamics model of a hypersonic vehicle, which achieves speed and altitude tracking convergence in a short time. A fault-tolerant control (FTC) system was created to further tackle the actuator defect problem, and RBFNN was employed to simulate the unknown actuator dynamics. The Lyapunov theorem verifies the closed-loop system’s uniformly ultimate stability. The FTC scheme is numerically simulated. The analysis and simulation findings reveal that the fault-tolerant control system described in this study can provide quick tracking response and has high resilience in the presence of actuator faults.
Data Availability
The model parameters of the numerical simulation used to support the findings of this study have been deposited in the “Design of robust control systems for a hypersonic aircraft” repository (doi:10.2514/2.4197).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (No. 12072090).