1. Introduction
Hypersonic vehicles [
1,
2] are a new generation of the national defense strategic development direction due to their high penetration probability, super speed and high accuracy. Hypersonic vehicles [
3,
4,
5] have many varieties from ballistic reentry to hypersonic cruise vehicles, including single-stage to orbit (SSTO), two-stage to orbit (TSTO), and space vehicles. As is known, when a hypersonic vehicle’s speed reaches more than Mach 5.0, aerothermoelastic problems can result [
6,
7,
8], including transient aerodynamic heating causing structural deformation of leading edges and the control surface, internal thermal gradients leading to the cumulative effects of residual stresses, creep, and degradation. The above-mentioned problems pose a threat to fabrication and environmental durability for hypersonic vehicles. Therefore, it is necessary to reproduce a real thermal environment for thermal protection systems (TPS) and hot structure examination [
9,
10,
11] of hypersonic vehicles in a time sequence.
The development of a thermal-structural test [
12,
13] for hypersonic vehicles establishes a real time ground simulation system of aerodynamic heating. So far, there are three main ways of heat transfer during the thermal-structural test: conduction, convection, and radiation. Among them, the wind-tunnel test [
14] is a typical forced convection heat transfer for TPS optimization design. A hypersonic wind tunnel is limited by shape and dimension of the part and can only test reduced-scale hypersonic models of a large size hypersonic vehicle. By contrast, a quartz lamp heater (QLH) [
15,
16] as an infrared radiation heating element is more widely applied in thermal-structural tests because of its small thermal inertia, compact structure and high efficiency. In [
17], quartz lamps were analyzed by the Monte Carlo method to demonstrate QLH with high heat flow. In [
15], the heat flux distribution of QLH and its array were studied, and the heat flux of the QLH array increased, corresponding to an increased load power for aerodynamic heating flux simulation.
We established a thermal-structural test with quartz lamp heaters (TSTQLH), including aerothermal data processing, a thermal-structural control system, and experimental feedback. As shown in
Figure 1, aerothermal data, called transient aerodynamic heating, is assessed by finite element analyses. There is also a link between aerothermal data and temperature trajectory, the temperature trajectory being a control system tracking target. The TSTQLH control system combined data acquisition with algorithm implementation, and acted on the QLH. Once the TSTQLH control system was obtained with effective dynamic performance when tracking the temperature trajectory, this indicates success in reproducing the real thermal environment for hypersonic vehicles, and the control system can be used to validate the TPS optimization. How to effectively design the control system is essential.
In recent years, some control methods have been developed for the TSTQLH, such as fuzzy logic control [
18,
19] and proportional-integral-derivative (PID) control [
20,
21,
22]. In [
18], the transient aerodynamic heating flow of a flying missile body was assessed using a QLH by the fuzzy control method with a complex membership function. In [
23], an aerodynamic heating transient test system with a QLH was designed to track real-time heat flux and temperature from a hypersonic vehicle’s trajectory. During the entire control process, fuzzy PID was adopted in a vacuum chamber as the test environment for the TPS. However, fuzzy logic control [
24] is mostly based on empirical formulae without a system dynamic model, and its controller cannot achieve a balance between control precision and decision-making effectively. In addition, PID control depends on simply linear superposition of tracking errors and has some errors related to rapidity and overshoot [
25]. At the same time, the TSTQLH is process monitoring and some parts of its control system may be out of control due resistance [
26], leading to an adverse effect on control capability. Moreover, the TSTQLH control system contains internal uncertainties and external disturbances, causing indeterminacy.
To solve the above-mentioned problems, model-free control (MFC) [
27,
28,
29] is required. For a single input single output (SISO) system, MFC primarily depends on the linear relationship between input and output, along with an ultra-local model. MFC does not rely on the system dynamic model heavily, reducing system complexity and compensating for dynamic disturbances. In general, MFC has four parts: an ultra-local model, a closed-loop controller, an observer, and an auxiliary controller. In [
26], an intelligent proportional-integral sliding mode control (iPISMC) was used to track the direct power of the doubly fed induction generator (DFIG). The proposed iPISMC consisted of an iPI (a closed-loop controller) and SMC (an auxiliary controller) in terms of an ultra-local model, along with an extended state observer (ESO, an observer). Under stochastic wind turbulences and parametric uncertainties, the proposed iPISMC was more robust than other classical PIs and iPIs. However, because of the linear superposition of tracking errors, there were some problems related to rapidity and overshoot. In [
30], a model-free fractional-order nonsingular fast terminal sliding mode control (MFF-TSMC) was proposed to track the robotic trajectory of a lower-limb robotic exoskeleton. Due to the proposed MFF-TSMC, high precision tracking, fast finite-time convergence, singularity-free and chattering-free operation could be simultaneously ensured in the whole control process. At the same time, because of lack of an auxiliary controller, some errors from the time delay observer and measurement noise still occurred.
Sliding mode control (SMC) [
31], as a variable structure control, is robust for system uncertainties and unknown disturbances because of its two distinct states: the switching phase and sliding phase. Before the system state is constrained to lie in a prescribed sliding mode surface, the system state is driven toward the sliding mode surface by the reaching law. During the reaching phase, the system dynamic performance is sensitive to system uncertainties and unknown disturbances resulting from high-frequency switching motions. Moreover, in traditional SMC, the convergence of the system state is asymptotic rather than in finite time. Therefore, some scholars have proposed a terminal SMC, a global SMC, a fractional-order SMC and other SMCs. In [
31], a nonsingular terminal SMC was applied for rigid manipulators and the time from any initial state to the equilibrium state was guaranteed to be in finite time. However, some uncertainties sometimes occur in the high frequent switching phase. In [
32], a global nonlinear integral SMC associated with a decay function was designed for chaotic synchronization systems to achieve non-integral saturation, fast response and insensitivity to uncertain parameters and disturbances; however, the tracking errors could not converge to zero in finite time.
Neural network control (NNC), as a branch of intelligent control, has the ability of proximation, as in a Hermite neural network (HNN) [
33], a layer recurrent neural network (LRNN) [
34], an extreme learning machine (ELM) [
35], and a radial basis function neural network (RBFNN) [
36]. The authors of [
33], compared a conventional super-twisting algorithm-based second-order sliding-mode control strategy (STA-SOSM) with a novel HNN-based SOSM control strategy that was found to be more feasible and superior for a synchronous reluctance motor (SynRM) drive system. In [
34], the authors used a super-twisting method with artificial neural networks (STA-ANN) for piezoelectric actuators (PEA) to solve nonlinear hysteresis positioning problem. In [
35], a fast nonsingular terminal SMC (FNTSMC) combined with an ELM was used with permanent-magnet linear motor systems. Because of FNTSMC and ELM, finite-time convergence, strong control robustness and system dynamics information independence are guaranteed. In [
36], a model-free and an NNC based on a time-delay estimator (TDE-MFNNC) was presented for five DOFs in a lower extremity exoskeleton. Compared to a PD controller, NNC and model-free iPD controller, the TDE-MFNNC is more stable and more effective. The RBFNN has a simple structure, faster convergence and better approximation properties, which can be considered as compensators.
This paper proposes a dynamic model of a TSTQLH based on energy conservation and a strategy for an adaptive neural network global nonsingular fast terminal sliding mode model-free control based on a linear extended state observer (ANN-GNFTSMMFC-LESO). The TSTQLH is based on infrared radiation heating of quartz lamp heaters and provides a heating environment by thermal radiation. In terms of the nonlinearity of the control process, the control loop is closed by a global nonsingular fast terminal sliding mode controller (GNFTSMC) instead of an intelligent proportional-integral-derivative (iPID) [
26,
28] controller because of problems related to rapidity and overshoot. On the basis of the traditional integral sliding mode control, the GNFTSMC eliminates the high-frequency switching phase [
32,
37] to suppress chattering phenomena and anchors the initial state on the sliding phase to enhance insensitive dynamic performance for internal uncertainties and external disturbances. Moreover, the GNFTSMC introduces a linear term and a nonsingular exponential term into the traditional integral sliding mode control to avoid the singular problem and improves the convergence rate to the equilibrium state in finite time rather than asymptotic convergence [
30,
31,
38]. The LESO is chosen as an observer of the GNFTSMC to estimate internal uncertainties and external disturbances. In addition, an ANN [
33,
34,
35,
36] shows good approximation properties in compensation for estimation errors by using a cubic B-spline function [
36,
39,
40]. In the NNC, the cubic B-spline function, as a basis function, has a similar shape to a Gaussian function with the same center positions, and has less inverse multiquadratic functions in terms of computational load. Moreover, the cubic B-spline function, as a piecewise cubic polynomial nonlinear function, has second-order smoothness, which is enough for continuous modeling.
In summary, this paper makes the following contributions:
- (1)
For hypersonic vehicles, real time ground simulation of aerodynamic heating is established and a thermal-structural test is devised based on an infrared radiation heating platform usinf quartz lamp heaters.
- (2)
By energy conservation, we establish a dynamic model of TSTQLH. In terms of TSTQLH system dynamic information, the control process of radiation heating, including internal uncertainties and external disturbances, is analyzed.
- (3)
ANN-GNFTSMMFC-LESO can provide an ultra-local model which can alleviate the dependence of system dynamics. Instead of an iPID controller, the GNFTSMC eliminates the high-frequency switching phase to suppress chattering phenomena, and combines a inear term with a nonsingular exponential term to avoid slow convergence rate and singularity.
- (4)
We chose a LESO for estimation of lumped disturbances and used a cubic B-spline function in the ANN for observation error compensation with low computational load and second-order smoothness.
- (5)
The fitting curve wall temperature expression (tracking reference temperature trajectory) of the surface contour from an X-43A wing in a time sequence was obtained by the finite element simulation. The superiority of our proposed control method is validated through comparative simulation results.
5. Simulation Results
Under optimized conditions, the ANN-GNFTSMMFC-LESO controller was further studied based on the TSTQLH. The parameters of QLH are listed as: ; ;; ; ; ; ; ; ; . The parameters of the ANN-GNFTSMMFC-LESO controller are listed as: ; ; ; ;; ; ;; ; ; ;
,
;
;
. Other parameters are the same. All parameters are shown in
Table 5,
Table 6,
Table 7,
Table 8,
Table 9 and
Table 10.
As is shown in
Figure 8 and
Figure 9, there are other four comparative control strategies, including the GNFTSMMFC-LESO controller, the TSMFC-LESO controller, iPIDMFC controller, and the PID controller. The GNFTSMMFC-LESO controller has the same sliding mode surface and LESO as the proposed scheme (ANN-GNFTSMMFC-LESO controller) with the auxiliary controller ANN. The TSMFC-LESO controller has a traditional TSM surface (
). The iPIDMFC controller combines the PID controller with the MFC, which is defined as:
As shown in
Figure 8a (the reference temperature trajectory of Equation (62)) and
Figure 8, the fitting curve wall temperature of the surface contour from an X-43A wing in time sequence is chosen as a reference trajectory, which is from 234.2 k in 0 s to 1711.79 k in 31 s. From
Figure 8b,c, these controllers have a silimar trend about the tracking dynamic performances. At the beginning, from 1.0 s to 3.0 s, the ANN-GNFTSMMFC-LESO controller has the shortest response time and other cotrollers have overshoot phenomena with different levels. From
Figure 8d,e, chattering phenomena occur in the TSMFC-LESO controller because of the traditional TSM surface with a high-frequency switching phase during the whole process. In
Figure 8f, chattering phenomena occur in the GNFTSMMFC-LESO controller at a later time from 24 s to 31 s, due to the lack of the auxiliary controller ANN, leading to estimation error accumulation. In
Figure 8g,h, it is apparent that the ANN-GNFTSMMFC-LESO controller has the smallest input and estimation error due to the combined effects of MFC, GNFTSMC and ANN.
Considering the QLH as an infrared radiation heating element, its resistance is sensitive to its temperature. The time-varying resistance of the quartz lamp filament is selected as a disturbance and its equation is:
Figure 9 reveals that the time-varying resistance has an adverse impact on performances which enlarges the fluctuation range and prolongs stabilization time. By contrast, the ANN-GNFTSMMFC-LESO controller has robustness to some disturbances. According to
Figure 9c,d, the tracking error of the ANN-GNFTSMMFC-LESO controller decreases dramatically at the beginning, then remains stable until 31 s. By contrast, the largest fluctuation occurred in the tracking error of the TSMFC-LESO controller at around 30 k at 0.2 s and reached a relative steady state at 3.0 s. However, at the later stage of the control process, the TSMMFC-LESO controller had more chattering as shown in
Figure 9e. The GNFTSMMFC-LESO controller had a steady state error because of the lack of the auxiliary controller ANN. If the time-varying resistance occurred, there were some tracking fluctuations for iPIDMFC controller and PID controller. In fact, the PID is based on output error feedback to eliminate error and its final state may be stable. However, the real output is an inertial variable without any sudden changes. Some nonlinear, unknown features and external uncertainties are involved in this control system, which cause sudden changes in the control process. This means that the non-jumping variable is used to control the jumping variable, leading to a conflict between rapidity and overshoot.
To make quantitative comparisons, calculations of simulation results from 10 s to 20 s are given in
Table 11, including the root mean square error
, the Maximum error
.
The above results show that the ANN-GNFTSMMFC-LESO controller possesses strong robustness in tracking temperature trajectory and its tracking error can rapidly converge to zero. Internal system parameters make no significant difference on its tracking precision over the period from 0 s to 31 s. Compared with GNFTSMMFC-LESO controller, TSMFC-LESO controller, iPIDMFC controller, and PID controller, the ANN-GNFTSMMFC-LESO control strategy has great application in real time ground simulation of aerodynamic heating produced by hypersonic vehicles.