A Control Method for Thermal Structural Tests of Hypersonic Missile Aerodynamic Heating

Abstract

This paper presents an intelligent proportional-derivative adaptive global nonsingular fast-terminal sliding-mode control (IPDAGNFTSMC) for tracking temperature trajectories of a hypersonic missile in thermal structural tests. Firstly, the numerical analyses on a hypersonic missile’s aerodynamic heating are based on three different external flow fields via the finite element calculation, which provides the data basis for the thermal structural test of hypersonic vehicles; secondly, due to temperature trajectory differences of a hypersonic missile and the thermal inertia and nonlinear characteristics of quartz lamps in thermal structural test, IPDAGNFTSMC is proposed, consisting of three components: (i) the mathematical model of the thermal structural test is established and further replaced via an intelligent proportional-derivative with a nonlinear extended state observer (NESO) for online unknown disturbances observation; (ii) compared with the traditional sliding-mode control method, the AGNFTSMC method eliminates the reaching phase and the initial control state is trapped on the sliding-mode surface. Therefore, it can alleviate chattering phenomenon, accelerate the convergence rate of the sliding mode, and ensure that there is no singular problem in the entire control process; (iii) the adaptive law is designed to effectively solve problems of convergence stagnation and chattering phenomenon. The Lyapunov stability theory is used to prove the stability of the proposed IPDAGNFTSMC-NESO. Finally, the advantages of the designed control method are verified by experimental simulation and comparison.

1. Introduction

The problem of the thermal barrier [1,2] will occur when a hypersonic vehicle [3,4,5] is flying at hypersonic speed. The high temperature, generated by a hypersonic vehicle, will not only affect material strength and the structural load-bearing capacity of a hypersonic vehicle, but also damage the high-precision electronic components inside a hypersonic vehicle leading to an abnormal flight. In response to this problem, a thermal structural test of a hypersonic vehicle (TSTHV) [6,7] is designed to determine whether the materials selected by a hypersonic vehicle can withstand the impact of the aerothermal environment. The main body of a thermal structural test is aerodynamic heating ground simulation in real time, including a wind tunnel test or a radiation heating test. In [8,9,10], the wind tunnel test theory of the aerothermal environment is analyzed, and the domestic application of the aerothermal environment, the wind tunnel test equipment, and its simulation capabilities are introduced, focusing on analysis of the developments and the tendencies of wind tunnel aerothermal environment technology. Compared with the wind tunnel test, the radiation heating of the quartz lamp has better implementation results and lower cost factors. Moreover, the quartz lamp heater is an ideal heating element for a TSTHV due to its small size, low thermal inertia, high power, and good controllability [11]. In [12], a high-performance quartz lamp heater system is designed to reproduce transient aerothermal environments experienced by a hypersonic vehicle. In [13], the thermal environment (radiation characteristics) of the flat quartz lamp heater system is analyzed to screen out the optimum material or evaluate material capability in the domain of hypersonic vehicle’s thermal protection systems (TPS). Therefore, the design of a TSTHV with quartz lamp heaters (TSTQLH) as a heating part is worth taking into consideration.

Apparently, a TSTQLH is a typical large thermal inertia, large time delay, and complex nonlinear system. Some control strategies are applied in the TSTQLH, such as fuzzy logic control [14], PID control [15], and iterative learning control (ILC) [16]. In [14], a fuzzy control strategy is used to track the transient aerodynamic heating simulation for a hypersonic missile; in [15], for an aerodynamic heating environment simulation system with PLC, a PID control is designed in each channel from the multiple temperature zones; in [16], the fractional order PD^α type ILC is presented to simulate the aerodynamic heating of aircrafts on the ground. These control methods can reproduce the aerodynamic heating environment of the hypersonic vehicle; however, there are still some disadvantages: fuzzy logic control has difficulty establishing fuzzy rules and membership function, depending only on empirical formulas; PID control depends on the simply linear superposition of tracking errors, leading to some conflicts between rapidity and overshoot; ILC improves the system performance from repetition to repetition, resulting in poor convergence and too much computational load.

In response to the above problems, the model-free control (MFC) based on an ultra-local model is proposed by Fliess [17], consisting of a closed loop controller, a disturbance observer, and an auxiliary controller, in which the system model is replaced by an ultra-local model. Thereby, MFC can simplify system complexity, being independent from the system model, while the main loop of an ultra-local model is closed by an intelligent PID (IPID), such as a proton exchange membrane fuel cell [18], lower extremity exoskeleton [19], or power system emergency frequency control [20]. In [18], an intelligent proportional-integral associated with adaptive interval type-2 fuzzy nonsingular fast-terminal sliding-mode control (MF-AIT2FSMC) is used to adjust the oxygen stoichiometry of the proton exchange membrane fuel cell in the MFC frame. In [19], for a 5 DOFs lower extremity exoskeleton, an intelligent proportional-derivative and a neural-network control are added into an ultra-local model along with a time delay estimation (TDE-MFNNC). In [20], MFC with emergency control schemes is obtained in the presence of reinforcement learning for power system applications. In these cases, they are independent on the system model and can track reference trajectories effectively.

For control problems of an uncertain system, Han proposes an active disturbance rejection control [21], which uses a nonlinear extended state observer (NESO) as a means to estimate the total disturbances in real time and to compensate them in feedback control. Then, ESO, a disturbance observer, is also applied to MFC, which is used as a tool to estimate all disturbances. For example, a NESO [22,23,24] can observe all disturbances in the ultra-local model, and the entire control process is formed into a closed loop through the IPID. In [22], a compensated model-free adaptive control (MFAC) scheme is proposed by combining an extended state observer (ESO) for an autonomous underwater vehicle. In [23], an ESO is utilized to estimate the dynamical behavior of the robot. In [24], based on an ESO, a model-free command-filtered backstepping controller is designed for the marine power systems. Therefore, this paper proposes an IPD control method with NESO for a TSTQLH.

A NESO has errors in its observation of disturbances, hence the IPD control method cannot guarantee that the tracking errors of the system converge to zero quickly. Then, sliding-mode control [25,26], as an auxiliary controller, can eliminate these errors because of its insensitivity to parameter uncertainties, anti-interference, and fast dynamic response. Thus, this paper considers predetermined sliding-mode control as an input control compensator to eliminate observation errors. In [27], a high-order terminal sliding- mode controller (TSMC) is developed for single-input single-output nonlinear systems in presence of external disturbances. The designed terminal-sliding surface can accelerate the convergence rate at the equilibrium point and eliminate errors in the finite time, but the convergence rate is slow, and the dynamic performance is poor when it is far away from the switching surface, and the singular phenomenon may happen in the convergence phase. Therefore, a global nonsingular fast-terminal sliding-mode surface (GNFTSMS) is proposed to ensure global stability and fast convergence of the system, which further shortens the convergence time of tracking error variables. Because the initial control state is trapped on the sliding-mode surface, the GNFTSMS can improve the robustness of the system and suppress the chattering phenomenon. In [28], a model-free terminal sliding- mode control method is designed for the lower limb exoskeleton robot, in which the upper bound of the estimation error is assumed to be a known value. However, in the actual TSTQLH, the upper bound of the estimation error is unknown. It is common to choose a large enough switching gain in the controller to cover the possible error values and to ensure the stability. An excessive switching gain will increase the chattering phenomenon. In this paper, an adaptive law is designed to approach the unknown upper bound disturbance and to reduce the chattering phenomenon. Based on the above work, an intelligent PD-adaptive global nonsingular fast-terminal sliding-mode control (IPDAGNFTSMC) method with a nonlinear extended state observer (NESO) method is proposed. The IPD control method based on a NESO ensures that the tracking errors are bounded and the designed AGNFTSMC method can eliminate these bounded errors in the finite time.

Compared with previous studies, the main contributions of this paper can be summarized as:

The main structure of a TSTQLH is introduced to reproduce a real aerodynamic heating environment of the hypersonic vehicle and the system dynamic model is established with respect to energy conservation.

The numerical analyses about a hypersonic missile’s aerodynamic heating are based on three different external flow fields and via the finite element calculation, and the fitting curve temperature expressions are regarded as the same real fight environment of a hypersonic missile, which are as the reference temperature trajectories.

For high nonlinearities and strong couplings of a TSTQLH, an IPDAGNFTSMC method is proposed, consisting of an IPD control and an AGNFTSMC. The proposed IPDAGNFTSMC method does not rely on the accurate mathematical model of TSTQLH heavily, in which the designed sliding mode surface (GNFTSMS) ensures the global fast convergence of the entire control system without singularity. At the same time, the chattering phenomenon is suppressed, and the robust performance is improved.

Based on the observation ability of a NESO for unknown terms, combined with the IPDAGNFTSMC method, some uncertainties and unknown disturbances from the system are closed-loop controlled.

According to the Lyapunov stability proof method, the stability proof processes of NESO and IPDAGNFTSMC are given.

The rest of this paper is organized as follows: in Section 2, an overview of TSTQLH is briefly presented; in Section 3, controller design and stability proof are given containing NESO, GNFTSMS; Section 4 gives some comparative simulation results. Finally, Section 5 briefly concludes this paper’s work.

2. An Overview of a TSTQLH

In this section, an overview of TSTQLH is given and further explained in detail. Firstly, it introduces some structural characteristics of TSTQLH, then explains some numerical analyses about a hypersonic missile and a mathematical model of TSTQLH.

2.1. The Introduction of a TSTQLH

The workflow of TSTQLH is given in Figure 1. It mainly consists of three parts: aerothermal data processing, an aerodynamic heating ground simulation control system, and a thermal protection system feedback module. The first part is to determine the kind of hypersonic vehicle and flight environment. Then, the finite element simulation is used to analyze its aerodynamic heating condition, and some numerical analyses are plotted as curves of the expected output temperature in the second part of the aerodynamic heating ground simulation control system. In the second part, through the design of the controller, the quartz lamp heaters can be used to track the temperature trajectory for the thermal protection system of the sample test. The third part is to observe and analyze whether the selected material can withstand the predetermined temperature of the hypersonic vehicles. If the requirements cannot be met, other materials will be selected for another round of experiment. If the requirements are met, it should be determined to be the materials of a hypersonic vehicle.

2.2. A Hypersonic Missile

A hypersonic missile is chosen as a calculated object during the finite element simulation. Figure 2 is a three-dimensional drawing and two-dimensional drawing of a hypersonic missile. The specific parameters of the hypersonic missile are a total length 7600 mm, body length 4270 mm, body diameter 1168.4 mm, angle of secondary warhead 7°, angle of warhead 12.84°, radius of warhead 30 mm.

2.3. Numerical Analyses

Based on the finite element simulation results, three calculated objects (wall 0, wall 1, wall 2) of the hypersonic missile warhead are chosen to plot some scatter graphs, with a relationship between positions and temperatures, in Figure 3. These 31 different groups act as an independent time interval (1.0 s), and Figure 3a–j is transformed into Figure 3j–l. Some details of the finite element calculation are explained in Appendix A.

Figure 3a,d,g, shows position and temperature relationships of wall 0 and it is obvious that wall 0 has the maximum temperature. Because of different attack angles, three scatter graphs have no overlapped positions and Figure 3j takes the average temperature values of each position to obtain the link of temperature in a time sequence. In addition, Figure 3k,l chooses three same positions (7.75 m, 8.5 m, 9.1 m) to obtain the temperature trajectory with an independent time interval (1.0 s). These three positions (7.75 m, 8.5 m, 9.1 m) can represent the whole body of a hypersonic missile warhead. In Figure 3j–l, the fitting curve temperature expressions of three calculated objects (wall 0, wall 1, wall 2) are as follows:

Wall 0_ average:

$${T_1}^{*}\left(t\right)=1.284\times {10}^{-5}{t}^6-7.775\times {10}^{-4}{t}^5+0.01862t^4-0.2734t^3+3.146t^2+0.7123t+201.6$$

(1)

Wall 1_ 7.75 m:

$${T_1}^{*}\left(t\right)=1.536\times {10}^{-5}{t}^6-1.169\times {10}^{-3}{t}^5+0.0383t^4-0.6755t^3+6.284t^2-7.783t+146.6$$

(2)

Wall 1_ 8.5 m:

$${T_1}^{*}\left(t\right)=1.381\times {10}^{-4}{t}^5-7.093\times {10}^{-3}{t}^4+0.09818t^3+0.3339t^2-4.099t+233.8$$

(3)

Wall 1_ 9.1 m:

$${T_1}^{*}\left(t\right)=-7.689\times {10}^{-6}{t}^6+9.043\times {10}^{-4}{t}^5-0.03555t^4+0.5952t^3-3.912t^2+12.42t+212.2$$

(4)

Wall 2_ 7.75 m:

$${T_1}^{*}\left(t\right)=3.131\times {10}^{-5}{t}^6-2.663\times {10}^{-3}{t}^5+0.08746t^4-1.344t^3+9.386t^2-11.78t+238.7$$

(5)

Wall 2_ 8.5 m:

$${T_1}^{*}\left(t\right)=-1.368\times {10}^{-7}{t}^8+1.752\times {10}^{-5}{t}^7-8.848\times {10}^{-4}{t}^6+0.02226t^5-0.2852t^4+1.62t^3-1.641t^2+0.5824t+229.7$$

(6)

Wall 2_ 9.1 m:

$${T_1}^{*}\left(t\right)=1.495\times {10}^{-8}{t}^9-2.338\times {10}^{-6}{t}^8+1.545\times {10}^{-4}{t}^7-5.576\times {10}^{-3}{t}^6+0.1185t^5-1.49t^4+10.59t^3-38.03t^2+66.68t+186.4$$

(7)

where ${{\dot{T}}_1}^{*}\left(t\right)$ is the temperature trajectory value of TSTQLH control system; $t$ is the flight time; the goodness of fit from different calculated objects (wall 0, wall 1, wall 2) is illustrated in

Table 1. Goodness of fit from different calculated objects (wall 0, wall 1, wall 2).

	SSE	R-Square	Adjusted R-Square	RMSE
wall 0	3927	0.9985	0.9982	12.79
wall 1_7.75 m	6282	0.9964	0.9954	16.53
wall 1_8.5 m	2441	0.9953	0.9942	10.53
wall 1_9.1 m	4606	0.9925	0.9905	14.47
wall 2_7.75 m	3443	0.9977	0.9971	11.98
wall 2_8.5 m	1404	0.9969	0.9958	7.989
wall 2_9.1 m	1231	0.9976	0.9966	7.659

. From Figure 1, the aim of TSTQLH is to reproduce the real aerodynamic heating environment of the hypersonic vehicle on the ground and via the finite element simulation, the fitting curve temperature expressions from Equation (1) to Equation (7) are regarded as the same real fight environment of the hypersonic vehicle, which are used as the reference temperature trajectories in next control module.

2.4. Control System

In Figure 1, via the aerothermal data processing module, the temperature trajectories are given from the mentioned-above numerical analyses, as the tracking aims of the control system. From the framework of the control system, electric energy is loaded on quartz lamp heaters and the conduction angle was changed to adjust the quartz lamp heaters’ power for tracking temperature trajectory. Then, the system dynamics of the TSTQLH’s control system can be described as follows:

2.4.1. Electric Energy

The expression of electric energy is defined:

$$W=P\Delta t$$

(8)

where $W$ is the total energy loaded on the quartz lamp heaters; $P$ is the input power; $\Delta t$ is the working time.

Round the circuit, the input voltage is:

$$U=U_{\mathrm{I}}\sqrt{\sin \left[2\alpha \left(t\right)\right]/2\pi +\left[\pi -\alpha \left(t\right)\right]/\pi }$$

(9)

where $U_{\mathrm{I}}$ is supply voltage; $U$ is input voltage; $\alpha \left(t\right)$ is conduction angle.

Then, $P$ is calculated:

$$P=U^2/R={\left\{U_{\mathrm{I}}\sqrt{\sin \left[2\alpha \left(t\right)\right]/2\pi +\left[\pi -\alpha \left(t\right)\right]/\pi }\right\}}^2/R=U_{\mathrm{I}}^2/R\left\{\sin \left[2\alpha \left(t\right)\right]/2\pi +\left[\pi -\alpha \left(t\right)\right]/\pi \right\}$$

(10)

where $R$ is the resistance of quartz lamp heaters.

2.4.2. Electrothermal Energy

The expression of electrothermal energy is deduced:

$$Q=cm\left[T_1\left(t\right)-T_0\right]+A\beta \left[T_1\left(t\right)-T_0\right]+A\lambda \left[T_1\left(t\right)-T_0\right]+A\epsilon \sigma F{T}_1^4\Delta t$$

(11)

where $Q$ is electrothermal energy; $T_1\left(t\right)$ is current temperature; $T_0$ is initial temperature; $c$ and $m$ are the specific heat capacity and mass of the quartz lamp filament, respectively. $A$ and $\epsilon$ are surface area of quartz lamp tube and blackness coefficient of quartz lamp filament, respectively. $\beta$, $\lambda$, $\sigma$ and $F$ are convective transmission coefficient, conductive transmission coefficient, Stephen Boltzmann’s constant, and angle coefficient, respectively.

Combining (8), (10), and (11), the expression is rewritten with respect to energy conservation:

$$U_{\mathrm{I}}^2/R\left\{\sin \left[2\alpha \left(t\right)\right]/2\pi +\left[\pi -\alpha \left(t\right)\right]/\pi \right\}\ast \Delta t=cm\left[T_1\left(t\right)-T_0\right]+A\beta \left[T_1\left(t\right)-T_0\right]+A\lambda \left[T_1\left(t\right)-T_0\right]+A\epsilon \sigma F{T}_1^4\left(t\right)\Delta t$$

(12)

Note that $\alpha \left(t\right)$ is input variable and $T_1\left(t\right)$ is output variable; via controlling $\alpha \left(t\right)$, it is possible to track temperature trajectory for the TSTHV so as to reproduce the real aerothermal environment.

3. Controller Design and Stability Proof

In this section, the NESO is introduced for the lumped disturbances estimation; the GNFTSMS, as a sliding-mode surface, is presented, including linear term and terminal term. It can guarantee global fast convergence without singularity and is insensitive to the lumped uncertainties and uncontrollable time-varying resistance. At the same time, the proposed IPDAGNFTSMC-NESO controller is given by contrast with the GNFTSMC controller and the IPD-NESO controller. The global stability and rapid convergence from GNFTSMS and the model-free nature from the ultra-local model of MFC are integrated into the IPDAGNFTSMC-NESO controller.

3.1. NESO

The derivative of (12) is the following:

$${\dot{T}}_1\left(t\right)=\left\{U_{\mathrm{I}}^2/R\left\{\sin \left[2\alpha \left(t\right)\right]/2\pi +\left[\pi -\alpha \left(t\right)\right]/\pi \right\}-A\epsilon \sigma F{T}_1^4\left(t\right)\right\}/\left[cm+A\left(\beta +\lambda \right)\right]$$

(13)

Equation (13) is rewritten:

$${\dot{T}}_1\left(t\right)=G\left(t\right)-\alpha \left(t\right)U_I^2∕\left\{R\pi \left[cm+A\left(\beta +\lambda \right)\right]\right\}$$

(14)

where $G\left(t\right)$ is unknown dynamics and system disturbances and is given as follows:

$$G\left(t\right)=\left[U_{\mathrm{I}}^2\sin \left[2\alpha \left(t\right)\right]/2R\pi +U_I^2/R-A\epsilon \sigma F{T}_1^4\left(t\right)\right]∕\left[cm+A\left(\beta +\lambda \right)\right]$$

(15)

Note that $G\left(t\right)$ contains local periodic oscillation and high-order nonlinear output.

By analyzing the system characteristics, it is of great importance to observe $G\left(t\right)$ in the practical application. Hence, the NESO [22,23,24,29] is applied to observe $G\left(t\right)$ and the NESO is defined:

$$\left\{\begin{array}{c}{e}_1\left(t\right)=z_1\left(t\right)-T_1\left(t\right)\\ e_2\left(t\right)=z_2\left(t\right)-G\left(t\right)\\ {\dot{z}}_1\left(t\right)=z_2\left(t\right)-{\beta }_1{e}_1\left(t\right)+\chi \alpha \left(t\right)\\ {\dot{z}}_2\left(t\right)=-{\beta }_2{\left|e_1\left(t\right)\right|}^{1/2}\mathrm{sign}\left[e_1\left(t\right)\right]\\ z_2\left(t\right)=\widehat{G}\left(t\right)\end{array}\right.$$

(16)

where $z_1\left(t\right)$ and $z_2\left(t\right)$ are the observation values of output temperature and unknown disturbances, respectively. $e_1\left(t\right)$ and $e_2\left(t\right)$ are estimation errors of output temperature and unknown disturbances, respectively. $\widehat{G}\left(t\right)$ is equal to $z_2\left(t\right)$; ${\beta }_1$ and ${\beta }_2$ are constants for gain adjustment and they satisfy the following:

$${\beta }_1>0, {\beta }_2>0, \mathrm{sign}\left[e_1\left(t\right)\right]=\left\{\begin{array}{c}1\\ 0\\ -1\end{array} \begin{array}{c}{e}_1\left(t\right)>0\\ e_1\left(t\right)=0\\ e_1\left(t\right)<0\end{array}\right.$$

(17)

The observation error of unknown disturbances is defined:

$$\tilde{G}\left(t\right)=G\left(t\right)-\widehat{G}\left(t\right)$$

(18)

where $\tilde{G}\left(t\right)$ is the observation error of unknown disturbances.

3.2. GNFTSMS

Define:

$$e\left(t\right)={T_1}^{*}\left(t\right)-T_1\left(t\right)$$

(19)

where $e\left(t\right)$ is the tracking error; ${T_1}^{*}\left(t\right)$ is reference temperature trajectory from Equations (1)–(7).

Then, the derivative of (19) is as follows:

$$\dot{e}\left(t\right)={{\dot{T}}_1}^{*}\left(t\right)-{\dot{T}}_1\left(t\right)$$

(20)

The GNFTSMS is defined:

$$s\left(t\right)=\varrho e\left(t\right)+{{\displaystyle \int }}_0^t\left(e\left(\gamma \right)+\mu e{\left(\gamma \right)}^{m/n}\right)\mathrm{d}\gamma -\varrho e\left(t_0\right)$$

(21)

where $\varrho >0$, that is the coefficient of linear term, and $\mu >0$, that is the coefficient of terminal term; $m$, $n$ are positive odd integers which satisfy $m<n<2m$; $e\left(t_0\right)$ is the initial state of the tracking error. Then, the GNFTSMS guarantees great control dynamics because of three factors: (1) the initial system state is confined on the sliding phase without high-frequency switching phase, which suppress the chattering phenomenon; (2) the integral component, which possesses the exponential stability, eliminates steady state error and avoids singularity problems; (3) the fast term and terminal term can alleviate convergence stagnation and improve the convergence rate at the equilibrium point.

The derivative of (21) is as follows:

$$\dot{s}\left(t\right)=\varrho \dot{e}\left(t\right)+e\left(t\right)+\mu e{\left(t\right)}^{m/n}$$

(22)

Via the Bernoulli equation and (22), the convergence time is calculated:

$$\varrho \dot{e}\left(t\right)+e\left(t\right)+\mu e{\left(t\right)}^{m/n}=0$$

(23)

$$\dot{e}\left(t\right)+e\left(t\right)/\varrho =-\mu e{\left(t\right)}^{m/n}/\varrho$$

(24)

$$\dot{e}\left(t\right)e{\left(t\right)}^{-m/n}+e{\left(t\right)}^{1-m/n}/\varrho =-\mu /\varrho$$

(25)

$$\mathrm{d}e{\left(t\right)}^{1-m/n}/dt\ast \left[1/\left(1-m/n\right)\right]+e{\left(t\right)}^{1-m/n}/\varrho =-\mu /\varrho$$

(26)

Let $E=e{\left(t\right)}^{1-m/n}$, the (26) is written:

$$\mathrm{d}E/dt+\left(1-m/n\right)E/\varrho =-\mu \left(1-m/n\right)/\varrho$$

(27)

$$E=\exp \left[-\int \left(1-m/n\right)/\varrho \mathrm{d}t\right]\ast \left\{\int -\mu \left(1-m/n\right)/\varrho \exp \left[\int \left(1-m/n\right)/\varrho \mathrm{d}t\right]+\psi \right\}$$

(28)

where $\psi$ is an arbitrarily constant.

Let $\psi =1$, (28) is calculated:

$$E=\exp \left[-\left(1-m/n\right)t/\varrho \right]\ast \left\{-\mu \exp \left[\left(1-m/n\right)t/\varrho \right]+1\right\}$$

(29)

$$e{\left(t\right)}^{1-m/n}+\mu =\exp \left[-\left(1-m/n\right)t/\varrho \right]$$

(30)

After logarithm (ln) calculation, (30) is calculated:

$$\ln \left[e{\left(t\right)}^{1-m/n}+\mu \right]=-\left(1-m/n\right)t/\varrho$$

(31)

$$t_{\mathrm{c}}=-\ln \left[e{\left(t\right)}^{1-m/n}+\mu \right]\varrho /\left(1-m/n\right)$$

(32)

where $t_{\mathrm{c}}$ is the convergence time.

3.3. Controller Design

Definition 1. 

Controller 1 is designed by GNFTSMC and NESO, which is indicated in Figure 4.

Define the constant reaching law:

$$\dot{s}\left(t\right)=-\eta \mathrm{sign}\left[s\left(t\right)\right]$$

(33)

$$\mathrm{sign}\left[s\left(t\right)\right]=\left\{\begin{array}{c}1\\ 0\\ -1\end{array} \right.\begin{array}{c}s\left(t\right)>0\\ s\left(t\right)=0\\ s\left(t\right)<0\end{array}$$

(34)

where $\eta$ is a gain parameter.

Substituting (14), (20), (33) to (22), $\alpha \left(t\right)$ is calculated:

$$\alpha \left(t\right)=\left\{\left[-\eta \mathrm{sign}\left[s\left(t\right)\right]-e\left(t\right)-\mu e{\left(t\right)}^{m/n}\right]/\varrho -{{\dot{T}}_1}^{*}\left(t\right)+G\left(t\right)\right\}\left\{R\pi \left[cm+A\left(\beta +\lambda \right)\right]\right\}/U_I^2$$

(35)

Eventually, the GNFTSMC controller is acquired for TSTQLH, which relies on the system model of Equation (14). While the TSTQLH system is high nonlinearities and strong couplings with time-varying parameters, it would be changeable and indeterminate. In the next step, an ultra-local model of MFC is expressed: the IPD-NESO controller and IPDAGNFTSMC-NESO controller.

Definition 2. 

Controller 2 is designed by IPD and NESO, which is described in Figure 5.

Equation (14) is rewritten:

$${\dot{T}}_1\left(t\right)=G\left(t\right)+\chi \alpha \left(t\right)$$

(36)

Defined:

$$\delta \left(e\right)=K_{\mathrm{P}}e\left(t\right)+K_{\mathrm{D}}\dot{e}\left(t\right)$$

(37)

Then, $\alpha \left(t\right)$ is obtained:

$$\alpha \left(t\right)=\left[-\widehat{G}\left(t\right)+{{\dot{T}}_1}^{*}\left(t\right)+\delta \left(e\right)\right]/\chi$$

(38)

where $\chi$ is a non-physical gain parameter, which is obtained from experimental simulation; $\delta \left(e\right)$ is PD controller; $K_{\mathrm{P}}$, $K_{\mathrm{D}}$ are gain parameters.

Substituting (18), (19), (38) to (36), the error function is:

$$\tilde{G}\left(t\right)+\dot{e}\left(t\right)+\delta \left(e\right)=0$$

(39)

The Laplace transform is applied in (39), that is solved as follows:

$$\tilde{G}\left(s\right)-\tilde{G}\left(0^{+}\right)+sE\left(s\right)+K_{\mathrm{P}}E\left(s\right)+K_{\mathrm{D}}sE\left(s\right)=0$$

(40)

Then, via the final value theorem [30], (40) is further calculated:

$$e\left(t_{\infty }\right)=\underset{s\to 0}{\lim } sE\left(s\right)=\underset{s\to 0}{\lim }\left\{s\left[\tilde{G}\left(0^{+}\right)-\tilde{G}\left(s\right)\right]/\left(s+K_P+K_{\mathrm{D}}s\right)\right\}$$

(41)

When the time tends to be infinite, the $e\left(t\right)$ is equal to zero. It reveals the fact that the designed controller, the IPD-NESO controller, combining IPD with ESO, can achieve a stable system dynamic. However, the IPD-NESO controller is deficient, when the observation values are not accurate in the existence of some measurement errors and unknown noises. Moreover, the IPD-NESO controller depends on the linear superposition of tracking errors from PD and observation values of unknown disturbances from NESO leading to some collections of observation errors. Hence, an auxiliary controller should be designed to reduce the extra errors.

Definition 3. 

An auxiliary controller is denoted by ${\alpha }_{aux}\left(t\right)$, and the improved IPDAGNFTSMC-NESO (controller 3) is designed as follows:

$$\alpha \left(t\right)=\left(-\widehat{G}\left(t\right)+{{\dot{T}}_1}^{*}\left(t\right)+\delta \left(e\right)\right)/\chi +{\alpha }_{\mathrm{aux}}\left(t\right)$$

(42)

where ${\alpha }_{\mathrm{aux}}\left(t\right)$ is an AGNFTMSC controller; ${\alpha }_{\mathrm{aux}}\left(t\right)={\alpha }_{\mathrm{eq}}\left(t\right)+{\alpha }_{\mathrm{sw}}\left(t\right)$, ${\alpha }_{\mathrm{eq}}\left(t\right)$ and ${\alpha }_{\mathrm{sw}}\left(t\right)$ are equivalent control and switching control, respectively. The control scheme of controller 3 is illustrated in Figure 6.

Substituting (20), (42) to (36), $\dot{e}\left(t\right)$ is rewritten:

$$\dot{e}\left(t\right)=-\tilde{G}\left(t\right)-\delta \left(e\right)-\chi {\alpha }_{\mathrm{aux}}\left(t\right)$$

(43)

Substituting (43) to (22), and $\dot{s}\left(t\right)=0$, ${\alpha }_{\mathrm{eq}}\left(t\right)$ is calculated:

$${\alpha }_{\mathrm{eq}}\left(t\right)=\left\{\varrho \left[-\tilde{G}\left(t\right)-\delta \left(e\right)\right]+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right\}/\varrho \chi$$

(44)

On the basis of the sliding mode accessibility, it satisfies:

$$s\left(t\right)\dot{s}\left(t\right)<-\kappa \left|s\left(t\right)\right|$$

(45)

where $\kappa >0$.

Substituting (22), (43), (44) and ${\alpha }_{\mathrm{aux}}\left(t\right)={\alpha }_{\mathrm{eq}}\left(t\right)+{\alpha }_{\mathrm{sw}}\left(t\right)$ to (45), (45) is further calculated:

$$\mathrm{sign}\left[s\left(t\right)\right]{\alpha }_{\mathrm{sw}}\left(t\right)>\left[\kappa -\varrho \tilde{G}\left(t\right)\mathrm{sign}\left(s\right)\right]/\varrho \chi$$

(46)

where $\left|\tilde{G}\left(t\right)\right|<g_{\mathrm{i}}$, $g_{\mathrm{i}}>0$, and $g_i$ is the unknown upper bound of $\tilde{G}\left(t\right)$.${\alpha }_{\mathrm{sw}}\left(t\right)$ can reduce the chattering $\tilde{G}\left(t\right)$. Then, the adaptive rate is chosen as the following:

$${\dot{\widehat{g}}}_{\mathrm{i}}=\varrho \left|s\left(t\right)\right|$$

(47)

Let $\left[\kappa -\varrho \tilde{G}\left(t\right)\mathrm{sign}\left[s\left(t\right)\right]\right]/\varrho \chi$ be equal to its maximum value $\left(\kappa +\varrho g_{\mathrm{i}}\right)/\varrho \chi$; ${\widehat{g}}_{\mathrm{i}}$ is an estimation value of $g_{\mathrm{i}}$, ${\alpha }_{\mathrm{sw}}\left(t\right)$ is rewritten:

$${\alpha }_{\mathrm{sw}}\left(t\right)=\left(\kappa +\varrho {\widehat{g}}_{\mathrm{i}}\right)\mathrm{sign}\left[s\left(t\right)\right]/\varrho \chi$$

(48)

The whole controller (IPDAGNFTSMC-NESO) is obtained:

$$\begin{gathered} \alpha \left(t\right)=\left[-\widehat{G}\left(t\right)+{{\dot{T}}_1}^{*}\left(t\right)+\delta \left(e\right)\right]/\chi +{\alpha }_{\mathrm{aux}}\left(t\right) \\ {\alpha }_{\mathrm{aux}}\left(t\right)={\alpha }_{\mathrm{eq}}\left(t\right)+{\alpha }_{\mathrm{sw}}\left(t\right) \\ {\alpha }_{\mathrm{aux}}\left(t\right)=\left[-\varrho \delta \left(e\right)+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right]/\varrho \chi +\left(\kappa +\varrho {\widehat{g}}_{\mathrm{i}}\right)\mathrm{sign}\left[s\left(t\right)\right]/\varrho \chi \end{gathered}$$

(49)

3.4. Stability Analyses

Proof. 

The stability of NESO is proved:

Substituting (36) to (16), the derivative of (16) is:

$$\begin{gathered} {\dot{e}}_1\left(t\right)={\dot{z}}_1\left(t\right)-{\dot{T}}_1\left(t\right) \\ {\dot{e}}_1\left(t\right)={\dot{z}}_1\left(t\right)-{\dot{T}}_1\left(t\right)=e_2\left(t\right)-{\beta }_1{e}_1\left(t\right) \\ {\dot{e}}_2\left(t\right)={\dot{z}}_2\left(t\right)-\dot{G}\left(t\right)=-{\beta }_2{\left|e_1\left(t\right)\right|}^{1/2}\mathrm{sign}\left[e_1\left(t\right)\right]-\dot{G}\left(t\right) \end{gathered}$$

(50)

Define:

$$\begin{gathered} {\phi }_1\left(t\right)=e_1\left(t\right) \\ {\phi }_2\left(t\right)=e_2\left(t\right)-{\beta }_1{e}_1\left(t\right) \end{gathered}$$

(51)

Substituting (50) to (51), the derivative of (51) is:

$$\begin{gathered} {\dot{\phi }}_1\left(t\right)={\phi }_2\left(t\right) \\ {\dot{\phi }}_2\left(t\right)=-{\beta }_2{\left|e_1\left(t\right)\right|}^{1/2}\mathrm{sign}\left[e_1\left(t\right)\right]-\dot{G}\left(t\right)-{\beta }_1{\phi }_2\left(t\right) \end{gathered}$$

(52)

Define the Lyapunov function:

$$V\left(t\right)={{\displaystyle \int }}_0^{{\phi }_1}2{\beta }_2{\left|\varphi \right|}^{1/2}\mathrm{sign}\left(\varphi \right)\mathrm{d}\varphi +{\phi }_2^2\left(t\right)$$

(53)

According to the Mean Value Theorem, (53) is further calculated:

$$V\left(t\right)=2{\beta }_2{\left|\aleph \right|}^{1/2}\mathrm{sign}\left(\aleph \right){\phi }_1\left(t\right)+{\phi }_2^2\left(t\right)$$

(54)

where $\exists \aleph \in \left[0,{\phi }_1\right]$ or $\exists \aleph \in \left[{\phi }_1,0\right]$. Because of ${\beta }_2>0$ and $\mathrm{sign}\left(\aleph \right)$ being an odd function, (45) satisfies $V>0$.

The derivative of (53) is:

$$\dot{V}\left(t\right)=2{\beta }_2{\left|{\phi }_1\right|}^{1/2}\mathrm{sign}\left[{\phi }_1\left(t\right)\right]{\dot{\phi }}_1\left(t\right)+2{\phi }_2\left(t\right){\dot{\phi }}_2\left(t\right)$$

(55)

Substituting (52) to (55), (55) is further calculated:

$$\begin{gathered} \dot{V}=2{\beta }_2{\left|{\phi }_1\left(t\right)\right|}^{1/2}\mathrm{sign}\left[{\phi }_1\left(t\right)\right]{\phi }_2\left(t\right)+2{\phi }_2\left(t\right)\left(-{\beta }_2{\left|{\phi }_1\left(t\right)\right|}^{1/2}\mathrm{sign}\left[{\phi }_1\left(t\right)\right]-\dot{G}\left(t\right)-{\beta }_1{\phi }_2\left(t\right)\right) \\ =2{\phi }_2\left(t\right)\left(-\dot{G}\left(t\right)-{\beta }_1{\phi }_2\left(t\right)\right) \end{gathered}$$

(56)

where $\dot{G}\left(t\right)>0,{\beta }_1>0,\left|{\phi }_2\left(t\right)\right|>\dot{G}\left(t\right)/{\beta }_1$ or $\dot{G}\left(t\right)<0,{\beta }_1>0,\left|{\phi }_2\left(t\right)\right|>-\dot{G}\left(t\right)/{\beta }_1$.

Hence, under parameter gain ${\beta }_1>0$ and inequation $\left|{\phi }_2\left(t\right)\right|>\left|\dot{G}\left(t\right)\right|/{\beta }_1$, (56) satisfies $\dot{V}<0$ and NESO is stable. Then, the observation states $z_1\left(t\right)$ and $z_2\left(t\right)$ converge to the actual states $T_1\left(t\right)$ and $G\left(t\right)$, respectively. □

Proof. 

The stability of controller 1 (GNFTSMC controller) is proven as follows:

Define the Lyapunov function:

$$V\left(t\right)=s^2\left(t\right)/2>0$$

(57)

Substituting (14), (20), (22), (35) to (57), the derivative of (57) is:

$$\begin{gathered} \dot{V}=s\left(t\right)\left[\varrho \dot{e}\left(t\right)+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right] \\ \dot{V}=s\left(t\right)\left\{\varrho \left[{{\dot{T}}_1}^{*}\left(t\right)-{\dot{T}}_1\left(t\right)\right]+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right\} \\ \dot{V}=s\left(t\right)\left\{\varrho \left\{{{\dot{T}}_1}^{*}\left(t\right)-G\left(t\right)-\alpha \left(t\right)U_I^2∕\left\{R\pi \left[cm+A\left(\beta +\lambda \right)\right]\right\}\right\}+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right\} \\ \dot{V}=s\left(t\right)\left\{-\varrho \tilde{G}\left(t\right)-\eta \mathrm{sign}\left[s\left(t\right)\right]\right\} \end{gathered}$$

(58)

Therefore, under assumption, the observation error of unknown disturbances is bounded and satisfies $\left|\tilde{G}\left(t\right)\right|<\eta /\varrho$, (57) and (58) satisfy $V>0$ and $\dot{V}<0$, then the GNFTSMC controller is stable. □

Proof. 

The stability of controller 3 (IPDAGNFTSMC-NESO) is proven as follows:

Define the Lyapunov function:

$$V\left(t\right)=s^2\left(t\right)/2+{{\tilde{g}}_{\mathrm{i}}}^2/2>0$$

(59)

where ${\tilde{g}}_{\mathrm{i}}=g_{\mathrm{i}}-{\widehat{g}}_{\mathrm{i}}$.

The derivative of (59) is as follows:

$$\dot{V}\left(t\right)=s\left(t\right)\dot{s}\left(t\right)+{\tilde{g}}_{\mathrm{i}}{\dot{\tilde{g}}}_{\mathrm{i}}$$

(60)

Substituting (22), (43), (47), (49) and ${\tilde{g}}_{\mathrm{i}}=g_{\mathrm{i}}-{\widehat{g}}_{\mathrm{i}}$ to (60), (60) is further calculated:

$$\begin{gathered} \dot{V}\left(t\right)=s\left(t\right)\left[\varrho \dot{e}\left(t\right)+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right]-\varrho \left|s\left(t\right)\right|\left(g_{\mathrm{i}}-{\widehat{g}}_{\mathrm{i}}\right) \\ \dot{V}\left(t\right)=s\left(t\right)\left\{\varrho \left[-\tilde{G}\left(t\right)-\delta \left(e\right)-\chi {\alpha }_{\mathrm{aux}}\left(t\right)\right]+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right\}-\varrho \left|s\left(t\right)\right|g_{\mathrm{i}}+\varrho \left|s\left(t\right)\right|{\widehat{g}}_{\mathrm{i}} \\ \dot{V}\left(t\right)=s\left(t\right)\left\{\varrho \left\{-\tilde{G}\left(t\right)-\delta \left(e\right)-\chi \left\{\begin{array}{c}\left[-\varrho \delta \left(e\right)+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right]/\varrho \chi \\ +\left(\kappa +\varrho {\widehat{g}}_{\mathrm{i}}\right)\mathrm{sign}\left[s\left(t\right)\right]/\varrho \chi \end{array}\right\}\right\}+e\left(t\right)+\mu e{\left(t\right)}^{m/n}\right\}-\varrho \left|s\left(t\right)\right|g_{\mathrm{i}}+\varrho \left|s\left(t\right)\right|{\widehat{g}}_{\mathrm{i}} \\ \dot{V}=s\left(t\right)\left[-\varrho \tilde{G}\left(t\right)-\left(\kappa +\varrho {\widehat{g}}_{\mathrm{i}}\right)\mathrm{sign}\left[s\left(t\right)\right]\right]-\varrho \left|s\left(t\right)\right|g_{\mathrm{i}}+\varrho \left|s\left(t\right)\right|{\widehat{g}}_{\mathrm{i}} \\ =-\varrho \tilde{G}\left(t\right)s\left(t\right)-\kappa \left|s\right|-\varrho \left|s\left(t\right)\right|g_{\mathrm{i}} \\ <-\varrho g_{\mathrm{i}}s\left(t\right)-\kappa \left|s\right|-\varrho \left|s\left(t\right)\right|g_{\mathrm{i}} \\ \le \varrho g_{\mathrm{i}}\left|s\left(t\right)\right|-\kappa \left|s\right|-\varrho \left|s\left(t\right)\right|g_{\mathrm{i}} \\ =-\kappa \left|s\right|<0 \end{gathered}$$

(61)

Therefore, (59) and (61) satisfy $V>0$, and $\dot{V}<0$; the stability is proved. □

4. Simulation Results

In order to clearly validate superiorities of this paper proposed control method (IPDAGNFTSMC-NESO controller) over other control methods (GNFTSMC controller and IPD-NESO controller), corresponding simulation results are given in Figure 7, Figure 8 and Figure 9. Specifically, the detailed parameters of TSTQLH are listed: $c=712 \mathrm{J}/\mathrm{kg}·\mathrm{K}$, $m=9.192 \mathrm{g}$, $A=3.77\ast {10}^{-2} {\mathrm{m}}^2$, $\beta =2743.9 \mathrm{w}/{\mathrm{m}}^2·\mathrm{K}$, $\lambda =110.6 \mathrm{w}/\mathrm{m}·\mathrm{K}$, $\epsilon =0.97$, $\sigma =5.67\ast {10}^{-8} \mathrm{w}/{\mathrm{m}}^2·{\mathrm{K}}^4$, $F=1$, $R=193 \mathrm{\Omega }$, $U_{\mathrm{I}}=220 \mathrm{V}$. The control parameters of IPDAGNFTSMC-NESO controller are selected: $\varrho =5$, $\mu =500$, $m=5$, $n=7$, $\kappa =20$, $\chi =-60$, ${\beta }_1=100$, ${\beta }_2=2000$, $K_{\mathrm{P}}=-80$, $K_{\mathrm{D}}=-2$, and $\eta =80$.

As is shown in Figure 7 and Figure 8, the fitting curves of reference temperature trajectory for a hypersonic missile in time sequence are chosen as the tracking targets, which include seven different fitting curves of wall 0_average, wall 1_7.75 m, wall 1_8.5 m, wall 1_9.1 m, wall 2_7.75 m, wall 2_8.5 m, wall 2_9.1 m. And line 1, line 2, line 3 are connected to controller 1 (GNFTSMC controller), controller 2 (IPD-NESO controller), controller 3 (IPDAGNFTSMC-NESO controller), respectively. Some details are explained in Section 3.

It can be seen from Figure 7 that the three control methods can track the target curves effectively. However, in Figure 7c,f,i,l,o,r,u, there are obvious overshoots in the GNFTSMC controller and the IPD-NESO controller within 0 s~4.0 s. The overshoot of the GNFTSMC controller is about 4%, and that of IPD-NESO control method is about 3.5%. Between 0 s and 4.0 s, there is no obvious overshoot in IPDAGNFTSMC-NESO controller, which has the shortest response time and the fastest convergence rate. It can be noticed clearly from Figure 8 that the chattering phenomenon of IPD-NESO controller is obvious from 0 s to 8.0 s, which is about 50 K. Although the chattering of the GNFTSMC controller is similar to that of the IPD-NESO controller, the amplitude of the GNFTSMC controller decreased significantly, so it is close to 1.0 K. By contrast, the IPDAGNFTSMC-NESO controller has great dynamic tracking performance in this process. Therefore, considering stability, rapidity, accuracy and overshoot, the IPDAGNFTSMC-NESO controller has the best control performance.

To further demonstrate the robustness and anti-interference capability of the proposed IPDAGNFTSMC-NESO controller, we designed additional simulations considering time-varying resistance in the system, specifically focusing on the resistance value of TSTQLH, which is known to fluctuate over time. In these tests, the variations in resistance (denoted as $aR$) were set as external disturbances, with values of $a=\left\{80\%, 100\%, 120\%\right\}$, representing both reduced and increased resistance compared to the nominal value. These disturbances were applied while maintaining the same IPDAGNFTSMC-NESO controller configuration. Figure 9 presents the simulation results of the system’s response under these external disturbances. Despite the variations in resistance, which would typically cause instability or performance degradation in traditional control systems, the IPDAGNFTSMC-NESO controller demonstrated remarkable resilience. The system output remained stable throughout the entire process, maintaining the desired dynamics with minimal deviation from the setpoint. This consistent performance, even in the face of significant external disturbances, clearly showcases the controller’s strong anti-interference capability. The results not only confirm the controller’s robustness against external disturbances but also emphasize its superior performance in terms of rapidity and stability. By effectively compensating for the time-varying resistance, the proposed IPDAGNFTSMC-NESO controller ensures that the system can maintain optimal performance under various uncertain conditions. This makes the controller a reliable solution for systems that are susceptible to environmental disturbances or parameter variations, further validating its practical applicability and robustness in real-world scenarios. The simulation results in Figure 9 illustrate that the IPDAGNFTSMC-NESO controller not only excels in terms of rapid response and system stability, but also exhibits exceptional anti-interference ability, ensuring the system operates effectively even when faced with external disturbances, such as time-varying resistance.

5. Conclusions

This paper provides an IPDAGNFTSMC-NESO control method for TSTQLH. Firstly, a kind of hypersonic missile is chosen as the calculated object in the finite element simulation. On the basis of analyses over the external flow field, the triangles method is employed with the same element size 10 mm to generate different element qualities of mesh metric. Under optimized conditions, seven fitting curves are obtained as reference temperature trajectories with different solvers. Secondly, the IPDAGNFTSMC-NESO controller is designed and it combines IPD with AGNFTSMC, which does not rely on the accurate mathematical model of TSTQLH. The AGNFTSMC, an auxiliary controller, has a GNFTSMS which can improve the convergence rate and avoid the singularity problem. In addition, an NESO, as an observer, can compensate the lumped disturbances from the local periodic oscillation, the high-order nonlinear output and time-varying uncertainties. The adaptive rate is designed to approach the upper bound of the uncertain part and suppresses the chattering caused by excessive switching gain. Finally, through the comparative simulations with other control methods, it was found that the proposed IPDAGNFTSMC-NESO method is superior in terms of stability, convergence speed, overshoot, and anti-interference ability.