Finite-Time Convergence Guidance Law for Hypersonic Morphing Vehicle

Abstract

Aiming at the interception constraint posed by defensive aircrafts against hypersonic morphing vehicles (HMVs) during the terminal guidance phase, this paper designed a guidance law with the finite-time convergence theory and control allocation methods based on the event-triggered theory, achieving evasion of the defensive aircraft and targeting objectives for a morphing vehicle in the terminal guidance phase. Firstly, this paper established the aircraft motion model; the relative motion relationships between HMV, defensive aircraft, and target; and the control equations for the guidance system. Secondly, a guidance law with finite-time convergence was designed, establishing a controller with the angle between the aircraft–target–defense aircraft triplet as the state variable and lift as the control variable. By ensuring the angle was non-zero, the aircraft maintained a certain relative distance from the defense aircraft, achieving evasion of interception. The delay characteristic of the aircraft’s flight controller was considered, analyzing its delay stability and applying control compensation. Thirdly, a multi-model switching control allocation method based on an event-triggered mechanism was designed. Optimal attack and bank angles were determined based on acceleration control variables, considering different sweep angles. Finally, simulations were conducted to validate the effectiveness and robustness of the designed guidance laws.

1. Introduction

Hypersonic glide vehicles typically refer to lift-type aircraft with flight speeds exceeding Mach 5 that can perform long-range unpowered gliding in near space using their aerodynamic capabilities [1]. These vehicles are characterized by high speed, wide flight airspace, and strong penetration capabilities, making them highly significant for development. To address the characteristics of large airspace, long time domain, wide speed range, and strong confrontation during the flight process, the concept of hypersonic morphing vehicles has emerged [2]. These vehicles can enhance their adaptability to different environments and conditions by changing their aerodynamic shapes, making them a key area for future research and improvement of hypersonic glide vehicles. Hypersonic morphing aircraft typically include two main methods of transformation: lower surface deformation [3,4] and wing morphing [5,6]. Lower surface deformation requires high demands on skin materials and can affect the payload capacity, posing engineering challenges in current applications. On the other hand, wing morphing is easier to engineer and effectively enhances the aerodynamic performance of the aircraft, making it a current focus of research.

When the hypersonic morphing vehicles approach a fixed ground target, they may face interception by defensive aircraft, which poses a risk to flight safety. To ensure that the vehicle reaches the target point safely and accurately, it is essential to study the following issue: guiding the vehicle to a specified terminal guidance target point while evading interception by defensive aircraft during the flight [7,8]. The complexity of this problem lies in ensuring that the vehicle can simultaneously evade intercepting aircraft while accurately guiding it towards the target, and in enhancing the vehicle’s flight performance through adjustments to the sweep angle.

Research on terminal guidance laws for hypersonic glide vehicles has always been a hot topic in the field of guidance. Due to the strong coupling and underactuated characteristics of these vehicles, the design of guidance laws is challenging. Current terminal guidance methods for hypersonic glide vehicles in addressing terminal penetration problems can be divided into three categories: optimal theory-based, nonlinear control-based, and machine learning-based approaches.

• Optimal theory-based. This guidance method category primarily utilizes the optimal theory to establish the Hamiltonian function based on flight constraints, obtaining the analytical solution of the control variables through the variational method and the principle of minimum. Kim [9] was the first to design an optimal angle guidance law for the longitudinal plane of a re-entry vehicle based on optimal control theory. Li [10] investigated a three-dimensional homing guidance problem of intercepting an evasive target with maneuverability comparable to the missile. Additionally, differential game guidance [11] and model predictive guidance [12] methods, derived from the optimal theory, have also become important research topics in the field of guidance. Liu [13] derived an optimal guidance law considering terminal interception and impact angle constraint.
• Nonlinear system control-based. This guidance method category originates from the modern control theory, establishing the system state equations based on the relative position relationships between the vehicles during the terminal guidance process. Control commands are designed using Lyapunov stability criteria to ensure the asymptotic stability of the guidance system, achieving target strikes by minimizing errors such as the line-of-sight angle. Anthony [14] designed a finite-time convergence guidance law under non-ideal conditions, reducing the required overload when the target maneuvers. Tang [15] designed a finite-time sliding mode guidance law, combining a disturbance observer to achieve terminal guidance with time constraints under disturbed conditions. Zhou [16] proposed a finite-time convergence sliding mode guidance method that can achieve terminal guidance with multiple constraints such as altitude, speed, and flight path angle. Zhang designed second-order sliding mode guidance laws considering actuator characteristics [17] and target movement characteristics [18], improving guidance accuracy under both conditions. Dong [19] proposed a guidance law considering terminal angle constraint and input saturation using the dynamic surface method.
• Machine learning-based. This guidance method category primarily utilizes learning models under specified conditions, designing reward functions based on flight missions to achieve vehicle guidance. Brian used reinforcement meta-learning methods to design an adaptive guidance law for terminal ballistic trajectories of hypersonic weapons, ensuring target hits and maneuver avoidance [20], and used reinforcement meta learning to optimize an adaptive guidance law [21]. Liu [22] designed a time-constrained guidance law combining deep neural networks within a predictive correction framework, optimizing guidance accuracy.

In summary, current research on terminal guidance for hypersonic glide vehicles mainly focuses on terminal constraints and guidance accuracy, with less consideration given to scenarios involving interception by defensive aircrafts. Due to the strong nonlinearity, high time variability, and uncontrollable speed characteristics of the flight process, researching such issues is challenging. The key to solving this problem lies in how to set interception constraints and achieve evasion. In the above methods, when considering the presence of defensive aircrafts on the battlefield, Method 1 can handle interception constraints through the design of interior point constraints in guidance problems, with a complex Hamiltonian function construction process requiring estimation of process variables. Method 2 can achieve evasion of defensive aircrafts by designing appropriate relative relationships between aircrafts, but the guidance process needs to be segmented. Although Method 3 can handle more complex environments, current algorithms primarily focus on scenarios with clear objectives and rewards. It requires relearning when problems or models change, and the method is heavily influenced by the designed reward function, indicating a need for further development.

Furthermore, during the terminal guidance process, it is still necessary to consider changes in the vehicle’s shape; a suitable configuration can enhance the vehicle’s flight performance. Designing an appropriate morphing method for the terminal guidance process is crucial to addressing this issue.

Aiming at the problem of morphing strategy in hypersonic morphing vehicles, Yao [23] designed a morphing method using Q-learning for fixed-sweep wing transformations, and realized trajectory planning for HMV by combining the predictive correction method. Hou [24] proposes an intelligent morphing decision method based on deep neural networking. Peng [2] solved the shape transformation scheme through multi-objective trajectory optimization using a decomposition-based multi-objective evolutionary algorithm. Dai [25] studied an offline integrated optimization problem of both wing morphing strategy and trajectory based on the Gauss pseudo-spectral method, which can fulfill actual task requirements and variable constraints.

It can be seen that the shape decision-making methods for hypersonic morphing vehicles are diverse, including but not limited to multi-objective optimization and machine learning. The multi-objective optimization method establishes the desired optimal flight performance and uses optimization techniques to determine the optimal shape change for different states. The machine learning method designs the required reward function based on the current flight environment, and, by taking the deformation of the vehicle as the learning action, it derives morphing strategies for different states. In order to adapt to the rapidity of the guidance law, this paper utilizes a multi-model event-triggered approach to implement shape transformation.

This study is divided into the following parts:

• Motion models for both the HMV and the defensive aircraft are established, and the relative motion relationships among the vehicle, the defensive aircraft, and the target point are developed. The control system equations for the guidance problem are also established.
• Guidance laws with finite-time convergence are designed and their stability is proven using the Lyapunov method. The impact of first-order actuator delay characteristics on the guidance laws is analyzed.
• A multi-model event-triggered switcher is designed to convert the acceleration commands derived from the guidance laws into commands for the angle of attack, bank angle, and sweep angle. Its stability is also proven.
• Simulations are conducted to validate the effectiveness and robustness of the guidance laws.

2. Guidance Model for Aircraft Vehicle

2.1. Aircraft Motion Model

The aircraft is composed of a body and foldable wings and adopts bank-to-turn (BTT) control with no thrust. The sweep angle of the wing can form three fixed sizes, χ₁ = 30°, χ₂ = 45°, and χ₃ = 80°, respectively, as shown in Figure 1.

The equations of the motion of the aircraft are established according to reference [26]. The following assumptions are considered:

• Earth is a homogeneous sphere;
• The aircraft is a mass point that satisfies the assumption of transient equilibrium;
• The sideslip angle β and the lateral force Z are both zero during flight;
• Earth’s rotation is not considered.

The equations of motion of the hypersonic morphing vehicle are given as follows:

$$\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{r}_M}{\mathrm{d}t}}=v_M\sin {\theta }_M\\ {\displaystyle \frac{\mathrm{d}{\lambda }_M}{\mathrm{d}t}}={\displaystyle \frac{v_M\cos {\theta }_M\sin {\psi }_M}{r_M\cos {\varphi }_M}}\\ {\displaystyle \frac{\mathrm{d}{\varphi }_M}{\mathrm{d}t}}={\displaystyle \frac{v_M\cos {\theta }_M\cos {\psi }_M}{r_M}}\\ {\displaystyle \frac{\mathrm{d}{v}_M}{\mathrm{d}t}}=-{\displaystyle \frac{D}{m_M}}-g\sin {\theta }_M=-a_{xM}\\ {\displaystyle \frac{\mathrm{d}{\theta }_M}{\mathrm{d}t}}={\displaystyle \frac{L\cos \sigma }{m{v}_M}}+\left({\displaystyle \frac{v_M}{r_M}}-{\displaystyle \frac{g}{v_M}}\right)\cos {\theta }_M=a_{yM}+\left({\displaystyle \frac{v_M}{r_M}}-{\displaystyle \frac{g}{v_M}}\right)\cos {\theta }_M\\ {\displaystyle \frac{\mathrm{d}{\psi }_M}{\mathrm{d}t}}={\displaystyle \frac{L\sin \sigma }{m{v}_M\cos {\theta }_M}}+{\displaystyle \frac{v_M}{r_M}}\cos {\theta }_M\sin {\psi }_M\tan {\varphi }_M=a_{zM}+{\displaystyle \frac{v_M}{r_M}}\cos {\theta }_M\sin {\psi }_M\tan {\varphi }_M\end{array}$$

(1)

where t is the time, r is the distance from the center of the Earth to the aircraft, λ is longitude, ϕ is latitude, v is the aircraft speed, θ is the flight path angle, ψ is the heading angle, L is lift, D is drag, α is the attack angle, σ is the bank angle, and g is the acceleration of gravity. The subscript M denotes a hypersonic morphing vehicle; a_xM, a_yM, and a_zM represent the aircraft’s accelerations in three directions, respectively. The equations of L, D, and g are as follows:

$$(\begin{array}{l}L=c_{l}qs\\ D=c_{d}qs\\ g=g_0{\left({\displaystyle \frac{r_0}{r}}\right)}^2\end{array}$$

(2)

where c_l and c_d are the lift and drag coefficients, respectively, and both are determined by α and χ; S is the reference area of the aircraft; q is the dynamic pressure; r₀ = 6371 km is the Earth’s radius; and g₀ = 9.8066 m/s² is the acceleration of gravity at the Earth’s surface.

The defense aircraft (DA) employs proportional guidance in both the longitudinal and lateral directions. The motion equations are as follows:

$$\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{r}_D}{\mathrm{d}t}}=v_D\sin {\theta }_D\\ {\displaystyle \frac{\mathrm{d}{\lambda }_D}{\mathrm{d}t}}={\displaystyle \frac{v_D\cos {\theta }_D\sin {\psi }_D}{r_D\cos {\varphi }_D}}\\ {\displaystyle \frac{\mathrm{d}{\varphi }_D}{\mathrm{d}t}}={\displaystyle \frac{v_D\cos {\theta }_D\cos {\psi }_D}{r_D}}\\ {\displaystyle \frac{\mathrm{d}{v}_D}{\mathrm{d}t}}=0\\ {\displaystyle \frac{\mathrm{d}{\theta }_D}{\mathrm{d}t}}={\displaystyle \frac{a_{yD}}{v_D}}=k_{yD}{\dot{q}}_{\theta MD}\\ {\displaystyle \frac{\mathrm{d}{\psi }_D}{\mathrm{d}t}}={\displaystyle \frac{a_{zD}}{v_D\cos {\theta }_D}}={\displaystyle \frac{k_{zD}{\dot{q}}_{\psi MD}}{\cos {\theta }_D}}\end{array}$$

(3)

where the subscript D denotes the defense aircraft, a_yD and a_zD represent the longitudinal and lateral control quantities of the defense aircraft, and k_yD and k_zD are the longitudinal and lateral proportional guidance coefficients. q_θMD and q_ψMD, respectively, denote the line-of-sight angles in pitch and yaw directions.

2.2. Relative Motion Model

The relative relationship among the aircraft, target, and defense aircraft is shown in Figure 2.

The expression for line-of-sight angular velocity as a function of aircraft-target distance in spherical coordinates is

$$\{\begin{array}{l}{q}_{\theta }=\arcsin \left(r_T/r_{MT}\cdot \sin A\right)-{\displaystyle \frac{\pi }{2}}\\ A=\arccos \left(\cos \left({\lambda }_M-{\lambda }_T\right)\cos {\varphi }_T\cos {\varphi }_M+\sin {\varphi }_T\sin {\varphi }_M\right)\\ r_{MT}=\sqrt{r_T^2+r_M^2-2r_T{r}_M\left(\cos \left({\lambda }_M-{\lambda }_T\right)\cos {\varphi }_T\cos {\varphi }_M+\sin {\varphi }_T\sin {\varphi }_M\right)}\\ q_{\psi }=\arcsin \left({\displaystyle \frac{\sin \left({\varphi }_T-{\varphi }_M\right)}{\sin \left(\arccos \left(\cos ({\varphi }_T-{\varphi }_M)\cos ({\lambda }_T-{\lambda }_M)\right)\right)}}\right)\end{array}$$

(4)

where q_θ and q_ψ represent the line-of-sight angles in pitch and yaw directions at time t, respectively.

According to the relative motion relationship between the aircraft and the target, the derivatives of relative distance and line-of-sight angle are given by Equation (5). The relative relationship between the aircraft and the defense aircraft (DA) is similar and is not further elaborated here.

$$\{\begin{array}{l}{\dot{r}}_{MD}=v_D\cos \left(q_{\psi }-{\psi }_D\right)\cos \left(q_{\theta }-{\theta }_D\right)-v_M\cos \left(q_{\psi }-{\psi }_M\right)\cos \left(q_{\theta }-{\theta }_M\right)\\ {\dot{q}}_{\theta }{r}_{MD}=v_D\cos \left(q_{\psi }-{\psi }_D\right)\sin \left(q_{\theta }-{\theta }_D\right)-v_M\cos \left(q_{\psi }-{\psi }_M\right)\sin \left(q_{\theta }-{\theta }_M\right)\\ {\dot{q}}_{\psi }{r}_{MD}\cos q_{\theta }=v_D\sin \left(q_{\psi }-{\psi }_D\right)\cos {\theta }_D-v_M\cos {\theta }_M\sin \left(q_{\psi }-{\psi }_M\right)\end{array}$$

(5)

Due to the short duration of engagement between the flight and defender, it is feasible to simplify the consideration to the longitudinal plane for the relative position model. For stationary targets, the relative positional relationship is shown as Figure 3.

In Figure 3, q_T represents the line-of-sight angle between the aircraft and the defense aircraft, with the aircraft pointing counterclockwise towards the defense aircraft being positive. For q_T, it is defined as

$$\{\begin{array}{l}{q}_T=q_{DT}-q_{MT}\\ {\dot{q}}_T={\displaystyle \frac{v_D}{r_{DT}}}\sin \left({\theta }_D-q_{DT}\right)-{\displaystyle \frac{v_M}{r_{MT}}}\sin \left({\theta }_M-q_{MT}\right)\\ {\ddot{q}}_T={\displaystyle \frac{{\dot{v}}_D}{r_{DT}}}\sin \left({\theta }_D-q_{DT}\right)+{\displaystyle \frac{v_D}{r_{DT}}}\cos \left({\theta }_D-q_{DT}\right)\left({\dot{\theta }}_D-{\dot{q}}_{DT}\right)-{\displaystyle \frac{v_D{\dot{r}}_{DT}}{r_{DT}}}\sin \left({\theta }_D-q_{DT}\right)\\ -{\displaystyle \frac{{\dot{v}}_M}{r_{MT}}}\sin \left({\theta }_M-q_{MT}\right)-{\displaystyle \frac{v_M}{r_{MT}}}\cos \left({\theta }_M-q_{MT}\right)\left({\dot{\theta }}_M-{\dot{q}}_{MT}\right)+{\displaystyle \frac{v_M{\dot{r}}_{MT}}{r_{MT}}}\sin \left({\theta }_M-q_{MT}\right)\end{array}$$

(6)

In the context of relative motion relationships, the third term in the equation can be written as

$$\{\begin{array}{l}{\ddot{q}}_T=L\left[{\displaystyle \frac{1}{L/D}}{\displaystyle \frac{\sin \left({\theta }_m-q_{MT}\right)}{m_M{r}_{MT}}}-{\displaystyle \frac{\cos \sigma \cos \left({\theta }_m-q_{MT}\right)}{m_M{r}_{MT}}}\right]+n_D{\displaystyle \frac{g\cos \left({\theta }_D-q_{DT}\right)}{r_{DT}}}+K_q\\ K_q=-{\displaystyle \frac{{\dot{q}}_{DT}{v}_D}{r_{DT}}}\cos \left({\theta }_D-q_{DT}\right)+{\displaystyle \frac{v_M}{r_{MT}}}\cos \left({\theta }_M-q_{MT}\right)\left[\left({\displaystyle \frac{v_M}{r_M}}-{\displaystyle \frac{g}{v_M}}\right)\cos {\theta }_M-{\dot{q}}_{MT}\right]\\ -{\displaystyle \frac{v_D{\dot{r}}_{DT}}{r_{DT}}}\sin \left({\theta }_D-q_{DT}\right)+{\displaystyle \frac{v_M{\dot{r}}_{MT}}{r_{MT}}}\sin \left({\theta }_M-q_{MT}\right)\end{array}$$

(7)

where n_D = a_yD/g represents the overload of the defense aircraft.

Let q_Td denote the desired q_T, and when q_Td varies linearly with a slope of k_qt. The tracking error and derivative of q_T are

$$\{\begin{array}{l}{e}_q=q_T-q_{Td}\\ {\dot{e}}_q={\dot{q}}_T-{\dot{q}}_{Td}={\dot{q}}_T-k_{qt}\\ {\ddot{e}}_q={\ddot{q}}_T-{\ddot{q}}_{Td}={\ddot{q}}_T\end{array}$$

(8)

The control model for the terminal guidance problem is derived as follows:

$$\begin{array}{l}\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=bu+d\end{array}\\ {\left[x_1,x_2\right]}^T={\left[e_q,e_q\right]}^T,u=L,\\ b={\displaystyle \frac{1}{L/D}}{\displaystyle \frac{\sin \left({\theta }_m-q_{MT}\right)}{m_M{r}_{MT}}}-{\displaystyle \frac{\cos \sigma \cos \left({\theta }_m-q_{MT}\right)}{m_M{r}_{MT}}},\\ d=n_D{\displaystyle \frac{g\cos \left({\theta }_D-q_{DT}\right)}{r_{DT}}}+K_q\end{array}$$

(9)

To ensure that the aircraft can evade the defense aircraft, it is sufficient to maintain q_T consistently at a certain angle δ.

3. Finite-Time Convergent Guidance Law Design

In Equation (9) of the control system, q_T changes very little but has a significant impact on the distance between the aircraft and the defense aircraft. It is necessary to ensure rapid tracking of q_T. Specifically, the steady-state error and convergence time of the second-order system model in Equation (9) need to be guaranteed. This paper will utilize finite-time control laws to achieve rapid and stable control of the system.

3.1. Design of the Control Law and Stability Proof

The controller designed for the model is as follows:

$$\{\begin{array}{l}u=b^{-1}\left(x_3^{*}-\mathrm{d}\right)\\ x_3^{*}=-{\beta }_{2}sig{\left({\xi }_2\right)}^{{\displaystyle \frac{1}{q_2}}},\\ {\xi }_2=sig{\left(x_2\right)}^{q_1}-sig{\left(x_2^{*}\right)}^{q_1}\\ x_2^{*}=-{\beta }_{1}sig{\left(x_1\right)}^{{\displaystyle \frac{1}{q1}}}\end{array}$$

(10)

where β_n is a positive definite function, 0 ≤ q₂ ≤ 1 ≤ q₁ ≤ 2, and sig(·)^a denotes the function sign(x)|x|^a. This control strategy ensures global finite-time convergence of the system. Next, stability will be proven using Lyapunov’s method.

Before proving stability, it is necessary to state the required lemma [27].

Lemma 1. 

For any system, if there exists a continuously differentiable function V satisfying the following conditions:

(1) V > 0;

(2) There exists a constant c > 0, a $\in$ (0,1), and an open neighborhood U₀ containing the origin in U, such that the following inequality holds:

$$\dot{V}\left(x\right)+c{V}^a\left(x\right)\le 0,x\in U_0$$

(11)

Then, the system is finite-time stable.

Lemma 2. 

If 0 < ρ < 1, ∀x,y $\in$ R, there exists

$$\left|x^{\rho }-y^{\rho }\right|\le 2^{1-\rho }{\left|x-y\right|}^{\rho }$$

(12)

Lemma 3. 

∀x,y $\in$ R, c > 0, d > 0, γ(x,y) > 0, there exists

$${\left|x\right|}^c{\left|y\right|}^d\le {\displaystyle \frac{c}{c+d}}\gamma {\left|x\right|}^{c+d}+{\displaystyle \frac{d}{c+d}}{\gamma }^{-{\displaystyle \frac{d}{c}}}{\left|y\right|}^{c+d}$$

(13)

Lemma 4. 

x_i ∈ R (i = 1, 2…, n), 0 < p ≤ 1, there exists

$${\left(\left|x_1\right|+\left|x_2\right|+\dots +\left|x_n\right|\right)}^p\le {\left|x_1\right|}^p+{\left|x_2\right|}^p+\dots +{\left|x_n\right|}^p\le n^{1-p}{\left(\left|x_1\right|+\left|x_2\right|+\dots +\left|x_n\right|\right)}^p$$

(14)

Next, prove the stability of the controller. Based on the lemma mentioned above, construct the Lyapunov function V = V₀ + V₁.

First, solve for V₀.

$$\{\begin{array}{l}{V}_0={\displaystyle \frac{1}{2}}{x}_1^2\\ {\dot{V}}_0={\displaystyle \frac{1}{2}}{\dot{x}}_1^2=x_1{x}_2=x_1{x}_2^{*}+x_1{x}_2-x_1{x}_2^{*}\end{array}$$

(15)

where

$$\begin{array}{l}{x}_1{x}_2^{*}=x_1{x}_2^{*}=-{\beta }_1{x}_{1}sig{\left(x_1\right)}^{{\displaystyle \frac{1}{q_1}}}\le -{\beta }_1{\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}\\ x_1{x}_2-x_1{x}_2^{*}=x_1\left(x_2-x_2^{*}\right)\\ \le \left|x_1\right|\left|x_2-x_2^{*}\right|=\left|x_1\right|\left|sig{\left(x_2^{q_1}\right)}^{{\displaystyle \frac{1}{q_1}}}-sig{\left(x_2^{*q_1}\right)}^{{\displaystyle \frac{1}{q_1}}}\right|\\ \le 2^{1-{\displaystyle \frac{1}{q_1}}}\left|x_1\right|{\left|{\xi }_2\right|}^{{\displaystyle \frac{1}{q_1}}}\le 2^{1-{\displaystyle \frac{1}{q_1}}}\left[{\displaystyle \frac{1}{1+\frac{1}{q_2}}}{\gamma }_0{\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}+{\displaystyle \frac{\frac{1}{q_2}}{1+\frac{1}{q_2}}}{\gamma }_0^{-{\displaystyle \frac{1}{q_1}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}\right]\\ =k_1{\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}+k_2{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}\end{array}$$

(16)

Then, solve for V₁, let

$$V_1\left(x_1,x_2\right)={\displaystyle {\int }_{x_2^{*}}^{x_2}sig{\left(sig{\left(s\right)}^{q_2}-sig{\left(x_2^{*}\right)}^{q_2}\right)}^{2-\frac{1}{q_2}}\mathrm{d}s}$$

(17)

Derive Equation (17) as

$${\dot{V}}_1={\displaystyle \frac{\partial V_1}{\partial x_1}}{\dot{x}}_1+{\displaystyle \frac{\partial V_1}{\partial x_2}}{\dot{x}}_2$$

(18)

For the first term in the equation, there is

$$\begin{array}{l}{\displaystyle \frac{\partial V_1}{\partial x_1}}{\dot{x}}_1={\displaystyle \frac{\partial V_1}{\partial x_2^{*q_2}}}{\displaystyle \frac{\partial x_2^{*q_2}}{\partial x_1}}{\dot{x}}_1=-\left(2-{\displaystyle \frac{1}{q_2}}\right){\displaystyle {\int }_{x_2^{*}}^{x_2}sig{\left(sig{\left(s\right)}^{q_2}-sig{\left(x_2^{*}\right)}^{q_2}\right)}^{1-\frac{1}{q_2}}\mathrm{d}s\left(-{\left({\beta }_1\right)}^{q_2}{x}_2\right)}\\ \le \left(2-{\displaystyle \frac{1}{q_2}}\right){\left({\beta }_1\right)}^{q_2}\left|sig\left(x_2\right)-sig\left(x_2^{*}\right)\right|{\left|sig{\left(x_2\right)}^{q_2}-sig{\left(x_2^{*}\right)}^{q_2}\right|}^{1-{\displaystyle \frac{1}{q_2}}}\left|x_2\right|\\ \le \left(2-{\displaystyle \frac{1}{q_2}}\right){\left({\beta }_1\right)}^{q_2}\cdot 2^{1-{\displaystyle \frac{1}{q_2}}}\left|{\xi }_2\right|\left|x_2-x_2^{*}+x_2^{*}\right|\\ \le \left(2-{\displaystyle \frac{1}{q_2}}\right){\left({\beta }_1\right)}^{q_2}\cdot 2^{1-{\displaystyle \frac{1}{q_2}}}\left(2^{1-{\displaystyle \frac{1}{q_2}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}+{\beta }_1{\left|x_1\right|}^{{\displaystyle \frac{1}{q_1}}}\left|{\xi }_2\right|\right)\\ \le \left(2-{\displaystyle \frac{1}{q_2}}\right){\left({\beta }_1\right)}^{q_2}\cdot 2^{1-{\displaystyle \frac{1}{q_2}}}\left(2^{1-{\displaystyle \frac{1}{q_2}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}+{\beta }_1\left({\displaystyle \frac{1}{1+\frac{1}{q_2}}}{\gamma }_1{\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}+{\displaystyle \frac{\frac{1}{q_2}}{1+\frac{1}{q_2}}}{\gamma }_1^{-{\displaystyle \frac{1}{q_1}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}\right)\right)\\ \le k_3{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}+k_4{\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}\end{array}$$

(19)

For the second term in the equation, there is

$${\displaystyle \frac{\partial V_1}{\partial x_2}}{\dot{x}}_2={\left(sig{\left(x_2\right)}^{q_1}-sig{\left(x_2^{*}\right)}^{q_1}\right)}^{2-{\displaystyle \frac{1}{q_1}}}\left(bu+\mathrm{d}\right)\le -{\beta }_2{\left|{\xi }_2\right|}^{2-{\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}}\le 0$$

(20)

According to Equations (15)~(20), there is

$$\dot{V}={\dot{V}}_0+{\dot{V}}_1=\left(k_1+k_4-{\beta }_1\right){\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}+\left(k_2+k_3\right){\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}+{\beta }_2{\left|{\xi }_2\right|}^{2-{\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}}$$

(21)

For the term ξ₂ in the equation, if |ξ₂| < 1, then it holds that

$$\{\begin{array}{l}{\left|{\xi }_2\right|}^{2-{\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}}\ge {\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}\\ -{\beta }_2{\left|{\xi }_2\right|}^{2-{\displaystyle \frac{1}{q_1}}+{\displaystyle \frac{1}{q_2}}}\le -{\beta }_2{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}\end{array}$$

(22)

Equation (20) can be written as

$$\dot{V}={\dot{V}}_0+{\dot{V}}_1\le \left(k_1+k_4-{\beta }_1\right){\left|x_1\right|}^{1+{\displaystyle \frac{1}{q_1}}}+\left(k_2+k_3-{\beta }_2\right){\left|{\xi }_2\right|}^{1+{\displaystyle \frac{1}{q_1}}}$$

(23)

Since γ can take any positive value, it is sufficient to choose an appropriate γ such that k₁ + k₄ − β₁ < 0 and k₂ + k₃ − β₂ < 0. Equation (23) can be written as

$$\begin{array}{l}\dot{V}\le k_5{V}^{1+{\displaystyle \frac{1}{q_1}}}\\ k_5=\max \left(\left(k_1+k_4-{\beta }_1\right),\left(k_2+k_3-{\beta }_2\right)\right)\end{array}$$

(24)

Equation (24) satisfies the second method of Lyapunov. From this, it is proven that the system under the current control action is asymptotically stable and can converge to the desired value in finite time.

The control variable in the controller can be written as

$$u=b^{-1}\left(-{\beta }_{2}sig{\left(sig{\left(x_2\right)}^{q_1}+{\left({\beta }_1\right)}^{q_1}sig\left(x_1\right)\right)}^{1/q_2}-d\right)$$

(25)

From the control variable, it can be observed, that under the same state conditions, the control variable increases with the increase in the lift-to-drag ratio (L/D). This means that aircraft operating at higher L/D ratios need to generate larger overload to track the state variables. Additionally, the control variable increases with the increase in the overload of the defense aircraft, with an impact coefficient of gcos(θ_d − q_DT)/b/r_DT.

3.2. The Influence of a First-Order Characteristic Pilot Controller

The influence of the first-order dynamic characteristics of the aircraft’s autopilot on the control law in Equation (25) is considered. The first-order dynamics of the autopilot are given by

$${\dot{n}}_m=-{\displaystyle \frac{1}{\tau }}{n}_m+{\displaystyle \frac{1}{\tau }}{n}_u$$

(26)

where n_m represents the actual overload command generated by the aircraft, n_u represents the overload command provided to the autopilot by the aircraft, and τ represents the time constant.

Treating the first-order characteristic autopilot as a delay element, the state equation of the autopilot can be written as

$${\dot{n}}_m\left(t\right)=-{\displaystyle \frac{1}{\tau }}{n}_m\left(t\right)+{\displaystyle \frac{1}{\tau }}{n}_m\left(t-t_d\right)$$

(27)

where t_d represents the delay time of the autopilot command.

Next, the stability of the system is analyzed using the Lyapunov functional. First, the required lemma is provided.

Lemma 5. 

For the constant time-delay system equation

$$\dot{x}\left(t\right)=A_{1}x\left(t\right)+A_{2}x\left(t-t_d\right),\forall t\ge 0$$

(28)

where t_d > 0 represents the constant time delay.

If there exist positive definite matrices P > 0, Q > 0, and R > 0 such that the following conditions are satisfied:

$$\left[\begin{array}{ccc}P{A}_1+A_1^{T}P+Q-R& P{A}_2+R& d{A}_1^{T}R\\ 0& -Q-R& d{A}_2^{T}R\\ 0& 0& -R\end{array}\right]<0$$

(29)

Then, the system (28) is uniformly asymptotically stable. Since the system is uniformly asymptotically stable, the origin of the system must also be in an asymptotically stable state. According to the second method of Lyapunov, there must exist a Lyapunov functional that satisfies the second method. That is, there exist c > 0, a > 0, and V(t,x) satisfying Lemma 1.

The input to the autopilot is typically the overload. The state equation of the autopilot’s control system is

$$\begin{array}{l}\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=b{x}_3+\mathrm{d}\\ {\dot{x}}_3=-{\displaystyle \frac{1}{\tau }}{x}_3\left(t\right)+{\displaystyle \frac{1}{\tau }}{x}_3\left(t-t_{\mathrm{d}}\right)\end{array}\\ b=-{\displaystyle \frac{g\cos \left({\theta }_m-q_{MT}\right)}{r_{MT}}},\\ \mathrm{d}=n_D{\displaystyle \frac{g\cos \left({\theta }_D-q_{DT}\right)}{r_{DT}}}+K_q+D{\displaystyle \frac{\sin \left({\theta }_m-q_{MT}\right)}{m_M{r}_{MT}}}\end{array}$$

(30)

where x₃ = n_M = Lcosσ/mg.

For the term x₃, using Lemma 5, choosing any Q = R > 0 and P > 0 satisfies Equation (29). Therefore, the autopilot component is stable. By choosing constants c > 0 and a > 0, there must exist a Lyapunov function that satisfies Lemma 1, denoted as V₃. That is

$${\dot{V}}_3\left(x\right)\le -c{V}_3^a\left(x\right)$$

(31)

According to Equations (23) and (31), there is

$$\begin{array}{l}{\dot{V}}_1+{\dot{V}}_2+{\dot{V}}_3\left(x\right)=\dot{V}=k_5{V}^{1+1/q_1}-c{V}_3^a\left(x\right)\\ \le k_5{V}^a-c{V}_3^a\left(x\right)\left(\mathrm{or}{k}_5{V}^{1+1/q_1}-c{V}_3^{1+1/q_1}\left(x\right)\right)\\ \Rightarrow \dot{V}\le -\min (-k_5,c){\left(V_1+V_2+V_3\right)}^a\end{array}$$

(32)

Thus, it can be proven that the control system remains stable even in the presence of autopilot delay characteristics.

For the autopilot system x₃, after adding the input u(t) = kx(t) (k > 0), the time-delay system (33) remains stable.

$$\dot{x}\left(t\right)=A_{1}x\left(t\right)+A_{2}x\left(t-d\right)+Bu(t),\forall t\ge 0$$

(33)

To satisfy condition (29), it is sufficient to ensure that

$$-\left(P\left(A_1+Bk\right)+{\left(A_1+Bk\right)}^{T}P+Q-R\right)<0$$

(34)

For the autopilot system, choosing any Q = R > 0, P > 0, there is

$$Bk>{\displaystyle \frac{1}{\tau }}$$

(35)

Its stability proof is the same as above.

4. Configuration Switching Method Design

4.1. Configuration Switching Method

In Section 3, the acceleration control variables a_x, a_y, and a_z are obtained. However, during actual flight, the guidance loop needs to obtain control variables expressed in terms of angle of attack and bank angle. For variable-geometry aircraft, it is also necessary to obtain the sweep angle command. Therefore, at each step, after calculating the acceleration, it must be converted into angle commands. Then, based on the angle commands, the aircraft’s motion equations are integrated to determine the next step’s optimized command. This subsection will use the multi-model switching concept and event-triggered mechanism to achieve the allocation of angles to accelerations. In fact, both methods are control strategies for nonlinear systems. The model dealt with in this section is the aerodynamic interpolation model, not a conventional system model. Thus, only the switching method within the control function needs to be considered. The framework of multi-model switching is shown in Figure 4.

In Figure 4, after obtaining the three-directional accelerations, the corresponding lift coefficient c_l and drag coefficient c_d are calculated. These coefficients are then processed through different aerodynamic models 1, 2, and 3 under various sweep angles, yielding the angles of attack for each model. The results from the three models are then fed into the switching strategy module to obtain the optimal angle of attack α* under the current conditions. The most crucial aspect is the design of the switching strategy.

The model switching needs to follow the following conditions:

• For the switching window, the window for the same model remains open at all times. After the aircraft completes one sweep change, the windows for other models are closed for a transition stabilization time t_s.
• The controller operates as a zero-order hold, meaning if two time points are not consecutive, the controller retains the command from the previous time point until the next control time point arrives.
• There is no delay in the execution process.

The key points of the switching algorithm are as follows:

• The purpose of the switching algorithm is to obtain the sweep angle, angle of attack, and bank angle.
• The switching of the sweep angle must ensure the stability of the aircraft’s wing deformation.
• The bank angle must satisfy the allowable range of variation.

In this section, the switching strategy consists of event triggers corresponding to three aerodynamic models. The output of the event trigger is the trigger performance, and the dynamic event trigger mechanism is designed as follows:

$$t_{k+1}=t_k+w\min \left\{t|J_{\alpha }\left(t_k\right)-a\left(t_k\right)>0\right\}$$

(36)

where t_k and t_k+1 are two consecutive time points; w denotes the window period, representing the stabilization time of the aircraft’s attitude after the last sweep angle change. The event trigger window can only open after the aircraft’s sweep angle stabilizes. When the trigger window is open, w = 1; when the window is closed, w = 0. J(t_k) represents the performance evaluation function of the aerodynamic model input at time t_k; a(t_k) > 0 is the trigger condition. In event-triggered mechanisms, the Zeno phenomenon may occur, leading to repeated fluctuations in the vehicle’s sweep angle within a short period, which can compromise the vehicle’s stability. In this section, the design of the trigger window introduces a time delay, ensuring that the vehicle’s swept wings do not change within a certain time frame. This approach prevents the occurrence of the Zeno phenomenon and ensures the stability of the vehicle during flight.

4.2. Design of Performance Function

In the trigger, the evaluation function Jiα represents the advantage of converting acceleration commands into angle of attack commands under different aerodynamic models. Let α_iy be the angle of attack obtained from α_y for module i. The evaluation function J considers the influence of n_D. Assuming the defense aircraft adopts a proportional guidance law, the derivative of n_D with respect to ignoring lateral maneuvers is given by

$$\begin{array}{l}{\dot{n}}_D={\dot{a}}_{yD}/g=\left(k_{yD}{v}_D{\ddot{q}}_{MD}\right)/g\\ ={\displaystyle \frac{v_D}{g{r}_{MD}}}\left({\dot{\theta }}_D-{\dot{q}}_{MD}\right)\cos \left({\theta }_D-q_{MD}\right)-{\displaystyle \frac{v_D{\dot{r}}_{MD}}{g{r}_{MD}}}\sin \left({\theta }_D-q_{MD}\right)\\ -{\displaystyle \frac{{\dot{v}}_M}{g{r}_{MD}}}\sin \left({\theta }_M-q_{MD}\right)-{\displaystyle \frac{v_M}{g{r}_{MD}}}\left({\dot{\theta }}_M-{\dot{q}}_{MD}\right)\cos \left({\theta }_M-q_{MD}\right)\\ +{\displaystyle \frac{v_M{\dot{r}}_{MD}}{g{r}_{MD}}}\sin \left({\theta }_M-q_{MD}\right)-{\displaystyle \frac{{\dot{q}}_{MD}{\dot{r}}_{MD}}{g}}\\ =n_D{\displaystyle \frac{\cos \left({\theta }_D-q_{MD}\right)}{r_{MD}}}+L\left[{\displaystyle \frac{1}{L/D}}{\displaystyle \frac{\sin \left({\theta }_M-q_{MD}\right)}{mg{r}_{MD}}}-{\displaystyle \frac{\cos \sigma \cos \left({\theta }_M-q_{MD}\right)}{mg{r}_{MD}}}\right]+K_a\\ K_a=-{\displaystyle \frac{{\dot{q}}_{MD}{v}_D}{g{r}_{MD}}}\cos \left({\theta }_D-q_{MD}\right)-{\displaystyle \frac{v_D{\dot{r}}_{MD}}{g{r}_{MD}}}\sin \left({\theta }_D-q_{MD}\right)+{\displaystyle \frac{v_M{\dot{q}}_{MD}}{g{r}_{MD}}}\cos \left({\theta }_M-q_{MD}\right)\\ +{\displaystyle \frac{v_M{\dot{r}}_{MD}}{g{r}_{MD}}}\sin \left({\theta }_M-q_{MD}\right)-{\displaystyle \frac{v_M}{g{r}_{MD}}}\left({\displaystyle \frac{v_M}{r_M}}-{\displaystyle \frac{g}{v_M}}\right)\cos \left({\theta }_M-q_{MD}\right)\cos {\theta }_M-{\displaystyle \frac{{\dot{q}}_{MD}{\dot{r}}_{MD}}{g}}\end{array}$$

(37)

It can be observed that the defense aircraft overload increases with the increase in the aircraft overload, which complies with the guidance law. At the same time, it can be seen that for the same lift magnitude, a smaller lift-to-drag ratio induces larger defense aircraft overloads, which is unfavorable for defense interception but advantageous for aircraft breakthrough. Therefore, lift-to-drag ratio-related terms can be added to the performance function J in the control allocation method. A smaller lift-to-drag ratio will yield a smaller performance index. Designing J in this scenario could be expressed as

$$J=e^{-k_{Di}{\displaystyle \frac{D\left({\alpha }_{iy}\right)}{mg}}}$$

(38)

where the weight k_Di > 0, D() represents the drag corresponding to the current angle of attack. After the acceleration commands pass through the triggers, they then go through the minimum performance function module, where the feasible configuration that produces the minimum drag is selected for the next flight simulation step.

4.3. Stability of Multi-Model Switching

Due to the continuous switching of aerodynamic models among three different sweep angles, the controller also changes whenever the model switches. Various aerodynamic shapes lead to different coefficient matrices, generating different subsystems. It is necessary to demonstrate the stability of the controller when switching between subsystems. Reference [28] indicates that, for a switchable system composed of s subsystems, if each subsystem is stable and remains within one subsystem T, the stability of the system can be ensured. For each stable subsystem i, there exists a Lyapunov function:

$$a_i{\Vert x\Vert }^2\le V_i\left(x\right)\le b_i{\Vert x\Vert }^2,\dot{V}\left(x\right)={\displaystyle \frac{\partial V}{\partial x}}{f}_i\left(x\right)\le -c_i{\Vert x\Vert }^2$$

(39)

where a_i > 0, b_i > 0, c_i > 0. Equation (39) can be written as

$${\displaystyle \frac{\partial V}{\partial x}}{f}_i\left(x\right)\le -{\displaystyle \frac{c_i}{b_i}}{V}_i\left(x\right)$$

(40)

The upper bound of V_i(x) is

$$V_i\left(x\left(t+T\right)\right)\le e^{-{\displaystyle \frac{c_i}{b_i}}\tau }{V}_i\left(x\left(t\right)\right)$$

(41)

As long as the conditions in Equation (42) are satisfied,

$$T\le \underset{i,j}{\sup }\left(-{\displaystyle \frac{b_i}{c_i}}\ln {\displaystyle \frac{a_i}{b_i}}\right)$$

(42)

For all subsystems, it is sufficient to decrease the Lyapunov function value to a small enough level, ensuring the stability of the entire system. Physically, this implies that, as long as a certain dwell time is allocated for each subsystem, the stability of the controller system can be guaranteed. In the context of multi-model switching strategies, the time window of event triggers serves the same purpose to ensure the stability of each subsystem, analogous to the dwell time in multiple Lyapunov functions.

5. Simulation

5.1. Algorithm Effectiveness Simulation

Firstly, verify the effectiveness of the controller. The controller parameters are q₁ = 0.9, q₂ = 1.5, β₁ = 2.3, β₂ = 4, t_d = 0.05, τ = 0.05, k = 1.05. The control performance under step response conditions is shown in Figure 5.

The simulation separately verifies the finite-time convergence controller (FCC), the finite-time controller with added delay elements, and the controller with added delay compensation. It can be seen that, under the criterion of 1% steady-state error, the finite-time convergence controller completes the step response of the controller after 1.12 s. After adding the control signal delay element, this time increases to 1.8 s. With further delay compensation, the response time is shortened to 1 s. This demonstrates the effectiveness of the controllers in Section 3.

Next, simulations conduct terminal guidance under the following four methods:

• The aircraft has fixed 45° swept wings and uses proportional navigation (PN) law to reach the target.
• The aircraft changes the sweep angle, using finite-time convergence controller commands to evade defensive aircrafts and then fly to the target.
• The aircraft changes the sweep angle, considering the time delay characteristic of the first-order pilot in the finite-time convergence controller (TDFCC) commands to evade the defensive aircraft and reach the target.
• The aircraft changes the sweep angle, adding delay compensation for the pilot time delay characteristic to the finite-time convergence controller (DCFCC).

The parameters for the simulation are as follows: the longitudinal and lateral proportional navigation coefficients for defensive aircrafts are k_yD = 5 and k_zD = 3, respectively. During the proportional navigation process, the aircraft uses longitudinal and lateral proportional navigation coefficients k_yM = 5 and k_zM = 3. The minimum separation distance between aircrafts is r_MDmin = 100 m. The conditions for executing controller commands are |q_t| < 0.1, r_md < 100 km, which avoid premature evasion interception to prevent energy waste. Assuming the starting time is t_q, the desired command q_td is

$$q_{td}\left(t\right)=0.01\left(q_t\left(t_q\right)-sign\left(q_t\left(t_q\right)\right)\right){\displaystyle \frac{t-t_q}{t_{fd}-t_q}}$$

(43)

where t_fd > t_q represents a time constant. This desired command ensures that the angle qt remains non-zero throughout the flight process, allowing the aircraft’s control commands to effectively evade defensive aircrafts. After successfully evading the defensive aircraft, the aircraft continues to use proportional navigation to reach the target. In the event trigger, k_L = 0.02.

The initial conditions for the aircraft at the start of terminal guidance are as follows: longitude λ₀ = 51°, latitude ϕ₀ = 5.1°, altitude h₀ = 30 km, initial velocity v_m0 = 3000 m/s, flight path angle θ₀ = 0°, heading angle ψ₀ = 86°. For sweep angles of 30°, 45°, and 80°, the corresponding overload constraints are approximately 15, 20, and 25, respectively. The defensive aircraft launch point is located at the target point, with a constant speed v_d = 1500 m/s. The longitudinal and lateral available overloads are 25 and 20. The target point is located at λ_T = 54°, ϕ_T = 5.3°, h_T = 0 km. After the aircraft-target distance falls below 100 m, the attack angle is set to 0°. The simulation results are as follows.

Figure 6 shows the three-dimensional trajectory. It can be seen that the aircraft can reach the target using all four methods. However, Trajectory 1 is intercepted by the defensive aircraft, while Trajectories 2~4 can evade the defensive aircraft successfully. The latter three methods achieve evasion by lowering the trajectory altitude. Although the trends are similar, there are still differences in the specific conditions.

Figure 7 shows the velocity curves. In Methods 2 to 4, the aircraft’s speed rapidly decreases after 45.6 s. Combined with the trajectory results, it can be seen that, under the controller’s commands, the aircraft quickly consumes its energy to lower the trajectory. The speed reduction slows down around 72 s, indicating that the aircraft has evaded the defensive aircraft and entered the final proportional navigation phase. In Method 1, the aircraft’s final speed is approximately 778 m/s. In Method 2, the final speed drops to 544 m/s, indicating that the evasion process consumes the aircraft’s energy. In Method 3, the final speed drops to 516 m/s, and the flight time increases from 164 s to 175 s. This shows that while the aircraft can safely reach the target when there is a lag characteristic, it requires more energy and time. After adding delay compensation, the speed variation is almost the same as in the no-delay case, with a final speed of 558 m/s, an 8% increase compared with the uncompensated case, and a flight time of 162 s (a 7% reduction).

Figure 8 shows the flight path angle curves. It can be seen that, after 45.6 s, the ballistic angle of the aircraft rapidly decreases under the controller’s action. After evading the defensive aircraft, the aircraft pulls up the trajectory again to reach the target. In Method 3, the descent angle is 46°, greater than the 32° and 34° in Methods 2 and 4. This is because, under the influence of the autopilot delay characteristic, the aircraft lowers its flight altitude more before pulling up, requiring a larger flight path angle to reach the target, resulting in a relatively larger descent angle. Figure 9 shows the variation of the heading angle. The lateral movement is small, and the heading angle varies within a small range. The figure also shows that the autopilot characteristics affect the lateral motion.

Figure 10 shows the variation of the distance between the HMV and the target. It can be seen that the distance ultimately converges to zero, reaching the target in all cases. Method 3 takes the longest time, while the proportional navigation case takes the shortest time, consistent with the speed variation patterns. Figure 11 shows the variation in the distance between the HMV and the defensive aircraft. It can be seen that, after adding the controller commands, the distance always remains outside the minimum safe distance. In Method 3, due to its delay characteristics, the minimum distance is the largest. This distance variation also demonstrates the effectiveness of the controller command method in evading the defensive aircraft.

Figure 12 and Figure 13 show the aircraft’s angle of attack and bank angle curves, both of which exhibit jumps due to shape changes. It can be seen that the angle of attack command after delay correction is nearly identical to the no-delay angle of attack command. Figure 14 shows the variation of the sweep angle. The initial sweep angle is 45°. After 10 s, when the switching window opens, the sweep angle immediately switches to 80° to generate minimal drag, which is beneficial for maintaining the aircraft’s energy. Between 40 s and 70 s, the aircraft switches its sweep angle four times to generate the required overload. At the time of 69 s, Method 3 produces a control allocation, resulting in a 30° sweep angle, while others switch to 45°. This indicates that the autopilot delay characteristic also affects shape change decisions.

Figure 15 shows the variation of the q_t angle. It can be seen that, when the aircraft uses PN to reach the target, q_t eventually becomes 0, indicating that it is collided by the defensive aircraft. Under the action of the controller, Methods 2~4 can ensure that the aircraft tracks near the desired angle, although some tracking error remains. This error is related to the changes in the environment and the defensive aircraft commands during the flight. Additionally, q_t, as a relational angle, has a sensitive variation characteristic, making stable tracking challenging. However, the commands generated by the controller enable q_t to change according to the desired trend, guiding the aircraft to evade the defensive aircraft. This demonstrates the effectiveness of the controller commands.

Figure 16 shows the aircraft’s longitudinal overload. It can be seen that different longitudinal overloads result in different shape commands. Particularly in Method 3, after 60 s when the shape switching window opens, a 30° sweep wing command is generated, resulting in a larger overload. Figure 17 shows the aircraft’s lateral overload. It varies within a small range and affects the heading angle. The overload increases rapidly at the last moment due to the rapid changes in terminal line-of-sight angle acceleration and the angle of attack command.

Figure 18 shows the defensive aircraft’s longitudinal overload. It can be seen that under the controller’s action, the defensive aircraft needs to maneuver quickly to collide the aircraft, requiring a high overload. Therefore, its command quickly reaches the maximum available overload and maintains it. Due to its limited flight capability, it cannot ensure that qt converges to 0, resulting in a miss. Figure 19 shows the defensive aircraft’s lateral overload. Similarly, as the two approach each other, the rapid changes in line-of-sight angle acceleration cause the aircraft’s lateral overload to increase.

The performance indicator variation of the event trigger in Method 4 is shown in Figure 20.

From the figure, it can be seen that the event trigger executed a total of five shape switches, corresponding to the sweep angle changes in Figure 14. At 10 s, the first event trigger window opened, and the aircraft’s shape switched for the first time to the 80° sweep wing corresponding to the maximum performance indicator. At 45.58 s, the aircraft entered the evasion state against the defensive aircraft, and, after initially lowering its altitude, the sweep wing switched for the second time to 30°. Subsequently, during the evasion process, there were two additional shape switches at 55.6 s and 65.6 s. At 75.6 s, following a successful evasion, the first window opened again, and the aircraft switched back to the 80° sweep wing, maintaining this configuration until reaching the target.

The Monte Carlo method is used to simulate 100 trajectories, where initial deviations are uniformly distributed between ±2% of the initial state (r, λ, ϕ, v, θ, ψ). The trajectories are shown in Figure 21, and the minimum distance distribution between HMV and DA is shown in Figure 22.

From the simulation, it can be seen that the minimum distance never falls below a safe distance of 600 m, indicating that all aircraft trajectories successfully evade defensive aircrafts and reach their targets. Additionally, trajectories vary slightly depending on the initial aircraft state during the evasion process, but all achieve evasion and strike objectives. These results validate the robustness of the method.

5.2. Comparison Simulation

5.2.1. Case 1

To verify the impact of the distance at which the aircraft begins to execute evasive maneuvers on the aircraft’s energy in the algorithm, this section conducts a comparison simulation of the algorithm’s performance. Using FCC, simulations are conducted for three trajectories with selected values of r_md < 80 km, r_md < 100 km, and r_md < 120 km. These are referred to as Trajectory 1 to Trajectory 3, respectively. The remaining parameters are kept constant. The simulation results are as follows.

Figure 23 shows the three-dimensional trajectory. It can be observed that both Trajectory 1 and Trajectory 2 successfully evade the defensive aircraft and reach the target point, while Trajectory 3 not only collides with the defensive aircraft’s trajectory but also fails to reach the target point. Figure 24 shows the velocity curves. Ignoring the collision during the process, the aircraft in Trajectory 3 stalls after 250 s, which prevents it from reaching the target point.

Figure 25 shows the distance between the HMV and target. Figure 26 show the distance between the HMV and DA. Combining the three-dimensional trajectories, it can be observed that the distances to the target for the first two trajectories eventually become zero, and the minimum distance to the defensive aircraft is greater than 100 m, indicating that the aircraft safely reach the target.

Figure 27 shows the aircraft’s longitudinal overload. This is the main factor affecting the aircraft’s energy. In Trajectory 3, the aircraft begins executing evasive maneuvers when it is 150 km away from the defensive aircraft, at 41 s. The premature evasive action consumes the aircraft’s energy, resulting in a decrease in speed, which leads to a collision with the defensive aircraft, preventing it from reaching the target point.

The above simulations indicate that executing evasive maneuvers too early can lead to a waste of the aircraft’s energy, potentially resulting in mission failure. Therefore, it is necessary to delay switching commands until the aircraft is within a certain distance.

5.2.2. Case 2

To investigate the performance of the guidance algorithm for HMV under limited available overload conditions, this section performs comparative simulations with different overload levels. The simulations include three scenarios, corresponding to Trajectory 1 through Trajectory 3. The overload values corresponding to different sweep angle configurations under varying conditions are shown in

Table 1. Available overloads under different sweep angle configurations.

Available Overload	χ1 = 30°	χ2 = 45°	χ3 = 80°
Trajectory 1	3	4	5
Trajectory 2	4.5	6	7.5
Trajectory 3	6	8	10

, r_md < 60 km; all other parameters held constant.

The simulation results are as follows.

Figure 28 shows the three-dimensional trajectories. It can be seen that none of the three trajectories collide with the defensive aircraft, but Trajectory 1 does not reach the target point, while Trajectory 2 and Trajectory 3 do reach the target. Figure 29 shows the velocity curves. Figure 30 and Figure 31 show the aircraft’s angle of attack and bank angle curves. Figure 32 shows the variation of the sweep angle. These three commands are obtained based on the vehicle’s guidance law and configuration switching method.

Figure 33 shows the variation of the distance between the HMV and target. Figure 34 shows the variation in the distance between the HMV and the defensive aircraft. Combining the three-dimensional trajectories, it can be seen that the minimum distance between the two aircraft is greater than zero, indicating no collision occurs. The distance to the target for Trajectory 2 and Trajectory 3 eventually becomes zero, indicating they reach the target. Trajectory 1 ends 19 km away from the target, meaning it does not reach the target.

Figure 35 shows the aircraft’s longitudinal overload. Figure 36 shows the aircraft’s lateral overload. Figure 37 shows the defensive aircraft’s longitudinal overload. Figure 38 shows the defensive aircraft’s lateral overload. Combining these four commands, it can be seen that when the ratio of the available overload of the HMV to that of the defensive aircraft is approximately 3:1, the finite-time convergence guidance law can ensure that the HMV avoids collision with the defensive aircraft and reaches the target, as shown by Trajectory 2. When the available overload of the HMV decreases, the aircraft can still avoid collision, but this process may cause a deviation in the trajectory, preventing the aircraft from reaching the target after evading interception, as shown by Trajectory 1. When the available overload of the HMV increases, the aircraft can still safely reach the target.

The above simulations demonstrate that the finite-time convergence controller can effectively guide aircraft to evade defensive aircrafts and reach targets. The principle involves maneuvering the aircraft to force defensive aircrafts to rapidly exceed their available overload capacity, making interception impossible and resulting in a miss. The method requires the aircraft to expend energy, but adjusting the aircraft’s sweep angle can reduce energy consumption during breakthrough maneuvers. The simulation results confirm the effectiveness and robustness of the methods, and the efficacy of the control allocation strategy employed during switching.

6. Conclusions

Aiming at the problem of penetration guidance for hypersonic morphing vehicles during terminal guidance when facing interception by defensive aircrafts, this paper conducts research on a guidance law based on finite-time convergence methods.

Firstly, based on the principle of finite-time convergence, a finite-time convergence guidance law is designed. The time-delay characteristic of the aircraft’s autopilot and implementing compensation are considered. Secondly, a control allocation method based on a multi-model event trigger is designed to solve optimal shape and angle control quantities according to acceleration commands. Finally, simulations are conducted to validate the algorithm, demonstrating the algorithm’s effectiveness and robustness.

The contributions and innovations of this paper are as follows:

• This paper introduces a guidance law based on finite-time convergence methods specifically designed for hypersonic morphing vehicles during terminal guidance. The finite-time convergence guidance law designed in this paper can evade interception by tracking the “aircraft–target–defensive aircraft” angle and reach the target, ensuring safety and efficiency. The development of a compensation algorithm that mitigates the impact of time delays on flight speed and mission performance is a significant advancement in enhancing the robustness of the guidance system.
• This paper proposes a control allocation method that utilizes a multi-model event-triggered mechanism to optimize shape and angle control quantities based on acceleration commands. The event-triggered multi-model switching controller designed in this paper can effectively map acceleration commands to angle and shape commands. The controller matches well with the guidance law, and the resulting “angle of attack–bank angle–sweep angle” commands. This method effectively translates acceleration commands into angle and shape adjustments, improving the vehicle’s performance in terms of energy efficiency and operational effectiveness.