Design of Head Pursuit Guidance Law Based on Fractional-Order Sliding Mode Theory

This paper addresses problems on the interception of hypersonic vehicles in near-space. The main contribution is to study a head pursuit guidance law based on fractional-order sliding mode theory and analyze the stability of the guidance law provided. Firstly, the fractional-order differential operator, which has characteristics such as fast convergence and memory, is introduced into the design of sliding mode surface, based on which a head pursuit guidance can be designed to improve the performance of the guidance system. The stability of this guidance law is proved by Lyapunov stability theory. Based on this, a head pursuit guidance law considering autopilot dynamic characteristic is designed and the stability is also analyzed. Finally, numerical simulations are presented and the results verify that the guidance laws designed in this paper can avoid overload saturation at the initial moment of the terminal guidance stage and improve the convergence speed.


INTRODUCTION
With the development of near-space technology, hypersonic vehicles have been further amplified because of the characteristics such as fast flight speed, strong maneuvering ability, long flight distance and strong penetration ability. The traditional tail-chase interception requires the missile to be faster than the target, which will cause severe aerothermal ablation, affect the performance of the seeker and thus reduce the interception accuracy. Although traditional head-on interception can reduce the speed requirement of missile and avoid aerothermal ablation, it causes the relative speed to be so fast that the remaining flight time would be too short.
In order to solve the problems above, Golan and Shima (2004) provided a head pursuit guidance law for the first time. Based on this, Xiao et al. (2013) proposed an optimal adaptive sliding mode guidance (SMG) law to intercept hypersonic vehicle, which adopted the optimal sliding mode theory. For the purpose of attenuating chattering, Liu et al. (2015) introduced double-power reaching law and provided a head pursuit guidance law. Si and Song (2017) introduced a fast double-power reaching law into the design of head pursuit guidance law, which can improve the approach speed of sliding mode variables. Taking into account the uncertain dynamics of missile, Zhang et al. (2018) presented a head pursuit guidance law based on time-scale separation and varied the robustness to system disturbance. Zhu and Mu (2019) designed a head pursuit guidance law based on adaptive sliding mode control theory to intercept targets with high speed. But the overload saturation problem exists in all guidance laws above.
Compared with traditional sliding mode control theory, fractional-order sliding mode theory introduces fractional differential operator on the basis of traditional sliding mode, so it not only has the robustness of traditional sliding mode control but also has the memory and genetic characteristics of information, which can improve the performance of control system effectively. Therefore, fractional-order sliding mode theory has been widely studied in recent years. Pashaei and Badamchizadeh (2020) presented a fractional-order extended disturbance observer to estimate the matched and mismatched disturbances and their derivatives, based on which a fractional-order sliding mode controller was designed. Simulation results indicated that the controller had features of fast response, chattering reduction, robust stability and so on. Fei and Wang (2020) designed a fractional-order current harmonic compensation controller by combining the fractional-order sliding mode theory and recurrent neural network, which improved the performance of the active power filter. For the purpose of modeling the magneto rheological damper based on semiactive vehicle suspensions more precisely and making sure the finite time convergence of the system, Nguyen et al. (2020) proposed a fractional-order derivative based sliding mode controller and the validity of this controller is verified by numerical simulation. In order to enhance the output power quality of permanent magnet synchronous generator, Xiong et al. (2020) proposed a fractional-order sliding mode control (FOSMC) method, which decided the controller coefficient boundaries and had stronger robustness, superior tracking precision and faster time response. Rabah and Ladaci (2020) adopted adjustable parameters in the designing of sliding surface and proposed an adaptive fractional-order sliding mode controller to solve the synchronization problem for nonlinear fractional-order systems with chaotic behavior. Rahmani and Rahman (2020) proposed a fractional-order sliding mode controller to improve track tracking performance of 7 degrees of freedom (DOFs) exoskeleton robot named ETS-MARSE, which incorporated the adaptive radial basis function neural network with fast fractional integral terminal sliding mode control and addressed the chattering phenomena. In order to mitigate sub-synchronous control interaction, Li et al. (2020) proposed a fractional-order sliding mode controller which adopted genetic algorithm to optimize control parameters. Simulation results manifested that the controller had strong robustness and better damping performance. Sharafian et al. (2020) studied the HIV mathematical dynamic model and proposed a fractional-order observer to estimate the uncertainties, which was verified to have strong robustness. With the continuous improvement of fractional-order sliding mode theory, it has been applied to the field of aerospace gradually. Wang and Lei (2010) proposed a fractional-order guidance law based on the proportion integration differentiation (PID) guidance law, which improves the accuracy of interceptor and has strong robustness. In order to solve the problem of attitude tracking control for flexible spacecraft, fractional sliding mode controllers are proposed to make the systems have strong robustness and rapid convergence (Deng and Song 2013). Until now, fractional-order sliding mode theory has not been widely used in aerospace field. Therefore, this paper proposed head pursuit guidance laws based on fractionalorder sliding mode theory for the interception of hypersonic vehicle.
The rest of this paper is organized as follows. In the upcoming section, some background information and preliminaries are introduced. Next, a head pursuit guidance law based on fractional-order sliding mode theory is designed and the stability is proved by Lyapunov stability theory. Furthermore, a fractional-order sliding mode head pursuit guidance law considering autopilot dynamic characteristics is designed and the stability is also analyzed. Finally, some numerical simulation comparisons in this study are summarized in the last part of the article. Figure 1 depicts the schematic view of head pursuit interception between target and missile in longitudinal plane, where points T and M denote the target and the missile, r and q, the relative missile-target range and the line-of-sight (LOS) angle, θ t and θ m the lead angles, V t and V m velocities, a t and a m the components of accelerations perpendicular to the velocities, the subscripts t and m denote the target and missile. The respective equations describing the relative motion dynamics become:

METHODOLOGY Problem formulation
Differentiating Eq. 2 with respect to time, it gets: According to the concept of head pursuit guidance law proposed by Golan and Shima (2004), if the lead angles θ t and θ m satisfy Eqs. 6 and 7, the target will be intercepted with zero miss-distance in the form of head pursuit interception: where n is a lead factor and satisfies . At the same time, it is proved that if the lead angles satisfied Eq. 7, the condition of Eq. 6 would be satisfied automatically (Golan and Shima 2004). Therefore, the purpose of the guidance law is to make the system satisfy Eq. 7. In the design of head pursuit guidance law, there are some assumptions as follows: Assumption 1. The target and missile are point masses and the respective accelerations only change the direction of the velocity (Zhou et al. 2018).
Assumption 2. There is a minimum homing guidance distance r 0 so that the time-varying parameter satisfies the following inequality (Zhou et al. 2018): Assumption 3. The target and missile accelerations and their first derivatives are bounded (Zhou et al. 2018). Assumption 4. According to the research presented by Golan and Shima (2004), the lead angles of target and missile satisfies the following constraints: where K is a ratio and satisfies .

Basic theories of fractional sliding mode
In the study of fractional-order theory, there are some definitions of fractional differential operators, such as Riemann-Liouville, Caputo and Grünwald-Letnikov definitions. In this paper, the Caputo definition is used.
Definition 1 where m is the smallest integer number which is larger than α, Γ(•) is the Gamma function. The notation D α has different meanings for different values of α. When α is bigger than zero, D α stands for fractional-order differentiation; when α is smaller than zero, D α stands for fractional-order integral. Fractional differential operators D α can be realized numerically by fractional-order toolbox (Tepljakov et al. 2011). Similar to the integer order calculus, the fractional-order integration and differentiation (Dadras and Momeni 2012): When the fractional-order calculus operator is introduced into the sliding mode control law or sliding mode surface, it is called fractional-order sliding mode control. It not only has the robustness of traditional sliding mode control, but also has the memory and genetic properties of fractional-order differential operators, which can improve the performance of control system effectively.

Design of head pursuit guidance law based on fractional-order sliding mode theory
In this part, a head pursuit guidance law based on fractional-order sliding mode theory is designed for near-space hypersonic vehicle and the stability of the closed-loop system is analyzed.

Design of head pursuit guidance law
According to the discussion above, the error variable e is defined as follows: Differentiating Eq. 13 with respect to time and substituting Eqs. 3 and 4 into the result gives: The fractional-order sliding mode surface is defined as: (15) where D α is a fractional calculus operator and . Differentiating Eq. 15 with respect to time yields: According to assumption 3, the target acceleration satisfies , where , then, a fractional-order sliding mode head pursuit guidance law can be designed as: (17) where . Theorem 1. Consider Eqs. 1 to 4, the fractional-order sliding mode head pursuit guidance law, Eq. 17, will make the closedloop system stable and the lead angles θ t and θ m will converge to zero.

Analysis of stability
In order to analyze the stability of the closed-loop system under the guidance law in Theorem 1, some lemmas are introduced here. Lemma 1. Consider a nonlinear time-varying system as the form of , x is the system state and t is the time. If there is a continuous positive-definite function V(x) which satisfies (Yu et al. 2005): where the system state will converge to zero in finite-time T, which has the following form: where is the initial time. Lemma 2. For the following fractional-order system (Aghababa 2013): where . If and only if the inequality (21) is true, the system (20) is asymptotically stable and the convergence rate of the system state is t -α : Proof Defining the following Lyapunov function candidate for Eq. 16: Computing time derivative of Eq. 22 along trajectories yields: Which means that: According to Lemma 1, the sliding variable S 1 will converge to zero in finite-time. So, the following equation can be given: According to Lemma 2, the error variable e converges to zero, therefore, the lead angles θ t and θ m will converge to zero.

Design of head pursuit guidance law considering the autopilot dynamic characteristic
Autopilot dynamic characteristic is also the main factor affecting interception accuracy, so it is necessary to consider it in the design of guidance law. Usually, the autopilot dynamic characteristic is approximated by the first-order inertia element as: (26) where is acceleration instruction and τ is the time constant of dynamic delay. Differentiating Eq. 14 with respect to time yields: Substituting Eq. 5 into Eq. 27 gives Eq. 28: Defining the intermediate variable y as: (29) The fractional-order sliding mode surface is defined as follows: where . Differentiating Eq. 30 with respect to time yields: The terms related to a t in Eq. 31 are regarded as total disturbance d, in which n, k and V t are positive constants. According to assumptions 2 to 4, and a t are bounded, the following inequality can be: where Therefore, Eq. 31 can be rewritten as: Designing the corresponding fractional-order sliding mode head pursuit guidance law as: where . Theorem 2. Considering Eqs. 1-4 and the autopilot dynamic characteristic as Eq. 26, the head pursuit guidance law Eq. 34, based on fractional-order sliding mode theory, will make the closed-loop system stable and the lead angles θ t and θ m will converge to zero.

Proof
Define the Lyapunov function as: (35) Computing time derivative of Eq. 35 along trajectories, it gets: (36) Similar to Theorem 1, the sliding variable S 2 will converge to zero in finite-time and the following equation can be given: According to Lemma 2, the intermediate variable y will converge to zero in finite time. Then, Eq. 29 can be rewritten as: Obviously, error variable e is asymptotically stable. Therefore, the lead angles θ t and θ m will converge to zero.

RESULTS AND DISCUSSION
The effectiveness of the head-guidance laws based on fractional-order sliding mode theory proposed in this paper is evaluated by numerical simulation in this section.
The initial positions of the target and missile are set as (0m, 0m) and (4000 m, 5000 m), the initial velocities V t =1600m/s and V m =1200m/s, the initial lead angles the design parameter n = 2. The guidance laws are compared in two cases: target with constant maneuver and target with cosine maneuver.
The guidance law ignoring autopilot dynamic characteristic is rewritten as:  Figures 2 and 6 show the relative motion orbits of the target and the missile, in both cases, the results indicate that both guidance laws can intercept the target in the form of head pursuit interception and have similar trajectories. Figures 3  and 7 show changing trend of lead angles, the results show that both guidance laws can adjust the leading angles to the states required in Eq. 7 and keep the state converging to zero. However, under the guidance law U 3 , the leading angle of the interceptor will deviate from the state required in Eq. 7 temporarily. Figures 4 and 8 are the sliding mode variables, which show that the guidance law U 1 can make the sliding mode variables converge to zero faster. Figures 5 and 9 are the overloads required for head pursuit interception, which show that overload saturations occur at the initial stage of terminal guidance stage guided by U 3 .
The miss distances of the two guidance laws are shown in Table 1, from which it can be seen that the miss distance under U 1 is better than that of U 3 . The guidance law considering the autopilot dynamic characteristic is rewritten as: Similarly, if β = 1, the integer order guidance law can be given as: The design parameter in U 2 and U 4 are k 2 = 20, k 3 = 15, ε 2 = 0.5, τ = 0.  Figures 10 and 14 show that both guidance laws can achieve head pursuit interception under the consideration of autopilot dynamic characteristics. Figures 11 and 15 show the trends of the variables associated with lead angles, the results indicate that both guidance laws can make θ m reach n times of θ t in a very short time and the lead angles will converge to zero. But when the target is guided by U 4 , the leading angle of the interceptor will deviate from the state required in Eq. 7 temporarily at the initial moment. Figures 12 and 16 are the sliding mode variables, which show that the sliding mode variables of both guidance laws can approach zero rapidly, but U 2 has a faster approach speed and without obvious chattering. Figures 13 and 17 are the overloads requirement of missile under two guidance laws. It is evident that the overload requirement of U 2 is lower than that of guidance law U 4 at the initial stage of terminal guidance stage.
The miss distances of the two guidance laws are shown in Table 2, from which it can be seen that the miss distance under U 2 is better than that of U 4 . Remarks: The symbol function in the guidance law causes system chattering, so it is replaced by a saturation function as follows: where ∆ is boundary layer thickness.

CONCLUSION
In order to intercept hypersonic vehicle in near-space, this paper proposes a head pursuit guidance law based on fractionalorder sliding mode, which can not only avoid the problems of conventional guidance methods in intercepting hypersonic vehicle but also lower the requirements of missile velocity and overload capability and reduce miss distance. Besides, a head pursuit guidance law considering autopilot dynamic characteristic which can affect the interception accuracy is designed. Finally, numerical simulations verify the effectiveness of the guidance laws designed in this paper.
However, this paper only studies the interception of hypersonic vehicle by one interceptor, which may not guarantee a 100% interception rate. Furthermore, guidance law in this paper is only theoretical research and has not been verified in practice. Therefore, the next major focus of the research will be to study guidance law for multiple missiles against a hypersonic vehicle and verify the guidance law in this paper by practical application.

DATA AVAILABILITY STATEMENT
All the datasets were generated during the current study.

FUNDING
Not applicable.