Design of Three-Dimensional Guidance Law With Impact Angle Constraints and Input Saturation

This thesis studies the problem of three-dimensional guidance considering maneuvering acceleration and input saturation based on the background of missile intercepting maneuvering target. On the basis of the three-dimensional guidance model with impact angle constraints, a three-dimensional guidance law is designed by using the integral sliding mode control and adaptive control. To solve the problem of input saturation, an adaptive anti-saturation integral sliding mode three-dimensional guidance law is designed by introducing an auxiliary system. The stability of the designed guidance law is proved by Lyapunov theory, which ensures that the sliding mode manifold converges to zero in finite time. The effectiveness of the designed guidance law is verified by simulation analysis.


I. INTRODUCTION
With the complexity of the war environment and the increasing improvement of the performance of high maneuvering targets, the missile must have the ability to intercept air maneuvering targets [1]. The design of robust threedimensional guidance law is the key link to ensure the successful interception of the target. In order to achieve the best damage effect, the missile is usually required to intercept the target at the desired impact angle [2]. Therefore, it is of great significance to study the three-dimensional guidance law with impact angle constraints.
Among the common guidance law design methods in automatic seeking guidance system, the main methods include classical control methods and modern control methods. In [3], [4], a bias pure proportional guidance law is designed for intercepting stationary or slow targets. In order to achieve the best damage effect, a proportional guidance law with angle constraint is designed for the relative motion model of three-dimensional missile and target [5], [6]. In [7], [8], the guidance law with impact angle constraint is designed based on the optimal control theory. Because the sliding mode guidance law is robust to external disturbances and the uncertainty of system parameters, it has been widely used in The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . aircraft guidance and control [9]- [15]. In [10], [11], based on the sliding mode control, an adaptive guidance law with impact angle constraints is designed. In order to obtain high precision guidance performance, the finite-time guidance law is designed by using the terminal sliding mode control in [12], [13]. In [14], the sliding mode three-dimensional guidance laws with impact angle constraints are designed for maneuvering target interception. In [15], a robust guidance law with impact angle constraints is designed for intercepting maneuvering targets based on the adaptive control and nonsingular terminal sliding mode control. In [16], the nonlinear disturbance observer is used to estimate the target acceleration on line, and the three-dimensional guidance law is designed by using the finite-time control. In the actual guidance process, the acceleration command of the missile has certain physical constraints. If the acceleration instruction constraint is not considered in the design of the guidance law, it may lead to the decline of the guidance performance and even cause the instability of the whole guidance system in [17]. In [18], [19], based on the adaptive control and optimal control, the adaptive three-dimensional guidance law and the optimal guidance law with input saturation constraints are designed. In [20], an anti-saturation three-dimensional guidance law with impact angle constraints is designed. In [21], to deal with the problem of input saturation, an adaptive dynamic surface three-dimensional guidance law is designed by using the smooth tangent function and Nussbaum function. In [22], [23], by introducing an adaptive algorithm to estimate the upper bound of target acceleration online, a threedimensional guidance law with acceleration constraint is designed by using the dynamic control, but the state of the guidance system can only be guaranteed to be uniformly bounded. In order to further improve the interception probability for high maneuvering targets, in this paper, a 3D guidance strategy with impact angle constraints is designed based on the integral terminal sliding mode theory and adaptive algorithm. Compared with the above references, the innovations are as follows (1) By designing the integral terminal sliding mode surface with adaptive gain and automatically adjusting the adaptive gain parameters according to the error information, the system error converges to the sliding mode surface quickly and the control performance of the system is improved.
(2) In this paper, the adaptive terminal sliding mode controller and anti-saturation adaptive terminal controller are designed respectively, so that the state of the system converges to the equilibrium point quickly in finite time.
(3) Compared with [15], this paper considers input saturation and finite-time stability, which is of more practical engineering significance.
The structures of this paper are as follows: Firstly, the guidance model of three-dimensional space with impact angle constraints is given. Secondly, based on the integral terminal sliding mode control theory and adaptive control method, the adaptive integral terminal sliding mode guidance law and the anti-saturation adaptive integral terminal sliding mode guidance law are designed respectively. Finally, the effectiveness of the designed guidance law is verified by digital simulation. Fig. 1 shows the three-dimensional guidance geometry of the missile intercepting the maneuvering target. The relative kinematic equation of the missile and target is given as follows [13] where ρ = V t /V m , V m and V t are the missile velocity and target velocity, respectively. R is the relative distance. θ m and φ m represent the direction of the velocity of the missile relative to the line-of-sight. θ t and φ t represent the direction of the velocity of the target relative to the line-of-sight coordinate system. θ L and φ L are the angles of sight. a ym and a zm are the missile accelerations in the pitch and yaw directions. a yt and a zt are the target accelerations in the pitch and yaw directions, respectively. Based on (5) and (7), the following equation holds (8) and (9), the threedimensional guidance system described as

II. SYSTEM DYNAMICS AND PROBLEM STATEMENT
where u is control input, D is external disturbances, in the following form To facilitate the terminal guidance law design, the following lemma and assumption are given. Assumption 1: Considering the system (10), assume that the external disturbances D is bounded, that is D ≤ D, where D is positive constant. VOLUME 8, 2020 Lemma 1 [13]: Lemma 2 [9]: Consider the following n order integral systemẋ If ε ∈ (0, 1) that satisfies for any α n ∈ (1 − ε, 1), and the design controller is is the polynomial of Hurwitz. Then, the system is globally finite time stable.

III. GUIDANCE LAW DESIGN
Based on the integral terminal sliding mode control and adaptive algorithm, the adaptive guidance law and anti-saturation guidance law are designed for the three-dimensional guidance system (10), respectively, to ensure the missile to intercept the maneuvering target successfully.
In order to deal with the unknown upper bound of target maneuvering, an integral sliding mode three-dimensional guidance law is designed using the adaptive control as follows where k 3 , k 4 , k 5 , γ , p 1 and p 2 are positive constants. Theorem 1: Considering system model (10) selecting the sliding modemanifold (13), under thedesigned three-dimensional guidancelaw (15), the s sconverges to zeroin finite time, then theline of sight angularrateθ L andφ L and will convergeto zero in finite time.
From (20), it can get that V 1 is bounded. Meanwhile, it can conclude that adaptive parameter estimateD is bounded, which means that there is positive constantD > 0, satisfyinĝ D ≤D.

Choose the Lyapunov function candidate as
whereD is positive constant, satisfyingD >D and D > D.
The time derivative of the V 2 can be written aṡ where ρ = min( √ 2k 4 , √ γ s ). Considering (23) and Lemma1, it can conclude that the sliding mode manifold s converges to zero in finite time.
According to (13), the inequality is satisfied As can be seen from the [9], x 1 and x 2 converges to zero in finite time, that is, the line of sight angular rateθ L andφ L will converge to zero in finite time. Theorem 1 is proved. Remark 1: In the guidance law (15), the design constraints of the guidance law are not considered, but in the actual guidance process, only the aerodynamic force provides maneuverability for the missile in the final guidance stage, which leads to the fact that the power actuator of the missile can only provide limited acceleration. So it is of great significance to design the guidance law with acceleration saturation.

B. DESIGN OF ANTI-SATURATION ADAPTIVE INTEGRAL SLIDING MODE CONTROLLER
Considering the input saturation, the (10) can be rewritten as In order to cope with input saturation, the auxiliary system (26) is introduced, as shown at the bottom of the page, where u = u − u c , u c is the actual control input, η is the state of the auxiliary system, σ, k η , k η1 and γ 1 are positive constants, 0 < γ 1 < 1. An adaptive anti-saturation three dimensional guidance law is designed as Theorem 2: Considering the system model (10) sujecting to input saturation, selecting the sliding mode surface (14), and under the adaptive anti-saturation integral sliding mode guidance law (27), the line of sight angular rateθ L andφ L converge to zero in finite time.
Proof : Choose the Lyapunov function V 3 as Applying (27), the time derivative of V 3 can be written aṡ As According to (33) and (34), then (32) can be rewritten aṡ From (35), it can get that V 3 is bounded. Meanwhile, it can conclude thatD is bounded, which means the existence of positive constantD > 0, satisfyingD ≤D. Proof : Choose the Lyapunov function candidate as Applying (27), the time derivative of (36)can be written aṡ According to (33) and (34), (37) can be rewritten aṡ From (38) and lemma1, it can be seen the sliding mode manifold s can converge to zero.

IV. SIMULATION RESULTS
In order to show the effectiveness of the designed guidance law, the simulation parameters is given in Table 1.

A. SIMULATION ANALYSIS OF ADAPTIVE INTEGRAL SLIDING MODE GUIDANCE LAW
In order to verify the effectiveness of the adaptive integral sliding mode guidance law (15), compared with the guidance law NTSMGL in reference [14] and external disturbances are considered as D =   The trajectory and relative distance curve of missile and target under the NTSMG and proposed guidance law are given in Fig.2 a)-(b), respectively. it can be seen that these two guidance laws can intercept the target successfully. Fig.2(c)-(d) gives the curves ofθ L andφ L of line-of-sight angular rate respectively. It can be seen from the simulation result that the line-of-sight angular rateθ L andφ L can converge to zero fast under the two kinds of guidance laws, and have faster convergence rate and higher convergence accuracy than the NTSMG. From the acceleration curve of the missile given in Fig.2(e), it can be seen that the acceleration saturation phenomenon of the guidance law designed in this paper and tends to be stable value after a period of time compared with that of NTSMG. From the adaptive parameter curve given in Fig.2(f), it can be seen that it can tend to a steady state in a short time, which shows that the adaptive law is effective in the guidance process.

B. SIMULATION ANALYSIS OF ANTI-SATURATION ADAPTIVE SLIDING MODE GUIDANCE LAW
In order to demonstrate the effectiveness of theantisaturation adaptive slidingmode guidance law (27), the following two forms of target maneuvering are simulated and analyzed. The guidance law parameters are chosen as: σ = 0.01, k η = 1.25, k η1 = 0.5 and γ 1 = 0.72, other guidance parameters are the same as in section 4.1, the simulation results are shown in Fig. 3. Fig.3 (a)-(b) gives the trajectory curve and relative distance curve of missile and target under two forms of target maneuverability. It can be seen that the missile can accurately intercept the target and satisfy the guidance accuracy. Fig.3(c)-(d) show the curves ofθ L andφ L of line-of-sight angular rate, which can fast converge to zero in finite time, and that ensures that the missile can hit the target accurately. From the curve of missile acceleration given in Fig.3 (e), it can be seen that after a period of time, the the acceleration curve tends to be steady value, and the control amplitude is always within the constraint range in the whole control process, which satisfies the input constraint. From the adaptive parameter curve given in Fig.3 (f), it can be seen that the adaptive estimation value can tend to a steady state value in a short time.

V. CONCLUSION
In this paper, a three-dimensional guidance scheme with impact angle constraints and input saturation is designed for intercepting maneuvering targets. The main results are as follows: (1) An adaptive algorithm is introduced to estimate the upper bound of the maneuvering acceleration of the target, which relaxes the requirement of the prior information of the maneuvering acceleration of the target.
(2) The adaptive three-dimensional guidance law and anti-saturation three-dimensional guidance law are designed based on the integral terminal sliding mode control, respectively, which can ensure that the line of sight angular rate converges to zero in finite time.
(3) The system state is proved to be finite time stable under the designed guidance strategy by using Lyapunov theory, and the effectiveness of the guidance scheme is verified by digital simulation.