Terminal Impact Angle Control Guidance Law Considering Target Observability

: The problem of the terminal impact angle control guidance law, considering the target observability for passive guidance with bearing-only measurement, is investigated in this paper. Modiﬁed line-of-sight (LOS) angle error dynamics and their closed-loop analytical solution are developed to enhance the target observability, and then their characteristics are studied, which makes the LOS angular rate oscillate in the early stage. The terminal impact angle control guidance law with the global sliding mode is designed to eliminate the approaching stage of sliding mode control, which makes the system robust throughout the entire process of control. Finally, numerical simulations are presented to demonstrate the performance of the proposed guidance law under various conditions, which achieves the desired results.


Introduction
Without the consideration of target maneuvers and the time delay between guidance and control systems, the well-known proportional navigation guidance (PNG) law, which can be regarded as an optimal guidance law when the navigation ratio is set as 3, has been used in various engineering practices successfully [1]. In some applications, in addition to hitting the target, it may also be desirable to shape the missile trajectory near impact. For example, it is expected to hit the weak part of the target to enhance the missile's attack effect, such as the anti-tank missile attacking the target with a large terminal impact angle to complete the high-efficiency damage [2,3]. Reference [4] introduced the Cauchy-Schwarz inequality to analyze the guidance problem and proposed an impact angle constrained optimal guidance law called the trajectory shaping guidance (TSG) law to achieve a better performance. Reference [5] proposed a new homing guidance law that can attack the target with a desired impact angle. Essentially, it is a variation of the PNG law, which includes a supplementary time-varying bias. Reference [6] studied the bias guidance law without the remaining flight time and added an angle constraint bias term based on the PNG law to control the terminal impact angle by changing the guidance command. Reference [7] proposed an optimal-control-based guidance law considering the field-of-view constraint for the practical implementation of impact angle control guidance. Reference [8] designed an optimal guidance law with a constraint on the impact angle and acceleration limit.
Owing to the time-varying aerodynamical coefficients, target maneuvers and external disturbances, the dynamical characteristics of missile guidance and control system are full of nonlinearities and uncertainties. To handle this problem, in the literature [9][10][11][12], many works design nonlinear guidance laws based on nonlinear control theories that hold the properties of a high precision, disturbance resistance and strong robustness. Among nonlinear control theories, the sliding mode control (SMC) theory has been widely used in guidance law design works. Reference [13] used the adaptive sliding mode method to design the guidance law, and the convergence speed of the sliding surface increases with the decrease in the relative distance between the missile and target. Reference [14] proposed a linear sliding mode guidance law that satisfied the stationary target and met the terminal impact angle constraint. Reference [15] designed the guidance law by the SMC method, which satisfied the terminal impact angle constraint and could improve the observability of a stationary or slow moving target. Although the above guidance laws can achieve an accurate interception, the target observability, one of the important factors for the passive seeker that can only measure the relative bearing or line-of-sight (LOS) angle, is not considered and discussed.
The target observability first proposed by Lingren and Gong [16] is determined by the relative target-missile geometry and motion relationships [17][18][19]. Specifically, for a passive bearing-only missile, the missile could maneuver under the guidance command to prevent the system state from being unobservable. It is known that the guidance law design, enhancing the target observability, can yield significant benefits for interceptors equipped with passive seekers [20]. A weaving guidance law was proposed in [21] for target observability improvement by introducing new virtual system states to be controlled. According to the principle of observability, because of its guidance principle, the target observability gradually decreases for passive seekers as the target approaches. Reference [22] proposed a biased PNG law to enhance the target observability for passive homing missile systems against a nonmaneuvering target. The target observability of the biased PNG law is related to the selection of the bias term. In order to adapt to the ballistic trajectory in order to control the terminal impact angle under the bias term, the biased PNG law has a certain target observability. However, the biased PNG law will become the PNG law when the LOS angular rate of the missile gradually converges to zero, and it will gradually decrease the target observability. Reference [23] developed a new optimal guidance law by maximizing the observability metric while minimizing the terminal miss distance, as well as the energy consumption. By intentionally utilizing the low damping ratio during the initial flight period, target observability improvement guidance for impact angle control can be found in [24]. Reference [25] suggested a linear quadratic guidance law that provides an oscillating trajectory to enhance the target observability, but it is formulated in a complicated guidance command and, thus, it is difficult to analyze its physical properties theoretically. Reference [26] proposed a new optimal guidance law for the passive guidance problem based on the relative bearing or LOS angle by the passive seeker, which considered the terminal miss distance, control energy and target observability, and took the integral of the LOS angular rate as the observability index of the target.
Considering the target observability, this paper proposes a new type of terminal impact angle control guidance law for the passive guidance problem. First, the observability criterion of the stationary target is given. Modified LOS angle error dynamics and their closed-loop analytical solution are studied to make the LOS oscillate to enhance the target observability. Next, the terminal impact angle control guidance law with the global sliding mode, considering the target observability, is designed. When the design parameter k 2 > 2 is obtained by analyzing the closed-loop analytical solution of the LOS angle error dynamics, the acceleration command of the guidance law can converge to zero at the guidance terminal.
The rest of this paper is organized as follows. First, missile-target relative kinematics are given in Section 2. Section 3 interprets the target observability and error dynamics, and the new type of guidance law for observability improvement is designed in Section 4. Then, some numerical simulation results are illustrated in Section 5. Finally, the conclusions of the whole paper are offered in Section 6.

Missile-Target Relative Kinematics
This paper considers a two-dimensional planar homing engagement geometry that is shown in Figure 1. As presented in the geometry, the inertial reference frame is denoted as (X, Y). Variables with subscripts of M and T denote those of the missile and target, respectively. λ and r represent the LOS angle and the missile-target relative range, respectively. θ represents the flight path angle of the missile defined in the inertial reference frame. The velocity and lateral acceleration of the missile are represented by V and a, respectively. Defining the target angle of view is the angle between the longitudinal axis of the missile and the line of sight of the missile. When the angle of attack is negligible, it can be expressed by the error angle σ of the velocity direction of the missile. θ f represents the angle between the x-axis and the velocity direction of the missile when the missile hits the target. For simplicity, assume that the missile is flying with a constant velocity and that the target is fixed. The relative kinematics of the missile to target can be formulated aṡ The complementary equation defining the relationship between the flight path angle and normal acceleration isθ For a stationary target, θ f = λ f . Therefore, the control of the terminal impact angle can be equivalently converted into the control of the terminal LOS angle. The control goal of the terminal impact angle constraint guidance law is to achieveλ → 0, λ → λ d in a finite time, where λ d is the terminal LOS angle.
Then, differentiating Equation (2) with respect to time yields Defining e = λ − λ d as the LOS angle error, we obtaiṅ e =λ (6) denoting the uncertainty of the guidance system, variable x 1 and variable x 2 as d, e andė, respectively. Among them, the uncertainty of the guidance system includes the model's model uncertainty and external disturbances. The LOS error dynamics equation, considering the uncertainty of the guidance system, can be expressed as Lemma 1. Suppose that, in the region U ⊂ R n , given a positive definite continuous function V(x), if there are real numbers satisfying a > 0 and b ∈ (0, 1) such thatV(x) + aV b (x) ≤ 0, then there exists a certain region U 0 ⊂ R n such that V(x) converges to the origin in a finite time, and the finite convergence time is there exists a certain region U 0 ⊂ R n such that V(x) converges to the origin in a finite time, and the finite convergence time is

Target Observability and Error Dynamics
The observability of the fixed target can be determined by the Gammian matrix [27]. When t > t 0 , the Gammian matrix is represented as (11) where M(t) = − sin λ(t) cos λ(t) . The target is observable when the matrix D(t) is positive. In order to judge the observability of the target more easily and intuitively, the matrix A(t) is defined as follows: When the matrix A(t) is not full rank, the target is unobservable. To determine whether the matrix A(t) is full rank, its determinant is Therefore, the condition where the fixed target is observable is that the LOS angle rate is not zero. However, from the viewpoint of reducing the amount of the terminal miss distance, it is desirable for the terminalλ(t) = 0. In order to satisfy both requirements at the same time, the LOS angle is oscillated near the desired terminal impact angle in the initial stage of guidance to ensure the observability of the target, and then the LOS angle rate is gradually converged to zero as the distance of the projectile decreases. Therefore, maximizing the value of |λ| is a practical way to increase the target observability. To realize this objective, this paper proposes the error dynamics equation by introducing a linear error term asë where t go is the time to go until the intercept and k 1 , k 2 and k 3 are design parameters.
To interpret the physical meaning of the introduced error term, we rewrite (14) as an instantaneous linear time-invariant system: where The boundaries of ζ and ω are obtained from (16) as where t f is the total flight time.
Analogous to the mass-spring-damper system, it follows from (16) that the initial natural oscillatory frequency ω increases and the initial damping ratio ζ decreases with the increase in k 3 . During the initial flight period, the large t go enforces the term k 3 t 2 go to play a dominant role that affects the value of the damping ratio. During the entire homing engagement, t go gradually converges to zero and, consequently, the term (k 1 + 1)k 2 will dominate over k 3 t 2 go , which means that the modified error dynamics gradually converges to the original dynamics. Therefore, the damping ratio gradually increases from a small value to a large one as the interceptor approaches the target to generate the oscillatory LOS motion by choosing the parameter k 3 properly.
To provide better insights of the introduced biased term and analyze the convergence of the LOS angle error, we seek to find the closed-form solution of (14) by using the Frobenius method. Let β k 1 + 1 − k 2 > 0, Θ = {x|x = 2k + 1, k ∈ Z} as the set of all odd numbers, Ω = {x|x = 2k, k ∈ Z} as the set of all even numbers; then, after some algebraic manipulations (please refer to Appendix A for the detailed derivation), the general closed-form solution of (14) can be readily obtained as (a) If β is not an integer, then where the coefficients a i and b i are determined by where the functions f (x, y) and g(x, y) are defined in (A10) and (A21), respectively, and C 1 , C 2 are two integration constants determined by the initial conditions. Observing (18) and (20), it is seen that the closed-form solutions have two different classes of LOS angle error trajectories depending on the difference in the root of the solution: one is the time-to-go polynomial error dynamics, and the other one is the error dynamics, which combine time-to-go polynomial with logarithmic functions. From these results, we have the following proposition to quantify the convergence of the LOS angle error and its rate.

Proposition 1.
The LOS angle error converges to zero at the time impact along the error dynamics in Equation (14) if k 1 > 0, and the LOS angle rate error converges to zero at the time impact along the error dynamics in (14) if k 2 > 1.
Proof. Please refer to Appendix B for details.
Since e converges to zero when k 2 > 0, the effect of the induced term k 3 e gradually vanishes, and thus the error dynamics in (14) converge to the original one when the interceptor approaches the targets. Proposition 1 also implies that it would be wise to obey the condition k 2 > 1 in the homing phase so as to make the terminal LOS angle rate converge to zero, thus leading to a miss distance reduction.

Guidance Law Design
For surface/air-to-surface missile against non-maneuvering targets, it follows from (6); then, (14) reduces toλ To force the system trajectory onto (21), we propose the following global sliding surface, which includes the LOS angle error and the LOS angle rate as where z = a t goλ where a = k 1 + k 2 and b = (k 1 + 1)k 2 are design parameters. Differentiating (22) with respect to time yieldṡ The terminal angle control guidance law considering the target observability consists of an equivalent control term and an additional control term, which can be expressed as where K > 0, 0 < α < 1 are design parameters of the guidance law. The term a eq M is known as the equivalent control part, which is derived by imposingṡ = 0, while the add-on term a add M is designed for discontinuous switching, which is used to guarantee the convergence of the sliding dynamics and the robustness against unexpected disturbances.
Considering the system (21) and the sliding surface (22), the guidance law (25) which satisfies the Lyapunov stability criteria.
Due to the fact that s(0) = 0, one can conclude that the global sliding manifold is achieved, i.e., s = 0 holds during the entire homing engagement. This means that the observability improvement during the initial flight stage is ensured and the LOS motion can be easily predicted by using the obtained closed-form solution. This property is different from previous observability improvement guidance laws [28,29], where only ad hoc maneuvers perpendicular to the LOS are generated and the convergence pattern is not easy to predict. Additionally, since there is no requirement of extra energy in the reaching phase, one can safely predict that the proposed guidance law requires less of a control effort than other sliding mode control guidance laws.

Proposition 2.
The guidance command of the proposed guidance law converges to zero at the time of impact, i.e., lim Proof. Please refer to Appendix C for details. Proposition 2 indicates that the limiting value of k 2 with the bounded acceleration command is 2. The lower gain will result in unbounded acceleration profiles, which will inevitably lead to a nonzero miss distance. A higher gain prevents the divergence of the guidance command near the interception of the target and, consequently, it reduces the terminal miss distance. Note that a stabilized acceleration for surface/air-to-surface tactical missiles is a crucial property to provide operational margins to cope with unexpected disturbances, especially near the interception.

Numerical Simulation
In this section, the performance of the proposed guidance law is investigated through numerical simulations under various conditions. To facilitate analysis, a point-mass missile model with a lag-free autopilot dynamics is used.
To verify the performance of the terminal angle control guidance law considering the target observability under different conditions, the parameters of the sliding surface and the guidance law changes. Table 1 provides the other related initial homing conditions that are used for simulations. First, the feasibility of different target terminal impact angle constraints is studied. The parameters of the sliding surface and the guidance law take k 1 = 3, k 2 = 2.5, α = 0.8, K = 5. The expected terminal impact angles are −40 • , −50 • and −60 • , respectively. The simulation results are shown in Figure 2. In Figure 2, it can be seen that the terminal angle control guidance law, considering the target observability, can achieve different expected terminal attack angles. In order to enhance the observability of the target, the initial stage of the guidance needs a large acceleration command to bend the trajectory, which causes the LOS angle rate to oscillate. The LOS angle rate and acceleration command at the end of the guidance converge to zero, which is beneficial to the improvement of the guidance performance and robustness of the guidance law. In order to further verify the validity of the proposed guidance law, the simulation compares the performance among the terminal angle control guidance law considering the target observability, trajectory shaping guidance (TSG) law [4] and the power function weighting optimal guidance (PFWOG) law [30]. The PFWOG law is obtained as the solution of a linear quadratic optimal control problem, with the energy cost weighted by a power of the remaining flight time. It produces different trajectories and command profiles depending on the choice of n. The acceleration command of the TSG and PFWOG law are determined in an expression as follows: (29) where the parameter n in the PFWOG law takes 1, the other parameters of the three guidance laws are the same as Table 1 and the desired terminal impact angle is −60 • .
The comparative simulation results of the three guidance laws are depicted in Figure 3. As can be seen from those figures, all guidance laws can achieve the desired terminal impact angle, and the terminal's acceleration commands are all zero. Compared with the TSG and PFWOG law, the proposed guidance law can generate a larger LOS angular rate in the initial stage. Figure 4 shows that the target observability under the proposed guidance law is higher than that of the TSG and PFWOG law, but the disadvantage is that it needs a higher acceleration in the initial stage and more control energy.  At the same time, to validate the observability enhancement property of the proposed guidance law, UKF-embedded closed-loop simulations are performed to demonstrate the advantage of observability improvement. The state variables of this model are the relative position and the velocity in the inertial reference frame. Let x k = x k , y k , v x,k , v y,k T and a M x ,k = a M x ,k , a M y ,k . The missile acceleration (i.e., a M x ,k and a M y ,k ) is measured from a body-fixed accelerometer, which contains noise. The state transition of this model is determined as follows: where T s = 0.1 s represents the sampling time and w k denotes the system noise, which is considered as Gaussian white noise with a zero mean and variance as Q k = diag σ 2 a , σ 2 a and σ a = 1 m/s 2 . To initialize EKF, the state estimates and covariance for the guidance filter are used:  Table 2. It can be clearly seen that the proposed guidance law can achieve a lower miss distance than the TSG and PFWOG law, which benefits from the improvement of target observability. Finally, the performance of the terminal angle control guidance law considering the target observability under different k 2 and k 3 values is studied. The desired terminal impact angle is −60 • . The simulation results are shown in Figure 5. The simulation results show that different k 2 and k 3 values cause different effects of ballistic oscillation. The design parameters of k 1 = 1.5, k 3 = 0.1 remain unchanged, and k 2 changes from 1, 2, 3 in Figure 5a,b. It can be found that, by comparing the simulation results, when k 2 < 2, the acceleration command of the guidance law cannot converge to zero at the guidance terminal. In Figure 5c,d, the design parameters of k 1 = 1.5, k 2 = 3.0 remain unchanged and k 3 changes from 0.05, 0.10, 0.15, where the k 3 value becomes increasingly larger. It is obvious that the larger the LOS angular rate, the higher the missile acceleration, the more energy that is required and the higher the observability by comparing the simulation result.

Conclusions
In this paper, the terminal impact angle control guidance problem of the stationary target, considering the target observability, is studied. Based on the sliding mode nonlinear control theory, which is robust to system uncertainty and external disturbance, the target observability is given under different constraints. Based on the criterion, new LOS angle error dynamics are proposed and their closed-loop analytical solution is solved. The terminal angle control guidance law with the global sliding model is designed by making the LOS oscillate in order to enhance the target observability. By analyzing the closedloop analytical solution of the LOS angle error dynamics to obtain the design parameter k 2 > 2, the acceleration command of the guidance law can converge to zero at the guidance terminal. The method outlined is also suitable for a three-dimensional scenario, which can be analyzed in more detail in future research.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The closed-form of the differential Equation (14), which can be analytically solved by the Frobenius method, and the obtained solutions, are in the form of a power series. To begin with, let the prime denote the derivative with respect to t go . Then, Equation (14) can be reformulated as Based on the concept of the Frobenius method, we seek a power series solution of the form where r may be any arbitrary number. Taking the first and second-order derivative of (A2) with respect to t go and substituting them into (A1) gives The indicial polynomial, which is the coefficient of the lowest power term t r go , is It is easy to obtain the solution of (A4) as Since r 1 > r 2 , we seek the first solution using r 1 . Substituting r = k 1 + 1 into (A3) and collecting the same power series results in Shifting the starting index from −2 gives For i = −1, the recurrence relation equation is (β + 1)a 1 = 0, which gives a natural selection of a 1 = 0. Without a loss of generality, we choose a 0 = 1.
For i ≥ 0, the recurrence relation equation is which leads to the following two natural cases: Case (1): i ∈ Θ&i > 0. Under this condition, it follows from (A8) and a 1 = 0 that a i = 0.
Case (2): i ∈ Θ&i ≥ 0. Under this condition, Equation (A8) gives a few coefficients as , a 4 = −k 3 2 2 · 4(β + 2) · (β + 4) , Define f (x, n) with n = 0, and with n being positive even numbers. Then, the coefficients for Case (2) can be rewritten as Therefore, the first solution can be obtained by combining Cases (1) and (2) as To derive the second solution using the Frobenius method, the following two conditions depending on the nature of the roots of (A4) are considered.
Condition (1): Distinct roots not differing by an integer, i.e., β is not an integer. Under this condition, the second solution can be obtained by substituting r = k 2 into (A3). Following the same lines as shown above, one can imply that Condition (2): Roots differing by an integer, i.e., β is an integer. Under this condition, the second solution has the following form: Taking the first and second-order derivative of (A14) and substituting them into (A1) gives C ln t go t 2 go e 1 t go + t go (k 1 + k 2 )e 1 t go + (k 1 + 1)k 2 + k 3 t 2 go e 1 t go + C 2t go e 1 t go − Since e 1 t go is a basic solution of (A1), the term multiplied by the logarithmic function equals zero. Shifting the starting index from −β − 2 for (A15) gives It is easy to verify that the lowest power series of the term 2t 2 go e 1 t go − (k 1 + k 2 + 1) e 1 t go is t go . Moreover, the term 2t 2 go e 1 t go − (k 1 + k 2 + 1)e 1 t go /t k 1 +1 go only has even power series. Rewriting (A16) in power series as Next, we consider the following two cases. Case (1): β is a positive even number. For this case, it follows from (A17) that b 0 = 0, b 1 = 0, and, therefore, the coefficients satisfy From (A17), the coefficient b β−2 can be obtained by equating the coefficient of the power series t k 1 +1 go from (A17) to zero as Then, for i ≤ −2, the coefficients can be calculated backward from b β−2 , whereas the coefficients with i ≥ 0 can be obtained forward from b β .
For i ≤ −2, the recurrence relation equation is Define g(x, n) with n = 0, and with n being positive even numbers. Then, combining (A18) and (A20) gives For i ≥ 0, the recurrence relation equation is which gives the following coefficients Substituting (A22) and (A24) into (A14) yields Since C and b β are two arbitrary constants, Equation (A25) is an incomplete equation. However, observing that the term multiplied by b β is the first basis solution, then, the second linearly independent solution of (A1) is e 2 t go = e 1 t go ln t go + β −2 ∑ i=−β,i∈Ω g(β, −2 − i)g(0, −2 − i) Case (2): β is a positive odd number. In this case, consider the following two subcases. Subcase (1): β = 1. Under this condition, equating the coefficient of the power series t k 1 +1 go from (A17) to zero gives Then, the basic solution e 2 t go reduces to which is the same as Condition (1) and thus shares the same solution. Subcase (2): β > 1. Under this condition, equating the coefficient of the power series t k 1 +1 go from (A17) to zero gives Since β > 1, it follows from (A17) that b 0 = 0, b 1 = 0 and the recurrence relation equation for i ≤ β − 2 is Combining (A29) and (A31) leads to C = 0. Then, the basic solution e 2 t go reduces to which is the same as Condition (1) and thus shares the same solution. Finally, by summarizing the above results, the closed-form solution of (A1) is (a) If β is not an integer, then where C 1 , C 2 are two integration constants determined by the initial conditions.

Appendix B
It is easy to verify that the lowest power terms of (18) are t k 1 +1 go , t k 2 go , whereas the lowest power terms of (20) are t k 1 +1 go , t k 1 +1 go ln t go , t k 2 go . For any ε > 0, using L'Hospital's rule, one can imply that Then, the lowest order in t go of e t go is larger than zero if k 1 > −1, k 2 > 0, whereas the lowest order in t go ofė t go is larger than zero if k 1 > 0, k 2 > 1. Next, in cooperation with the condition k 1 + 1 > k 2 , the proof is completed.