Nonsingular Continuous Finite-Time Convergent Guidance Law with Impact Angle Constraints

This paper documents a novel nonsingular continuous guidance which can drive the line-of-sight (LOS) angular rate to converge to zero in finite time in the presence of impact angle constraints. More specifically, based on the second-order sliding mode control (SMC) theory, a second-order observer (2-OB) is presented to estimate the unknown target maneuvers, while a super twisting algorithm(STA-) based guidance law is presented to restrict the LOS angle and angular rate. Compared with other terminal sliding mode guidance laws, the proposed guidance law absorbs the merits of the conventional linear sliding mode (LSM) and terminal sliding mode (TSM) and uses switching technique to avoid singularity. In order to verify the stability of the proposed guidance law, a finite-time bounded (FTB) function is invited to prove the boundedness of the proposed observer-controller system and a Lyapunov approach is presented to prove the finite-time convergence (FTC) of the proposed sliding system. Rigorous theoretical analysis and numerical simulations demonstrate the mentioned properties.


Introduction
Facing the modern battleground, zero miss distance is not the only operational effect index of accurately guided weapon guidance [1].The complex combat circumstance requires some new properties, such as energy optimization, interference resistance, and impact angle constraints.Among these requirements, impact angle constraints are one of the prerequisites for some real cases.For example, for a missile against a tank, it is easy to pierce the top rather than the armored body; this process requires that the missile intercepts the tank within a limited impact angle range that can be realized by using a reasonable guidance law design.
Because of their effectiveness, ease of implementation, and successful history of application, the well-known proportional navigation guidance (PNG) law [2][3][4][5] and its variants have been widely used in engineering practice in the last few decades.The key point of PNG is to drive the LOS angular rate to zero to guide the missile to intercept the target with zero miss distance.However, because of the terrible disturbance resulting from target maneuvers, environment noise, and measurement error, PNG cannot provide the qualified performance in some extreme cases.For example, for missiles with maneuvering targets, such as interceptors or airto-air missiles, the LOS angular rate usually varies with the target maneuvers; this variation always results in perturbations that PNG or argument PNG cannot cope with.Hence, it is necessary to design a novel robust guidance law to resist this perturbation.
With the development of the modern control theory in recent years, many scholars proposed many advanced guidance laws for intercepting maneuvering targets.To cite some typical works, to attenuate the effect of bounded uncertain and unknown interference, Yang and Chen [6] proposed a robust H-infinity guidance law by regarding unpredictable target maneuvers as bounded unknowns.Based on the results of their research, Liu and Shen [7] presented a novel three-dimensional spherical H-infinity guidance law that takes missile acceleration constraints into account.By formulating interception as a nonlinear perturbation attenuation H-infinity problem, Yang and Chen [8] presented a novel robust guidance law.Zhou et al. [9] presented a L 2 gain performance guidance law with the aid of a Lyapunov-like approach.In the work of Yan and Ji [10], a novel input-to-state stability-based guidance (ISSG) law was proposed to limit bounded target maneuvers and estimate the bounded LOS angular rate.In addition to the mentioned guidance laws and their variants, SMCbased guidance laws are alternatives to the most popular guidance laws because of their inherent strong robustness and FTC properties.
Second-order SMC is a useful method to allow guided missiles to intercept maneuvering targets with the desired impact angles.Because of its robustness, global convergence, and order reduction, the sliding mode controller has drawn great attention from research communities and has been used in a variety of applications in various fields, such as missile guidance, autopilots of unmanned air vehicles (UAV), robotic manipulator control, and microelectromechanical system (MEMS) gyroscopes [11][12][13][14].Based on the SMC theory, many advanced guidance laws were coined for maneuvering targets.For example, by using the SMC technique, Zhou et al. [15] proposed a novel FTC guidance law to guide the LOS angular rate to converge to zero or within a small neighborhood of zero in finite time.Facing planar interception, Moon et al. [16] provided an adaptive sliding mode guidance approach based on the reaching law, in which the key point is that the convergent rate should increase in proportion to the decrease in the distance between the missile and the target.By using the so-called inertial delay control (IDC) technique to estimate the target acceleration, Phadke and Talole [17] proposed an SMCbased PN guidance law that requires no knowledge of the bounds of the target acceleration.Kumar et al. [18] proposed a planar nonsingular terminal sliding mode guidance law in the presence of impact angle constraints; however, they did not consider the FTC property.
For a LOS angle constrained maneuvering target interception, some challenges should be taken into account.First, the knowledge of a LOS angle is difficult to be obtained, especially when the target is moving with a large velocity and a large acceleration.Next, in real practice, the properties of the actuator, such as the response rate and efficiency margin, are another two important factors which also should be taken into consideration.In addition, some perturbations such as measurement error and environmental noise may result in performance degradation.Considering these problems, this paper discusses a new nonsingular continuous impact-angle-constraint SMC guidance law that forces the system state to converge to a sliding manifold in finite time as well as drives the sliding manifold to converge to zero in finite time.By using a switchable design of the sliding manifold, this control approach avoids the singularity that a conventional TSM can encounter.Furthermore, this guidance law utilizes the STA algorithm to reduce the chattering phenomenon.Moreover, a 2-OB which can track the unknown trajectory is employed to estimate the target maneuvers.Combining with the proposed observer and guidance algorithm, the LOS angle error (the difference between the desired LOS angle and the real LOS angle) and LOS angular rate can converge to zero in finite time.
The rest of this paper is organized as follows: the planar and spherical mathematical model of the engagement phase is provided, and the guidance strategy is discussed in Section 2; the 2-OB and the finite-time convergent guidance approach is proposed in Section 3; some simulation results are given and discussed in Section 4; a brief conclusion of this paper is given in Section 5; some proofs of the proposed propositions are given in Appendices A, B, and III.

Problem Formulation
2.1.Planar Mathematical Model of the Engagement Phase.Throughout the paper, the research object is assumed to satisfy the following hypothesizes: (i) the mass of the missile is in uniform distribution, (ii) the no roll or roll rate is small enough, and (iii) the mechanical properties of the missile are similar in every direction perpendicular to the roll axis.As a result, the missile can be regarded as a point mass.To simplify the problem, the relationship between the missile and the target during the engagement phase can be taken into account as a planar interception, which is illustrated in Figure 1.
From Figure 1, one can conclude that where M denotes the missile and T denotes the target; V T and V M denote the velocities of the target and the missile, respectively; a T and a M denote the accelerations of the target and the missile, respectively; r and r denote the distance and its derivative, respectively, between the missile and the target; λ and λ denote the LOS angle and the LOS angular rate, respectively; γ M and γ M denote the heading angle and the heading angular rate of the missile, respectively; γ T and γ T denote the heading angle and the heading angular rate of the target, respectively.Taking the derivative of equation ( 1) and equation ( 2) with respect to time yields r = rλ 2 + a Tr − a Mr , 5 Here, a Tr = a T sin λ − γ T and a Mr = a M sin λ − γ M denote accelerations following the LOS direction of the target and the missile, respectively.a Tλ = a T cos λ − γ T and a Mλ = a M cos λ − γ M denote accelerations perpendicular to the LOS direction of the target and the missile, respectively.This completes the planar mathematical model.

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International Journal of Aerospace Engineering Remark 1.For a missile controlled by active forces, equation ( 5) and equation ( 6) must be considered simultaneously during the process of guidance law design.However, for a missile only controlled by aerodynamic forces, the velocity in the LOS direction can be regard as constant.Hence, using only equation ( 6) to design the guidance law is adequate.

Strategy of Interception.
According to the zeroing LOS angular rate principle, assuming that the LOS angular rate will converge to zero when collision occurs, from equation (2), one can infer that where the subscript f denotes the values of the relevant parameters at terminal time.Denote impact angle θ imp as the following equation: Substituting equation (8) into equation ( 7) yields By assuming that V T < V M , solving equation ( 9) yields From equation (10), one can conclude that, for any given impact angle θ imp , there must exist one and one only relevant LOS angle.In other words, the problem of impact angle constraints can be transformed as the tracking LOS angle problem.We can design a Lyapunov approach to drive the LOS angle λ to converge to the desired LOS angle λ d and the LOS angular rate λ to converge to zero in finite time to solve this problem.

Some Assumptions and Lemmas.
The following assumptions and lemmas are used throughout this paper.
Assumption 1.There must exist a minimum distance r min between the missile and the target.For any given r, the following is required: r ≥ r min .
Remark 3.Because of the length of the missile and the length of the target from the shells to the point masses, Assumption 1 is reasonable.
Assumption 2. Because of the physical limits, the target maneuvers are unknown but continuous and bounded.Assumption 3. Actuators can respond to guidance commands in real time.
Assumption 4. The vertical acceleration of a missile afforded by actuators is no more than 300 m/s 2 .
Lemma 1 [19].Suppose that V x is a C 1 smoothing positive function defined on U ∈ ℝ n .For any parameters β 1 > 0 and β 2 ∈ 0, 1 , if there always exists a function defined on U ∈ ℝ n that satisfies then there must exist a region U 0 ∈ ℝ n such that any V x that starts from U 0 ∈ ℝ n will approach to V x ≡ 0 in finite time T reach .Moreover, this time T reach is governed by where V x 0 is the initial value of V x .
Lemma 2 [20].Suppose a function can be described as where x = x 1 , x 2 ,⋯,x n T ∈ ℝ n and A is a positive definite nonsingular matrix that is radially unbounded.The proposed function f x satisfies where λ min A and λ max A denote the minimum and the maximum eigenvalues, respectively.Moreover, x = x T x ∈ R is the Euclid norm.
Lemma 3 [21].For any given x ∈ ℝ and y ∈ ℝ, with c > 0 and d > 0, the following is satisfied: The planar model of the relationship between the missile and the target.

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Lemma 4 [22].Consider a cascaded system which is formulated as where Assume that the subsystems x 1 = f 1 t, x 1 and x 2 = f 2 t, x 2 are uniformly globally finite-time stable (UGFTS).For arbitrary-fixed and bounded x 2 , if there exists a positive definite FTB function B t, x 1 : for some constant a > 0, cascaded system ( 16) is UGFTS.To estimate the target acceleration, based on second-order SMC theory, an observer is proposed as follows:

Guidance Law Design and Stability Analysis
where q = rλ, h 1 > 0, h 2 > 0, and υ > 2 are the design parameters; q and âTλ are the estimated values of q and a Tλ , respectively.Proposition 1. Considering system equations ( 1)-( 4) and observer equation (17), the estimated errors denoted as e 1 = q − q and e 2 = a Tλ − âTλ will approach the following area in finite time: where e = e 1 , e 2 T and 19 λ min M is denoted as the minimum eigenvalue of matrix M; similarly, λ max M is denoted as the maximum eigenvalue of matrix M.
Proof of Proposition 1. See Appendix A.

Sliding Manifold Design. By denoting
where k 1 > 0 and k 2 > 0 are the design parameters and α x 1 is governed by τ 1 is an even integer, τ 2 is an even integer, μ is a very small constant, and sign ⋅ denotes the sigmoid function, which is governed by where parameter k is a positive constant.
Remark 4. The design parameters b 1 and b 2 are properly selected according to the above discussion such that the switching manifold equation ( 20) is continuous and differentiable.
Remark 5. Note that to avoid singularity, this switching manifold will switch to a novel linear sliding mode (LSM) switching manifold when the system state ε 1 is very small and the conventional fast terminal sliding mode (TSM) switching manifold is not equal to zero [23].In addition to this switching manifold, another nonlinear terminal sliding mode (NTSM) switching manifold is governed by [24].
Remark 6.In comparison with the conventional NTSM switching manifold equation ( 22), the proposed switching manifold uses a switchable design technique to avoid the singularity phenomenon.When the system trajectory is close to zero, the switching manifold will switch from the terminal switching manifold to the linear switching manifold.Moreover, when the system trajectory is far away from the switching manifold, the linear term k 1 ε 1 will accelerate the convergent rate and the nonlinear term k 2 α ε 1 will play a leading role.Hence, compared with the conventional NTSM switching manifold equation (22), the proposed switching manifold shows better convergent performance if the parameters are properly selected.
Proposition 2. For an arbitrary second-order certain system (such as system equation ( 6)) or uncertain system, if the proposed NTSM switching manifold s = 0 is achieved, then the system states ε 1 and ε 2 will converge to zero in finite time.

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Proof of Proposition 2. See Appendix B.
3.3.Finite-Time Convergent Guidance Law Design.Taking the derivative of equation ( 19) with respect to time yields where the third term α ε 1 is governed by For the convenience of following research, equation ( 20) can be rewritten as with By replacing a Tλ with âTλ , the estimated state function f ε 1 , ε 2 , t can be described as The proposed switchable controller can be described as where c 1 and c 2 are positive constants and γ > 2 is a constant.

Proposition 3.
Considering the proposed second-order system equation ( 16), the proposed controller equation (28) can drive the system states to converge to the switching manifold in finite time.
Proof of Proposition 3.For an "observer-controller" system, the conventional separation principle is not reasonable.The proof should be divided into two steps.The system's boundedness will be illustrated via the FTB [22] function method in Step I, and the feature FTC of the LOS angular rate will be illustrated in Step II.
Step I. Substituting equation (29) into equation ( 26) yields For simplicity, denote two auxiliary variables as Taking the derivatives of equation (31) with respect to time yields and taking the following FTB function into account yield: It is easy to verify that V 1 ∈ ℝ 2 is continuous within its domain of definition, except for the set Ψ = ω 1 , ω 2 ∈ ℝ 2 ω 1 = 0 .However, it follows from ω 1 = 0 and ω 2 ≠ 0 that ω 1 = ω 2 ≠ 0 and the trajectories of system states will pass through Ψ rather than stay in, until system states reach zero.Thus, the proposed Lyapunov function V 1 can be used to verify the property of FTB of the proposed guidance.
Taking the time derivative of According to Lemma 3 and the average value of the inequality, one can conclude that Consider the following two cases.

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with

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with According to Proposition 1, a Tλ − âTλ is globally bounded.In other words, there exists a positive constant κ that satisfies a Tλ − âTλ ≤ κ.Combined with the results of Case 1 and Case 2, one can conclude that where Hence, in any time range of 0, t , solving inequality (35) yields where V 1 0 denotes the initial value of V 1 .This completes the proof of the boundedness of the system.
Step II.According to Lemma 1, the estimated target acceleration âTλ will converge into a small region around the real target acceleration âTλ in finite time.Hence, equation (32) can be rewritten as International Journal of Aerospace Engineering To prove the finite-time stability, the selected Lyapunov function is governed by 0 and can be used to verify the property of FTC.
Taking the derivative of equation ( 32) with respect to time yields Equation (42) can be rewritten as Since parameters c 1 and c 2 are positive constants and γ > 2, it is easy to verify that matrix P is positive definite.
Equation can be rewritten as Since c 1 and c 2 are positive constants and γ > 2, it is easy to verify that matrix Q is positive definite and radial unbounded.According to Lemma 2, one can obtain Combining equation ( 43) and equation ( 45), one can conclude that Since 2γ − 3 / 2γ − 2 ∈ 0, 0 5 , according to Lemma 1, the system states of equation ( 6) can converge to zero in finite time.This completes the proof.Remark 7. A FTB function of system equation ( 6) and observer equation ( 17) with a certain upper bound is presented in Step I, and the UGFTS property of the proposed guidance law is demonstrated in Step II.From Assumption 2, one can imply that the observer is UGFTS.Finally, according to Lemma 4, system equation ( 6) with observer equation (17) and controller equation ( 29) is UGFTS.

Case Study
In this section, numerical simulations are performed to demonstrate efficiency and effectiveness of the proposed guidance law for missiles that intercept three types of targets during terminal guidance.The simulations are performed with the MATLAB platform by using the fourth-order Runge-Kutta solver with a fixed step size of 0.001 s.

Simulation Setup.
Verify the effectiveness of the proposed guidance law in the presence of a two-dimension environment.The required initial conditions are selected as follows: (1) the initial range is r 0 = 7071 m, (2) target velocity is   7) values of the design parameters are given in Table 1.

Simulation Results
. Verify the effectiveness of the proposed guidance law in the presence of a two-dimension environment.To strengthen the universality of the proposed planar guidance law, three performances of the target maneuvers are considered, as described in the following: case 1: the target maneuver is constant, a T = 50 m/s 2 , case 2: the target maneuver satisfies the sinusoidal components, a T = 50 sin 0 5t m/s 2 , and case 3: the target maneuver is random, as illustrated in Figure 2.
The simulation results include trajectories of the missile and the target, missile accelerations, LOS angles and angular rates, sliding manifolds, and estimated target maneuvers.Figure 3 is for case 1, Figure 4 is for case 2, and Figure 5 is for case 3.As shown in Figures 3(a), 4(a), and 5(a), one can see clearly that the missile can intercept the target accurately in the presence of different target maneuvers and different impact angles under the proposed guidance law.In Figures 3(b), 4(b), and 5(b), the missile accelerations are limited to 300 m/s 2 and are chattering free.Figures 3(c  -300     24).These results verify the efficiency and effectiveness of the proposed planar guidance law.Note that there exists a divergent phenomenon in some figures, such as in Figures 4(d) and 5(d).This phenomenon results from the inherent property of guidance.When the interceptor is very close to the target, r is very small or even regarded as zero.Considering that r is contained in denominators in the system of equation ( 6), the terrible influence from r → 0 is difficult to affect.However, by considering that r has a minimum value r min (Assumption 1), the divergent phenomenon at the end of the engagement phase can be ignored; thus, the proposed guidance law still meets the requirements in actual practice.
Next, for the purpose of further demonstration of the efficiency and effectiveness of the proposed guidance law, Time (s) Time (s) (d) LOS angular rates Time (s) (g) The third term in equation ( 24) Figure 4: Simulation results case 2. 9 International Journal of Aerospace Engineering the NTSM guidance law in [20] shown below is employed for comparison:

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Here, assume that knowledge of the target acceleration a Tλ in the NTSM guidance law can be obtained in real time.Moreover, the design parameters α, β, p, v 1 , and v 2 are chosen to be the same as those in [22].In this comparison case, the target maneuver is chosen as a T = −50 sin 0 5t m/s 2 and the desired final LOS angles are selected as λ d = 30 deg and λ d = 60 deg.
The simulation results of the comparison are illustrated in Figure 6.Similar to the above simulation cases, Figures 6(a   Sliding manifold (g) The third term in equation ( 24) Remark 8.In real practice, considering that the central processing units (CPUs) on the interceptors can only identify and generate simple instant signal, it is important to design discrete-time guidance laws [25].Compared with the continuous sliding mode guidance law, however, the chattering phenomenon is much easier to occur and much more difficult to avoid.Thanks to the works in [26,27], this problem has been addressed.

Conclusion
This paper presented a chattering-free nonsingular finitetime convergent guidance law in the presence of impact angle constraints; more specifically, the target maneuvers are obtained by proposing a second-order SMC-based observer and singularity is avoided by proposing a switchable sliding manifold.A super twisting type of guidance law was proposed to force the LOS angle and angular rate to converge to the desired values in finite time.Furthermore, the proposed planar guidance was extended in three-dimension cases.Numerical simulation results demonstrated the aforementioned properties.

Appendix A
Proof of Proposition 1.By denoting q = rλ, equation ( 6) can be rewritten as Substituting equation (A.1) into equation ( 17) yields Regarding system equation ( 17), consider the following Lyapunov function: Equation (A.3) can be rewritten in the following form: Because h 2 > 0 and υ > 2, V 3 is positive and unbounded in the radial direction.According to Lemma 2, Note that V 3 is continuous, except Ψ = ω 1 , ω 2 ∈ ℝ 2 ω 1 = 0 .Similar to V 1 , V 3 can be used to evaluate the stability of system equation (43).Taking the derivative with respect to time yields Because h 1 > 0, h 2 > 0, and υ > 2, it is easy to verify that M is Hurwitz.If λ min M e υ−2 / υ−1 − a max Tλ B > 0, equation (A.9) can be transformed as where β = λ min M e υ−2 / υ−1 − a max Tλ B > 0. According to Lemma 1, system equation (43) will converge into the following region: This completes the proof.

3. 1 . 2 -
OB Design and Stability Analysis.According to Assumption 2, the acceleration of the target is unknown.

X
) and 6(b) demonstrate that the missile intercepts the target with high accuracy and that the actuator can provide enough acceleration during the engagement phase.Figures6(c) and 6(d) show the superiority of the proposed 7000 LOS angular rate (rad/s)

Table 1 :
Values of the design parameters.