# Strictly Finite-Time-Convergent Missile Guidance Law Based on Adaptive-Gain Observer.

1. Introduction

As a classical guidance method, proportional navigation guidance law (PNGL) [1-6] has the advantage of easy implementation in engineering. However, a series of theoretical researches and engineering practices have shown that PNGL has insufficient effect in the presence of maneuvering target. To improve the robustness of guidance system in allusion to maneuvering target, many modern control theories have been applied to design guidance laws, such as nonlinear [H.sub.[infinity]] robust guidance law [7], L2 gain guidance law [8], Lyapunov-theory-based nonlinear guidance law [9], and sliding-mode guidance law [10]. However, the missile guidance problem considered in [7-10] is solved by the asymptotic stability analysis which implies that the system trajectories converge to the equilibrium with infinite time. Actually, in many applications, the time of termination is really quite short. For example, in the space interception where a missile is intercepting a ballistic target, the whole process of terminal guidance usually lasts for only a few seconds. Thus, the finite-time control for the guidance system is necessary in many practical guidance cases.

Since the 1990s, with the development of finite-time stability theories [11-14], the study on the finite-time-convergent (FTC) control method has increasingly became a research hotspot. Some guidance laws based on FTC control have been developed. The most representative one is the FTCG law proposed in [15]. The authors in [16, 17] adopted terminal-sliding-mode method to design FTCG laws. However, in order to guarantee the stability of guidance system under the condition of maneuvering target, the FTCG laws in [15-17] contain a discontinuous control term, which brings undesirable chattering phenomenon.

It is well known that the chattering phenomenon may reduce the performance of system and cause the instability of whole system. Thus, research on chatter-free FTCG laws has the important practical and theoretical significance. In [15], to alleviate the chattering phenomenon, a saturation function was utilized to replace the symbolic function of the guidance input. However, to do this, the disturbance rejection performance is sacrificed. To alleviate the chattering phenomenon and hold the disturbance rejection performance, some FTCG laws were designed in [18], by employing a non-smooth disturbance observer (NSDOB) to estimate the target acceleration. The FTCG law in [18] eliminates the effect of target acceleration without using the saturation function and the symbolic function. Thus, the chattering problem is eliminated in [18].

However, for the above-mentioned NSDOB-based FTCG laws in [18], the upper bound of derivative of target acceleration must be known. In reality, the maneuvering characteristic of the target is complex; thus it is difficult to know the upper bound in advance. So far, fewer FTCG laws have been developed in the absence of the upper bound. It is well known that the extended state observer (ESO) is a powerful DO. Compared with NSDOB used in [18], ESO requires no information on the target acceleration. In [19], the target acceleration was estimated by ESO, and then a FTCG law was designed without using the upper bound of derivative of target acceleration.

The ESO-based FTCG (ESOFTCG) laws in [19], however, still have limitation: ESO cannot guarantee the estimation error fully converges to zero when the target acceleration is time-varying. Thus, ESOFTCG law only can guarantee the sliding surface converges to a neighbourhood of zero. In particular, if the target acceleration is fast varying, the estimation error is very large. And then the finite-time convergent feature of ESOFTCG law in [19] maybe destroyed by the large estimation error of ESO. Actually, most of the target acceleration instances are time-varying in practical engineering. Thus, the ESOFTCG law in [19] may not work well in the practical situations.

In this paper, a new adaptive finite-time disturbance observer (AFDO) based on ASTW is proposed to estimate the time-varying target acceleration. And then a novel FTCG law is designed based on the output of AFDO. The main contributions of this paper lie in the following aspects:

(1) In the absence of the upper bound information of time-varying target acceleration, the proposed AFDO can fully estimate the target acceleration. Compared with NSDOB used in [18], AFDO requires no a priori information on the target acceleration. Unlike ESO used in [19], the advantage of AFDO is that the estimation error of AFDO can fully converge to zero in finite time when the target acceleration is time-varying.

(2) The proposed FTCG law is strictly finite-time convergent in the presence of time-varying target acceleration. When the target acceleration is time-varying, the proposed guidance law can strictly guarantee the sliding surface converges to zero in finite time rather than a neighbourhood of zero in [19]. Moreover, with the help of AFDO, the proposed FTCG law does not require the a priori information on the target acceleration which is needed in [18].

The remaining parts of this paper are as follows. In Section 2, the guidance model, the design objective, and the design idea of this paper are expounded. The main results are presented in Section 3. In Section 3, a novel observer AFDO is developed and the stability proof of the estimation with AFDO is presented. Then, a AFDO-based FTCG law is proposed, and the finite-time stability of the law is also obtained. In Section 4, a simulation verifies the effectiveness of both AFDO and the proposed FTCG law. In Section 5, the conclusion of the whole paper is presented.

Notations. The following notations will be used in this paper: [t.sub.0] denotes the initial time. Let [parallel]*[parallel] denote the Euclidean norm of a vector and its induced norm of a matrix.

2. Problem Formulation

Consider a standard 2D dimensional geometry of interception shown in Figure 1. The origin M is the missile and T the target. The positions of the missile are [x.sub.M] and [y.sub.M]. The positions of the target are [x.sub.T] and [y.sub.T]. q is the LOS angle, r is the range along LOS, [A.sub.M] and [A.sub.T] are the normal acceleration instances of the missile and target, and [V.sub.M] and [V.sub.T] are the velocities of missile and target. [[phi].sub.m] and [[phi].sub.T] are the flight path angles of missile and target. The relative motion between the missile and its target can be expressed by the following equations [19]:

[mathematical expression not reproducible]. (1)

Let [V.sub.r] = [V.sub.T] cos(q - [[phi].sub.T]) - [V.sub.M] cos(q - [[phi].sub.M]), [V.sub.[lambda]] = -[V.sub.T] sin(q - [[phi].sub.T]) + [V.sub.M] sin(q - [[phi].sub.M]), [A.sub.Tr] = [A.sub.T] sin(q - [[phi].sub.T]), and [A.sub.T[lambda]] = [A.sub.T] cos(q - [[phi].sub.T]). Differentiating (1) yields [19]

[mathematical expression not reproducible]. (2)

To achieve the hit-to-kill interception, a direct interception guidance strategy is given as [19]

[V.sub.[lambda]] [right arrow] [c.sub.0], (3)

where [c.sub.0] is a constant. Thus for the guidance strategy (3), the guidance error is [V.sub.[lambda]] - [c.sub.0] [square root of r]. In order to satisfy condition (3), the sliding surface is chosen as [19]

S = [V.sub.[lambda]] - [c.sub.0][square root of r]. (4)

If S = 0 can be satisfied in finite time, the objective of FTCG will be achieved [19].

Using the method in [16], S = 0 can be satisfied in finite time by the following CFTCG law:

[mathematical expression not reproducible], (5)

where [tau] > 0, [sigma] > 0, 0 < [gamma] < 1. sign(*) represents the sign function, and its expression can be found in [16]. From [16], it can be known that [sigma][[absolute value of S].sup.[gamma]] sign(S) play a very important role in the CFTCG law (5). The convergence time of the sliding surface S decreases as the value of [sigma] is increased and the value of [gamma] is decreased. From [16], it also can be known that the finite-time convergent feature of CFTCG law may be destroyed if the target acceleration [A.sub.T[lambda]] is not equal to zero. Thus d sign(S) in (5) is used to eliminate the effect of target acceleration. And the constant d must be selected as the upper bound of the target acceleration:

[absolute value of [A.sub.T[lambda]]] [less than or equal to] d. (6)

However, d sign(S) is discontinuous and brings chattering problem. To avoid the chattering, NSDOB is used to estimate the target acceleration in [18], a non-smooth-control-based finite-time convergent guidance (NSCFTCG) law can be designed by using the method in [18]:

[mathematical expression not reproducible], (7)

where [p.sub.1] is the estimation of [A.sub.T[lambda]] and given by following non-smooth disturbance observer (NSDOB):

[mathematical expression not reproducible], (8)

where [[lambda].sub.p0] = 2[L.sub.1/3], [[lambda].sub.p1] = 1.5[L.sup.1/2], and [[lambda].sub.p2] = 1.1L. L must be selected as the upper bound of [[??].sub.T[lambda]].

[absolute value of [[??].sub.T[lambda]]] [less than or equal to] L. (9)

However, the upper bound information d and L may not be easily obtained because the maneuvering characteristic of the target is complex. If d and L are unknown, the CFTCG law (5) and NSCFTCG law (7) are no longer available. To avoid using the upper bound information and eliminate the chattering problem, ESO is used to estimate the target acceleration in [19]. The guidance laws (5) and (7) are modified as the following ESOFTCG [19]:

[mathematical expression not reproducible], (10)

where the estimation of target acceleration [Z.sub.2] is given by following ESO:

[mathematical expression not reproducible], (11)

[mathematical expression not reproducible] (12)

From [19], it can be known that the ESOFTCG law (10) can drive the trajectory of the closed-loop system (2) into a neighbourhood of the sliding surface in finite time:

[mathematical expression not reproducible], (13)

where [E.sub.2] is the estimation error of ESO (11) and satisfied [E.sub.2] = [Z.sub.2] - [A.sub.T[lambda]]. And g(t) is the varying rate of the target acceleration; that is, g(t) = [[??].sub.T[lambda]]. [[beta].sub.01], [[beta].sub.02], [[alpha].sub.1], and [delta] are constant.

Motivation of This Paper. Unlike the CFTCG law (5) and NSCFTCG law (7), the ESOFTCG law (10) does not need the upper bounds d and L. However, from (13), it is clear that ESOFTCG law (10) cannot strictly guarantee the sliding surface S converges to zero in finite time if the varying rate of the target acceleration g(t) [not equal to] 0. Moreover, the upper bound of [absolute value of S] will increase progressively as [absolute value of g(t)]] become bigger. Thus the requirement of FTCG cannot be guaranteed by the FTCG law (10) if the varying rate g(t) is very large (in Section 4 of this paper, the simulation result also demonstrates that the performance of existing ESOFTCG law is poor when the target acceleration is fast time-varying). This motivates the research topic of this paper, that is, for the missile in the presence of time-varying target acceleration, designing a new FTCG law to strictly guarantee S converge to zero in finite time without using the upper bounds d and L.

Like [19], the following assumption should be assumed to be valid throughout this paper.

Assumption 1. The target acceleration ATX is unknown bounded disturbance and satisfied the following condition:

[mathematical expression not reproducible], (14)

where [A.sup.max.sub.T[lambda]] and [[??].sup.max.sub.T[lambda]] are unknown positive constant.

Assumption 1 implies that it is unnecessary to know the upper bound information d of (6) and L of (9).

3. Main Result

3.1. Observer Design and Stability Analysis. In this section, a new adaptive finite-time disturbance observer (AFDO) will be proposed to estimate the target acceleration based on ASTW algorithm. The performance of AFDO will not be affected by the varying rate of target acceleration. And AFDO requires no a priori information on the target acceleration.

Before giving the AFDO and guidance law of this paper, the following Lemmas 2 and 3, which play important role in the subsequent analysis, are recalled here for convenience.

Lemma 2 (see [14]). Provided that V(x,t) is a differentiable and nonnegative scalar function, and V(x,t) satisfies the differential inequality [??](x, t) [less than or equal to] -[alpha][V.sup.[gamma]](x, t), where [alpha] > 0 and 0 < [gamma] < 1, [alpha] and [gamma] are constants, then we have

V (x, t) = 0, [for all]t > [t.sub.r], (15)

where [t.sub.r] [less than or equal to] [t.sub.0] + [V.sup.1-[gamma]] ([x.sub.0], [t.sub.0])/[alpha](1 - [gamma]), V([x.sub.0], [t.sub.0]) is the initial value and [t.sub.0] is the initial time, and V([x.sub.0], [t.sub.0]) is bounded.

The ASTW algorithm is given by the following lemma.

Lemma 3 (ASTW algorithm [20]). Consider the following differential inclusion:

[mathematical expression not reproducible], (16)

where [lambda](t) and k(t) are given as

[lambda](t) = 2[square root of l(t)], k(t) = 41 (t), (17)

where l(t) is a positive time-varying scalar. The adaptive law of l(t) is given by

[mathematical expression not reproducible], (18)

where [bar.q] is a positive constant. If the following condition can be satisfied,

[absolute value of [??]] [less than or equal to] [bar.c], (19)

where [bar.c] is an unknown constant, then e and [??] will converge to zero infinite time.

Other similar ASTW algorithms can be seen in [21-24]. Then AFDO and the stability analysis are given in Theorem 4.

Theorem 4. Taking guidance system (2) into consideration, the AFDO (20) is constructed.

[mathematical expression not reproducible]. (20)

The adaptive law of [l.sub.1] is given by

[mathematical expression not reproducible], (21)

where [l.sub.1]([t.sub.0]) and [[bar.q].sub.1] are positive constants. Assumption 1 is valid. The estimation error of AFDO is defined as [H.sub.2] = [J.sub.1] - [A.sub.T[lambda]]. Then the estimation error [H.sub.2] will converge to zero infinite time.

Proof. Differentiating [H.sub.1] gives

[mathematical expression not reproducible]. (22)

Substituting the expression of [[??].sub.1] in (20) and [[??].sub.[lambda]] in (2) into (22) yields

[mathematical expression not reproducible]. (23)

Let [[phi].sub.1] = [J.sub.1] + AT[lambda]. Differentiating [[phi].sub.1] gives

[mathematical expression not reproducible]. (24)

Substituting the expression of [[??].sub.1] in (20) into (24) yields

[mathematical expression not reproducible] (25)

Combining (23) and (25), we have

[mathematical expression not reproducible], (26)

where

[mathematical expression not reproducible] (27)

[mathematical expression not reproducible]. (28)

It is not difficult to note that (26), (27), and (28) have the same structure as (16), (17), and (18) in Lemma 3. Moreover, since Assumption 1 is valid, condition (19) can be satisfied. Then, according to Lemma 3, the following equations can be satisfied:

[mathematical expression not reproducible], (29)

where [t.sub.1] is a finite time.

Substituting (29) into (23), we have

[A.sub.T[lambda]] = [J.sub.1], [for all]t [greater than or equal to] [t.sub.1]. (30)

Then it is clear that the estimation error [H.sub.2] = [J.sub.1] - [A.sub.T[lambda]] will converge to zero in finite time [t.sub.1.] The demonstration of Theorem 4 is completed. ?

In order to implement the AFDO (20), the following assumption is needed.

Assumption 5. r, q, [??], [??], [A.sub.M], and [[phi].sub.M] are measurable.

Remark 6. Note that the third formula [J.sub.1] in (20) is most important. Theorem 4 shows that [J.sub.1] can estimate the target acceleration [A.sub.T[lambda]]. Unlike [Z.sub.2] in ESO (11), which only can estimate the target acceleration with the estimation error [E.sub.2], [J.sub.1] can fully estimate the target acceleration in finite time. And like ESO (11), AFDO (20) do not need the upper bound of [A.sub.T[lambda]] and [[??].sub.T[lambda]], which only need [A.sub.T[lambda]] and [[??].sub.T[lambda]] to be bounded.

3.2. Guidance Law Design and Stability Analysis. After estimating the target acceleration with AFDO, a novel FTCG law is designed as follows:

[mathematical expression not reproducible], (31)

where [J.sub.1] is given by the AFDO (20). [[tau].sub.1] > 0, [[sigma].sub.1] > 0, and 0 < [[gamma].sub.1] < 1. Then Theorem 7 will prove the finite-time-convergent feature of the closed-loop system under the AFDO-based guidance law (31).

Theorem 7. Consider the guidance system (2) adopts AFDO-based guidance law (31). If Assumption 1 is valid, then the trajectory of system (2) can be driven onto the sliding surface (S = 0) infinite time.

Proof. From (2) and (4), we have

[mathematical expression not reproducible]. (32)

Substituting the proposed AFDO-based guidance law (31) into (32), we have

[mathematical expression not reproducible]. (33)

Construct Lyapunov function [V.sub.2] as

[V.sub.2] = [1/2][S.sup.2]. (34)

Then calculating the time derivative of [V.sub.2] along the trajectories of (33), we get

[mathematical expression not reproducible]. (35)

From (35), it denotes that [V.sub.2] is affected by the estimation error [H.sub.2] = [J.sub.1] - [A.sub.T[lambda]]. Thus, in the following, the proof of Theorem 7 consists of two steps. In the first step, it will be proved that [V.sub.2] will not escape to infinity before ([J.sub.1] - [A.sub.T[lambda]]) converges to zero. In the second step, it will be proved that [V.sub.2] will converge to zero in finite time after ([J.sub.1] - [A.sub.T[lambda]]) converges to zero. And the total convergence time of [V.sub.2] will be calculated.

Step 1. From (25) and Assumption 1, we have

[mathematical expression not reproducible]. (36)

From the expression of [k.sub.1] in (20), (36) can be rewritten as

[mathematical expression not reproducible]. (37)

Then considering (21) and (37), it can be deduced that [absolute value of [[??].sub.1]] is bounded by an unknown positive constant for t [less than or equal to] [t.sub.1]:

[mathematical expression not reproducible]. (38)

From (38), we have

[mathematical expression not reproducible]. (39)

From (39), it is clear that [[phi].sub.1] is bounded by unknown positive constant [[bar.c].sub.2] in finite time [T.sub.1]:

[mathematical expression not reproducible], (40)

where [mathematical expression not reproducible] is the convergence time of AFDO (20) and given in Theorem 4.

Combining (40) with (35), we have

[mathematical expression not reproducible]. (41)

From (41), it is clear that [mathematical expression not reproducible]. Thus, it can be known that [V.sub.2] is bounded in finite time [t.sub.1]:

[mathematical expression not reproducible]. (42)

Thus [V.sub.2] will not escape to infinity before ([A.sub.T[lambda]] + [J.sub.1]) converges to zero; that is, [V.sub.2]([t.sub.1]) is bounded.

Step 2. Since Assumption 1 is valid, (30) will be satisfied in finite time [t.sub.1]. Then combining (30) with (35), we have

[mathematical expression not reproducible]. (43)

[V.sub.2]([t.sub.1]) is bounded and has been proved in Step 1. As [V.sub.2]([t.sub.1]) is bounded, [[sigma].sub.1] > 0 and 0 < [[gamma].sub.1] < 1, and [V.sub.2] and S will converge to zero in finite time [t.sub.2] based on Lemma 2:

S = [V.sub.2] = 0, [for all]t [greater than or equal to] [t.sub.2]. (44)

The convergence time [t.sub.2] satisfies the following equation:

[mathematical expression not reproducible]. (45)

The demonstration of Theorem 7 is completed.

Remark 8. From the result of Theorem 7, it is clear that the problem that ESOFTCG law (10) cannot strictly guarantee S converge to zero is solved by the proposed AFDO-based FTCG law.

Remark 9. From Remark 4.3 in [19], it can be known that the boundary layer of the sliding surface in [19] is determined by the estimation error of the ESO. Thus, the parameter selection of the ESO is more important since it not only determines the performance of the ESO but also impacts the behavior of the sliding surface. However, in this paper, the estimation error of AFDO will converge to zero in finite time as soon as the parameters satisfy [l.sub.1]([t.sub.0]) > 0 and [[bar.q].sub.1] > 0. Thus, the parameter selection in this paper is much simpler.

Remark 10. From [21], if the final miss distance is less than 0.25 m, the hit-to-kill interception also can be satisfied.

Remark 11. Condition [H.sub.1] = 0 is difficult to be satisfied in practice due to numerical approximations and measurement noise. From [25], it can be known that the condition [H.sub.1] = 0 can be modified by the following dead-zone technique:

[mathematical expression not reproducible], (46)

where [eta] is a sufficiently small positive value.

4. Simulation Results

This subsection shows the performances of the AFDO and the proposed AFDO-based guidance law. The initial positions of the missile are [x.sub.M]([t.sub.0]) = 0 and [y.sub.M]([t.sub.0]) = 0. The initial positions of the target are [x.sub.T]([t.sub.0]) = 2000 m and [y.sub.T]([t.sub.0]) = 2000 m. The initial path angles are [[phi].sub.M]([t.sub.0]) = [pi]/4 rad and [[phi].sub.T]([t.sub.0]) = [pi]/3.8 rad. Seeker measurement delays for 30 ms. In addition, the maximum limit of the missile acceleration command is selected as 200 m/[s.sup.2].

For the comparison, the ESOFTCG law (10) given in [19] are also considered in this section. The parameters of ESO (8) are chosen as [[beta].sub.01] = 50, [[beta].sub.02] = 100, [[alpha].sub.1] = 0.2, and [delta] = 0.15. The parameters of ESOFTCG law (10) are chosen as [tau] = 10, [sigma] = 1, and [gamma] = 0.5. Note that, in this paper, the parameters of ESOFTCG law and ESO are the same as those in [19] and used here to ensure the fairness of comparison.

The parameters of AFDO (20) are chosen as [eta] = 0.0005, [l.sub.1]([t.sub.0]) = 5, and [[bar.q].sub.1] = 1000. The parameters of the proposed law (31) are chosen as [[tau].sub.1] = 10, [[sigma].sub.1] = 1, and [[eta].sub.1] = 0.5.

Like [19], the parameter of the sliding surface S in the ESOTFCG law and the proposed law is selected as [c.sub.0] = 0.1.

Case 1 (constant target acceleration). The target acceleration [A.sub.T[lambda]] is given as

[A.sub.T[lambda]] = -35m/[s.sup.2]. (47)

From (47), it can be known that the target acceleration is constant in Case 1. Figures 2(a)-2(d) and Table 1 show the simulation results for Case 1. From Figure 2(a), it is clear that the proposed law and the ESOFTCG law can guarantee the sliding surface converges to zero in finite time. Figure 2(b) shows that AFDO and ESO ensure the estimation error converges to zero. From Figure 2(c) and Table 1, it can be known that the proposed law and ESOFTCG law guarantee the final miss distances are less than 0.1m, which means that the proposed law and ESOFTCG law can guarantee the missile accurately attacks the target (see Remark 10). From Figure 2(d), it is clear that the acceleration commands of the proposed law and ESOFTCG law are chatter-free. Thus the proposed method and method in [19] exhibit good performance in the presence of constant target acceleration.

Case 2 (slowly varying target acceleration). The target acceleration [A.sub.T[lambda]] with a frequency 1/(6n) HZ is given as

[A.sub.T[lambda]] = 175 cos ([t/3]) m/[s.sup.2] (48)

From (48), it can be known that the target acceleration is time-varying in Case 2. But the varying rate in Case 2 is small. Figures 3(a)-3(d) and Table 1 show the simulation results for Case 2. From Figure 3(a), it is clear that the proposed law guarantees the sliding surface converges to zero, but ESOFTCG law only can guarantee the sliding surface converges to a neighbourhood of zero. The reason for Figure 3(a) is that ESO cannot guarantee the estimation error converges to zero when the target acceleration is time-varying, while AFDO can guarantee the estimation error of AFDO converges to zero. The reason can be observed from Figure 3(b). Figure 3(c) and Table 1 show that that the proposed law and ESOFTCG law guarantee the final miss distances are less than 0.25 m, which means that the proposed law and ESOFTCG law can accomplish hit-to-kill interception (see Remark 10). And, from Table 1, it also can be known that the proposed guidance law can achieve a smaller final miss distance.

Case 3 (fast varying target acceleration). The target acceleration [A.sub.T[lambda]] with a frequency 2/[pi] HZ is given as

[A.sub.T[lambda]] = 175 cos (41) m/[s.sup.2]. (49)

From (49), it can be known that the varying rate of target acceleration in Case 3 is much larger than that in Case 2. It is noted that the method in [19] does not consider the target acceleration of this type. Figures 4(a)-4(d) and Table 1 show the simulation results for Case 3. From Figure 4(a), it is clear that the proposed law can guarantee the sliding surface converges to zero in finite time. But the ESOFTCG law cannot guarantee the sliding surface converges to zero. From Figure 4(b), it is clear that the AFDO can fully estimate the target acceleration, but the estimation error of ESO is very large. From Figure 4(c) and Table 1, it is clear that the proposed guidance law still can guarantee the final miss distance is less than 0.1 m. Thus, the missile with the proposed law can accomplish hit-to-kill interception, while the final miss distance of ESOFTCG law is 3.2 m, which means that the missile with ESOFTCG law cannot accomplish the hitto-kill interception (see Remark 10). The reason for Figures 4(a), 4(b), and 4(c) is that the estimation error of ESO is large when target acceleration is fast varying. And then the finite-time convergent feature of ESOFTCG law is destroyed by the large estimation error (the relationship between sliding surface, estimation error of ESO, and varying rate of target acceleration is shown in (13)). And the hit-to-kill guidance strategy cannot be accomplished by ESOFTCG law. Since AFDO can fully estimate target acceleration, the proposed law can still accomplish hit-to-kill interception in the present of the fast varying target acceleration.

According to the simulation results, the following can be concluded:

(1) AFDO can achieve a good estimation effect on the condition of the target acceleration instances with either low or high varying rate (Figures 2-4). But ESO can only have a good estimation effect on the condition of constant acceleration (Figure 2). If the target acceleration is varying, the estimation error of ESO will increase with the increase of the varying rate of target acceleration (Figures 3 and 4).

(2) The proposed guidance law can strictly guarantee the sliding surface converges to zero in finite time when the target acceleration is constant or time-varying (Figures 2-4). But ESOFTCG law only can guarantee the sliding surface converges to a neighbourhood of zero if the target acceleration is time-varying (Figures 3 and 4). In particular, if the target acceleration is fast varying, the estimation error of ESO is very large (Figure 4). Then the finite-time convergent feature of ESOFTCG law will be destroyed by the large estimation error of ESO (Figure 4).

(3) Unlike the CFTCG law, the proposed law does not need the upper bound of target acceleration.

(4) Since the target acceleration has been fully estimated by AFDO, the proposed guidance law has no discontinuous control term. Thus the chattering problem in CFTCG law is solved (Figures 2(d), 3(d), and 4(d)).

5. Conclusion

(1) In this paper, a novel adaptive finite-time disturbance observer (AFDO) based on ASTW algorithm was proposed, which does not need to know the upper bound of the target acceleration in advance. Moreover, the estimation error of AFDO strictly converges to zero in finite time even if the target acceleration is time-varying.

(2) Subsequently, a novel FTCG law based on AFDO was proposed. The newly proposed FTCG law has several advantages over existing FTCG laws. First, for the time-varying target acceleration, the proposed guidance law can strictly drive the trajectory of the closed-loop system onto the sliding-mode surface rather than a neighbourhood of sliding-mode surface in the ESOFTCG law. Second, unlike the CFTCG law, the proposed method requires no information on the target acceleration. Third, the chattering problem in the CFTCG law is completely avoided in this paper.

(3) Finally, mathematical simulation result demonstrated that the performances of the AFDO and the proposed guidance law are excellent.

http://dx.doi.org/10.1155/2017/3985396

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61304238), the National High-Tech R&D Program of China (2012AA120602, 2015AAxxx7033, and 2015AAxxx1008), and the Chinese Aerospace Supported Fund (2015-HT-XGD).

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Jun Zhou and Yang Wang

Institute of Precision Guidance and Control, Northwestern Polytechnical University, Xi'an 710072, China

Correspondence should be addressed to Yang Wang; 2013100280@mail.nwpu.edu.cn

Received 8 November 2016; Revised 2 January 2017; Accepted 10 January 2017; Published 1 February 2017

Academic Editor: M. Djemai

Caption: Figure 1: Interception geometry.

Caption: Figure 2: Simulation result for Case 1 (constant target acceleration).

Caption: Figure 3: Simulation result for Case 2 (slowly varying target acceleration).

Caption: Figure 4: Simulation result for Case 3 (fast varying target acceleration).
```Table 1: Performance of guidance laws in Section 4.

Case   Guidance law   Final miss distance (m)

1      Proposed law            0.033
ESOFTCG law             0.054
2      Proposed law            0.079
ESOFTCG law             0.237
3      Proposed law            0.087
ESOFTCG law             3.23
```
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