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

In the absence of the upper bound of time-varying target acceleration, the finite-time-convergent guidance (FTCG) problem for missile is addressed in this paper. Firstly, a novel adaptive finite-time disturbance observer (AFDO) is developed based on adaptivegain super twisting (ASTW) algorithm to estimate the unknown target acceleration. Subsequently, a new FTCG law is proposed by using the output of AFDO. The newly proposed FTCG law has several advantages over existing FTCG laws. First, for timevarying target acceleration, the proposed method can strictly guarantee the trajectory of the closed-loop system is driven onto the sliding surface rather than a neighbourhood of sliding surface in the extended-state-observer-based FTCG (ESOFTCG) law. Second, the proposed method requires no upper bound information on the target acceleration. Third, the chattering problem in the conventional FTCG (CFTCG) law is completely avoided in this paper. Simulation result demonstrates the effectiveness of the proposed AFDO and the proposed FTCG law.


Introduction
As a classical guidance method, proportional navigation guidance law (PNGL) [1][2][3][4][5][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  ∞ robust guidance law [7], L2 gain guidance law [8], Lyapunovtheory-based nonlinear guidance law [9], and sliding-mode guidance law [10].However, the missile guidance problem considered in [7][8][9][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 finitetime control for the guidance system is necessary in many practical guidance cases.
Since the 1990s, with the development of finite-time stability theories [11][12][13][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 terminalsliding-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][16][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] may be 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 timevarying.
(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:  0 denotes the initial time.Let ‖ ⋅ ‖ denote the Euclidean norm of a vector and its induced norm of a matrix.

Problem Formulation
Consider a standard 2D dimensional geometry of interception shown in Figure 1.The origin  is the missile and  the target.The positions of the missile are   and   .The positions of the target are   and   . is the LOS angle,  is the range along LOS,   and   are the normal acceleration instances of the missile and target, and   and   are the velocities of missile and target.  and   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]: ( To achieve the hit-to-kill interception, a direct interception guidance strategy is given as [19]   →  0 √, where  0 is a constant.Thus for the guidance strategy (3), the guidance error is   −  0 √.In order to satisfy condition (3), the sliding surface is chosen as [19]  =   −  0 √.
If  = 0 can be satisfied in finite time, the objective of FTCG will be achieved [19].
Using the method in [16],  = 0 can be satisfied in finite time by the following CFTCG law: where  > 0,  > 0, 0 <  < 1. sign(⋅) represents the sign function, and its expression can be found in [16].From [16], it can be known that ||  sign() play a very important role in the CFTCG law (5).The convergence time of the sliding surface  decreases as the value of  is increased and the value of  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   is not equal to zero.Thus  sign() in ( 5) is used to eliminate the effect of target acceleration.And the constant  must be selected as the upper bound of the target acceleration: However,  sign() 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]: where  1 is the estimation of   and given by following nonsmooth disturbance observer (NSDOB): where  0 = 2 1/3 ,  1 = 1.5 1/2 , and  2 = 1.1. must be selected as the upper bound of Ȧ .
However, the upper bound information  and  may not be easily obtained because the maneuvering characteristic of the target is complex.If  and  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]: where the estimation of target acceleration  2 is given by following ESO: fal 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: where  2 is the estimation error of ESO (11) where  max  and Ȧmax  are unknown positive constant.
Assumption 1 implies that it is unnecessary to know the upper bound information  of ( 6) and  of (9).

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.
The ASTW algorithm is given by the following lemma.
Lemma 3 (ASTW algorithm [20]).Consider the following differential inclusion: where () and () are given as where () is a positive time-varying scalar.The adaptive law of () is given by where  is a positive constant.If the following condition can be satisfied, where  is an unknown constant, then  and ė will converge to zero in finite time.
The adaptive law of  1 is given by where  1 ( 0 ) and  1 are positive constants.Assumption 1 is valid.The estimation error of AFDO is defined as  2 =  1 −  .Then the estimation error  2 will converge to zero in finite time.
Then, according to Lemma 3, the following equations can be satisfied: where  1 is a finite time.Substituting (29) into (23), we have Then it is clear that the estimation error  2 =  1 −   will converge to zero in finite time  1 .The demonstration of Theorem 4 is completed.
In order to implement the AFDO (20), the following assumption is needed.(11), AFDO (20) do not need the upper bound of   and Ȧ , which only need   and Ȧ to be bounded.

Guidance Law Design and Stability Analysis.
After estimating the target acceleration with AFDO, a novel FTCG law is designed as follows: where  1 is given by the AFDO (20). 1 > 0,  1 > 0, and 0 <  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 AFDObased guidance law (31).
If Assumption 1 is valid, then the trajectory of system ( 2) can be driven onto the sliding surface ( = 0) in finite time.
Proof.From ( 2) and ( 4), we have Substituting the proposed AFDO-based guidance law (31) into (32), we have Construct Lyapunov function  2 as Then calculating the time derivative of  2 along the trajectories of (33), we get From (35), it denotes that  2 is affected by the estimation error  2 =  1 −   .Thus, in the following, the proof of Theorem 7 consists of two steps.In the first step, it will be proved that  2 will not escape to infinity before ( 1 −   ) converges to zero.In the second step, it will be proved that  2 will converge to zero in finite time after ( 1 −   ) converges to zero.And the total convergence time of  2 will be calculated.
Step 1. From (25) and Assumption 1, we have From the expression of  1 in ( 20), (36) can be rewritten as Then considering ( 21) and (37), it can be deduced that | φ 1 | is bounded by an unknown positive constant for  ≤  1 : where (20) and given in Theorem 4.
Step 2. Since Assumption 1 is valid, (30) will be satisfied in finite time  1 .Then combining (30) with (35), we have 2 ( 1 ) is bounded and has been proved in Step 1.As  2 ( 1 ) is bounded,  1 > 0 and 0 <  1 < 1, and  2 and  will converge to zero in finite time  2 based on Lemma 2: The convergence time  2 satisfies the following equation: 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  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  1 ( 0 ) > 0 and  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  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  1 = 0 can be modified by the following dead-zone technique: where  is a sufficiently small positive value.

Simulation Results
This subsection shows the performances of the AFDO and the proposed AFDO-based guidance law.The initial positions of the missile are   ( 0 ) = 0 and   ( 0 ) = 0.The initial positions of the target are   ( 0 ) = 2000 m and   ( 0 ) = 2000 m.The initial path angles are   ( 0 ) = /4 rad and   ( 0 ) = /3.8rad.Seeker measurement delays for 30 ms.In addition, the maximum limit of the missile acceleration command is selected as 200 m/s 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  01 = 50,  02 = 100,  1 = 0.2, and  = 0.15.The parameters of ESOFTCG law (10) are chosen as  = 10,  = 1, and  = 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.
Like [19], the parameter of the sliding surface  in the ESOTFCG law and the proposed law is selected as  0 = 0.1.

Case 1 (constant target acceleration). The target acceleration 𝐴 𝑇𝜆 is given as
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  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   with a frequency 1/(6) HZ is given as 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 timevarying, 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   with a frequency 2/ HZ is given as 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.

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.

( 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)).
[19]satisfied  2 =  2 −   .And () is the varying rate of the target acceleration; that is, () = Ȧ .01 ,  02 ,  1 , and  are constant.Thus the requirement of FTCG cannot be guaranteed by the FTCG law(10)if the varying rate () 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  converge to zero in finite time without using the upper bounds  and .Like[19], the following assumption should be assumed to be valid throughout this paper.
(10)vation of This Paper.Unlike the CFTCG law(5)and NSCFTCG law(7), the ESOFTCG law(10)does not need the upper bounds  and .However, from(13), it is clear that ESOFTCG law(10)cannot strictly guarantee the sliding surface  converges to zero in finite time if the varying rate of the target acceleration () ̸ = 0.Moreover, the upper bound of || will increase progressively as |()| become bigger.
(11)mption 5. , , ṙ , q ,   , and   are measurable.Note that the third formula  1 in (20) is most important.Theorem 4 shows that  1 can estimate the target acceleration   .Unlike  2 in ESO(11), which only can estimate the target acceleration with the estimation error  2 ,  1 can fully estimate the target acceleration in finite time.And like ESO , it is clear that  1 is bounded by unknown positive constant  2 in finite time  1 :      1     =     − 1 +       =      1 −       ≤  2 , ∀ ≤  1 ,

Table 1 :
Performance of guidance laws in Section 4.