Abstract

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 adaptive-gain 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 time-varying 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.

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 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] 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 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 1. The following notations will be used in this paper: denotes the initial time. Let 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 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]:Let , , , and . Differentiating (1) yields [19]

Figure 1 

Interception geometry.

To achieve the hit-to-kill interception, a direct interception guidance strategy is given as [19]where is a constant. Thus for the guidance strategy (3), the guidance error is . In order to satisfy condition (3), the sliding surface is chosen as [19]If can be satisfied in finite time, the objective of FTCG will be achieved [19].

Using the method in [16], can be satisfied in finite time by the following CFTCG law:where , , . represents the sign function, and its expression can be found in [16]. From [16], it can be known that 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 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, 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 is the estimation of and given by following non-smooth disturbance observer (NSDOB):where , , and . must be selected as the upper bound of .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 is given by following ESO: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 is the estimation error of ESO (11) and satisfied . And is the varying rate of the target acceleration; that is, . , , , and 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 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 . Moreover, the upper bound of will increase progressively as become bigger. 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.

Assumption 1. The target acceleration is unknown bounded disturbance and satisfied the following condition:where and are unknown positive constant.

Assumption 1 implies that it is unnecessary to know the upper bound information of (6) and 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 is a differentiable and nonnegative scalar function, and satisfies the differential inequality , where and , and are constants, then we havewhere , is the initial value and is the initial time, and is bounded.

The ASTW algorithm is given by the following lemma.

Lemma 3 (ASTW algorithm [20]). Consider the following differential inclusion:where and are given aswhere is a positive time-varying scalar. The adaptive law of is given bywhere is a positive constant. If the following condition can be satisfied,where is an unknown constant, then and will converge to zero in finite 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.The adaptive law of is given bywhere and are positive constants. Assumption 1 is valid. The estimation error of AFDO is defined as . Then the estimation error will converge to zero in finite time.

Proof. Differentiating givesSubstituting the expression of in (20) and in (2) into (22) yieldsLet . Differentiating givesSubstituting the expression of in (20) into (24) yieldsCombining (23) and (25), we havewhereIt 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:where is a finite time.
Substituting (29) into (23), we haveThen it is clear that the estimation error will converge to zero in finite time . The demonstration of Theorem 4 is completed.

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

Assumption 5. , , , , , and are measurable.

Remark 6. Note that the third formula in (20) is most important. Theorem 4 shows that can estimate the target acceleration . Unlike in ESO (11), which only can estimate the target acceleration with the estimation error , can fully estimate the target acceleration in finite time. And like ESO (11), AFDO (20) do not need the upper bound of and , which only need and 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:where is given by the AFDO (20). , , and . 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 () in finite time.

Proof. From (2) and (4), we haveSubstituting the proposed AFDO-based guidance law (31) into (32), we haveConstruct Lyapunov function asThen calculating the time derivative of along the trajectories of (33), we getFrom (35), it denotes that is affected by the estimation error . Thus, in the following, the proof of Theorem 7 consists of two steps. In the first step, it will be proved that will not escape to infinity before converges to zero. In the second step, it will be proved that will converge to zero in finite time after converges to zero. And the total convergence time of will be calculated.
Step 1. From (25) and Assumption 1, we haveFrom the expression of in (20), (36) can be rewritten asThen considering (21) and (37), it can be deduced that is bounded by an unknown positive constant for :From (38), we haveFrom (39), it is clear that is bounded by unknown positive constant in finite time :where and is the convergence time of AFDO (20) and given in Theorem 4.
Combining (40) with (35), we haveFrom (41), it is clear that if . Thus, it can be known that is bounded in finite time :Thus will not escape to infinity before converges to zero; that is, is bounded.
Step 2. Since Assumption 1 is valid, (30) will be satisfied in finite time . Then combining (30) with (35), we have is bounded and has been proved in Step . As is bounded, and , and and will converge to zero in finite time based on Lemma 2:The convergence time 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 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 and . 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 is difficult to be satisfied in practice due to numerical approximations and measurement noise. From [25], it can be known that the condition can be modified by the following dead-zone technique:where 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 and . The initial positions of the target are  m and  m. The initial path angles are  rad and  rad. Seeker measurement delays for 30 ms. In addition, the maximum limit of the missile acceleration command is selected as 200 .

For the comparison, the ESOFTCG law (10) given in [19] are also considered in this section. The parameters of ESO (8) are chosen as , , , and . The parameters of ESOFTCG law (10) are chosen as , , and . 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 , , and . The parameters of the proposed law (31) are chosen as , , and .

Like [19], the parameter of the sliding surface in the ESOTFCG law and the proposed law is selected as .

Case 1 (constant target acceleration). The target acceleration is given asFrom (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 0.1 m, 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.

(a) Sliding surface

(b) Estimation error of observer

(c) Time history of miss distance

(d) Acceleration command

(e) Adaptive gain

(a) Sliding surface
(b) Estimation error of observer
(c) Time history of miss distance
(d) Acceleration command
(e) Adaptive gain

Figure 2 

Simulation result for Case 1 (constant target acceleration).