Min GONG, Di ZHOU,*, Xingung ZOU

a School of Astronautics, Harbin Institute of Technology, Harbin 150001, China

b School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China

KEYWORDS Highly maneuvering target;Input saturation;Missile guidance control;Robust control;Sliding mode control;Super-twisting algorithm

Abstract This paper concentrates on developing a missile terminal guidance law against a highly maneuvering target whose maneuvering acceleration is very close to that of the missile or even exceeds the missile normal acceleration in a finite period of time. A new saturated super-twisting algorithm is proposed and applied to the design of missile guidance law. The proposed algorithm has the advantages of simple structure,easy parameter tuning rules and a full utilization of the limit control input.The designed saturated super-twisting sliding mode guidance law is then employed in a missile guidance system.Simulation and its superior performance against strong maneuvering targets is demonstrated.

1. Introduction

The terminal guidance law of missile is mainly for accurate interception, even in the presence of external environment interference, parameter uncertainty, or target maneuvering.Proportional Navigation (PN) guidance law is generally used due to its easy implementation.1Although PN guidance law has the ability to intercept relatively less maneuverable targets,it may lead to unacceptable miss distance when confronted with highly maneuvering targets.The augmented proportional navigation guidance law is capable of dealing with maneuvering target.However,it requires target acceleration information in real-time,which is often difficult to obtain in practical applications.In order to intercept maneuvering targets,some robust control and nonlinear control methods are applied to the design of guidance law. Sliding mode guidance law is one of the hot topics in research.2-7

Sliding mode control is invariant to parameter variations and external disturbances. However, chattering with high frequency is a barrier for the application to the guidance law design. In order to alleviate the chattering, in Refs. 5 and 6,the high frequency switching control function is replaced with its continuous approximation function but resulting in a loss of robustness. To reduce the chattering but keep the robustness against the target maneuver, there are currently three kinds of methods in literature. The first kind of method is using high (second) order sliding mode control to develop guidance laws.6,7Moreover, Super-Twisting Algorithm(STA),a kind of Second-Order Sliding Mode(SOSM) control algorithms, was also applied in the design of guidance law.8,9Although the second-order sliding mode control methods can avoid chattering without losing robustness, it may exhibit a windup phenomenon due to integral implementation.The second broadly applied approach is to estimate the target acceleration and external and internal disturbance in real time using a disturbance observer. For example, some sliding mode guidance laws were designed with high-order disturbance observer,6,7,10extended state observer,11,12extended high-gain observer,13and so on. These observers can actively observe the target maneuvers and compensate them in the guidance law. However, from a practical point of view, these observers are sensitive to the sensor noise and time delay of measurements.More importantly,the observers cannot converge when the system input is saturated.The third method is applying an adaptive law to adjust the parameters according to the target acceleration to weaken the chattering phenomenon.14-16

From a practical standpoint, in the above mentioned guidance laws,the saturation of guidance command may result in a loss of intercept accuracy,or even lead to the instability of the guidance system. Therefore, it is quite significant to study a missile guidance law considering saturated acceleration commands,especially for intercepting a highly maneuvering target whose normal acceleration is very close to that of a missile.For example,the maximum normal accelerations of some near space hypersonic vehicles are very close to the missile maximum normal acceleration. Considering this strict case, in this paper, a novel STA accounting for input saturation is proposed and used to resist the target maneuver. Super-twisting algorithm, initially proposed in Ref. 17, is one of the SOSM control algorithms. Recently, there has been rapidly growing interest in saturated STA to overcome the windup phenomenon such as Refs. 18-21. However, in Refs. 18 and 20,the modifications are based on switching from different conditions, leading to a discontinuity in the control signal or complexity in applications. Moreover, the capabilities of the saturated STAs in Refs. 18-21 are all limited in dealing with perturbations, and the parameter tuning rules for upper bounds of the perturbation are severely strict. This paper presents a modification for saturated STA to overcome these drawbacks. Using this approach, the designed guidance law can robustly enforce the interception for highly maneuvering targets while the acceleration command constraint is considered.

The rest of this paper is arranged as follows. Section 2 is dedicated to a formulation of target-missile engagement. In Section 3,the saturated super-twisting control law is proposed.In Section 4, the design of saturated STA guidance laws is described. In Section 5, numerical simulations are carried out to demonstrate the effectiveness of the proposed saturated STA and guidance laws. Section 6 concludes the whole paper.

2. Missile-target engagement equations

In planar interception, as shown in Fig. 1, the scenario between the target and missile is mainly described in terms of relative distance R and the Line-of-Sight (LOS) angle q.22

The relative motion equation can be described as follows

where V denotes the velocity and denotes the flight path angle,the subscript ‘‘T” and ‘‘M” represent target and missile,respectively. Substituting VR= ˙R and Vq=R ˙q respectively into Eqs. (1) and (2), and differentiating VRand Vqwith respect to time produce

where wRand uRare the target and missile acceleration along the LOS, respectively; wqand uqrespectively represent the components of target and missile acceleration perpendicular to the LOS.22Considering Eqs. (1) and (2), and substituting wR, uR, wqand uqinto Eqs. (3) and (4), the equations of missile-target engagement are obtained

In the terminal guidance phase, uRshould guarantee that the target-missile relative velocity VRis negative, while uqshould guarantee the LOS stable. Usually, uRis not available in a practical terminal guidance process and the initial negative VRis ensured by the middle course guidance of the missile.22The following work will be concentrated on designing the control variable uqto send Vqto zero in Eq. (10), where wqis regarded as a disturbance variable.

3. Saturated super-twisting control

3.1. Saturated super-twisting controller design

Consider the following first-order system with input saturation

where W represents the upper bound of perturbation amplitude and L is the perturbation’s Lipschitz constant.21The control task is to design the input u for system (11) to send the sliding variable σ to zero in finite time.In practice,the control input must be stronger than the perturbation, and so it is reasonable to assume that

where k1and k2are positive parameters. When the control input is saturated, the original STA has an integral element and hence may suffer from a so called ‘‘windup problem”.Inspired by the Variable-Structure PID (VSPID) method in Ref. 23, an additional saturation error feedback term is employed in this paper,to prevent the windup effect.However,unlike Ref. 23, the method proposed in this paper does not require conditional switching. The proposed saturated STA is obtained as

Remark 1. The proposed saturated STA is easy to implement,because, compared with the VSPID method in Ref. 23 and recent Refs.20 and 18,no switching between different modes is required and only the sign function of the saturation error needs to be incorporated into the original STA. Fig. 2 depicts the block diagram of the original STA and that of the proposed STA.

Remark 2. Maximum perturbation amplitude W of Ref. 21 should fulfill the following inequality

3.2. Stability analysis

When the system is not exceeding the control bound U, i.e.,u = satU(u), the controllers (17) and (18) are equivalent to the original STA. Introducing the abbreviation δ= ˙w, and the state variables

Stability of this case is investigated in Refs. 24, 25 and 26.Herein, the Lyapunov based approach proposed in Ref. 26 is used in the following theorem.

Proof. Given in Ref. 26, Section 2.□

Remark 3. From the above analysis, it is found that the proposed saturated STA preserves the properties of the original STA. While the control laws in Ref. 19 will never behave like the original STA when the control input is not saturated.

Theorem 1. Suppose w and ˙w satisfies Eq. (13). If the controller parameters k1, k2and k3in Eqs. (17) and (18) fulfill the following inequalities

Therefore, the condition in Lemma 1 is fulfilled and so the system consisting of Eqs. (11), (17) and (18), or rather, the closed loop system (22) and (23), converges to zero in finite time.

Case 2. If the amplitude of control input is exceeding the control bound U, we have

there clearly exists such a T1, because v decreases faster than a negative constant rate until Eq. (28) eventually holds. If u ＜-U ＜0, similarly, it follows from Eq. (18)and (27) that

It will be shown that there exists T ＞T1such that |u |＜U.Consider the derivative of |u | with respect to time for t ＞T1:

4. Saturated STA guidance law

It is widely known that keeping the LOS angle constant can eventually lead to a successful interception. For the guidance system (10) we choose a sliding surface as

Remark 4. The term -VRVq/R=-VR˙q obtained from the known dynamics is similar to a PN guidance term.The integral windup will not occur even if the acceleration bound is exceeded.

Though the initial value of Vqis taken into consideration,the maneuvering of the target may also cause the abrupt change of Vq. All the strategies are designed to avoid a large guidance command, i.e. the saturation, in the initial phase,but sacrificing the robustness of the guidance system. In our paper, the proposed saturated STA guidance law is able to work in a saturation status.Therefore,the robustness of guidance system will not be lessened because of the avoidance of saturation in the initial phase.

5. Numerical results

In order to illustrate the performance of the proposed Saturated STA Guidance (SSTAG) law, a three-dimensional interception problem is considered. This problem can be decoupled into two planar loops, i.e., the elevation loop and the azimuth loop, in the guidance law design,2because it was proven that planar guidance laws are able to ensure the convergence of angular rates as they are directly applied to the three-dimensional guidance environment.27,28

An inertial reference coordinate system is defined as follows.The origin is at the launch site, and the X-axis is in the circle around the earth which contains the launch site and the targeting point and tangential to the circle,positively pointing to the targeting point. The Y-axis is also in the circle and vertical to the X-axis, positively pointing to the sky. The direction of the Z-axis is determined according to the right-hand rule.

To demonstrate the effectiveness of the SSTAG,the Adaptive Sliding-Mode Guidance(ASMG)law in Ref.3 and a typical Second-Order Sliding Mode Guidance (SOSMG) law in Ref. 6 are introduced for comparison. The ASMG law has been verified to be effective to intercept maneuvering targets.3However, the performance remains to be tested when the target is highly maneuvering. The ASMG law is given by

The parameters of the ASMG are set as N=4 and ε=160.The variable structure item sgn(˙q ) in ASMG can be replaced with an approximate function ˙q/|˙q |+δ2, where δ2is set as 0.05.

Like STA, the SOSMG law increases robustness by introducing integral terms. However, the SOSMG law lacks antisaturation terms compared with SSTAG. The SOSMG law is given by

with c0=0.1. In order to ensure the fairness of the comparative test,the parameters k1and k2in SOSMG law are the same as those in SSTAG. The advantages of the proposed guidance law are that it can work in saturated state and has high robustness.The advantage of SSTAG can only be seen when the target’s maneuvering ability is very strong or the handover error is very large to lead to saturation.Thus three scenarios will be investigated.

5.1. Scenario 1

In this scenario, we consider a near space interception problem. The initial states of the target and interceptor are listedin Table 1. It can be calculated from Table 1 that the initial target-missile relative range and velocity are R0=150 km and VR0= 6.2774 km/s. The initial LOS angles and its angular rate are qε=-0.1910。 and ˙qε=-0.0123。/s in pitch,qβ=0。 and ˙qβ=0。/s in yaw, respectively. In this scenario,target accelerations which are normal to LOS in the target translation coordinate system are chosen as constants aTy=C1g and aTz=C1g, where C1=2.5, 3.5, 4.5, respectively. It is observed that the maneuverability of the target is close to that of the interceptor as the value of C1increases.

Table 1 Initial states of target and interceptor in Scenario 1.

Table 2 Miss distances in Scenario 1.

The final miss distances under the three guidance laws are respectively shown in Table 2. As is shown in Table 2, with the enhancement of target maneuvering ability, the miss distance under ASMG law increases gradually, i.e., 0.0012 m,55.7055 m and 1526.7834 m while SSTAG can maintain a very small miss distance. The ASMG law has been verified to be effective to intercept maneuvering targets.3But it requires the interceptor to have an absolute advantage in normal acceleration. The interceptor with SOSMG law can approach the target quickly. With a proper parameter c0=0.02 in the SOSMG law, the miss distance can be reduced to 0.0032 m,0.0032 m and 0.0052 m. As is shown in Fig. 4, when C1=4.5,the SSTAG law can guarantee a successful interception while the ASMG law cannot. The LOS angular rates and guidance commands when C1=4.5 under three guidance laws are plotted in Figs. 5 and 6. Fig. 5 illustrates that under the SOSMG and SSTAG,the LOS angular rates strictly remained around zero as compared with its counterparts.Fig.6 indicates that the transient responses of SOSMG and SSTAG’s acceleration commands are better than the ASMG in the initial phase. SSTAG’s acceleration commands do not saturate during the whole guidance process.The SOSMG’s acceleration command became saturated in the final stage while ASMG’s acceleration commands remain saturated after 10 s. As a result, ASMG yields a much larger miss distance.

5.2. Scenario 2

In the second scenario, the initial conditions of the target and interceptor are listed in Table 3. The initial relative range and velocity are calculated as R0=150 km and VR0= 6.2774 km/s. In this scenario, the target flies without maneuvering. But the handover error from the midcourseguidance process to the terminal guidance process is relatively large. The initial LOS angular rates in pitch and yaw are the following three initial conditions: Condition 1.˙qε=-0.1262。/s, ˙qβ=-0.1289。/s, Condition 2.˙qε=-0.1917。/s, ˙qβ=-0.1955。/s, Condition 3.˙qε=-0.2446。/s, ˙qβ=-0.2604。/s, respectively. As is shown in Table 4, in scenario 2 the ASMG law and the SSTAG law can guarantee successful interceptions under three conditions while the SOSMG law leads to a much larger miss distance as the initial error increases i.e., 0.1094 m, 18.1313 m, and 391.2587 m. Figs. 7, 8 and 9 show the engagement trajectory,LOS angular rate and guidance commands under condition(3)). Fig. 8 illustrates that the LOS angular rates are well remained around zero under the ASMG law and the SSTAG law while that under the SOSMG law diverges fast.Fig.9 indicates that the guidance commands under the SOSMG law cannot recover from saturation due to the lack of anti-saturation term, resulting in a large miss distance.

Table 3 Initial states of target and interceptor in Scenario 2.

Table 4 Miss distances in Scenario 2.

5.3. Scenario 3

Next, we consider an exo-atmospheric interception problem.The initial conditions of the target and interceptor are listed in Table 5. The initial relative range and velocity between the target and the interceptor can be calculated as R0=230.4886 km and VR0=10.7958 km/s. The initial LOS angles and its angular rate are qε=3.7314。 and ˙qε=-0.0130。/s in pitch,qβ=0。 and ˙qβ=0。/s in yaw,respectively. In this scenario, the target begins a maneuver with aTy=C2g and aTz=C2g at t=3 s and lasts for t=12 s,

Table 5 Initial states of target and interceptor in Scenario 3.

where C2=6, 7, 8, respectively. After t=12 s, the target accelerations are chosen as constants aTy=-2g and aTz=-2g. Obviously, the target acceleration has overpassed the maximum acceleration of the interceptor from 3 s to 12 s.From Table 6, when C2=8, only the SSTAG law achieves a successful interception with the smallest miss distance(0.0033 m). The engagement trajectories of the interceptor under different guidance laws and the trajectory of target when C2=8 can be clearly seen in Fig.10.Moreover,it is illustrated in Fig.11 that under the SSTAG,when C2=8,the LOS angular rates are strictly remained around zero while that under ASMG and SOSMG cannot be nullified. From Fig. 12, it is seen that when the target changes its direction of maneuvering,the SSTAG can quickly recover while ASMG law cannot respond in time.Therefore,large miss distance is yielded under ASMG law (see Table 6). Moreover, due to the lack of antisaturation term,the SOSMG law cannot work over a long period of saturation.As a consequence,much larger miss distance is also yielded under the SOSMG law when C2increases (see Table 6).

Table 6 Miss distances in Scenario 3.

6. Conclusions

For intercepting a highly maneuvering target with the maximum normal acceleration close to that of the missile, the designed guidance law is required to work in a saturation state and have strong robustness. To solve this problem,first, a new saturated super-twisting algorithm has been proposed. The proposed algorithm has the advantages of simple structure, easy parameter tuning rules, and a full utilization of the limit control input. By using this algorithm, a saturated super-twisting sliding mode guidance law with strong robustness to a maneuvering target is designed for the terminal guidance of missile. Three simulation scenarios demonstrate the superior performance of the proposed guidance law.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 61773142).