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Impact time control guidance law with field of view constraint
Aerospace Science and Technology 39 (2014) 361–369
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Impact time control guidance law with field of view constraint Youan Zhang, Xingliang Wang ∗ , Huali Wu Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, 264001, China
a r t i c l e
i n f o
Article history: Received 16 July 2014 Received in revised form 19 September 2014 Accepted 10 October 2014 Available online 18 October 2014 Keywords: Anti-ship missile Impact time control Field-of-view constraint Missile guidance
a b s t r a c t The problem of Impact Time Control Guidance (ITCG) with field-of-view (FOV) constraint is investigated. The proposed ITCG law is a combination of the well-known Proportional Navigation Guidance (PNG) law and an additional biased term of impact time error, which is defined as the difference between the impact time by PNG and the prescribed one. A rule of the cosine of weighted leading angle in the biased term is used to guarantee that the FOV constraint is not violated during the engagement. Stability of the closed-loop guidance system is proved strictly. Numerical simulation results are presented to demonstrate the effectiveness of the proposed guidance law. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction Modern battleships are equipped with various defensive systems against anti-ship missiles, such as anti-air defense missile systems and close-in weapon systems. These defensive systems have been great obstacles for anti-ship missiles to complete their missions. To penetrate the formidable defensive systems, on the one hand, sea-skimming flight or terminal evasive maneuvering has been developed for high performance missiles. On the other hand, simultaneous attack, which is viewed as another cost effective and efficient cooperative attack strategy to enhance survivability of conventional missiles, attracts much attention in these years. Works in simultaneous attack can be classified into two categories. The first category is the individually simultaneous attack, in which a common impact time is commanded to all members of the group of missiles a priori, and thereafter each missile tries to impact the target at the designated time independently. In [4], which is the initial effort in this direction, an impact time constrained guidance law which is designed for anti-ship missiles consists of two terms, one being the conventional proportional navigation guidance (PNG) term and the other being a feedback term of the impact time error. As an extension of [4], a guidance law to control both impact time and angle was presented in [8]. An improvement of [4] can also be found in [1], using the numerical optimizing method to determine the designated impact time. In [3], the nonlinear engagement dynamics is transferred into linear form using the flight path angle as the independent variable, based on which an impact time and impact angle control guidance
*
Corresponding author.
http://dx.doi.org/10.1016/j.ast.2014.10.002 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
(ITIACG) law is designed. Unfortunately, the resultant guidance law is singular at some points. In [2], under the assumption that the information of the target is known beforehand, the desired line of sight (LOS) rate trajectory, which satisfies the desired impact time and impact angle constraints simultaneously, is calculated offline first, and then a sliding mode guidance law is designed to track the desired LOS rate trajectory. In [12], a three dimensional impact time and impact angle control guidance law is designed using a two-stage control approach. In [13], a biased PNG (BPNG) law is designed to control the impact time and impact angle. In [7], sliding mode impact time guidance laws are designed based on the conventional and improved formulas for time-to-go estimation. However, both the time-to-go formulas used are not derived under the proposed guidance law. Hence the resultant time-to-go dynamics is inaccurate, which may lead to unexpected results. The second category is cooperatively simultaneous attack, in which the missiles communicate among each other during the whole guidance phase to synchronize the impact time. In [14], a cooperative guidance scheme for salvo attack (i.e., simultaneous attack) is proposed, in which, the impact time control guidance (ITCG) law presented in [4] is used as the local guidance law and then a coordination algorithm is used to synchronize the impact time of all the missiles. In [5], by introducing the concept of the time-to-go variance of multiple missiles, a cooperative PNG law is proposed to achieve a simultaneous attack by decreasing the time-to-go variance cooperatively until the interception. In [15], a cooperative guidance scheme is proposed by integrating the ITCG law (presented in [4] as well) with the decentralized consensus algorithm for multiple missile groups. In [11], by modeling the communication topology of the multimissiles system as a leader-followers network, the impact time
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synchronization of multi-missiles is achieved with time-delays and switching topologies. In [9], without using a leader, each missile is modeled as a subsystem with time-to-go as the output, and the communication topology of the multi-missiles is modeled as a peer-to-peer network. The theory of cooperative control is then applied to reduce the relative error of time-to-go among the subsystems to synchronize the impact time. All the guidance laws aforementioned didn’t consider the limitation of the seeker’s field-of-view (FOV) of the missile explicitly. In practice, the ITCG law may produce large additional command to maneuver the missile to adjust the impact time. This may lead to the target moving out of the missile seeker’s FOV. In such case, the seeker fails to maintain a lock on the target and the missile cannot intercept the target. To cope with this challenge, two methods can be used. One is the active method, in which a proper designated impact time is selected within some interval determined by the missile’s maneuvering capacity and the FOV, see [10]; the other is the passive method, in which a switching logic is used to keep the target within the missile’s FOV, see [10] and [6]. The active method cannot utilize the maneuvering capacity of the missile sufficiently since it just simply ensures the maximum look angle does not exceed the FOV limit. Although the passive way can utilize the maneuvering capacity sufficiently, it suffers abrupt jumping of guidance command due to the switching among different guidance modes. It is well known that the problem of intercepting a stationary target in the plane is already well studied. In this paper, the planner guidance problem is studied as well. But we emphasize on the impact time control when the seeker’s FOV is limited, and aims to overcome the drawbacks resulting from switching logic mentioned above. It will be shown later that the proposed guidance law has the form of an = aPNG + aξ , where aPNG is the conventional PNG law term, and aξ is a biased term to control the impact time. Different from the switching logic method, a rule of the cosine of weighted leading angle in the biased term will be used to guarantee that the FOV constraint not be violated during the engagement. When the impact time error is reduced to zero, the biased term aξ equals zero as well, and the proposed ITCG law degrades to the conventional PNG law. It will also be proved that the closed-loop guidance system under the proposed ITCG law is finite time stable with respect to leading angle and impact time error. While the rangeto-go decreases monotonically, thus interception can be guaranteed. The remainder of this paper is organized as follows. In Section 2, the problem of impact time control subject to seeker’s FOV constraint is formulated. Then, an ITCG law with the form of biased PNG (BPNG) law is designed in Section 3. The stability of the closed-loop guidance system under the proposed ITCG law is proved strictly. In Section 4, comparisons between the proposed ITCG law and the scheme in [10] are presented. Application of the proposed ITCG law to the salvo attack scenario is verified as well. Finally, Section 5 summarizes the conclusions. 2. Problem formulation Consider a planar engagement between an anti-ship missile (denoted as M) and a ship (denoted as T ) as shown in Fig. 1. The ship is modeled as being stationary since the maneuver capacity and the speed of ships are not comparable with those of anti-ship missiles of high subsonic or supersonic speed. It is assumed that the missile speed V is constant during the engagement and the autopilot lag is negligible. The heading angle and leading angle are denoted by θ and ϕ , respectively; the range between the ship and the anti-ship missile is R, q is the line-of-sight (LOS) angle relative to a fixed reference axis.
Fig. 1. Homing guidance geometry.
From Fig. 1, the engagement dynamics can be given as
R˙ = − V cos ϕ
(1a)
R q˙ = V sin ϕ
(1b)
θ˙ = an / V
(1c)
q=θ +ϕ
(1d)
where, an is the missile’s lateral acceleration, i.e., the guidance command. The impact time error, ξ , is defined as
ξ = t¯go − t go
(2)
where, t¯go = t d − t is the designated time-to-go, t d is the designated impact time, t go is the real time-to-go at the current time moment, which is usually not known a priori and need to be estimated. Besides, the seeker’s FOV is usually defined as the angle between the missile body axis and the LOS (i.e., the sum of the leading angle and the missile’s angle of sideslip. We consider the lateral plane here). Under the assumption that the missile’s angle of sideslip is small, we can use the leading angle limitation for handling the seeker’s FOV constraint, just as that of [10]. That is, the seeker’s FOV constraint can be equivalently expressed as |ϕ (t )| ≤ ϕmax , where, ϕmax is a positive constant determined by the seeker’s FOV limitation. The objective of this paper is to design a guidance law such that the missile can impact the target at the designated time t d while keeping ϕ (t ) ∈ [−ϕmax , −ϕmax ] for all t ∈ [0, t d ]. 3. Impact time control guidance with FOV constraint 3.1. Derivation of impact time error dynamics To design an ITCG law, we first derive the dynamics of impact time error. As mentioned in Section 1, the ITCG law we are going to design in this paper would have the form of
an = aPNG + aξ
(3)
where,
aPNG = N V 2 sin ϕ / R ,
(4)
N > 1 is the navigation constant, and aξ is the biased term to control the impact time. Since the ITCG law to be designed is of the form given by (3), the time-to-go of PNG should be used to calculate the impact time error. Let us assume that, without loss of generality, the time-to-go of PNG can be expressed as
t go =
R V
1+
ϕ2 4N − 2
+ t ( R , ϕ )
(5)
Y. Zhang et al. / Aerospace Science and Technology 39 (2014) 361–369
where, the first term of the right hand side of Eq. (5) coincides with the t go formula of PNG (assuming the leading angle is small) given in [5], and the second term is the portion to compensate for the error caused by potential abusing of small angle assumption. Assume, without loss of any generality, that the time derivative of t ( R , ϕ ) can be expressed as
dt ( R , ϕ ) dt
= f ( R , ϕ ) + b(a, ϕ )aξ
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leading angle. In fact, though no strict proof is given, we found through simulations that the inequality b( R , ϕ ) · b0 ( R , ϕ ) > 0 holds for all |ϕ | ≤ π /2, which is a large enough range to include that determined by the FOV constraint in practice. On the other hand, b( R , ϕ ) would finally equal zero as long as ϕ (t ) converges, thus h(t ) can be viewed as a vanishing uncertainty and brings no influence on the control precision.
(6) 3.2. ITCG law with FOV constraint
Using Eqs. (5) and (6), the time derivative of t go can be given as
t˙go = f ( R , ϕ ) + f ( R , ϕ ) + b0 ( R , ϕ )aξ + b( R , ϕ )aξ where,
f ( R , ϕ ) = − cos ϕ 1 +
4N − 2
To meet the impact time as well as the FOV constraints, the biased term aξ can be designed as
ϕ2
(7)
−
N −1 2N − 1
aξ = K cos
ϕ sin ϕ
(8)
πϕ /(2ϕmax ) |ξ |1/2 sign(ξ )/b0 ( R , ϕ )
where K > 0 is an adjustable coefficient. And hence the resultant ITCG law can be given as
and
b0 ( R , ϕ ) = − R ϕ / (2N − 1) V
2
(14)
an = N V 2 sin ϕ / R + K cos
πϕ /(2ϕmax ) |ξ |1/2 sign(ξ )/b0 ( R , ϕ )
(9)
(15)
Obviously, when there is no need to adjust the impact time (that is, when aξ = 0), the decreasing rate of t go should coincide with that of t¯go , which equals −1 since t go decreases as t increases. It follows that f ( R , ϕ ) + f ( R , ϕ ) = −1. Substituting this equality into Eq. (7) yields
To prove the stability of the closed-loop guidance system, an assumption is given as follows.
t˙go = −1 + b0 ( R , ϕ )aξ + b( R , ϕ )aξ
(10)
Differentiating Eq. (2) and taking into account Eq. (10) yields the dynamics of impact time error
ξ˙ = h(t ) − b0 ( R , ϕ )aξ
(11)
where,
h(t ) = −b( R , ϕ )aξ
Assumption 1. The initial value of leading angle |ϕ (0)| < ϕmax , and ϕ (0) = 0. Remark 2. It is reasonable to assume that the target is within the seeker’s FOV at the initial time, which implies that |ϕ (0)| < ϕmax . However, when ϕ (0) = 0, the control input coefficient b0 (0) = 0, which means that additional control energy cannot be injected into the guidance system to adjust the impact time. In this case, an additional mechanism is needed to perturb the leading angle so that the impact time becomes controllable.
(12)
is the uncertainty term caused by the potential abusing of small angle assumption. Note that, when the leading angle ϕ (t ) is confined within a relatively small range due to the requirement for maintaining the FOV constraint condition, it is reasonable to use the t go estimation formula proposed in [5] directly. Thus, in the case of ITC problem under FOV constraint, we can assume that h(t ) = 0. Substituting the guidance law Eq. (3) into Eq. (1), and using R and ϕ as the state variables, the engagement dynamics including the impact time error can be given as
R˙ = − V cos ϕ
(13a)
ϕ˙ = −( N − 1) V sin ϕ / R − aξ / V
(13b)
ξ˙ = −b0 ( R , ϕ )aξ
(13c)
where, the control input coefficient b0 ( R , ϕ ) is defined by Eq. (9). Remark 1. To get a simple form of the impact time error dynamics, the use of the relationship of f ( R , ϕ ) + f ( R , ϕ ) = −1 is the key. Besides, the complete expression for the impact time error dynamics should be ξ˙ = −b( R , ϕ )aξ , where, b( R , ϕ ) = b0 ( R , ϕ ) + b( R , ϕ ). We assumed in the above derivation that b( R , ϕ ) = 0. This is reasonable when ϕmax is relatively small and ϕ (t ) is confined within [−ϕmax , ϕmax ] during the whole engagement. When ϕmax is relatively large, b( R , ϕ ) = 0 is no longer valid during the whole engagement. However, as long as b( R , ϕ ) · b0 ( R , ϕ ) > 0, that is, b0 ( R , ϕ ) dominates b( R , ϕ ), the control direction is determined by b0 ( R , ϕ ). And the results given in the subsequent subsections work, except that new algorithm should be developed to obtain reasonable time-to-go estimation for the case of large
Theorem 1. Consider the guidance system (13) satisfying Assumption 1. Given the desired impact time t d and the largest allowable leading angle ϕmax < π /2, with properly chosen coefficients N > 1 and K > 0, the closed-loop guidance system under the ITCG law, Eq. (15), is finite time stable in the sense that (1) R (t d ) = 0; (2) |ϕ (t )| < ϕmax holds for all t ∈ [0, t d ], and ϕ (t d ) = 0; (3) ξ(t d ) = 0. Proof. We first show that S 1 = {ϕ ||ϕ | ≤ ϕmax } is a positively invariant set of the closed-loop system. Consider the ϕ -subsystem (13b). Choose the Lyapunov function candidate as
V 1 (ϕ ) =
1 2
ϕ2
(16)
The time derivative of V 1 (ϕ ) along the trajectory of Eq. (13b) can be written as
V˙ 1 (ϕ ) = ϕ · ϕ˙ = −( N − 1) V sin ϕ · ϕ / R − aξ · ϕ / V
(17)
From Eq. (14), we have aξ = 0 when |ϕ | = ϕmax . It follows that
V˙ 1 (ϕ )||ϕ |=ϕmax = −( N − 1) V sin ϕmax · ϕmax / R < 0
(18)
Hence, it can be concluded that S 1 is a positively invariant set of the closed-loop system, since V˙ 1 is negative on the boundary of S 1 . That is, |ϕ (t )| ≤ ϕmax holds for all t ∈ [0, t d ], if |ϕ (0)| ≤ ϕmax . Furthermore, we assume, without loss of any generality, that |ϕ (0)| ≤ ϕmax − ε0 , where ε0 is some small positive constant. We can show that |ϕ | cannot reach ϕmax for all t ≥ 0. It is obviously so for the case of aξ · ϕ ≥ 0, since the right-hand side of
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Eq. (17) is negative, which implies that |ϕ | decreases monotonically. For the case of aξ · ϕ ≥ 0, if V˙ 1 (ϕ ) > 0, there must exist some constant ε1 , such that V˙ 1 (ϕ )||ϕ |=ϕmax −ε1 = 0, since V˙ 1 (ϕ ) is continuous with respect to ϕ and V˙ 1 (ϕ )||ϕ |=ϕmax < 0. Therefore, we can conclude that ϕ is restricted within the interval of [−ϕmax + ε1 , ϕmax − ε1 ]. It follows that, there exists a positive constant ε2 , such that cos(πϕ /(2ϕmax )) > ε2 holds for all t > 0. Now we are going to show the finite time convergence of ξ(t ). Substituting Eq. (14) into Eq. (13c), yields
ξ˙ = −c |ξ |1/2 sign(ξ )
(19)
where, c = K cos(πϕ /(2ϕmax )). Though c is time varying, it is a positive value and uniformly bounded from below by β = K ε2 since cos(πϕ /(2ϕmax )) > ε2 holds for all t > 0. Here, we simply assume that c is some positive constant in order to investigate the property of the solution of Eq. (19). Integrating Eq. (19) over the time interval [0, t ] with boundary condition ξ(0) = ξ0 yields
ξ(t ) =
sign(ξ0 )(|ξ0 |1/2 − 2c t )2 ,
0≤t <
0,
t≥
2|ξ0 |1/2 ; c
2|ξ0 |1/2 . c
(20)
It is clear from Eq. (20) that ξ(t ) is finite time stable, and the settling time T 0 = 2|ξ(0)|1/2 /c. Note that, T 0 is proportional to 1/ K . By choosing K properly large, T 0 < t d can be guaranteed, i.e., ξ(t d ) = 0. Note that, we have proved that cos ϕ ≥ cos(ϕmax − ε1 ) > 0 holds for all t > 0. It follows from Eq. (13a) that R decreases monotonically, and there always exists a finite time moment t f such that R (t f ) = 0. Since we have proved that ξ(t d ) = 0, which implies that t f = t d , it can be concluded that R (t d ) = R (t f ) = 0. Finally, we show that ϕ (t d ) = 0 by contradiction. Assume that ϕ (t d ) = 0. We continue working with the Lyapunov function defined by Eq. (16). By Eq. (17), we have V˙ 1 (t d ) = limt →td − ( N − 1) V sin ϕ · ϕ / R = −∞ since R (t d ) = 0 and aξ (t d ) = 0 (because ξ(t d ) = 0). Note that, V˙ 1 (t ) is continuous with respect to t. Hence, there exists a time t 1 , 0 < t 1 < t d , such that V˙ 1 (t ) < 0 holds for all t ∈ [t 1 , t d ]. On the other hand, V 1 (t 1 ) is finite since we have proved that ϕ (t ) is uniformly bounded with respect to t for all t ∈ [0, t d ]. Again, by the fact that V˙ 1 (t d ) = −∞ and the continuity of V˙ 1 (t ), we can conclude that there exists a time t 2 , t 1 < t 2 < t d , such that V˙ 1 (t ) ≤ − V 1 (t 1 )/(t d − t 2 ) holds for all t ∈ [t 2 , t d ]. It follows that V 1 (t d ) ≤ 0. This together with the fact that V 1 (t d ) ≥ 0 gives V 1 (t d ) = 0, which implies that ϕ (t d ) = 0. This is a contradiction, and the fact ϕ (t d ) = 0 follows. The proof is completed. 2 A few comments are in order on the use of the ITCG law (15). First, note that Theorem 1 shows that, with the guidance law equation (15), a perfect interception (i.e., intercept the target at the designated impact time without violation of FOV constraint) can be achieved by choosing proper coefficients N and K , as long as the missile can supply enough control energy. However, due to the limited capacity of dynamic actuators, the missile acceleration cannot be arbitrarily large. Therefore, in practice, an is saturated as
an = sat B N V 2 sin ϕ / R + K cos
× sign(ξ )/b0 ( R , ϕ )
πϕ /(2ϕmax ) |ξ |1/2 (21)
where, the saturation function sat B (·) is defined as
⎧ x > B; ⎨ B, |x| ≤ B ; sat B (x) = x, ⎩ −B, x < −B.
When the saturated control, Eq. (21), is introduced due to the limited maneuver capacity of missile, for given range-to-go and
leading angle, there exist two nonnegative constants ξ and ξ¯ , determined by B and ϕmax , such that Λ = Λ0 × Λ1 = {ϕ ||ϕ | ≤ ϕmax } × {ξ | − ξ ≤ ξ ≤ ξ¯ } is a positively invariant set of the closedloop system. The conclusions of Theorem 1 would hold in a relatively large domain Λ, if the saturation level B is high. That is, given the engagement geometry and maneuvering capacity level B, the designated impact time t d should be chosen such that t d ≤ t d ≤ t¯d , where t d = t + tˆgo − ξ , t¯d = t + tˆgo + ξ¯ , tˆgo is the estimation of time-to-go. In the scenario of salvo attack, t d should be chosen such that max1≤i ≤n {t di } ≤ t d ≤ min1≤i ≤n {t¯di }, where n is the
number of missiles involved in the salvo attack, t di and t¯di are the lower bound and upper bound of designated time for the i-th missile, respectively. The method to evaluate the feasible interval of t d for given engagement geometry, saturation level and FOV constraint can be found in Ref. [10]. It should also be noted that two coefficients, i.e., N and K , must be properly tuned in the guidance law (15). The navigation constant N can be simply chosen as N ≥ 3. While the coefficient K , which determines the convergence rate of impact time error, must be tuned carefully. If K is too small, the impact time error may not converge to zero before the missile impacts the target, leading to failure of impact time control. In this work, instead of setting K to be a constant, we set K to be time varying as K = k0 V 2 / R (here, k0 > 0 is a constant), which indicates that the convergence rate is relatively slow as R is relatively large, but it increases sharply as R approaches zero so that the impact time error converges to zero rapidly when the missile has been very close to the target. Also noted that, by Eq. (14), when b0 ( R , ϕ ) is small, the biased term aξ would be very large even if the impact time error ξ is small. This can lead to chattering when the impact time error approaches zero, especially when the estimation of impact time is not so accurate due to external disturbance. On the other hand, in the scenario of salvo attack, if the group of missiles participating the mission can impact the target ship within a short time interval, which is short enough to jam the defensive system of the ship, the tactical objective is achieved. Hence, it is sufficient to ensure |ξ(t d )| ≤ δ1 , where δ1 is a small positive value (for example, δ1 = 0.05 s). Therefore, to avoid the potential chattering, the ITCG law (15) is modified by introducing a term w 1 (ξ ) as follows
an = N V 2 sin ϕ / R + K w 1 (ξ ) cos
πϕ /(2ϕmax ) |ξ |1/2
× sign(ξ )/b0 ( R , ϕ )
(22)
where, w 1 (ξ ) is defined as
w 1 (ξ ) =
⎧ ⎪ ⎨ 0, ⎪ ⎩
|ξ |−δ1 δ2 −δ1 ,
1,
|ξ | < δ1 ; δ1 ≤ |ξ | ≤ δ2 ; |ξ | > δ2 .
(23)
δ1 and δ2 are small positive constants. The reason for selecting w 1 (ξ ) in this way is to ensure that the magnitude of aξ does not change drastically during implementation. The constant δ1 is chosen according to the requirement of impact time control precision as well as the magnitude of the external disturbance. While, the constant δ2 should be chosen to be larger than δ1 . Finally, it should be pointed out that the ITCG law (15) can be extended to cater to the constant velocity targets by using the idea of predicted interception point (PIP), defined as the point at which the missile is expected to intercept the target or the point of closest approach of the missile–target engagement. Assuming that the missile intercepts the target at the desired impact time, the predicted target position can be calculated online using the current designated time-to-go t¯go as X TP = X T + V T cos θ T t¯go , Y TP = Y T + V T sin θ T t¯go , where, ( X T , Y T ), V T , and θ T are the current position, speed, and heading angle of the target, respectively. Then the predicted range-to-go and leading angle of the missile
Y. Zhang et al. / Aerospace Science and Technology 39 (2014) 361–369
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Fig. 2. Comparison with existing guidance law: a) trajectory, b) guidance command, c) a close-up view of guidance command, d) leading angle, e) time-to-go, f) impact time error.
can be obtained by using the predicted target position ( X TP , Y TP ), which are needed to implement the ITCG law (15). 4. Numerical simulation In this section, simulation results of the proposed ITCG law (22) are presented for a variety of scenarios. For each simulation, the initial conditions are given in Table 1. And the coefficients in guidance law (22) are chosen as N = 3, K = 0.5V 2 / R, δ1 = 0.005, and δ2 = 0.1. The simulation step is set to be 0.01 s, and the simulation is terminated when the range-to-go is less than 0.3 m or the closing speed becomes positive. In the first engagement scenario, the missile’s initial leading angle is taken as 30 deg. By virtue of Ref. [10], the feasible interval of t d can be given as [33.7641 s, 44.6797 s]. Hence, the designated impact time can be chosen as t d = 40 s. The target is assumed to
Table 1 Parameter values used in simulation. Parameters
Values
Initial missile position Target position Missile speed Maximal acceleration Maximal leading angle
(−10 000, 1000) m (0, 0) m 300 m/s 5g 45 deg
be stationary. The results for this simulation are shown in Fig. 2. For comparison, the simulation results of the scheme proposed in [10], i.e., OGL-ITC plus switching logic, are included as well. As can be seen in Fig. 2, both the proposed ITCG law (22) and the guidance law proposed in [10] can drive the missile to impact the target at the designated time without violation of FOV constraint. The time histories of the key variables under the two different
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Fig. 3. Results for different initial heading angles: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
guidance laws are very similar. The main difference exists in the time history of the guidance command, as shown in Fig. 2c. Under the guidance law proposed in [10], the guidance command suffers a jumping due to the switching of guidance mode. This is not preferable in practice, for it may lead to instability. While under the ITCG law (22), the guidance command is continuous. The next set of simulations involves testing the robustness of the proposed guidance law with respect to the initial heading angle and the designated impact time. A simulation was first conducted by choosing a particular impact time of 40 s and allows the initial heading angle to vary, with values of −35◦ , −10◦ , 10◦ , and 35◦ . (Note that, the corresponding feasible intervals of impact time are given by [33.6442 s, 44.5598 s], [33.5000 s, 43.6886 s], [33.5214 s, 44.1488 s], and [33.8938 s, 44.7329 s], respectively.) The results for this simulation are shown in Fig. 3. Another simulation was then performed for a range of impact times of 33.6 s, 38 s, 40 s, and 43 s for a specific initial heading angle of 20◦ , with the results shown in Fig. 4. Note that, the initial time-togo t go (0) for this scenario is 34.1741 s, and the corresponding feasible interval of impact time is [33.5968 s, 44.4703 s]. Thus, the feasible interval of initial impact time error can be given by [−ξ (0), ξ¯ (0)] = [−0.5773 s, 10.2962 s]. It is easy to check that, for the designated impact times of 33.6 s, 38 s, 40 s, and 43 s, the corresponding ξ(0) (given by −0.5741 s, 3.8259 s, 5.8259 s, and 8.8259 s, respectively) all fall into this feasible interval. For each specific initial heading angle, Fig. 3 indicates that the proposed ITCG law generates a feasible trajectory which satisfies the impact time constraint while keeping the leading angle within the prescribed range. Furthermore, from Fig. 4, it is observed that the proposed ITCG law can be used to generate trajectories which satisfy a range of impact times. Two points are worth making here. First, the initial impact time error ξ(0) can be either positive or negative, as shown Fig. 4d, but ξ is usually much smaller
than ξ¯ . This is because, in the case of negative ξ (as shown in Fig. 4d, t d = 33.6 s), which implies that the missile must home to the target along a shorter route than the PNG trajectory, the biased term must have the same sign with the PNG term to accelerate the convergence of leading angle. When the leading angle converges to zero, which indicates the missile flies towards the target without heading error and the trajectory is straight already, the impact time cannot be decreased anymore. This process can be very fast, since the biased term has the same sign with the PNG term. Therefore, ξ is relatively small. While, in the case of positive ξ (as shown in Fig. 4d, t d = 38, 40, 43 s), the biased term has different sign with the PNG term, and tends to diverge the leading angle to extend the trajectory. And the leading angle cannot converge to zero before the designated impact time is achieved, which admits a relatively large ξ¯ . Of course, both ξ and ξ¯ are determined by engagement geometry, the acceleration limit, and the FOV constraint (i.e., R (0), ϕ (0), amax , and ϕmax ). Second, to get a zero guidance command at the final time, which is preferable in practice, ξ(0) < ξ¯ (or equivalently, t d < t¯d ) must be satisfied. This is obvious by noting the fact that, in Ref. [10], t¯d is evaluated under the assumption that the missile flies with its maximum maneuvering capacity (i.e., |an | = amax ) at the terminal phase of engagement. (The saturation of guidance command at the terminal phase in the case of t d = 43 s in Fig. 4b reveals this to some extend.) According to the above analysis, to save the control energy and taking the potential external disturbance into account, in practice, we can set t d to be slightly larger than t go (0) (or, to be slightly larger than i max1≤i ≤n {t go (0)} in the scenario of salvo attack), if possible. As mentioned in Section 3, the proposed ITCG law can also be utilized to intercept constant velocity target by using the idea of PIP. In order to demonstrate the ITCG law’s effectiveness, a simulation was conducted for a slowly moving target with constant velocity. The initial conditions are the same as those in the first
Y. Zhang et al. / Aerospace Science and Technology 39 (2014) 361–369
Fig. 4. Results for different impact times: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
Fig. 5. Results for a constant velocity target: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
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Fig. 6. Results for salvo attack: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
5. Conclusions
Table 2 Scenario for salvo attack.
Target Missile 1 Missile 2 Missile 3
Position
Heading angle
Speed
(0, 0) m (−9000, 5000) m (−6000, −6000) m (5000, −9000) m
0 −15 85 120
0 280 250 300
deg deg deg deg
m/s m/s m/s m/s
simulation, only now the target is assumed to be moving at a velocity of 15 m/s with a heading angle of 135◦ . The velocity of 15 m/s was chosen to correspond to roughly 30 knots, which is a typical cruise speed for a modern warfare ship. Fig. 5 shows the simulation results for a range of impact times of 36 s, 38 s, and 40 s. For each specific impact time, the desired impact time and FOV constraints are satisfied. A final simulation was performed with the proposed ITCG law applied to a scenario of salvo attack. In this scenario, the requirement is that three missiles, each with different initial conditions and speed, must impact the target at a common designated time. The initial conditions for this scenario are given in Table 2. In case of PNG with N = 3, the impact times (namely, t go (0)) are 37.0 s, 35.7 s, and 34.3 s, respectively, as shown in Fig. 6a. On the other hand, by virtue of Ref. [10] again, the feasible interval of impact time for each missile can be evaluated and the feasible interval of impact time for the whole group of missiles can be given by [36.7845 s, 44.700 s]. Here, we choose t d = 40 s as well, which i is slightly larger than maxi =1,2,3 {t go (0)} = 37 s. As expected, the three missiles impact the target simultaneously at the designated time. Fig. 6 also shows that, for each missile, the leading angle is within the constraint of FOV, and the guidance command is proper during the whole guidance phase.
We designed an ITCG law under FOV constraint, which belongs to the class of biased PNG law. To handle the FOV constraint, a rule of the cosine of weighted leading angle is used in the biased term. Compared with the existing switching logic method, which suffers abrupt jumping of guidance command when the guidance mode is switched to maintain the FOV constraint, the proposed guidance law provides a continuous guidance command. Thus it is more preferable in the engineering practice, and feasible for implementation. The closed-loop system under the proposed ITCG law is proved to be finite time stable in some sense, thus interception at the desired impact time without violation of FOV constraint can be guaranteed theoretically. Though the guidance law in this work was designed under the assumption that the target is stationary, it can also be used for engaging a constant velocity target by combining with the idea of PIP. Simulation results show that, the proposed guidance law can be used for salvo attack of a group of missiles with different initial conditions and speed effectively. In future study several factors, such as impact angle constraint, and external disturbances should be considered. Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Impact time control guidance law with field of view constraint”.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 61273058). Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ast.2014.10.002. References [1] Shunsaku Arita, Seiya Ueno, Improvement of guidance law for simultaneous attack, in: SICE Annual Conference 2011, Tokyo, Japan, 2011, pp. 1807–1812. [2] N. Harl, S.N. Balakrishnan, Impact time and angle guidance with sliding mode control, IEEE Trans. Control Syst. Technol. 20 (6) (Nov. 2012) 1436–1449, http:// dx.doi.org/10.1109/TCST.20011.2169795. [3] H.Q. Huang, J. Zhou, J.G. Guo, Design and simulation of an optimal missile trajectory with constraints of impact time and impact angle, J. Northwestern Polytech. Univ. 28 (2) (Apr. 2010) 165–170, http://dx.doi.org/10.1109/ TAES.2007.357135. [4] I.S. Jeon, J.I. Lee, M.J. Tahk, Impact-time-control guidance law for anti-ship missiles, IEEE Trans. Control Syst. Technol. 14 (2) (Mar. 2006) 260–266, http:// dx.doi.org/10.1109/TCST.2005.863655. [5] I.S. Jeon, J.I. Lee, M.J. Tahk, Homing guidance law for cooperative attack of multiple missiles, J. Guid. Control Dyn. 33 (1) (Jan. 2010) 275–280, http://dx.doi. org/10.2514/1.40136.
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[6] T.H. Kim, B.G. Park, M.J. Tahk, Bias-shaping method for biased proportional navigation with terminal-angle constraint, J. Guid. Control Dyn. 36 (6) (Nov. 2013) 1695–1708, http://dx.doi.org/10.2514/1.59252. [7] S.R. Kumar, D. Ghose, Sliding mode control based guidance law with impact time constraints, in: American Control Conference, Washington, DC, 2013, pp. 5760–5765. [8] J.I. Lee, I.S. Jeon, M.J. Tahk, Guidance law to control impact time and angle, IEEE Trans. Aerosp. Electron. Syst. 43 (1) (Jan. 2007) 301–310, http://dx.doi.org/ 10.1109/TAES.2007.357135. [9] G.X. Ma, Y.A. Zhang, Time-cooperative distributed guidance law design for multi-missiles, Control Decis. 29 (5) (May 2014) 843–847, http://dx.doi.org/ 10.13195/j.kzyjc.2013.0039. [10] D.K. Sang, M.J. Tahk, Guidance law switching logic considering the seeker’s field-of-view limits, Proc. Inst. Mech. Eng., G J. Aerosp. Eng. 223 (8) (Jul. 2009) 1049–1058, http://dx.doi.org/10.1243/09544100JAERO614. [11] X.J. Sun, R. Zhou, D.L. Hou, J. Wu, Consensus of leader-followers system of multi-missile with time-delays and switching topologies, Optik 125 (2014) 1202–1208, http://dx.doi.org/10.1016/j.ijleo.2013.07.159. [12] Y.G. Zhang, Y.A. Zhang, Three-dimensional guidance law to control impact time and impact angle: a two-stage control approach, Control Theory Appl. 27 (10) (Oct. 2010) 1429–1434. [13] Y.A. Zhang, G.X. Ma, A.L. Liu, Guidance law with impact time and impact angle constraints, Chin. J. Aeronaut. 26 (4) (Aug. 2013) 960–966, http://dx.doi.org/ 10.1016/j.cja.2013.04.037. [14] S.Y. Zhao, R. Zhou, Cooperative guidance for multi-missile salvo attack, Chin. J. Aeronaut. 21 (6) (Jan. 2008) 533–539. [15] L. Zou, R. Zhou, Q.X. Ding, Decentralized cooperative guidance for multiple missile groups in salvo attack, Acta Aeronaut. Astronaut. Sin. 32 (2) (Jan. 2011) 281–290.
Contents lists available at ScienceDirect
Aerospace Science and Technology www.elsevier.com/locate/aescte
Impact time control guidance law with field of view constraint Youan Zhang, Xingliang Wang ∗ , Huali Wu Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, 264001, China
a r t i c l e
i n f o
Article history: Received 16 July 2014 Received in revised form 19 September 2014 Accepted 10 October 2014 Available online 18 October 2014 Keywords: Anti-ship missile Impact time control Field-of-view constraint Missile guidance
a b s t r a c t The problem of Impact Time Control Guidance (ITCG) with field-of-view (FOV) constraint is investigated. The proposed ITCG law is a combination of the well-known Proportional Navigation Guidance (PNG) law and an additional biased term of impact time error, which is defined as the difference between the impact time by PNG and the prescribed one. A rule of the cosine of weighted leading angle in the biased term is used to guarantee that the FOV constraint is not violated during the engagement. Stability of the closed-loop guidance system is proved strictly. Numerical simulation results are presented to demonstrate the effectiveness of the proposed guidance law. © 2014 Elsevier Masson SAS. All rights reserved.
1. Introduction Modern battleships are equipped with various defensive systems against anti-ship missiles, such as anti-air defense missile systems and close-in weapon systems. These defensive systems have been great obstacles for anti-ship missiles to complete their missions. To penetrate the formidable defensive systems, on the one hand, sea-skimming flight or terminal evasive maneuvering has been developed for high performance missiles. On the other hand, simultaneous attack, which is viewed as another cost effective and efficient cooperative attack strategy to enhance survivability of conventional missiles, attracts much attention in these years. Works in simultaneous attack can be classified into two categories. The first category is the individually simultaneous attack, in which a common impact time is commanded to all members of the group of missiles a priori, and thereafter each missile tries to impact the target at the designated time independently. In [4], which is the initial effort in this direction, an impact time constrained guidance law which is designed for anti-ship missiles consists of two terms, one being the conventional proportional navigation guidance (PNG) term and the other being a feedback term of the impact time error. As an extension of [4], a guidance law to control both impact time and angle was presented in [8]. An improvement of [4] can also be found in [1], using the numerical optimizing method to determine the designated impact time. In [3], the nonlinear engagement dynamics is transferred into linear form using the flight path angle as the independent variable, based on which an impact time and impact angle control guidance
*
Corresponding author.
http://dx.doi.org/10.1016/j.ast.2014.10.002 1270-9638/© 2014 Elsevier Masson SAS. All rights reserved.
(ITIACG) law is designed. Unfortunately, the resultant guidance law is singular at some points. In [2], under the assumption that the information of the target is known beforehand, the desired line of sight (LOS) rate trajectory, which satisfies the desired impact time and impact angle constraints simultaneously, is calculated offline first, and then a sliding mode guidance law is designed to track the desired LOS rate trajectory. In [12], a three dimensional impact time and impact angle control guidance law is designed using a two-stage control approach. In [13], a biased PNG (BPNG) law is designed to control the impact time and impact angle. In [7], sliding mode impact time guidance laws are designed based on the conventional and improved formulas for time-to-go estimation. However, both the time-to-go formulas used are not derived under the proposed guidance law. Hence the resultant time-to-go dynamics is inaccurate, which may lead to unexpected results. The second category is cooperatively simultaneous attack, in which the missiles communicate among each other during the whole guidance phase to synchronize the impact time. In [14], a cooperative guidance scheme for salvo attack (i.e., simultaneous attack) is proposed, in which, the impact time control guidance (ITCG) law presented in [4] is used as the local guidance law and then a coordination algorithm is used to synchronize the impact time of all the missiles. In [5], by introducing the concept of the time-to-go variance of multiple missiles, a cooperative PNG law is proposed to achieve a simultaneous attack by decreasing the time-to-go variance cooperatively until the interception. In [15], a cooperative guidance scheme is proposed by integrating the ITCG law (presented in [4] as well) with the decentralized consensus algorithm for multiple missile groups. In [11], by modeling the communication topology of the multimissiles system as a leader-followers network, the impact time
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synchronization of multi-missiles is achieved with time-delays and switching topologies. In [9], without using a leader, each missile is modeled as a subsystem with time-to-go as the output, and the communication topology of the multi-missiles is modeled as a peer-to-peer network. The theory of cooperative control is then applied to reduce the relative error of time-to-go among the subsystems to synchronize the impact time. All the guidance laws aforementioned didn’t consider the limitation of the seeker’s field-of-view (FOV) of the missile explicitly. In practice, the ITCG law may produce large additional command to maneuver the missile to adjust the impact time. This may lead to the target moving out of the missile seeker’s FOV. In such case, the seeker fails to maintain a lock on the target and the missile cannot intercept the target. To cope with this challenge, two methods can be used. One is the active method, in which a proper designated impact time is selected within some interval determined by the missile’s maneuvering capacity and the FOV, see [10]; the other is the passive method, in which a switching logic is used to keep the target within the missile’s FOV, see [10] and [6]. The active method cannot utilize the maneuvering capacity of the missile sufficiently since it just simply ensures the maximum look angle does not exceed the FOV limit. Although the passive way can utilize the maneuvering capacity sufficiently, it suffers abrupt jumping of guidance command due to the switching among different guidance modes. It is well known that the problem of intercepting a stationary target in the plane is already well studied. In this paper, the planner guidance problem is studied as well. But we emphasize on the impact time control when the seeker’s FOV is limited, and aims to overcome the drawbacks resulting from switching logic mentioned above. It will be shown later that the proposed guidance law has the form of an = aPNG + aξ , where aPNG is the conventional PNG law term, and aξ is a biased term to control the impact time. Different from the switching logic method, a rule of the cosine of weighted leading angle in the biased term will be used to guarantee that the FOV constraint not be violated during the engagement. When the impact time error is reduced to zero, the biased term aξ equals zero as well, and the proposed ITCG law degrades to the conventional PNG law. It will also be proved that the closed-loop guidance system under the proposed ITCG law is finite time stable with respect to leading angle and impact time error. While the rangeto-go decreases monotonically, thus interception can be guaranteed. The remainder of this paper is organized as follows. In Section 2, the problem of impact time control subject to seeker’s FOV constraint is formulated. Then, an ITCG law with the form of biased PNG (BPNG) law is designed in Section 3. The stability of the closed-loop guidance system under the proposed ITCG law is proved strictly. In Section 4, comparisons between the proposed ITCG law and the scheme in [10] are presented. Application of the proposed ITCG law to the salvo attack scenario is verified as well. Finally, Section 5 summarizes the conclusions. 2. Problem formulation Consider a planar engagement between an anti-ship missile (denoted as M) and a ship (denoted as T ) as shown in Fig. 1. The ship is modeled as being stationary since the maneuver capacity and the speed of ships are not comparable with those of anti-ship missiles of high subsonic or supersonic speed. It is assumed that the missile speed V is constant during the engagement and the autopilot lag is negligible. The heading angle and leading angle are denoted by θ and ϕ , respectively; the range between the ship and the anti-ship missile is R, q is the line-of-sight (LOS) angle relative to a fixed reference axis.
Fig. 1. Homing guidance geometry.
From Fig. 1, the engagement dynamics can be given as
R˙ = − V cos ϕ
(1a)
R q˙ = V sin ϕ
(1b)
θ˙ = an / V
(1c)
q=θ +ϕ
(1d)
where, an is the missile’s lateral acceleration, i.e., the guidance command. The impact time error, ξ , is defined as
ξ = t¯go − t go
(2)
where, t¯go = t d − t is the designated time-to-go, t d is the designated impact time, t go is the real time-to-go at the current time moment, which is usually not known a priori and need to be estimated. Besides, the seeker’s FOV is usually defined as the angle between the missile body axis and the LOS (i.e., the sum of the leading angle and the missile’s angle of sideslip. We consider the lateral plane here). Under the assumption that the missile’s angle of sideslip is small, we can use the leading angle limitation for handling the seeker’s FOV constraint, just as that of [10]. That is, the seeker’s FOV constraint can be equivalently expressed as |ϕ (t )| ≤ ϕmax , where, ϕmax is a positive constant determined by the seeker’s FOV limitation. The objective of this paper is to design a guidance law such that the missile can impact the target at the designated time t d while keeping ϕ (t ) ∈ [−ϕmax , −ϕmax ] for all t ∈ [0, t d ]. 3. Impact time control guidance with FOV constraint 3.1. Derivation of impact time error dynamics To design an ITCG law, we first derive the dynamics of impact time error. As mentioned in Section 1, the ITCG law we are going to design in this paper would have the form of
an = aPNG + aξ
(3)
where,
aPNG = N V 2 sin ϕ / R ,
(4)
N > 1 is the navigation constant, and aξ is the biased term to control the impact time. Since the ITCG law to be designed is of the form given by (3), the time-to-go of PNG should be used to calculate the impact time error. Let us assume that, without loss of generality, the time-to-go of PNG can be expressed as
t go =
R V
1+
ϕ2 4N − 2
+ t ( R , ϕ )
(5)
Y. Zhang et al. / Aerospace Science and Technology 39 (2014) 361–369
where, the first term of the right hand side of Eq. (5) coincides with the t go formula of PNG (assuming the leading angle is small) given in [5], and the second term is the portion to compensate for the error caused by potential abusing of small angle assumption. Assume, without loss of any generality, that the time derivative of t ( R , ϕ ) can be expressed as
dt ( R , ϕ ) dt
= f ( R , ϕ ) + b(a, ϕ )aξ
363
leading angle. In fact, though no strict proof is given, we found through simulations that the inequality b( R , ϕ ) · b0 ( R , ϕ ) > 0 holds for all |ϕ | ≤ π /2, which is a large enough range to include that determined by the FOV constraint in practice. On the other hand, b( R , ϕ ) would finally equal zero as long as ϕ (t ) converges, thus h(t ) can be viewed as a vanishing uncertainty and brings no influence on the control precision.
(6) 3.2. ITCG law with FOV constraint
Using Eqs. (5) and (6), the time derivative of t go can be given as
t˙go = f ( R , ϕ ) + f ( R , ϕ ) + b0 ( R , ϕ )aξ + b( R , ϕ )aξ where,
f ( R , ϕ ) = − cos ϕ 1 +
4N − 2
To meet the impact time as well as the FOV constraints, the biased term aξ can be designed as
ϕ2
(7)
−
N −1 2N − 1
aξ = K cos
ϕ sin ϕ
(8)
πϕ /(2ϕmax ) |ξ |1/2 sign(ξ )/b0 ( R , ϕ )
where K > 0 is an adjustable coefficient. And hence the resultant ITCG law can be given as
and
b0 ( R , ϕ ) = − R ϕ / (2N − 1) V
2
(14)
an = N V 2 sin ϕ / R + K cos
πϕ /(2ϕmax ) |ξ |1/2 sign(ξ )/b0 ( R , ϕ )
(9)
(15)
Obviously, when there is no need to adjust the impact time (that is, when aξ = 0), the decreasing rate of t go should coincide with that of t¯go , which equals −1 since t go decreases as t increases. It follows that f ( R , ϕ ) + f ( R , ϕ ) = −1. Substituting this equality into Eq. (7) yields
To prove the stability of the closed-loop guidance system, an assumption is given as follows.
t˙go = −1 + b0 ( R , ϕ )aξ + b( R , ϕ )aξ
(10)
Differentiating Eq. (2) and taking into account Eq. (10) yields the dynamics of impact time error
ξ˙ = h(t ) − b0 ( R , ϕ )aξ
(11)
where,
h(t ) = −b( R , ϕ )aξ
Assumption 1. The initial value of leading angle |ϕ (0)| < ϕmax , and ϕ (0) = 0. Remark 2. It is reasonable to assume that the target is within the seeker’s FOV at the initial time, which implies that |ϕ (0)| < ϕmax . However, when ϕ (0) = 0, the control input coefficient b0 (0) = 0, which means that additional control energy cannot be injected into the guidance system to adjust the impact time. In this case, an additional mechanism is needed to perturb the leading angle so that the impact time becomes controllable.
(12)
is the uncertainty term caused by the potential abusing of small angle assumption. Note that, when the leading angle ϕ (t ) is confined within a relatively small range due to the requirement for maintaining the FOV constraint condition, it is reasonable to use the t go estimation formula proposed in [5] directly. Thus, in the case of ITC problem under FOV constraint, we can assume that h(t ) = 0. Substituting the guidance law Eq. (3) into Eq. (1), and using R and ϕ as the state variables, the engagement dynamics including the impact time error can be given as
R˙ = − V cos ϕ
(13a)
ϕ˙ = −( N − 1) V sin ϕ / R − aξ / V
(13b)
ξ˙ = −b0 ( R , ϕ )aξ
(13c)
where, the control input coefficient b0 ( R , ϕ ) is defined by Eq. (9). Remark 1. To get a simple form of the impact time error dynamics, the use of the relationship of f ( R , ϕ ) + f ( R , ϕ ) = −1 is the key. Besides, the complete expression for the impact time error dynamics should be ξ˙ = −b( R , ϕ )aξ , where, b( R , ϕ ) = b0 ( R , ϕ ) + b( R , ϕ ). We assumed in the above derivation that b( R , ϕ ) = 0. This is reasonable when ϕmax is relatively small and ϕ (t ) is confined within [−ϕmax , ϕmax ] during the whole engagement. When ϕmax is relatively large, b( R , ϕ ) = 0 is no longer valid during the whole engagement. However, as long as b( R , ϕ ) · b0 ( R , ϕ ) > 0, that is, b0 ( R , ϕ ) dominates b( R , ϕ ), the control direction is determined by b0 ( R , ϕ ). And the results given in the subsequent subsections work, except that new algorithm should be developed to obtain reasonable time-to-go estimation for the case of large
Theorem 1. Consider the guidance system (13) satisfying Assumption 1. Given the desired impact time t d and the largest allowable leading angle ϕmax < π /2, with properly chosen coefficients N > 1 and K > 0, the closed-loop guidance system under the ITCG law, Eq. (15), is finite time stable in the sense that (1) R (t d ) = 0; (2) |ϕ (t )| < ϕmax holds for all t ∈ [0, t d ], and ϕ (t d ) = 0; (3) ξ(t d ) = 0. Proof. We first show that S 1 = {ϕ ||ϕ | ≤ ϕmax } is a positively invariant set of the closed-loop system. Consider the ϕ -subsystem (13b). Choose the Lyapunov function candidate as
V 1 (ϕ ) =
1 2
ϕ2
(16)
The time derivative of V 1 (ϕ ) along the trajectory of Eq. (13b) can be written as
V˙ 1 (ϕ ) = ϕ · ϕ˙ = −( N − 1) V sin ϕ · ϕ / R − aξ · ϕ / V
(17)
From Eq. (14), we have aξ = 0 when |ϕ | = ϕmax . It follows that
V˙ 1 (ϕ )||ϕ |=ϕmax = −( N − 1) V sin ϕmax · ϕmax / R < 0
(18)
Hence, it can be concluded that S 1 is a positively invariant set of the closed-loop system, since V˙ 1 is negative on the boundary of S 1 . That is, |ϕ (t )| ≤ ϕmax holds for all t ∈ [0, t d ], if |ϕ (0)| ≤ ϕmax . Furthermore, we assume, without loss of any generality, that |ϕ (0)| ≤ ϕmax − ε0 , where ε0 is some small positive constant. We can show that |ϕ | cannot reach ϕmax for all t ≥ 0. It is obviously so for the case of aξ · ϕ ≥ 0, since the right-hand side of
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Eq. (17) is negative, which implies that |ϕ | decreases monotonically. For the case of aξ · ϕ ≥ 0, if V˙ 1 (ϕ ) > 0, there must exist some constant ε1 , such that V˙ 1 (ϕ )||ϕ |=ϕmax −ε1 = 0, since V˙ 1 (ϕ ) is continuous with respect to ϕ and V˙ 1 (ϕ )||ϕ |=ϕmax < 0. Therefore, we can conclude that ϕ is restricted within the interval of [−ϕmax + ε1 , ϕmax − ε1 ]. It follows that, there exists a positive constant ε2 , such that cos(πϕ /(2ϕmax )) > ε2 holds for all t > 0. Now we are going to show the finite time convergence of ξ(t ). Substituting Eq. (14) into Eq. (13c), yields
ξ˙ = −c |ξ |1/2 sign(ξ )
(19)
where, c = K cos(πϕ /(2ϕmax )). Though c is time varying, it is a positive value and uniformly bounded from below by β = K ε2 since cos(πϕ /(2ϕmax )) > ε2 holds for all t > 0. Here, we simply assume that c is some positive constant in order to investigate the property of the solution of Eq. (19). Integrating Eq. (19) over the time interval [0, t ] with boundary condition ξ(0) = ξ0 yields
ξ(t ) =
sign(ξ0 )(|ξ0 |1/2 − 2c t )2 ,
0≤t <
0,
t≥
2|ξ0 |1/2 ; c
2|ξ0 |1/2 . c
(20)
It is clear from Eq. (20) that ξ(t ) is finite time stable, and the settling time T 0 = 2|ξ(0)|1/2 /c. Note that, T 0 is proportional to 1/ K . By choosing K properly large, T 0 < t d can be guaranteed, i.e., ξ(t d ) = 0. Note that, we have proved that cos ϕ ≥ cos(ϕmax − ε1 ) > 0 holds for all t > 0. It follows from Eq. (13a) that R decreases monotonically, and there always exists a finite time moment t f such that R (t f ) = 0. Since we have proved that ξ(t d ) = 0, which implies that t f = t d , it can be concluded that R (t d ) = R (t f ) = 0. Finally, we show that ϕ (t d ) = 0 by contradiction. Assume that ϕ (t d ) = 0. We continue working with the Lyapunov function defined by Eq. (16). By Eq. (17), we have V˙ 1 (t d ) = limt →td − ( N − 1) V sin ϕ · ϕ / R = −∞ since R (t d ) = 0 and aξ (t d ) = 0 (because ξ(t d ) = 0). Note that, V˙ 1 (t ) is continuous with respect to t. Hence, there exists a time t 1 , 0 < t 1 < t d , such that V˙ 1 (t ) < 0 holds for all t ∈ [t 1 , t d ]. On the other hand, V 1 (t 1 ) is finite since we have proved that ϕ (t ) is uniformly bounded with respect to t for all t ∈ [0, t d ]. Again, by the fact that V˙ 1 (t d ) = −∞ and the continuity of V˙ 1 (t ), we can conclude that there exists a time t 2 , t 1 < t 2 < t d , such that V˙ 1 (t ) ≤ − V 1 (t 1 )/(t d − t 2 ) holds for all t ∈ [t 2 , t d ]. It follows that V 1 (t d ) ≤ 0. This together with the fact that V 1 (t d ) ≥ 0 gives V 1 (t d ) = 0, which implies that ϕ (t d ) = 0. This is a contradiction, and the fact ϕ (t d ) = 0 follows. The proof is completed. 2 A few comments are in order on the use of the ITCG law (15). First, note that Theorem 1 shows that, with the guidance law equation (15), a perfect interception (i.e., intercept the target at the designated impact time without violation of FOV constraint) can be achieved by choosing proper coefficients N and K , as long as the missile can supply enough control energy. However, due to the limited capacity of dynamic actuators, the missile acceleration cannot be arbitrarily large. Therefore, in practice, an is saturated as
an = sat B N V 2 sin ϕ / R + K cos
× sign(ξ )/b0 ( R , ϕ )
πϕ /(2ϕmax ) |ξ |1/2 (21)
where, the saturation function sat B (·) is defined as
⎧ x > B; ⎨ B, |x| ≤ B ; sat B (x) = x, ⎩ −B, x < −B.
When the saturated control, Eq. (21), is introduced due to the limited maneuver capacity of missile, for given range-to-go and
leading angle, there exist two nonnegative constants ξ and ξ¯ , determined by B and ϕmax , such that Λ = Λ0 × Λ1 = {ϕ ||ϕ | ≤ ϕmax } × {ξ | − ξ ≤ ξ ≤ ξ¯ } is a positively invariant set of the closedloop system. The conclusions of Theorem 1 would hold in a relatively large domain Λ, if the saturation level B is high. That is, given the engagement geometry and maneuvering capacity level B, the designated impact time t d should be chosen such that t d ≤ t d ≤ t¯d , where t d = t + tˆgo − ξ , t¯d = t + tˆgo + ξ¯ , tˆgo is the estimation of time-to-go. In the scenario of salvo attack, t d should be chosen such that max1≤i ≤n {t di } ≤ t d ≤ min1≤i ≤n {t¯di }, where n is the
number of missiles involved in the salvo attack, t di and t¯di are the lower bound and upper bound of designated time for the i-th missile, respectively. The method to evaluate the feasible interval of t d for given engagement geometry, saturation level and FOV constraint can be found in Ref. [10]. It should also be noted that two coefficients, i.e., N and K , must be properly tuned in the guidance law (15). The navigation constant N can be simply chosen as N ≥ 3. While the coefficient K , which determines the convergence rate of impact time error, must be tuned carefully. If K is too small, the impact time error may not converge to zero before the missile impacts the target, leading to failure of impact time control. In this work, instead of setting K to be a constant, we set K to be time varying as K = k0 V 2 / R (here, k0 > 0 is a constant), which indicates that the convergence rate is relatively slow as R is relatively large, but it increases sharply as R approaches zero so that the impact time error converges to zero rapidly when the missile has been very close to the target. Also noted that, by Eq. (14), when b0 ( R , ϕ ) is small, the biased term aξ would be very large even if the impact time error ξ is small. This can lead to chattering when the impact time error approaches zero, especially when the estimation of impact time is not so accurate due to external disturbance. On the other hand, in the scenario of salvo attack, if the group of missiles participating the mission can impact the target ship within a short time interval, which is short enough to jam the defensive system of the ship, the tactical objective is achieved. Hence, it is sufficient to ensure |ξ(t d )| ≤ δ1 , where δ1 is a small positive value (for example, δ1 = 0.05 s). Therefore, to avoid the potential chattering, the ITCG law (15) is modified by introducing a term w 1 (ξ ) as follows
an = N V 2 sin ϕ / R + K w 1 (ξ ) cos
πϕ /(2ϕmax ) |ξ |1/2
× sign(ξ )/b0 ( R , ϕ )
(22)
where, w 1 (ξ ) is defined as
w 1 (ξ ) =
⎧ ⎪ ⎨ 0, ⎪ ⎩
|ξ |−δ1 δ2 −δ1 ,
1,
|ξ | < δ1 ; δ1 ≤ |ξ | ≤ δ2 ; |ξ | > δ2 .
(23)
δ1 and δ2 are small positive constants. The reason for selecting w 1 (ξ ) in this way is to ensure that the magnitude of aξ does not change drastically during implementation. The constant δ1 is chosen according to the requirement of impact time control precision as well as the magnitude of the external disturbance. While, the constant δ2 should be chosen to be larger than δ1 . Finally, it should be pointed out that the ITCG law (15) can be extended to cater to the constant velocity targets by using the idea of predicted interception point (PIP), defined as the point at which the missile is expected to intercept the target or the point of closest approach of the missile–target engagement. Assuming that the missile intercepts the target at the desired impact time, the predicted target position can be calculated online using the current designated time-to-go t¯go as X TP = X T + V T cos θ T t¯go , Y TP = Y T + V T sin θ T t¯go , where, ( X T , Y T ), V T , and θ T are the current position, speed, and heading angle of the target, respectively. Then the predicted range-to-go and leading angle of the missile
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Fig. 2. Comparison with existing guidance law: a) trajectory, b) guidance command, c) a close-up view of guidance command, d) leading angle, e) time-to-go, f) impact time error.
can be obtained by using the predicted target position ( X TP , Y TP ), which are needed to implement the ITCG law (15). 4. Numerical simulation In this section, simulation results of the proposed ITCG law (22) are presented for a variety of scenarios. For each simulation, the initial conditions are given in Table 1. And the coefficients in guidance law (22) are chosen as N = 3, K = 0.5V 2 / R, δ1 = 0.005, and δ2 = 0.1. The simulation step is set to be 0.01 s, and the simulation is terminated when the range-to-go is less than 0.3 m or the closing speed becomes positive. In the first engagement scenario, the missile’s initial leading angle is taken as 30 deg. By virtue of Ref. [10], the feasible interval of t d can be given as [33.7641 s, 44.6797 s]. Hence, the designated impact time can be chosen as t d = 40 s. The target is assumed to
Table 1 Parameter values used in simulation. Parameters
Values
Initial missile position Target position Missile speed Maximal acceleration Maximal leading angle
(−10 000, 1000) m (0, 0) m 300 m/s 5g 45 deg
be stationary. The results for this simulation are shown in Fig. 2. For comparison, the simulation results of the scheme proposed in [10], i.e., OGL-ITC plus switching logic, are included as well. As can be seen in Fig. 2, both the proposed ITCG law (22) and the guidance law proposed in [10] can drive the missile to impact the target at the designated time without violation of FOV constraint. The time histories of the key variables under the two different
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Fig. 3. Results for different initial heading angles: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
guidance laws are very similar. The main difference exists in the time history of the guidance command, as shown in Fig. 2c. Under the guidance law proposed in [10], the guidance command suffers a jumping due to the switching of guidance mode. This is not preferable in practice, for it may lead to instability. While under the ITCG law (22), the guidance command is continuous. The next set of simulations involves testing the robustness of the proposed guidance law with respect to the initial heading angle and the designated impact time. A simulation was first conducted by choosing a particular impact time of 40 s and allows the initial heading angle to vary, with values of −35◦ , −10◦ , 10◦ , and 35◦ . (Note that, the corresponding feasible intervals of impact time are given by [33.6442 s, 44.5598 s], [33.5000 s, 43.6886 s], [33.5214 s, 44.1488 s], and [33.8938 s, 44.7329 s], respectively.) The results for this simulation are shown in Fig. 3. Another simulation was then performed for a range of impact times of 33.6 s, 38 s, 40 s, and 43 s for a specific initial heading angle of 20◦ , with the results shown in Fig. 4. Note that, the initial time-togo t go (0) for this scenario is 34.1741 s, and the corresponding feasible interval of impact time is [33.5968 s, 44.4703 s]. Thus, the feasible interval of initial impact time error can be given by [−ξ (0), ξ¯ (0)] = [−0.5773 s, 10.2962 s]. It is easy to check that, for the designated impact times of 33.6 s, 38 s, 40 s, and 43 s, the corresponding ξ(0) (given by −0.5741 s, 3.8259 s, 5.8259 s, and 8.8259 s, respectively) all fall into this feasible interval. For each specific initial heading angle, Fig. 3 indicates that the proposed ITCG law generates a feasible trajectory which satisfies the impact time constraint while keeping the leading angle within the prescribed range. Furthermore, from Fig. 4, it is observed that the proposed ITCG law can be used to generate trajectories which satisfy a range of impact times. Two points are worth making here. First, the initial impact time error ξ(0) can be either positive or negative, as shown Fig. 4d, but ξ is usually much smaller
than ξ¯ . This is because, in the case of negative ξ (as shown in Fig. 4d, t d = 33.6 s), which implies that the missile must home to the target along a shorter route than the PNG trajectory, the biased term must have the same sign with the PNG term to accelerate the convergence of leading angle. When the leading angle converges to zero, which indicates the missile flies towards the target without heading error and the trajectory is straight already, the impact time cannot be decreased anymore. This process can be very fast, since the biased term has the same sign with the PNG term. Therefore, ξ is relatively small. While, in the case of positive ξ (as shown in Fig. 4d, t d = 38, 40, 43 s), the biased term has different sign with the PNG term, and tends to diverge the leading angle to extend the trajectory. And the leading angle cannot converge to zero before the designated impact time is achieved, which admits a relatively large ξ¯ . Of course, both ξ and ξ¯ are determined by engagement geometry, the acceleration limit, and the FOV constraint (i.e., R (0), ϕ (0), amax , and ϕmax ). Second, to get a zero guidance command at the final time, which is preferable in practice, ξ(0) < ξ¯ (or equivalently, t d < t¯d ) must be satisfied. This is obvious by noting the fact that, in Ref. [10], t¯d is evaluated under the assumption that the missile flies with its maximum maneuvering capacity (i.e., |an | = amax ) at the terminal phase of engagement. (The saturation of guidance command at the terminal phase in the case of t d = 43 s in Fig. 4b reveals this to some extend.) According to the above analysis, to save the control energy and taking the potential external disturbance into account, in practice, we can set t d to be slightly larger than t go (0) (or, to be slightly larger than i max1≤i ≤n {t go (0)} in the scenario of salvo attack), if possible. As mentioned in Section 3, the proposed ITCG law can also be utilized to intercept constant velocity target by using the idea of PIP. In order to demonstrate the ITCG law’s effectiveness, a simulation was conducted for a slowly moving target with constant velocity. The initial conditions are the same as those in the first
Y. Zhang et al. / Aerospace Science and Technology 39 (2014) 361–369
Fig. 4. Results for different impact times: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
Fig. 5. Results for a constant velocity target: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
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Fig. 6. Results for salvo attack: a) trajectory, b) guidance command, c) leading angle, d) impact time error.
5. Conclusions
Table 2 Scenario for salvo attack.
Target Missile 1 Missile 2 Missile 3
Position
Heading angle
Speed
(0, 0) m (−9000, 5000) m (−6000, −6000) m (5000, −9000) m
0 −15 85 120
0 280 250 300
deg deg deg deg
m/s m/s m/s m/s
simulation, only now the target is assumed to be moving at a velocity of 15 m/s with a heading angle of 135◦ . The velocity of 15 m/s was chosen to correspond to roughly 30 knots, which is a typical cruise speed for a modern warfare ship. Fig. 5 shows the simulation results for a range of impact times of 36 s, 38 s, and 40 s. For each specific impact time, the desired impact time and FOV constraints are satisfied. A final simulation was performed with the proposed ITCG law applied to a scenario of salvo attack. In this scenario, the requirement is that three missiles, each with different initial conditions and speed, must impact the target at a common designated time. The initial conditions for this scenario are given in Table 2. In case of PNG with N = 3, the impact times (namely, t go (0)) are 37.0 s, 35.7 s, and 34.3 s, respectively, as shown in Fig. 6a. On the other hand, by virtue of Ref. [10] again, the feasible interval of impact time for each missile can be evaluated and the feasible interval of impact time for the whole group of missiles can be given by [36.7845 s, 44.700 s]. Here, we choose t d = 40 s as well, which i is slightly larger than maxi =1,2,3 {t go (0)} = 37 s. As expected, the three missiles impact the target simultaneously at the designated time. Fig. 6 also shows that, for each missile, the leading angle is within the constraint of FOV, and the guidance command is proper during the whole guidance phase.
We designed an ITCG law under FOV constraint, which belongs to the class of biased PNG law. To handle the FOV constraint, a rule of the cosine of weighted leading angle is used in the biased term. Compared with the existing switching logic method, which suffers abrupt jumping of guidance command when the guidance mode is switched to maintain the FOV constraint, the proposed guidance law provides a continuous guidance command. Thus it is more preferable in the engineering practice, and feasible for implementation. The closed-loop system under the proposed ITCG law is proved to be finite time stable in some sense, thus interception at the desired impact time without violation of FOV constraint can be guaranteed theoretically. Though the guidance law in this work was designed under the assumption that the target is stationary, it can also be used for engaging a constant velocity target by combining with the idea of PIP. Simulation results show that, the proposed guidance law can be used for salvo attack of a group of missiles with different initial conditions and speed effectively. In future study several factors, such as impact angle constraint, and external disturbances should be considered. Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Impact time control guidance law with field of view constraint”.
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